University of Iowa
Iowa Research Online
Theses and Dissertations
Fall 2016
Numerical simulation of the impact of a steel ball
with a rigid foundation
Benjamin M. Dill
University of Iowa
Copyright © 2016 Benjamin M. Dill
This dissertation is available at Iowa Research Online: http://ir.uiowa.edu/etd/2204
Recommended Citation
Dill, Benjamin M.. "Numerical simulation of the impact of a steel ball with a rigid foundation." PhD (Doctor of Philosophy) thesis,
University of Iowa, 2016.
http://ir.uiowa.edu/etd/2204.
Follow this and additional works at: http://ir.uiowa.edu/etd
Part of the Applied Mathematics Commons
NUMERICAL SIMULATION OF THE IMPACT OF A STEEL
BALL WITH A RIGID FOUNDATION
by
Benjamin M. Dill
A thesis submitted in partial fulfillment
of the requirements for the Doctor of Philosophy degree
in Applied Mathematical and Computational Sciences
in the Graduate College of the University of Iowa
December 2016
Thesis Supervisor: Professor David E. Stewart
Graduate College
The University of Iowa
Iowa City, Iowa
CERTIFICATE OF APPROVAL
Ph.D. Thesis
This is to certify that the Ph.D. thesis of
Benjamin M. Dill
has been approved by the Examining Committee for the thesis requirement for the
Doctor of Philosophy degree in Applied Mathematical and Computational Sciences
in the December 2016 graduation.
Thesis Committee:
David E. Stewart, Thesis Supervisor
Keith Stroyan
Colleen Mitchell
Gerhard Strohmer
Bruce Ayati
Acknowledgements
I want to thank my advisor David Stewart for his help and support in completing this
work. I would also like to thank Jessica Forbes for her support and encouragement as well as
for reading and providing feedback on this thesis. Finally, I would like to ackknowledge the
Graduate College and AMCS programs that provided me with dedicated time for completion
of this thesis.
ii
Abstract
We simulate the behavior of a steel ball bearing as it impacts a rigid foundation by solving
a discretized version of the dynamic equations of linearized elasticity for a homogeneous,
isotropic material. Space is discretized using the finite element method and time is discretized
using the implicit trapezoidal method. Impact with a fixed foundation is incorporated into
the model using a complementarity condition. This ensures that we have normal forces
acting on the bearing only when and where the bearing is in contact with the foundation.
After discretization in space, this condition becomes a linear complementarity problem (LCP)
which is solved using an iterative method for solving LCPs that is similar to the Gauss-Seidel
method for solving linear systems. The LCP is solved at each time step to determine the
normal forces due to contact. By assuming cylindrical symmetry, we are able to simulate the
impact of a three-dimensional ball using only two spatial coordinates and two-dimensional
finite elements. This decreases the computational cost of a highly refined three-dimensional
simulation dramatically. Using this model, we investigate the deformations that occur during
and after contact. We hypothesized that dropping a steel ball from even a small height
causes plastic deformation. We tested this hypothesis using our model by computing the
state of stress inside the ball at various times during the simulation. By comparing the
computed maximum shear stress to the yield strength of the material, we can determine if
the threshold for plastic deformation is reached. We found that with an impact speed of 2
m/s the stresses induced in the ball are large enough to cause plastic deformation. Because
plastic deformation requires energy and is irreversible, it is an important consideration when
investigating how high the ball will bounce after contact. To quantify the energy loss due
to plastic deformation, we estimate the rate of energy loss due to plastic deformation using
Perzyna’s law. Our model predicts the total energy lost to plastic deformation during impact
is substantial. This provides evidence that plastic deformation is likely the primary cause of
energy dissipation in the collision. Further investigations in which plastic behavior is added
to the simulation are also discussed.
iii
Public Abstract
Can you drop the same ball bearing twice? When a ball bearing impacts a rigid surface, huge
forces act on the surface of the ball. These forces prevent the ball from passing through the
surface and ultimately cause the ball to change direction. Such collisions are surprisingly
violent, even at low speeds. They result in the production of light, heat, sound, elastic
vibrations, and possibly plastic deformation. Plastic deformation refers to the permanent
change of the shape of the ball in response to forces. The reality of these phenomena
explains the fact that a dropped ball can never bounce to the height from which it was
dropped. To further investigate this fact, we generated high-resolution simulations of steel
ball bearing impacts. Our primary goal was to determine the relative importance of plastic
deformation in low speed collisions. These simulations show that a drop from a modest
height is enough to permanently deform a steel ball bearing. We present visualizations
of the forces acting on and within the ball during impact and compare the results of our
simulations with classical investigations of contact mechanics. Our model largely agrees with
classical theory and provides further evidence of the usefulness of the predictions made by
Hertzian contact theory. Additionally, we provide an estimate of the amount of energy lost
to plastic deformation. This estimate and the small amount of vibrational energy observed
in our simulations points to plastic deformation as the primary energy sink in low speed
steel ball collisions. Further investigations in which complex interactions such as friction
and plastic deformation are added to the model are planned for the future.
iv
Table of Contents
List of Figures
vii
1 Preliminaries
1
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.2.1
Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.2.2
Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.2.3
Yield and Plastic Deformation . . . . . . . . . . . . . . . . . . . . . .
16
1.2.4
Contact Conditions and Friction Laws . . . . . . . . . . . . . . . . .
20
1.2.5
Linear Complementarity Problems . . . . . . . . . . . . . . . . . . .
23
1.2.6
Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . .
25
1.2.7
Hertzian Contact Theory . . . . . . . . . . . . . . . . . . . . . . . . .
25
2 An Elastic Contact Model and Its Discretization
30
2.1
Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.2
Complementarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
2.3
Cylindrical Coordinates
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
2.4
Energy Considerations in the Discrete Model . . . . . . . . . . . . . . . . . .
38
3 Software Implementation
40
3.1
Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
3.2
Forming the Stiffness and Mass Matrices . . . . . . . . . . . . . . . . . . . .
42
v
3.3
Forming the Boundary Matrix . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.4
Computing Displacements and Stresses . . . . . . . . . . . . . . . . . . . . .
44
3.4.1
Solving the Linear Systems . . . . . . . . . . . . . . . . . . . . . . . .
44
3.4.2
Solving the LCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
3.4.3
Computing and Visualizing Stresses . . . . . . . . . . . . . . . . . . .
45
Computing Hertzian Distributions . . . . . . . . . . . . . . . . . . . . . . . .
46
3.5
4 Numerical Results
47
4.1
Parameter Values and Simulation Description . . . . . . . . . . . . . . . . .
47
4.2
Results for Dynamic Model and Hertzian Comparisons . . . . . . . . . . . .
49
4.3
Estimation of Energy Loss to Plastic Deformation . . . . . . . . . . . . . . .
55
5 Conclusion
57
References
61
vi
List of Figures
1.1
Displacement function for material points. . . . . . . . . . . . . . . . . . . .
8
1.2
Depiction of stress tensor and stress vector on infinitessimal parallelpiped.
.
10
1.3
Depiction of internal forces acting on a set ω ⊂ Ω. . . . . . . . . . . . . . . .
11
1.4
Schematic showing Kelvin-Voigt model of viscoelasticity. . . . . . . . . . . .
14
1.5
Simplified stress-strain curves for elastic and plastic behavior in a slow loading
experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
1.6
Creep strain response to suddenly applied constant stress. . . . . . . . . . .
18
1.7
Stress relaxation at constant strain. . . . . . . . . . . . . . . . . . . . . . . .
18
1.8
The Tresca and von Mises yield surfaces in the space of principal stresses. . .
20
1.9
Contours of scaled maximum shear stress for Hertzian contact.
. . . . . . .
27
1.10 Maximum shear stress contours for predicted contact area. . . . . . . . . . .
29
2.1
The physical setting for the contact problem. . . . . . . . . . . . . . . . . . .
31
2.2
Basis function for one and two-dimensional FEM using piecewise linear elements. 34
2.3
The gap between the ball and the foundation as a function in the continuous
problem (left) and as a vector in the discrete problem (right). . . . . . . . .
2.4
35
Representative half-disk for describing displacements under non-rotational
displacement assumption. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.1
Flowchart describing the major steps of the MATLAB implementation. . . .
40
3.2
Flowchart describing the computation of the displacement and stresses at each
time step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
41
3.3
Triangular meshes generated by DistMesh on a half disk. . . . . . . . . . . .
42
4.1
Kinetic, elastic potential, and total energy during the simulation. . . . . . .
50
4.2
Propagation of elastic stress wave. . . . . . . . . . . . . . . . . . . . . . . . .
50
4.3
Comparison of simulation contact pressures and Hertzian contact pressures
for computed contact radius at various times during impact. . . . . . . . . .
51
4.4
Simulated pressure distribution over time. . . . . . . . . . . . . . . . . . . .
52
4.5
Maximum shear stress and yield stress during the simulation. . . . . . . . . .
52
4.6
Maximum shear stress contours 2.6µs and 5µs after impact. . . . . . . . . .
53
4.7
Maximum shear stress contours at 10µs and 15µs after impact.
54
4.8
Estimated scaled rate of energy change due to plastic deformation during the
. . . . . . .
simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
56
Chapter 1
Preliminaries
1.1
Introduction
From everyday experience, we know that certain objects bounce upon impacting surfaces.
Objects such as rubber balls bounce quite well, but we observe that a dropped object never
returns to its original height after impacting a rigid surface. Some energy is dissipated during the collision by the generation of sound, light, and heat. Depending on the materials
and velocities involved, plastic deformation may also occur. Plastic deformation is the phenomenon wherein an object’s shape is permanently altered by the application of force. Up
to a certain threshold, deformation of an object occurs elastically. That is, when the forces
causing the deformation are removed, the deformation is reversed and no permanent changes
take place in the object.
During an elastic process, energy is stored as elastic potential energy while forces are
applied to an object and released when the forces are removed. When forces reach a critical
material-dependent threshold called the yield strength, objects cease to behave elastically.
Beyond this threshold, objects are said to behave plastically: some deformations become
permanent and energy is dissipated in the process. Under the correct conditions, forces
applied to an object during a collision may exceed the threshold for plastic deformation.
Since this process requires energy and is irreversible, it is an important consideration when
investigating energy dissipation during collisions.
Our goal is to determine whether plastic deformation occurs during the collision of a 1 cm
1
steel ball bearing and a rigid foundation at a speed attainable by dropping the ball from a
small height. We also seek to quantify the importance of plastic deformation in determining
the bounce height of the ball. To accomplish this, we sought to simulate the impact as
realistically and in as much detail as possible. In reality, complex behaviors are observed in
response to impact. Detailed simulations of impact problems can help us understand basic
phenomenological questions and provide insight into processes that are difficult to observe in
reality. Additionally, contact and impact models are widely used in engineering applications.
Increasingly realistic models of impact behavior can inform engineering decisions in a variety
of settings.
Modeling the behavior of bouncing balls has a long history, beginning with Isaac Newton.
Newton developed the coefficient of restitution (COR), which is defined as the ratio of the
relative speed of two objects after they collide to their relative speed immediately prior
to collision. A COR of 1 corresponds to a perfectly elastic collision. The COR serves as a
means to describe the behavior of objects after a collision based on previously experimentally
determined values. The treatment of the COR as an a priori known constant is problematic
as it has been shown that the COR can vary significantly for the same two bodies depending
on impact angle, impact speed, and many other factors. Ideally, an impact model will
not require the prior assumption of a COR, but will naturally dissipate energy in a way
that mirrors reality as much as possible. A model using the COR may accurately predict an
object’s behavior in a simplistic way, but will not describe the collision itself or independently
consider the physical causes of energy dissipation in inelastic collisions.
The field of contact mechanics takes a more sophisticated approach, modeling the deformation of colliding objects using the principles of continuum mechanics. Many of these
models consider the displacements and stresses induced at each point of each object and so
provide a much more detailed description of behavior during collisions. The ability to mathematically describe the stresses within an object subjected to varied forces is necessary if we
are to investigate plastic deformation. This is because experiments show that the onset of
2
plastic deformation occurs at a specific level of stress in solids. The field of contact mechanics
is seen to have begun in 1882 when Henrich Hertz published his work “On the contact of
elastic solids” [1]. Classical contact mechanics provides descriptions of contact areas, indentation depths, and pressure distributions resulting from contact of elastic objects with simple
shapes and known material properties. These models are useful in engineering applications
and have been generalized and adapted to many problems. However, the predictions made
by classical contact mechanics are based on significant simplifying assumptions. The goal of
these models is to understand and predict the behavior of elastic objects in equilibrium during contact. Focusing on Hertzian contact theory, we note the following assumptions. First,
the analysis makes the assumption that both objects in a collision can be modeled as elastic
half-spaces. This is reasonable if the contact area is small compared with the size of the
contacting spheres. Second, the pressures and stresses described in Hertzian theory depend
on the compression force between the colliding objects. When realistically describing our impact, this compressive force will change over time and is not generally known. Even knowing
these forces, the applicability of Hertzian contact relies on the system being in equilibrium.
To model an inherently dynamic process like the bouncing of a ball, this would require us to
make a quasistatic assumption. That is, we would have to assume that the forces balance at
all times and the system is always in equilibrium. This is a reasonable assumption for slowly
evolving physical processes and processes for which transient behavior can be ignored. In
modeling the behavior of an object during a collision, the quasistatic assumption may not
be reasonable as the system evolves rapidly. Additionally, collisions may take place on short
enough timescales that elastic vibrations do not have time to significantly dampen. In these
cases, these elastic waves may be important in describing the behavior of the ball during
and after the collision. Models that include the transient behavior and do not make the
quasistatic assumption are called dynamic models. More information on Hertzian contact is
presented in section 1.2.7.
In [11] an equilibrium solution for a heavy sphere on a rigid plate found by Bondareva
3
in [12] is adapted and used to describe normal impact of an elastic sphere with a rigid
foundation. Similar to the above discussion, the approach used in [11] to model the dynamic
contact problem relies on the assumption that the collision lasts long enough for many elastic
wave traversals to take place. In this case the results obtained are similar to the predictions
obtained using the Hertzian solution to simulate contact. Our numerical approach naturally
models elastic wave generation and its effects and so we do not require this assumption.
In [5], Antonio Signorini published a description of an equilibrium problem for an elastic
ball in the language of partial differential equations. The problem is to determine the configuration of an elastic ball resting on a rigid frictionless surface and subject to prescribed forces.
Because the surface is rigid and the final configuration of the ball is unknown, the boundary
conditions on the portion of the ball’s surface that may contact the surface are unknown.
Earlier theory on boundary value problems was not applicable to these ambiguous boundary
conditions and the solution of this problem (which is known as the Signorini Problem) by
Fichera in [6] gave rise to the field of variational inequalities. Variational inequalities can
be used to describe and solve boundary value problems. The general variational inequality
problem is to find x ∈ K such that hF (x), y − xi ≥ 0 ∀ y ∈ K where h·, ·i : E ∗ × E → R
is the duality pairing between Banach space E and it s dual space E ∗ and F : K → E ∗ is
a functional on K ⊂ E. For the Signorini problem, K is the set of all displacement vectors
for the points comprising the ball that satisfy the ambiguous boundary conditions. Fichera
was able to show that within this set there is a unique solution.
To describe dynamic problems with the same types of hard inequality constraints, the
concept of the differential variational inequality (DVI) was introduced by Pang and Stewart
in [7]. The DVI problem is to find u(t) ∈ K such that hF (t, x(t), u(t)), v − u(t)i ≥ 0 ∀
v ∈ K and almost all t, where K is a closed convex set and
dx
dt
= f (t, x(t), u(t)), x(t0 ) = x0 .
Much more about DVIs and their applications can be found in [8]. The theoretical model
on which our simulation is based can be interpreted as a DVI that allows us to include a
complementarity condition enforcing unilateral contact constraints between the steel ball and
4
foundation while using the equations of linearized elasticity to describe the behavior of the
ball. The system describes the time evolution of the configuration of the ball as it is subjected
to body and surface forces, including surface forces due to contact. The displacement vector
u(t) for the points comprising the ball is the primary unknown, though any normal forces
due to contact are also a priori unknown. Formally, the behavior of the ball occupying set
Ω with boundary ∂Ω and subject to body forces described by f and normal forces due to
contact on ΓN ⊂ ∂Ω described by N is modeled by the system
ρü = f + Divσ for every x ∈ Ω and every t > 0
(1.1)
˙
σ = C(u) + D(u)
(1.2)
1
∇u + (∇u)T
2
(1.3)
=
σ(x)ν = N (x, t) for each x ∈ ΓN
(1.4)
0 ≤ ϕ(x, t) ⊥ N (x, t) ≥ 0 for all t.
(1.5)
Here σ and are the stress and strain tensors, C and D describe the elastic and viscous
behavior of specific materials, and ϕ describes the gap between the ball and the foundation.
More information about elasticity is provided in section 1.2.1 and more information about the
tensors C and D is provided in section 1.2.2. Equation (1.5) is a complementarity condition
that serves to prevent the overlap of the body with the rigid foundation.
Significant effort has been made to solidify the mathematical justification for theoretical
dynamic contact models using complementarity conditions. In [2] it is shown that in the onedimensional case, solutions exist for the frictionless contact problem with complementarity
ü = f + ∇2 u for every x ∈ Ω
N (x, t) = −
∂u
(x, t) for each x ∈ ∂Ω
∂n
5
(1.6)
(1.7)
0 ≤ ϕ(x, t) ⊥ N (x, t) ≥ 0.
(1.8)
However, this result relies on the div-curl lemma of compensated compactness and does not
generalize to higher dimensions. In [3], it is shown that even in the case above solutions are
not necessarily unique. In higher dimensions both existence and uniqueness of solutions for
frictionless contact problems in elasticity are open questions. By including viscosity via the
Kelvin-Voigt model, solutions are generally smoother and the existence of solutions to these
frictionless viscoelastic systems has been proved in [4]. The uniqueness of these solutions is
an open question. The theoretical model in equations (1.1)-(1.5) is three-dimensional and
includes Kelvin-Voigt viscoelasticity, hence the existence of solutions is established, but the
uniqueness of solutions to the system is an open question. The results in this paper assume
no viscosity and so are purely elastic. In this case, neither existence nor uniqueness has been
established for the continuous model.
Complementarity approaches to the numerical solution of contact problems have been
used to simulate complex systems in a number of different contexts. In [18], the authors
use a complementarity scheme to simulate the longitudinal impact of two elastic rods, the
oblique impact of a rectangular plate, and the oblique impact of a plate with a round boundary. These are all two-dimensional simulations which include frictional effects. In [19], the
behavior of a vibrating string undergoing unilateral contact with various objects is simulated
using a complementarity condition and an Euler-Bernoulli beam with similar conditions is
investigated in [20]. In [21], a more general dynamic contact problem in viscoelasticity is
formulated and solved numerically. The complementarity condition is solved using a nonsmooth Newton’s method. This paper is not concerned with yielding or an investigation
of plastic deformation energy losses, and instead focuses on energy losses due to viscous
properties of materials.
In this paper we consider the dynamic impact model with Signorini contact conditions
in equations (1.1)-(1.5). We numerically solve this boundary value problem while enforcing
complementarity via solution of an LCP. We simulate the normal impact of a steel ball
6
with a rigid foundation at low speed by discretizing this system in space with the finite
element method and in time with the implicit trapezoidal rule. We solve the resulting
system of equations numerically using a preconditioned conjugate gradient method. By
making use of cylindrical symmetry, we simulate the impact in three dimensions and in
high resolution. Near the contact area, we use a mesh size of 40 µm. Because of the
difficulties establishing theoretical results for this type of system, detailed numerical solutions
are particularly interesting. To the best of our knowledge, no previous numerical results are
available in such high resolution for dynamic three-dimensional ball impact problems.
By considering the stress and strain states in our model and a constitutive law in plasticity, we attempt to quanitify the energy loss we would expect to see due to plastic deformation.
By using Perzyna’s law to describe the rate of change of stress for stress states above the
yield limit, we compute the rate of energy loss due to plasticity at each time in the simulation. More information about the elastoplastic constitutive equations can be found in section
1.2.2 and 1.2.3. This represents a first step toward a model capable of accurately describing both elastic and plastic behavior during collisions. Simulations of elastoplastic behavior
have been carried out for a number of impact problems, though again a high-resolution,
three-dimensional, dynamic simulation would be novel.
1.2
Background
The following sections provide background information describing the derivation of equations
(1.1)-(1.5), a description of the conditions necessary for plastic deformation, and the tools
we use to simulate our contact problem. Basic information on the theory of elasticity,
constitutive equations, yield criteria, linear complementarity problems, the finite element
method, and Hertzian contact theory is provided.
7
1.2.1
Elasticity
Many quality references exist which describe the theory of elasticity, constitutive equations
and yield criteria. Here we provide a brief introduction following [15]. More detailed descriptions of basic quantities and concepts are available in [16] and numerous textbooks on
elasticity. We consider a continuous medium which occupies Ω ⊂ Rd at time t = 0. We will
refer to this as the body. At time t > 0, the body may change shape and it will then occupy
the set Ωt ⊂ Rd . Note that generally d ∈ {1, 2, 3}, and here we focus on the case d = 3.
A point x ∈ Ω is called a material point and the function y(x, t) describes the location of
each material point at each time. As illustrated in Figure 1.1, we also define a displacement
function u(x, t) = y(x, t) − x.
Figure 1.1: Displacement function for material points.
As parts of the body change position relative to each other, we need a way to quantify
this deformation. To that end, we begin by defining
∇u =
∂ui
∂xj
,
(1.9)
the deformation gradient. Using the deformation gradient, we define the finite strain tensor
E=
1
∇u + (∇u)T + (∇u)T ∇u .
2
8
(1.10)
This is a measure of the relative deformation of the body. In the linear theory of elasticity,
we assume that ∇u is small enough that we can neglect the order 2 term (∇u)T ∇u. Making
this assumption, we obtain a simplified version of E which will henceforth be called the
strain tensor. We denote the strain tensor and define it as
=
1
∇u + (∇u)T .
2
(1.11)
Three types of forces are typically included in elastic models. First, we can have body
forces, which can act on all parts of the body. Gravity is an example of a body force. Second,
we can have surface forces, which act only on the surface of the body. Of particular interest
to us will be forces generated due to the surface of the body contacting some other object.
Finally, we can have internal forces. These are forces which occur within the body to oppose
deformations induced by surface and body forces. The density of body and surface forces
are described by the functions f(x, t) and g(x, t) respectively. Internal forces are described
by the stress tensor, σ, and the stress vector, t.
The stress vector t(y,n) describes the force per unit area at a point y of the body across
a surface with normal n passing through y. For a given surface element, we can relate the
unit normal n to t(y, n) through σ according to
t(y, n) = σ(y)n.
(1.12)
Thus we can see that the stress tensor provides a complete description of the stresses at a
given point of the body as we can determine the pressure in any direction through equation
(1.12). In three dimensions, the tensor has a unique representation as a 3 × 3 matrix for any
9
choice of coordinates. As we will soon see, the stress tensor is symmetric and can be written

 σx τxy

σ=
τyx σy

τzx τzy



τxz  σ11 σ12 σ13 
 

σ

=
τyz 
σ
σ
  21 22 23 
 

σz
σ31 σ32 σ33
(1.13)
where σx , σy , and σz are normal stresses which work perpendicular to the cross section being
considered and the τ components are shear stresess which are the result of forces parallel to
the cross section on which they act. Figure 1.2 illustrates that we can compute the stress
vector for any normal direction by considering the stress vectors on mutually perpendicular
planes. This is known as Cauchy’s stress theorem. Because σ is a tensor quantity, the values
Figure 1.2: Depiction of stress tensor and stress vector
on infinitessimal parallelpiped.
of the stress components will vary depending on the choice of coordinates. The eigenvalues
of the stress tensor are called the principal stresses, and they serve as important invariants.
They will later allow us to easily determine the maximum shear stress experienced at a given
point, which is important in determining if the material will behave elastically or plastically.
By applying the principle of conservation of momentum to a part of the body occupying
10
the region ω ⊂ Ω, we can derive equations describing the elastic behavior of the body. We
can compute the net force that the material outside ω exerts on the material inside ω by
integrating the stress vector over the boundary ∂ω. That is,
Z
net internal force on ω =
t(y, ν)ds,
(1.14)
∂ω
where ν is the outward normal at y ∈ ∂ω. Figure 1.3 shows the set ω and the stress vector
on the boundary ∂ω. Conservation of linear momentum tells us that for any ω ⊂ Ω and any
Figure 1.3: Depiction of internal forces acting on a set
ω ⊂ Ω.
t > 0, we must have
d
dt
Z
Z
ρu̇dx =
Z
fdx +
ω
ω
t(x, ν)ds.
∂ω
Since ω is independent of t,
d
dt
Z
Z
ρu̇dx =
ω
ρüdx.
ω
Further, using Green’s Theorem, we can write
Z
Z
t(x, ν)ds =
∂ω
Z
σ(x)νds =
∂ω
Divσdx,
ω
11
(1.15)
where Divσ is a vector field given by
(Divσ)i =
d
X
∂σij
∂xj
j=1
.
Thus, we can rewrite equation (1.15) as
Z
Z
Z
ρüdx =
fdx +
ω
ω
Divσdx.
(1.16)
ω
Since the choice of ω was arbitrary, we have the equation of motion
ρü = f + Divσ for every x ∈ Ω and every t > 0.
(1.17)
In three dimensions, this is a system of 3 PDEs with 9 unknowns.
Applying conservation of angular momentum to the region ω, we have
d
dt
Z
Z
Z
x × t(x, ν).
(1.18)
Div (x × σ) dx.
(1.19)
x × f dx +
x × ρu̇ dx =
∂ω
ω
ω
Similar to above we can simplify this to
Z
Z
Z
x × f dx +
x × ρü dx =
ω
ω
ω
Using the fact that
Div (x × σ) = ε : σ + x × Divσ
where ε is the permutation tensor, we can further simplify to
Z
Z
x × (ρü − f − Divσ) dx =
ω
ε : σ dx.
ω
12
(1.20)
Applying equation (1.15) and observing again that the choice of ω was arbitrary, we have
ε : σ = 0.
(1.21)
For this to be true, we must have σ = σ T at all times and material points. This symmetry
reduces the number of unknowns in the equation of motion from 9 to 6. Knowledge of the
attributes of the materials composing the body is necessary to continue our description.
1.2.2
Constitutive Equations
We complete the picture with constitutive equations, equations describing the relationship
between the stress and strain tensors, σ and for a specific material. Many constitutive
relations have been proposed and used to describe the behavior of materials under various
conditions. Many materials exhibit a linear relationship between stress and strain up to a
certain threshold, beyond which a more complex nonlinear relationship is observed. In linear
elasticity, we assume there is a linear relationship between stress and strain at each point.
The elastic response to forces at any point of a linearly elastic material can be imagined as
that of a linear spring obeying Hooke’s Law. Thus, we have a tensor, C, which relates stress
and strain according to
σ = C(u).
(1.22)
Every component of the stress depends linearly on the elements of the strain. Most metals
can be accurately modeled as isotropic, homogeneous materials. Isotropic materials have
the same elastic response in all directions. Homogeneous materials have the same stressstrain relationship at all points. In the case of isotropic, homogeneous materials, symmetry
considerations allow us to simplify the relationship between stress and strain to
σ(u) = λe tr((u))I + 2µe (u)
13
(1.23)
where λe and µe are material constants called the Lamé parameters.
Viscous behavior of a material can be modeled by assuming the stress is linearly related
to the strain rate ˙ through another tensor D according to
˙
σ = D(u).
(1.24)
Again we are able to simplify this relationship when considering isotropic, homogeneous
materials. For such materials, we have
˙
˙
σ(u) = λv tr((u))I
+ 2µv (u).
(1.25)
These models of the elastic and viscous behavior of a material can be combined in many
ways to form viscoelastic constitutive equations describing the material’s overall behavior.
The elastic portion of a material’s behavior is modeled as a spring, and the viscous portion
is modeled as a dashpot. Combining a spring and dashpot in parallel as in Figure 1.4, we
form the Kelvin-Voigt model of viscoelasticity. In this model we combine the elastic and
Figure 1.4: Schematic showing Kelvin-Voigt
model of viscoelasticity.
viscous properties of a material in the constitutive equations
˙
σ = C(u) + D(u).
14
(1.26)
The elasticity and viscosity parameters are related in homogeneous isotropic materials, with
the ratio depending on the specific material.
When stresses become too large, stress and strain cease to be linearly related. Beyond
this limit, permanent deformations occur in solids. This is the domain of plastic behavior.
Solids experiencing plastic deformation can exhibit phenomena called creep and relaxation.
Creep refers to an increase in strain over time for a constant stress. Relaxation is a decrease
in stress over time while strain is held constant. Both of these behaviors can be included in a
constitutive model by allowing time derivatives of stress and strain to enter the stress-strain
relationship. To model plastic behavior, we can assume the displacements and strains will
have an elastic part and a plastic part. For strain, we can write
= e + p .
(1.27)
The elastic part of the strain is still linearly related to the stress according to
e = Eσ.
(1.28)
To determine p , we assume that the plastic strain rate will depend on the current stressstrain state according to
˙ p = ψ(σ, ).
(1.29)
Then differentiating with respect to time, we have
˙ = ˙ e + ˙ p = E σ̇ + ψ(σ, ).
(1.30)
Rearranging in terms of the stress rate we have
σ̇ = C˙ + G(σ, ).
15
(1.31)
In equation (1.31) we see that the material’s behavior will be elastic when ψ(σ, ) = 0.
Perzyna’s law gives a natural choice of the function ψ which allows us to easily have ψ(σ, ) =
0 whenever the current state of stress is within the domain of elastic behavior. Perzyna’s
law defines
ψ=
1
(σ − Pk (σ))
µ
(1.32)
where µ is a viscosity parameter and Pk (σ) is the projection of σ onto a yield surface
describing the limit of elastic behavior when the stress state σ is outside of the realm of
elastic behavior. For other stress states, Pk (σ) = σ. In the following section on yield and
plastic deformation, we will discuss a specific version of this constitutive equation for one
yield criterion.
This constant µ in equation (1.32) depends on many factors including temperature, the
exact composition of the material, and the strain rate. As discussed in [25], the µ also depends
on the physical scale of the plastic deformation being considered. Values of µ on the orders
of magnitude 104 and 105 have been calculated for steel in different experiments. When
considering plastic deformation on a macroscopic scale over a volume of material, values on
the order of 105 have typically been observed. Determination of the most appropriate value
of µ for a given situation is challenging.
1.2.3
Yield and Plastic Deformation
As we saw in equation (1.22), purely elastic materials have a linear relationship between
stress and strain. When applied stresses are removed from an elastic material, deformations
will disappear and the material will return to its original shape. Many materials behave
elastically when responding to relatively small stresses. However, all such materials have a
limit to their ability to deform elastically. When stresses reach a critical threshold called
the yield strength, plastic deformation begins to occur. This means that some part of the
deformation becomes permanent and an object’s shape will be irreversibly changed.
If we imagine an experiment where we gradually increase the stress on an object until
16
we reach a predefined stress and then gradually decrease the stress back to zero, we can
visualize the stress-strain relationship as in Figure 1.5. The left diagram shows perfectly
Figure 1.5: Simplified stress-strain curves for elastic and plastic behavior in a slow
loading experiment.
elastic behavior. Stress and strain are perfectly linearly related during loading and unloading
and no permanent deformation takes place. The right diagram shows a simplified plastic
stress-strain relationship. We see that beyond the yield strength σy , the stress and strain
cease to be linearly related. After unloading, some amount of strain remains. This is due to
the permanent plastic deformation.
More complex stress and strain behaviors are also observed in experiments. Two phenomena that can be important in describing viscosity and plastic deformation in models are
creep and relaxation. Creep refers to an increase in strain over time while stress remains
constant. Relaxation refers to a decrease in stress while strain is held constant. Simplified
illustrations of creep and relaxation can be seen in Figures 1.6 and 1.7. This dependence
of the stress-strain state on time can be included in models to make them more realistic.
The Kelvin-Voigt constitutive equations can effectively model creep, but do not accurately
describe relaxation. Perzyna’s law is capable of modeling both creep and relaxation.
Because we are investigating the presence and prevalence of plastic deformation in our
impact problem, it is important that we understand how to identify conditions that would
17
Figure 1.6: Creep strain response to suddenly applied constant stress.
Figure 1.7: Stress relaxation at constant strain.
lead to plastic deformation. Our model is not sophisticated enough to describe plastic
deformation, but we can observe the stresses developed during contact to determine if plastic
deformation would occur in reality. In three dimensions, it is more difficult to determine if a
material is in a stress state that will cause plastic deformation. A yield criterion is used to
determine if a given stress state is within the elastic limit or not. Different yield criteria are
appropriate for different types of materials and even for a particular material various criteria
may exist based on different assumptions or theoretical backing. We describe the Tresca
18
yield criterion and the von Mises yield criterion, which are both well known and commonly
used criteria for isotropic materials that are not brittle.
The Tresca yield criterion assumes that a material will deform plastically only in response
to excessive shear stresses. Shear stress is easily described in terms of principal stresses
according to
1
τij = (σi − σj ).
2
(1.33)
The criterion states that as long as the magnitudes of all of the shear stresses are below the
shear yield strength of the material, no plastic deformation will occur. We can express this
compactly as
1
1
max(|σ1 − σ2 |, |σ2 − σ3 |, |σ3 − σ1 |) ≤ σy
2
2
(1.34)
where σy is the tensile yield strength of the material. As seen in Figure 1.8, we can visualize
this criterion in the space of principal stresses. For stress states within the surface
1
1
max(|σ1 − σ2 |, |σ2 − σ3 |, |σ3 − σ1 |) = σy ,
2
2
plastic deformation will not occur. Similarly, the von Mises yield surface is described by
σy =
q
1
2
[(σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 ]
(1.35)
and stress states within this surface are assumed to be elastic when using Von Mises’ criterion.
We define the set
q
1
2
2
2
K = σ | 2 [(σ1 − σ2 ) + (σ2 − σ3 ) + (σ3 − σ1 ) ] < σy .
(1.36)
If we know the stress state at a point within an object, we can compute the principal
stresses and use equation (1.34) or (1.35) to determine if behavior would be purely elastic or whether plastic deformation would occur. We use this approach to determine if we
19
Figure 1.8: The Tresca and von Mises yield surfaces in the space of principal stresses.
have reached the threshold for plastic deformation in our model. Additionally, we use von
Mises’ criterion when determining the projection necessary when applying Perzyna’s law as
~ = (σ1 , σ2 , σ3 )T onto
described in equation (1.32). The projection of a principal stress state σ
its closest point on the yield surface in equation (1.35) is easily accomplished and results in









 
Pk (~
σ ) = R  q

 


R + 23 σy


 



 R
if σ ∈ K
~
σ


,
σ1 − R σ1 − R

 

σ2 − R σ2 − R

 

σ3 − R
σ3 − R 

(1.37)
if σ ∈
/ K.
Using Perzyna’s law with Pk defined as in equation we later attempt to estimate the theoretical energy loss due to plastic deformation in our simulation.
1.2.4
Contact Conditions and Friction Laws
As a ball bearing impacts a rigid object, normal forces keep the ball from passing through
or overlapping with the object. The ball deforms in response to these surface forces and
20
ultimately rebounds. As the ball deforms, an area of contact forms. During this process
we may have frictional forces acting tangential to the contact surface and normal forces
acting perpendicular to the contact surface. These forces are described by the tangential
stress σ T (x, t) and the normal stress N (x, t) on the contact surface. Conditions describing
the normal forces are commonly called contact conditions and conditions describing the
formation of frictional forces are called friction conditions. We follow section 5.4 of [15]
in describing some established contact and friction conditions. It is assumed that N will
depend only on the displacement and velocity in the normal direction, uν (x, t) and vν (x, t).
Frictional forces are assumed to depend only on the tangential velocity vT .
The first and simplest contact condition we discuss is the Signorini contact condition.
This model describes contact with a perfectly rigid foundation. In this case, we can ensure
nonpenetration by enforcing the simple condition
uν (x, t) ≤ ϕ0 (x, t)
(1.38)
where ϕ0 (x, t) describes the initial gap between the body and the foundation. When and
where we have uν < ϕ0 , there can be no normal forces and so N = 0. When and where
uν = ϕ0 , normal forces may exist. Letting normal forces acting toward the interior of the
ball be positive, we have N ≥ 0. We can summarize the Signorini condition as
0 ≤ ϕ(x, t) ⊥ N (x, t) ≥ 0
(1.39)
where ϕ(x, t) describes the distance from points on the surface of the ball to the rigid
foundation at each time.
Normal compliance contact conditions describe contact with a foundation that is deformable rather than perfectly rigid. In this case we assume N satisfies
N = Pν (uν − ϕ0 )
21
(1.40)
where Pν is a function with Pν (r) = 0 if r ≤ 0. When part of the surface of the ball has
not displaced far enough to contact the foundation, we will have no normal force there as
uν − ϕ0 ≤ 0. When and where we do have contact, the amount of normal force will be
controlled by the function Pν . Notice that we can have displacements that are larger than
the initial gap, indicating deformation of the foundation. A similar contact condition can be
defined to model a damped response in the foundation. In this case we relate N to vν in a
way similar to above.
Friction laws vary greatly in complexity and may introduce significant difficulty in computation models as well as theoretical work. Frictional forces resist motion in the direction
tangential to a surface of contact. Up to a certain threshold, tangential forces can be applied
without causing any tangential displacement. This is known as a stick state. Beyond the
threshold, tangential displacements begin and we say that we have a slip state.
We begin with the simplest friction model. In frictionless contact, we assume contact
forces act only in the normal direction, and so have
σ T = 0.
(1.41)
This model is easy to work with, and in many situations where friction is not expected to
be important it may be used.
Classical Coulomb friction is another friction law that is commonly used. In classical
Coulomb friction, we assume that the upper limit of the frictional force is proportional to
the normal force at each point. Once the force of friction is overcome and slip begins, the
frictional force opposes the motion. Formally, we have
||σ T || ≤ µ|N |,
(1.42)
where µ is called the coefficient of friction. When the friction forces are below the threshold
22
µ|N | we have no tangential velocity. The tangential velocity is described by



vT = 0 if ||σ T || < µ|N |
(1.43)


vT = −λσ T if ||σ T || = µ|N |.
Here λ ≥ 0 is a constant. Depending on the current normal and frictional forces, some parts
of the boundary may be slipping while others are sticking.
1.2.5
Linear Complementarity Problems
Following the development in [10], we provide some basic information about linear complementarity problems (LCPs). An LCP is the problem of finding a vector z ∈ Rn given a
vector q ∈ Rn , a matrix M ∈ Rn×n and the condition that
0 ≤ z ⊥ q + Mz ≥ 0.
(1.44)
This problem is referred to as the LCP (q, M). LCPs are a generalization of quadratic
programming problems and also include bimatrix games as a special case. To see the connection between LCPs and quadratic programs, let’s begin by defining the general quadratic
program. Given symmetric matrix Q ∈ Rn×n , c ∈ Rn , A ∈ Rm×n , and b ∈ Rm , a quadratic
program is the optimization problem
minimize f (x) = cT x + 21 xT Qx
(1.45)
subject to Ax ≥ b
(1.46)
and x ≥ 0.
(1.47)
In the case that Q is positive semi-definite, the function f (x) is convex and the KarushKuhn-Tucker conditions provide necessary and sufficient conditions for x to be a globally
23
optimal solution. That is, x is a globally optimal solution to the program in equations (1.45)
- (1.47) if and only if there exists a vector y ∈ Rm such that the pair (x, y) satisfies
0 ≤ u ⊥ x ≥ 0 and 0 ≤ v ⊥ y ≥ 0
(1.48)
u = c + Qx − AT y and v = −b + Ax.
(1.49)
where
Notice that equations (1.48) and (1.49) can be interpreted as the LCP (q, M) with



T

 c 
Q −A 
q =   and M = 
.
−b
A
0
A solution of this LCP would be a vector z of the form
 
x
z =  ,
y
and thus we see that the LCP and the original quadratic program are equivalent when Q is
symmetric and positive semi-definite.
Depending on the properties of the matrix M, various theorems have been proved regarding the existence and multiplicity of solutions to the LCP (q, M). We will make use of
the following theorem, which establishes existence and uniqueness of solutions when M is
positive definite.
Theorem 1. If M ∈ Rn×n is positive definite, then the LCP (q, M) has a unique solution
for all q ∈ Rn .
Proof of this theorem and further details are available in [10].
24
1.2.6
Finite Element Method
The finite element method (FEM) is a commonly used method for finding approximate
numerical solutions to boundary value problems. Detailed theory and mathematical justification can be found in many textbooks. [23] or [24] are two quality resources, with the first
also offering applications in solid mechanics. The method consists of breaking the domain of
the problem into small elements and looking for solutions to a variational restatement of the
problem over a finite subspace of the true, continuous solution space. FEM uses piecewise
continuous polynomials as basis functions for the finite-dimensional solution space. In the
case of elliptic boundary value problems, FEM reduces the initial problem to a system of
linear equations. Due to the small support of the FEM basis functions, this linear system
will be sparse and generally easy to solve. In the case of hyperbolic boundary value problems
FEM allows discretization in the space variables, but time derivatives still remain. Application of the FEM reduces the initial problem to a system of ordinary differential equations in
this case. To complete a numerical scheme for solving hyperbolic boundary value problems,
we must use another method to discretize in time. The small support of the basis functions
is still beneficial when solving the fully discretized system.
1.2.7
Hertzian Contact Theory
Hertzian contact theory has proved to be very useful in practice. Despite the somewhat unrealistic assumptions used in deriving the results, it is often the case that the theory provides
a good approximation even in situations where the assumptions are violated. Chapters 6
and 7 in [13] provide a good description of Hertzian contact and serves as a quality reference
for the practical use of the Hertzian theory. We restate some basic results regarding the
contact of two spheres which we will later use to compare some of our model predictions to
Hertzian predictions.
Let R1 and R2 be the radii of the spheres and assume they are being pushed together by
a compression force F . As a result of this compression, the spheres form a common interface
25
known as the contact area. We assume a circular contact area and denote its radius a. The
material properties of the spheres are described by their Poisson ratios and shear moduli,
ν1 , ν2 , µ1 , and µ2 respectively. We can then define
A=
1 − ν1 1 − ν2
+
µ1
µ2
(1.50)
1
1
+
.
R1 R2
(1.51)
and
k=
The compression force and contact radius are related by
a3 =
3F A
.
8k
(1.52)
Further, we can determine the pressure distribution in the contact area with the function
r
p(r) = −p0
1−
r2
a2
(1.53)
where
p0 =
4ka
.
πA
(1.54)
Letting z be the distance below the contact plane, the state of stress within each sphere
is given by
"
3 #
√
z
z
(1 − ν)u
1
σθθ
1 − 2ν a2
−1
√
√
+ √ 2ν +
=
1−
− (1 + ν) u tan
−p0
3 r2
1+u
a u
a u
u
(1.55)
σzz
=
−p0
z
√
a u
3
26
u2
u
+ z 2 /a2
(1.56)
"
3 # 3
1 − 2ν a2
z
σrr
z
u
√
√
=
1−
+
+
2
2
p0
3 r
a u
a u u + z 2 /a2
√
1
z
(1 − ν)u
−1
√
√
+ (1 + ν) u tan
−2
1+u
a u
u
√
τrz
rz 2
u
= 2 2
2
−p0
a u +z 1+u
(1.58)
τzθ = τrθ = 0
(1.59)
where
2
(1.57)
2
r +z
2u =
−1+
a2
s
2
r2 + z 2
4z 2
−
1
+
.
a2
a2
(1.60)
From these stress equations, we can compute the maximum shear stress at each point of the
colliding bodies. Figure 1.9 shows the contours of the maximum shear stress immediately
below the contact surface. Note that the distances have been scaled by a and the stresses
have been scaled by p0 . We will later compare the stresses in our contact model with the
Figure 1.9: Contours of scaled maximum shear stress for
Hertzian contact.
27
stresses predicted by the Herztian theory.
In [22], Hertzian theory is used to estimate contact times and maximum contact radii for
impacts between a ball and a semi-infinite block. The Hertzian contact time is
Tc = 3.21 (1 −
νs2 )
+ (1 −
2/5
νp2 )Es /Ep
rv
1/5
Es
ρs
2/5
(1.61)
where νs , Es , and ρs are the shear modulus, elastic modulus, and density of the material
composing the sphere, νp and Ep are the shear modulus and elastic modulus for the block
material, r is the radius of the sphere, and v is the impact velocity. Applying this estimate
to our impact problem, we obtain an estimated contact time of 38 µs.
The maximum contact radius is given by
am ax = r
2/5
15π(Ks + Kp )mv 2
16
1/5
,
(1.62)
where m is the mass of the ball and
Ki =
1 − νi2
Ei
.
The estimated contact radius for our impact problem is .7 mm. From this contact radius, we
can determine the stresses within the ball and compute the maximum shear stresses. Figure
1.10 shows the expected maximum shear stress contours within the ball when the contact
radius is .7 mm. We see that the Hertzian model predicts we will exceed that threshold for
plastic deformation.
28
Figure 1.10: Maximum shear stress contours for predicted contact area.
29
Chapter 2
An Elastic Contact Model and Its Discretization
In this chapter we introduce the numerical scheme we use to solve our contact problem. We
begin with the equations of linearized elasticity, constitutive equations for an isotropic homogeneous material, and the complementarity condition in equation (1.39). We discretize the
equations in space using the finite element method and in time using the implicit trapezoidal
method. In the discretized model, the complementarity condition is enforced by solving an
LCP, and so at each discrete time step we solve an LCP and a system of linear equations.
This results in a vector describing the displacement of each discrete material node at each
time step. From these displacements, we are able to compute the stress state at each node
and determine whether plastic deformation would occur.
We model the results of body and surface forces acting on a material body occupying set
Ω with the system
ρü = f + Divσ for every x ∈ Ω and every t > 0
=
1
∇u + (∇u)T
2
(2.1)
(2.2)
σ(u) = λtr((u))I + 2µ(u)
(2.3)
σ(x)ν = g for each x ∈ ΓN .
(2.4)
Here g describes the density of surface forces acting on ΓN ⊂ ∂Ω and f describes the density
of body forces acting on the body. To model contact with a rigid foundation, we use the
30
Signorini contact condition
0 ≤ ϕ(x, t) ⊥ N (x, t) ≥ 0
(2.5)
and we assume frictionless contact. We assume the only body forces are normal forces due
to contact, and so we can amend equation (2.4) to
σ(x, t)ν = N (x, t) for each x ∈ ΓN .
(2.6)
In our simulation, the material body is a steel ball and it is given an initial velocity v0 toward
the foundation. The physical setting for the contact problem can be seen in Figure 2.1.
Foundation
Figure 2.1: The physical setting for the contact problem. ΓN is the part of the boundary
that may come into contact with the foundation during the collision.
2.1
Discretization
We begin by introducing the variational form of the equation of motion. Finding a function
u which solves the equation of motion is equivalent to finding a function u satisfying the
requirement that
Z
Z
ρü · wdx =
Ω
Z
Divσ(u) · wdx +
Ω
f · wdx
Ω
31
(2.7)
for all test functions w ∈ H 1 (Ω). Here H 1 (Ω) is the space of weakly differentiable functions
over the ball. We now modify this by first considering the divergence term. Using the
divergence theorem and the boundary conditions σν = −N ν, we have
Z
Divσ(u) · w dx =
Z X
∂σij
wi dx
∂xj
XZ ∂
∂wi
(σij wi ) − σij
dx
=
∂xj
∂xj
Ω
j,i
Z X
Z X
∂
∂wi
=
(σji wi )dx −
σij
dx
∂xj
Ω j,i ∂xj
Ω j,i
Z
Z X
Div(σw)dx −
σij ij dx
=
Ω
Ω j,i
Ω
Ω i,j
Z
Z
σw · νds −
=
σ(u) : (w) dx
Z
w · N νds − σ(u) : (w)dx.
Ω
∂Ω
Z
=−
∂Ω
Ω
Thus the variational form of the PDEs can be rewritten as
Z
Z
ρü · wdx = −
Ω
Z
w · N νds −
Z
f · wdx
σ(u) : (w)dx +
∂Ω
Ω
(2.8)
Ω
and we are now ready to apply the finite element method.
Letting φ denote basis functions specific to the mesh to be used, we can define finite
dimensional analogs of u, w, N, and f. We define
uh =
X
ui φi ,
(2.9)
wi φi ,
(2.10)
Ni φi ,
(2.11)
i
wh =
X
i
Nh =
X
i
32
∂Ω
and
X
fh =
fi φi .
(2.12)
i
Inserting these finite dimensional approximations in equation (2.7) and simplifying, we get
the system of ODEs
M ü = −Au − BN + f
(2.13)
where
Z
M=
ρφi φj dx ,
(2.14)
Ω
Z
A=
σ(φi ) : (φj )dx ,
(2.15)
Ω
Z
B=
νφi φj ds ,
(2.16)
∂Ω
and
Z
fφi dx .
f=
(2.17)
Ω
We use piecewise linear elements, and so the basis function φi has a value of 1 at node i
and decreases linearly to a value of 0 at all adjacent nodes and the edges connecting them.
The function has a value of 0 over the rest of the domain. In one and two dimensions, these
are tent functions as depicted in Figure 2.2. Note also that N is a vector containing the
normal forces acting perpendicular to the foundation on the potential contact nodes on the
surface of the ball. Because there will be no normal force outside of this region of potential
contact, we only need to compute B for basis functions corresponding to potential contact
nodes.
We then complete our discretization by applying the implicit trapezoidal method
yk+1 = yk +
h k+1
ẏ
+ ẏk .
2
(2.18)
Because we are simulating a mechanical system, energy should be conserved over time. The
33
Figure 2.2: Basis function for one and two-dimensional FEM using piecewise linear
elements.
implicit trapezoidal method is commonly used for numerically solving ODEs in mechanics
because of its stability properties and its ability to preserve the amplitude of oscillations.
Breaking our system into first-order ODEs and applying this method, we arrive at the fully
discrete iterative scheme
h2
h2
k+1
= M − A vk − hAuk − hBNk+1 + hf
M+ A v
4
4
uk+1 = uk +
h k
v + vk+1 .
2
Here uk describes the displacement of each node at time k and vk =
(2.19)
(2.20)
d k
u .
dt
The time step is
given by h.
2.2
Complementarity
After discretizing our system, the condition (2.5) is transformed into a linear complementarity problem or LCP. Instead of continuous functions describing normal forces and the gap,
we instead have vectors N and ϕ which describe the normal forces on discrete nodes on
the boundary of the ball and the gap between the position of these discrete nodes and the
34
foundation. Then the complementarity condition is stated as
0 ≤ ϕ ⊥ N ≥ 0.
Foundation
(2.21)
Foundation
Figure 2.3: The gap between the ball and the foundation as a function in the
continuous problem (left) and as a vector in the discrete problem (right).
At each time step, we must compute Nk+1 before we can compute uk+1 by our previous
scheme. These normal forces must be complementary to a vector describing the gap between
each potential contact node and the point on the foundation where it may make contact.
Letting ϕ0 represent the distances from each potential contact node to the foundation in the
normal direction at t = 0, we express the complementarity condition
0 ≤ Nk+1 ⊥ ϕ0 − Cuk+1 ≥ 0.
(2.22)
Here Cuk+1 describes the distance each boundary node has displaced in the downward vertical direction at time k + 1, and so ϕ0 − Cuk+1 is the remaining gap between each node
and the foundation at time k + 1. Solving for uk+1 in our iterative scheme, we have
k+1
u
−1 h2
h2
h2
h2
k+1
k
k
= M+ A
M − A u + hM v − BN
+ f
4
4
2
2
(2.23)
and plugging this into our complementarity condition in equation ( 2.22) we arrive at the
35
LCP
0 ≤ Nk+1 ⊥ Mc Nk + q ≥ 0
(2.24)
−1
h2
h2
B
Mc = C M + A
2
4
(2.25)
−1 h2
h2
h2
k
k
q = ϕ0 − C M + A
M − A u + hM v + f .
4
4
2
(2.26)
where
and
We solve this LCP at each time step to determine the normal forces during our simulation.
2.3
Cylindrical Coordinates
As the ball approaches the foundation, contact will occur first at one point. The momentum
of the ball will continue to carry the ball into the foundation as normal forces prevent the
ball from passing through it. The normal forces will deform the ball and this will result in
a further set of points on the boundary of the ball contacting the foundation. For a perfect
ball composed of isotropic, homogeneous material impacting a perfectly rigid foundation,
this contact area is expected to be a perfect disk. We assume that the normal forces applied
to this disk will be rotationally symmetric. Within the contact disk, the normal forces at
all points on a circle of radius r centered at the initial contact point will be equal. Because
of this and our previous assumptions of isotropy and homogeneity and the absence of any
asymmetric body or surface forces, the entire problem has rotational symmetry. The forces,
displacements, and velocities are all invariant under rotation about the ball’s central axis
normal to the foundation. The PDEs we are solving describe the contact problem in three
dimensions, but the unknown displacements can be described with only two coordinates: the
radial displacement ur and the vertical displacement uz . The rotational displacement uθ is
assumed to be 0, and so we have u = ur r̂ + 0θ̂ + uz ẑ. This is reasonable because the only
surface forces are being applied in the direction normal to the foundation. In the cylindrical
coordinate system, these forces are in the z direction and have no rotational component. In
36
our model, this means that there are no rotational displacements.
Figure 2.4: Representative half-disk for describing displacements under non-rotational displacement assumption.
Physical considerations impose a further constraint on our displacements. Any radial
displacements along the central axis of the sphere (or straight edge of the half disk) would
correspond to the formation of a hole in the center of the ball or overlap between different
regions of the ball. These represent nonphysical behaviors, and so we add the Dirichlet
condition
ur = 0 for all nodes with r = 0.
(2.27)
In order to compute the matrix A from equation (2.15), we will need to compute the
quantities (u) and then σ(u) : (w) in cylindrical coordinates. We begin by computing
1 ∂
∂
∂
(ur r̂ + uz ẑ)
+ ẑ
∇u = r̂ + θ̂
∂r
r ∂θ
∂z
∂ur
∂ur
1
∂uz
∂uz
=
r̂r̂ +
ẑr̂ + ur θ̂ θ̂ +
r̂ẑ +
ẑẑ.
∂r
∂z
r
∂r
∂z
The strain tensor is the symmetric part of ∇u and so we have
∂ur
(u) =
r̂r̂ +
∂r
∂ur ∂uz
+
∂z
∂r
1
1
∂uz
(ẑr̂ + r̂ẑ) + ur θ̂ θ̂ +
ẑẑ.
2
r
∂z
37
(2.28)
Here again we can see the importance of equation (2.27) as the θ̂ θ̂ term could approach
infinity as r approached 0 without this condition. From equation (2.3) we can then compute
1
∂uz
∂ur
∂ur ∂uz 1
+
(ẑr̂ + r̂ẑ) + ur θ̂ θ̂ +
σ(u)= λ
r̂r̂ +
ẑẑ
∂r
∂z
∂r 2
r
∂z
∂ur 1
∂uz
+2µ
+ ur +
(r̂r̂ + θ̂ θ̂ + ẑẑ).
∂r
r
∂z
(2.29)
For u = ur r̂ + uz ẑ and w = wr r̂ + wz ẑ we can compute
T 

 ur 
 
u 
 z
 
 ∂ur 
 ∂r 

σ(u) : (w) = 
 ∂uz 
 
 ∂r 
 
 ∂ur 
 ∂z 
 
∂uz
∂z
λ+2µ
 r2

 0


 λ
 r


 0


 0


λ
r
0
λ
r
0 0
λ
r
0
0
0 0
0
0 λ + 2µ 0 0
λ
0
0
µ µ
0
0
0
µ µ
0
0
λ
0 0 λ + 2µ


  wr 
 
w 
 z 
 
  ∂wr 
  ∂r 
 .
  ∂wz 
 
  ∂r 
 
  ∂wr 
  ∂z 
 
(2.30)
∂wz
∂z
Equation (2.30) will be useful when computing the stiffness matrix A.
2.4
Energy Considerations in the Discrete Model
At each time step, we can easily compute the total system energy by summing the kinetic
energy, elastic potential energy, and the potential energy due to body forces. Ignoring body
forces, we have
1
1
E t = KE t + EP E t = (vt )T M v + (ut )T Aut .
2
2
Recalling that according to our discretization scheme we have
ut+1 = ut +
h
2
vt+1 + vt ,
vt+1 = vt +
h
2
v̇t+1 + v̇t ,
38
(2.31)
and M v̇t = −Aut − CNt ,
we can analyze the evolution of the total energy by computing the change in energy E t+1 −E t
as follows.
E t+1 − E t = 12 (vt+1 )T M vt+1 + 12 (ut+1 )T Aut+1 − 12 (vt )T M v − 12 (ut )T Aut
T
2
= h2 (vt )T M v̇t+1 + v̇t + h2 (ut )T A vt+1 + vt + h8 v̇t+1 + v̇t M v̇t+1 + v̇t
T
2
+ h8 vt+1 + vt A vt+1 + vt
T
= − h4 vt+1 + vt
CNt+1 + CNt
T
= 21 Nt+1 + Nt
C T ut − C T ut+1
T
= 21 Nt+1 + Nt
ϕ0 − C T ut+1 − (ϕ0 − C T ut ) .
Because of the complementarity condition (2.22), we know that
(Nt+1 )T (ϕ0 − C T ut+1 ) = (Nt )T (ϕ0 − C T ut ) = 0.
Thus, we have
E t+1 − E t = 12 (Nt+1 )T (ϕ0 − C T ut ) − 12 (Nt )T (ϕ0 − C T ut+1 ).
(2.32)
Equation (2.32) tells us that during the first half of the impact (when normal forces may
be expanding to new boundary nodes), we may see a decrease in energy from one time
step to the next. During the restitution phase of the simulation, we may see an increase in
energy. For small time steps the energy changes were observed to be small and energy was
approximately conserved. Energy analysis for completed simulations is available in chapter
4.
39
Chapter 3
Software Implementation
We implement our discretized description of the impact problem in MATLAB code. The
major parts of this process can be seen in Figures 3.1 and 3.2 and are described in the
following sections. Figure 3.2 is a further explanation of the final step of the flowchart in
Figure 3.1. The process in Figure 3.2 continues until we reach a predetermined end time
after contact between the ball and foundation has ceased.
Figure 3.1: Flowchart describing the major steps of the MATLAB implementation.
The software written specifically for this work is available for download from GitHub at
[26].
40
Figure 3.2: Flowchart describing the computation of the displacement and stresses at
each time step.
3.1
Mesh Generation
Because we are assuming our displacements will be nonrotational, we only need to use
two-dimensional finite elements. For our desired finite element approximation, we must
generate a triangular mesh for a half disk. To accomplish this, we used DistMesh, MATLAB
code written by Per-Olof Persson that is available for download online. This software is
described in [9]. This code iteratively produces high quality triangular meshes based on
input geometry and element size. Geometries are described by signed distance functions,
and code is provided with DistMesh to easily combine simple geometries into more complex
ones. To describe the desired half disk shape, we took the intersection of an appropriate
rectangle and circle. DistMesh also requires a description of desired element size. This is
described as a function, allowing the user to adjust the element density at different parts
of the mesh based on knowledge about the specific application. We used meshes which had
smaller triangles near the contact area as this was the area where violent forces were to be
applied. This allowed greater detail and accuracy near the contact region without some of
the added computational cost of decreasing the mesh size in a uniform mesh. Examples of
uniform and nonuniform meshes as used in our simulations can be seen in Figure 3.3.
41
Figure 3.3: Triangular meshes generated by DistMesh on
a half disk. The left mesh has uniform element size and the
right mesh has smaller elements near the contact area.
3.2
Forming the Stiffness and Mass Matrices
The matrices in equations (2.14), (2.15), and (2.17) are computed using software written in
MATLAB for numerically integrating basis functions over triangular meshes. This software
was written prior to this work by David Stewart and only minor modifications were made
in adapting it for use with cylindrical coordinates. In our three-dimensional problem, the
entries of A and M are volume integrals as we are integrating over the entire ball. Because
we assume displacements have no rotational component, these integrals easily simplify to
integrals over the half disk. In cylindrical coordinates, (2.14) and (2.15) become
Z
M=
ρφi φj dx = 2π
Ω
Z
A=
Z
ρφi φj rdrdz
and
(3.1)
ω
Z
σ(φi ) : (φj )dx = 2π σ(φi ) : (φj )rdrdz ,
Ω
ω
42
(3.2)
where ω is the half disk and equation (3.2) can be further simplified through equation (2.30).
A MATLAB function takes in a representation of the triangulation of the half disk and the
PDE to be solved and returns the appropriate matrices. The two-dimensional integrals are
computed in this function by finding the affine transformation mapping each triangle of
the triangulation to a reference triangle, computing integrals numerically on the reference
triangle, and then using the affine transformation to appropriately modify the resulting
values so that they are correct for the original triangle. It is important to note that integrals
over triangles further from the central axis of the ball will have much larger values because
of the increased volume generated by rotating these elements about the central axis. This
causes poor conditioning in these matrices and necessitates the use of preconditioners when
later solving the linear system. Note that both M and A are symmetric and positive definite
and will also be very sparse.
3.3
Forming the Boundary Matrix
We must also form the matrix B in equation (2.16). The entries in this matrix are integrals
over the boundary of the ball and the integrand includes the outward normal on the sphere,
ν. This integral can be simplified similar to above by our assumption that displacements
have no rotational component. In cylindrical coordinates, equation (2.16) becomes
Z
Z
νφi φj ds = 2π
B=
∂Ω
νφi φj rdrdz ,
(3.3)
∂ω
where ∂ω is the boundary of the triangular mesh representation of the half disk.
Outward normal vectors are computed for each boundary edge that connects potential
contact nodes. This is accomplished by first computing a vector perpendicular to the edge
and then checking to make sure it points outside of the triangle that includes the edge. We
verify this by computing the dot product between the perpendicular vector and another edge
of the triangle. A positive result means that the perpendicular is pointing into the disk and
43
needs to be reversed. The integrals in equation (3.3) are computed in MATLAB again using
code written by David Stewart. We find the affine transformation which maps each line
segment of ∂ω to the reference interval [0, 1]. Integrals for each boundary edge are computed
and then transformed back to account for scaling.
3.4
Computing Displacements and Stresses
With the triangular mesh and necessary matrices constructed, we now focus on the computation of the unknown displacements, normal forces, and stresses. Before we can take the
first time step, we must compute the matrix Mc from equation (2.25). This being done, we
follow the flowchart in Figure 3.2 to sequentially compute our unknowns at each time step
until we reach a predefined end time.
3.4.1
Solving the Linear Systems
In the numerical scheme, two linear systems need to be solved at each time step. We must
determine the vector q from equation (2.26) and the vector vk+1 from equation (2.19). In
both cases, we have systems of the form Ax = b, where A is symmetric, positive definite, and
sparse. Because of this, we use MATLAB’s implementation of the conjugate gradient method
to efficiently solve these systems. As mentioned earlier, the use of cylindrical coordinates
means that the linear systems are poorly conditioned. To decrease the condition number,
we apply diagonal preconditioning through the MATLAB function pcgm.
3.4.2
Solving the LCP
We use an iterative scheme as described in chapter 5 of [10] to solve the LCP. We describe
the method for the general LCP (q, M ). We split the matrix M = B + C where B = L + D.
Here L is the lower diagonal part of M and D is the diagonal. This leads to a solver that is
akin to the Jacobi method for solving systems of linear equations. Starting from an initial
44
guess z0 , we can generate the next iteration zk+1 recursively according to
!!
zik+1 = max 0, zik − Mii−1 qi +
X
Mij zjk+1 +
j<i
X
Mij zjk
.
(3.4)
j≥i
To complete the method, we must define a tolerance and a measure of how close our current
iterate is to being a solution to the LCP. We use the function
r(zk , q, M) = ||min(zk , q + Mzk )||∞ .
3.4.3
(3.5)
Computing and Visualizing Stresses
After computing the displacements for each node according to 2.20, we must compute the
state of stress at each node to determine if and where plastic deformation would occur. To do
this, we first compute the strain tensor at each node. This requires computing derivatives
of the deformation function with respect to our spatial variables according to
=
1
∇u + (∇u)T .
2
(3.6)
Because we are using piecewise linear basis functions, these derivatives do not exist at the
nodes. To estimate the derivatives at a node, we average the derivative values on all faces
adjacent to the node. Once we have computed , it is easy to compute σ by equation (2.3).
To determine whether elastic or plastic behavior would occur at each point, we compute the
principal stresses and compare the maximum shear stress to the yield strength using Tresca’s
criterion (1.34).
To visualize the maximum shear stress contours, we use a MATLAB function called
tricontour. In computes and displays contours for a quantity defined on points comprising a
triangular mesh. This function is available online and was written by Darren Engwirda. It
can be downloaded on the MathWorks file exchange at [27].
45
3.5
Computing Hertzian Distributions
We compare the stress distributions computed in our model with those predicted by Hertzian
contact theory. We use the results of our simulation to determine the contact radius at each
time and then compute the appropriate Hertzian stress distribution for that contact radius.
We determine the times when the number of nodes contacting the foundation changes in our
model. At these times, the contact radius is easy to calculate based on the number of contact
nodes. In between these times, we linearly interpolate to determine the contact radius used
to compute Hertzian stresses. Because we use the number of contact nodes in our simulation
to determine the Hertzian stresses, we are essentially comparing the relationship between
contact area and stresses determined in our model with that predicted by Hertzian theory.
We use a similar method to compare the pressure distribution over the contact area in the
simulation to the elliptical Hertzian pressure distribution.
46
Chapter 4
Numerical Results
We now present the main results of our impact simulation. We begin by explicitly outlining
the parameter values and conditions used in our simulation. We then present numerical
results of the induced normal forces, maximum shear stress field, and total energy in our
system during impact. We compare these results to the predictions of Hertzian contact
theory at different times by estimating the contact area in our simulation and computing
the expected Hertzian quantities for this contact area and our material parameters. Finally,
we present an estimate of the amount of energy that would be dissipated due to plastic
deformation in the collision.
4.1
Parameter Values and Simulation Description
The body is assumed to be a steel ball of radius 1 cm. Since the ball is steel, we take
λe = 121 GPa, µe = 80.8 GPa and ρ = 7850 kg/m3 . We take the yield strength σy = 2GPa.
We designate the part of the boundary of the ball which may contact the foundation ΓN .
We give the ball an initial downward velocity of 2 m/s, assume no body forces exist, and
assume normal forces due to contact as the only surface force.
Because we are using the implicit trapezoidal method for time stepping, we do not have
to strictly enforce a CFL condition. However, the nonsmooth and impulsive nature of the
contact forces means that the propagation of shock waves may be possible in the simulation.
To preserve the possibility of capturing this in the simulation, it is necessary to ensure that
47
our time step is small enough that waves cannot travel further than the width of the mesh
in one iteration. This leads us to choose
∆t ≤
∆x
,
c
(4.1)
where c is the speed of wave propagation in the material being considered. For steel, we
have c ≈ 6400 and in our simulations we set
∆t =
∆x
.
25600
(4.2)
The results that follow are for a simulation with ∆x = 40 × 10−6 m. The LCP is solved with
a tolerance of 10−18 and the conjugate gradient method is used to solve the linear systems
to a tolerance of 10−10 .
In preliminary simulations, some unrealistic numerical oscillations were seen near the
central axis of the ball. Additionally increases in the total amount of energy in the system
were possible from one time step to the next. As the mesh size was decreased, these issues
became less significant. To further decrease these undesirable model features, we added a
small amount of numerical viscosity by modifying equation (2.19) to
h
h2
h
h2
k+1
M + D+ A v
= M − D − A vk − hAuk − hBNk+1 + hf.
2
4
2
4
(4.3)
Here D is the viscosity tensor described in chapter 1 and we take the viscosity parameters
to be λv = 12.1 KPa·s and µv = 8.08 KPa·s.
To ensure uniqueness of solution in our discrete problem, we must have unique solutions to
the LCP enforcing complementarity. Upon constructing the matrix Mc described in equation
(2.25), we found that it is positive definite and so Theorem 1 tells us that the LCP has a
unique solution.
48
4.2
Results for Dynamic Model and Hertzian Comparisons
Figure 4.1 shows that the total system energy is nonincreasing in our simulation. The kinetic
energy and elastic potential energy are easily computed from the mass and stiffness matrices
according to
1
KE = vT M v,
2
(4.4)
1
EPE = uT Au.
2
(4.5)
During the first stages of impact, kinetic energy is transformed to elastic potential energy.
During the restitution phase, this energy is then converted back to kinetic energy as the
ball bounces. The small losses in total energy are attributable to the numerical viscosity
discussed above. Additionally, we see that the amount of elastic energy remaining in the ball
is negligible after separation. This provides evidence that significant energy must be lost to
plastic deformation in reality as experimental steel ball collisions exhibit significant energy
losses. Some energy is dissipated as heat, sound, and light, but these losses are expected
to be minimal. The fact that we do not see substantial energy trapped in vibrations leaves
plastic deformation as the primary reason for energy losses.
Elastic shock waves were observed shortly after impact. These are generated in response
to the sudden application of normal forces and rapid change in velocity at the initial contact
point. Figure 4.2 shows the progression of a shock wave through the ball after impact
at t = 5µs. We see that the wavefront progresses from z = −.01 m to z = .008 m in
approximately 3.2µs, giving a wave speed of 5625 m/s. This is in agreement with the
observed speed of longitudinal sound waves in steel, which ranges from 5500 to 6500 m/s
depending on exact composition. Given that the total contact time in the simulation was
about 45µs, relatively few elastic wave transits are possible. Because of this there is the
potential for some energy to remain trapped in the ball as elastic vibration energy. As
stated before we see only minimal evidence of this, partially as a result of the dampening
49
Figure 4.1: Kinetic, elastic potential, and total energy during the
simulation.
Figure 4.2: Propagation of elastic stress wave at .2µs, 1.2µs, 2.2µs, and 3.2µs after
impact.
effects of numerical viscosity.
In Figure 4.3, we see the evolution of the radial pressure distribution in our simulation
and the Hertzian prediction of the pressure distribution given the contact radius observed
50
in our simulation. These distributions are quite similar, with the only significant deviation
Figure 4.3: Comparison of simulation contact pressures and Hertzian contact pressures
for computed contact radius at various times during impact.
occurring along the central axis of the ball at r = 0. This may be due to the discrete
nature of the simulation or numerical issues with cylindrical coordinates as r approaches 0.
A visualization of the contact pressure over time in the simulation is presented in figure 4.4.
Figure 4.5 shows the computed maximum shear stress during the simulation. At each time
the maximum shear stress was computed by first computing the stress tensor at each node
from the displacements at the current time and then using this to compute the maximum
shear stress at each node. The value presented in Figure 4.5 for the maximum shear stress
is the maximum of these node stresses. We see that soon after contact is established, the
51
Figure 4.4: Simulated pressure distribution over time.
Figure 4.5: Maximum shear stress and yield stress during the simulation.
52
maximum shear stress exceeds the yield strength of steel and so our model predicts that
plastic deformation would occur during impact.
Figures 4.6 and 4.7 show the computed maximum shear stress distribution and the
Hertzian maximum shear stress distribution expected for the computed contact area at
various times of the simulation.
We see that generally the two stress distributions are
Figure 4.6: Maximum shear stress contours 2.6µs and 5µs after impact.
53
Figure 4.7: Maximum shear stress contours at 10µs and 15µs after impact.
quite similar. The computed maximum shear stresses are slightly higher in most locations
than those predicted by the Hertzian model. We can see that the contours for the simulated
stresses spread further than those for the Hertzian stresses. This is particularly important
when we consider the contour describing the yield strength because this describes the portion
of the ball where plastic deformation is possible. We see that the stress contours take dif-
54
ferent shapes in the simulation than in the Hertzian prediction. The simulated contours are
wider and so correspond to larger volumes of material where plastic deformation is predicted
to occur. The larger the plastically deforming region is, the more energy we expect to be
dissipated by plastic deformation.
4.3
Estimation of Energy Loss to Plastic Deformation
Because the maximum shear stress exceeds the yield strength of steel over a portion of
the ball during impact, we know that plastic deformation would occur. Using the plastic
constitutive relation described by equations (1.31) and (1.32), we compute an estimate of the
amount of energy that would be lost to plastic deformation. We let Ep denote the amount
of energy used in plastic deformation. We can write
1
Ep =
2
Z
p : σ
(4.6)
Ω
and so the rate that energy is used in plastic deformation is given by
∂Ep
1
=
∂t
2
Z
˙ p : σ.
(4.7)
Ω
Then using equations (1.31) and (1.32) we can write
∂Ep
1
=
∂t
2
By computing the value of
∂Ep
∂t
Z
1
G(σ, ) : σ =
2µ
Ω
Z
(Pk σ − σ) : σ.
(4.8)
Ω
at each time step from equation (4.8), we obtain the curve in
Figure 4.8. This curve describes the estimated rate that energy is used for plastic deformation
during impact.
−1.25
× 104
µ
joules due to plastic deformation. As mentioned in chapter 1, µ is a viscosity constant that
Numerically integrating this, we get an estimated total energy change of
varies considerably for a given material depending on the exact conditions to which it is
55
Figure 4.8: Estimated scaled rate of energy change due to plastic
deformation during the simulation.
being applied. Taking µ to be between 105 and 106 , we get a range of estimated total energy
lost to plastic deformation from −.125 to −.0125 joules. The total amount of energy at the
start of our simulation is .068 joules, and so even a loss of .0125 joules respesents a significant
part of the total energy. Thus even the lower end of this range is considerably more energy
loss than is expected for our collision based on experiments. This is not unexpected because
once plastic flow begins, its progression will serve to decrease the stresses in the ball. This
lowering of the stress level will lead to lower rates of energy lost to plastic deformation. We
discuss plans to construct a model capable of including the effects of plastic deformation in
the following chapter. This would enable us to more accurately determine the energy lost to
plastic deformation.
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Chapter 5
Conclusion
A number of effects contribute to the decrease in energy observed when a ball is dropped onto
a rigid object. Because of the short timescales involved and the potential for complex internal
behavior, determining the relative importance of these effects is difficult in experiments. The
effects of plastic deformation are particularly difficult to quantify as they take place internally
where their observation is problematic. Classical Hertzian contact theory predicts that the
conditions necessary for plastic deformation would exist during the collision of a 1 cm ball
bearing with a rigid foundation at low velocity. Because of the significant assumptions
required in using the Hertzian theory, we devised a more applicable way of determining
whether plastic deformation occurs during this type of collision. By creating a dynamic
model, we were able to more realistically model impact and investigate elastic waves as
another possible source of energy loss.
We devised a numerical scheme for solving the equation of motion of linear elasticity
in three dimensions while enforcing frictionless Signorini contact conditions and making
minimal assumptions. Because of our assumption that no rotational displacements will exist
during contact, we were able to use two-dimensional finite elements in simulating a threedimensional problem. This reduced computational cost dramatically and as a result we were
able to simulate impact at a resolution of 40 µm near the contact area. To avoid smearing
shock waves generated on impact, we took time steps of 1.56 ns.
We used our model to simulate the impact of a 1 cm steel ball bearing with a perfectly
rigid foundation at 2 m/s. To describe unilateral contact with a rigid object in our model,
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we discretized the complementarity condition preventing the ball from penetrating into the
object. By doing this, we were able to determine the appropriate normal forces for maintaining nonpenetration by solving an LCP at each time step. In summary, we were able to
create a high resolution dynamic three-dimensional elastic impact simulation with minimal
unrealistic assumptions. The model naturally includes dynamic behavior such as shock wave
formation while also determining the displacement and stress states at each node.
Comparing our simulation to the predictions of Hertzian contact theory, we found that the
two approaches give similar results. Our simulation predicted a somewhat longer contact
time with lower pressures and stresses, though the differences were not dramatic. The
pressure distribution for a given contact area in our simulation very closely matched that
predicted by Hertzian theory. The level of agreement between our simulation results and the
Hertzian predictions provides evidence that our approach to modeling contact is reasonably
accurate as Hertzian theory is generally seen to be a good approximation of reality in many
situations. Our model has the advantage that it makes fewer assumptions and includes
dynamic responses in impact.
Using our model, we determined that plastic deformation would occur during impact.
Using Perzyna’s law to estimate the rate of plastic deformation expected in our mode, we
computed an estimate of the amount of energy that would be lost in plastic deformation.
Because of the wide range of dynamic viscosity constants for steel measured in different
experiments, it is difficult to precisely estimate this energy loss. Additionally, when materials
begin yielding, this serves to reduce stress and change future behavior. Because our model
does not account for the actual behavior during plastic deformation, our estimate is likely to
predict more energy loss than what would be observed in reality. We found that negligible
energy remained in the ball as elastic vibrational energy after impact. This points to plastic
deformation as an important part of energy dissipation in collisions. We plan to build a
more sophisticated model capable of simulating plastic effect in order to get a more accurate
estimate of the energy lost to plastic deformation.
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To include plastic deformation in our model, we would need to include constitutive equations describing plastic behavior. Perzyna’s law takes the form of a differential equation
describing the relationship between stress rate, strain rate, and the current stress-strain
state. This equation would be discretized in time for inclusion in our numerical model. This
would result in a time-stepping method for stress and strain such as
σ k+1 − σ k
=C
h
k+1 − k
h
+ G(σ k + 1).
(5.1)
Instead of being calculated purely from current deformations of the body, stress and strain
would become unknowns subject to their own dynamic responses. Up to the elastic limit,
the model would behave as in the elastic model.
Successful inclusion of plasticity would allow a simulation that naturally dissipates energy
as plastic deformation occurs. Such a simulation could be used to provide a more realistic
estimate of the amount of energy used in plastic deformation. Simulations would show the
internal structure of a ball as it deforms during impact. This could also provide insight into
long term accumulations of deformation from repeated contact, an important consideration
in engineering applications. Further work would also be required to determine the most
appropriate value of the viscosity constant µ. It is probable that in reality µ depends on the
current state of deformation, stress, and strain.
To further investigate the bounce height of a steel ball impacting a rigid surface, the
amount of energy converted to heat and light should be estimated. By systematically addressing all the sources of energy loss, we could provide more evidence that plastic deformation is key. We expect that the energy lost to sound and light production is small compared
to the total energy lost. Heating due to external and internal friction should be investigated
further.
In our elastic model and the proposed elastoplastic model, the ultimate purpose is to
provide insights that are difficult to obtain in other ways. Physically measuring the internal
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deformation and stress-strain state of a ball in impact presents difficulties. Additionally,
the lack of complete theoretical understanding of the important processes governing the
behavior of a steel ball in impact means that simulations can provide a needed means of
testing theories and explaining phenomena.
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