ACOUSTO-PLASTIC DEFORMATION OF METALS BY
NONLINEAR STRESS WAVES
DISSERTATION
Presented in Partial Fulfillment of the Requirements of the Degree Doctor of Philosophy
in the Graduate School of The Ohio State University
By
Minghao Cai, B.S., M.S.
*****
The Ohio State University
2006
Dissertation Committee:
Prof. Sheng-Tao (John) Yu, Advisor
Approved by
Prof. Stephen E. Bechtel
Prof. Ray Jahn
_________________________
Advisor
Prof. M.-H. Herman Shen
Graduate Program in Mechanical Engineering
ABSTRACT
The present dissertation summarizes the results of research and development for elasticplastic deformation of metals by nonlinear stress waves. The focus of the present work is
the development of a new theoretical and numerical framework to simulate nonlinear
stress waves in solids. The application of the new method focuses on modeling the
plateau stage of a novel ultrasound welding process. Superimposed by ultrasonic
vibrations, metal specimens under static loading experience remarkable transitory
softening, which has been called the Blaha effect or the Acousto-Plastic Effect (APE).
Previous microscopy studies showed that the metals experience significant plastic
deformation and the morphology resembles the Kelvin- Helmholtz instability waves in the
presence of shear stresses. The present research addresses the need to understand the APE
beyond phenomenological descriptions in the literature. Theoretical and numerical
studies have been performed to understand the unusual metal deformation induced by
ultrasonic vibrations.
A comprehensive continuum mechanics model for macroscopic description of
nonlinear stress waves in solids has been developed, including conservation laws of mass,
momentum, and energy in the Eulerian frame, in conjunction with a set of transport
equations for stress components, which have been derived based on the elastic-plastic
constitutive relation employed. The present approach aims to directly address the
dynamic nature of the processes rather than treating transitory metal softening by
proposing “effective constitutive relations” to relate the nominal “static stress” to the
nominal “static strain.” A hierarchy of theoretical model equations have been systematic
developed, including (i) one-dimensional isothermal elastic model for longitudinal
extensional waves in a thin rod, (ii) one-dimensional isothermal elastic model for
longitudinal plane waves in a bulk material, (iii) one-dimensional isothermal elastic
model for shear waves in a bulk material, (iv) one-dimensional elastic-plastic model for
longitudinal plane waves in a bulk material with and without considering the thermal
effect, and (v) two-dimensional elastic-plastic model for coplanar waves in solids with
and without considering the thermal effect. Various forms of the governing equations
have been systematically derived, including the conservation form, the non-conservative
form, and the characteristic form. The eigensystem of each set of model equations has
been ana lyzed in details with clear presentation of the analytical forms of eigenvalues and
the associated eigenvector matrices. As a part of the eigenvalues, the speed of sound for
various wave phenomena can be clearly discerned.
To solve these governing equations for the nonlinear stress waves, the space-time
Conservation Element and Solution Element (CESE) method has been used. Based on a
unified treatment of space and time in calculating flux conservation, the CESE method is
a novel numerical method for time-accurate solutions of nonlinear hyperbolic systems. In
particular, no Riemann solver is used as a building block of the time marching scheme.
Thus, the operation count and the logic of the CESE method are more efficient and
simpler than that of the modern upwind methods. Numerical results of one- and twoii
dimensional elastic and elastic-plastic stress waves are reported in the dissertation. The
numerical results are validated by a series of comparison between the one–dimensional
numerical results of elastic and elastic-plastic waves and the available analytical solutions,
experimental results and published numerical works.
Successful development of the present theoretical and modeling capabilities
demonstrates a new paradigm for high- fidelity simulation of nonlinear stress waves in
solids. With accurate modeling capabilities, one can explore processing parameter space,
thereby predicting performance properties, and the necessary process adjustments to
achieve successful implementation of high-power ultrasounds to various metal
forming/joining processes.
iii
Dedicated to my parents, my wife, and my son.
iv
ACKNOWLEDGMENTS
I wish to express my sincere thanks to Professor Sheng-Tao (John) Yu, my
dissertation advisor, for his generosity in spending a lot of time with me and precise
guidance throughout the research project, upon which this dissertation is based. Moreover,
I am grateful for his unique suggestions on my mental attitude towards both research
works and people around me. I am in debt to Drs. Moujin Zhang and Hao He, whose help
in numerical methods and parallel computation has been invaluable. I also thank
Professors Stephen E. Bechtel and M.-H. Herman Shen, and Dr. Ray Jahn of the Ford
Motor Company, for serving as members in my doctoral committee.
I am eternally grateful to me parents, who gave me life and raised me up to be
who I am now. Both of them are fighting against serious illness now. I have done and
will do my best to make them proud of me.
Finally, I thank my wife, Fang Wang, and my mother-in- law, whose constant
support and unfailing love have propelled me through this arduous period of my life.
As a marathon runner, no matter how hard the situation I have to face, I will keep
running towards the goal with the motto in my mind: No walking, no stopping, and no
thought of giving up.
v
VITA
October, 1971…..………………Born in Shanxi, China
1994……………………………B.S., Huazhong University of Science and Technology
1999……………………………M.S., Harbin Institute of Technology
2001 – present …………………Graduate Research Assistant, The Ohio State University
PUBLICATIONS
1.
Minghao Cai, S.-T. John Yu and Moujin Zhang, “Theoretical and Numerical
Solutions of Linear and Nonlinear Elastic Waves in a Thin Rod,” Journal of Wave
Motion, 2006, to appear.
2.
Minghao Cai, S.-T. John Yu and Moujin Zhang, “Theoretical and Numerical
Solutions of Linear and Nonlinear Elastic Waves in a Thin Rod,” AIAA Paper,
2006-4778, collected in the proceeding for the 42nd AIAA/ASME/SAE/ASEE Joint
Propulsion Conference & Exhibit, July 2006, San Francisco, CA.
3.
S. E. Bechtel, M. Cai, F. J. Rooney, Q. Wang, “Investigation of simplified thermal
expansion models for compressible Newtonian fluids applied to nonisothermal plane
Couette and Poiseuille flows,” PHYSICS OF FLUIDS, 2004, Vol. 16, No.11, pp
3955-3974.
FIELDS OF STUDY
Major Field: Mechanical Engineering
vi
TABLE OF CONTENTS
Abstract .................................................................................................................................i
Dedication...........................................................................................................................iv
Acknowledgments................................................................................................................v
Vita......................................................................................................................................vi
List of Figures .....................................................................................................................ix
List of Tables ....................................................................................................................xiii
CHAPTER 1 INTRODUCTION ........................................................................................ 1
1.1 Motivation and Objectives ........................................................................................ 1
1.2 Literature Review...................................................................................................... 4
1.3 The Approaches of the Present Work ..................................................................... 15
1.4 Numerical Methods for Hyperbolic System ........................................................... 17
CHAPTER 2 THE SPACE-TIME CESE METHOD ....................................................... 21
2.1 One-dimensional CESE method ............................................................................. 24
2.2 Two-dimensional CESE method............................................................................. 37
2.2.1 Conservation Elements and Solution Elements ............................................... 37
2.2.2 Approximations with a Solution Element ........................................................ 40
2.2.3 Evaluation of um ............................................................................................... 42
2.2.4 Evaluation of umx and umy................................................................................. 45
CHAPTER 3 THE FIRST ORDER HYPERBOLIC MODELS OF ELASTIC
EXTENSIONAL WAVE IN THIN ROD......................................................................... 48
3.1 Introduction............................................................................................................. 48
3.1.1 Model Equations of Stress Waves ................................................................... 48
3.1.2 Model Equations and Analytical Solutions ...................................................... 51
vii
3.1.3 The Objectives of the Current Chapter ............................................................ 52
3.2 The Second-Order Linear Wave Equation and Analytic Solution.......................... 55
3.3 The Two-Equations Model of Elastic Extensional Wave in Thin Rod .................. 60
3.4 Three-Equation Model-I of Elastic Extensional Wave in Thin Rod....................... 70
3.5 Three-Equation Model-II of Elastic Extensional Wave in Thin Rod ..................... 76
3.6 Numerical Results................................................................................................... 85
3.7 Conclusions ............................................................................................................. 90
CHAPTER 4 APPLYING THREE-EQUATION MODEL-I OF ELASTIC WAVE IN
THIN ROD TO ONE-DIMENSIONAL MULTI-BAR IMPACT PROBLEMS AND
APPROXIMATED HOPKINSON BAR IMPACT PROBLEM ...................................... 91
4.1 Introduction............................................................................................................. 91
4.2 Modeling Equations ................................................................................................ 96
4.3 Description of Cases in Computation ..................................................................... 97
4.4 Results ................................................................................................................... 100
4.4.1 Results of Case-I ............................................................................................ 101
4.4.2 Results of Case-II........................................................................................... 109
4.4.3 Results of Case-III ......................................................................................... 117
4.4.4 Results of Case-IV ......................................................................................... 121
4.5 Conclusions ........................................................................................................... 127
CHAPTER 5 THE FIRST ORDER HYPERBOLIC MODELS OF LONGITUDINAL
PLANE WAVE IN ELASTIC-PLASTIC BULK MATERIAL AND ISOTHERMAL
ONE-DIMENSIONAL IMPACT PROBLEM ............................................................... 129
5.1 Introduction........................................................................................................... 129
5.2 Models of Elastic Longitudinal Plane Wave in Bulk Material............................. 135
5.2.1 The Second-Order Linear Wave Equation..................................................... 135
viii
5.2.2 The Two-Equations Model of Elastic Longitudinal Plane Wave in Bulk
Material................................................................................................................... 136
5.2.3 The Isothermal Model-I of Elastic Longitudinal Plane Wave in Bulk Material
................................................................................................................................. 142
5.2.4 The Isothermal Model-II of Elastic Longitudinal Plane Wave in Bulk Material
................................................................................................................................. 148
5.3 The Isothermal Model of Longitudinal Plane Wave in Elastic-Plastic Bulk Material
..................................................................................................................................... 155
5.3.1 Infinitesimal Plasticity ................................................................................... 155
5.3.2 Radial Return Maping.................................................................................... 163
5.3.3 Modeling Equations ....................................................................................... 165
5.4 Computation Settings ............................................................................................ 168
5.5 Numerical Results................................................................................................. 169
5.6 Conclusions ........................................................................................................... 172
CHAPTER 6 THE COMPLETE THERMAL DYNAMIC MODEL OF
LONGITUDINAL PLANE WAVE IN ELASTIC-PLASTIC BULK MATERIAL AND
ONE-DIMENSIONAL IMPACT PROBLEM ............................................................... 173
6.1 Introduction........................................................................................................... 173
6.2 The Modeling Equations ....................................................................................... 174
6.3 Numerical Results................................................................................................. 185
6.4 Conclusions ........................................................................................................... 192
CHAPTER 7 THE TWO-DIMENSIONAL THERMAL MODEL OF STRESS WAVE
IN ELASTIC-PLASTIC BULK MATERIAL AND ULTRASOUND WELDING
PROBLEM ..................................................................................................................... 194
7.1 Introduction........................................................................................................... 194
7.2 Modeling Equations .............................................................................................. 199
7.3 Stress Boundary Condition by FEA...................................................................... 216
7.4 Parallel Computation ............................................................................................ 227
ix
7.5 Numerical Results................................................................................................. 230
7.6 Conclusions ........................................................................................................... 234
CHAPTER 8 THE TWO-DIMENSIONAL ISOTHERMAL MODEL OF STRESS
WAVE IN ELASTIC-PLASTIC BULK MATERIAL AND TWO-DIMENSIONAL
CRACK PROBLEM ...................................................................................................... 236
8.1 Introduction........................................................................................................... 236
8.2 Modeling Equations .............................................................................................. 240
CHAPTER 9 CONCLUSIONS AND FUTURE WORKS............................................. 251
9.1 Conclusions ........................................................................................................... 252
9.2 Future Works......................................................................................................... 254
REFERENCE……………… …………………………………………………………..257
x
LIST OF FIGURES
Figure
Page
Figure 1.1: Blaha and Langenecker [1] reported the first APE by a compression
experiment. ................................................................................................................... 1
Figure 1.2: The Acousto-Plastic Effect (APE) reported by Kirchner et al. [2]. ................. 2
Figure 2.1: A schematic of space-time integral of the CESE method. ............................ 25
Figure 2.2: Schematics of the CESE method in one spatial dimension: (a) zigzagging
SEs; (b) integration over CE to solve ui and (ux)i at the new time level. ................... 28
Figure 2.3: Schematics of the modified CESE method in one spatial dimension: (a) the
staggered space-time mesh; (b) SE (j, n), shown as the yellow part, and CE (j, n). .. 35
Figure 2.4: The space-time mesh in two spatial dimensions: (a) grid points in the x-y
plane; (b) SE and CE for the two-dimensional scheme. ............................................ 38
Figure 2.5: Spatial translation of the quadrilateral A1* A2* A3* A4* ........................................... 45
Figure 3.1: A schematic for linear and nonlinear waves in a thin rod .............................. 55
Figure 3.2: (a) The stress variation at the vibrating end of the rod and (b) the speed of
sound profile at t=0.312 ms calculated by the CESE method. ................................... 86
Figure 3.3: The snapshot (t=0.312ms) of stress wave propagation generated by (a)
numerical solution of nonlinear wave model Eq. (3.54) and (b) theoretic solution of
linear wave Eq. (3.10). ............................................................................................... 87
Figure 3.4: A snapshot (t=0.312ms) of density profile predicted by two-equations model,
Eq. (3.54). ................................................................................................................... 88
Figure 3.5 Snapshots of the normal stress profiles at t = 0.312 ms by using 301, 151, and
75 grid points.............................................................................................................. 89
Figure 4.1: The sketch of split Hopkinson bar apparatus ................................................. 92
Figure 4.2: Typical time history of the strain at the mid point of the incident bar. .......... 94
xi
Figure 4.3: Typical time history of the strain at the mid point of the transmitter bar....... 94
Figure 4.4: Sketches of cases in computation includes (a) Case-I: Aluminum striker bar
hits aluminum pressure bar, (b) Case-II: Aluminum striker bar hits aluminum-copper
pressure bars, (c) Case-III: Aluminum striker bar hits aluminum-copper-aluminum
bar combination and (d) Case-IV: C-350 striker bar hits C-350-copper-C-350 bar
combination................................................................................................................ 99
Figure 4.5: Snapshot-I of stress wave propagation in Case-I ......................................... 102
Figure 4.6: Diagram of stress wave superposition for snapshot-I in Case-I ................... 103
Figure 4.7: A snapsho t of a left-running stress wave after all waves reflect from the right
end of the target bar in Case-I. ................................................................................. 105
Figure 4.8: Stress wave superposition for wave snapshot shown Figure 4.7 for Case-I 106
Figure 4.9: The strain time history at the position of gauge-I in Case-I ......................... 108
Figure 4.10: The strain time history at the position of gauge-II in Case-I...................... 108
Figure 4.11: Snapshot-I of stress wave propagation in Case-II ...................................... 109
Figure 4.12: Snapshot-II of the stress wave propagation after the right-running wave
reaches the aluminum-copper interface in Case-II................................................... 110
Figure 4.13: A stress free aluminum bar with initial speed 5m/s impacts a stress free and
static copper bar ....................................................................................................... 111
Figure 4.14: A static aluminum bar with known initial stress 68MPa contacts a static and
stress free copper bar ................................................................................................ 111
Figure 4.15: The snapshot of stress wave in the problem shown by Figure 4.13 ........... 112
Figure 4.16: The snapshot of stress wave in the problem shown by Figure 4.14 ........... 112
Figure 4.17: Snapshot-III of stress wave propagation in Case-II ................................... 113
Figure 4.18: Snapshot-IV of stress wave propagation in Case-II ................................... 114
Figure 4.19: Snapshot-V of stress wave propagation in Case-II .................................... 114
Figure 4.20: Snapshot-VI of stress wave propagation in Case-II ................................... 115
Figure 4.21: The time history of the strain at Gauge-I in Case-II................................... 116
Figure 4.22: The time history of the strain at Gauge-II in Case-II. ................................ 116
xii
Figure 4.23: Snapshot-I of stress wave propagation in Case-III..................................... 117
Figure 4.24: Snapshot-II of stress wave propagation in Case-III ................................... 118
Figure 4.25: Snapshot-III of stress wave propagation in Case-III .................................. 118
Figure 4.26: Snapshot-IV of stress wave propagation in Case-III .................................. 119
Figure 4.27: Snapshot-V of stress wave propagation in Case-III ................................... 119
Figure 4.28: Snapshot-IV of stress wave propagation in Case-III .................................. 120
Figure 4.29: The strain time history at position of gauge-I in Case-III .......................... 120
Figure 4.30: The strain time history at position of gauge-II in Case-III ......................... 121
Figure 4.31: Snapshot-I of stress wave propagation in Case-IV .................................... 122
Figure 4.32: Snapshot-II of stress wave propagation in Case-IV ................................... 122
Figure 4.33: Snapshot-III of stress wave propagation in Case-IV.................................. 123
Figure 4.34: Snapshot-IV of stress wave propagation in Case-IV.................................. 123
Figure 4.35: Snapshot-V of stress wave propagation in Case-IV................................... 124
Figure 4.36: Snapshot-VI of stress wave propagation in Case-IV.................................. 124
Figure 4.37: The time history of the strain data at Gauge-I in Case-IV. ....................... 125
Figure 4.38: The time history of the strain data at Gauge-II in Case-IV ........................ 125
Figure 5.1: Elastic plane shear wave propagate in x direction........................................ 130
Figure 5.2: The longitudinal plane wave in the bulk material ........................................ 135
Figure 3 Radian return for finite plasticity ..................................................................... 164
Figure 5.4: Initial condition of the one-dimensional impact problem. ........................... 168
Figure 5.5: A snapshot of density at t = 0.17ms in initial static copper bulk. The CESE
numerical result by using the isothermal model is compared to the exact solution by
Udaykumar et al. [55]............................................................................................... 170
Figure 5.6: A snapshot of pressure at t = 0.17 ms in the initial static copper bulk. The
numerical result of the isothermal model by the CESE method is compared with the
exact solution by Udaykumar et al. [55]. ................................................................. 171
xiii
Figure 6.1: A snapshot of density in the right copper bulk (initially stationary) at t = 0.17
ms. ............................................................................................................................ 187
Figure 6.2: A snapshots of the pressure profile in the right copper block at t = 0.17 ms.
.................................................................................................................................. 188
Figure 6.3: The snapshots of density in the right copper bulk at t = 0.17ms at three
different initial impact speeds: (a) u = 80m/s, (b) u = 200m/s, and (c) u = 1000m/s.
.................................................................................................................................. 189
Figure 6.4: Snapshots of pressure profiles in the right copper bulk at t = 0.17 ms at three
different initial impact speeds: (a) u = 80 m/s, (b) u = 200 m/s, and (c) u = 1000 m/s.
.................................................................................................................................. 191
Figure 7.1: The USW process with normal pressure and transverse vibrations. ............ 195
Figure 7.2: The nonlinear wave pattern of plastic deformation textures at the contact
interface in commercial AA6111-T4 alloys produced by the USW process, from [82].
.................................................................................................................................. 197
Figure 7.3: Dimensions of aluminum coupon with deformation, units are mm. ........... 217
Figure 7.4: Tip vertically penetrates into the metal coupon. (a) Initial shapes and position
of the coupon and the tips. (b) A magnified view of the deformed mesh. ............... 219
Figure 7.5: The motion of one tooth in one vibration cycle: (a) the initial condition, (b)
1/4 cycle, (c) 3/4 cycle, and (d) at the end of one cycle........................................... 221
Figure 7.6: The stress profiles around one top tooth at one time point in one vibration
cycle: (a) mesh points along one tooth, (b) the pressure profile, (c) the profile of
stress component S11 , (d) the profile of stress component S12 , and (e) the profile of
stress component S22 ................................................................................................. 224
Figure 7.7: The stress profiles around one bottom tooth at one time point in one vibration
cycle: (a) mesh points along one tooth, (b) the pressure profile, (c) the S11 profile. (d)
the S12 profile, and (e) the S22 profile. ...................................................................... 226
Figure 7.8: The detail of mesh and mesh decomposing for parallel computation, (a)
original mesh of whole domain, (b) mesh of one tooth, and (c) decomposed mesh. 229
Figure 7.9: Snapshots of the evolving interface of two joined aluminum plates. Wavy
pattern and roll- up occur as time progresses. The interface is tracked by solving a
Level Set Method (LSM) equation passively coupled with the continuum mechanics
equations................................................................................................................... 231
Figure 7.10: Snapshot of velocity vectors superimposed on the deformed interface. .... 232
xiv
Figure 7.11: Effective stress distribution inside the metal coupon. ................................ 232
Figure 7.12: Snapshots of evolving nonlinear waves: (a) pressure, and (b) density. ..... 234
Figure 8.1: A two-dimensional strip with a semi- infinite crack .................................... 237
Figure 8.2: Lin [52] conducted numerical computation and compared his results with the
experimental results. (a) Simulated wave pattern, and (b) comparison with the
experimental data for velocity on the sample boundary. ......................................... 238
Figure 8.3: Giese and Fey [72] conducted similar computation with conditions shown in
(a) and presented results shown in (b)...................................................................... 239
xv
LIST OF TABLES
Table
Page
Table 1: The lengths of the bars used in the calculated bar- impact problems ................ 100
Table 2: Material parameters .......................................................................................... 100
Table 3: Material properties of copper............................................................................ 169
xvi
CHAPTER 1
INTRODUCTION
1.1 Motivation and Objectives
When superimposing ultrasonic vibrations on a static compression or tensile loading, the
metal specimen being tested would experience remarkable transitory softening. This
phenomenon is called the Blaha effect, or the Acousto-Plastic Effect (APE).
Figure 1.1: Blaha and Langenecker [1] reported the first APE by a compression
experiment.
In 1955, Blaha and Langenecker [1] conducted a compression test by using a zinc
crystal. They first applied a static compression to the metal. Then they applied
ultrasounic vibrations to the metal at a frequency of 800 kHz. When the vibrations were
1
turned on, the reading of the static compression force reduced 40%. Refer to curve A in
Figure 1.1. In other words, with the applying ultrasounic vibrations, a much lower normal
compression stress, around 60% of regular yield stress, could cause the plastic
deformation. In another test, they simultaneously applied ultrasound vibrations with the
static compression from the beginning. As such, the stress-strain curve followed that of
Curve B in Fig. 1.1, instead of Curve A, and the yielding stress of curve B is about 40%
less than that of Curve A. They concluded that the applied ultrasounic vibrations incurred
a much lower yield stress than the regular yield stress of the zinc crystal.
Figure 1.2: The Acousto-Plastic Effect (APE) reported by Kirchner et al. [2].
After Blaha and Langeneck[1] reported the APE in 1955, many researchers have
independently conducted similar experiments to study the phe nomenon. Figure 1.2 shows
a typical APE test reported by Kirchner et al. [2]. The tested specimen was an hourglassshaped aluminum alloy 6061. A double resonator subjected the specimen to a cyclic
2
loading of 20 kHz. The background loading is a quasi-static compression, increasing at a
strain rate of 1×10-4 sec-1 . Shown in Figure 1.2, when the vibratory loading was turned on
at Point B, the measured mean stress dropped to a lower value at Point C. After cessation
of vibrations at Point D, the response of the metal returned to Point E on the original
static stress-strain curve. At Point F, the ultrasonic vibrations were applied again and the
measure mean stress dropped to Point G, and so forth.
To date, many experimental works have been performed on various materials.
The data consistently showed significant stress decreases up to 80%. Owing to the
advantageous stress reduction, ultrasound-aided metal forming/joining processes have
been developed, including forging, extrusion, drawing, welding, etc. In almost all
ultrasound-aided metal processing applications, reduction in the forming forces and/or
increases in the formability of the work pieces were reported. For example, in forging,
the force was significantly reduced [3-7]. Early works on aluminum [3] showed that not
only the forging force could be reduced to zero, but the barreling could be eliminated. By
applying ultrasounds to metal extrusion, the force could be reduced up to 40% and the
rate of extrusion would increase up to 300%. In wire drawing and tube-drawing processes,
ultrasounds were applied along the wire axis and the drawing forces could be reduced up
to 60% [8-10]. Moreover, the applied ultrasounds improved the surface finish, and the
wire production of more complex sections were achieved [11, 12]. In deep drawing, the
applied ultrasounds improved the drawing depth up to about 15% [13-19]. In wall ironing,
the ultrasounds increased the maximum area reduction from 63 to 80% [20].
3
In spite of the widely known APE and its obverious advantages in metal forming
and joining applications, the basic understanding of the APE mechanisms is far from
complete. All ultrasound–aided processes for metal joining and forming fall into a
category of technologies, for which applications preceded basic understanding of the
physical processes.
1.2 Literature Review
This section will summarize the basic findings related to the APE in the literature.
The content will focus on the following aspects of the processes: (i) Metal softening
during the application of acous tic energy; and (ii) Strain hardening induced by the
applied ultrasounic vibrations, which appears after a period of time, following the
applications. In particular, we will focus on the following operational parameters:
1. Dependence of the APE on temperature.
2. Dependence of the APE on the frequency of the imposed vibrations.
3. Dependence of the APE on the amplitude of the imposed vibrations.
4. Dependence of the APE on the energy input.
In the past, several hypotheses have been proposed to explain experimental results
related to the APE. These hypotheses can be categorized into the following three groups:
(i) the potential- well hypothesis, (ii) the stress-superposition hypothesis, and (iii) the
energy-superposition hypothesis. In what follows, we provide the background
information about these works.
4
The potential-well hypothesis assumes that dislocations in metal grains are
activated by the imposed sound energy. Essentially, the energy of the sound waves is
assumed to be absorbed by metal grains. Thus, the average yield strength, or the apparent
static yield strength, of metal becomes lower. In 1959, Blaha and Langenecker [21]
proposed that sound waves would lift the energy level of the absorbing dislocations from
their equilibrium positions. When the static stress is applied to the metal sample, these
dislocations would move, and the onset of plastic deformation would occur.
Consequently, the apparent “static stress” needed for plastic deformation is reduced. In
the literature, the potential well model was illustrated through phenomenological
arguments. No mathematical formula has been developed.
To further investigate the potential-well theorem, Langenecker [22] conducted
experiments on zinc, aluminum, beryllium, tungsten, low-carbon steel and stainless steel.
He concluded that a certain relation existed among (i) the lattice imperfection, (ii) applied
static stresses, and (iii) the imposed ultrasonic vibrations. For instance, the experiment on
zinc showed that the ultrasonic energy input of only 1012 ev/unit caused about the same
level of “static stress” reduction (nearly 40%) in a zinc crystal as that by a thermal energy
input of 1020 ev/unit. The huge difference in the energy inputs between the two means for
the same yielding effect was regarded as the evidence that sound energy was absorbed
only by lattice imperfections, whereas thermal energy was absorbed homogenously by
the whole material.
Langenecker further proposed that the APE is independent of vibration frequency
or the temperature of the material being tested. On the other hand, the APE strongly
5
depends on the amplitude of the imposed vibrations. Langenecker [23] conducted a
extension test on zinc crystals at room temperature (295k) with the static loading at a
strain rate of 2×10-4 1/sec. The frequency of the applied ultrasound was 25 kHz. For
vibrations at low amplitudes, e.g., P < 1.2×107 dynes/cm2 , he found that no work
hardening after the ultrasounic processes. Work hardening became detectable when
sound pressure amplitude was high enough, e.g., P > 1.2×107 dynes/cm2 , and increased
with increasing pressure amplitudes up to P ≈ 2.7×107 dynes/cm2 . According to his
experiments, Langenecker reported that the ultrasound-activated work hardening could be
superior to that aschieved by conventional straining processes. He also confirmed that the
applied ultrasonic vibrations induced metal softening and thus made plastic deformation
much easier.
In 1966, Langenecker [24] proposed that dislocation theory and the theory on
mechanical wave propagation were inadequate to describe the interactions between highamplitude ultrasonic waves and the dislocations in metal crystalline grains. The theories
seemed to provide a reason of the APE during the static loading in conjunction with the
superimposed ultrasounds. However, the theorem could not explain the metal hardening
which would appear after the ultrasound actions. He continued comparing the effects of
the applied ultrasounds and the applied heat in reducing the apparent shear stress. Under
specific conditions, both ultrasounds and heat could reduce the apparent shear stress to
null. The difference was that the acoustic softening would take place immediately when
ultrasonic irradiation is applied. On the other hand, a significant time period is needed for
homogeneously increasing temperature to activate the plastic deformation.
6
Furthermore, Langenecker [25] reported that the level of ultrasounic energy input
is directly related to the permanent change of material properties. If the energy input by
the ultrasound is lower than a threshold value, which depends on the metals and other
conditions, there is no permanent change in material properties. When the energy exceeds
the critical value, a permanent change in material always occurred.
In 1972 Langenecker [25] reported more experiments on the effect of the
ultrasound energy level by applying high-power ultrasonic oscillations at 20 kHz on
aluminum, copper and steel. The experimental results showed that Young’s modulus of
specimen changed by the applied ultrasounds. He also proposed an energies superposition
hypothesis : Eth + E th + E a = E def , where Eth is the thermal energy of a dislocation at the
test temperature, Eth denotes the heat produced through internal frictio n of oscillating
dislocations, Ea is the energy resulting from acoustic strain, and Edef is the energy
required for plastic deformation. He proposed to use this new energy theorem to replace
the potential- well hypothesis. The equation that he presented, however, was only for
illustrating the concept and it has not been used to provide quantitative analyses.
In 1969 Baker and Carpenter [26] carried out APE experiments by using
tantalum. They reported that the dislocation motions in the APE were thermally activated.
They also proposed a linear relation between the stress variation and oscillating stress
amplitude. In a similar effort, Endo in 1979 [27] reported experimental results of tension
tests using a metal Fe-3%Si. They reported that the influence of the temperature effect
on the stress reduction for the onset of the plastic flows could be neglected. To date, the
7
temperature effect on the APE remains controversial. More works on both experiments
and theorems are needed to clarify the APE mechanism.
In 1984, Kirchner et al. [2] conducted compression tests on aluminum alloys, i.e.,
compacted powder aluminum alloys 7090 and 7091, and aluminum alloy 6061. In those
experiments, they simultaneously applied a static loading, at the strain rate is 10-4 sec-1 ,
and a vibratory loading, at 0.5, 1.0, 10.0, 50.0, and 20 kHz, to the metal samples.
Different from previous works on the APE, they reported the first experimental study of
the APE at low frequencies. They emphasized the importance of studying the actual
stresses inside the samples. They conjectured that a stress dip could occur due to elastic
relaxation, as shown in Figure 1.2. Their experimental results showed no frequency
influence on the APE at room temperature. They did not observe the microscopic
structural changes such as rearrangement of dislocation structures. They believed that
apparent stress reduction caused by ultrasonic vibrations could be explained within the
framework of elastic-plastic behavior of the material. However, they recognized that it
was extremely difficult to determine the stress level inside the specimen by experimental
methods because the stress distribution inside the material is inhomogeneous, and the
ultrasonic frequency exceeds the frequency response of mechanical extensometers. They
expected that a suitable numerical model could offer certain advantage s in this area. They
also suggested the use of destructive ultrasonic test, a timesaving method, to shorten the
duration of experiments for characterizing the fatigue behavior of the tested metals.
In 1987, Ohgaku and Takeuchi [28] carried out compression tests on crystals of
KCl, NaCl, and NaBr. They applied oscillatory unidirectional stresses at 20 kHz on the
8
sampels. They confirmed that the effect of temperature on the APE was negligible. They
also reported that higher amplitudes of the imposed vibrations led to reduction in the
“static- loading stress.” However, they disagreed with the theorem of stress superposition.
Kozlov and Seliyser [29, 30] also studied APE at low frequency at about 1Hz. By
using (i) the exponential relationship between the plastic strain rate and the stress, and (ii)
a theory of thermally activated motion of dislocations, they proposed a theoretical model
to describe the process of plastic deformation of a crystal under alternating stresses. By
solving the model equation, they obtained theoretical solutions of the APE processes for
imposed vibrations at small and large amplitudes. The results correlated well with the
experimental data reported by Kirchner et al. in 1984 at both low and high frequencies.
Contrast to previous ly reported APE works, they found that the APE processes depend
not only on the amplitudes of the imposed vibrations but also on the frequencies. While
the dependence of the APE on the amplitudes of imposed vibrations has been generally
accepted, the role of frequency of the imposed vibrations in the APE has not been clear.
Tanibayashi [31] proposed an empirical equation to relate the stresses to the
dislocation velocity v:
v = v0 (σ e σ c ) , for σ e > 0, and σ e = σ a − σ i ,
m
(1.1)
where v0 , σ c are constants, σ e is the effective stress, σ a is the applied stress, σ i is the
internal stress, and m is a positive number. He proposed the following equation to
calculate the strain rate:
9
  σ − ∆σ + σ sin ϖ t − σ m
a
v
i
ε&
 for (σ a − ∆σ + σ v sinϖ t − σ i ) > 0 ,
ε& ( t ) =  0 
σc


0
otherwise

(1.2)
where ε& ( t ) is the strain rate, which is a function of time, ε&0 is the initial strain rate, σ v is
the amplitude of oscillatory stresses, and ∆σ is the decrease of the applied static stress.
Tanibayashi proposed that dislocations are motionless when the applied stress is lower
than a certain threshold level. This threshold stress level is independent of the frequency
of applied stress when the applied vibrations are not close to the frequency of vibrating
dislocations.
Note that the internal stress was neglected in both the thermodynamic equation by
Baker and Carpenter [26] and the empirical model equation by Kaiser and Pechhold [32].
These two equations cannot correctly predict dislocation velocity when the amplitudes of
applied alternating stress is larger than the effective stress. Tanibayashi used the
superposition theory to explain the mechanism of APE. Although Ohgaku and Takeuchi
[28] disagreed with the stress superposition theory, Tanibayashi thought the experimental
data obtained by Ohgaku and Takeuchi could still be analyzed by using the superposition
model if stress exponent m for a high strain rate was greater than that for a low strain rate.
In 2000, Malygin [33] proposed that the key feature of the APE was not the
vibratory characteristics of the applied alternating stresses, but whether the alternating
stresses increased or decreased the overall effective stress. He developed a stress
superposition model and studied the kinetics of the AP E process. He focused on the
relations between the magnitudes of the imposed vibrations and the amount of plastic
10
deformation, temperature, and the strain rate. He proposed that the effective stress
influenced by the imposed vibratory stresses as
σ *~ ( t ) = σ * + σ m cos ϖ t ,
(1.3)
where σ * is the effective stress without considering the acoustic stress, ϖ = 2π f , f is
the frequency of the oscillatory stresses, and σ m is the magnitude of the oscillatory
stresses. In Eq.(1.3), σ * is defined as:
σ * = σ − σ f −σ µ ,
(1.4)
where σ is the stress applied to the specimen, σ f is the thermal component of friction
stress due to the interaction of dislocations with impurity and their clusters, and σ µ ( ε ) is
the strain hardening of the specimen incurred by the interacting dislocations. The stain
rate in the model is expressed as:
( )
 H σ* 
ε& = ε&v exp −
,
kT 

(1.5)
where ε&v is the vibratory strain rate, H ( σ * ) is the activation energy, K is the Boltzmann
constant, and T is temperature. Based on the property of elastic-plastic material, a relation
between stress increment and strain rate can be given as
dσ
= ε&0 − ε& ,
Edt
(1.6)
where E is Young’s modulus, and ε&0 is the constant strain rate determined by the
machine used in experiment. The activation energy can be expressed by a linear function:
11
( )
H σ * = H 0 − Vσ * ,
(1.7)
where V = dH dt is the activation volume. The stress could be calculated by integrating
Eq. (1.6). The stress without the applied vibrations is
σ ( t ) = Eε&0t −
kT
ln 1 + q ( t )  ,
V
(1.8)
where
VE
V

q(t) =
ε&0 ( 0) ∫ exp   Eε&0t −σ f − σ µ ( ε )   dt ,
kT
 kT

0
(1.9)
 − H0 
ε&0 (0 ) = ε&v exp
.
 kT 
(1.10)
t
and
The stress influenced by the applied alternating stresses is expressed as
σ ~ ( t ) = Eε&0t −
kT
ln 1 + q~ ( t )  ,
V
(1.11)
where
q~ ( t ) =
t
VE
V

ε&0 ( 0 ) ∫ exp   Eε&0t − σ f − σ µ ( ε ) + σ m cosϖ t   dt .
kT
kT


t1
(1.12)
In an APE process, the decrease of the “static stress” is given by
∆σ (t ) =
k T 1 + q (t )
ln
V 1 + q~ (t )
12
(1.13)
In the past decades, various research works have been carried out to investigate
the APE phenomena. However, due to the complexity of the processes, clear illustration
about the mechanism of the APE process has been scant. Many experimental results,
instead of aiding the illustration, have been supplying more confusive pictures to the
basic understanding of the processes. Nevertheless, in what follows, a brief summary of
illustration about the APE reported by previous researchers is provided.
During the application of ultrasound vibration, the acoustic energy softening
effect for solids is distinctly different from that in a thermal softening process, in which
an excessive amount of energy is homogeneously absorbed by the solid materials. The
APE is a more effective means to cause metal yielding, mainly because only a small
amount of energy is needed and the effect is instantaneous. Some of previous studies
conjectureed that, in the APE, the vibrational energy is absorbed by the dislocations of
grains [22, 24]. Moreover, most of research works acknowledged that the influence of
temperature on the APE is negligible [2, 26-28, 33].
Occurrence of strain hardening in an ultrasonic aided process [23, 24] is much
quicker and more effective as compared to that in conventional strain hardening
processes.
In an APE process, no apparent change of material can be detected if the power of
the imposed vibrations is lower than a threshold value [2].
High energy input by
ultrasonic vibrations would cause permanent changes in material properties [23, 25].
Since the amplitude of the imposed vibrations is proportional to the input energy, it is
13
widely accepted that the amplitude of the vibratons is an important factor for the APE
processes [2, 25-33].
Contrast to the general consensus regarding the role of the vibrational amplitude
on the APE, two contradicting opinions exist about the frequency effect on the APE.
Most of researchers thought that the frequency effect is negligible [2, 26-28, 33]. Others
opposed the conjecture [29, 30].
In more than a half of century, various theories have been proposed to describe
the basic mechanisms of the APE process, but a comprehensive mathematical model is
unavailable. The potential-well hypothesis [21] and the energy superposition hypothesis
[25] have be formulated. However, the formualitons have not been used to perform
quantitatively analyses on the APE processes.
The stress superposition hypothesis or the superposition model has been
developed and used to analyze experimental results [2, 29-31, 33]. As shown in Eqs. (1.3)
-(1.13), Malygin [33] provided the formulas to calculate the stress decrease caused by the
stresses induced by the imposed vibrations. However, in the process of derivation, he had
to use empirical equations and certain assumptions, which cannot be easily validated.
Consequently, the existing superposition model is very specific for certain processes with
many inherent limitations.
As stated by Kirchner et al.[2], current devices for measuring stresses cannot
provide detailed information of dynamic stress profiles inside the tested specimen, to
which alternating stresses are applied. However, instantaneous stress profiles are
14
critically important to understand the APE. Various spots inside the metal sample could
experience that the local stresses sporadically exceed the yiled stress limit and thus local
plastic flow occurs. Thus, high- fidelity simulation of nonlinear stress waves inside solid
specimens by numerical modeling could be a fruitful exercise to gain basic understanding
about the APE.
1.3 The Approaches of the Present Work
To study the APE, a new theoretical framework for nonlinear stress waves in metals will
be developed. The model equations will be solved numerically to simulate evolving
nonlinear stress wave inside solids. The modeling capabilities will be able to capture the
morphology of metal deformation as well as the propagation of nonlinear stress wave in
solids. The envisioned modeling tool will be a synergy of a set of advanced theoretical
equations and the CESE method for numerical solution. The primary goal is to
understand the APE through observing time-accurate solutions of nonlinear stress waves
in solids.
The theoretical model will be developed to directly address the nonlinear stress
wave and large-scale deformation in ultrasound aided metal joining/forming processes.
The model equations will be composed of a set of convection-diffusion partial differential
equations based on the conservation laws, including conservation of mass, momentum,
and energy. These conservation laws are universal for any continuum medium. Note that,
nonlinear stress waves cause large-amplitude changes not only in forces, motions, and
deformations, but also in energies and temperatures. Therefore, significant temperature
15
gradients inside the metal specimens will also be considered. As such, the energy
equation will be considered in the present work. As such, an equation of state must also
be employed to relate density, pressure, and temperature (or internal energy). For
deformations with small amplitude, isothermal models without considering the energy
equation will also be developed in the present work.
The second part of the model equations will be derived based on the constitutive
equation. For linear elasticity, conventional Hooke’s law will be used. For plasticity, a
suitable relation between the deviatoric stress and the rate of deformation will be used. I
then apply a time derivative to the adopted constitutive relation, usually an algebraic
relation. The resuls is a set of convection-diffusion equation for the evolving stress
components. These stress equations will be fully coupled with the partial equations for
the conservation laws.
The theoretical model includes a set of fully coupled partial differential equations
formulated in terms of primitive variables, i.e., density, velocities, internal energy, and
Cauchy (or devitoric) stresses. The equations are defined in the Eulerian frame. To close
the equation, an equation of state will be employed. As will be shown in the following
chapters, the model equations are hyperbolic in time with nonlinear convection terms
suitable for the simulation of evolving nonlinear stress waves.
Contrast to previous works on the APE, The model equations will be able to
model a wide range of deformations, strains, strain rates, and stresses. In particular, the
advantages of the present model equations are twofold: (i) The theoretical model will be
16
built based on the conservation laws for nonlinear stress waves; they will not be derived
based on a certain solution structure with inherent mathematical restrictions. (ii) The
model equations will be an open framework, readily to be extended to include new
constitutive relations for complex elastic-plastic deformations as well as suitable
microscopic models in the future.
1.4 Numerical Methods for Hyperbolic System
The present research work is a synergy of analytical and numerical effort to study
the APE process. The above hyperbolic model will be solved by using the CESE method
[34-44], which is an advanced numerical framework to obtain high- fidelity solution of
the nonlinear hyperbolic equations.
In numerical solution of the equation set, one has to directly address the inherent
Riemann problems [40, 45] that naturally arise as the solution of a continuum mechanics
model equations in describing the nonlinear wave propagations.
Commercial Finite
Element Analysis (FEA) tools have been widely used to provide numerical solutions to
various elastic/plastic deformations in solid mechanics. For deformation in plastic
regimes, e.g., crash-worthiness tests and stamping processes, advanced features have
been developed and extensively used, including the use of the Eulerian-Lagrangian
formulation, moving meshes, explicit time- marching methods, etc. However, FEA
packages are ill-equipped for time-accurate simulation of nonlinear stress wave
propagation, because the conventional FEA tools do not attempt to provide the Riemann
solution of the hyperbolic systems.
In other words, the FEA solutions provide no
17
resolution to track the stress waves. As will be shown in the following chapters, the
eigenvalues of the Jacobian matrices of the hyperbolic equation represent the wave
speeds. One has to employ a suitable numerical method to provide numerical resolution
in both space and time for faithful numerical solutions of the propagating waves.
In contrast to conventional FEA tools based on solving the elliptic and/or
parabolic equation for elastic/plastic deformations, predicting nonlinear stress waves
requires the use of a time-accurate hyperbolic solver. Moreover, because large-scale
deformations occur, both elastic and plastic deformations must be included in the wave
model equations. Therefore, the wave dynamics model to be developed is very different
from the standard vibration problems in which the solutions of linear elastic waves are
calculated.
The space-time Conservation Element and Solution Element (CESE) method [3444] will be used in the present work. The CESE method is a novel numerical method for
high- fidelity solution of nonlinear hyperbolic systems. Successful development of the
proposed research will demonstrate a new paradigm for high- fidelity simulation for
nonlinear stress waves in solid mechanics. The proposed approach directly simulates the
complex nonlinear waves in metals based on time-accurate solution of the hyperbolic
partial differential equations.
The ultrasound aided metal forming and joining processes could be greatly
enhanced by the theoretical and numerical capabilities to be presented in the present
dissertation. In addition to the new theoretical and numerical capabilities, one can apply
18
the simulation tool to the APE and gain in-depth understanding of the processes. In
general, the number of compositions and processing parameters in the ultrasound-aided
processes are impossible to navigate experimentally. By testing the new theoretical and
mathematical models, and by identifying necessary new physics to be added to the
existing theory, this research work will build the key ingredients to an eventual control
and optimization of the ultrasound aided metal forming and joining processes. With
accurate models, manufacturers will be able to explore processing parameter space,
thereby predict material behavior and the necessary process adjustments to achieve
successful implementation. The experience and understanding gained will be
indispensable for the further development and controlling of the APE, leading to efficient
mass production of ultrasound-aided metal forming/joining processes. Further extension
of the numerical tool will widen the scope of potential applications of the APE to other
manufacturing processes.
The rest of the dissertation is organized as follows. Chapter 2 presents a brief
review of the CESE method for solving hyperbolic system. The time marching schemes
of the CESE method in one and two spatial domains will be presented. Chapter 3 reports
the development of a one-dimensional hyperbolic model for elastic waves in a thin rod.
Detailed analyses of the eigen-structure of the hyperbolic model equations will be
provided. Numerical solution of elastic longitudinal wave in a thin rod by the CESE
method will be compared with the analytical solution. Chapter 4 presents a three-equation
model for elastic wave in a thin rod. To demonstrate the capabilities of the model
equations and the numerical solution, One-dimensional computations simulate the waves
19
propagation in multicomponent impact problems. Chapter 5 describes the development
of one-dimensional isothermal models for longitudinal waves in an bulk material.
Analyses of the eigen-structure of the equations will be provided. The model will be
applied to a one-dimensional impact problem to study elastic-plastic longitudinal plane
waves in a bulk material. In Chapter 6, I will extend the above one-dimensional models
by adding the energy equation and an equation of state for the material employed. This
new one-dimensional model will be applied to an impact problem and analyze the effect
of the energy equation on the numerical solutions. Chapter 7 reports the numerical
solutions of the ultrasound welding processes in two spatial dimensions. The complete
model equations will be employed, including the energy equations and a suitable
equation for state for the metal. The calculated nonlinear elastic-plastic stress wave inside
aluminum specimens under a static compression loading and the lateral ultrasonic
vibrations will be reported. The simulated interface morphology will also be reported. In
Chapter 8, one classical problem of two-dimensional wave papagation, crack growth in a
strip, is introuduced. Moreover, the model of isothermal elastic-plastic waves in twodimensional plates is presented. Finally, the conclusions and suggestions of future works
are provided in Chapter 9.
20
CHAPTER 2
THE SPACE-TIME CESE METHOD
The main challenge in adopting the hyperbolic equations to model stress waves in solids
is numerically solving these coupled nonlinear equations for the temporally evolving
processes. Conventionally, a finite-volume approach in conjunc tion with a Riemann
solver has been used to solve this type of systems of equations. When the stress level is
extremely high and shock waves and contact discontinuity are of interest, a limiter
functions is employed to treat the jump conditions. The original framework for
hyperbolic nonlinear equations was developed by Godunov [46] and the so-called upwind
method has been successfully used to solve various nonlinear wave problems. Recent
development of modern upwind schemes is a direct extension to the classical Godunov
method in three aspects: (i) extension to second-order or higher in both spatial an
temporal resolution; (ii) development of various approximate Riemann solvers for more
efficient calculations; and (iii) development of advanced limiter functions for crisp
resolution of jump conditions.
However, in extending this scheme from one-dimensional to multi-dimensional
problems, a major theoretical difficulty has been encountered. In the setting of ideal gases
for aerodynamics, the analytical solution of one-dimensional Riemann problem is readily
available. For flows in multiple spatial domains, the Riemann solution is not available.
21
Thus, one would simply employ directional splitting when treating two- and threedimensional problems. In other words, the solution of a two- or three-dimensional
Riemann problem is approximated by a superposition of two or three one-dimensional
Riemann solutions along the coordinate axes. While the practice is commonplace and
quite successful in solving gas dynamics equations, its application to more complex
Riemann problems such as plasma flows and solids mechanics with complex constitutive
relations could pose serious problems.
To overcome this difficulty, Colella [47] introduced the corner transport
upwinding method. The goal was to do away with the operator splitting. The numerical
fluxes were obtained by solving the characteristic form of the multi-dimensional
equations at the zo ne edge. Solutions of the Riemann problem were used to correct
discontinuous solutions.
Miller and Colella [48] presented an explicit second-ordered Godunov method for
solid mechanics problems in one and multiple spatial dimensions. Operator splitting was
not applied in the two- and three-dimensional problems. The equations were written in
non-conservative form, and deformation tensor was solved directly. Numerical results
showed that plastic/elastic shock waves could be captured by about five nodes.
Trangenstein [49, 50], and Trangenstein and Pember [51] developed a secondorder extension of the Godunov method. The idea was to handle the impact problem of
multiple materials as a Riemann problem. Lin and Ballmann [52] used a CFD method by
Zwas [53] and reported numerical solutions of wave propagation around a crack. They
22
combined the method of bi-characteristics and a finite-difference scheme for a secondorder Godunov scheme [52]. They validated the scheme by applying it to multidimensional dynamic problems in elastic-plastic solids, including anti-plane shear
problems, plane strain problem [52] and axisymmetric stress wave propagation in linear
elastic solids [52]. They also provided a comprehensive review of simulation of stress
wave in solids [52].
LeVeque [54] developed a general solver, CLAWPACK, for generic hyperbolic
systems. He used the toolkit to solve the solid mechanics problems by solving the
Riemann problem and applying limiter functions. He reported numerical solutions of
multi-dimensional waves in solids.
Udaykumar et al. [55] employed a high-order ENO scheme and an interface
tracking technique to calculate a
multi- material impact problem. To automatically
capture shock as well as immersed material boundaries, Tran and Udaykumar [55]
employed a high-order accurate ENO and hybrid particle level set technique in problem
of a Tungsten rod impact and penetrate into steel plate.
Contrast to the above upwind methods, Fey [56, 57] proposed a new approach to
solve the hyperbolic equations for solids in multiple spatial dimensions. Referred to as
the Method of Transport, the method does not rely on Riemann solvers. Instead, the
method is designed by a more general definition of the waves in conjunction with the
concept of consistency of a set of wave vectors.
23
Contrast to the modern upwind schemes, Chang [34] introduced the Conservation
Element and Solution Element (CESE) method, a novel numerical framework for
hyperbolic conservation laws, without using Riemann problem solution at all.
Hence, CESE method does not have the trouble of expanding scheme from onedimensional to multidimensional. Chang, Yu and their coworkers [34, 37-41, 43, 44, 5860] already showed the high-resolution shock-capturing capability of CESE, easily
expanded this scheme to 2D and 3D problems, and have reported a wide range of highly
accurate solutions of hyperbolic systems, including detonations, cavitations, complex
shock waves, turbulent flows with embedded dense sprays, dam breaking flows, MHD
flows, aeroacoustics. CESE has never been applies in the case of stress wave propagation
in solids. Validating CESE method in a problem with analytical solution will be a good
attempt.
2.1 One-dimensional CESE method
Finite volume methods are formulated according to a flux balance over a fixed
spatial domain. The conservation laws state that the rate of change of the total amount of
a substance contained in a fixed spatial domain, i.e., the control volume V, is equal to the
flux of that substance across the boundary of V, denoted as S(V). Consider the differential
form of a conservation law as follows:
∂u
+ ∇ ⋅f = 0
∂t
24
(2.1)
where u is density of the conserved flow variable, f is the spatial flux vector. By applying
Reynolds’s transport theorem to the above equation, one can obtain the integral form as:
∂
udV + Ñ∫ f ⋅ ds = 0
∂t ∫V
S (V )
(2.2)
where dV is a spatial volume element in V, ds=ds n with ds and n being the area and the
unit outward normal vector of a surface element on S(V) respectively. By integrating
Eq.(2.2), we have
 udV  −  udV  + f
 ∫V
t =t f  ∫V
 t= t s ∫ts
t
( Ñ∫
S (V )
)
f ⋅ ds dt = 0
(2.3)
The discretization of Eq.(2.3) is the focus of the finite-volume methods.
t
ds
V
dr
r+dr
r
S(V)
x
Figure 2.1: A schematic of space-time integral of the CESE method.
Let’s consider the one dimensional case first. Let time and space be the two orthogonal
coordinates of a space-time system, i.e., x1 = x and x2 = t . They constitute a two25
dimensional Euclidean space E2 . Define h @ ( f , u ) , then by using the Gauss divergence
theorem, Eq.(2.1) becomes
Ñ∫
h ⋅ ds = 0
(2.4)
S (V )
Eq.(2.4) states that the total space-time flux h leaving the space-time volume V through
S(V) vanishes. Refer to Figure 2.1 for a schematic of Eq.(2.4).
To solve this kind of conservative- form equation, i.e. Eq. (2.4), we employed the CESE
method [34], a novel numerical framework for hyperbolic conservation law. The tenet of
the CESE method is the uniform treatment of space and time in calculating flux
conservation.
Based on the CESE method, a suite of computer one-, two-, and threedimensional codes using structured and unstructured meshes have been developed. The
two- and three-dimensional codes have been parallelized and can be used to perform
large-scaled simulations of nonlinear stress waves in fluids and solids. In the present
paper, only basic ideas of the CESE method in one and two spatial domain will be
illustrated.
In the CESE method, separated definitions of Solution Element (SE) and
Conservation Element (CE) are introduced. In each SE, solutions of unknown variables
are assumed continuous and a prescribed function is used to represent the profile. In the
present calculation, a linear distribution is used. Over each CE, the space-time flux in the
integral form, Eq.(2.4), is imposed.
26
Figure 2.2 shows the space-time mesh and the associated SEs and CEs. Solutions
of variables are stored at mesh nodes which are denoted by filled circular dots. Since a
staggered mesh is used, solution variables at neighboring SEs leapfrog each other in timemarching calculation. The SE associate with each mesh node is a yellow rhombus. Inside
the SE, the solution variables are assumed continuous. Across the interfaces of
neighboring SEs, solution discontinuities are allowed. In this arrangement, solution
information from on SE to another propagates only in one direction, i.e., toward the
future through the oblique interface as denoted by the red arrows. Through this
arrangement of space-time staggered mesh, the classical Riemann problem has been
avoided. Figure 2.2(b) illustrates a rectangular CE, over which the space-time flux
conservation is imposed. This flux balance provides a relation between the solutions of
three mesh nodes:
( j, n ) , ( j − 1 2 , n − 1 2 ),
and ( j + 1 2 , n − 1 2 ) . If the solutions at time
step n − 1 2 are known, the flux conservation condition would determine the solution at
( j, n ) .
27
(a)
(b)
Figure 2.2: Schematics of the CESE method in one spatial dimension: (a) zigzagging
SEs; (b) integration over CE to solve ui and (ux)i at the new time level.
28
In the present research, most of problems are expressed by the differential equations
which have the source terms on the right side. Thus, we consider the one-dimensional
equations with source terms:
∂um ∂f m
+
= µ m , m = 1, 2.
∂t
∂x
(2.5)
Note the source term µm is the function of um . Let x 1 = x, x2 = t and h m @ ( f m , um ) . By
using Gauss’ theorem, we have
Ñ∫
S (V )
hm ⋅ ds = ∫ µ mdV
V
(2.6)
For any (x, t) ∈ SE (j, n), um (x, t), fm (x, t) and hm (x, t), are approximated by u* ( x, t ; j ,n ) ,
f * ( x , t ; j ,n ) , and h* ( x , t ; j ,n ) , respectively. By assuming linear distribution in SEs, we
have
u*m ( x, t ; j , n ) = (um )nj + (umx ) nj ( x − x j ) + (umt ) nj (t − t n )
(2.7)
Let ( f m ) nj and ( f m, l ) nj denote the value of f m and ?f m /?ul, m, l = 1, 2, respectively, when
um assumes the value of (u m ) nj . Let
2
( f mx ) j @ ∑ ( f m, l ) j (ulx ) j ,
n
n
n
(2.8)
l =1
and
2
( f mt ) j @ ∑ ( f m ,l ) j (ult ) j .
n
n
l =1
Because
29
n
(2.9)
2
2
∂f m
∂f m ∂ul
∂f m
∂f ∂u
=∑
, and
=∑ m l ,
∂x l=1 ∂ul ∂x
∂t
l =1 ∂ul ∂t
(2.10)
( f mx ) nj and ( f mt )nj can be considered as the numerical ana logues of the value of ?f m /?x
and ?f m /?t at (x j, t n ), respectively. As a result, we assume
f m* ( x , t ; j , n) = ( f m ) nj + ( f mx ) nj ( x − x j ) + ( f mt ) nj ( t − t n ) ,
(2.11)
h*m ( x, t ; j , n) = ( f m* ( x, t ; j , n), um* ( x, t ; j , n)) ,
(2.12)
and
Note that, by their definitions, for any m = 1, 2, ( f m ) nj and ( f m, l ) nj are functions of (u m ) nj ;
( f mx ) nj are functions of (u m ) nj and ( umx ) nj ; and ( f mt )nj are functions of (u m ) nj and (u mt ) nj .
Assume that, for any (x, t) ∈ SE (j, n), um = u*m ( x , t ;j ,n ) and f m = f m* ( x , t ; j ,n ) satisfy
Eq.(2.5), i.e.,
∂um* ( x, t ; j , n) ∂f m* ( x , t ;j ,n )
+
= µm* ( x, t ;j ,n )
∂t
∂x
(2.13)
According to Eqs. (2.7) and (2.11), and further assume that µ *m is constant within SE(j, n),
i.e., µm* ( x , t ;j ,n )= ( µ m )nj , the above equation is equivalent to
(u mt ) nj = −( f mx ) nj + ( µ m )nj .
30
(2.14)
Since ( f mx ) nj are functions of (u m ) nj and ( umx ) nj ; and ( µ m )nj are also functions of
(u m ) nj , Eq. (2.14) implies that (u mt ) nj are also functions of (u m ) nj and ( umx ) nj . Thus, the
only independent discrete variables needed to be solved are (u m ) nj and ( umx ) nj .
To proceed, we employ local space-time flux balance over CE(j, n) to solve the
unknowns. Refer to Figure 2.2(b). Assume that u*m and u*mx at mesh points
( j −1 2 , n −1 2)
and
( j + 1 2 , n −1 2 )
are known and used to calculate (u m ) nj and ( umx ) nj
at the new time level n. By enforcing the flux balance over CE(j, n), i.e.,
Ñ∫
h m ⋅ ds = ∫
*
S ( CE ( j , n ))
CE (j ,n )
*
µ m dV ,
(2.15)
one obtains
(u m ) nj −
∆t
( µm ) nj =
4
1
∆t
∆t
n −1/2
n −1/2
n −1/2
n −1/2
n −1/2
n −1 / 2 
(um ) j−1/2 + ( um ) j+1/2 + ( µm ) j−1/2 + ( µ m ) j +1/2 + ( sm ) j −1/2 − ( sm ) j +1 / 2  ,

2
4
4

(2.16)
( sm ) nj = ( ∆x /4)(u mx ) nj +( ∆t / ∆x )( f m )nj + ( ∆t 2 / 4∆x)( f mt ) nj .
(2.17)
where
Given the values of the marching variables at mesh points
( j + 1 2 , n −1 2 ) , the RHS of Eq.
( j −1 2 , n −1 2)
and
(2.16) can be readily calculated. Since ( µ m )nj on the
LHS of Eq. (2.16) is a function of (u m ) nj , we use Newton’s method to solve for (u m ) nj .
To solve (umx)jn at point (j, n), central differencing is performed:
31
(u x ) nj = [(ux+ ) nj + (ux− ) nj ] / 2 ,
(2.18)
(u x± ) nj = ±( u nj±1 / 2 −u nj )/( ∆x /2) ,
(2.19)
n−1 / 2
u nj±1/2 = u nj ±−1/2
1/2 + ( ∆t /2)(ut ) j ±1 / 2 ,
(2.20)
where
For flows with discontinuities, Eq. (2.18)is replaced by a re-weighting procedure to add
artificial damping:
(u x ) nj = W (( ux− ) nj , (u +x ) nj ,α ) ,
(2.21)
where the function W is defined as
W ( x− , x+, α) =
x+
α
x− + x−
α
x+ + x−
α
α
x+
,
(2.22)
α is an adjustable constant, and usually α = 1 or 2. The above method with CE and SE
defined as in Figure 2.2 is useful for treating the conservation laws with non-stiff source
terms.
To proceed, we consider the condition when the source term in Eq. (2.5) is stiff. We first
normalize the source term and let µm = 1 κ ⋅ µm* , where the order of the magnitude of µm*
is comparable with that of ∂um ∂t and ∂f m ∂x . Aided by this normalized source term, Eq.
(2.5) becomes
∂um ∂f m 1 *
+
= µm ,
∂t
∂x κ
32
for
m = 1, 2.
The source terms is stiff when κ = 1 . In other words, the time scale of the source term is
much smaller than that of the convection term. In numerical calculations, the magnitude
of the source term would directly impact the calculation of (u mt ) nj (Eq.(2.14)), and hence
the calculation of ( f mt )nj (Eq.(2.9)). Essentially, small difference between the value of
(u m ) nj−−11 // 22 and (u m ) nj−+11 // 22 which will be amplified by the stiffness factor 1 κ , leading to
amplified differences between ( µ m ) nj −−11 // 22 and ( µ m ) nj −+11 // 22 . This difference in turn would lead
to the amplified differences between (u mt ) nj−−11 // 22 and (u mt ) nj−+11 // 22 , and between ( f mt ) nj −−11 // 22 and
( f mt ) nj −+11 // 22 . As a result, Newton’s method would fail to converge when solving this stiff
relaxation system.
Due to the above difficulty, a modification to the original method has been
developed to avoid the amplification effects. The new treatment was based on
redistributing the space-time regions such that the source term effect is hinged on the
mesh point at the new time level. Figure 2.3 shows the new layout of CEs and SEs
associated with mesh nodes. Shown in Figure 2.3(b), the new SE is constituted by the
rectangle ABB'A', the line segments QQ?, and the immediate neighborhood of QQ?. The
CE is rectangle ABB'A'. Note that Q, Q' and Q? share the same spatial projection, so do A
and A', and B and B'. The superscript prime and quotation mark denote the time level n1/2 and n+1/2, respectively. Besides the SE(j, n), two more neighboring SEs are also
shown to illustrate the belonging (to SEs) of the three parts of CE(j, n).
33
With this new construction of SEs, we proceed to perform integration as in the
original CESE method, and have Eqs.(2.7), (2.11) and (2.12). However, the evaluation of
(u mt ) nj−−11 // 22 and (u mt ) nj−+11 // 22 differs from the original method by simply using
n−1 / 2
(u mt ) nj −±1/2
1/2 = −( f mx ) j ±1 / 2 ,
(2.23)
and no source term effect is included. Note that these temporal derivatives are used only
along the line segment protruding from the top of AA’B’B. The area of the line segment is
zero due to the geometry of new SEs. This al yout excludes the influence of the stiff
source term from mesh nodes at time level n − 1 2 in the overall space-time flux
conservation.
34
(a)
(b)
Figure 2.3: Schematics of the modified CESE method in one spatial dimension: (a) the
staggered space-time mesh; (b) SE (j, n), shown as the yellow part, and CE (j, n).
35
Therefore, in the calculation of the overall flux balance over CE(j, n), i.e.,
Eq.(2.15), the integration of the source term is based on the flow properties at the mesh
point (j, n), and one obtains
(u m ) nj −
∆t
1
n − 1/2
n − 1/2
n −1 / 2

( µ m )nj =  (um )n j −−1/2
1/2 + ( um ) j+ 1/2 + ( s m ) j− 1/2 − ( sm ) j+1 / 2  ,
2
2
(2.24)
where
( sm ) nj = ( ∆x /4)(u mx ) nj +( ∆t / ∆x )( f m )nj + ( ∆t 2 / 4∆x)( f mt ) nj ,
(2.25)
Similar to that in the original method, for any m = 1, 2, ( f m ) nj and ( µ m )nj are
functions of (u m ) nj ; ( f mt )nj is a function of (u m ) nj and ( umx ) nj . Thus, given the values of
the marching variables at t = t n-1/2 , Eq. (2.24) is a nonlinear equation of (u m ) nj . Again,
Newton’s iteration method is used. The algorithm for solving ( umx ) nj in the modified
method remains unchanged, i.e., by Eqs. (2.18)-(2.21).
Based on the above treatment, we avoid the amplification effect of the stiff source
term in numerical calculations and thus stabilize the iterative procedure of Newton’s
method. Essentially, the effect of the source-term calculation on the flow properties from
the time level n − 1/2 was not involved. The layout of the modified SEs helps to
eliminate the stiffness problem. Nevertheless, the union of all the CEs still covers the
whole space-time domain without overlapping and the source-term effect would satisfy
the local and global space-time flux balance in an integral sense.
36
2.2 Two-dimensional CESE method
The CESE method also has been developed to solve problem, which could be expressed
by the following standard conservation fo rm in two spatial dimensions :
∂um ∂ f m ∂ gm
+
+
= µm , m = 1, 2, … , 7
∂t
∂x
∂y
(2.26)
where f m and gm , are functions of the independent conservative variables um . Let x 1 = x, x2
= y, and x 3 = t be the coordinates of a three-dimensional Euclidean space E3 . By using
Gauss’ divergence theorem, we have
Ñ∫
S (V )
hm ⋅ ds = ∫ µ mdV , m = 1, 2, … , 7
V
(2.27)
where S(V) and ds were defined by Eq. (2.4) and Figure 2.1, and h m @ ( f m , gm , um ) .
2.2.1 Conservation Elements and Solution Elements
In two spatial dimensions, the computational domain on the x–y plane is divided into
non-overlapping convex quadrilaterals and any two neighboring quadrilaterals share a
common side. Refer to Figure 2.4(a). Vertices and centroids of quadrilaterals are marked
by dots and circles, respectively. Q is the centroid of a typical quadrilateral B1 B2 B3 B4 .
Note the underscore differentiates the points in the computational domain on the x–y
plane with those in the space-time domain as to be introduced below. Points A1 , A2 , A3 ,
and A4 , respectively, are the centroids of the four neighboring quadrilaterals of the
quadrilateral B1 B2 B3 B4 . Point Q* (marked by a cross in Figure 2.4(a)), is the centroid of
the polygon A1 B1 A2 B2 A3 B3A4 B4 .
37
(a)
(b)
Figure 2.4: The space-time mesh in two spatial dimensions: (a) grid points in the x-y
plane; (b) SE and CE for the two-dimensional scheme.
38
Hereafter, point Q* , which generally does not coincide with point Q, is referred to as the
solution point associated with the centroid Q. Similarly, points A1 * , A2* , A3* , and A4* ,
which are also marked by crosses, are the solution points associated with the centroids A1 ,
A2 , A3 , and A4 , respectively.
To proceed, consider the space-time mesh shown in Figure 2.4(b). Here t = n∆t at
the nth time level, where n = 0, 1/2, 1, 3/2, …. For a given n, Q, Q', and Q?, respectively,
denote the points on the time levels n, n-1/2, and n+1/2 with point Q being their common
spatial projection. Other space-time mesh points in Figure 2.4(b) are defined similarly. In
particular, Q* , A1 * , A2 * , A3* , and A4 * , by definition, lie on the nth time level and are the
space-time solution mesh points associated with points Q, A1 , A2 , A3 , and A4 , respectively.
Q’* , A1'* , A2'* , A3'* , and A4'* , lie on the time level n-1/2 and are the space-time solution
mesh points associated with points Q', A1 ', A2 ', A3', and A4 ', respectively.
With the above preliminaries, we are ready to discuss the geometry of the CE and
the SE associated with point Q* . The numerical solution of the flow variables um at the
nth time level are calculated based on the known flow solution at points in the time level
n-1/2, denoted by superscript prime. To integrate Eq.(2.27), four Basic Conservation
Elements (BCEs) of point Q* are constructed and denoted by BCEl (Q), with l = 1, 2, 3,
and 4. These four BCEs are defined to be the space-time cylinders A1 B1 QB4 A1 'B1 'Q'B4',
A2 B2 QB1 A2 'B2'Q'B1 ', A3 B3 QB2 A3 'B3'Q'B2 ', and A4 B4 QB3 A4 'B4 'Q'B3', respectively. The
compounded conservation element of point Q, denoted by CE(Q), is defined to be the
space-time cylinder A1 B1 A2 B2 A3 B3A4 B4 -A1 'B1 'A2'B2 'A3'B3 'A4 'B4' , i.e., the union of the
39
above four BCEs. Moreover, the SE of point Q* , denoted by SE(Q* ), is defined as the
union of CE(Q) and four plane segments QQ?B1 ?B1 , QQ?B2 ?B2 , QQ?B3 ?B3 , and
QQ?B4 ?B4 as well as their immediate neighborhoods.
2.2.2 Approximations with a Solution Element
To proceed, denote the set of the space-time mesh points whose spatial projections are
the centroids of quadrilaterals and the set of the space-time mesh points whose spatial
projections are the solution points, depicted in Figure 2.4(a), by O and O* , respectively.
For any Q* ∈ O* and any (x, y, t) ∈ SE(Q* ), the flow variables and flux vectors, i.e., um (x, y,
t), fm (x, y, t), gm (x, y, t), and hm (x, y, t), are approximated to their numerical counterparts,
i.e., um * (x, y, t; Q* ), fm * (x, y, t; Q* ), gm* (x, y, t; Q* ), and hm * (x, y, t; Q* ), respectively, based
on the first-order Taylor series expansion with respect to Q* (x Q*, yQ*, tn ). Specifically, for
any m = 1, 2, 3, 4, 5, 6, 7, let
u*m ( x , y, t ; Q*) @ (u m ) Q* + (umx ) Q* ( x − xQ * ) + ( umy )Q* ( y − yQ * ) + (umt ) Q* ( t − t n ) ,
(2.28)
where (x Q*, yQ*, tn ) are the coordinates of the space-time solution mesh point Q* and
(u m ) Q* , (u mx )Q* , (u my )Q* , and (u mz ) Q* , which are constant in SE(Q* ), are the numerical
analogues of the values of um , ∂um / ∂x , ∂um / ∂y , and ∂um / ∂t at point Q* , respectively.
Based on the chain rule, ( f mx ) Q* , ( gmx ) Q* , ( f my ) Q* , ( gmy ) Q* , ( f mt )Q* , and ( gmt ) Q*
are defined in a similar way as ( f mx ) nj and ( f mt )nj are defined in the 1D case. Refer to
Eqs.(2.8) and (2.9). We then define
f m* ( x , y , t ;Q*) @ ( f m )Q* + ( f mx )Q* ( x − xQ * ) + ( f my ) Q* ( y − yQ * ) + ( f mt )Q* ( t − t n ) , (2.29)
40
g *m ( x , y , t ;Q*) @ (gm ) Q* + ( g mx )Q* ( x − xQ* ) + ( gmy ) Q* ( y − yQ * ) + (g mt ) Q* (t − t n ) , (2.30)
and
h*m ( x , y , t ;Q *) @ ( f m* ( x , y , t ;Q*), g m* ( x , y , t ;Q*),u m* ( x , y , t ;Q*)) .
(2.31)
similarly for any m = 1, 2, 3, 4, 5, 6, 7.
Note that by definitions ( f m )Q* , ( gm )Q* , ( f mx ) Q* , ( gmx ) Q* , ( f my ) Q* ; ( f mt )Q* and
( gmt ) Q* are functions of (u m ) Q* , (u mx )Q* , (u my )Q* and ( umt )Q* only, for any m = 1, 2, 3, 4,
5, 6, 7.
To proceed, we assume that for any (x, y, t) ∈ SE(Q* ), and any m = 1, 2, 3, 4, 5, 6,
7
∂um* ( x, y, t; Q* ) ∂ f m* ( x , y, t ; Q* ) ∂ g *m ( x, y, t; Q* )
+
+
= µ m ( Q* ) .
∂t
∂x
∂y
(2.32)
which is the numerical analogue of Eq.(2.26). Note that the source term µ *m is assumed
constant within SE(Q* ) and the value of µ *m is determined by (u m ) Q* only. With the aid
of Eqs. (2.28)-(2.30), Eq.(2.32) implies that for any m = 1, 2, 3, 4, 5, 6, 7
(u mt ) Q* = −( f mx )Q* −( gmy )Q* + ( µ m ) Q* .
(2.33)
However, as in the 1D case, to solve Euler equations with stiff source term, for any l = 1,
2, 3, 4, (u mt ) A*' are evaluated excluding the source term effect, i.e.,
l
(u mt ) A'* = − ( f mx ) A'* − ( gmx ) A'* .
l
l
41
l
(2.34)
Using the above equations, it can be shown that the only independent discrete
solution variables associated with the space-time solution point Q* are (u m ) Q* , (u mx )Q*
and (u my )Q* , m = 1, 2, 3, 4, 5, 6, 7 analogous to the 1D case.
2.2.3 Evaluation of um
To evaluate space-time flux in E3 , consider the following preliminaries first. Let G be a
space-time plane segment lying within SE(Q* ). Let A be the area of G, (x c, yc, tc) be the
coordinates of the centroid of G, and n be a unit vector normal to G. Then, because
u*m ( x , y, t ; Q* ) , f m* ( x , y, t ; Q * ) , and g *m ( x , y, t ; Q* ) are linear in x, y, and t, Eq. (2.26)
implies that
∫h
Γ
*
m
⋅ ds = h*m ( xc , yc , tc ; Q* ) ⋅ An .
(2.35)
where ds = ds n with ds being the area of a surface element on G.
The boundary of CE(Q) belongs to the union of SE(Q* ) and SE( Al'* ), l = 1, 2, 3, 4.
Specifically, (i) the octagon A1 B1 A2 B2 A3 B3A4 B4 belongs to SE(Q* ); (ii) the quadrilaterals
A1 'B1'Q'B4 ', A1 'B4'B4 A1 , and A1 'B1 'B1 A1 belong to SE( A1'* ); (iii) the quadrilaterals
A2 'B2'Q'B1 ', A2 'B1'B1 A2 , and A2 'B2 'B2 A2 belong to SE( A2'* ); (iv) the quadrilaterals
A3 'B3'Q'B2 ', A3'B2 'B2 A3 , and A3'B3 'B3 A3 belong to SE( A3'* ); and (v) the quadrilaterals
A4 'B4'Q'B3 ', A4 'B3'B3 A4 , and A4 'B4 'B4 A4 belong to SE( A4'* ).
42
To proceed, we evaluate the surface vector (i.e., the unit outward normal vector
multiplied by the area) of every boundary face of CE(Q). Let S denote the area of the
octagon A1 B1 A2 B2 A3 B3 A4 B4 , then the surface vector of the top face of CE(Q) is (0, 0, S)
because the unit outward normal vector of this face is (0, 0, 1).
Then we consider the bottom face of CE(Q), which is constituted by four
quadrilaterals, namely, A1 'B1 'Q'B4 ', A2'B2 'Q'B1', A3'B3 'Q'B2', and A4 'B4 'Q'B3 '. Let (x l, yl)
and Sl, l = 1; 2; 3; 4, denote the spatial coordinates of the centroids and the areas of the
above four quadrilaterals, respectively. Then for any l = 1, 2, 3, 4, (x l, yl, tn-1/2 ) are the
coordinates of the above four centroids, and (0, 0, -Sl) are the surface vectors of the above
four quadrilaterals, respectively. Furthermore, because the area of the bottom face of
CE(Q) is identical to that of the top face, one concludes that S = ∑ l =1 S l .
4
Finally consider the side faces of CE(Q), i.e., A1 'B4 'B4 A1 , A1 'B1'B1 A1 , A2'B1 'B1 A2 ,
A2 'B2'B2 A2 , A3'B2 'B2 A3 , A3 'B3'B3 A3 , A4'B3 'B3 A4 , and A4 'B4 'B4 A4 , which belong to SE( Al'* ), l
= 1, 2, 3, 4, respectively. Let the eight side faces be assigned the indices (k, l),
respectively. Then the (1, l) and (2, l) side faces belong to SE( Al'* ). Note the spatial
projection of each side face is a line segment on the x-y plane. Let λkl , ( nlkx , nkyl ) , and
( xlk , ylk ) , respectively, denote the length, the unit outward normal, and the coordinates of
the midpoint of the spatial projection (on the x-y plane) of the (k, l) side face. Then,
43
because each side face is sandwiched between the (n-1/2)th and the nth time levels, one
concludes that the surface vector and the coordinates of the centroid of the (k, l) side face,
are given by ( ∆t /2)λkl (nkxl ,nkyl ,0) and ( xlk , ylk , t n − ∆t /4) , respectively.
By using Eq.(2.35), the flux of h*m leaving each face of CE(Q) can then be
evaluated in terms of the independent marching variables at points Q* and Al'* , l = 1, 2, 3,
4. For example, because (x Q* , yQ*, tn ) are the coordinates of the centroid Q* of the top face
of CE(Q); u*m ( xQ* , y Q*, t n ; Q* ) = (u m ) Q* (see Eq.(2.28)); and the surface vector of the top
face is (0, 0, S), the flux of h*m leaving CE(Q) through its top face is (u m ) Q* S.
To proceed, we employ local space-time flux balance over CE(Q):
Ñ∫
h m ⋅ ds = ∫
*
S ( CE (Q ))
C E( Q)
µ m dV , m = 1, 2, 3, 4, 5, 6, 7.
(2.36)
With the aid of the above discussion, it can be shown that
(u m ) Q*-
∆t
1 4

( µm )Q*=  ∑ Rml  , m = 1, 2, 3, 4, 5, 6, 7.
2
S  l =1 
(2.37)
where, for any l = 1, 2, 3, 4,
Rml = S l u*m (x l ,yl , tn −1/2 ; Al'* )-
∆t l
λk
k =1 2
2
∑
 l *  l l n ∆t '* 
∆t '*  
l
*  l
l
n
 nkx f m  xk , y k , t − 4 ; Al  + nky g m  xk , y k , t − 4 ; Al   .





(2.38)
Note that the functions um* ( x , y, t; Al'* ) , f m* ( x , y, t; Al'* ) , and g *m ( x , y, t; Al'* ) are
defined using Eqs.(2.28)-(2.30), respectively, with the symbols Q* and t n in these
44
equations being replaced by Al'* and t n-1/2 , respectively. Given the values of the marching
variables at t = tn-1/2 , the RHS of Eq. (2.37) can be readily calculated. Note ( µ m )Q* is a
function of (u m ) Q* . Therefore, (u m ) Q* can then be solved with Newton’s method.
2.2.4 Evaluation of umx and umy
Like in the 1D case, (u mx )Q* and (u my )Q* are evaluated with a similar finite-difference
approach. First, we perform a spatial translation of the quadrilateral A1* A2* A3* A4* so that the
centroid of the resulting new quadrilateral A1o A2o A3o A4o coincides with Q* . Refer to Figure
2.5. Let the centroid of the quadrilateral A1* A2* A3* A4* and its spatial coordinates be denoted
by A* and ( x A* , y A* ), respectively. Then for any l = 1, 2, 3, 4, ( x Ao , y Ao ), the spatial
l
l
coordinates of Alo are given by
x Ao = xA * + xQ * − xA * , and y Ao = y A * + yQ * − y A * .
l
l
l
(2.39)
l
Figure 2.5: Spatial translation of the quadrilateral A1* A2* A3* A4* .
To proceed, let
(u m ) Ao @ u*m ( x Ao ,y Ao ,t n ; Al'* ) , m = 1, 2, 3, 4, 5, l = 1, 2, 3, 4.
l
l
l
45
(2.40)
Next, for any m = 1, 2, 3, 4, 5, 6, 7 consider the three points in the x–y–u space
with the coordinates ( xQ* , yQ* , (u m ) Q* ), ( x Ao , y Ao , (u m ) Ao ), and ( x Ao , y Ao , (u m ) Ao ),
1
1
1
2
2
2
respectively. The values of ∂u / ∂x and ∂u / ∂y on the plane that intercepts the three
points are given by
(u )
( l)
mx Q*
( )
( l)
@ ∆ x / ∆ and umy
@ ∆ y / ∆ ( ∆ ≠ 0 ),
Q*
(2.41)
where
∆@
x Ao − xQ*
yAo − yQ *
x Ao − xQ*
yAo − yQ *
1
2
∆x @
1
,
2
(u m ) Ao − (u m )Q*
yA o − yQ*
(u m ) Ao − (u m )Q*
yA o − yQ*
(um ) Ao − ( um ) Q*
xA o − xQ *
(um ) Ao − ( um ) Q*
xA o − xQ *
1
2
(2.42)
1
,
(2.43)
.
(2.44)
2
and
∆y @
1
2
1
2
Note that ? = 0 if and only if the spatial projections of A1o , A2o and Q* are
collinear. Similarly,
(u )
(k)
mx Q*
( )
(k)
and umy
Q*
, k = 2, 3, 4, are defined, respectively, by
replacing the points A1o and A2o in the above operations with A2o and A3o , A3o and A4o ,
and A4o and A1o , respectively.
With the above preliminaries, for each m = 1, 2, 3, 4, 5, 6, 7, (u mx )Q* and (u my )Q*
can then be evaluated by
46
( )
1 4 (k)
(u mx )Q*= ∑ umx
4 k =1
, and (u my )Q*=
Q*
( )
1 4 (k )
∑ umy
4 k =1
Q*
.
(2.45)
For a flow with discontinuities, the above equation may be replaced by a reweighting procedure, i.e.,
4
(u mx )Q*=
(
 W (k)
∑
 m
k =1 
)( )
α
(k )
umx

Q* 

k =1
and (u my )Q*=
(k) α
m
k =1
(k )
my Q*


,
(2.46)
 , m = 1, 2, 3, 4, 5, 6, 7, k = 1, 2, 3, 4,

(2.47)
∑ (W )
4
∑ (W ) ( u )
4
(k) α
m
∑ (W )
4
k =1
(k) α
m
where α ≥ 0 is an adjustable constant,
( )
θ mk @  u (mxk )

2
Q
*
( )
 +  u(k)
  my
Q
2
*
and
Wm(1) @ θ m 2θ m 3θ m 4 , Wm(2) @ θ m3θ m4θ m1 , Wm(3) @ θ m 4θ m 1θ m 2 , Wm(4) @ θ m1θ m2θ m3 .
(2.48)
In particular, if for any k = 1, 2, 3, 4, ?mk = 0, then (u mx )Q* and (u my )Q* are set to
be 0. Usually a = 1 or a = 2, and Eqs. (2.46) reduce to Eqs. (2.45) if a = 0. Note that in
practice a small positive number like 10-60 is added to the denominator of Eqs. (2.46) to
avoid dividing by zero.
47
CHAPTER 3
THE FIRST ORDER HYPERBOLIC MODELS OF
ELASTIC EXTENSIONAL WAVE IN THIN ROD
3.1 Introduction
3.1.1 Model Equations of Stress Waves
In dealing with stress wave propagation in solids, most of standard text books [61-70]
focus on the discussions of small material deformation in the elastic range, i.e., linear
elastic waves [61, 62, 65, 68-70]. The model equation generally appears as a secondorder wave equation with the displacement as the only unknown. For nonlinear waves [63,
64, 66, 67], the discussions have been by-and- large about nonlinear material response to
elastic waves. Efforts have been devoted in constructing complex constitutive relations,
and small material deformations in the elastic range have been the main focus.
However, a second category of the nonlinear waves exist, in which the material
deformation is significant, sometimes even with plastic flows. As such, the above
approach of using a second-order wave equation formulated in term of displacement is
inapplicable. Instead, one has to model the waves based on solving the conservation laws,
including conservation of mass, momentum, and even energy if severe impact problems
are of concern. In particular, the nonlinear convective terms in the convection-diffusion
48
equations of the conservation laws must be included. The resultant modeling equations
are fully coupled and highly nonlinear. In general, the model equations are a set of firstorder, fully coupled, nonlinear hyperbolic partial differential equations. Their timeaccurate numerical solutions are extremely challenging problems.
The simplest first-ordered, hyperbolic equations for stress waves in solids have
been presented by Bedford and Drumheller [63], and Drumheller [64]. The equation set
include the mass and momentum conservation equations for elastic waves in onedimensional medium:
∂ρ ∂ ( ρ u )
+
=0
∂t
∂x
∂ ( ρu )
∂t
+
∂ ( ρuu − σ )
∂x
(3.1)
=0
(3.2)
where ρ is density, u is velocity and Cauchy stress σ could be expressed by Young’s
Modulus E, current density ρ and initial density ρ0 by equation
σ = E ( ρ 0 ρ − 1)
(3.3)
Alternatively, the hyperbolic equations for stress wave in solids could include the
momentum conservation equation and a linear elastic constitutive equation [52, 71, 72]:
∂u 1 ∂σ
−
=0
∂t ρ ∂ x
(3.4)
∂σ
∂u
−E
=0
∂t
∂x
(3.5)
49
In this form, the mass conservation equation is eliminated based on the
assumption of constant density. Equation (3.4) is identical to Eq. (3.1) with ρ = constant.
Equation (3.5) is obtained by first applying the time derivative to the constitutive
equation σ = Eε for a linear elastic material, and then let ∂ε ∂t = ∂u ∂x . Both forms of
nonlinear wave equations could be recast into the following vector form:
∂U
∂U
+A
=0
∂t
∂x
(3.6)
The unknown vector is U = ( ρ , ρu ) . The eigenvalues of the Jacobian matrix A
T
for Eqs. (3.1) and (3.2) are
λ1,2 = u ±
E ρo
.
2
ρ
Both eigenvalues are real and the speed of sound of solid is Eρo ρ 2 . Thus, the
equation set is hyperbolic in time. Moreover, the equations are nonlinear because the
eigenvalues are functions of the unknowns ρ and ρu . For Eqs. (3.4) and (3.5), the
eigenvalues of matrix A are
λ1,2 = ±
50
E
.
ρ
Again, both eigenvalues are real and the speed of sound of solid is E ρ . We
note that one cannot easily justify the assumption of ρ = constant in deriving Eqs. (3.4)
and (3.5) for stress wave propagation, and their eigenvalues are different to that of Eqs. of
(3.1) and (3.2) with ρ = constant.
The above modeling equations in the first-order hyperbolic form are ve rsatile and
they could be used to describe a wide range of wave propagation problems in solids
including seismic waves in the earth and ultrasonic waves in biological tissues. Since the
model equations directly include the nonlinear convective terms, in which the Riemann
problem is embedded, one could also use these equations to model shock waves in solids.
In particular, the same set of the model equations could be used to describe nonlinear
waves with large deformation as well as linear elastic waves, which tend to coexist with
nonlinear deformations in solids.
3.1.2 Model Equations and Analytical Solutions
Owing to its simplicity and available analytical solution, one-dimensional elastic wave
propagation often were solved by using a new numerical method for code and method
validation. For examples, shown in Eqs. (3.1) and (3.2), (3.4) and (3.5), Bedford and
Drumheller [63], Drumheller [64] and LeVeque [71] discussed the characteristic and the
eigenstructure of one-dimensional hyperbolic equations for linear stress waves.
51
However, as will be shown in this chapter, there are various one-dimensional
problems in solid mechanics and each of them has its own eigenstructure. In particular,
when solving these one-dimensional nonlinear hyperbolic equations for linear elastic
waves, the solutions should be coincided with the classical solutions of the second-order
linear wave equations.
Several classical analytical solutions exist for stress wave propagation in solids.
By using approximate Mindlin-Herrmann theory, Miklowitz [73] showed the analytical
solution of elastic compressional waves propagation in a rod. He solved the problem of
semi- infinite rod subject to a step pressure in the axial direction. He also solved waves in
an infinite rod subject to an infinite axial pressure load at the initial condition.
In the present reserach, we plan to focus on the elastic extensional wave in thin
rod. The first problem is one of the most popular wave propagation problems and the
fundamental one-dimensional distribution parameter vibration system with analytical
solution. One end of thin rod is fixed and another end is applied with a force loading.
Graff [65], Kolsky [66] and Meirovitch [74], present the analytic solution by solving
second-order wave equation.
3.1.3 The Objectives of the Current Chapter
In this chapter, we will present detailed discussion of characteristic and eigenstructure of
first-order hyperbolic equations for wave in thin rod, as well as validate the model and
1D CESE solver by solving the elastic extensiona l wave in thin rod and elastic-plastic
wave in one-dimensional impact problem.
52
The discussion about characteristic and eigenstructure of hyperbolic system will
show process of deriving speed of sound in thin rod E ρ . In previous discussion about
one-dimensional problem, Lin [52] presented Hook’s law as σ = Eε at the beginning and
then speed of sound as E ρ . Giese and Fey [72] used constant density assumption as a
necessary condition to realize the Cauchy stress integration. The advantage brought by
constant density assumption is eliminating the uncertainties arose by implementing a
specific equation of state to calculate pressure, and simplify the model by cutting off
mass conservation equation and energy conservation equation. However, they have to
determine current yield stress in plasticity by calculating effective stress with Cauchy
stresses.
For the elastic wave in thin rod, we compare the analytical solution provided by
Meirovitch [74] and numerical solutions from two first-order hyperbolic equations, they
are mass conservation equation and momentum conservation equation. This twoequations wave model is different from the works presented by Lin [52], Giese and Fey
[72]. With a constant density assumption, their models include momentum conservations
and constitutive equations for one-dimensional, two-dimensional or three-dimensional
problems. The numerical solutions are obtained by using CESE method, which is a novel
high-resolution finite volume solver.
Different from the common idea in the developing process of those upwind
schemes, Chang [34] introduced a novel idea to develop CESE method, a novel
numerical framework for hyperbolic conservation laws, without using Riemann problem
53
solution at all. Hence, CESE method does not have the trouble of expanding scheme from
one-dimensional to multidimensional. Chang, Yu and their coworkers [34, 37-41, 43, 44,
58-60] already showed the high-resolution shock-capturing capability of CESE, easily
expanded this scheme to 2D and 3D problems, and have reported a wide range of highly
accurate solutions of hyperbolic systems, including detonations, cavitations, complex
shock waves, turbulent flows with embedded dense sprays, dam breaking flows, MHD
flows, aeroacoustics. CESE has never been applies in the case of stress wave propagation
in solids. Validating CESE method in a problem with analytical solution will be a good
attempt.
In this chapter, the two equations model also is extended to be three first-order
hyperbolic equations where constitutive equation is expressed by Cauchy stress and three
first-order hyperbolic equations where the Cauchy stress in constitutive equation is
separated to be pressure and deviatoric stress. The last set of equations could be extended
from describing small deformation elastic problems to the large deformation elasticplastic problems. All of numerical solutions are obtained by using CESE method, which
is a novel high-resolution finite volume solver.
54
3.2 The Second-Order Linear Wave Equation and Analytic Solution
Figure 3.1: A schematic for linear and nonlinear waves in a thin rod
As shown in the Figure 3.1, we consider a metal rod in a horizontal position. At x=0, the
material is fixed without motion. At x=L, an arbitrary force f ( x , t ) is imposed in the
following manner:
f ( x, t ) = F ( t ) δ ( x − L )
(3.7)
F ( t ) = −σ B Aa cos ( 2π ft ) ,
(3.8)
where F ( t ) assumes the form of
and δ ( x − L ) is a Dirac delta function:
δ ( x − L ) = 0 for x ≠ L and
∫ δ ( x − L ) dx = 1
L
(3.9)
0
The line elastic wave equation and boundary conditions for this problem are given by
∂H ( x, t ) 
∂ 2 H ( x, t )
∂ 
,
 EA
 + F ( t ) δ ( x − L ) = mL
∂x  a
∂x 
∂t 2
55
0 < x< L
(3.10)
H ( 0, t ) = 0,
EAa
∂H ( x , t )
∂x
x=L
=0
(3.11)
where h(x, t) is the axial displacement, E is the Young’s modulus, Aa is the constant cross
section area of the rod, and ml is the constant mass per unit length.
Meirovitch [74] presented the analytical solution of this problem with a boundary
force in the form of step function. The process here is adapted from normal modes
analysis presented by Meirovitch [74].
The solution of displacement H(x,t) governed by Eqs. (3.10) and (3.11), which
form a distributed-parameters system possessing an infinite number of natural
frequencies and modes, can be expressed as a linear combination of natural motions with
amplitudes and phase angles depending on the initial conditions. Hence, we express the
solution of displacement as:
∞
H ( x, t ) = ∑ H r ( x ) ηr ( t ), r = 1,2,...
(3.12)
r =1
where the displacement H(x,t), a function of one-dimensional space and time, is
approximated as a linear combination of infinite products of one special function H r ( x )
and one time function η r ( t ) . The H r ( x ) , r = 1,2,... are the natural modes and
η r ( t ) , r = 1,2,... represent the time-dependent functions, which indicate how the
amplitude of H r ( x ) , r = 1,2,... vary with time t.
56
To obtain each H r ( x ) of the distributed-parameters system described by Eqs. Eqs. (3.10)
and (3.11), we first have to solve the eigenvalue problem defined by the following
differential equation
−
dH ( x ) 
d 
2
 EAa
 = ϖ ml H ( x ) ,
dx 
dx 
0<x<L
(3.13)
=0
(3.14)
with the boundary conditions
H r ( 0) = 0,
EAa
dH r ( x )
dx
x=L
where the ϖ r2 is eigenvalue and its square root ϖ r is recognized as the natural frequency
of the system; corresponding to each natural frequency ϖ r , there is a eigenfunction
H r ( x ) , which is defined as natural mode. The natural frequencies and natural modes
represent a characteristic of the system, because they are determined by the Young’s
modulus, cross section area, mass distribution and boundary conditions. The
eigenfunctions H r ( x ) , r = 1,2,... of a system are orthogonal and are assumed to be
normalized, thus they must satisfy the ortho-normal conditions
∫
L
0
mL H r ( x ) H s ( x ) dx = δ rs , r , s = 1,2,...
(3.15)
and
− ∫0 Hs ( x )
L
dHr ( x ) 
d 
2
 EAa
 dx = ϖ r δ rs , r , s = 1,2,...
dx 
dx 
57
(3.16)
Inserting Eq. (3.12) into Eq.(3.10), multiplying by H s ( x ) , integrating over the length of
rod and considering the orthonormality conditions, i.e. Eqs.(3.15) and (3.16), we have the
modal equations
η&&r ( t ) + ϖ r2ηr ( t ) = N r ( t ) , r = 1,2,...
(3.17)
N r ( t ) = ∫ H r ( x ) F ( t ) δ ( x − L ) dx = H r ( L ) F ( t ), r = 1,2,...
(3.18)
where
L
0
are the modal forces. Then, the solution of the modal equations can be written in the form
of the convolution integrals
η r (t ) =
1
ϖr
∫
t
0
Nr ( t − τ ) sin (ϖ rτ )d τ =
Hr (L )
ϖr
∫ F ( t − τ ) sin (ϖ τ ) dτ ,
t
0
r
r = 1,2,... (3.19)
The displacement response of the rod to the boundary force could be calculated by the
Eqs.(3.12) -(3.14) and (3.18). For the uniform rod shown in Figure 3.1, the eigenvalue
problem are governed by the differential equations
d H r ( x)
2
dx
2
+ β 2 H r ( x ) = 0,
β2 =
ϖ r2 mL
, 0 < x < L, r = 1,2,...
EAa
(3.20)
with the boundary conditions
H r ( 0) = 0,
dHr ( x )
dx
x= L
= 0, r = 1,2,...
(3.21)
The ortho-normal modes in the solutions of these eigenvalue problems are
H r ( x) =
 ( 2r − 1) π x 
2
sin 
 , r = 1,2,...
mL L
2
L


58
(3.22)
And natural frequencies are
ϖr =
( 2r − 1) π
2
EAa
, r = 1,2,...
mL L2
(3.23)
When the specific force boundary condition expressed by Eq.(3.8) is applied in Eq.(3.18),
we could calculate the stress response of rod by following steps. We first use Eqs. (3.8),
(3.19), (3.22), and (3.23) to get time functions as
η r (t ) =
Hr ( L ) t
− A σ cos  2π f ( t − τ )  sin (ϖ rτ ) dτ
ϖ r ∫0 a B
 ( 2r −1) π 
2
Aaσ B
=−
sin 
cos (ϖ r t ) − cos ( 2π ft )  , r = 1,2,...
 2
2 
mL L
2

 ϖ r − ( 2π f )
(3.24)
Then we insert Eqs.(3.22) and (3.24) into Eq. (3.12)to obtain the displacement response
of the rod. With known displacement response H(x,t), we can calculate the stress
response as:
σ ( x,t ) = E
=−
∂h (x , t )
∂x
Eσ Bπ
ρ L2
 ( 2r − 1)
 ( 2r −1) π 
sin 
 2

∑
2
2
r =1 ϖ r − ( 2π f )



∞
 ( 2r − 1) π x 

cos 
  cos (ϖ r t ) − cos ( 2π ft ) 
2L



Eσ Bπ
=−
ρ L2
(3.25)
( −1) ( 2 r − 1)  ( 2r − 1) π x  
cos 
  cos (ϖ rt ) − cos ( 2π ft ) 
∑
2
2
2L
r =1 ϖ r − ( 2π f )


∞
r −1
where
ϖr =
( 2r − 1) π
2
m
E
, and ρ = l
2
ρL
Aa
59
(3.26)
The speed of sound is given by
c=
EAa
=
ml
E
ρ
(3.27)
3.3 The Two-Equations Model of Elastic Extensional Wave in Thin
Rod
In this section, we present detailed derivation of the one-dimensional model equations for
nonlinear elastic waves in a thin rod. We will show that although the problem is onedimensional, we have to consider lateral contraction and expansion in thin rod. To
proceed, we consider the differential form of the mass and momentum conservation laws
in three spatial dimensions :
∂ρ ∂ ( ρ u ) ∂ ( ρ v ) ∂ ( ρ w )
+
+
+
=0
∂t
∂x
∂y
∂z
∂ ( ρu )
∂t
∂ ( ρv )
∂t
∂ ( ρ w)
∂t
+
+
+
∂ ( ρuu − T11 )
∂x
∂ ( ρuv − T21 )
∂x
∂ ( ρ uw −T31 )
∂x
+
+
+
(3.28)
∂ ( ρ uv − T12 ) ∂ ( ρ uw − T13 )
+
=0
∂y
∂z
∂ ( ρ vv − T22 )
∂y
∂( ρ wv −T32 )
∂y
60
+
+
∂ ( ρ vw − T23 )
∂z
=0
∂( ρ ww − T33 )
∂z
=0
(3.29)
(3.30)
(3.31)
For the isotropic material, Hook’s law in general form is
Tij = λ ( ε ii ) δ ij + 2 µε ij
(3.32)
where λ and µ are two Lame parameters, and µ is also called shear modulus, Tij is
Cauchy stress components, ε ij the strain components.
For the problem described in the Figure 3.1, the Cauchy stress tensor T and strain tensor
e are given by
T11 0 0
T =  0 0 0 ,
 0 0 0
ε 11 0
e =  0 ε 22
 0 0
0  ε11
0


0  =  0 −vε11
ε 33   0
0
0 
0 
−vε11 
(3.33)
where the Poisson’s ratio is defined by
ν=
λ
2 (λ + µ )
(3.34)
Based on the Hook’s law (3.32) and strain tensor in Eq. (3.33), we have
T11 = λ ( ε11 + ε 22 + ε 33 ) + 2 µε11
0 = λ ( ε 11 + ε22 + ε 33 ) + 2µε 22
0 = λ ( ε 11 + ε22 + ε 33 ) + 2µε 33
Three strain components are not independent, they have relation as
ε 22 + ε 33 = −
λ
ε11
λ+µ
For the thin rod, the Cauchy stress component T11 always is given by
61
(3.35)
T11 =
µ ( 3λ + 2 µ )
λ+µ
ε11 = Eε11
(3.36)
To be consistent with the strain shown in Eq. (3.33) with lateral contraction/expansion,
the velocity components must satisfy the following relations:
∂v
∂u
= −ν
,
∂y
∂x
∂w
∂u
= −ν
∂z
∂x
(3.37)
Therefore, v and w are functions of y and z, respectively, and the problem is not strictly
one-dimensional. To recap, for wave propagation in thin rod, we could assume the
following
ρ = ρ ( x, t ) , u = u (x , t ), v = v ( x , y, t ), w = w( x, z, t )
(3.38)
As we known, the strain is defined by
1  ∂d ∂d j
ε ij =  i +
2  ∂x j ∂xi



i , j = 1,2,3
(3.39)
where d i is a displacement component.
And the components of symmetric part of gradient of velocity tensor D are defined by
1  ∂v ∂v j 
Dij =  i +

2  ∂x j ∂xi 
For this problem, we have v1 → u, v2 → v , v3 → w, x1 → x , x2 → y , x3 → z .
Observing Eqs.(3.39) and (3.40), we have
62
(3.40)
Dij =
dε ij
dt
(3.41)
Since the Eq.(3.33) shows ε12 = 0 , therefore we have D12 = 0 . Considering the velocity
component u(x,t) in Eq.(3.38) and following definition
D12 =
1  ∂u ∂v 
 + ,
2  ∂y ∂x 
(3.42)
we have ∂v ∂ x = 0 , which means that velocity component v defined in Eq.(3.38) is not a
function about x. This velocity component should be redefined as v = v( y, t ) . Similarly,
we w = w ( z, t) . Thus, we have new definitions about density and velocity components as
ρ = ρ ( x, t ) , u = u (x , t ), v = v ( y, t ), w = w( z, t )
(3.43)
To proceed, we substitute Eqs. (3.33) and (3.37) into Eqs. (3.28)-(3.31), and arrive at the
following model equations:
 µ

∂
ρu 
∂ρ
λ+µ
 = − λ u ∂ρ
+ 
∂t
∂x
λ + µ ∂x
 µ

∂
ρuu − T11 
∂ ( ρu )
λ+µ
 = − λ u ∂ ( ρu)
+ 
∂t
∂x
λ+µ
∂x
 2µ − λ

∂
ρ uv 
 2 (λ + µ )

∂ ( ρv )
∂ ( ρ v)
 = − 3λ
+ 
u
∂t
∂x
2 (λ + µ )
∂x
63
(3.44)
(3.45)
(3.46)
 2µ − λ

∂
ρ uw 
 2( λ + µ )

∂ ( ρ w)
∂ ( ρ w)
 = − 3λ
+ 
u
∂t
∂x
2( λ + µ )
∂x
(3.47)
As shown in [75], the initial density ρ0 , the current density ρ , and the gradient of
deformation F are connected by
ρ0
= det F
ρ
(3.48)
where gradient of deformation F for thin rod is
0
0 
1 + ε11

F= 0
1 + ε 22
0 
 0
0
1 + ε 33 
(3.49)
Aided by Eqs. (3.35) and (3.48), we have
det F = 1 +
µ
µ
ε 11 +
ε112 + ε113
λ+µ
λ +µ
(3.50)
We neglect the second- and third-order terms in Eq. (3.48). Aided by Eqs. (3.48) and
(3.50), we have the following relation between ε11 and ρ :
ε11 =
λ + µ  ρ0 
 − 1
µ ρ

(3.51)
With Eq.(3.51), T11 in Eq. (3.45) can be recast as a function of ρ only:
T11 =
µ ( 3λ + 2 µ ) λ + µ  ρ0

 ρ0

 ρ0

 − 1  = (3 λ + 2 µ )  − 1  = 3k  −1 
λ +µ
µ  ρ

 ρ

 ρ

where the bulk modulus and Young’s Modulus are
64
(3.52)
µ ( 3λ + 2µ )
2
k = λ + µ , and E =
3
λ+µ
(3.53)
Because T11 in Eq. (3.45) is a function of ρ only, the y and z momentum
equations, Eqs. (3.46) and (3.47), are decoupled from the continuity equation and the xmomentum equation, Eqs. (3.44) and (3.45). Thus Eqs. (3.44) and (3.45) form a closed
system of two equations and two unknowns ρ and u.
Substitute Eq. (3.52) in Eq. (3.45) and rewrite Eqs. (3.44) and (3.45) into a vector
form:
∂U ∂E
+
=H
∂t ∂ x
(3.54)
where the conservative variables vector is U = ( ρ , ρu ) ,
T
 µ

∂ ( ρu) 
µ
ρ

λ
∂ρ
λ
E=
ρu ,
ρ uu − 3k  0 − 1   and H =  −
u
,−
u

λ+µ
∂x 
 ρ

λ+µ
 λ + µ ∂x λ + µ
T
T
Equation (3.54) could be transformed as
∂U
∂U
+A
=H
∂t
∂x
(3.55)
where the matrix A is defined by

0
∂E 
A=
=
∂U 
µ
3k ρ0
2
− λ + µ u + ρ 2

65
µ

λ +µ 

µ

2
u
λ + µ 
(3.56)
By directly calculating the eigenvalues of matrix A, we have the eigenvalues are:
λ1,2 =
µ
u±
λ +µ
3k µρ0
(λ + µ ) ρ 2
(3.57)
The conservative form of this model is not unique. The one presented here is one
of many options. Since these eigenvalues are real, this conservative form is a hyperbolic
system, and it could be solved by explicit solver. Moreover, we will study this hyperbolic
system by following analysis of eigenstructure of non-conservative form.
T
To rewrite Eq. (3.55) with non-conservative variables vector U% = ( ρ , u ) , we
multiply both sides of Eq. (3.55) with matrix M
M
∂U
∂U
+ MAM −1M
= MH
∂t
∂x
(3.58)
where M is defined by
1
% 
∂U
M=
=
u
∂U  −
 ρ
0
1 
ρ 
(3.59)
With Eq. (3.59), we rewrite Eq.(3.58) as
%
%
∂U
% ∂U = 0
+A
∂t
∂x
where
66
(3.60)

u

% =
A
 ( 3λ + 2 µ ) ρ0

ρ3

µ

ρ
λ +µ


u 

(3.61)
% are
The eigenvalues of matrix A
λ1,2 = u ± c = u ±
E ρ0
2
ρ
(3.62)
The speed of sound is
E ρ0
2
ρ
c=
(3.63)
The eigenvalues are real and distinct. Thus, the system equations are hyperbolic.
The two eigenvalues are functions of both unknowns, ρ and u, and the equation system is
nonlinear. The associated eigenvectors could be obtained by solving the equation
( A% − λ I ) m%
i
i
=0
i = 1,2
(3.64)
These two column eigenvectors are
% 1 = m% 11 , m
% 12
m
(
)
(
)
% = m
% , m%
m
2
2
1
2
2
T
 λ +µ
= 1,

µ

T

λ +µ
=  1, −

µ

E ρ0
ρ2
T



E ρ0
ρ2
% is composed of the two column vectors:
The right eigen- matrix M
67
T



1


% =
M
 λ + µ E ρ0
 µ
ρ2

1
−
λ+µ
µ


E ρ0 
ρ 2 
% −1 can be readily found as
Its inverse denoted M
1
2
−1
%
M = 
1

 2
µ
ρ2 
2 ( λ + µ ) E ρ0 
µ
ρ2 

−
2 ( λ + µ ) E ρ 0 
To proceed, we derive the characteristic form of Eq. (3.60) by pre- multiplying the
% −1 :
equation by M
%
%
% −1 ∂U + M
% −1AMM
% % % −1 ∂U = 0 .
M
∂t
∂x
(3.67)
Note that we insert an identity matrix I = MM −1 in the convective term. Equation (3.67)
can be expressed as
ˆ
ˆ
∂U
ˆ ∂U = 0 ,
+A
∂t
∂x
(3.68)
where the diagonal matrix  is
ˆ = M% −1AM
% % =  λ1
A
0


E ρ0
u +
ρ2
0 
=
λ2  

0

The characteristic variables are
68


.
E ρ0 
u−
ρ 2 
0
(3.69)
1
µ
 ρ+
2
2 (λ + µ )
ˆ
% −1U
% =  u1  = 
Uˆ = M

µ
 uˆ 2   1 ρ −
2
2 (λ + µ )

ρ 2u 

Eρ0 
ρ 2u 
E ρ 0 
(3.70)
In the setting of the Method Of Characteristics (MOC), Eqs. (3.68)-(3.70)
constitute the analytical solution of nonlinear elastic waves in a thin rod. In the x-t plane,
along the characteristic lines dx dt = λ1,2 , the Riemann invariants û1,2 are constant:
duˆ i  ∂
∂ 
=  + λi  uˆi = 0,
dt  ∂ t
∂x 
i = 1,2
(3.71)
The right running wave:
along
dx
Eρ0 1
µ
ρ 2u
=u+
,
ρ
+
= constant .
dt
ρ2
2
2 ( λ + µ ) E ρ0
The left running wave:
along
dx
=u−
dt
E ρ0 1
µ
ρ 2u
,
ρ
−
= constant.
2
ρ
2
2 ( λ + µ ) E ρ0
Direct integration along the above right and left characteristic lines allows
analytical solution of nonlinear elastic waves in a thin rod. Moreover, since the analytical
form of the MOC equations are derived, the boundary condition treatments at the two
ends of the thin rod can also be readily obtained by following the characteristic lines in
conjunction with the specified conditions in terms of ρ and u.
69
Based on the previous analysis, the eigenvalues of conservative form equations
are artificial and they are different from that of the characteristic forms, which is the true
analytical solution of the hyperbolic system. In numerical solution, one has to use this
artificial eigenvalue to pace the CFL constraint in the calculations. Nevertheless, this
conservative- form equations are suitable for apply the modern upwind methods for
numerical solution.
3.4 Three-Equation Model-I of Elastic Extensional Wave in Thin Rod
In this section, we present alternative formula for elastic expansion wave in a thin rod. To
proceed, we differentiate Hook’s law (3.32) with respect to time to get ,
DTij
Dt
= λ tr ( D) δ ij + 2µ Dij
(3.72)
For the objective time derivative of Cauchy stress tensor at left side of Eq.(3.72),
Bechtel et al. [52] presented a general form as
D T ∂T
∂T
∂T
∂T
=
+u
+ v + w + TW − WT − a ( TD + DT )
Dt
∂t
∂x
∂y
∂z
(3.73)
where a is a constant, normally − 1 ≤ a ≤ 1; when it equals to zero, this time derivative
becomes to be the Jaumann derivative. And matrix W is the skew part of gradient of
velocity tensor and its components is defined by
70
Wij =
1  ∂vi ∂v j 
−


2  ∂x j ∂xi 
(3.74)
Consider Eqs. (3.33) (3.36) (3.37) (3.72) (3.73), and apply a = 0 in Eq.(3.73), we have the
time derivative of Cauchy stress component T11 as
 µ

∂
uT11 
∂T11
λ+µ
 = E ∂u − λ u ∂T11
+ 
∂t
∂x
∂ x λ+ µ ∂ x
(3.75)
Using flux ( ρT11 ) and applying Eq.(3.44), we transform Eq.(3.75) to be
 µ

∂
ρuT11 
∂ ( ρT11 )
λ +µ
 = E ρ ∂u − λ u ∂ ( ρT11 )
+ 
∂t
∂x
∂x λ + µ
∂x
(3.76)
When nonlinear elastic stress wave propagation in the rod is assumed as an
isothermal process, the solid dynamics model-I for the elastic wave in thin rod includes
the Eqs. (3.44) (3.45) and (3.76). Note that in the previous two-equation mode, we solve
Eqs. (3.44) and (3.45) for the unknowns dens ity ? and velocity u. The normal stress T11 is
then calculated as a post processing procedure by using Eq. (3.52).
To proceed, we recast Eqs. (3.44) (3.45) and (3.76) into the vector form:
∂U ∂E
+
=H
∂t ∂ x
(3.77)
where the conservative variables vector is U = ( ρ , ρu , ρT11 ) , conservative flux
T
variables
71
T
 µ
µ
µ

E=
ρu ,
ρuu − T11,
ρ uT11 
λ+µ
λ +µ
λ+µ

and the source term on the right side of equation is

∂ ( ρu )
∂ ( ρT11 ) 
λ
∂ρ
λ
∂u
λ
H = −
u
,−
u
, Eρ
−
u
 .
∂x
∂x λ + µ
∂x 
 λ + µ ∂x λ + µ
T
Aide by the chain rule, Eq. (3.77) is transformed to be
∂U
∂U
+A
=H
∂t
∂x
(3.78)
where the matrix A is defined by

0


∂E 
µ
T
A=
= −
u 2 + 11
∂U
λ +µ
ρ


µ
 − λ + µ uT11

µ
λ +µ
µ
2
u
λ +µ
µ
T
λ + µ 11



1 
−
ρ 

µ

u
λ +µ 
0
(3.79)
The eigenvalues of this matrix A could be readily calculated. They are
λ1,2,3 =
µ
u
λ+µ
(3.80)
Since all eigenvalues of this matrix A are real, the convective terms of the
equation set are hyperbolic and in general the equations can be solved by a timemarching method. We remark that the true characteristics of this equation set cannot be
determined due to the complexity of the source term. In what follows, we present the
non-conservative form of the same equations and the analysis of its eigen-system.
72
To proceed, we recast the equation set into the non-conservative form by using the
T
variables vector U% = ( ρ , u, T11 ) . We multiply both sides of Eq. (3.78) with matrix M
M
∂U
∂U
+ MAM −1M
= MH
∂t
∂x
(3.81)
where


1

%
∂U  u
M=
=−
∂U
ρ

 T11
−
 ρ
0
1
ρ
0


0


0

1

ρ
(3.82)
Its inverse is given by
 1 0 0
M =  u ρ 0 
T11 0 ρ 
−1
(3.83)
By using Eqs.(3.82)-(3.83), we write Eq. (3.81) as
%
%
∂U
% ∂U = H
%
+A
∂t
∂x
(3.84)
where
% = MAM −1
A
µ

u λ + µ ρ


= 0
u

0
0

73

0 

1
− 
ρ

u 

(3.85)
and
% =  0,0, E ∂u 
H


∂x 

T
Move the source term to the left side of equations, we have Eq. (3.84) in the form of
%
%
∂U
∂U
+A
=0
∂t
∂x
(3.86)
where
µ

u λ + µ ρ


A = 0
u

−E
0


0 

1
− 
ρ

u 

(3.87)
The eigenvalues of matrix A could be straightly calculated, they are
λ1 = u,
λ2,3 = u ± c = u ±
E
ρ
Where the speed of sound is
c=
E
ρ
(3.88)
This is the extensional wave speed in the thin rod. Corresponding to those three
eigenvalues, the eigenvectors could be calculated by solving the equation
( A − λ I ) m%
i
i
=0
74
i = 1,2,3
(3.89)
T
% 1 = ( m% 11, m% 21 , m% 31 ) = (1,0,0) ,
These three eigenvectors are m
T
(
2
1
2
2
E λ +µ E 
,−

ρ
µ ρ 

λ +µ 1
=  1, −
µ ρ

E λ+µ E
,−

ρ
µ ρ 
T
2
3
T
 λ+µ 1
=  1,
µ ρ

)
% = m% , m% , m%
m
2
and
(
% = m% , m% , m%
m
3
3
1
3
2
3
3
)
T
T
% is given by
The eigenvector matrix M

1

% =  0
M


0

1
λ+µ 1
µ ρ
λ +µ
−
µ
E
ρ
E
ρ


1

λ+µ 1 E 
−

µ ρ ρ
λ +µ E 
−

µ ρ 
(3.90)
% −1 is
and its inverse M

0
1


µ ρ ρ
% −1 =  0
M
λ+µ 2 E


0 − µ ρ ρ
λ +µ 2 E

µ ρ 
λ + µ E 
µ ρ 
−

λ + µ 2E 
µ ρ 
−
λ + µ 2E 
(3.91)
The characteristic form of Eq. (3.86) could be given by multiplying the both sides of the
% −1
equation with M
75
%
%
% −1 ∂U + M
% −1AMM
% % −1 ∂U = 0
M
∂t
∂x
(3.92)
ˆ
ˆ
∂U
ˆ ∂U = 0
+A
∂t
∂x
(3.93)
Then we have
where the matrix  is defined by
 λ1
ˆ = M% AM
% = 0
A

 0
−1
0
λ2
0


u
0
0  
E
0  =  0 u +
ρ

λ3  
0
0



0 

0 


E
u−
ρ 
(3.94)
The variables vector for Eq. (3.93) is given by


µ T11 ρ
ρ+


λ+µ E




µ
ρ
u
ρ
µ
T
ρ
11
% −1U
% =
Uˆ = M
−

 λ + µ 2 E λ + µ 2E 


− µ ρu ρ − µ T11ρ 
 λ + µ 2 E λ + µ 2 E 
(3.95)
3.5 Three-Equation Model-II of Elastic Extensional Wave in Thin
Rod
The Hook’s law also could be expressed as the relation between deviatoric stress
components and strain components by
76
2
Sij = − µ tr ( e )δ ij + 2 µε ij
3
(3.96)
To proceed, we apply time derivative to Eq. (3.96) ,
DSij
2
= − µ tr ( D) δ ij + 2µ Dij
Dt
3
(3.97)
where Sij is the components of deviatoric stress tensor, which relate the pressure and
Cauchy stress components by
Tij = − p + Sij and p = −
1 3
∑ Tii
3 i =1
(3.98)
When the Cauchy stress is separated to deviatoric stress and pressure as in Eq.(3.98), the
momentum conservation equation (3.45) becomes
 µ

∂
ρuu + p − S 11 
∂ ( ρu )
λ +µ
 = − λ u ∂ ( ρu)
+ 
∂t
∂x
λ +µ
∂x
(3.99)
Considering Eq.(3.98), the Cauchy stress tensor could be separated into two parts like
T11
0

 0
0 0 − p 0
0   S11
0 0  =  0 − p 0  +  0
0 0   0
0 − p   0
where
77
0
S22
0
0
0 
S33 
(3.100)
S11
0

 0
0
S 22
0
2
 Eε11
0  3
0 = 0
 
S33  
 0

0
1
− Eε11
3
0



0 


1
− Eε11 

3
0
(3.101)
Considering Eqs. (3.96) and (3.101), and following the similar steps in obtaining Eq.
(3.76), we have
 µ

∂
ρuS11 
∂ ( ρ S11 )
λ+µ
 = 2 E ρ ∂u − λ u ∂ ( ρ S11 )
+ 
∂t
∂x
3
∂x λ + µ
∂x
 µ

∂
ρ uS22 
∂ ( ρ S22 )
λ+µ
 = − 1 E ρ ∂u − λ u ∂ ( ρ S22 )
+ 
∂t
∂x
3
∂x λ + µ
∂x
 µ

∂
ρ uS33 
∂ ( ρ S33 )
λ +µ
 = − 1 E ρ ∂u − λ u ∂ ( ρ S 33 )
+ 
∂t
∂x
3
∂x λ + µ
∂x
(3.102)
(3.103)
(3.104)
Note that the solution of S22 and S33 are decoupled from the continuity and
momentum equations. Therefore, we only need to consider density ?, velocity u, pressure
p and the normal stress component S11 as the main unknown variables. Moreover, we can
relate pressure to density by using an equation of state:
p = k ln
78
ρ
ρ0
(3.105)
As such, the three-equation model-II for elastic expansion wave in a thin rod includes
Eqs. (3.44)(3.99), and (3.102). To proceed, we rewrite Eqs. (3.44)(3.99), and (3.102) into
a vector form:
∂U ∂E
+
=H
∂t ∂ x
(3.106)
where the conservative variables vector is U = ( ρ , ρ u, ρ S11 ) , the conservative flux
T
variables is
T
 µ

µ
ρ
µ
E=
ρu ,
ρuu + k ln
− S11,
ρ uS11 
λ+µ
ρ0
λ +µ
λ+µ

and source term at the right side of equation is

∂ ( ρu ) 2
∂ ( ρ S11 ) 
λ
∂ρ
λ
∂u
λ
H = −
u
,−
u
, Eρ
−
u
 .
∂x 3
∂x λ + µ
∂x 
 λ + µ ∂x λ + µ
T
Aided by the chain rule, we transform Eq. (3.106) into a non-conservative form:
∂U
∂U
+A
=H
∂t
∂x
(3.107)
where

0


∂E 
µ
k S
A=
= −
u 2 + − 11
∂U
λ +µ
ρ ρ


µ
−
uS

λ + µ 11

79
µ
λ+µ
µ
2
u
λ+µ
µ
S
λ + µ 11



1 
−
ρ 

µ

u
λ +µ 
0
(3.108)
With the direct calculation, the eigenvalues of matrix A are given by:
λ1 =
µ ( k − 2S11 )
µ
µ
u and λ2,3 =
u±
λ +µ
λ+µ
(λ + µ ) ρ
(3.109)
Generally, the bulk modulus k is much higher than the deviatoric stress
component S11 . For instance, the bulk modulus of aluminum 6061 is 77GPa and initial
yield stress is 240MPa. In principal, S11 is lower than the initial yield stress even
considering hardening. Therefore
( k − 2 S11 )
always is a positive. As such, the
eigenvalues in Eq. (3.109) are real. Hence, the conservative form Eq.(3.107) is a
hyperbolic system.
Similar to the previous analysis for the two-equation model and the three-equation
Model-I, the genuine eigenstructure of the system of equations can only be obtained
T
reformulated the equations by using non-conservative variables U% = ( ρ , u, S11 ) . To
proceed, we pre- multiply Eq. (3.107) by a matrix M, i.e.,
M
∂U
∂U
+ MAM −1M
= MH
∂t
∂x
(3.110)
where M is defined by


1

%  u
∂U
M=
= −
∂U
ρ

 S11
−
 ρ
80
0
1
ρ
0


0


0

1

ρ
(3.111)
Its inverse is
1
M =  u
 S11
−1
0
ρ
0
0
0 
ρ 
(3.112)
Aided by Eqs.(3.111)-(3.112), we rewrite Eq. (3.110) as:
%
%
∂U
% ∂U = H
%
+A
∂t
∂x
(3.113)
where
% = MAM −1
A

 u

 k
= 2
ρ

 0

µ
ρ
λ +µ
u
0

0 

1
− 
ρ

u 

(3.114)
% =  0,0, 2 E ∂u  By moving the source term from right side of
and source term is H


3 ∂x 

T
equation to left side, we change the Eq. (3.113) into the form as
%
%
∂U
∂U
+A
=0
∂t
∂x
where
81
(3.115)
µ
ρ
λ +µ

u

 k
A= 2
ρ


 0
u
2
− E
3

0 

1
− 
ρ


u 

(3.116)
The eigenvalues of matrix A are
λ1 = u,
λ2,3 = u ± c = u ±
E
ρ
where the speed of sound is
E
ρ
c=
(3.117)
Obviously, this speed of sound is the extensional wave speed in the thin rod, and
exactly same as the one provided by previous model. For those three real eigenvalues,
corresponding eigenvectors could be obtained by solving the equations:
( A − λ I ) m%
i
(
i
=0
% 1 = m% 11, m% 21 , m% 31
These three eigenvectors are m
(
% = m
m
% ,m
% , m%
2
(
2
1
2
2
% = m% , m% , m%
and m
3
3
1
3
2
3
3
2
3
)
T
)
T
 λ +µ 1
=  1,
µ ρ


λ +µ 1
=  1, −
µ ρ

)
T
i = 1,2,3
(
= 1,0, k ρ
),
T
T
E 2 λ+µ E
,−
 ,
ρ 3 µ ρ 
T
E 2 λ +µ E 
,−
 .
ρ 3 µ ρ 
82
(3.118)
% and its inverse M
% −1 are given by
Therefore, the transform matrix M

1

% =  0
M


0

1



λ+µ 1 E 
−

µ ρ ρ
2 λ+µ E 
−

3 µ ρ 
1
λ+µ 1
µ ρ
2λ +µ
−
3 µ
E
ρ
E
ρ
(3.119)
and

0
1


µ ρ ρ
% −1 =  0
M
λ+µ 2 E


0 − µ ρ ρ
λ +µ 2 E

3 µ ρ 
2 λ + µ E 
µ 3ρ 
−

λ + µ 4E 
µ 3ρ 
−
λ + µ 4E 
(3.120)
To obtain characteristic form of Eq.(3.115), we multiplying the both sides of the equation
% −1
with M
%
%
% −1 ∂U + M
% −1AMM
% % −1 ∂U = 0
M
∂t
∂x
(3.121)
ˆ
ˆ
∂U
ˆ ∂U = 0
+A
∂t
∂x
(3.122)
to have
where the matrix  is defined by
83
 λ1
ˆ = M% AM
% = 0
A

 0
−1
0
λ2
0


u
0
0  
E
0  =  0 u +
ρ

λ3  
0
0



0 

0 


E
u−
ρ 
(3.123)
The variables vector in the characteristic form Eq. (3.122) is given by

2
ρ+

3


% −1U
% =  µ ρu
Uˆ = M
 λ+µ 2

 − µ ρu
 λ + µ 2



ρ
µ 3S11ρ 
−

E λ + µ 4E 
ρ
µ 3S11ρ 
−
E λ + µ 4E 
µ S11ρ
λ+µ E
(3.124)
Aided by the equations of state, Eq. (3.105), we have shown that the isothermal
solid dynamics models expressed by Model-I in terms of Cauchy stress as well as in the
Model-II in terms of the deviatoric stress have the same eigenvalues and the speed of the
sound. Moreover, the speed of the sound is also identical to that derived by using the two
equations model without using stress components as the unknowns. All models presented
above could be used to catch the elastic wave in a thin rod. We remark, however, the
isothermal solid dynamics model expressed by using the deviatoric stress is more useful
for complex problems. In particular, it has advantage to be used in modeling elasticplastic problems as will be illustrated in the following chapters.
84
3.6 Numerical Results
By Gauss’ theorem, the conservative- form equations, i.e. Eqs. (3.54), (3.77) and (3.106),
can be recast into an integral form:
∫ h ⋅ ds = ∫ hdV
i
∂V
i
where h i = ( f i , ui ) , i = 1,2
T
(3.125)
V
where V is a space-time domain, ∂V = S ( V ) is the surface of V, and f i and ui are the ith
component of F and U, respectively. To solve the those conservative-form equations, i.e.
(3.54), (3.77) and (3.106), we emp loyed the CESE method [34], a novel numerical
framework for hyperbolic conservation law.
In the present paper, we consider elastic longitudinal wave propagating in an
aluminum thin rod. For aluminum, ρ0 = 2700kg / m3 , λ = 60.5GPa , µ = 26GPa ,
E = 70GPa , and k = 77GPa . The length of the rod L = 1.0m .
To be consistent with the forced boundary condition on the right end of the thin
rod, Eq.(3.8), we assume the displacement applied at the right end of rod is a cosine
function of time:
d = −CA ⋅ cos ( 2π ft ) ,
(3.126)
where d is the displacement, CA is the amplitude of the imposed vibrations, and f is the
frequency of the imposed vibrations. In the present calculations, CA = 50 µ m and
f = 20kHz . Corresponding to the given vibration in Eq. (3.8), the velocity at right
boundary is
85
uB = 2π f ⋅ CA ⋅ sin ( 2π ft ) ,
(3.127)
It is noted that the amplitude of stress at boundaryσ B in stress boundary Eq. (3.8)
must be provided to obtain the analytical solution of the second order wave equation,
Eq.(3.10). To this end, we first use the CESE method to solve the conservative-form
first-order equations, i.e., Eq. (3.54) to obtain σ B , which is then used to obtain the
analytical solution, Eq. (3.25), of the second-order linear wave equation.
(b)
(a)
Figure 3.2: (a) The stress variation at the vibrating end of the rod and (b) the speed of
sound profile at t=0.312 ms calculated by the CESE method.
By solving the nonlinear equations, Eq. (3.54), by the CESE method, we obtained
the dynamics stress boundary condition and speed of sound. Figure 3.2(a) shows the time
history of σ B . About 6 cycles of vibrations were calculated. The solution shown in Figure
3.2(a) reveals that at a high frequency, imposed vibrations even with small amplitude
could generate higher stresses, which then propagate in the specimen at the speed of
86
sound. Figure 3.3(b) shows a snapshot of the speed of sound profile at t = 0.312 ms. At
this time, the wave initiated from the right end of the rod has reached and rebounded
from the left end of the rod. Due to wave reflection, the wave amplitude increases. Since
we solve the first order nonlinear equations, the speed of sound, which is a part of the
eigenvalues, depends on the instantaneous solution of the primary unknown
U = ( ρ, ρu )
T
.
The initial speed of sound at 5092 m/s is denoted by a blue line. Because the
density is not a constant in this process, the speed of sound shown in Figure 3.3(b)
presents the fluctuation around the initial value at 5092m/s.
(a)
Figure 3.3:
(b)
The snapshot (t=0.312ms) of stress wave propagation generated by (a)
numerical solution of nonlinear wave model Eq. (3.54) and (b) theoretic solution of linear
wave Eq. (3.10).
Figure 3.3 shows the side-by-side comparison of the normal stress between (a) the
CESE solution of Eq. (3.54), and (b) the analytical solution of Eq. (3.10). Figure 3.3
87
demonstrate that for linear elastic wave problem, the nonlinear wave model, Eq. (3.54),
solved by the CESE method can faithfully catch the linear elastic waves. These figures
also show that the magnitude of the stress wave doubles after wave reflection from the
left end of the rod. Figure 3.4 shows the snapshot of density profile of the thin rod at the
same time. Because of the nonlinear wave model, Eq. (3.54), density is not a constant. As
wave propagates, density at a spatial location would fluctuate accordingly. Except of
stress wave, the nonlinear wave model (3.54) also could catch other evolving physical
parameters in this process, for instance, the density shown in Figure 3.4.
Figure 3.4: A snapshot (t=0.312ms) of density profile predicted by two-equations model,
Eq. (3.54).
88
The capability of the CESE method in capturing stress waves could be assessed
by the number of grid points needed for resolving the propagating waves. Figure 3.5
shows snapshots of profiles of norma l stress at t = 0.312 ms. Three sets of solutions using
301, 151, and 75 grid points. When 75 mesh nodes are used, dissipation effect can be
observed as compared to the solutions by using 151 and 301 mesh nodes. Although not
shown, further mesh refinement does change the solution. In all three cases, no obvious
dispersive effect can be discerned. Approximately, the one meter aluminum rod contents
3 cycles of wave. Therefore, we need at least about 25 nodes per wave length to resolve
the stress wave.
Figure 3.5 Snapshots of the normal stress profiles at t = 0.312 ms by using 301, 151, and
75 grid points.
89
3.7 Conclusions
This chapter reported detailed analyses of the first-order model equations and its
eigen-system for nonlinear elastic stress waves in a thin rod. Based on the conservation
laws of mass and momentum, model equations in non-conservative form, characteristic
form, and conservative form were derived. Analytical solutions of the eigenvalues,
eigenvector matrices, and the Riemann invariants along characteristic lines were provided.
Due to the contraction/expansion effect of the cross section area of the thin rod, the
derived model equations were quite complex. In particular, when put into a conservative
form, a stiff source term appeared on the right hand side of the first-order hyperbolic
equations. A suitable conservative form for numerical solutions was then solved by the
space-time CESE method. The treatment for stiff source term was designed by redistributing the space-time areas of SE such that the amplification effect by the stiff
source term was avoided and the numerical integration was stabilized. For linear waves,
favorable comparison between the numerical results and the classical solutions of the
second-order wave equation was found. The result here is a steppingstone for the further
development of the model equations and the CESE method for stress waves of large
material deformation.
90
CHAPTER 4
APPLYING THREE-EQUATION MODEL-I OF ELASTIC
WAVE IN THIN ROD TO ONE-DIMENSIONAL MULTIBAR IMPACT PROBLEMS AND APPROXIMATED
HOPKINSON BAR IMPACT PROBLEM
4.1 Introduction
In 1913, Bertram Hopkinson developed a technique to determine the pressure – time
relations, which is specific to the impact condition generated by a bullet or explosive. The
apparatus Hopkinson used is mainly composed of a device of generating impact, a long
steel rod, a short steel billet, and a ballistic pendulum. A compressive pressure wave
inside the rod is generated by impacting one end of the rod. By using a thin layer of
grease, a short steel billet is attached at the far end of the rod. As the compressive wave
propagates along the bar, passes through the greased joint, and transmits into the billet,
part of the compressive wave will be reflected at the far end as a pulse of tension.
Because the grease could not hold up any palpable tensile loads, the billet would fly off
with a definite momentum, which could be measured by a ballistic pendulum. The round
trip time of the wave propagating in the billet equals to the time over which the
momentum acts. By conducting several tests with identical magnitude but varying length
91
of cylindrical billets, one may obtain a series of pressure – time curves, which could be
used to describe the impact event. Hopkinson was always able to determine the peak
value of pressure and total duration of these impact cases, but just provide an
approximation of exact pressure-time curves.
In 1949, Kolsky [66] reported further improvement of the apparatus by adding a
second pressure bar. Hence, the specimen is sandwiched by two pressure bars. The
original pressure bar, in which the impact is incurred, is also called incident bar. The
second pressure bar added by Kolsky is called the transmitter bar. The new apparatus is
referred to as the split Hopkinson bar or the Kolsky bar, which is shown in Figure 4.1.
Figure 4.1: The sketch of split Hopkinson bar apparatus
To start a split-Hopkinson-bar impact test, the striker bar would hit the left end of
incident bar. Refer to Figure 4.1. A compressive stress wave is then generated and
immediately begins to propagate towards the specimen. Once the wave reaches the
specimen, part of the wave is reflected back towards the impact end as a tension wave.
The remainder of the wave passes through the specimen and transmits into the transmitter
92
bar. If the strength of the stress wave is significant, it would cause irreversible plastic
deformation in the specimen. Usually, two strain gauges are applied to the midpoints of
the incident and transmitter bars. The strain gauges are away from the interface to reduce
noises in the signals. By the strain- gauge data, one may calculate the stress-strain
properties of the tested specimen.
To ensure that the experimental data are correctly related with the properties of
specimen, several requirements have to be followed. Typically, the length of incident bar
should be much longer than that of the specimen to ensure uniform strain in the specimen,
and one-dimensional equilibrium condition in the specimen.
Moreover, the length-to-diameter ratios of the incident and the transmitter bars are
far above ten to ensure a one-dimensional axial impact problem. Moreover, the bar length
must be as least twice the length of the impact pulse. As far as the effects of geometry
and length-to-diameter ratio of specimen on testing results, Woldesenbet and Vinson [76]
compared the results from many experiments and concluded that no statistically
significant effects of either length-to-diameter ratio or geometry could be found. Frantz et
al [77] suggested using length-to-diameter ratio of 0.5~1.0 to minimize the errors caused
by pressure bar/specimen friction and radial inertia. To make sure that the stress waves in
pressure bars are elastic while plastic deformation appears only in specimen, the pressure
bars are generally made of high strength maraging steels. Kaiser [32] used Vassomax C93
350 as incident and transmitter bars. The length of the two bars is 6 feet and the diameter
is 3 quarters of an inch. The specimens used in his work were one quarter of an inch in
both length and diameter. Figure 4.2 and Figure 4.3 show Kaiser’s data [32] for the time
histories of the strains. The data have the unit of voltages.
Figure 4.2: Typical time history of the strain at the mid point of the incident bar.
Figure 4.3: Typical time history of the strain at the mid point of the transmitter bar.
94
The split Hopkinson bar test is commonly used to determine material properties at
intermediate strain rates (102 -104 s-1 ). By using the strain time histories and following the
force equilibrium and continuity, Kolsky [66] developed the following relations to
calculate the specimen stress.
σs = E
A0
εT ( t )
As
(4.1)
where E is the Young’s modulus of transmitter bar, A0 the cross section area of
transmitter bar, As the cross section area of specimen, and εT ( t ) is the transmitter strain
history like shown in Figure 4.3. Specimen strain rate could be calculated by
d ε s (t )
dt
=−
2C0
εR (t )
L
(4.2)
where ε R ( t ) is the reflected incident bar strain history, L the initial specimen length, and
C0 is the wave speed in thin rod, give by
C0 =
E
ρ
(4.3)
where ? is density.
Integrate Eq.(4.2), we have specimen strain given by
εs ( t ) = −
2C0
L
∫ ε ( t )dt
t
0
R
(4.4)
Based on Eqs.(4.1)-(4.4), one may build up the stress-strain curve for specimen with two
strain time histories.
95
Owing to the widely available Hopkinson-bar test data, we will use this impact
problem as a testing bed to validate the isothermal hyperbolic model of elastic waves in
thin rods and the CESE method for solving the model equations. In the future, the
simulation capabilities developed here could be used to aid the data analysis process to
deduce the mathematic form of the constitutive equation as well as the values of the
parameters in the equation for the specimen being tested.
In the present chapter, we will focus on elastic waves propagation in a
Hopkinson-bar like setup. We note that the impact strength in most of Hopkinson bar
tests is strong enough to result in plastic deformatio n of specimen tested. The present
effort however will focus on elastic wave only. Moreover, the calculations in the present
chapter will be one-dimensional only. Thus, the change of the cross-sectional areas
between the testing bars and the sample cannot be modeled by the one-dimensional
calculations. Nevertheless, the calculations in this chapter will lay the foundation for
further modeling development of elastic-plastic deformation of a specimen in a
Hopkinson bar apparatus in the future.
4.2 Modeling Equations
We apply the isothermal model developed in CHAPTER 3 for elastic wave in a thin rod
to study the elastic wave propagation in the bar impact problem. Since we will deal with
elastic wave only, we plan to use the three-equation Model-I, in which the constitutive
equation is formulated by using the Cauchy stress. The model applied in this chapter
includes:
96
 µ

∂
ρu 
∂ρ
λ+µ
 = − λ u ∂ρ
+ 
∂t
∂x
λ + µ ∂x
(4.5)
 µ

∂
ρuu − T11 
∂ ( ρu )
λ+µ
 = − λ u ∂ ( ρu)
+ 
∂t
∂x
λ+µ
∂x
(4.6)
 µ

∂
ρuT11 
∂ ( ρT11 )
λ +µ
 = E ρ ∂u − λ u ∂ ( ρT11 )
+ 
∂t
∂x
∂x λ + µ
∂x
(4.7)
Recast Eqs. (4.5)-(4.7) into the vector form, we have:
∂U ∂E
+
=H
∂t ∂ x
(4.8)
where the conservative variables vector is U = ( ρ , ρu , ρT11 ) , conservative flux
T
T
 µ
µ
µ

variables E = 
ρu ,
ρuu − T11,
ρ uT11  and source term on the right side
λ+µ
λ +µ
λ+µ


∂ ( ρu )
∂ ( ρT11 ) 
λ
∂ρ
λ
∂u
λ
of equation is H = −
u
,−
u
, Eρ
−
u
 .
∂x
∂x λ + µ
∂x 
 λ + µ ∂x λ + µ
T
4.3 Description of Cases in Computation
In the present section, we present the numerical results of a series of one-dimensional bar
impact problems. The purpose is to understand the mechanisms of the wave
superposition/cancellation in the bars and wave reflection at the interfaces and the free
ends of the bars. From simple to complex, we will present the following four cases: (i)
an aluminum striker bar hits an aluminum pressure bar; (ii) an aluminum striker bar hits
97
an aluminum pressure bar, which is connected with a copper bar with the same length; (iii)
an aluminum striker bar hits an aluminum incident bar, which sandwiches a short copper
bar with another identical aluminum transmitter bar; (iv) a Vascomax C-350 steel striker
bar hits a long Vascomax C-350 steel incident bar, which connects with a small copper
specimen, and then connects with another Vascomax C-350 steel transmitter bar.
For all cases, the initial speed of striker bar is 10m/s and all other bars including
the specimens are static. In each case, numerical strain gauges are placed at selected
positions, where the time histories of strain will be recorded in the calculations. Figure
4.4 shows the schematics of all four cases. The lengths of the bars used in these four
cases are listed in Table 1. For all three materials used in the calculations, the material
parameters, which are used in modeling Equations (4.5)(4.5)-(4.7), are shown in Table 2.
98
(a)
(b)
(c)
(d)
Figure 4.4: Sketches of cases in computation includes (a) Case-I: Aluminum striker bar
hits aluminum pressure bar, (b) Case-II: Aluminum striker bar hits aluminum-copper
pressure bars, (c) Case-III: Aluminum striker bar hits aluminum- copper-aluminum bar
combination and (d) Case-IV: C-350 striker bar hits C-350-copper-C-350 bar
combination
99
Case
Lengths
Case-I
L1 =600mm, L2 =3400mm
Case-II
L1 =600mm, L2 =1700mm, L3 =1700mm
Case-III
L1 =600mm, L2 =1600mm, L3 =200mm, L4 =1600mm
Case-IV
L1 =354mm, L2 =1524mm, L3 =6mm, L4 =1524mm
Table 1: The lengths of the bars used in the calculated bar- impact problems
Li, i = 1, 2, 3, 4, shown in Figure 4.4.
? (kg/m3 )
E (GPa)
? (GPa)
µ (GPa)
Aluminum
2700
70
60
26
Copper
8230
130
108
48
Vascomax C-350
8080
200
116
77
Table 2: Material parameters
4.4 Results
For each case, the presented results include several snapshots to show the stress wave
propagation, and two time histories of the strain at the selected locations.
100
4.4.1Results of Case-I
Case-I is the simplest one among the four cases. Therefore, we could obtain clear
analyses of the stress wave propagation in this case. The solutions he re will serve as the
reference for the other three cases.
The calculation starts when the striker bar hits the other bar. As shown in Figure
4.5, the interface of the two bar remains at the location of x = 0. At the instance of the
initial impact, two compression waves were formed: one is right running into the target
bar and the other is left running into the striker bar. While the right-running wave would
continue moving toward the right, the left running wave would reach the left end of the
striker bar, located at x = -0.6 m, and then reflect from the free end of the rod as a rightrunning expansion wave.
Figure 4.5 shows a snapshot of stress wave after the striker bar has hit the target
bar and the left running compression wave has reflected from the left end of the striker
bar. In Figure 4.5, one wave of rectangular profile is shown. The wave is actually
composed of two right-running waves with the same wave speed. These two waves
however are different. The wave on the left hand side, i.e., the trailing wave, is an
extension wave. Across this propagating wave front, the normal stress of the material
increases from a negative value to become null. The wave on the right hand side is a
compression wave, across which the material experiences a normal stress decrease from
zero to be a negative value. The left extension wave is generated by the reflection of leftrunning compression from the left end of the striker bar.
101
Figure 4.5 illustrates the formation of this wave pattern. Figure 4.6 (a) shows that
two compression waves are formed after the striker bar hits the pressure bar. One
compression wave runs to the left end of the striker bar, and another compression wave
runs to the right end of the pressure bar.
At a free end of a metal bar, the solution must satisfy the following two conditions:
(i) density and velocity at a free end must be equal to that of the immediately adjacent
area, which is close to the free end, and (ii) the value of the material stress at the free end
must be null, i.e., the stress- free condition at a free end.
Figure 4.5: Snapshot-I of stress wave propagation in Case-I
102
(a)
(b)
(c)
Figure 4.6: Diagram of stress wave superposition for snapshot-I in Case-I
103
As shown in Figure 4.6(b), a positive right-running extension wave is generated
when the left running compression wave reaches the free end of the striker bar and is
reflected from it. This positive extension wave would superimpose and cancel the
compression wave in front of it, and the resultant stress value becomes to zero. As shown
in Figure 4.6(c), this wave superposition process results in a right-running positive
extension wave counteracts the existing negative stress value generated by previously
passing compression wave. Because pressure bar in Case-I does not have any change or
discontinuity in material properties, this right-running extension wave would have the
same wave speed as that of the leading right-running compression wave. The gap
between these two right running waves will not change until right running compression
wave reaches the free end of the target bar.
After these two right running waves reach the right end of the target bar, waves
reflect and interfere in a similar manner as shown in Figure 4.6. Finally, two left-running
waves are formed as shown in Figure 4.7.
104
Figure 4.7: A snapshot of a left-running stress wave after all waves reflect from the right
end of the target bar in Case-I.
105
(a)
(b)
(c)
(d)
Figure 4.8: Stress wave superposition for wave snapshot shown Figure 4.7 for Case-I
106
These two left running waves shown in Figure 4.7 are not identical. The left one
is an extension positive stress wave and the right one is a compression negative stress
wave. The result of this wave form is illustrated in the diagrams shown in Figure 4.8.
Figure 4.8 is constructed to illustrate the reflected wave motion. We first show
that Figure 4.8(a) is the same wave as that in Figure 4.6(b). Figure 4.8(a) shows one
right-running extension wave and one right-running negative compression wave. In
Figure 4.8(b), when the compression wave reaches the right free end of the target bar, it
reflects and a left-running positive extension wave is formed. Hence, there are two
positive extension waves running in two opposite directions. When these two positive
extension waves meet, they create a net positive stress value after cancelling the existing
negative stress produced by the previous compression wave.
Figure 4.8(c) shows this positive stress value of two extension waves moving in
the opposite directions. When the right-running extension wave hits the right end of the
target bar, it reflects and generates a left-running negative compression wave. This
compression wave would cancel the existing positive stress value when it propagates
along the bar. Finally, it follows the left-running extension wave at the same wave speed
and the distance between the two left-running wave fronts remains constant. Figure 4.8 (d)
shows this wave pattern, which is identical as that in Figure 4.7.
107
Figure 4.9: The strain time history at the position of gauge-I in Case-I
Figure 4.10: The strain time history at the position of gauge-II in Case-I
108
Since Case-I has only one striker bar and one target bar with the same material
properties, the wave pattern is clear and simple. For the elastic wave, a linear relationship
exists between the stress and the strain. Based on the numerical results of the stress, the
strain time histories could be readily calculated. Figure 4.9 and Figure 4.10 show at the
time histories of strain at gauge-I and gauge-II, respectively. Because gauge-II is closer to
right free end than gauge-I, the time interval between two step wave fronts is shorter than
that in the data of gauge-I.
4.4.2 Results of Case-II
Since there is a jump in material properties at the interface between the aluminum bar and
the copper pressure bar, the stress wave will partially reflect back at the interface. The
wave pattern caused by superposition and cancellation are quite different from those in
Case-I.
Figure 4.11: Snapshot-I of stress wave propagation in Case-II
109
As shown in Figure 4.11, after wave reflection at the left end of the striker bar and
before the wave reach the interface between aluminum and copper, two right-running
waves at the same wave speed can be observed. This wave pattern is identical to that
shown in Figure 4.5 of Case-I.
These two waves continue running to the right and reaches the aluminum-copper
interface. The change of material properties result in the change of wave pattern. As
shown in Figure 4.12, the compression wave becomes stronger after passing through the
interface and into the copper bar. In the meantime, a left-running compression wave
reflects from the interface back to the aluminum bar.
Figure 4.12: Snapshot-II of the stress wave propagation after the right-running wave
reaches the aluminum-copper interface in Case-II.
The change of stress wave at the aluminum-copper interface is caused by (i) the
existing compression stress in the aluminum bar, and (ii) wave speed change across the
110
aluminum-copper interface. When the stress wave reaches the interface, the problem
could be considered as an aluminum bar with initial stress and speed hitting a static and
stress free copper bar. To understand the wave change, we separate this problem into two
independent problems: (i) a stress- free aluminum bar with an initial speed impacting on a
stress-free static copper bar, and (ii) putting a static aluminum bar with an initial stress in
contact with a static and stress-free copper bar. Figure 4.13 and Figure 4.14 show these
two processes.
Figure 4.13: A stress free aluminum bar with initial speed 5m/s impacts a stress free and
static copper bar
Figure 4.14: A static aluminum bar with known initial stress 68MPa contacts a static and
stress free copper bar
111
Figure 4.15: The snapshot of stress wave in the problem shown by Figure 4.13
Figure 4.16: The snapshot of stress wave in the problem shown by Figure 4.14
Figure 4.15 and Figure 4.16 show the snapshots of stress profiles for the two
problems illustrated in Figure 4.13 and Figure 4.14, respectively. By observing these two
112
figures, we could make following conclusions : (i) when an aluminum bar with initial
speed hits a static copper bar, two compression waves in the opposite directions are
generated. (ii) when an aluminum bar with an initial stress is put in contact with a stressfree copper bar, a part of the initial compression stress would pass through the interface,
and the rest of the wave energy would reflect back and become a left-running extension
wave. By superposition of the stress waves in Figure 4.15 and Figure 4.16, the overall
stress wave pattern is presented in Figure 4.17.
Figure 4.17: Snapshot-III of stress wave propagation in Case-II
113
Figure 4.18: Snapshot-IV of stress wave propagation in Case-II
Figure 4.19: Snapshot-V of stress wave propagation in Case-II
114
Figure 4.20: Snapshot-VI of stress wave propagation in Case-II
Figure 4.17 to Figure 4.20 show the evolution process of the stress wave. One
could follow the illustration of that of Figure 4.12 to understand the stress wave
propagation and influence of interface on the change of wave pattern. Based on these
snapshots, we found that the discontinuity of material properties results in more complex
wave pattern in Case-II then that in Case-I.
115
Figure 4.21: The time history of the strain at Gauge-I in Case-II
Figure 4.22: The time history of the strain at Gauge-II in Case-II.
The time history of the strain at Gauge I is presented in Figure 4.21. Obviously,
the complicated stress wave pattern due to the interface leads to the complex time history
of the strain values at the position of Gauge-I. Since Gauge-II is located inside the copper
bar, which has a free right end, the interface does not produce a complex time history of
the strain dada. Instead, the profile is similar to that in Case-I as shown in Figure 4.10.
116
4.4.3 Results of Case-III
In Case-III, there are two interfaces between the central copper bar sandwiched by two
aluminum bars. Compared to Case-II, the wave pattern of Case-III is much more complex.
However, the mechanism of wave reflection at interfaces and the free end is identical to
that in Cases I ans II. No new physics is involved here.
We present six snapshots of stress wave propagation in Figure 4.23-Figure 4.28
and two time histories of the strain data in Figure 4.29 and Figure 4.30. Since the
structure of Case-III is similar to that of the Hopkinson-bar test, Case-III is a good
reference to understand the wave pattern in the Hopkinson-bar problem.
Figure 4.23: Snapshot-I of stress wave propagation in Case-III
117
Figure 4.24: Snapshot-II of stress wave propagation in Case-III
Figure 4.25: Snapshot-III of stress wave propagation in Case-III
118
Figure 4.26: Snapshot-IV of stress wave propagation in Case-III
Figure 4.27: Snapshot-V of stress wave propagation in Case-III
119
Figure 4.28: Snapshot-IV of stress wave propagation in Case-III
Figure 4.29: The strain time history at position of gauge-I in Case-III
120
Figure 4.30: The strain time history at position of gauge-II in Case-III
4.4.4 Results of Case-IV
In Case-IV, we apply the real material parameters and bar lengths in a typical Hopkinsonbar experiment. The calculation is one-dimensiona l. Therefore, we do not consider the
multi-dimensional effect due to the change of the cross-sectional areas at the interfaces
between the steel bars and the tested sample. We present the snapshots of the
instantaneous stress profiles in Figure 4.31-Figure 4.36 and two time histories of the
strain data in Figure 4.37 and Figure 4.38.
121
Figure 4.31: Snapshot-I of stress wave propagation in Case-IV
Figure 4.32: Snapshot-II of stress wave propagation in Case-IV
122
Figure 4.33: Snapshot-III of stress wave propagation in Case-IV
Figure 4.34: Snapshot-IV of stress wave propagation in Case-IV
123
Figure 4.35: Snapshot-V of stress wave propagation in Case-IV
Figure 4.36: Snapshot-VI of stress wave propagation in Case-IV
The basic mechanism of the evolving wave pattern presented by these snapshots
is similar to that in Case-I, Case-II and Case-III. Essentially, the wave pattern is the result
124
of wave reflection and superposition. Because the material properties of high-carbon steel
used in for the incident and transmitter bars are much stronger then those of the copper
specimen, the wave reflection at the steel-copper interfaces with material discontinuities
does not cause obvious drastic change in the wave pattern as that in Cases II and III.
Moreover, the length of the two steel bars is also much longer then the copper specimen.
As a result, the overall evolution of wave pattern is very similar to that in Case-I.
Figure 4.37: The time history of the strain data at Gauge-I in Case-IV.
Figure 4.38: The time history of the strain data at Gauge-II in Case-IV
125
We remark that we have used the real conditions of material properties and the
lengths of the Hopkinson bars and the specimen in the calculation. The only difference
between the present model calculation and the real Hopkinson-bar testing condition is
that our one-dimensional simulation does not consider the change of cross-sectional area
at the interfaces between differential metals. Although the time history of the strain data
in Figure 4.37 resembles that of the typical experimental results as shown in Figure 4.2,
in the experiments, the wave reflection from the interface between the incident bar and
the copper sample is very different from the reflected wave at the same place in the real
Hopkinson bar test. As a result, the time history of the strain data cannot capture the
effect of the cross-sectional area change at interface. Thus, the time history of the strain
data in Figure 4.38 resembles more of that in Figure 4.2 instead of Figure 4.3.
The diameter of specimen is one quarter of an inch, which is one third of that of
the incident and transmitter bars, i.e., three quarters of an inch. Thus, the cross section
area of the tested specimen is only one ninth of that of the incident and the transmitter
bars. The eight- ninth of the cross sectional area at the right end of incident bar, which is
in contact to the sample, is free boundary condition. Only one- ninth of the cross sectional
area is connected with the specimen. In a real Hopkinson-bar test, variations of cross
sectional areas and connection status prevent the elastic wave with large amplitude in the
incident bar to be transmitted into the transmitter bar. As such, most of the wave energy
from the incident bar would reflect back to the incident bar at the interface to the tested
sample.
126
Thus, the wave pattern shown in Figure 4.2 cannot be modeled by the present
one-dimensional simulation, in which the effect of the cross sectional area change is not
considered.
As shown in the triangular profile of the wave pattern in Figure 4.3, wave
transmission from the specimen to the transmitter bar, i.e., from a smaller cross section to
a larger cross section, could involve wave deflection. By analyzing and comparing
computational results and experimental results, we believe that one must use the twodimensional axisymmetric simulation to model the effect of the cross sectional area
variation in real test.
4.5 Conclusions
In this section, we summarize the results presented in the present paper. We
solved the hyperbolic equations Eq. (4.8) for nonlinear stress waves in isothermal solids
by using the one-dimensional CESE method. The setup mimicked that of classical
Hopkinson-bar test. Four sets of one-dimensional bar impact problems were studied. We
found that the developed model equations of elastic waves in a thin rod can be accurately
solved by using the one-dimensional CESE solver. The numerical results captured all
salient features of elastic wave propagation, reflection, and transmission. All wave
features were automatically captured by the CESE method based on integrating the
space-time control volume formulation in enforcing the mass and momentum
conservation. Overall wave pattern could be understood by wave reflection at the free
ends of the rods, at the material interfaces, and wave superposition and cancellation.
127
The one-dimensional simulation however is inadequate for modeling the
Hopkinson-bar test because the no cross sectional area variation was assumed. With
considering the area change effect, the time history of the stress wave signal would be
similar to that shown in Case I. Nevertheless, one-dimensional computations provide
fundamental understanding of wave propagation in multi-components bar impact
problems. The model calculation could be used as a reference to understand the
Hopkinson bar test data. Although Hopkinson bar impact problem has been generally
categorized as a one-dimensional problem, we found that two-dimensional axisymmetric
numerical computation is necessary to capture wave features in a real Hopkinson-bar test.
128
CHAPTER 5
THE FIRST ORDER HYPERBOLIC MODELS OF
LONGITUDINAL PLANE WAVE IN ELASTIC-PLASTIC
BULK MATERIAL AND ISOTHERMAL ONEDIMENSIONAL IMPACT PROBLEM
5.1 Introduction
In one-dimensional domain, a solid material could have three different wave
speeds associated with three different elastic waves. In CHAPTER 3, the longitudinal
extension wave in a thin rod has been discussed. The other two elastic waves are the
longitudinal plane wave and the shear plane wave in an one-dimesnional bulk material.
For instance, consider the three elastic wave speeds of aluminum: (i) The speed of
longitudinal extension wave in a thin rod is c = E ρ = 5100m/s. As discussed in
CHAPTER 3, the speed of longitudinal extension wave in a thin rod is a part of the
eigenvalues of the Jacobian matrix of the governing equations. (ii) The speed of
longitudinal plane wave in a bulk aluminum is 6300m/s. (iii) The speed of the shear
plane wave is about 3000m/s.
129
The speed of shear plane wave c = µ ρ could be derived either from the
second-order linear wave equation, or from the eigenvalue analysis of the first-order
nonlinear wave equations.
Figure 5.1: Elastic plane shear wave propagate in x direction
As shown in Figure 5.1, we consider a bulk of material and a shear force in the y
direction is applied to the x-y plane. As a result, the material displaces in the y direction
and the shear plane wave propagates along the x axis, which is perpendicular to the
direction of material displacement. According to the classical theorem of bulk plane
waves, for an isotropic medium, the linear elastic shear wave equation is given by
ρ dx
∂2y
∂2 y
=
µ
dx
∂t 2
∂x 2
(5.1)
Based this second-order wave equation, i.e. Eq.(5.1), the shear wave speed is
c=
130
µ
ρ
(5.2)
Alternatively, the same wave spped can be derived by using the first-order
hyperbolic nonlinear wave equations. For the problem described in the Figure 5.1, the
Cauchy stress tensor T and strain tensor e are given by
 0 T12
T = T21 0
 0 0
0
0  ,
0 
 0 ε12
e = ε 21 0
 0
0
0
0
0
(5.3)
Three velocity components in the x, y and z driecitons are
u = 0, v = v( x), w = 0
(5.4)
Thus, the velocity field is solinoidal, and the mass conservation equation becomes
∂ρ ∂t = 0 . In other words, the density is constant.
In the setting of the conservation laws, the nonlinear elastic shear wave equations
include only the momentum conservation equation and a constitutive equation:
∂v 1 ∂S12
−
=0
∂t ρ ∂x
(5.5)
∂S12
∂v
−µ
=0
∂t
∂x
(5.6)
By writing these two non-conservative equations in a vector form, we have:
∂U
∂U
+A
=0
∂t
∂x
(5.7)
where the non-conservative variables vector is U = ( v , S12 ) , and Jacobin matrix A is
T
defined by
131

0
A=

 −µ
1
ρ

0 
−
(5.8)
By solving the equation det ( A − λ I ) = 0 , we obtain the eigenvalues of A:
µ
ρ
λ1,2 = ±
(5.9)
which is the speed of the shear wave :
c=
µ
ρ
(5.10)
This results is identical to that of Eq. (5.2). The above elastic shear plane wave is one of
two main waves in two-dimensional materials. The other one is the longitudinal plane
wave.
To proceed, we consider the wave speed of longitudinal plane wave in a bulk
material. In the following, we will show that the wave speed of elastic wave is
c=
( k + (4 / 3 ) µ )
ρ . Moreover, the discussion here will not be limited to elastic wave.
It will also include elastic-plastic problems. The longitudinal plane wave in bulk material
can be modeled by a set of governing equations, which are different from that for
longitudinal waves in a thin rod as illustrated in previous chapters. In the following, we
will first present the governing equations for the elastic longitudinal plane wave in bulk
material. We then provdie detailed derivation for the eigenstructure of the equations.
132
We will consider three sets of modeling equations of elastic wave in bulk material:
(i) a two-equation model composed of the mass and the momentum conservation
equations; (ii) a three-equation model, including the mass and the momentum equations,
and the constitutive relation for elasticity, formulated in terms of Cauchy stresses; and (iii)
a second three-equation model similar to that of (ii) but the constitutive equations is
formulated in terms of pressure and deviatoric stresses. All three models assume tha the
material is isothermal. Thus, the energy conservation equation is not considered.
Previously, in the setting of elastic waves in a thin rod, we have clearly
demonstrated that the governing equations are hyperbolic in CHAPTER 3 and
CHAPTER 4. As will be shown in the following sections, the three sets of the governing
equations for longitudinal waves in a bulk material are also hyperbolic because the
eigenvalues of the jacobina matrix are real. Moreover, the derivation of the eigenvalues
of these three hyperbolic systems will show that the speed of the sound of the elastic
material in the three sets of equations are identical and it is c =
( k + (4 / 3 ) µ )
ρ.
In this chapter, by using the deviatoric stresses, we will also show that the firstorder hyperbolic model equations can be used not only for elastic waves but also for
elastic-plastic waves, as well as validate the model and the one-dimensional CESE solver
by solving the elastic-plastic wave in one-dimensional impact problem.
133
Since the modeling equations do not include energy equation, we assume that the
impact process of concerned is isothermal. As such, the model equations are referred to
as the isothermal model. The isothermal assumption is valid for low-speed impact
problem. The equation of state, Eq. (3.105), applied to this model relates pressure with
density only.
Previously, Udaykumar et al. [55] have studied the one-dimensional impact
problem involving elastic-plastic deformation. In their paper, numerical solution of
elastic-plastic wave propagation in one-dimensional copper bulk was provided. Their
model did not assume the isothermal condition. Thus, they included the energy equation
and an equation of state, which defines the relationship between pressure, density and
internal energy. In this chapter, we will compare the numerical results between our
isothermal model and Udaykumar’s model with the energy equation. The comparison
will determine if a simplified model witout considering the energy equation could be
safely applied to analyze nonlinear stress waves in the ultrasound aided manufacture
processes, such as ultrasound welding, in which the magnitudes of material motions are
much lower than that of the impact condition. All of numerical solutions are obtained by
using the CESE method, which has been presented in CHAPTER 3.
134
5.2 Models of Elastic Longitudinal Plane Wave in Bulk Material
Figure 5.2: The longitudinal plane wave in the bulk material
As shown in the Figure 5.2, we consider a bulk material, which extended
indefinitely in all direction. We will focus on the longitudinal wave propagation in the x
direction.
5.2.1 The Second-Order Linear Wave Equation
The equation of longitudinal plane wave is given by
∂2h
∂2h
ρ dx 2 = ( λ + 2 µ ) 2 dx
∂t
∂x
(5.11)
where h(x, t) is the axial displacement, ρ is density, λ and µ are two Lame parameters,
and µ is also called shear modulus. According to the above second order wave equatons,
the speed of the wave is
c=
λ + 2µ
=
ρ
135
4
µ
3
ρ
k+
(5.12)
where c is speed of sound, and the bulk modulus k is defined by
k = λ+
2
µ
3
(5.13)
5.2.2 The Two-Equations Model of Elastic Longitudinal Plane Wave in Bulk
Material
Based on Hook’s law(3.32), the Cauchy stress tensor T and strain tensor e of the
problem described in the Figure 5.2 are given by
T11 0

T =  0 T22
 0
0
0  ( λ + 2µ ) ε11
0
 
0 =
0
λε 11
T33  
0
0
0 

0 ,
λε11 
ε 11

e= 0
 0
0 0

0 0
0 0
(5.14)
The velocity components along the three axes are:
u = u ( x, t ) , v = v( x, t ), w = w(x, t )
(5.15)
Based on the Eq.(5.15), the mass conservation equation and momentum conservation
equation are given by
∂ρ ∂ ( ρ u )
+
=0
∂t
∂x
∂ ( ρu )
∂t
+
∂ ( ρuu − T11 )
∂x
(5.16)
=0
(5.17)
The gradient of deformation F for bulk material is defined by
1 + ε 11
F =  0
 0
136
0 0
1 0
0 1 
(5.18)
Considering the constitutive relations, Eq. (3.48), we have the relation between the strain
ε11 and density ? as
ε11 =
ρ0
−1
ρ
(5.19)
The Cauchy stress component T11 in Eq.(5.17) could be expressed by a function of
density as
ρ
 
4  ρ

T11 = ( λ + 2µ )  0 − 1 =  k + µ   0 − 1
3  ρ
 ρ
 

(5.20)
Substitute Eq.(5.20) into Eq.(5.17) and rewrite Eqs.(5.16) and (5.17) into a vector form,
we have
∂U ∂E
+
=0,
∂t ∂ x
(5.21)
where the conservative variables vector is U = ( ρ , ρu ) , and the flux vector is
T
ρu




E=
4   ρ0   .

 ρ uu −  k + 3 µ   ρ − 1 



To proceed, we apply the chain rule to Eq. (5.21), and get
∂U
∂U
+A
=0
∂t
∂x
where the matrix A is the Jacobian matrix:
137
(5.22)
0


∂E
4 

A=
=
k + µ  ρ0

∂U  2 
3 
 −u +
ρ2

1



2u 

(5.23)
The eigenvalues of matrix A are
λ1,2
4 

 k + 3 µ  ρ0

=u± 
2
ρ
(5.24)
The wave speed is
4 

 k + 3 µ  ρ0

c= 
ρ2
(5.25)
which is slightly different from the wave speed defined in Eq. (5.12).
Different from the conservative-form equations, Eq. (3.54), in the discussion
about the elastic wave in a thin rod, the conservative-form equation, Eq. (5.21), does not
have any source term on the right side of the equation. Therefore, the eigenvalues and the
wave speed of this conservative form equatons are exactly same as those derived from the
non-conservative- form equations.
T
To rewrite Eq. (5.22) with non-conservative variables vector U% = ( ρ , u ) , we
multiply both sides of Eq. (5.22) with matrix M
M
∂U
∂U
+ MAM −1M
=0
∂t
∂x
138
(5.26)
where M is defined by
1
% 
∂U
M=
=
u
∂U  −
 ρ
0
1 
ρ 
(5.27)
With Eq.(5.27), we rewrite Eq.(5.26) as
%
%
∂U
% ∂U = 0
+A
∂t
∂x
(5.28)
where
% = MAM −1
A
u


4 

=   k + µ  ρ0

3 

ρ3

ρ



u

(5.29)
% are
The eigenvalues of matrix A
λ1,2
4 

 k + 3 µ  ρ0

= u ± c =u ± 
2
ρ
(5.30)
The speed of sound is
4 

 k + 3 µ  ρ0

c= 
2
ρ
(5.31)
Corresponding to those two eigenvalues, the eigenvectors could be obtained by solving
the equation
( A% − λ I ) m%
i
i
=0
139
i = 1,2
(5.32)
These two eigenvectors are
(
% 1 = m% 11, m% 12
m
(
T
)
 
4 
  k + µ  ρ0
3 
= 1, 

ρ2


)

4 


 k + 3 µ  ρ0

=  1, − 

ρ2


% 2 = m% 12 , m% 22
m
T
T






T






% and its inverse M
% −1 are given by
Therefore, the transform matrix M
1


% =  k + 4 µ  ρ
M
 
3  0

ρ2



4  

 k + 3 µ  ρ0 


−

2
ρ

1
(5.33)
and
1

2


% −1 = 
M
1
2







1
ρ2

−

2 
4  
k
+
µ
ρ

 0
3  

1
ρ2
2 
4 
 k + µ  ρ0
3 

(5.34)
The characteristic form of Eq. (5.28) could be given by multiplying the both sides of the
% −1
equation with M
140
%
%
% −1 ∂U + M
% −1AMM
% % % −1 ∂U = 0
M
∂t
∂x
(5.35)
ˆ
ˆ
∂U
ˆ ∂U = 0
+A
∂t
∂x
(5.36)
We have
where the matrix  is defined by
ˆ = M% −1AM
% % =  λ1
A

0

4 


 k + µ  ρ0
3 
u + 
ρ2
0 
=
 
λ2  


0




0


4  

 k + µ  ρ0 
3 

u− 
ρ2

(5.37)
The variable s vector for Eq.(5.36) is given by
1
1
 ρ+
2
2


% −1U
% =
Uˆ = M
1
1
ρ
−
2
2




4  

 k + µ  ρ0 
3 


ρ 2u


4  

k
+
µ
ρ

 0
3  

ρ 2u
(5.38)
In the setting of the Method Of Characteristics (MOC), Eqs. (5.36)-(5.38)constitute the
analytical solution of nonlinear elastic waves in bulk material. In the x-t plane, along the
characteristic lines dx dt = λ1,2 , the Riemann invariants û1,2 are constant:
duˆ i  ∂
∂ 
=  + λi  uˆi = 0,
dt  ∂ t
∂x 
141
i = 1,2
(5.39)
The right running wave:
(k + 43 µ) ρ
dx
along
=u+
dt
0
ρ2
,
1
1
ρ+
2
2
,
1
1
ρ−
2
2
(
ρ 2u
)
= constant .
)
= constant .
k + 4 µ ρ0
3
The left running wave:
( k + 43 µ ) ρ
dx
along
=u−
dt
0
ρ2
(
ρ 2u
k + 4 µ ρ0
3
5.2.3 The Isothermal Model-I of Elastic Longitudinal Plane Wave in Bulk
Material
Consider Eqs. (5.14) (3.72) (3.73), and apply a = 0 in Eq.(3.73), we have the time
derivative of Cauchy stress component T11 as
∂T11
∂T
∂u 
4  ∂u
+ u 11 = ( λ + 2 µ )
= k + µ 
∂t
∂x
∂x 
3  ∂x
(5.40)
Aided by the continuity equation, we transfer the equation into the conservative form
with ρT11 as the unkno wn:
∂ ( ρT11 )
∂t
+
∂ ( ρT11u )
∂x
4  ∂u

= k + µ  ρ
3  ∂x

(5.41)
When the wave propagation in bulk material is assumed isothermal, the present model for
the elastic wave in bulk material includes Eqs. (5.16), (5.17), and (5.41). By solving these
three coupled equations, we obtain the solution of density ρ , velocity u and stress
component T11 simultaneously.
142
To proceed, we rewrite Eqs. (5.16), (5.17), and (5.41) into a vector form:
∂U ∂E
+
=H
∂t ∂ x
(5.42)
where the unknown vector, conservative flux components and source term on the right
side of equation
U = ( ρ , ρu , ρT11 ) ,
T
E = ( ρu , ρ uu − T11, ρT11u )
T
T

4  ∂u 

H =  0,0,  k + µ  ρ
 .
3  ∂x 


Aide by the chain rule, Eq. (5.42) is transformed to
∂U
∂U
+A
=H
∂t
∂x
(5.43)
where the matrix A is defined by

0

∂E  2 T11
A=
= −u +
∂U 
ρ

 −T11u
0 

1
2u −
ρ

T11 u 
1
(5.44)
The eigenvalues of matrix A could be calculated directly, and they are
λ1,2,3 = u
(5.45)
Since the eigenvalues of this matrix A are real, this conservative form is a
hyperbolic system. However, the calculation of these three eigenvalues does not consider
the source term on the right side. Thus, they are not the real eigenvalues of the hyperbolic
143
system. In what fo llows, we use the model equations in the non-conservative form to
derive the eigenvalues. To proceed, we rewrite Eq. (5.43) to be in terms of the nonT
conservative variables vector U% = ( ρ , u, T11 ) . We pre multiply both sides of Eq. (5.43) by
matrix M and have
M
∂U
∂U
+ MAM −1M
= MH
∂t
∂x
(5.46)
where matrix M is defined by


1

%
∂U  u
M=
=−
∂U
ρ

 T11
−
 ρ
0
1
ρ
0


0


0

1

ρ
(5.47)
The inverse matrix is given by
 1 0 0
M =  u ρ 0 
T11 0 ρ 
−1
(5.48)
Using Eqs. (5.47) and (5.48), we rewrite Eq.(5.46) as
%
%
∂U
% ∂U = H
%
+A
∂t
∂x
where
144
(5.49)
% = MAM −1
A
(
)
µ

u λ + µ ρ


= 0
u

0
0


0 

1
− 
ρ

u 

(5.50)
% =  0,0, k + 4 µ ∂u  . To proceed, we move the source term from the right hand
and H

3 ∂x 

T
side of the equation to the left hand side. As a result, Eq. (5.49) has the following form:
%
%
∂U
∂U
+A
=0
∂t
∂x
(5.51)


µ
ρ
0 
u
λ+µ



1
A = 0
u
− 
ρ



4 

0 −  k + µ  u 
3 



(5.52)
where
After directly calculation, the eigenvalues of matrix A are given by
λ1 = u,
4 

k + 3 µ 

λ2,3 = u ± c = u ± 
ρ
The speed of sound is defined by
4 

k + 3µ

c= 
ρ
This speed is same as the plane longitudinal wave speed presented by Eq.(5.12).
145
(5.53)
Corresponding to those three eigenvalues, the eigenvectors could be obtained by solving
the equation
( A − λ I ) m%
i
=0
T
,
i
i = 1,2,3
(5.54)
These three eigenvectors are
(
% 1 = m% 11, m% 21 , m% 31
m
) = (1,0,0)
(
% 2 = m% 12 , m% 22 , m% 32
m
(
% 3 = m% 13 , m% 23 , m% 33
m
T
)

 1
=  1,
 ρ

)

1

=  1, −
ρ


T
T
(
) (
k + 43 µ
k + 43 µ
,−
ρ
ρ
(
)
) (
k + 43 µ
k + 43 µ
,−
ρ
ρ
T





)
T





% and its inverse M
% −1 are given by
Therefore, the transform matrix M

1



%
M = 0




0

1
1
ρ
4 

k + µ 
3 

ρ
4 

k + µ 
3 
−
ρ
and
146


1

4 

k + µ 

1 
3 
−

ρ
ρ

4  

k + µ  
3 
−

ρ

(5.55)


1
0




ρ
% −1 =  0 ρ
M
4 
2 

k + µ 

3 


ρ
ρ

0 −

4 
2 

k + µ 
3 




ρ

4  

k + µ 
3 



ρ
−

4 

2k + µ 
3 


ρ

−
4 

2k + µ 
3  

(5.56)
The characteristic form of Eq.(5.51) could be given by multiplying the both sides of the
% −1
equation with M
%
%
% −1 ∂U + M
% −1AMM
% % −1 ∂U = 0
M
∂t
∂x
(5.57)
ˆ
ˆ
∂U
ˆ ∂U = 0
+A
∂t
∂x
(5.58)
or
where the matrix  is defined by
 λ1
−1

ˆ
%
%
A = M AM =  0
 0
0
λ2
0
0

0
λ3 


0
u

4 


k + µ 
3 


= 0 u +
ρ



0
0

The variables vector for Eq.(5.58) are given by
147


0




0


4 

k
+
µ

3 

u−

ρ
(5.59)




T11ρ


ρ+

4 


k + µ 


3 





ρ
T11 ρ
% −1U
% =  ρu
Uˆ = M
−

 2  k + 4 µ  2 k + 4 µ  





3 
3  




ρ
T11ρ
 ρu

−
−
 2 
4 
4 

k
+
µ
2
k
+
µ 




3 
3  



(5.60)
5.2.4 The Isothermal Model-II of Elastic Longitudinal Plane Wave in Bulk
Material
When the Cauchy stress components are divided into the deviatoric stresses and pressure
as shown in Eq. (3.98), the momentum conservation equation, Eq. (5.17), becomes
∂ ( ρu )
∂t
+
∂ ( ρuu + p − S11 )
∂x
=0
(5.61)
To proceed, we consider Eq. (3.98), in whic the Cauchy stress tensor can be divided into
the devitoric stresses and pressure:
0  − p 0
0   S11
T11 0
0 T


0  =  0 − p 0  +  0
22

 0 0 T33   0
0 − p   0
where the devitoric stresses can be expressed as
148
0
S22
0
0
0 
S33 
(5.62)
S11
0

 0
0
S22
0

S
0   11
0 =0
 
S33  
0

0
−
  4 µε
  3 11
 
0 = 0
 

1 
− S11   0
2  
0
0
1
S11
2
0
2
− µε11
3
0



0 


2
− µε11 

3
0
(5.63)
Aided by Eqs. (3.73) and (5.63), and similar derivation steps for Eq. (3.76), we have
∂ ( ρ S11 )
∂t
∂ ( ρ S22 )
∂t
∂ ( ρ S33 )
∂t
+
+
+
∂ ( ρuS11 )
∂x
∂ ( ρ uS33 )
∂x
∂ ( ρuS33 )
∂x
=
4
∂u
µρ
3
∂x
(5.64)
2
∂u
= − µρ
3
∂x
(5.65)
2
∂u
= − µρ
3
∂x
(5.66)
Since we consider the wave propagation one-dimensional, S22 and S33 are directly related
to S11 , as shown in the above equations. Thus the unknowns of the model equations are
density ρ , velocity u, pressure p, and stress component S11 . The isothermal solid
dynamics model-II is composed of Eqs. (5.16), (5.61), (5.64) and the equation of sate,
Eq.(3.105). To proceed, we recast Eqs. (5.16), (5.61) and (5.64) into a vector form:
∂U ∂E
+
=H
∂t ∂ x
(5.67)
where
U = ( ρ , ρ u, ρ S11 ) ,
T
E = ( ρu , ρ uu + p − S11 , ρ S11u ) ,
T
149
4
∂u 

H =  0,0, µρ
 .
3
∂x 

T
Aided by the chain rule, Eq.(5.67) becomes
∂U
∂U
+A
=H
∂t
∂x
(5.68)
where

0

∂E  2 k S11
A=
= −u + +
∂U 
ρ ρ

− S11u

0 

1
− 
ρ

u 
1
2u
S11
(5.69)
The eigenvalues of matrix A can be readily derived:
λ1 = u and λ2,3 = u ±
k
ρ
(5.70)
All eigenvalues are real. Thus, the equation system is hyperbolic. However, the above
derivation of the eigenvalues did not consider the source term. Therefore, they do not
represent the real wave speeds of the wave propagation process.
Alternatively, we rewrite the above hyperbolic system by using the nonconservative variables vector:
T
U% = ( ρ , u, S11 ) .
We premultiply both sides of Eq. (5.68) by a matrix M :
150
M
∂U
∂U
+ MAM −1M
= MH
∂t
∂x
(5.71)
where


1

%  u
∂U
M=
= −
∂U
ρ

 S11
−
 ρ
0
1
ρ
0


0


0

1

ρ
(5.72)
Its inverse matrix is
1
M =  u
 S11
−1
0
ρ
0
0
0 
ρ 
(5.73)
Aided by Eqs. (5.72) and (5.73), we rewrite Eq. (5.71) to be
%
%
∂U
% ∂U = H
%
+A
∂t
∂x
(5.74)
where
u

% = MAM − 1 =  k
A
ρ2

0
% =  0,0, 4 µ ∂u  .
and H


3 ∂x 

T
151
ρ
u
0
0 

1
− 
ρ

u 
(5.75)
By moving the source term from right side to the left side, we could transform Eq. (5.74)
into the form as
%
%
∂U
∂U
+A
=0
∂t
∂x
(5.76)
where

u

k
A =  2
ρ

 0


ρ
0 

1
u
− 
ρ

4
− µ u 
3

(5.77)
The eigenvalues of matrix A can be readily derived and they are
λ1 = u,
4 

k + 3 µ 

λ2,3 = u ± c = u ± 
ρ
As a part of the eigenavlues, the speed of sound is
4 

k + 3µ

c= 
ρ
(5.78)
This speed of sound is identical to that in Eqs. (5.12) and (5.53). To proceed, the left and
right eigenvectors of the Jacobian matrix can be derived by solving the equation
( A − λ I ) m%
i
i
=0
These three eigenvectors are
152
i = 1,2,3
(5.79)
(
% 1 = m% 11, m% 21 , m% 31
m
(
%2= m
% 12 , m
% 22 , m% 32
m
(
(
),
)

 1
=  1,
 ρ

(
)

1

=  1, −
ρ


)
T
% 3 = m% 13 , m% 23 , m% 33
m
= 1,0, k ρ
T
T
T
)
T

k + 43 µ
4µ
,−
and
ρ
3 ρ 

(
)
T

k + 43 µ
4 µ
,−
ρ
3 ρ 

% and its inverse M
% −1 are given by
Then, the transform matrix M
1


% =  0
M


0

1
1
ρ
4 

k + 3 µ 


ρ
−


4 

k + µ
1 
3 
−

ρ
ρ


4µ
−

3ρ

1
4µ
3ρ
(5.80)
and




0
1


ρ
% −1 =  0 ρ
M
4 
2 

k + µ 

3 


ρ
ρ

0 −

4 
2 

k + µ
3 


153



3ρ
4µ 

3ρ 
−

8µ 


3ρ 
−
8µ 


(5.81)
The characteristic form of Eq.(5.76) could be given by multiplying the both sides of the
% −1
equation with M
%
%
% −1 ∂U + M
% −1AMM
% % −1 ∂U = 0 ,
M
∂t
∂x
(5.82)
ˆ
ˆ
∂U
ˆ ∂U = 0
+A
∂t
∂x
(5.83)
we have
where the matrix  is defined by
 λ1
ˆ = M% AM
% = 0
A

 0
−1
0
λ2
0
0

0
λ3 


0
u

4 


k + µ 
3 


= 0 u +
ρ



0
0



0




0


4 

k + µ  
3 
u− 

ρ
(5.84)
The variables vector for Eq. (5.83) is given by







% −1U
% =  ρu
Uˆ = M
 2


 ρu
−
 2





3 S11ρ

ρ+

4 µ

ρ
3S ρ 
− 11 
4 

8µ 
k + µ 
3 



ρ
3S11 ρ 
−
4 

8µ 

k + µ
3 


154
(5.85)
To recap, with the equations of state, Eq. (3.105), the isothermal solid dynamics
models can be expressed by using the Cauchy stresses as well as the deviatoric stresses.
Both formulas can catch the correct elastic wave in bulk material. As will be shown in the
following sections, the isothermal solid dynamics model expressed by deviatoric stress is
more useful in modeling elastic-plastic problems.
5.3 The Isothermal Model of Longitudinal Plane Wave in ElasticPlastic Bulk Material
By using a suitable constitutive equation to model the material response of a
elastic-plastic material, we proceed to extend the isothermal model-II for modeling stress
wave propagation in elastic-plastic media. Before presenting the modeling equations, the
description about plasticity is needed. In this work, we will focus on the infinitesimal
plasticity.
5.3.1 Infinitesimal Plasticity
In the plasticity for infinitesimal deformation, whose magnitude of elastic deformation is
assumed comparable to that of plastic deformation, the Prandtl-Reuss Equation could
express the total strain as,
dε ij = dε ije + dε ijp
(5.86)
where the elastic strain rate dε ije and plastic strain rate dε ijp are defined as,
dε ije =
1+ν
ν
dTij − dTkk δ ij
E
E
155
(5.87)
d ε ijp = d λ p
∂f p
(5.88)
∂Tij
where f p = f p ( Tij , ε ijp ) , a plastic flow potential, is a function of yield surface; λ p is a nonnegative function which may depends on stress, stress rate, strain, and history of the
deformation. Equation (5.88) satisfies the requirement that the plastic strain rate vector is
normal to the yield surface.
The stress point, which remains on the yield surface, has a consistency condition requires,
(
)
f p Tij , ε ijp = 0
∂f p
df p =
∂Tij
dTij +
∂f p
∂ε ijp
(5.89)
dε ijp
(5.90)
For the material with hardening behavior during deformation, we have
∂f p
∂ε
p
ij
dε ijp ≠ 0 ,
and then
∂f p
∂ε
p
ij
d ε ijp = −
∂f p
∂Tij
dTij .
(5.91)
By using Eq. (5.88), we change Eq. (5.91) in the form as,
∂f p
∂ε
p
ij
dλp
∂ fp
∂Tij
and then we have
156
=−
∂ fp
∂Tij
dTij ,
(5.92)
∂f p
dTij
∂Tij
d λp = −
≥0
∂f p ∂f p
(5.93)
∂ε klp ∂Tkl
Using Eq. (5.93) to rewrite Eq. (5.88), we have
∂f p
dT
∂Trs rs ∂f p
p
d ε ij = −
∂f p ∂f p ∂Tij
∂ε klp ∂Tkl
(5.94)
In the present work, the flow potential is taken as
f p = J2 =
1
Sij Sij
2
(5.95)
Aided by the above J2 flow potential, Eq.(5.95), we will show that ∂f p ∂T ij = S ij
in following. To proceed, we use the chain rule and have
∂f p
∂Tij
=
∂f p ∂S kl
.
∂Skl ∂Tij
(5.96)
According to J2 flow potential shown in Eq.(5.95), we have
∂f p
∂S kl
= S kl .
(5.97)
Based on the definition of deviatoric stress component shown by Eq.(3.98), we have
∂S kl
∂ 
1

=
 Tkl − Tmmδ kl 
∂Tij ∂Tij 
3

.
1
= δ ki δ lj − δ kl δ ij
3
Aided by Eqs. (5.97) and (5.98), the Eq. (5.96) becomes
157
(5.98)
∂f p
1


=  δ kiδ lj − δ klδ ij  S kl
∂Tij 
3

1
= δ kiδ lj Skl − δ klδ ij Skl
3
1
= Sij − Skk δ ij
3
= Sij
(5.99)
When the effective stress is assumed as a function of plastic work, the flow potential is
defined by,
fp =
( )
2
1
1
S ij S ij − T W p 

2
3
(5.100)
Using this function, we will show the steps of derivation of d λp as following.
∂f p
∂Trs
dTrs = SrsdTrs
d 
1

S rs + Tkk δ rs 

dt 
3

1
= S rsdS rs + dTkk δ rs S rs
3
= S rs dSrs
1d
=
( Srs Srs )
2 dt
1 d 2 2
=
T
2 dt  3 
2
= TdT
3
= S rs
158
(5.101)
∂f p
∂ε klp
=
∂f p ∂T ∂W p
∂T ∂W p ∂ε klp
(
p
2 ∂T ∂ Tklε kl
=− T
3 ∂W p ∂ε klp
)
2 ∂T
=− T
T
3 ∂W p kl
∂f p ∂f p
2 ∂T
=− T
Tkl S kl
∂ε ∂Tkl
3 ∂W p
p
kl
2 ∂T 
1

=− T
S + Tkk δ kl  Skl
p  kl
3 ∂W 
3

2
∂T
=− T
Skl Skl
3 ∂W p
2 ∂T 2 2
=− T
T
3 ∂W p 3
4
∂T
= − T3
9 ∂W p
(5.102)
Aided by Eqs. (5.101) and (5.102), d λp is given by
2
TdT
d λp = − 3
4
∂T
− T3
9 ∂W p
3 dT
=
2 2 ∂T
T
∂W p
And with definition dW
p
(5.103)
= T dε p , we can get d λ in the case that effective stress is a
function of effective plastic strain.
159
dT
∂T
2
T
T ∂ε p
3 dT
=
2 ∂T
T
∂ε p
dλ =
3
2
(5.104)
By assuming J2 flow potential and plastic strain hardening, and aided by Eqs. (5.87),
(5.88), (5.99), and (5.104), we recast Eq. (5.86) to be
dε ij =
1 +ν
ν
3
dTij − dTkk δ ij +
E
E
2
dT
S ij
∂T
T
p
∂ε
(5.105)
Multiplying Sij with both sides of Eq. (5.105), then we have,
S ij dε ij =
=
=
1 +ν
ν
3
S ij dTij − S ij dTkk δ ij +
E
E
2
1 +ν 2
3
T dT − 0 +
E 3
2
dT
S ij S ij
∂T
T
∂ε p
dT 2 2
T
∂T 3
T
∂ε p
(5.106)
1+ν 2
dT
T dT +
T
E 3
∂T
∂ε p
Using Eq. (5.106), we have effective stress increment:
dT =
S ij dε ij
2 1 +ν
T
T+
∂T
3 E
∂ε p
Rewriting Eq. (5.105), then we get
160
(5.107)
1 +ν
ν
3
dTij = dε ij + dTkk δ ij −
E
E
2
dT
S ij
∂T
T
p
∂ε
(5.108)
Substituting Eq. (5.107) into Eq. (5.108), we have
S ij
S kl dε kl
1 +ν
ν
3
dTij = dε ij + dTkk δ ij −
E
E
2 2(1 + ν )
T
∂T
T+
T
dT
3E
∂ε p
dε p
(5.109)
According to the definition of the normal strain increment component, we have
d ε kk = dε kkE =
1 − 2ν
dTkk
E
(5.110)
we could change Eq.(5.109) to be


3
S kl dε kl


E
ν
2
dTij =
dε kk δ ij −
S ij 
dε ij +
2 1 + ν 2 dT
1 +ν 
1 − 2ν
T
+T 2 
p


3 E
dε
(5.111)
1
By using relation of T = tr (T)I + S and Eq. (5.110), we transform Eq. (5.111) as
3


3
S kl dε kl


1 E
E
ν
2
dS ij +
dε kk δ ij =
dε kk δ ij −
S ij  (5.112)
 dε ij +
2 1 + ν 2 dT
3 1 − 2ν
1 +ν 
1 − 2ν
T
+T2 
p


3 E
dε
Rewriting Eq. (5.112), we have deviatoric stress increment comonents
161


E 
1
dS ij =
dε ij − dε kk δ ij −
1 +ν 
3
1 + ν


 E


3

S kl dε kl
2
S ij 

dT
3
+ (S kl S kl ) 
p
dε
2

(5.113)
Recast Eq. (5.113) into a tensor form and we have




DS
E 
1
3
1
(
)
(
)
=
D − tr D I −
S ⋅ D S

Dt 1 + ν 
3
2  1 + ν dT
3

+

(
S
⋅
S
)


p
2 
 E dε


Using relation of µ =
(5.114)
E
, we rewrite equation (5.114) as,
2(1 + ν )




DS
1
3
1

= 2 µ D − tr (D)I −
(S ⋅ D)S

Dt
3
2  1 dT
3

+ (S ⋅ S )


p
2
 2 µ dε


2
1
(S ⋅ D)S
= 2 µD − µtr (D)I − 3µ
3
 1 dT
3

+ (S ⋅ S )
p
2
 2µ dε
(5.115)
Aided by the assumption of linear strain hardening, i.e.,
( )
T = A + BSH ε
we have
p n
,
n =1
(5.116)
dT
= BSH , then Eq. (5.115) becomes
dε p
DS
2
1
= 2 µ D − µ tr ( D) I − 3µ
( S ⋅ D )S
Dt
3
 1
3
 2µ BSH + 2  ( S ⋅ S )


162
(5.117)
To recap, in this research work, we assume J2 flow potential and linear strain
hardening. The above derivation shows that the constitutive equation of linear elastic
linear strain hardening plastic solid is
DS 2
+ µtr(D )I − 2 µD + θ ( s )(S ⋅ D )S = 0
Dt 3


if s = S ⋅ S < 2k 2
0,

θ ( s) = 
0,
if s = S ⋅ S ≥ 2k 2
for unloading

3
µ if s = S ⋅ S ≥ 2k 2
for loading

,
2
 ( BSH / µ + 3 ) k
(5.118)
When plasticity of material is perfect plasticity ( BSH = 0 ), the Eq. (5.118) will be
changed to be the constitutive equation for linear elastic perfect plastic solid material, i.e.,
DS 2
~
+ µtr(D )I − 2 µD + θ (s )(S ⋅ D )S = 0
Dt 3

if s = S ⋅ S < 2k 2
0
,

~

2
for unloading
(
)
θ s =  0, if s = S ⋅ S ≥ 2k
 µ if s = S ⋅ S ≥ 2k 2
for loading
 k 2 ,
(5.119)
5.3.2 Radial Return Maping
Based on the consistent condition, the effective stress must be constrained to always fall
either within or on the yield surface. Different from the solving the equation for
infinitesimal plasticity, in solving finite plasticity problem, the typical numerical method
is radian return algorithm.
163
Figure 3 Radian return for finite plasticity
When the plastic deformation is considered as finite plasticity, we use Radial
Return Algorithm [55, 78] to calculate stress of material. This algorithm includes two
steps: (i) predict a trial stress by assuming purely elastic deformation of the material by
following Eq.,
∂ ( ρ S11, tr )
∂t
+
∂ ( ρ uS11, tr )
∂x
=
4
∂u
µρ
3
∂x
(5.120)
and (ii) correct the trial stress to be true elastic-plastic stress by pulling the trial stress
back to the yield surface.
S11, tr = S11, tr −
= S11, tr −
S11, tr
S11, tr ⋅ S11,tr
S11,tr
S11,tr
⋅
S11, tr ⋅ S11,tr − S11, pre ⋅ S11,pre
⋅

B 
1+

 3µ 
S11, tr − S11,pre
1+
B
3µ
164
(5.121)
where S11,tr is the trial stress predicted by assuming purely elastic deformation, S11, pre the
true elastic-plastic stress calculated at the previous time step. Here, the shear modulus µ
appeared in this Radial Return Algorithm is assumed as a constant instead of a variable,
which normally is updated for each time step according to the deformation.
5.3.3 Modeling Equations
Similar to the modeling equations for the stress wave propagation in elastic material, the
modeling equations of the elastic-plastic stress waves include the mass and momentum
conservation equation. For one-dimensional cases, the mass and momentum equations
are,
∂ρ ∂ ( ρ u )
+
=0
∂t
∂x
∂ ( ρu )
∂t
+
∂ ( ρuu + p − S11 )
∂x
(5.16)
=0
(5.61)
Moreover, an elastic-plastic constitutive equation in one spatial dimension is


∂ ( ρ S11 ) ∂ ( ρuS11 ) 4
θ
+
= µρ  1 −
∂t
∂x
3
 1 + BSH

3µ

2
0, if s = S ⋅ S < 2k

θ = 0, if s = S ⋅ S ≥ 2k 2
1,
 if s = S ⋅ S ≥ 2k 2

 ∂u

 ∂x


(5.122)
for unloading
for loading
Based on the previous analysis, Eqs. (5.16),(5.61) and (5.122) could generate a
hyperbolic system, which could be written in vector form as:
165
∂U ∂E
+
=H
∂t ∂ x
(5.123)
where
U = ( ρ , ρ u, ρ S11 ) ,
T
E = ( ρu , ρ uu + p − S11 , ρ S11u ) ,
T
4
∂u 

H =  0,0, µρ (1 − θ (1 + BSH 3µ ) )  .
3
∂x 

T
To close the system of equations, we employ the following equation of state to relate
pressure to density:
p = k ln
ρ
ρ0
(3.105)
By analyzing the eigenstructure of this hyperbolic system, we directly calculate the speed
of sound in the solid with plastic deformation.
Alternatovely, we rewrite the above hyperbolic system by using the nonconservative variables vector:
T
U% = ( ρ , u, S11 ) .
We have non-conservative form as:
%
%
∂U
% ∂U = H
%
+A
∂t
∂x
where
166
(5.124)
u

% = MAM − 1 =  k
A
ρ2

0
ρ
u
0
0 

1
− 
ρ

u 
(5.125)
% =  0,0, 4 µ (1 − θ (1 + B 3µ ) ) ∂u  .
and H

SH

3
∂x 

T
By moving the source term from right side to the left side, we could transform Eq. (5.124)
into the form as
%
%
∂U
∂U
+A
=0
∂t
∂x
(5.126)
where




u

 k
A= 2
ρ



 0


ρ
u

4 
θ
− µ 1−
3  1 + BSH

3µ


 ∂u

 ∂x






0 

1
− 
ρ



u 


(5.127)
For the plastic wave, i.e. θ = 1 , the eigenvalues of matrix A can be readily derived and
they are
167
λ1 = u,


4 
1

k + µ 1−

3  1 + BSH
3µ 

λ2,3 = u ± c = u ±
ρ
(5.128)
which is slower than elastic wave speed shown in Eq.(5.78). In particular, for the elasticperfect plastic material, i.e. BSH = 0 in Eq.(5.122), the plastic wave speed is given by:
c=
k
ρ
(5.129)
In the rest of the present chapter, the above formulation will be numerical solved by the
CESE method. In Section 5.4, the computational conditions will be illustrated. Section
5.5 shows the numerical results. We then offer the conclusion remarks about the present
chapter in Section 5.6.
5.4 Computation Settings
We consider a one-dimensional copper bulk with an initial speed u = 40 m/s hitting a
stationary copper bulk. Refer to Figure 5.4
Figure 5.4: Initial condition of the one-dimensional impact problem.
The initial pressures p and deviatoric stress component S11 in both copper bulks are null.
The material properties of copper are listed in Table 3. We assume the material is elasticperfect plastic, i.e. the yield stress always equals to the initial yield stress without
hardening. Or, BSH = 0 in Eq. (5.122).
168
k (GPa)
ρ0 (kg/m3 )
µ(GPa)
E (GPa)
σ y (MPa)
140
8930
45
122
90
Table 3: Material properties of copper
The boundary conditions of at the left end of the initially moving copper bulk and
the right end of initially static copper bulk are set as the non-reflective boundary
conditions. This boundary condition allows waves exit the copper bulks without ant
reflection. The focus of the present impact problem is the interactions between the
moving copper block and initially static block. The non-reflective boundary condition at
the two far ends allows clear observation of wave evolution initiated from the impact.
The computational domain is 2 meters, which is uniformly discretized into 400
numerical cells. The time step for the time marching calculations is 0.6 µs. Based on the
known size of spatial grid, the time increment, and the longitudinal plane wave speed in
copper bulk, the CFL number in computation is controlled to be about 0.6. The physical
duration of wave propagation in computation is 0.17 ms.
5.5 Numerical Results
In this section, we present the computation results of pressure wave and density wave in
right initially static copper bulk.
169
Figure 5.5: A snapshot of density at t = 0.17ms in initial static copper bulk. The CESE
numerical result by using the isothermal model is compared to the exact solution by
Udaykumar et al. [55].
170
Figure 5.6: A snapshot of pressure at t = 0.17 ms in the initial static copper bulk. The
numerical result of the isothermal model by the CESE method is compared with the exact
solution by Udaykumar et al. [55].
In both Figure 5.5 and Figure 5.6, red lines with symbols present the numerical
solutions of density and pressure by the CESE method. The blue solid lines in these two
figures represent the exact solutions by Udaykumar et al. [55]. They used the MieGruneisen equation as the equation of state to relate internal energy, pressure and density.
Figure 5.5 and Figure 5.6 show that the numerical solutions by solving the isothermal
model equations compare well with the analytical solution [55] in terms of the wave
locations and strength for both the plastic wave and the precursive elastic wave. Since the
analytical solution was calculated by using the Mie-Gruneisen equation of state [79], that
agreement between numerical solutions and the analytical solution shows that in the
171
range of low- impact force, the material response simulated by the simple equation of
state with isothermal assumption asymptotically approaches that simulated by the MieGruneisen equation.
Both the exact solution and the numerical solution show that the elastic wave is
faster than the plastic wave. This is consistent with elastic-plastic wave speed shown in
Eq. (5.128). In a solid with pure elastic deformation, the wave speed is
c =  k + ( 4 3) µ  ρ . When the deformation involves perfect plasticity, the wave speed
is c = k ρ , which is lower than elastic wave speed.
5.6 Conclusions
The isothermal hyperbolic model of stress wave in elastic-plastic solid does not include
energy conservation equation, and equation of state employed relates pressure and
density, without considering internal energy. We applied the isothermal model to
simulate low speed impact problem, e.g., the impact speed at u = 20 m/s. The numerical
results were validated by comparing to the analytical solution, which was derived by
using a more comprehensive equation state with the thermal effect.
The above results show that the isothermal model developed in the present
chapter could correctly predict the elastic-plastic wave propagation. Thus, if temperature
is not of concern in a low- impact-speed problem, process which might be able to
assumed as a isothermal problem due to slight temperature change and low material
particle speed, one may use the isothermal model to simulate process instead of complete
model including the thermal effect.
172
CHAPTER 6
THE COMPLETE THERMAL DYNAMIC MODEL OF
LONGITUDINAL PLANE WAVE IN ELASTIC-PLASTIC
BULK MATERIAL AND ONE-DIMENSIONAL IMPACT
PROBLEM
6.1 Introduction
The present chapter extends the isothermal model in the last chapter for modeling elasticplastic stress wave problems with significant temperature change s. To this end, we
include the energy conservation equation as a part of the model equation. Moreover, the
equation of state also needs to include the temperature effect. As discussed in CHAPTER
5, previously, Udaykumar et al. [55] reported a dynamic model for stress waves with the
energy equation. For one-dimensional impact problem, they also provided the exact
solution of the elastic-plastic wave propagation in a copper bulk impact problem.
The model equations to be presented in this chapter include the mass, momentum,
and energy conservation equations. Moreover, the time rate of the elastic/plastic
constitutive equation in the form of a convection-diffusion equation will also be included.
Finally, the equation set will be closed by the Mie-Gruneisen equation, an equation of
state to relate pressure, density, and internal energy.
173
The model will be validated by numerical solution of one-dimensional impact
problem, similar to that in the last chapter. The CESE method will be employed and the
results will be compared with the theoretical solution provided by Udaykumar et al. [55].
We also conduct a series of comparison between the solutions by the new model with
those generated by using the isothermal model presented in CHAPTER 5. Comparison as
such will allow clear assessment for the influence by the energy equation and the MieGrune isen equation.
One of the conclusions from this effort will determine if a simplified isothermal
model could be safely applied to analyze nonlinear stress wave propagation in the
manufacture processes aided by ultrasonic alternating stresses, e.g., Ultra-Sound
Welding (USW), in which the magnitudes of material motions are much lower than that
of the impact condition.
6.2 The Modeling Equations
As described in the section of introduction, to model the thermal effect, we add the
energy conservation equation to the model equations. For waves in the one-dimensional
space, the current model includes four equations :
∂ρ ∂ ( ρ u )
+
=0
∂t
∂x
∂ ( ρu )
∂t
+
∂ ( ρuu + p − S11 )
∂x
(5.16)
=0
∂e ∂ u ( e + p ) − uS11 
+
=0
∂t
∂x
174
(5.61)
(6.1)


∂ ( ρ S11 ) ∂ ( ρuS11 ) 4
θ
+
= µρ  1 −
∂t
∂x
3
 1 + BSH

3µ


 ∂u

 ∂x


(5.122)
where
2
0, if s = S ⋅ S < 2k

θ = 0, if s = S ⋅ S ≥ 2k 2
1,
2
 if s = S ⋅ S ≥ 2k
for unloading
for loading
and BSH = 0 for the elastic-perfect plastic material. The specific total energy per unit
volume e is defined by
e = ρ ein + ρ ( v ⋅ v ) 2 + ( S ⋅ D) 2
(6.2)
which includes the internal energy ein , kinematical energy ρ ( v ⋅ v ) 2 and the shear
elastic strain energy ( S ⋅ D) 2 . For linear elasticity, we have
( S ⋅ D)
2 = ( S ⋅ S ) 4µ
(6.3)
In an one-dimensional space, velocity vector v = ( u ,0,0) . Thus, we have the total
T
energy per unit volume as:
ρ
3S 2
e = ρ ein + u 2 + 11
2
8µ
(6.4)
As one of three components of the specific total energy, the elastic strain energy
per unit volume is defined as [80]
V0 =
1
K T T
2 ijkl ij kl
175
(6.5)
where K ijkl , a forth order tensor, is elasticity tensor. When material is homogeneous, this
tensor is a constant tensor, which has 81 components. Considering the symmetric of
strain, the number of components reduces from 81 to 54; when stress tensor is symmetric,
then number reduces from 54 to 36; furthermore, because the strain energy is a positive
definite, therefore, the number reduces from 36 to 21. When material is isotropic, the
number reduces to 2. For linear isotropic elasticity, we have Lame’s parameter-I ? and
Lame’s parameter-II µ, i.e. shear modulus. Using Lame’s parameters, Poisson’s ratio ?
and Young’s Modulus E, the strain energy of linear isotropic elastic material could be
expressed as [80]
e0 =
1
ν
1
T112 + T222 + T332 − ( T11T22 + T11T33 + T22T33 ) +
T122 + T132 + T232
2E
E
2µ
(
)
(
)
(6.6)
where shear modulus µ, Poisson’s ratio ν and Young’s Modulus E are related by:
µ=
E
2(1 + ν )
(6.7)
This strain energy could be split into two parts: one due to the change in volume and the
other due to the distortion. The first part due to the volume change is proportional to the
sum of the three normal stresses and it can be defined as
αp 3(1 − 2ν ) 2 1 − 2ν
=
p =
(T11 + T22 + T33 )2
2
2E
6E
(6.8)
where α is the volume expansion. By subtracting this part of strain energy from total
strain energy, the shear strain energy is [80],
176
e0 −
1 − 2ν
2
1 +ν 
2
2
2
( T11 + T22 + T33 ) =
(T11 − T22 ) + (T22 − T33 ) + (T33 − T11 ) 

6E
6E
1
+
T122 + T132 + T232
2µ
(
)
(6.9)
For linear isotropic elastic material, the strain energy (S ⋅ D ) / 2 is equal to (S ⋅ S ) / 4µ . To
verify this expression, we extend the inner product of deviatoric stress tensor
3
3
(
2
2
S ⋅ S = ∑∑ S ji S ij = S 112 + S222 + S 332 + 2S122 + 2S 13
+ 2S 23
i =1 j=1
)
(6.10)
According to Eqs.(3.98), we have
2T11 − T22 − T33
3
(6.11)
2T22 − T11 − T33
2T − T11 − T22
, and S 33 = 33
.
3
3
(6.12)
S11 =
similarly,
S 22 =
Therefore, by inserting Eqs.(6.11) and (6.12) into Eq.(6.10), we have
3
3
S ⋅ S = ∑∑ S ji S ij
i =1 j =1
 2T − T − T  2  2T − T − T  2  2T − T − T  2

22
33
11
33
22
11
=  11
 +  22
 +  33
 + 2T122 + 2T132 + 2T232 
3
3
3







2 2
2
2
2
2
2
2
= T11 + T22 + T33 − (T11T22 + T11T33 + T22T33 ) + 2 T12 + T13 + T23
3
3
(6.13)
(
)
(
)
The shear elastic strain energy of linear isotropic elastic material could be expressed as,
177
(
)
(
)
1
1 2 2
2
 1
S ⋅S =
T11 + T222 + T332 − (T11T22 + T11T33 + T22T33 ) +
T122 + T132 + T232

4µ
4µ  3
3
 2µ
(6.14)
1
1
2
2
2
2
2
2
=
(T11 − T22 ) + (T22 − T33 ) + (T33 − T11 ) + T12 + T13 + T23
12 µ
2µ
[
]
(
)
Aided by Eq.(6.7), the shear elastic energy becomes
[
]
1
1 +ν
S⋅S=
(T11 − T22 )2 + (T22 − T33 )2 + (T33 − T11 )2 + 1 T122 + T132 + T232 (6.15)
4µ
6E
2µ
(
)
Base on above expression, the (S⋅ S ) / 4 µ exactly is the linear isotropic elastic shear strain
energy. Generally, the elastic shear strain energy is (S ⋅ D) / 2 .
To close the equation set, which include Eqs.(5.16), (5.61), (6.1) and (5.122), we employ
the Mie-Gruneisen equation (EOS) [79] to relate pressure, density and internal energy as:
2


1 − ρ0 ρ )
c02
(


p = p ( ein , ρ ) = ρ c
+
ρ
Γ
e
−
0 0  in
2
2
2 1 − s (1 − ρ0 ρ )  
1 − s (1 − ρ0 ρ ) 


 
2
0 0
1− ρ0 ρ
(6.16)
where Γ0 is the Gruneisen parameter, c0 and s are coefficients that relate the shock speed
Us and the particle velocity up as,
U s = c 0 + su p
(6.17)
The complete thermal dynamic model includes Eqs. (5.16), (5.61), (6.1) and (5.122), and
they can be expressed in the conservation form as
∂Q ∂J
+
=K
∂t ∂ x
where the conservative variables vector is
178
(6.18)
Q = ( ρ , ρu , e, ρ S11 ) = ( q1 , q2 , q3 , q4 ) ,
(6.19)
q2




2
q2
q4


ρu

  j1 
+ p−


q
q
 ρ uu + p − S   j 
1
1


11
2
 =  = q
J =
q2 q4  ,

2
 u ( e + p) − uS11   j3 
( q3 + p ) −
q1 q1 

    q1
ρ uS11


  j4  
q2 ⋅ q4




q

1

(6.20)
T
T
conservative flux vector is
and source term at right side is
0


0


0

K =

4 
θ
 µρ  1 −
 3  1 + BSH

3µ







 .
 ∂u 
 
 ∂x 
 
 
(6.21)
Aided by the chain rule, we rewrite Eq. (6.18) as
∂Q
∂Q
+B
=K
∂t
∂x
where the Jacobin matrix B is defined by:
179
(6.22)
B=
∂J
∂Q
0
1
0
0




2
q
∂p q4
2q2 ∂p
∂p
1 ∂p 

− 22 +
+ 2
+
− +

q1 ∂q1 q1
q1 ∂q2
∂q3
q1 ∂q4 


=  q2 ( q3 + p ) q2 ∂p 2q2 ⋅ q4 ( q3 + p ) q2 ∂p q4 q2 
∂p  q2 ∂p q2 
+
+
+
−
−
1 +

−
q12
q1 ∂q1
q13
q1
q1 ∂ q2 q12 q1  ∂q3  q1 ∂q 4 q12 




q2 ⋅ q4
q4
q2
− 2
0


q1
q1
q1


0
1
0
0




∂p S11
∂p
∂p
1 ∂p 

− u2 +
+
2u +
− +
∂q1 ρ
∂q2
∂q 3
ρ ∂q 4 

=

(6.23)
 − u e + p + u ∂p + 2uS11 ( e + p ) + u ∂p − S11 u  1+ ∂p  u ∂p − u 
(
)


 ρ
∂q1
ρ
ρ
∂q2 ρ
∂q4 ρ 
 ∂q3 


−uS11
S11
0
u


Aided by the definition of the total energy, i.e., Eq.(6.4), we can calculate the internal
energy as
ein =
e 1 2 3 S112
− u −
ρ 2
8 µρ
(6.24)
Using this equation for the internal energy and aided by the equation of the state, i.e., Eq.
(6.16), we can calculate pressure by :
p = pref + ρ0Γ0 ( ein − Eref
)
(6.25)
where pref and Eref are the pressure and energy at the reference point respectively, and
they are defined by
180
1−
pref = ρ0 c0
2
ρ0
ρ

ρ0 
1 − s + s ρ 


(6.26)
2
and
Eref =
 ρ0 
1 − 
ρ

2
c0
2
2 
ρ0 
1
−
s
+
s


ρ 

(6.27)
The derivatives of pref and Eref respect to density are give by
−3
∂pref

ρ 
= ρ 0c0 1 − s + s 0   ρ + s ( ρ − ρ0 ) 
∂ρ
ρ 

2
(6.28)
and
∂Eref
−3

ρ 
= ρ0 c0 ( ρ − ρ0 ) 1 − s + s 0 
∂ρ
ρ 

2
(6.29)
With these two known derivatives, four partial derivatives of pressure appeared in Eq.
(6.23) are given by
∂Eref
 e u 2 9 S112 
∂p ∂pref
=
− ρ0 Γ 0
+ ρ 0Γ 0  − 2 + +

∂q1
∂ρ
∂ρ
ρ 8 µρ 2 
 ρ
(6.30)
∂p
u
= − ρ0 Γ0
∂q2
ρ
(6.31)
∂p ρ 0 Γ 0
=
∂q3
ρ
(6.32)
181
∂p
3
S
= − ρ0 Γ0 112
∂q4
4
µρ
(6.33)
To rewrite Eq. (6.22) in the non-conservative form with non- conservative variables
% = ( ρ , u, e, S )T , we multiply both sides of Eq. (6.22) with matrix N
vector Q
11
N
∂Q
∂Q
+ NBN −1N
= NK
∂t
∂x
(6.34)
where N is defined by
1


q

− 22

q1
% 
∂Q
N=
=
q
q 2 9 1 q42
∂Q  − 32 + 23 +
 q1
q1 8 µ q14

q

− 42

q1
1


u

−

ρ

=  e u 2 9 1 S112
− + +
 ρ 2 ρ 8 µ ρ2

S

− 11

ρ
0
1
q1
0
q2
q12
1
q1
0
0
−
0
1
ρ
−
u
ρ
0
0
0
0
1
ρ
0



0


3 1 q4 
−
4 µ q13 

1


q1
0



0


3 1 S11 
−
4 µ ρ2 

1


ρ
0
(6.35)
The non-conservative form model equations can be expressed in the following vector
form:
%
%
∂Q
∂Q
%
+ B%
=K
∂t
∂x
where the non-conservative variables vector is
182
(6.36)
 q q 1  q 2 3 1 q 2 q 
T
4
%
Q = ( ρ , u, ein , S11 ) =  q1 , 2 , 3 −  2  −
, 4 ,
 q1 q1 2  q1  8 µ q13 q1 


the Jacobin matrix B% is given by
 u
p
 ρ
ρ
−
1
B% = NBN = 

 0

 0
ρ
0
pein
u
p+
ρ
3 S112
− S11
8 µ
ρ
0
u
0
0 

1
−
ρ



0 

u 
(6.37)
and source term at right side is
0


0


S ∂u
− 11

ρ ∂x
% = NK = 
K


4 
 µ 1 − θ
BSH
3 
 1 + 3µ



%
∂Q
=K
∂x
 0
 0
 
 
 0
=
  
 ∂u  
  0
 ∂x  
  
  
0
0
S
− 11
ρ

4 
θ
µ 1 −
3  1 + BSH

3µ







0 0
0 0 
 ρ
0 0  
u
 ∂  
 ∂x  ein 
 S 
0 0   11 



(6.38)
To proceed, we move the source term at the right hand side of the equation to the left
hand side. Thus Eq. (6.36) becomes
%
%
∂Q
∂Q
+B
=0
∂t
∂x
where the new Jacobin matrix B is given by:
183
(6.39)
ρ
 u
p
 ρ
ρ



B = NBN −1 − K =  0



 0



u
p+
0 

1
− 
ρ



0 



u 



0
pein
ρ
2
11
3S
8 µ
ρ

4 
θ
− µ 1−
3  1 + BSH

3µ

u






0
(6.40)
The above equation is ready to be analyzed for its eigen-system. By solving the
equation det ( B − λ I ) = 0 , we directly derive the eigenvalues of matrix B :
λ1 = λ2 = u and λ3,4 = u ±

2
pein 

3 S11
4 µ
θ
pρ + 2  p+
+
 1−

BSH
ρ 
8 µ  3 ρ
 1 + 3µ







(6.41)
As a part of the eigenvalues, the speed of sound is
c=

2
pein 

3S
4µ
θ
pρ + 2  p + 11  +
1 −
BSH
ρ 
8 µ  3 ρ
 1 + 3µ







(6.42)
Based on Eq. (6.16), we have
pein = ρ0Γ0
p ρ = ρ 02 c02
ρ + (ρ − ρ 0 )(s − Γ0 )
[ρ − s (ρ − ρ 0 )]3
184
(6.43)
(6.44)
Substitute the Eqs. (6.43) and (6.44) into the Eq.(6.42), we have the speed of sound
expressed by
c = ρ02c02
ρ + ( ρ − ρ0 )( s − Γ0 )
 ρ − s ( ρ − ρ0 ) 
3

4
θ
+ 1−
BSH
3
 1 + 3µ


µ ρ Γ 
3 S112 
0 0
 + 2  p+

ρ 
8 µ 
ρ


(6.45)
For the elastic-perfect plastic material, i.e. BSH = 0 , the elastic wave speed, i.e. θ = 0 is
c = ρ02c02
ρ + ( ρ − ρ0 )( s − Γ 0 )
 ρ − s ( ρ − ρ0 ) 
+
3
ρ 0Γ0 
3 S112  4 µ
p
+

+
ρ2 
8 µ  3ρ
(6.46)
and plastic wave speed i.e. θ = 1 is
c = ρ02c02
ρ + ( ρ − ρ0 )( s − Γ0 )
 ρ − s ( ρ − ρ0 ) 
3
+
ρ 0Γ0 
3 S112 
p
+


ρ2 
8 µ 
(6.47)
Compare these two speeds in Eqs.(6.46) and (6.47), obviously, the elastic wave is faster
then plastic wave. Compare elastic wave speeds in Eqs.(6.46) and (5.78), we note that the
term k ρ is replaced by:
ρ02c02
ρ + ( ρ − ρ0 )( s − Γ0 )
 ρ − s ( ρ − ρ0 ) 
3
+
ρ 0Γ0 
3 S112 
p
+


ρ2 
8 µ 
Obviously, energy equation and EOS, i.e. Mie-Gruneisen equation implements influence
in the speed of wave propagation.
6.3 Numerical Results
We will use the same one-dimensional impact problem as that in Chapter 5. The
description of impact problem is shown in Figure 5.4. The initial pressures p and
185
deviatoric stress component S11 in both copper bulks are null. The material properties of
copper are listed in Table 3. We assume the material is elastic-perfect plastic, i.e. the
yield stress always equals to the initial yield stress without hardening. Or, BSH = 0 in Eq.
(5.122).
The initial conditions for setting up the computation have been described in Section 5.4.
For the Mie-Gruneisen equation, the following data, specific for copper, are
used: c0 = 3940 m s , ρ0 = 8930 kg m 3 , s = 1.49 , and Γ0 = 2 . Furthermore, we have to
set up the initial condition for the internal energy. By using the known initial
condition ρ = ρ0 , and aided by Eqs. (6.26) and (6.27), pref = 0 and Eref = 0 . Since the
initial pressure is null, aided by Eq. (6.25), the initial internal energy ein = 0 . With this
information, we can calculate the initial total energy by Eq.(6.4) and the initial stresses
inside the two copper bulks.
To proceed, we will first repeat the low speed impact problem with the impact
velocity at u = 40 m/s in CHAPTER 5 as the baseline case, which was calculated by
using the isothermal model, Eq. (5.123). We then will study the influence by including
the energy equa tion and the Mie-Gruneisen equation on the numerical solution of wave
propagation. In this effort, we will apply the isothermal model, i.e. Eqs. (5.123) and the
present the rmal model, Eq. (6.18), to solve three cases with different initial speeds of left
bulk material: u = 80 m/s, u = 200 m/s, and u =1000 m/s.
Figure 6.1 shows a snapshot of the density profile of the right copper bulk after a
low-speed impact with the impact velocity at u = 40 m/s. Figure 6.2 shows the snapshot
186
of the pressure profile. In both figures, blue lines without symbol represent the exact
solution [55]. Green lines with square symbols represent the numerical solutions by
solving the complete model, Eq. (6.18). Red lines with circle symbols are the solutions of
the isothermal model, Eq. (5.123). In this impact problem with a low initial speed of 40
m/s, solutions of both models compared well with the analytical solution by Udaykumar
et al. [55].
Figure 6.1: A snapshot of density in the right copper bulk (initially stationary) at t = 0.17
ms.
Solutions by both models show a precursive elastic waves followed by a much
stronger plastic wave s. The wave speeds compared well with that shown in Eqs.(5.78),
(5.128), (6.46) and (6.47).
187
Figure 6.2: A snapshots of the pressure profile in the right copper block at t = 0.17 ms.
To observe the difference between the isothermal model, Eq. (5.123), and the thermal
model, Eq. (6.18), we perform a series of calculations and they are presented in the
following.
(a)
188
(b)
(c)
Figure 6.3: The snapshots of density in the right copper bulk at t = 0.17ms at three
different initial impact speeds: (a) u = 80m/s, (b) u = 200m/s, and (c) u = 1000m/s.
189
(a)
(b)
190
(c)
Figure 6.4: Snapshots of pressure profiles in the right copper bulk at t = 0.17 ms at three
different initial impact speeds: (a) u = 80 m/s, (b) u = 200 m/s, and (c) u = 1000 m/s.
Figure 6.3 and Figure 6.4 show several main differences of predictions produced
by two models. With increasing initial impact speeds, the plastic wave speeds by thermal
dynamic model are faster than the plastic wave speeds by isothermal model; the peak
values of pressures by thermal dynamic model are higher those that by isothermal model;
in contrast, the peak values of densities by thermal model are lower than those by
isothermal model. All of these differences are obvious in the cases with low impact
speeds: u=40m/s and u=80m/s, they become apparent in the case with middle high impact
speed u=200m/s, and they are amplified in high speed case u=1000m/s.
Compare two plastic wave speeds in Eqs.(5.129) and (6.47), the difference of two
plastic wave speeds might be explained by that the inequality
191
ρ02c02
ρ + ( ρ − ρ0 )( s − Γ0 )
 ρ − s ( ρ − ρ0 ) 
3
+
ρ 0Γ0 
3 S112 
p
+

>k
ρ2 
8 µ 
(6.48)
will be amplified by increasing impact speed. The developments of differences of
pressure and density consist with this trend and support this inequality. When the thermal
dynamic model is used to describe an impact problem, part of the initial kinematic energy
of moving copper bulk will be transferred to be internal energy and shored in the material,
therefore the strain energy will be lower than that predicted by isothermal model.
Consequently, the deformation induced by compression will be smaller than that in
isothermal process, and density increment also is smaller than that in isothermal process.
In Eqs. (6.26) and (6.27), pref is proportional to (1 − ρ0 ρ ) , and Eref is proportional
to (1 − ρ0 ρ ) . The slower trend of density increasing will results in the faster growing of
2
pressure increasing. And this trend will be amplified by an increasing impact speed.
6.4 Conclusions
In the impact problem with a low initial speed of 40 m/sec, solutions of both
models compared well with the analytical solution by Udaykumar et al. [55]. For the
many manufacture processes involving nonlinear stress wave propagation, whose strain
rates and stress values are comparable with the low- impact problem of 40 m/sec, the
above solutions show that perhaps it is not necessary to include the energy equations in
modeling the stress waves. We note that we could still append a heat conduction equation
to be solved passively with the wave equations to simulate the temperature profile in the
metal in those manufacture processes to address the heat transfer effect. The conclusion
192
here is that the energy equation does not affect the wave propagation in the impact
problem at this low speed level. However the isotherma l model is not applicable to the
high speed impact problem, because it can not deal with a reasonable distribution of
energy, which is in the capability of thermal dynamic model with the energy equation.
193
CHAPTER 7
THE TWO-DIMENSIONAL THERMAL MODEL OF
STRESS WAVE IN ELASTIC-PLASTIC BULK MATERIAL
AND ULTRASOUND WELDING PROBLEM
7.1 Introduction
Previous chapters lay the foundation of theories and numerical methods for
modeling nonlinear stress waves in solids. A wide range of one- and two-dimensional
wave problems have been considered, including linear and nonlinear elastic wave in a
thin rod, linear and nonlinear elastic wave in a bulk material, elastic and elastic-plastic
wave in bulk materials. In modeling these waves, we have developed a suite of model
equations. These model equations could be divided into two main groups: (i) isothermal
models, no temperature effect is considered, and (ii) thermal models, in which the energy
conservation equation is included such that the model equations are suitable for modeling
energy mode ramification, including transition from internal stress potential energy to
thermal energy in severe impact problems.
Based on the fundamental understanding so gained, we will apply the theories and
numerical method to model the elastic-plastic waves in the UltraSound Welding (USW)
process in this chapter. Our goal is to track down nonlinear stress waves inside of metal
194
specimens. Moreover, we also plan to investigate the basics mechanism of AcoustoPlastic Effect (APE) in metals under imposed ultrasonic vibrations. One of the unique
contributions here is that both theoretical model and the numerical methods for the
solution have been developed in the setting of continuum mechanics.
In the USW process, the ultrasonic vibrations are applied in the direction parallel
to the welding interface. Refer to Figure 7.1. Observations of experiments show that no
external heat is added to the specimens in the USW and no melting occurs, as well as the
USW can be used to weld different metals, e.g., copper to aluminum. The USW has been
widely used to bond small-scale nonferrous metals for non-structural uses, e.g., bonding
electrical wires, electronic chip packaging, etc. The power input of the conventional
USW machines is usually a fraction of 1kw.
Figure 7.1: The USW process with normal pressure and transverse vibrations.
195
Recent development of high-power ultrasound machines (>10kW) provides a new
prospect of the ultrasound aided metal forming/joining processes. For example,
researchers at Ford Research and Advanced Engineering (R&AE) have shown a USW
process [81, 82] that aluminum alloy AA6111-T4 sheets could be effectively joined
through the process of direct contact between two metallic surfaces under combined static
normal pressure and in-plane vibrations in the ultrasound range, with the two metal sheets
to be joined ultrasonically oscillated relative to another in shearing motions. It has been
recognized that the APE plays a key role in the USW processes. Essentially, imposed
lateral vibrations in conjunction with the normal pressure would induce metal yielding
and the subsequent material plastic flow, while the norminal static loading is well within
the elastic range of the tested metals. This transitory plastic flow would allow the metals
fingering into other and thus drastically increase the size of the contact/adhesive surface.
That is considered as a part of reason of metal bonding in USW. The plastic deformation
in material will cease once the imposed vibrations are stopped even though static loading
is still applied.
In the USW process, the normal pressures and vibrations are applied to the metal
sheets by a sonotrode tip and an anvil. At the end of the process, typical temperatures on
the specimen surface are about 80 to 100 o C, which are well below the melting point of
aluminum. The temperature increase is mainly due to internal friction.
196
(a)
(b)
Figure 7.2: The nonlinear wave pattern of plastic deformation textures at the contact
interface in commercial AA6111-T4 alloys produced by the USW process, from [82].
By using a marker to show interface, the microscopic images of sectioned and
polished specimens showed that significant plastic deformation with large-scale rollups
occurred at the interface of the two adjoined aluminum plates. Refer to Figure 7.2. The
morphology of the welded interface resembled those of the Kelvin-Helmholtz instability
waves, which is a continuum mechanics phenomenon in the presence of shear stresses.
The corrugated interface provided additional surface for adhesion. As one of the
important reasons, the rollup structure allowed the solid aluminums from two metal
sheets fingering into each other and form a zipper- like lockup bond.
197
Further examinations revealed that grain orientation changed and grain sizes were
much smaller than that of the untreated specimens. The clusters of submicron cavities and
porosities were observed around interface and along deformation flow lines. There was
no evidence of any material melting at micrometer scale such as dendrite formation in
conventional welding. Similar waky mophology (with less intensity if the power input of
ultrasound
machines
is
less)
occurred
in
numerous
ultrasound-aided
metal
forming/joining processes as discussed in Chapter 1. It has been clear that the applied
ultrasounds are responsible for the transitory plastic flow and crystalline refinement in
these metal forming and bonding processes.
In the present chapter, we report the theoretical and numerical studies on the APE
phenomenon. We will report our basic understanding of its underpinning physics. In the
past, the lack of basic understanding and the associated process control are the main
obstacles hindering the further development of the APE processes including USW. The
most notable feature of wake formation and the microstrutcure of plastic deformation at
the welding interface were examined. Such examinations present the macrostructure
morphology and irreversible microstructure changes. Nevertheless, theoretical studies are
needed to understand and describe the mechanics. Moreover, recent development of highpower ultrasound machines has rekindled the intense interests of the industrial
manufacturers
in
ultrasound-aided
metal
joining/forming
processes.
Thorough
understanding of the APE and accurate modeling capabilities will be critical to improve
product quality and process control of the existing manufacturing processes, and even for
new process development.
198
7.2 Modeling Equations
To describe the modeling equations about two-dimensional problem, we first extend the
modeling equations developed in the previous chapter to the following vector form.
∂ρ
+ ∇ ⋅(ρ v ) = 0
∂t
(7.1)
∂ρ v
+ ∇ ⋅ ( ρ vv − T ) = 0
∂t
(7.2)
∂e
+ ∇ ⋅ ( ev − T ⋅ v ) = 0
∂t
(7.3)
DS 2
+ µ tr ( D ) I − 2µ D + θ ( s )( S ⋅ D ) S = 0
Dt 3
(7.4)
where ρ is density, v = (u, v, w) is the velocity vector, D is the symmetric part of the
velocity gradient matrix, T is the Cauchy stress tensor, S is the deviatoric part of T. The
relation between T, S and pressure p is
1
T = tr ( T) I + S = − pI + S
3
(7.5)
where
tr ( S ) = 0
3
tr ( T ) = ∑ Tii = T ⋅ I
(7.6)
i =1
1
p = − tr ( T )
3
In the energy equation,
e ≡ ρ ein +
ρ 2
1
u + v 2 + w 2 + (S ⋅ D )
2
2
(
)
199
(7.7)
is the total energy per unit volume. This quantity includes three parts as formulated in the
three terms on the right hand side of the above equation: (i) ein the specific thermal
internal energy, (ii) the specific kinetic energy, and (iii) the shear elastic strain energy per
unit volume.
To proceed, we assume that the stress status in the cross section of aluminum
plate in the USW process is a two-dimensional plane problem. Refer to Figure 7.2. The
direction of x or 1 is in the plane of paper and points to right, y or 2 points to up, z or 3 is
perpendicular to the paper plane and points out. As such, the Cauchy stress tensor and the
deviatoric stress tensor are
 T11 T12
T = T21 T22
 0
0
0
 S11

0 and S =  S21
 0
0

S12
S22
0


0
.
− ( S11 + S 22 ) 
0
(7.8)
The velocity components in the cross section are
v = ( u , v, w) = ( u , v , 0) .
(7.9)
Using Eqs.(7.5) (7.8) and (7.9), we expand Eqs. (7.1)-(7.3) into the following scalar
equations:
∂ρ ∂( ρu ) ∂( ρv )
+
+
=0
∂t
∂x
∂y
(7.10)
∂(ρu ) ∂ (ρuu + p − S11 ) ∂( ρuv − S 12 )
+
+
=0
∂t
∂x
∂y
(7.11)
∂(ρv ) ∂ (ρuv − S12 ) ∂( ρvv + p − S 22 )
+
+
=0
∂t
∂x
∂y
(7.12)
200
∂e ∂ u ( e + p ) − uS11 − vS12  ∂  v ( e + p) − uS12 − vS 22 
+
+
=0
∂t
∂x
∂y
(7.13)
To proceed, aided by the two-dimensional stresses and velocities as defined in
Eqs. (7.8) and (7.9), we reformulate the equation for the total energy per unit volume e,
defined by Eq.(7.7), and we have
e = ρ ein +
ρ 2 2
s
u + v + se
2
4µ
(
)
(7.14)
where
(
sse = S ⋅ S = 2 S112 + S222 + S122 + S11S22
)
(7.15)
Follow the definition of objective time derivative DT Dt in Eq.(3.73), we calculate the
objective time derivative of the deviatoric stress tensor, i.e., DS Dt :
D S ∂S
∂S
∂S
∂S
=
+u
+v
+w
+ SW − WS − a ( SD + DS )
Dt ∂t
∂x
∂y
∂z
With the known velocities, Eq. (7.9), we have

∂u
1  ∂u ∂v  
 +  0

∂x
2  ∂y ∂x  

 1  ∂v ∂u 

∂v
D =   + 
0
∂y
 2  ∂x ∂y 

0
0
0





201
(7.16)

0


 1  ∂v ∂ u 
W =   − 
 2  ∂x ∂y 
0



1  ∂u ∂v  

−  0
2  ∂y ∂x  

0
0

0
0


(7.17)
Aided by the formulation of D and W in Eq.(7.17), and S in Eq. (7.8), we have
1
 ∂v ∂u 
 S12  − 
 ∂x ∂y 
2
1
 ∂v ∂u 
SW =  S 22  − 
 ∂x ∂y 
2
0



1
 ∂u ∂v 
− 
 S 21 
2
∂
y
∂x 


1
 ∂v ∂u 
WS =  S 11  −

 ∂x ∂y 
2
0



1
 ∂u ∂v  
S11 
−  0
2  ∂y ∂ x  
 ∂u ∂v  
1
S 21
−  0
2
 ∂y ∂x  
0
0


1
 ∂u ∂v  
S 22 
−  0 
2
 ∂y ∂x  

 ∂v ∂u 
1
S 12  −  0 
2
 ∂x ∂y 

0
0


 ∂u 1  ∂v ∂u 
∂v 1  ∂u ∂v 
+ S12  +  S12
+ S11  + 
 S11
∂y 2  ∂y ∂x 
 ∂x 2  ∂x ∂y 
 ∂u 1
 ∂v ∂u 
∂v 1
 ∂u ∂v 
SD = S 21
+ S 22  +  S 22
+ S 21  + 
∂y 2
 ∂x ∂y 
 ∂y ∂x 
 ∂x 2
0
0



 ∂u 1  ∂v ∂u 
∂u 1
 ∂u ∂v 
+ S 21 +  S12
+ S 22  + 
 S11
∂x 2
 ∂y ∂x 
 ∂x 2  ∂x ∂y 

∂v 1  ∂v ∂u 
∂v 1  ∂u ∂v 
DS =  S 21
+ S11  +  S 22
+ S 12  + 
∂
y
2
∂
x
∂
y
∂
y
2  ∂y ∂x 



0
0




0


0

0



0


0

0


Aided by these expressions, we expand the equations Eq. (7.16) for three
deviatory stress components into the following scalar forms:
202

DS 11 ∂S11
∂S
∂S
 ∂u ∂v 
∂u
 ∂v ∂u 
 (7.18)
=
+ u 11 + v 11 − S12 
−  − a  2S11
+ S 12  +
Dt
∂t
∂x
∂y
∂
x
∂
x
∂
y
 ∂y ∂x 




DS 22 ∂S 22
∂S
∂S
 ∂u ∂v 
∂v
 ∂v ∂u 
 (7.19)
=
+ u 22 + v 22 + S12 
−  − a  2S 22
+ S 12  +
Dt
∂t
∂x
∂y
∂y
 ∂y ∂x 
 ∂x ∂y 

DS 12 DS 21 ∂S12
∂S
∂S
 ∂u ∂v 
1
=
=
+ u 12 + v 12 + ( S11 − S 22 ) − 
Dt
Dt
∂t
∂x
∂y 2
 ∂y ∂x 
  ∂u ∂v  1
 ∂v ∂u 

− a  S12  +  + (S11 + S 22 ) +
∂
x
∂
y
2
∂
x
∂
y





(7.20)
Substituting Eqs.(7.18)-(7.20) into the constitutive equation, Eq. (7.4), we obtain the
following three constitutive equations for USW as:
∂S11
∂S
∂S
 ∂u ∂v  
∂u
 ∂v ∂u  
+ u 11 + v 11 − S 12  −  − a  2S11
+ S12  +  
∂t
∂x
∂y
∂x
 ∂y ∂x  
 ∂x ∂y  
4 ∂u 2 ∂v
− µ
+ µ
+ θ (s )(S ⋅ D)S 11 = 0
3 ∂x 3 ∂y

∂S 22
∂S
∂S
 ∂u ∂v 
∂v
 ∂v ∂u 
+ u 22 + v 22 + S12  −  − a 2 S 22
+ S12  +

∂t
∂x
∂y
∂
y
∂
x
∂
y
 ∂y ∂x 



2 ∂u 4 ∂v
+ µ
− µ + θ (s )(S ⋅ D)S 22 = 0
3 ∂x 3 ∂y
∂S12
∂S
∂S
 ∂u ∂v 
1
+ u 12 + v 12 + (S11 − S 22 ) −  −
∂t
∂x
∂y
2
 ∂y ∂x 
  ∂u ∂v  1
 ∂v ∂u 
∂u
∂v
 − µ
a  S12  +  + (S11 + S 22 ) +
− µ + θ (s )(S ⋅ D )S12 = 0
∂y
∂x
 ∂x ∂y 
  ∂x ∂y  2
where θ ( s ) is defined by Eq.(5.118),
203
(7.21)
(7.22)
(7.23)

if s = S ⋅ S < 2k 2

0,

if s = S ⋅ S ≥ 2k 2
θ ( s) = 
0,

3
µ if s = S ⋅ S ≥ 2k 2

,
2
 ( BSH / µ + 3 ) k
for unloading
(5.118)
for loading
and ( S ⋅ D) is given by


(S ⋅ D) = S11 ∂u + S12  ∂u + ∂v  + S 22 ∂v
∂x
 ∂y
∂x 
∂y
(7.24)
For an elastic-perfect plastic material, the definition of θ ( s ) becomes:
2
 0, if s = S ⋅ S < 2k

2
θ ( s ) =  0, if s = S ⋅ S ≥ 2k
µ k 2 , if s = S ⋅ S ≥ 2k 2

for unloading
(7.25)
for loading
Let a = 0 and we have the Jaumann rate for deviatoric stress. As such, Eqs. (7.21)-(7.23)
become
 ∂u ∂v  4 ∂ u 2 ∂ v
∂S11
∂S
∂S
+ u 11 + v 11 − S12  − − µ
+ µ
+θ ( s )( S ⋅ D) S11 = 0
∂t
∂x
∂y
 ∂y ∂x  3 ∂x 3 ∂y
(7.26)
 ∂ u ∂ v  2 ∂ u 4 ∂v
∂S 22
∂S
∂S
+ u 22 + v 22 + S12  −  + µ
− µ
+θ ( s )( S ⋅ D) S22 = 0 (7.27)
∂t
∂x
∂y
 ∂y ∂x  3 ∂x 3 ∂y
 ∂u ∂v 
∂S12
∂S
∂S
1
∂u
∂v
+ u 12 + v 12 + ( S11 − S 22 )  −  − µ − µ
+θ ( s )( S ⋅ D) S12 = 0 (7.28)
∂t
∂x
∂y 2
∂y
∂x
 ∂y ∂x 
Equations (7.26)-(7.28) are in non-conservative form with the unknowns in the nonconservative form, i.e.,
( S11, S22 , S12 )
T
. To be consistent with the model equations
formulated in the conservative form, i.e., Eqs.(7.10)-(7.13), we use conservative variables
( ρ S11 , ρ S22 , ρ S12 )
T
and rewrite Eqs. (7.26)-(7.28) to be
204
∂ ( ρ S11 ) ∂ ( ρuS11 ) ∂ ( ρ vS11 )  ∂ ( ρuS12 )
∂ ( ρ S12 )   ∂ ( ρ vS12 )
∂ ( ρ S12 ) 
+
+
=
−u
−v
− 

∂t
∂x
∂y
∂y   ∂x
∂x 
 ∂y
(7.29)
4  ∂ ( ρu )
∂ρ  2  ∂ ( ρ v )
∂ρ 
+ µ
−u  − µ 
− v  + θ ( s )( S ⋅ D) ρ S11
3  ∂x
∂x  3  ∂y
∂y 
∂ ( ρ S22 ) ∂ ( ρ uS22 ) ∂ ( ρ vS 22 )  ∂ ( ρ vS12 )
∂ ( ρ S12 )   ∂( ρuS12 )
∂ ( ρS12 ) 
+
+
=
−v
−u
− 

∂t
∂x
∂y
∂x   ∂y
∂y 
 ∂x
(7.30)
4  ∂ ( ρ v)
∂ρ  2  ∂ ( ρ u )
∂ρ 
+ µ
−v − µ
− u  + θ ( s )( S ⋅ D ) ρ S22
3  ∂y
∂y  3  ∂x
∂x 
∂ ( ρ S12 ) ∂ ( ρ uS12 ) ∂ ( ρvS12 ) 1   ∂ ( ρvS11 )
∂ ( ρS11 )   ∂( ρuS11 )
∂ ( ρS11 )  
+
+
= 
−v
−u
− 

∂t
∂x
∂y
2   ∂x
∂x   ∂ y
∂ y  
∂ ( ρ S22 )   ∂( ρuS22 )
∂ ( ρS22 )  
1  ∂ ( ρ vS22 )
− 
−v
−u
− 

2 
∂x
∂x  
∂y
∂y  
∂ ( ρ u)
 ∂ ( ρv)
∂ρ 
∂ρ 
+µ
−u
− v  + θ ( s )( S ⋅ D) ρ S12
+µ
∂y 
∂x 
 ∂y
 ∂x
(7.31)
To close Eqs.(7.10)-(7.13) and (7.29)-(7.31), we have to provide a relation between the
internal energy, pressure, and density, i.e., an Equation Of State (EOS). To this end, we
employ the Mie-Grunesian equation [79] as the EOS for the aluminum alloy:
2


1 − ρ0 ρ )
c02
(


p = p ( ein , ρ ) = ρ c
+
ρ
Γ
e
−
0 0  in
2
2
2 1 − s (1 − ρ0 ρ )  
1 − s (1 − ρ0 ρ ) 


 
2
0 0
1− ρ0 ρ
(6.16)
where s and c0 are two thermodynamic coefficients, G is the Gruneisen constant, and ρ0
the density at zero pressure.
To proceed, those governing equations (7.10)-(7.13) and (7.29)-(7.31) in the
conservative form could be cast into the following vector form:
205
∂Q ∂ F ∂ G
+
+
= Ss
∂t ∂x ∂y
(7.32)
where the conservative variable vector is defined by
Q = ( q1, q2 , q3 , q4 , q5 , q6 , q7 ) = ( ρ , ρ u, ρ v, e, ρ S11 , ρ S22 , ρ S12 )
the conservative flux vectors are
T
T
T
 q22
q qq q q
qq qq qq qq qq 
F =  q2 , + p − 5 , 2 3 − 7 , 2 ( q4 + p ) − 2 2 5 − 3 2 7 , 2 5 , 2 6 , 2 7 
q1
q1 q1
q1 q1
q1
q1
q1
q1
q1 

= ( ρ u, ρuu + p − S11 , ρuv − S12 , u ( e + p ) − uS11 − vS12 , ρuS11 , ρuS 22 , ρuS12 )
T
T
 q q q q2
q q
qq qq qq qq qq 
G =  q3 , 2 3 − 7 , 3 + p − 6 , 3 ( q4 + p ) − 2 2 7 − 3 2 6 , 3 5 , 3 6 , 3 7 
q1
q1 q1
q1 q1
q1
q1
q1
q1
q1 

= ( ρ v, ρuv − S12 , ρvv + p − S22 , v ( e + p) − uS12 − vS 22 , ρ vS11 , ρvS22 , ρ vS12 )
T
the source terms in the right side is
206
0




0




0


0


   ∂ ρuS

∂ (ρ S12 )   ∂ ( ρvS12 )
∂ ( ρ S12 ) 
12 )
  (
−u
−v

− 

  ∂y
∂y   ∂x
∂x 



  4 ∂ ( ρ u )
∂ρ  2  ∂ ( ρ v )
∂ρ 

−u  − µ 
− v  + θ ( s )( S ⋅ D ) ρ S11  
 + µ 
∂x  3  ∂y
∂y 

  3  ∂x



   ∂ ( ρ vS12 ) − v ∂ ( ρS12 )  −  ∂ ( ρuS12 ) − u ∂ ( ρS12 ) 


 

S s =   ∂x

∂x   ∂y
∂y 





  4  ∂ ( ρv )
∂ρ  2  ∂ ( ρ u )
∂ρ 

−v  − µ
− u  + θ ( s )( S ⋅ D) ρ S22  
 + µ 
∂y  3  ∂x
∂x 
  3  ∂y



 
∂ ( ρ S11 )   ∂( ρuS11 )
∂( ρS11 )  
  1  ∂ ( ρ vS11 )
−v
−u
 
− 

  2  ∂x
∂x   ∂y
∂y  


 






 ∂ ( ρ vS22 )
∂ ( ρS 22 )   ∂( ρuS22 )
∂ ( ρS 22 )    
1

 − 
−v
−u
− 
  
  2  ∂x
∂x   ∂y
∂y    
 
 
 
 ∂ ( ρu )



∂
ρ
v
∂ρ
( ) − v ∂ρ + θ s S ⋅ D ρ S  
−u
( )( ) 12  
 + µ 
 +µ 


∂y
∂y 
∂x
∂x 



 
 
To proceed, aided by the chain rule, we rewrite Eq.(7.32) as:
∂Q
∂Q
∂Q
+A
+B
= Ss
∂t
∂x
∂y
where Jacobian matrixes A and B are defined as:
207
(7.33)
A=
∂F
∂Q
0



A21


 − q2q3 + q7
 q12
q12


A41
=

 − q2q5

q12

 − q2q6

q12

 − q2q7

q12

1
2
q2 ∂p
+
q1 ∂ q2
q3
q1
0
∂p
∂q3
q2
q1
0
∂p
∂q4
∂p 1
−
∂q5 q1
0
∂p
∂q6
0
0
0
A42
A43
A44
q2 ∂p q2
−
q1 ∂q 5 q12
q2 ∂p
q1 ∂q6
q5
q1
0
0
q2
q1
0
0
0
0
q2
q1
0
0
0
0
q6
q1
q7
q1
0
where
A21 = −
A41 =
q22 ∂p q5
+
+
q12 ∂q1 q12
q2 ∂ p q2
qq
qq
− 2 ( q4 + p ) + 2 2 3 5 + 2 3 3 7
q1 ∂q1 q1
q1
q1
A42 =
q2 ∂p q4 + p q5
+
− 2
q1 ∂q2
q1
q1
A43 =
q2 ∂ p q7
− 2
q1 ∂ q3 q1
A44 =
q2 
∂p 
1 +

q1  ∂q4 
To proceed, we have final form of matrix A,
208
0


∂p

∂q7


1

−

q1

q2 ∂p q3 
−
q1 ∂q 7 q12 


0



0


q2


q1
 (7.34)
0


 A21


 −uv + S12
ρ
∂F 
A=
=
∂Q 
A41


 −uS11
 −uS
22

−
uS

12
1
0
0
0
0
∂p
∂q3
∂p
∂ q4
∂p 1
−
∂q5 ρ
∂p
∂q6
v
u
0
0
0
A42
A43
A 44
S11
0
0
u
0
S22
0
0
0
u
S12
0
0
0
0
2u +
∂p
∂q2
u
∂p u
−
∂ q5 ρ
where
A21 = −u 2 +
A41 = u
∂p S11
+
∂q1 ρ
∂p u
uS
vS
− ( e + p ) + 2 11 + 2 12
∂q1 ρ
ρ
ρ
A42 = u
∂p e + p S11
+
−
∂q2
ρ
ρ
A43 = u
∂p S12
−
∂q3 ρ

∂p 
A44 = u  1 +

 ∂q4 
Similarly, the matrix B could be calculated by following form:
209
u
∂p
∂ q6


∂p

∂ q7 

1 
−
ρ 
 (7.35)
∂p v 
u
−
∂q7 ρ 

0


0

u

0
B=
∂G
∂Q
0

 qq
 − 2 2 3 + q72
 q1
q1


B31



B41
=

 − q3q5

q12

 − q3q6

q12

 − q3q7

q12

0
q3
q1
∂p
∂q2
0
0
0
0
0
0
q3 ∂p
+
q1 ∂q3
∂p
∂q4
B 42
B 43
B44
∂p
∂q5
q3 ∂p
q1 ∂q5
∂p 1
−
∂q6 q1
q3 ∂p q3
−
q1 ∂q 6 q12
0
q5
q1
0
q3
q1
0
0
0
q3
q1
0
0
0
0
0
1
q2
q1
2
q6
q1
q7
q1
where
B31 = −
B41 =
q32 ∂p q6
+
+
q12 ∂q1 q12
q3 ∂ p q3
qq
qq
− 2 ( q4 + p ) + 2 2 3 7 + 2 3 3 6
q1 ∂ q1 q1
q1
q1
B42 =
B43 =
q3 ∂p q7
− 2
q1 ∂q 2 q1
q3 ∂ p q4 + p q6
+
− 2
q1 ∂q3
q1
q1
B44 =
q3 
∂p 
1+

q1  ∂q4 
The final form of matrix B is given by
210
0


1

−
q1


∂p


∂q7

q3 ∂p q2 
−
q1 ∂q 7 q12 


0



0


q3


q1

(7.36)
B=
∂G
∂Q
0


 −uv + S12

ρ

 B31

=
 B41


 −vS11
 −vS22

 −vS12
0
1
0
0
0
v
u
0
0
0
∂p
∂q4
∂p
∂q5
∂p
v
∂q5
∂p 1
−
∂q6 ρ
∂p v
v
−
∂q6 ρ
v
0
0
0
v
0
∂p
∂q2
2v +
∂p
∂q3
B 42
B43
B44
0
0
0
S11
S 22
S12
0
0
0
0

1 
−
ρ 
∂p 
∂q7 
∂p u 
v
−
∂q7 ρ 

0


0

v

(7.37)
where
B31 = −v 2 +
B41 = v
∂p S22
+
∂q1 ρ
∂p v
uS
vS
− ( e + p ) + 2 12 + 2 22
∂q1 ρ
ρ
ρ
B42 = v
B43 = v
∂p S12
−
∂ q2 ρ
∂p e + p S22
+
−
∂ q3
ρ
ρ

∂p 
B44 = v  1 +

 ∂q4 
In A and B, the derivative of pressure with respect to the unknown variable
∂p ∂qi , = 1,2,... is calculated by using the Mie-Gruneisen equation:
p = pref + ρ0Γ0 ( ein − Eref
211
)
(7.38)
where pref and Eref are the pressure and energy at the reference point respectively, and
they are defined by
1−
pref = ρ0 c0
2
ρ0
ρ

ρ0 
1 − s + s ρ 


(7.39)
2
and
Eref =
 ρ0 
1 − 
ρ

2
c02
2
2 
ρ0 
1 − s + s 
ρ 

(7.40)
where
∂pref
∂ρ
∂Eref
∂ρ
= ρ 02c02
= ρ 0c0
2
ρ + s ( ρ − ρ0 )
3
(7.41)
3
(7.42)
( ρ − s ρ + s ρ0 )
ρ
3
( ρ − ρ0 )
( ρ − s ρ + s ρ0 )
 ∂Eref
∂p ∂pref
e u 2 + v2 
=
− ρ0 Γ 0 
+ 2−

∂q1
∂ρ
ρ
ρ 
 ∂ρ
(7.43)
∂p
u
= − ρ0 Γ0
∂q2
ρ
(7.44)
∂p
v
= − ρ0 Γ0
∂q2
ρ
(7.45)
∂p ρ 0 Γ 0
=
∂q4
ρ
(7.46)
212
∂p
=0
∂q5
(7.47)
∂p
=0
∂q6
(7.48)
∂p
=0
∂q7
(7.49)
To study the mathematical structure of the equation system, we proceed to
calculate the eigenvalues of the Jacobian matrices A and B. We first rewrite the equation
system into the non-conservative form:
∂ρ
∂ρ
∂u
∂ρ
∂v
+u
+ρ
+v
+ρ
=0,
∂t
∂x
∂x
∂y
∂y
(7.50)
∂u pρ ∂ρ
∂u p ∂e
1 ∂S11
∂u 1 ∂S12
+
+ u + ein in −
+v −
= 0,
∂t ρ ∂x
∂x ρ ∂x ρ ∂x
∂y ρ ∂y
(7.51)
∂v
∂v 1 ∂S12 pρ ∂ρ
∂v p ∂e
1 ∂S22
+u −
+
+ v + ein in −
=0,
∂t
∂x ρ ∂x
ρ ∂y
∂y ρ ∂y ρ ∂y
(7.52)
∂ein
∂e
∂e
1
s
+ u in + v in +  p + se
∂t
∂x
∂y ρ 
4µ
  ∂ u ∂v 
 +  = 0 .
 ∂x ∂y 
(7.53)
For elastic media, the constitutive equations in non-conservative form are
∂S11 4 ∂u
∂v
∂S
∂ u 2 ∂v
∂S
− µ
+ S12
+ u 11 − S12
+ µ + v 11 = 0 ,
∂t
3 ∂x
∂x
∂x
∂y 3 ∂y
∂y
(7.54)
∂S 22 2 ∂u
∂v
∂S
∂ u 4 ∂v
∂S
+ µ
− S12
+ u 22 + S12
− µ + v 22 = 0 ,
∂t
3 ∂x
∂x
∂x
∂y 3 ∂y
∂y
(7.55)
∂S12 1
∂v
∂v
∂S
1
∂u
∂u
∂S
− ( S11 − S 22 ) − µ + u 12 + ( S11 − S22 ) − µ
+ v 12 = 0 .
∂t
2
∂x
∂x
∂x 2
∂y
∂y
∂y
(7.56)
213
Equations (7.50)-(7.56) can be expressed in vector form as
%
%
%
∂Q
% ∂Q + B
% ∂Q = 0
+A
∂t
∂x
∂y
(7.57)
% = ( ρ , u, v, e , S , S , S )T and the coefficient
where non-conservative vector variable is Q
in
11
22
12
matrixes are
 u

 pρ
ρ

 0



% = 0
A


 0

 0


 0










B% = 









ρ
0
u
0
0
u
0
0
0
u
0
S12
0
u
−S12
0
0
0
0
( p + sse
4µ )
ρ
4µ
3
2µ
3
−
−µ −
0
0
pein
ρ
( S11 − S 22 )
2
0 

0
0 


1
0 −
ρ


0 0 


0
0 

u
0 


0 u 

0
−
0
1
ρ
v
0
ρ
0
0
0
0
v
0
0
0
0
ρ
0
v
0
0
0
− S12
0
S12
pρ
0
−µ +
( p + s se
ρ
( S11 − S22 )
2µ
3
4µ
−
3
0
2
214
pein
ρ
4µ )
0 −
1
ρ
v
0
0
0
v
0
0
0
v
0
0
0
0 

1
− 
ρ

0 



0 


0 

0 


v 

% as
To simplify the calculation of eigenvalues, we rewrite the matrix A
u ρ

 z0 u


0 0
%
A=

 0 a1
 0 b1

 0 c1
 0 δ1
0
0
0
0
0
z1 −
u
0
0
0
a2
u
0
0
b2
0
u
0
c2
0
0
u
δ1
0
0
0
1
ρ
0
0 

0 


1
− 
ρ

0 
0 

0 
u 
% are,
and eigenvalues of matrix A
λ1,2,3 = u ,
λ4,5 = u ± c1 , λ6,7 = u ± c2
(7.58)
where
c1 = r 2 + r 2 4 − q and c2 = r 2 − r 2 4 − q
r2
 r δ  (b − z a ρ )δ1
− q =  + 2  + 2 1 2 2
4
ρ
2 ρ 
b + δ2
r = ρz 0 + a1 z1 − 1
ρ
2
Since δ 1 is zero, we have
c1 =
r r δ2
δ
+ +
= r+ 2 =
2 2 ρ
ρ
c2 =
Similarly,
(
we
could
pρ +
pein 
s  4µ
p + se  +
2 
ρ 
4µ  3 ρ
(7.59)
µ ( S11 − S22 )
+
ρ
2ρ
calculate
)
equation det B% − λ I = 0 ,
215
the
eigenvalues
(7.60)
of
B%
by
solving
λ1,2,3 = v ,
λ4,5 = v ± c1 , λ6,7 = v ± c2
(7.61)
pein 
s  4µ
p + se  +
2 
ρ 
4 µ  3ρ
(7.62)
µ ( S22 − S11 )
+
ρ
2ρ
(7.63)
where
c1 =
pρ +
c2 =
Aided by the Mie-Grunesian equation as the EOS employed, pein and pρ are defined by
pein = ρ0Γ0
p ρ = ρ 02 c02
ρ + (ρ − ρ 0 )(s − Γ0 )
[ρ − s (ρ − ρ 0 )]3
(7.64)
(7.65)
7.3 Stress Boundary Condition by FEA
To conduct the simulation of the USW process, we use the method of coupling
the Finite Element Ana lysis (FEA) and the CESE method. In the USW processes, tip
motions of anvil include indentation at –y direction and ultrasonic vibrations along x
axial with typical amplitudes about 10µm and at frequencies about 20 kHz. Due to lateral
vibrations at high frequencies, if one only considered several hundreds cycles of
vibrations, the vertical indentation of the sonotrode tips could be assumed frozen. As
such, we model the whole process of tip penetration and vibration by several consecutive
steps. In each step, we model the motion of the tip by first a vertical indentation, which is
then followed by ultrasonic vibrations.
216
We use ABAQUS, a FEA commercial code, to model the vertical indentation of
the sonotrode tips. In the FEA calculations, the computational domain includes both the
sonotrode tips and the metal specimens. The sonotrode tips are treated as rigid solids and
metal specimens are plastic. Then, the calculated stress profiles between the sonotrode
tips and the metal specimens are extracted from the FEA results. These stress profiles are
then used for the CESE computation, which is carried out in a fix domain with the
deformed metal specimens calculated by the FEA. In the FEA, vibrations and indentation
were alternatively applied to the metal.
To recap, the modeling efforts for the USW processes include the following tasks:
(i) FEA analyses of dynamic loading on the metal coupons; (ii) parallel computation by
the space-time CESE method for the solution of the continuum mechanics equations for
aluminum alloy morphology; and (iii) analysis of stress propagation within coupons and
wavy features at the coupon interface.
Figure 7.3: Dimensions of aluminum coupon with deformation, units are mm.
217
In the practical computation, we have used several tip designs with different
dimensions. As one of them, Figure 7.3 shows the geometry of deformed aluminum
coupon, which is the final shape of the metal coupons calculated by the FEA. The shape
of the metal coupons is then used for the CESE simulation. Based on the dimensions
shown in Figure 7.3, we design the geometries of coupon and tips for the FEA by using
ABAQUS.
ABAQUS is a suite of commercial FEA computer programs. It consists of two
main modules: (i) ABAQUS/Standard and (ii) ABAQUS/Explicit. Both modules can be
used to model dynamic problems. ABAQUS/Standard uses implicit time integration
method, while ABAQUS/Explicit uses explicit time integration method. In the explicit
time integration method, the size of the step time employed is constrained by the smallest
element size and the local wave propagation speed. The advantages of the explicit time
integration method are (i) more efficient computation for each time step, and (ii) no
convergence issue for complex problems. The explicit time integration method require
that the users must know how to input appropriate restrains in order to obtain correct
results.
We used the ABAQUS/Explicit module to solve the dynamic loading problem, in
which tip is vibrating at 20 kHz. The deformation of coupon is shown in Figure 7.4.
218
(a)
(b)
Figure 7.4: Tip vertically penetrates into the metal coupon. (a) Initial shapes and position
of the coupon and the tips. (b) A magnified view of the deformed mesh.
In the FEA, we consider the larger-scale deformation of metal done by the vertical
indentation of the sonotrode tips. We employ the adaptive element and the explicit
dynamic solver in ABAQUS. The mesh is composed of about 9,000 adaptive linear
quadrilateral elements. The nonlinear waves in the metal coupons cannot be tracked
down by the FEA calculations. Therefore, the calculations cannot mimic the nonlinear
stress superposition which might be the mechanisom of APE.
When original yield stress of Al alloy and normal stress measured in experiments
are applied in FEA calculation, the results of simulation show that tip cannot penetratie
into the metal coupons. Therefore, we have to modify the material responses in the FEA
219
calculations. According to experimental results of the APE observed by Kirchner [2], and
given the vibration frequency around 20KHz, we have artificially reduced the yielding
stress in the FEA calculation to be 60% of the aluminum alloy employed.
In the simulation of vertical indentation, the tip was set at a constant speed in the
vertical direction. We chose the indentation speed and the time duration to match the
experimental conditions. Following the penetration of tip at –y direction, the second stage
of simulation is vibration along x axial. In this part of the simulation, we use the
sinusoidal function to describe the vibrations of the tip:
x = Av sin ( 2π ft ) ,
(7.66)
where x is the horizontal position of tip, t is the time, f is the vibrating frequency, and Av
is the vibrating amplitude. In the following calculations, the amplitude is set at 10
microns and the frequency is 20 kHz.
To simplify the process of implementing stress boundary conditions, we assumed
that every tooth has the same stress distribution. Furthermore, we assumed that the stress
distribution around one tooth does not change between any two vibrating cycles.
Therefore, we use the evolving stress boundary conditions of one vibration cycle for
multiple cycles in the USW process. The motion of tip in one vibration cycle is shown in
Figure 7.5.
220
(a)
(b)
(c)
(d)
Figure 7.5: The motion of one tooth in one vibration cycle: (a) the initial condition, (b)
1/4 cycle, (c) 3/4 cycle, and (d) at the end of one cycle.
221
In simulation, for each vibration cycle, the time period of 0.5×10-4 sec is divided
into 20 time steps. At each time step, stress profiles around one tooth are extracted. They
are pressure and three deviatoric stress components, S11 , S22 and S12 . As such, we obtain
the stress boundary conditions as functions of space and time. Figure 7.6 shows the stress
boundary conditions around one upper tooth at one particular time step during the
vibrations. Similar stress distributions around one bottom tooth are in Figure 7.7. All
stress profiles presented are the results after applying a smooth procedure to the profiles.
(a)
(b)
222
(c)
(d)
223
(e)
Figure 7.6: The stress profiles around one top tooth at one time point in one vibration
cycle: (a) mesh points along one tooth, (b) the pressure profile, (c) the profile of stress
component S11 , (d) the profile of stress component S12 , and (e) the profile of stress
component S22 .
(a)
224
(b)
(c)
225
(d)
(e)
Figure 7.7: The stress profiles around one bottom tooth at one time point in one vibration
cycle: (a) mesh points along one tooth, (b) the pressure profile, (c) the S11 profile. (d) the
S12 profile, and (e) the S22 profile.
226
When the original yield stress was used in ABAQUS calculation, the obtained
pressure for the Al alloy is about 160MPa. The experimental normal stress, however, is
only about 60MPa. The experimental pressure was calculated by dividing the
experimental clamping force by the tip area. In the field of continuum mechanics, the
difference might be induced by nonlinear stress superposition, and this needs to be
verified by further effort of numerical computation. Essentially, a new stress-strain curve
is needed to be developed for the USW processes. To tune up stress boundary condition
to be close to experimental magnitude, we artificially reduce the normal stresses and
shear stresses from the ABAQUS results for the boundary conditions for the CESE
calculation.
7.4 Parallel Computation
Equations (7.10)-(7.13) and (7.29)-(7.31) could be cast into the following standard
conservation form in two spatial dimensions :
∂um ∂ f m ∂ gm
+
+
= µm , m = 1, 2, … , 7
∂t
∂x
∂y
(7.67)
where f m and gm , are functions of the independent conservative variables um . Let x 1 = x, x2
= y, and x 3 = t be the coordinates of a three-dimensional Euclidean space E3 . By using
Gauss’ divergence theorem, we have
227
Ñ∫
S (V )
hm ⋅ ds = ∫ µ mdV , m = 1, 2, … , 7
V
(7.68)
where S(V) and ds were defined by Eq. (2.4) and Figure 2.1, and h m @ ( f m , gm , um ) . Then
we could use two-dimensional CESE method solve Eqs. (7.10)-(7.13) and (7.29)-(7.31).
In order to track wave propagation and the evolving roll- ups features, highresolution computation with dense meshes is needed. To speed up computation, we
employed parallel computation based on domain decomposition. The computational is
conducted on a 27- node PC cluster at OSU. In parallel computation, balanced loads
among computer nodes and efficient data communication between computer nodes are
important. Figure 7.8 shows the decomposed computational domain for the parallel
computation.
In our present study, 23,800 triangle elements are used for the computational
domain as shown in Figure 7.8. The domain is 8×1.9 mm2 with five teeth on the top and
-9
the bottom boundaries. With a time internal of 1.25x10
sec, 80,000 time steps are
needed for one cycle. For 16 cycles of vibrations, it takes about 10 hours of continuous
computation by using our parallel PC cluster.
228
(a)
(b)
(c)
Figure 7.8: The detail of mesh and mesh decomposing for parallel computation, (a)
original mesh of whole domain, (b) mesh of one tooth, and (c) decomposed mesh.
For velocity boundary condition, sinusoidal horizontal oscillations are imposed on
the top and bottom boundaries. Their motions are specified by
Utop = 2π fAv cos ( 2π ft ) .
(7.69)
Ubottom = 2π fAv cos ( 2π ft + π ) .
(7.70)
229
and f = 20 KHz , Av = 10µ m . The vertical velocity is set to be zero. Non-reflecting
boundary conditions are applied on the two vertical boundaries of computational domain.
Except of the boundaries along the teeth, the rest part of domain is applied initial
conditions ( ρ , ρu, ρ v, ρ S11 , ρ S22 , ρ S12 ) = ( 2730kg / m3 ,0,0,0,0,0) .Based on Eqs. (7.39)
T
T
and (7.40), and the initial condition ρ = ρ0 ,we have
pref = 0, and Eref = 0 .
(7.71)
By using the given initial condition p = 0 , and based on Eq. (7.38), we have
ein = 0 .
(7.72)
Aided by the total energy per unit volume defined in Eq.(7.7) and the known initial
conditions of velocity and stresses, we have the initial condition of the total energy per
unit volume as null:
e =0.
(7.73)
The initial condition is ( ρ , ρu, ρ v, e, ρ S11 , ρ S22 , ρ S12 ) = ( 2730kg / m3 ,0,0,0,0,0,0) . For
T
T
aluminum alloy 6061, the parameters in the Mie-Gruneisen equation (6.16) c0 =5350 m/s,
ρ0 = 2730 kg/m3 , s = 1.34 and Γ0 = 2.0.
7.5 Numerical Results
In this section, the numerical results of deformation, effective stress, density and
pressure are presented.
230
Figure 7.9: Snapshots of the evolving interface of two joined aluminum plates. Wavy
pattern and roll- up occur as time progresses. The interface is tracked by solving a Level
Set Method (LSM) equation passively coupled with the continuum mechanics equations.
Since we focus on the plateau strain stage of the USW process, we assume that
there is no sliding between the two metal coupons to be bonded together. To simplify this
computation, we assume no heat generated by friction at the interface. In Figure 7.9, the
roll- up features of the interface are qualitatively similar to that observed in the USW
microscopy results, shown in Figure 7.2. The interface shown in Figure 7.9 was tracked
by passively solving a Level Set Method (LSM) equation, which denotes two aluminum
plates by two different colors. Figure 7.10 shows the velocity vectors superimposed on
the interface plots, which clearly discern the wave propagation denoted by the green
velocity arrows. These velocity vectors clearly present the existence of vortexes, which
are gnereated by nonlinear wave motion due to the ultrasound vibration. These vorteses
also are observed in the image of cross section of specimen used in experiments. Refer to
Figure 7.2. Especially, the vortex at the head of roll- up is clearer and stronger than other
area, that propably imply that roll- up always comes up with vortex.
231
Figure 7.10: Snapshot of velocity vectors superimposed on the deformed interface.
Figure 7.11: Effective stress distribution inside the metal coupon.
The most striking feature of Figure 7.11 is that the magnitudes of effective stress
in the interior areas are higher than that around the tips of the ultrasound machine on the
top and bottom boundaries. Although we applied the boundary stress at lower values of
that produced by the ABAQUS results for the CESE calculation with about 40% artificial
reduction, the calculated peak values of the effective stresses at many areas inside the
coupons are actually very close to the yield stress. This result indicates that through wave
232
superposition and cancellation, nonlinear stress waves are indeed responsible for creating
transitory high-stress regions in the coupons. The nonlinear stress waves propagate,
reflect, interference, and result in the high values at some areas inside the material. By
tracking down nonlinear stress waves, the numerical solutions show that stress
superposition theory is reasonable.
Based on the analysis of effective stress and wavky interface, the material
behaviours in USW could be explained clearly. First, the nonlinear stress wave pragation
results in stress superposition, thus the effective stress of some area inside material
exceeds the yield stress and occurrence of plastic deformation results material flow. Once
the material starts flowing, the boundary vibration applied by the tip will generate the
vibrating material flow at boundary, and this boundary material motion will produce a lot
of vortexes inside the material. Finaly, these vortexes will drive the motion of material
around interface and then result in the wavky deformation at interface.
Since the equations in model (7.32) are fully coupled, and all of variables are
evolving and solved simultaneously, we can also observe other variables behaviors.
Figure 7.12 shows snapshots of nonlinear waves of pressure and density.
233
(a)
(b)
Figure 7.12: Snapshots of evolving nonlinear waves: (a) pressure, and (b) density.
Similar to the nonlinear stress wave, the above two snapshots also show obvious
nonlinearility of pressure and density.
7.6 Conclusions
This chapter summarizes the direct calculations of the USW process by a
combined approach of the FEA and the CESE method. By solving a dynamic loading
problem, the FEA analysis provided the geometry of the deformed metal coupons and the
stress profiles on the top and bottom boundaries. This information is then used as the
boundary conditions for the CESE solution. Governing equations for the metal plastic
effects in the USW process have been developed.
In order to resolve the wave
propagation, a very large mesh and fine time steps are used for the computation. To
perform the calculation, we use the parallelized CESE code and a PC cluster computer
234
for parallel computations. The numerical results have the following highlights. With a
reasonable stress profiles as the boundary conditions, wavy patterns were formed and
developed at the coupon interface.
Owing to stress wave superposition, the effective stresses have their maximum at
the interior areas of the computational domain − around the coupon interface, instead of
in the vicinity of the tips at the top and bottom boundaries. This finding supports the
theorem of wave superposition for the transitory metal softening effect in the APE.
During the USW process or other manufacture processes involving application of
ultrasonic vibrations, stress waves superposition would lead to sporadic and local
yielding in a transitory manner. The collective effect of the dynamic and transient
yielding would lead to the apparent metal softening as shown in the experiments.
The overall approach strategy here is applying the CESE method to solve the
hyperbolic governing equations, derived by using the conservation laws in conjunction
with the elastic-plastic constitutive equation of solid material. This is to a novel approach
to study nonlinear stress waves in solids.
235
CHAPTER 8
THE TWO-DIMENSIONAL ISOTHERMAL MODEL OF
STRESS WAVE IN ELASTIC-PLASTIC BULK MATERIAL
AND TWO-DIMENSIONAL CRACK PROBLEM
8.1 Introduction
To validate computer code and model in two-dimensional elastic-plastic problems,
we simulate a crack initiation and growth problem. Shown in Figure 8.1 and Figure 8.2,
initial plane waves would interact with and wrap around the crack and change to various
wave modes. This is a complex problem of two-dimensional elastic-plastic wave
propagation with rich physics and most importantly experimental data for comparison.
Ravichandran and Clifton [83], Prakash and Clifton [84] have investigated this problem
for many years and they have reported experiment results. Lin [52] presented numerical
solution and compared his results with the experiment data. Recently, Giese and Fey [72]
also reported numerical simulation of the problem to validate their numerical method.
As shown in Figure 8.1, a two-dimensional solid strip has a semi- infinite crack on
the negative x-axis. As the initial condition, a sudden tensional traction is applied to the
boundary at y = H, causing a plane longitudinal wave to propagate in the negative ydirection. After reaching the line y = 0, i.e., the bottom of the computational domain, the
236
wave in the region of x > 0 will continue travel through the body, while the wave in the
region x < 0 will be reflected back. The reflected wave doubles the magnitude of the
velocity on one side of the crack surface, which would open up the crack. A circular
scattered wave around the crack tip would smoothly connect the reflected wave and the
two crack surfaces. In the experiments, changes of the velocity components on the
boundary of the metal sample were measured.
Figure 8.1: A two-dimensional strip with a semi- infinite crack
237
(a)
(b)
Figure 8.2: Lin [52] conducted numerical computation and compared his results with the
experimental results. (a) Simulated wave pattern, and (b) comparison with the
experimental data for velocity on the sample boundary.
238
(a)
(b)
Figure 8.3: Giese and Fey [72] conducted similar computation with conditions shown in
(a) and presented results shown in (b).
239
Lin [52] conducted numerical simulations of the wave/crack interaction problem.
He used the bi-characteristic scheme, a semi-analytical method, to solve the nonlinear
wave equations. His theoretical model equations were quite different from ours. He
successfully predicted various wave forms, i.e. pressure wave, shear wave and Rayleigh
wave. The prediction of velocity components on the sample boundary compare
reasonably well with experimental results. Refer to Figure 8.2(b). As shown in Figure 8.3,
Giese and Fey [72] studied the similar crack problem to validate their multi-dimensional
numerical scheme for nonlinear stress waves.
Figure 8.3(b) clearly shows the waves pattern, which is close the one shown in
Figure 8.2(a). The capability of capturing these waves in this two-dimensional problem
could be used to evaluate a multi-dimensional numerical scheme.
8.2 Modeling Equations
To describe the modeling equations about isothermal two-dimensional problem, we have
the modeling equations in general vector form, i.e. mass conservation law, momentum
conservation law and elastic-plastic constitutive equation of solid material:
∂ρ
+ ∇ ⋅(ρ v ) = 0
∂t
(8.1)
∂ρ v
+ ∇ ⋅ ( ρ vv − T ) = 0
∂t
(8.2)
DS 2
+ µ tr ( D ) I − 2µ D + θ ( s )( S ⋅ D ) S = 0
Dt 3
(8.3)
240
where ρ is density, v = (u, v, w) is the velocity vector, D is the symmetric part of the
velocity gradient matrix, T is the Cauchy stress tensor, S is the deviatoric part of T, the
relation between them could be defined as,
1
T = tr ( T) I + S = − pI + S
3
(8.4)
and they have following attributes:
tr ( S ) = 0
3
tr ( T ) = ∑ Tii = T ⋅ I
(8.5)
i =1
1
p = − tr ( T )
3
For a plane stress problem, we express Cauchy stress tensor and deviatoric stress tensor
respectively as:
 T11 T12
T = T21 T22
 0
0
0
 S11

0 and S =  S21
 0
0

S12
S22
0


0

− ( S11 + S 22 ) 
0
(8.6)
The velocity components in the cross section are given by
v = ( u , v, w) = ( u , v , 0)
(8.7)
By using Eq.(8.4), (8.6) and (8.7), we expand Eqs.(8.1) and (8.2) from tensor form to a
detail form:
∂ρ ∂( ρu ) ∂( ρv )
+
+
=0
∂t
∂x
∂y
(8.8)
∂(ρu ) ∂ (ρuu + p − S11 ) ∂( ρuv − S 12 )
+
+
=0
∂t
∂x
∂y
(8.9)
241
∂(ρv ) ∂ (ρuv − S12 ) ∂( ρvv + p − S 22 )
+
+
=0
∂t
∂x
∂y
(8.10)
We write the three constitutive equations for plane stress problem as:
∂S11
∂S
∂S
 ∂u ∂v  
∂u
 ∂v ∂u  
+ u 11 + v 11 − S 12  −  − a  2S11
+ S12  +  
∂t
∂x
∂y
∂x
 ∂y ∂x  
 ∂x ∂y  
4 ∂u 2 ∂v
− µ
+ µ
+ θ (s )(S ⋅ D)S 11 = 0
3 ∂x 3 ∂y
(8.11)

∂S 22
∂S
∂S
 ∂u ∂v 
∂v
 ∂v ∂u 
+ u 22 + v 22 + S12  −  − a 2 S 22
+ S12  +

∂t
∂x
∂y
∂y
 ∂y ∂x 
 ∂x ∂y 

2 ∂u 4 ∂v
+ µ
− µ + θ (s )(S ⋅ D)S 22 = 0
3 ∂x 3 ∂y
(8.12)
∂S12
∂S
∂S
 ∂u ∂v 
1
+ u 12 + v 12 + (S11 − S 22 ) −  −
∂t
∂x
∂y
2
 ∂y ∂x 
  ∂u ∂v  1
 ∂v ∂u 
∂u
∂v
 − µ
a  S12  +  + (S11 + S 22 ) +
− µ + θ (s )(S ⋅ D )S12 = 0
∂y
∂x
 ∂x ∂y 
  ∂x ∂y  2
(8.13)
where θ ( s ) is defined by Eq.(5.118),

2

if s = S ⋅ S < 2k
0,

if s = S ⋅ S ≥ 2k 2
θ ( s) = 
0,

3
µ if s = S ⋅ S ≥ 2k 2

,
2
 ( BSH / µ + 3 ) k
for unloading
(5.118)
for loading
and ( S ⋅ D) is given by


(S ⋅ D) = S11 ∂u + S12  ∂u + ∂v  + S 22 ∂v
∂x
 ∂y
∂x 
∂y
For an elastic-perfect plastic material, the definition of θ ( s ) becomes:
242
(8.14)
2
 0, if s = S ⋅ S < 2k

2
θ ( s ) =  0, if s = S ⋅ S ≥ 2k
µ k 2 , if s = S ⋅ S ≥ 2k 2

for unloading
(5.118)
for loading
By assuming a = 0 , we apply Jaumann rate for deviatoric stress and rewrite Eqs.(8.11) (8.13) like:
 ∂u ∂v  4 ∂ u 2 ∂ v
∂S11
∂S
∂S
+ u 11 + v 11 − S12  − − µ
+ µ
+θ ( s )( S ⋅ D) S11 = 0
∂t
∂x
∂y
 ∂y ∂x  3 ∂x 3 ∂y
(8.15)
 ∂ u ∂ v  2 ∂ u 4 ∂v
∂S 22
∂S
∂S
+ u 22 + v 22 + S12  −  + µ
− µ
+θ ( s )( S ⋅ D) S22 = 0 (8.16)
∂t
∂x
∂y
 ∂y ∂x  3 ∂x 3 ∂y
 ∂u ∂v 
∂S12
∂S
∂S
1
∂u
∂v
+ u 12 + v 12 + ( S11 − S 22 )  −  − µ − µ
+θ ( s )( S ⋅ D) S12 = 0 (8.17)
∂t
∂x
∂y 2
∂y
∂x
 ∂y ∂x 
Equations (8.15)-(8.17) are expressed in the non-conservative form, which uses nonconservative variables ( S11, S22 , S12 ) . To consist with the conservative form shown by
T
Eqs.(8.8)-(8.10), we use conservative va riables ( ρ S11 , ρ S22 , ρ S12 ) and rewrite Eqs.(8.15)
T
-(8.17) in the conservative fo rm as:
∂ ( ρ S11 ) ∂ ( ρuS11 ) ∂ ( ρ vS11 )  ∂ ( ρuS12 )
∂ ( ρ S12 )   ∂ ( ρ vS12 )
∂ ( ρ S12 ) 
+
+
=
−u
−v
− 

∂t
∂x
∂y
∂y   ∂x
∂x 
 ∂y
(8.18)
4  ∂ ( ρu )
∂ρ  2  ∂ ( ρ v )
∂ρ 
+ µ
−u  − µ 
− v  + θ ( s )( S ⋅ D) ρ S11
3  ∂x
∂x  3  ∂y
∂y 
∂ ( ρ S22 ) ∂ ( ρ uS22 ) ∂ ( ρ vS 22 )  ∂ ( ρ vS12 )
∂ ( ρ S12 )   ∂( ρuS12 )
∂ ( ρS12 ) 
+
+
=
−v
−u
− 

∂t
∂x
∂y
∂x   ∂y
∂y 
 ∂x
(8.19)
4  ∂ ( ρ v)
∂ρ  2  ∂ ( ρ u )
∂ρ 
+ µ
−v − µ
− u  + θ ( s )( S ⋅ D ) ρ S22
3  ∂y
∂y  3  ∂x
∂x 
243
∂ ( ρ S12 ) ∂ ( ρ uS12 ) ∂ ( ρvS12 ) 1   ∂ ( ρvS11 )
∂ ( ρS11 )   ∂( ρuS11 )
∂ ( ρS11 )  
+
+
= 
−v
−u
− 

∂t
∂x
∂y
2   ∂x
∂x   ∂ y
∂ y  
∂ ( ρ S22 )   ∂( ρuS22 )
∂ ( ρS22 )  
1  ∂ ( ρ vS22 )
− 
−v
−u
− 

2 
∂x
∂x  
∂y
∂y  
∂ ( ρ u)
 ∂ ( ρv)
∂ρ 
∂ρ 
+µ
−u
− v  + θ ( s )( S ⋅ D) ρ S12
+µ
∂y 
∂x 
 ∂y
 ∂x
(8.20)
To make Eqs.(8.8) -(8.10) and (8.18)-(8.20) to be a close form, we use general bulk
modulus
 ρ 
p = k ln  
 ρ0 
(8.21)
to relate pressure and density in an isothermal process.
Those governing equations (8.8) -(8.10) and (8.18)-(8.20) in conservative form
could be written in the vector form as:
∂Q ∂ F ∂ G
+
+
= Ss
∂t ∂x ∂y
(8.22)
where the conservative variable vector is defined by
Q = ( q1, q2 , q3 , q4 , q5 , q6 ) = ( ρ , ρ u, ρ v , ρ S11 , ρ S22 , ρ S12 )
T
T
the conservative flux vectors are
T
 q22
q qq q qq qq qq 
F =  q2 , + p − 4 , 2 3 − 6 , 2 4 , 2 5 , 2 6 
q1
q1 q1
q1 q1
q1
q1 

= ( ρ u, ρuu + p − S11, ρuv − S12 , ρuS11 , ρuS 22 , ρuS12 )
T
244
T
 q q q q2
q qq qq qq 
G =  q3 , 2 3 − 6 , 3 + p − 5 , 3 4 , 3 5 , 3 6 
q1
q1 q1
q1 q1
q1
q1 

= ( ρ v, ρ uv − S12 , ρ vv + p − S22 , ρvS11 , ρvS 22 , ρvS12 )
T
and the source terms on the right side is
0




0




0



∂ ( ρ S12 )   ∂ ( ρ vS12 )
∂ ( ρ S12 ) 
  ∂ ( ρ uS12 )
−u
−v

 −

  ∂y
∂y   ∂ x
∂x 





 4  ∂ ( ρu)

∂ρ  2  ∂ ( ρ v )
∂ρ 

 + µ 
−
u
−
µ
−
v
+
θ
s
S
⋅
D
ρ
S
(
)(
)



11 
∂x  3  ∂y
∂y 
  3  ∂x




  ∂ ( ρ vS12 )
∂ ( ρ S12 )   ∂( ρuS12 )
∂ ( ρS12 ) 
−v
−u

 
− 

  ∂x
∂x   ∂y
∂y 
 
Ss = 


 4  ∂ ( ρ v)
∂ρ  2  ∂ ( ρ u )
∂ρ 

− v  − µ
− u  + θ ( s )( S ⋅ D) ρ S 22  
 + µ 
∂y  3  ∂x
∂x 
  3  ∂y



 
∂ ( ρS11 )   ∂( ρuS11 )
∂( ρS11 )  
  1   ∂ ( ρvS11 )
−v
−u
 
− 

  2   ∂x
∂x   ∂y
∂y  
 
   
 
 
  ∂ ( ρvS 22 )

∂
ρ
S


∂
ρ
uS
∂
ρ
S

(
)
(
)
(
)
1



 
22
22
22
 −
−v
−
−u








∂x   ∂y
∂y    
 2   ∂x
 
 
 

 ∂ ( ρu )
 ∂ ( ρv )
∂ρ 
∂ρ 
−u
+µ 
− v  + θ ( s )( S ⋅ D ) ρ S12  
 +µ



∂y 
∂x 
 ∂y
 ∂x
 
 
The Eq.(8.22) could be rewritten as:
∂Q
∂Q
∂Q
+A
+B
= Ss
∂t
∂x
∂y
where matrixes A and B are defined as:
245
(8.23)
A=
∂F
∂Q
0

 2
 − q2 + ∂ p + q4
 q12 ∂q1 q12

 − q2 q3 + q6

q12
q12

=
qq
− 224

q1

qq

− 225

q1

q2 q6

−

q12

0


∂p S11
 −u 2 +
+
∂q1 ρ


S
=  −uv + 12

ρ

−uS11


−uS 22

−uS12

2
1
0
0
0
q2 ∂ p
+
q1 ∂ q2
∂p
∂ q3
∂p 1
−
∂q4 q1
∂p
∂ q5
q3
q1
q2
q1
0
0
q4
q1
0
q2
q1
0
q5
q1
0
0
q2
q1
q6
q1
0
0
0
1
0
0
0
∂p
∂ q3
∂p 1
−
∂ q4 ρ
∂p
∂ q5
v
u
0
0
S11
0
u
0
S 22
0
0
u
S12
0
0
0
2u +
∂p
∂ q2
246
0 

∂p 
∂ q6 

1
−
q1 

0 


0 


q2 
q1 
0 

∂p

∂q6 

1
−
ρ

0 
0 

u 
(8.24)
B=
∂G
∂Q
0


 − q2 q3 + q6
q12
q12

 2
 − q3 + ∂ p + q5
 q12 ∂q1 q12

=
qq
− 3 24

q1

qq

− 3 25

q1

q3q6

−

q12

0


S
 −uv + 12

ρ

∂p S22
2
+
=  −v +

∂q1 ρ

−vS11


−vS 22

−vS12

0
1
0
0
q3
q1
q2
q1
0
0
q3 ∂p
+
q1 ∂q3
∂p
∂q4
∂p 1
−
∂q5 q1
0
q4
q1
q3
q1
0
0
q5
q1
0
q3
q1
0
q6
q1
0
0
0
1
0
0
v
u
0
0
∂p
∂q4
∂p 1
−
∂q5 ρ
∂p
∂q2
∂p
∂ q2
2
2v +
∂p
∂ q3
0
S11
v
0
0
S22
0
v
0
S12
0
0
0 

1
−
q1 

∂p 
∂ q6 

0 



0


q3 
q1 
0 

1
− 
ρ

∂p

∂ q6 

0 
0 

v 
(8.25)
Considering the EOS defined by Eq.(8.21), we have
∂p k
=
∂q1 ρ
(8.26)
∂p
= 0, i = 2,...,6
∂qi
(8.27)
To study the hyperbolic structure of this system and calculate the eigenvalues in
the elastic media, we rewrite this system in the non-conservative form. We have
conservation laws in non-conservative form as:
247
∂ρ
∂ρ
∂u
∂ρ
∂v
+u
+ρ
+v
+ρ
=0
∂t
∂x
∂x
∂y
∂y
(8.28)
∂u k ∂ρ
∂u 1 ∂S11
∂u 1 ∂S12
+ 2
+u
−
+v −
=0
∂t ρ ∂x
∂x ρ ∂x
∂y ρ ∂y
(8.29)
∂v
∂v 1 ∂S12 k ∂ρ
∂v 1 ∂S 22
+u −
+ 2
+v −
=0
∂t
∂x ρ ∂x ρ ∂y
∂y ρ ∂y
(8.30)
For elastic media, the constitutive equations in non-conservative form are given by
∂S11 4 ∂u
∂v
∂S
∂ u 2 ∂v
∂S
− µ
+ S12
+ u 11 − S12
+ µ + v 11 = 0
∂t
3 ∂x
∂x
∂x
∂y 3 ∂y
∂y
(8.31)
∂S 22 2 ∂u
∂v
∂S
∂ u 4 ∂v
∂S
+ µ
− S12
+ u 22 + S12
− µ + v 22 = 0
∂t
3 ∂x
∂x
∂x
∂y 3 ∂y
∂y
(8.32)
∂S12 1
∂v
∂v
∂S
1
∂u
∂u
∂S
− ( S11 − S 22 ) − µ + u 12 + ( S11 − S22 ) − µ
+ v 12 = 0
∂t
2
∂x
∂x
∂x 2
∂y
∂y
∂y
(8.33)
The equations in the non-conservative form, i.e. Eqs. (8.28)-(8.33) can be
expressed in vector form as
%
%
%
∂Q
% ∂Q + B
% ∂Q = 0
+A
∂t
∂x
∂y
(8.34)
% = ( ρ , u, v, S , S , S )T and matrixes are
where non-conservative vector variable is Q
11
22
12
defined by
248
u
 k
 2
ρ

 0

%
A=
 0


 0


 0

 v

 0

 k
 2
ρ
B% = 
 0


 0


 0

(
ρ
0
u
0
0
4µ
3
2µ
3
−
−µ −
0
0
1
−
ρ
0
u
0
0
S12
u
0
−S12
0
u
0
0
( S11 − S 22 )
2
0
ρ
0
0
v
0
0
0
0
v
0 −
− S12
S12
−µ +
0
( S11 − S22 )
2
2µ
3
4µ
−
3
0
1
ρ
v
0
0
v
0
0
0 

0 

1
− 
ρ

0 


0 


u 

0 
1 
−
ρ

0 


0 


0 


v 

)
% − λ I = 0 , we have
By solving equation det A
(u − λ )
2

k 4µ  
1
S11 − S22   
2
2
u
−
λ
−
µ
+
(
)
 ( u − λ ) − −

 = 0
ρ 3ρ  
ρ
2


(8.35)
% are
the eigenvalues of matrix A
λ1,2 = u, λ3,4 = u ± c1 = u ±
k + 4µ 3
µ S11 − S22
, λ5,6 = u ± c2 = u ±
+
ρ
ρ
2ρ
249
(8.36)
where c1 =
c2 =
( k + 4µ 3)
(µ + (S
11
(
ρ is longitudinal elastic wave component in bulk material, and
− S 22 ) 2) ρ is shear elastic wave component in bulk material. By solving
)
equation det B% − λ I = 0 , we have
(v − λ )
2

k 4µ  
1
S − S 
2
2
( v − λ ) −  v + 22 11    = 0
 ( v − λ ) − −


ρ 3ρ  
ρ
2
 

(8.37)
% are
the eigenvalues of matrix A
λ1,2 = v, λ3,4 = v ± c1 = v ±
where c1 =
c2 =
( k + 4µ 3)
(µ+ (S
22
k + 4µ 3
µ S22 − S11
, λ5,6 = v ± c2 = v ±
+
ρ
ρ
2ρ
(8.38)
ρ is longitudinal elastic wave component in bulk material, and
− S11 ) 2) ρ is shear elastic wave component in bulk material.
250
CHAPTER 9
CONCLUSIONS AND FUTURE WORKS
The present research is a balanced theoretical and numerical effort for modeling
nonlinear stress wave in solids. A novel theoretical framework based on the conservation
laws in conjunction with the elastic-plastic constitutive relation has been developed.
Various one- and two-dimensional model equations for waves in thin rids, bulk materials,
and two-dimensional plane have been reported. For each special application, the
equations have been cast into a set of first-order, strongly coupled and nonlinear partial
differential equations. For each set of equations, we have shown the conservative form,
the non-conservative form, and the characteristic form. We have also provided detailed
analyses of the eigen-structure of the governing equations, including the eigenvalues of
the Jacobian matrices, the speed of the sound, and the Riemann invariants along the
characteristic lines.
To solve the complex system of equations in both one and two spatial dimensions,
we have used the space-time CESE method, a novel numerical framework for solving
nonlinear conservation laws. The combined approach of in-depth studies of the
theoretical model equations and the numerical method for solving the equations was used
to model linear and nonlinear elastic waves in thin rods, in bulk materials, in a Hopkinson
bar impact problem, and in a severe impact problem in bulk materials involving plastic
251
deformation. Finally, the approach was employed to model nonlinear elastic-plastic
waves in the USW processes. In particular, the calculation results have lead to basic
understanding of the APE process based on the phenomenon of stress wave superposition.
9.1 Conclusions
The present approach consists of two developments: (i) a series of fundamental
hyperbolic models, which directly addresses nonlinear stress waves in various situations,
and (ii) the CESE method, an advanced numerical framework for high- fidelity solution of
the nonlinear hyperbolic equations. To recap, unique contributions by the present
research include:
(i)
Development of hyperbolic nonlinear wave models for various cases,
including elastic longitudinal extension in a thin rod, isothermal elasticplastic longitudinal plane wave in one-dimensional bulk materials,
thermal dynamic elastic-plastic longitudinal plane wave in onedimensional bulk material, isothermal two-dimensional elastic-plastic
waves in solids, and thermal dynamic two-dimensional elastic-plastic
waves in the USW processes.
(ii)
We have provided detailed analysis of the eigen-structure of each
hyperbolic model equation set. The nature of the wave propagation in
each case could be clearly discerned by the formulations of eigenvalues,
the characteristic form, and the Riemann invariants.
252
(iii)
The role of the equation of state and the energy equation as a part of the
hyperbolic equation system has been analyzed. Their influences on the
behaviors of nonlinear waves in solids have been demonstrated in the
numerical solutions.
(iv)
We have presented the dynamic features of the nonlinear stress wave
profiles inside the metal specimen of the USW process. The present
research is also the first numerical effort for manufacture processes
aided by ultrasonic alternating stresses.
(v)
We have further developed the CESE method for simulating the very
stiff systems of equations for nonlinear stress waves. The use of the
CESE method, a novel numerical framework, to study nonlinear stress
wave propagating in solids is a new approach, which is out of the scope
of the current modeling capabilities.
We have successfully developed the present research approach of solving the
conservation laws by a new numerical method for waves in solids. As such, we have
demonstrated a new paradigm for high- fidelity simulation for nonlinear stress waves in
solid mechanics. In contrast to conventional FEA approaches, this approach points to a
new direction to directly simulate complex nonlinear waves in metals based on timeaccurate solution of the hyperbolic partial differential equations for mass, momentum,
and energy conservations, supplemented by advanced constitutive models and
thermodynamic relationships.
253
In addition to the above academic results, the outcome of efforts presented is indepth understanding of the physics involved in the USW processes. The remarkable
potential of the APE processes for metal processing will be greatly enhanced by the
further development of the numerical tool. By testing the new theoretical and
mathematical models, and by identifying necessary new physics to be added to the
system presented, this research work has built the key ingredients to an eventual control
and optimization of the USW processes. With accurate models, manufacturers will be
able to explore processing parameter space, thereby predict performance properties and
the necessary process adjustments to achieve successful implementation. The experience
and understanding gained will be indispensable for the further development and
controlling of the USW processes, leading to efficient mass production of ultrasoundaided metal forming/joining processes. Further extension of the numerical tool will widen
the scope of potential applications of the APE to other manufacturing processes.
9.2 Future Works
Based on our current results and experience, we have the following suggestion for
the future works:
(i)
For Hopkinson split bar simulation, a two-dimensional symmetric
numerical computation is necessary. In the future work, the
deformation should involve in elastic-plastic deformation instead of
linear elastic deformation. When the elastic-plastic properties of a new
material could be written by an equation in the conservative form, one
254
can evaluate the correctness of this equation by us ing it to conduct the
Hopkinson bar simulation and comparing the numerical results with
experimental observations.
(ii)
Since the gradients of velocities, which appear in the hyperbolic system
and they are solved at each time step in computation process, equal to
the strain rates, one could obtain the strain at each time step by
integrating strain rate over time period. In the future work, this
integration probably generates the solution of strain and deformation
based on the current frame work of hyperbolic system.
(iii)
To obtain higher fidelity in the numerical computation of elastic-plastic
waves, we have to possess more option of elastic-plastic constitutive
equations in conservative form than current elastic-perfect plastic and
elastic- linear hardening plastic relations. If the strain and plastic strain
could be solved in current hyperbolic system, then we will have
abundant selections of elastic-plastic constitutive equations in the
future.
(iv)
In the field of continuum mechanics, we already have done numerical
work to show the stress superposition phenomena. However, this work
did not get in the microstructure of material yet. Since previous works
on APE tried to use dislocations to explain this phenomenon, thus we
255
have to extend current numerical computation by considering
microstructure.
(v)
The further development of integrating the commercial FEA software
packages with CESE method and hyperbolic system will be helpful to
apply this approach to solve many real industrial cases with
complicated geometry.
256
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