FailuremodelinginLS‐DYNA
ofballisticimpact
simulations
Erik Strömberg Master Thesis LIU‐IEI‐TEK‐A‐‐14/02106‐‐SE Department of Management and Engineering Division of Solid Mechanics FailuremodelinginLS‐DYNA
ofballisticimpact
simulations
Author: Erik Strömberg Supervisor at LiU: Bo Torstenfelt Examiner at LiU: Daniel Leidermark Supervisors at GKN: Dennis Rikemanson and Robert Tano Master Thesis LIU‐IEI‐TEK‐A‐‐14/02106‐‐SE Department of Management and Engineering Division of Solid Mechanics
Abstract
The aim of this master thesis is to perceive if the outcome from a ballistic impact can be
predicted beforehand with the help of material testing and finite element simulations. It is also
to check if a new failure behavior has to be calculated for only small differences in material
properties, changes like heat treatment and grain size.
A failure criterion called modified Mohr-Coulomb will be presented and used in this thesis.
The failure criterion will determine when failure occurs in the struck material depending on
stress state. The needed inputs to create failure criterion is difficult to determine with only
experiments, therefore finite element-analysis of the same test specimens are needed to extract
required data. Tensile tests data from a material lab are used to determine displacement until
failure.
Six material test specimen geometries are used in this report to calibrate the failure criterion.
Five of them are used to get different stress states from the tests and the last one is used to
find the stress-strain curves for quasistatic and high strain rate.
A failure locus was calculated with the history of stress state and plastic strain up to fracture,
which was found in simulations. The calculated failure locus fit all material test specimens
within 4-11% depending on element size. Because the failure strain differs depending on
element size three different failure loci were created. One element size was used as a
reference and an element size compensation function was calculated. This makes it possible
for a final simulation of fracture to not be mesh dependent and therefore different element
sizes can be used in a final model.
The failure behavior, element size compensation, stress-strain curves depending on strain rate
and temperature were used to get a good material description and therefore a good failure
prediction.
Simulations of a ballistic impact were conducted to test the material description. The results
show that the lowest predicted velocity found where a crack appeared is slightly higher than
the ballistic test results. A crack is found in the ballistic tests for a velocity somewhere
between 50-60 m/s and the velocity predicted with simulation lie around 63 m/s.
This thesis show that in this case a better failure prediction is found if the material description
and failure behavior is calculated for the new material compilations, that even small changes
like a slightly different grain size and a different heat treatment gives great changes when
predicting failure.
Preface
This report is a result of my master thesis in the field of mechanical engineering done with the
help of GKN aerospace and the research program, National Aviation Engineering Programme
(NFFP). NFFP which is the collaboration between Swedish Armed Forces, Swedish Defense
Materiel Administration and Swedish Governmental Agency for Innovation Systems.
I would like to acknowledge and thank the people involved in my master thesis, and to thank
all the people around me at GKN for all the help given.
I would also like to thank Luleå University of Technology for the testing they have done
making this thesis possible. And a special thanks to my two supervisors, Dennis Rikemanson
and Robert Tano at GKN who have given me the knowledge and support to make this thesis
possible and for the help they have offered me throughout this project. Lastly I would like to
offer my gratitude and appreciation to my supervisor Bo Torstenfelt at Linköping University
for the help and support he has shown me during this project.
Linköping, November 2014
Erik Strömberg
Nomenclature
,
,
, ,
̅
Abbreviations
MMC – Modified Mohr-Coulomb
MC – Mohr-Coulomb
QS – Quasistatic
HSR – High strain rate
FE – Finite element
E1, E2, E3, E4 – One, two, three and four elements through the thickness after symmetry
planes were used.
Contents
1 2 Introduction ........................................................................................................................ 1 1.1 Background .................................................................................................................. 1 1.2 Objective ...................................................................................................................... 1 1.3 Object definition .......................................................................................................... 2 1.4 Approach ..................................................................................................................... 2 1.5 Restrictions .................................................................................................................. 2 1.6 Other considerations .................................................................................................... 3 Teori ........................................................................... Fel! Bokmärket är inte definierat. 2.1 The MC criterion ......................................................................................................... 4 2.1.1 The stress tensor ................................................................................................... 4 2.1.2 The Mohr-coulomb fracture criterion .................................................................. 7 2.2 Modification of the Mohr-Coulomb criterion ............................................................. 7 2.2.1 MMC for plane stress ........................................................................................... 9 2.2.2 MMC practical use and properties ....................................................................... 9 2.2.3 Effect of parameters on the fracture criterion .................................................... 10 2.3 Stress strain curve ...................................................................................................... 10 2.4 Required tests ............................................................................................................ 10 2.4.1 Variation of geometry to generate different stress states ................................... 10 2.4.2 Geometries and triaxiality used in this work ...................................................... 11 2.4.3 The material analyzed in this thesis (Inconel 718) ............................................. 12 2.4.4 Available test data .............................................................................................. 12 2.4.5 The use of ARAMIS system .............................................................................. 14 2.5 Regularization ............................................................................................................ 14 2.6 Ballistic testing .......................................................................................................... 15 2.6.1 3 Just cracked ........................................................................................................ 15 Method ............................................................................................................................. 16 3.1 Geometry ................................................................................................................... 16 3.1.1 Geometry for the QS FE-simulations ................................................................. 16 3.1.2 Geometry for the HSR FE-simulations .............................................................. 16 3.2 Meshing of the specimens ......................................................................................... 18 3.2.1 Element type ....................................................................................................... 18 3.2.2 Shear mesh ......................................................................................................... 19 3.3 Stress-strain curve...................................................................................................... 20 3.3.1 QS ....................................................................................................................... 20 3.3.2 HSR curve .......................................................................................................... 21 3.3.3 The calculated HSR curve used in a simulation and compared to tensile test
results 22 3.4 FE-calculations for the QS-case ................................................................................ 22 3.5 FE-calculations for the HSR case .............................................................................. 23 3.5.1 The displacement used in simulations ................................................................ 23 3.5.2 Mass scaling ....................................................................................................... 24 3.6 4 3.6.1 A and n ............................................................................................................... 24 3.6.2 Fracture locus ..................................................................................................... 25 3.7 Element size regularization ....................................................................................... 25 3.8 Ballistic simulations .................................................................................................. 25 Result ................................................................................................................................ 26 4.1 5 Calculation of MMC fracture parameters .................................................................. 24 Stress strain curve ...................................................................................................... 26 4.1.1 QS ....................................................................................................................... 26 4.1.2 The curve for a strain rate of 1000 s-1 ................................................................ 27 4.1.3 The found HSR curve compared to a tensile test ............................................... 28 4.2 Mass scaling .............................................................................................................. 29 4.3 Simplified geometry for the HSR case ...................................................................... 30 4.4 Fracture parameters ................................................................................................... 31 4.4.1 A and n ............................................................................................................... 31 4.4.2 Calculations of
4.4.3 Failure locus ....................................................................................................... 31 ,
and
........................................................................... 31 4.5 Triaxialitys for different geometries during loading ................................................. 33 4.6 Failure strain for different element sizes ................................................................... 34 4.7 Stress through the thickness ...................................................................................... 37 4.8 Ballistic results .......................................................................................................... 37 4.8.1 Damage from the ballistic impact ...................................................................... 37 4.8.2 Impact force........................................................................................................ 39 Discussion ........................................................................................................................ 41 5.1 Failure prediction ....................................................................................................... 41 5.2 The need for a new failure locus when the material properties change..................... 41 5.3 Sources of error ......................................................................................................... 41 5.3.1 Stress strain curves ............................................................................................. 41 5.3.2 Number of tensile test ........................................................................................ 42 5.4 6 The use of ARAMIS results ...................................................................................... 42 Conclusion ........................................................................................................................ 42 6.1 Answers to questions in object definition.................................................................. 43 7 Future work ...................................................................................................................... 43 8 References ........................................................................................................................ 44 Tableoffigures
Figure 1: Airplane engine ........................................................................................................... 1 Figure 2: Different coordinate system in the space of principal stresses. Cartesian
,
,
, Cylindrical
, ,
and Spherical
, ,
....................................................... 6 Figure 3: All the geometries used in this thesis in the lode-triaxiality plane. The names given
to the geometries are the same as the ones used at LTU. ......................................................... 12 Figure 4: Strain measured with ARAMIS. The figure is for the Optus geometry at the strain
rate 80 s-1 .................................................................................................................................. 14 Figure 5: The geometry used for the shear specimen............................................................... 16 Figure 6: Example of a geometry used in the HSR case. The geometry seen is called FFA in
Figure 2 .................................................................................................................................... 16 Figure 7: Position of the gauge length measurement during testing ........................................ 17 Figure 8: The smaller FFA geometry used so that boundary conditions were easier to apply 17 Figure 9: The models used in simulations. The red line shows the symmetry plane used ...... 18 Figure 10: Lines created to get a better mesh........................................................................... 19 Figure 11:Mesh for Optus geometry ........................................................................................ 19 Figure 12: Shows the mesh for the shear specimen. Between the yellow dots help lines were
made to get a better mesh. ........................................................................................................ 20 Figure 13: Separation of the elastic and plastic region ............................................................ 21 Figure 14. Tensile test results for QS and the HSR test at 1000 s-1. Points to calculate the scale
factor is marked in the figure. .................................................................................................. 22 Figure 15: Plastic strain found in simulations for different displacements plotted on the
simulated force displacement curve. The simulated curve is run up to the fracture strain found
with ARAMIS. The Data is for Optus at a strain rate of 80 s-1. The simulation is for the
element size E2......................................................................................................................... 24 Figure 16 : The new stress-strain curve for QS compared to tensile test and the old curve. The
plot is for Central hole at the strain rate 80 s-1 ......................................................................... 27 Figure 17: New stress-strain curve compared to tensile test and old curve. The figure is for
FFA at QS................................................................................................................................. 27 Figure 18: Stress strain curve comparison between QS and HSR scaled from the QS curve.
The tensile test for both cases is included. ............................................................................... 28 Figure 19: Comparison between results found from the tensile test and simulation of the
Optus geometry at a strain rate of 1000 s-1............................................................................... 29 Figure 20: Displacement comparison between the full geometry and the simplified .............. 30 Figure 21: The best fit curve made with A and n ..................................................................... 31 Figure 22: 3D failure locus with the plane stress case plotted in the plane. ............................ 32 Figure 23: The failure locus in the plane stress case. This is the two dimensional curve seen in
the plane in Figure 20 ............................................................................................................... 33 Figure 24: Triaxiality and failure strain history for the different geometries. The figure also
shows how the geometries lay in the plane stress case. The results are taken for the element
size E4 ...................................................................................................................................... 34 Figure 25: The OPTUS geometry with the same displacement in every figure above. The
displacement used in the figures is the one found from tensile tests. The same legend has been
used in all three element sizes. ................................................................................................. 35 Figure 26: The failure locus for different element sizes at plane stress. .................................. 36 Figure 27: The stress in the direction of the displacement through the thickness of the
specimen ................................................................................................................................... 37 Figure 28: The schematic of the ballistic test at LTU .............................................................. 37 Figure 29: The damage for a plate fired at an initial velocity of 60 m/s .................................. 38 Figure 30: The damage for a plate fired at an initial velocity of 70m/s ................................... 38 Figure 31: Damage levels for the figure to the right, where the value 1 is when an element is
removed .................................................................................................................................... 38 Figure 32: The damage depending on velocity. The damage at the just cracked velocity
predicted with the old failure model is also shown. ................................................................. 39 Figure 33. The force under the duration of impact. The simulated force is compared to impact
test ............................................................................................................................................ 40 Figure 34. The force under the duration of impact. The initial velocity of the plate was 60 m/s
.................................................................................................................................................. 40 Figure 35 The stress over time result from simulations with the full size geometry and the
simplified geometry. The same force has been applied to both specimens. The geometry used
is FFA ....................................................................................................................................... 46 Figure 36: Impact force for 40 m/s........................................................................................... 46 1
1 Introduction
1.1 Background
A modern gas turbine aero engine operates at 1500 – 12000 rpm depending on engine size
and turbine stage, therefore each turbine blade has a high kinetic energy due to the great
rotation speed. In case of a blade failure which may be caused by material defects or foreign
objects in the engine, the casing of the engine need to prevent high energy blade debris from
penetrating and damaging other parts of the airplane. Calculations in this field require a
material description taking the high strain, the high operation temperature of the engine, strain
rate dependency and differences in stress states into account.
The field is called containment and all moving parts in the engine need to be contained. At
GKN the interesting area lies after the combustion chamber where parts made by them are.
This can be seen as the "cone" at the right in Figure 1.
A method using dynamic finite element simulations in the software LS-DYNA exists at GKN
today and the method correlates well to experiments. However, test results from ballistic
experiments performed at Luleå University of Technology (LTU) showed less containment
capability than predicted by GKN. A theory was that the material properties were slightly
different due to small differences in heat treatment and grain size.
LTU has used optical measurement called ARAMIS to measure the strain directly from
tensile tests; this is done by filming the test and calculate the relative position of known
points. Results from ARAMIS have not been frequently used at GKN.
Figure 1: Airplane engine
1.2 Objective
The goal of this thesis is to determine the fracture behavior for a material used by LTU in
ballistic tests; this is done using material tests and simulations. The failure model used in this
report is called modified Mohr-Coulomb (MMC).
The found fracture behavior will then be used to simulate ballistic tests with different initial
velocities. The simulation results in the form of force, damage and fracture velocity will be
evaluated against ballistic experiments. The final goal is to obtain fracture results from
simulations that coincide with experiments, thus confirming the work method.
2
It will be examined if a failure locus based on material testing with the same heat treatment
and grain size as in the ballistic test gives a better result than the existing failure locus for the
same material but for a slightly different heat treatment and grain size.
1.3 Objectdefinition
Is it possible to get a good prediction of an impact test by use of a method based on only
results from material testing?
Is it necessary to calculate a new fracture criterion for the same material, but with a different
heat treatment and grain size?
The used fracture criteria will be examined to answer the following questions:
Does the used failure criterion give a good prediction at every stress state?
If the failure strain is predicted badly for one geometry or some geometries do this give a bad
failure plane?
1.4 Approach
The impact prediction is to be done by use of material tests done at LTU. This data is to be
used with an existing method to determine fracture characteristics.
Failure strain is mesh dependent, this will be handled using scale factors for different element
sizes. Three different sizes will be included in this study.
The stress strain curve for the material will be determined by use of data from tensile tests of
a smooth specimen. To get the behavior after necking the curve will be tuned with FEsimulations of a notched specimen where the curve found from the tensile test will be used as
input.
A best fit will be done to find the fracture parameters. For this function the history of plastic
strain and principal stresses for the element closes to fracture will be used. To know at what
displacement fracture occurs, test data from the LTU testing will be used.
When all the fracture parameters are known these results will be used to calculate the
behavior of fracture. When the complete behavior of the material is known this data will be
used to simulate ballistic tests. The simulated results for force and crack initiation will then be
compared to experimental results.
1.5 Restrictions
Scripts used to perform best fit operations will not be created. Existing scripts at GKN will be
used.
The heat from plastic work will change the temperature of the material and therefore give the
material another behavior. This is taken into account, but how this is done will not be further
explained in this thesis. The compensation done will be based on earlier compensations done
by GKN.
3
Geometry and boundary conditions of the ballistic simulation exist at GKN and the existing
model will be used. No work will be done to evaluate the validity of the existing model.
The simulations indicate that some contradictions exist in the material test data between the
force-displacement and optical strain measurements; this is only noted as further studies
would require firsthand information from the material testing.
The focus in this work is not of how the data from tests are found but on the use of this data,
therefore no deeper analysis of how the ARAMIS-system work will be presented, only a short
summery.
1.6 Otherconsiderations
No gender-based questions have arisen by the work.
This work is done to get a better material description of the engine casing. With a good
material description finite element simulations can be used to calculate a more precise
thickness that is needed and weight on the engine can be saved. With reduced weight of the
engine the wings will be put under less stress and the thickness of the wings and wing
attachment can be reduced. Therefore does a small weight reduction on the engine lead to a
large total reduction? A lighter plane uses less fuel. The reduced weight also means less
material used and therefore contributing to sustainable development.
When it comes to ethical aspects this is done for the civil aircraft industry but the company
also do military material, therefore no ethical questions have arisen.
4
2 Theory
2.1 TheMCcriterion
The Modified Mohr-Coulomb (MMC) fracture criterion is based on Mohr-Coulomb (MC)
which is an old failure model dating back to 1773 and created by Charles-Augustin de
Coulomb. It was later generalized by Mohr around the end of the 19th century and given a
name after them both, Khoei (2005). The original MC method is used to predict fracture in
brittle materials such as concrete and is still used today (Zhao (2000)).
2.1.1 Thestresstensor
To define the stress state at a certain point the Cauchy stress tensor or simply the stress tensor
is used. The tensor is denoted [ and can be written as follow,
.
Three invariants of the stress tensor
are used. The invariants are described with the
principal stresses denoted, , . A variant of the first invariant to the stress tensor is
as follows,
.
Variants of the second and third invariants of the deviatoric stress tensor
:
(1)
can be written,
(2)
and
∙
where
:
det
, (3)
is,
.
(4)
In equation (4)
is the identity tensor and
is assumed. The invariants
described above are ones used extensively for MMC in the literature, examples of this is
found in Bai, Wierzbicki (2010) and Li, Wierzbicki (2010).
A commonly used parameter is the so called triaxiality, which is the dimensionless
hydrostatic pressure. This parameter is denoted and is defined by,
.
(5)
This parameter is used because it is a good way to describe the stress state. Different
geometries will give different ratios between the principal stresses, thus giving different
5
triaxiality. Triaxiality has been shown to be of good use when describing ductile fracture and
will therefore be used in this report.
The Mohr-Coulomb criterion also depends on the lode angle, which is denoted . The lode
angle is related to the invariants through
cos 3
,
with the range 0
(6)
. It is practical to normalize the lode angle to the range
1
this is done with,
̅
1
1
arccos
,
where ̅ denotes the lode parameter.
(7)
̅
1,
6
σ2
z
Deviatoric plane
(π plane)
O’
P
θ (Lode angle)
A
ϕ
ρ
σ1
O
σ3
Figure 2: Different coordinate system in the space of principal stresses. Cartesian
and Spherical , , .
,
,
, Cylindrical
, ,
The state of stress can be described using the triaxiality and the lode parameter. Different
geometries will give different stress states and the parameters will differ. An example is found
in Bai and Wierzbicki (2010) where the authors say that ̅ 0 correspond to the shear
condition.
Later in the report the plane stress case will be used to calculate element size compensation,
0
this because the plane stress relates lode and triaxiality in a unique way. In plane stress
which gives the relation,
cos 3
cos
1
̅
.
(8)
In the case of plane stress it is possible to calculate the failure strain depending element size
and on either lode parameter or triaxiality.
7
2.1.2 TheMohr‐coulombfracturecriterion
To predict fracture the following equation is used
max
where
plane.
,
,
(9)
are material constants and ,
are shear and normal stress acting at a certain
According to the Mohr-Coulomb fracture criterion fracture will occur when equation (9)
reaches a maximum value.
It has been shown by Bai and Wierzbicki (2010) that this equation can be expressed with the
principal stresses in the following way,
1
1
2 .
(10)
When modifying the Mohr-Coulomb criterion, the principal stresses are transformed to the
space of lode-, triaxiality parameter and failure strain. How this is done can be read in detail
in Bai and Wierzbicki (2010).
, ,
The three principal stresses are expressed in terms of
1
and is as follows
,
cos
1
,
cos
1
.
(11)
Equation (10) and (11) then gives the MC criterion expressed with , ,
cos
sin
.
seen below,
(12)
2.2 ModificationoftheMohr‐Coulombcriterion
The Mohr-Coulomb method was extended to not only be able to calculate brittle fracture but
also ductile fracture. Bai and Wierzbicki (2010).
To describe a material undergoing plastic deformation the hardening function caused by
deformation needs to be described. Bai and Wierzbicki (2008) suggested using a generalized
hardening rule with pressure and Lode angle dependency. This has the form
̅ 1
where
,
(13)
8
̅
̅
1
√
sec
√
0
,
0
(14)
1.
(15)
In equation (13) is a material constant, is the strain hardening constant, , , ,
are parameters to describe lode and triaxiality dependency. In equation (15) it is seen that is
related to the lode parameter.
To limit the general yield function, some parameters can be chosen so it becomes well known
0,
1 this
yield conditions. For example if the parameters are put to
corresponds to the von Mises yielding function.
Under monotonic loading, equations (12), (13) and (15) can be put together to form an
expression on the failure strain ( ̅ , this is seen in equation (16),
̅
1
√
∙
sin
sec
√
1
∙
cos
.
(16)
For application purposes the term of the triaxiality depending on the yield surface can be
neglected, this can be done because the effects of the parameters
and are similar. For a
detailed parameter study see Bai and Wierzbicki (2010). The final expression of the modified
Mohr-Coulomb then becomes,
̅
√
∙
sin
sec
√
.
1
∙
cos
(17)
9
2.2.1 MMCforplanestress
Inthespecialcasewhereplanestresscanbeassumed,thelodeanglerelatetotriaxiality
seeninequation(8).AccordingtoLi and Wierzbicki (2010)thefracturemodel getsthe
followingappearanceintheplanestresscase(
1,
0),
∙
̅
,
(18)
(19)
where,
cos
, sin
,
√
√
1
1 .
(20)
(21)
2.2.2 MMCpracticaluseandproperties
In equation (17) there are six unknown parameters , , , , , present. and can be
found using a smooth specimen tensile test, this is described more in detail in chapter 3.6.1.
are two basic MC parameters and needs to be determined with test done up to
fracture Bai and Wierzbicki (2010). The two parameters left ( , ) controls the lode angle
dependency and asymmetry of the fracture locus. Different methods exist to calculate ( , )
and in this report they are left undetermined from tensile test and the last four of the six
parameters are determine from a best fit function of the fracture data, according to Bai and
1 is put,
is described in eq
Wierzbicki (2010). A simplified method exist where
(14), this is called MMC3 in the literature. It is the method used in this report. When all four
parameters are calculated the method is called MMC4, Dunand and Mohr (2011).
Although the modification of the failure criteria is fairly new it is one of the more popular
methods used in the industry. This model is one fracture criterion used at GKN Trollhättan.
The criterion have been confirmed in a lot of different papers, (example Li, Luo Gerlach,
Wierzbicki (2010), Luo, Wierzbicki (2010) and Dunand, Mohr (2011)) for many different
loading cases.
In Bai and Wierzbicki (2010) tests for round bars are presented and it is shown that MMC
predicts these fractures relatively bad, this is because the fracture is more dependent of void
growth and linkages. It is concluded that MMC captures plane stress with high accuracy but
with a little less accurate in thicker test specimen.
10
2.2.3 Effectofparametersonthefracturecriterion
In Bai and Wierzbicki (2010) a parameter study was conducted, the authors found the
following results.
: The parameter
down.
: Higher value of
locus.
scales the fracture locus. If
is increased then the fracture locus is scaled
decreases the triaxiality and Lode parameter dependence of the fracture
: As
increases, fracture strain becomes more dependent on triaxiality and the fracture
locus becomes more asymmetric.
: The parameter scales the fracture in opposite direction compeered to . The parameter
only changes the "height" of the locus and not the shape.
:
introduces asymmetry between negative and positive Lode parameters. Lowering
will increase the fracture strain for lode values below zero.
: With a decrease of
dependence.
the fracture locus will raise, it will also decrease the lode parameter
2.3 Stressstraincurve
The input to LS-DYNA need to be expressed in true stress and true strain, this means that the
force- displacement curve given from tests need to be recalculated. The known force and
displacement can be used to find the engineering stress and strain; this is done with the help of
Eq. (22, 23). Following equations is found in William and Callister (2006):
∙
→
(22)
(23)
In(22) istheforce, istheinitialarea,Δ isthechangeinlengthand istheinitial
length.Equations(22)and(23)arethenconvertedtotruestress‐strainwith:
1
(24)
ln 1
(25)
2.4 Requiredtests
Experiments on specimens with different stress states giving different triaxiality are needed to
be able to determine the fracture parameters, Gruben, Hopperstad Borvik (2012). The stress
states are easiest determined by FE simulations performed on data from tensile tests.
Tests with many different stress states are needed to get a good prediction of the fracture. The
desired result is a best fit plane for fracture strain vs. triaxiality and lode.
2.4.1 Variationofgeometrytogeneratedifferentstressstates
In Bai and Wierzbicki (2010) it is described what geometries and type of tests are needed to
obtain a large variety of triaxiality, this can be seen in Figure 3. The different ranges of
triaxiality are presented below.
11
2.4.1.1 Negativetriaxiality,‐0.3–0
To reach the negative side of the triaxiality range, compression tests are needed. To vary the
triaxiality, different relations between the diameter and height are used. With a lower / ,
where is diameter and height, a lower triaxiality is received. A problem with these tests is
that no fracture will occur and if fracture occurs during compression the stress states will
change before fracture and therefore the triaxiality won’t remain to the desired value.
Gradually a shear fracture will occur through the specimen but this is partly due to the friction
between the specimen and the plates compressing it. It is also hard to simulate compression
tests. The negative triaxialitys are sometimes not calculated and the element distortion
becomes significant unless adaptive methods are used. The values in this range are chosen
manually and are put to approximate and often high values for
0.33with lower failure
strain closer to
0.
2.4.1.2 Lowtriaxialitys0‐0.4
It is important to have at least one test to determine failure strain close to zero triaxiality; this
is because the failure strain from compression is much higher. Without a test to determine
failure strain at zero triaxiality the best fit function will give a much higher value in this area
than in the real case. To have more than just the pure shear will give a little more accurate
failure behavior but is not essential for the total result.
Specimens exposed to pure shear are used to get zero triaxiality. For slightly higher triaxiality
tensile test with combined shear and tensile load are used. Pure shear test will cover triaxiality
close to zero and combined test can cover the low triaxialities.
An important test in the low triaxiality section is the so called central hole, as seen in Figure 3
this is a plate with a hole in the middle. Often materials have a local peek in failure strain in
this triaxiality range and the central hole test is therefore important to find the right height on
the peek.
2.4.1.3 Hightriaxiality0.4–0.95
Higher triaxialities are obtained by different tensile tests. To vary the high triaxiality notched
specimens with different radiuses are used. A notch with a smaller radius gives a higher
triaxiality. The tests can be done with flat or round specimens. A triaxiality of 0.667 can be
found with a biaxial tensile test.
2.4.2 Geometriesandtriaxialityusedinthiswork
In this thesis data from five different tests have been used to get a variety of triaxialities, and
data from a smooth specimen has been used to find the stress strain curve and the parameters
, . Figure 3 shows all geometries and the names used in this thesis; it also explains where
every geometry places themselves in the lode-triaxiality plane.
A test included in Figure 3 but without its geometry displayed is the punch test, this is a
biaxial test that can be done with a ball pushing slowly at a plate until fracture, more about
this can be read in Beese, Luo, Li, Bai, Wierzbicki (2010).
12
Seen in the Figure 3 is that the punch test lay closest or in the area where the fracture assume
to occur for an impact test, this makes the punch test very important. The material model can
give a good prediction on fewer tests (the material model requires a minimum of two tests
according to Bai and Wierzbicki (2010) but when predicting failure for an impact tests, the
punch test should not be excluded. More examples of geometries that are possible to use can
be found in Bai and Wierzbicki (2010).
Figure 3: All the geometries used in this thesis in the lode-triaxiality plane. The names given to the geometries are the
same as the ones used at LTU.
2.4.3 Thematerialanalyzedinthisthesis(Inconel718)
Thomas, El-Wahabi, Cabrera and Prado (2006), say that Inconel 718 is a nickel-chromium
based superalloy. The material has good resistance to corrosion, oxidation, carburizing and
other damage mechanisms at high temperatures. The material has a high weldability and has
high mechanical properties. It is used in gas turbines, rocket engines and turbine blades.
2.4.4 Availabletestdata
The test data available for this thesis was in the form of force-displacement data and optically
measured strain data found with a system denoted ARAMIS. The available data is seen in
13
Table 1 where x denotes existing data. The data available for this work was lacking
information of at what exact displacement fracture occur. In the data given was however,
information of plastic strain at fracture for the optical measurement length used. Triaxiality
and failure strain at fracture was available for the biaxial (punch) test at QS condition.
14
Table 1: Test data available. In the table x marks if data is available for that test geometry and strain rate.
Geometry
FFA
OPTUS
Central hole
Shear
Smooth
Punch
QS
x
x
HSR at 1 s-1
x
x
x
HSR at 80 s-1
x
x
x
HSR at 1000 s-1
x
x
x
x
x
x
2.4.5 TheuseofARAMISsystem
The ARAMIS system is a software for making optical measurements. The technique is often
denoted Digital Image Correlation (DIC) or Digital Speckle Photography (DSP). ARAMIS
calculate the strain by filming the movement of tiny given squares and the relative position
between them, and then according to the positions of the squares a value for the surface strain
is calculated. An example on result from ARAMIS can be seen in Figure 4. The frame rate of
the camera was high enough to get an accurate result even for the specimen that is pulled
during an interval of 500 .
31,6%
30,9%
Figure 4: Strain measured with ARAMIS. The figure is for the Optus geometry at the strain rate 80 s-1
2.5 Regularization
According to Buyuk (2014) the following can be said about regularization.
The plastic description in a non-linear analysis is often not good enough to alone give an
accurate answer for every element size, when simulations are made close to fracture. Often in
these cases an additional mesh regularization algorithm is needed. If the number of elements
in a linear analysis is increased convergence is found, but for non-linear problems this is often
not the case.
15
Different regularization methods exist. In this thesis a method where the result is calculated
for several different element sizes, one of them was chosen as a reference and scaling factors
was calculated for the other mesh sizes. More about regularization can be found in Buyuk
(2014)
2.6 Ballistictesting
When a turbine blade hits the containing shell there is a so called structural impact problem.
An impact is a complicated process and to describe it properly tests alone are often not a
reliable method. Tests are therefore complimented by numerical methods for a more accurate
and dynamic answer according to (Zhang, Chen, Guan, Gao, Li 2013). Numerical calculations
done by modern FE-programs are capable of giving an accurate answer for complex impact
problems. The largest source of error is often not the calculation method but the material
properties used. An issue with dynamic problems is that the material properties often behave
different than in the static state.
An example of this can be found in section 4.1.2 where the stress is higher for a given strain
when the strain rate is higher. Due to the dynamics of an impact it is important to give the
material model a strain rate dependency.
2.6.1 Justcracked
The velocity that gives the first visible crack when a projectile hits a surface is called just
crack velocity or, just cracked. Ballistic tests have to be made to find the velocity where
cracks are visible. These tests are often used to verify the material description.
16
3 Method
The following chapter will describe what geometries were used and how the mesh was done.
It will also mention how the stress-strain curves was determined and how the FE-calculations
were made for both the quasistatic (QS) and high strain rate (HSR) load speed. In short this
chapter contains what has been done to get the final result.
3.1 Geometry
The specimen drawings that the models were made from were supplied by LTU and no
alterations of these geometries have been made in this thesis. The different geometries were
modeled in Hypermesh 10.0.
3.1.1 GeometryfortheQSFE‐simulations
The only test geometry not tested at HSR was the shear specimen. This geometry can be seen
in Figure 5. For the shear specimen a symmetry plane through the thickness was used. The
use of this plane through the thickness was the only modification of the geometry.
Figure 5: The geometry used for the shear specimen
3.1.2 GeometryfortheHSRFE‐simulations
One example of the geometries modeled is seen in Figure 6. In this figure a long top part of
the specimen is seen. The top part was used by the machine to grip the specimen when the
desired speed is reached. Symmetry is used in the xy-plane and xz-plane to reduce calculation
time. No symmetry has been used in the yz-plane. However, other simplifications used are
seen in 3.1.2.1.
Y
z
Z
Top p
X
Figure 6: Example of a geometry used in the HSR case. The geometry seen is called FFA in Figure 3
17
3.1.2.1 Geometrysimplificationforeasierapplicationofboundarycondition
The relative displacement of the two measurement points in Figure 7 was used to find the
boundary condition to implement in simulations. However, when modeling the entire
specimen the displacement to implement at the top part of the specimen was difficult to define
when only the movements of the measurement points were known. A method was trial and
error; this work method is not practical for many different geometries and strain rates.
50mm
Measurement points
Figure 7: Position of the gauge length measurement during testing
If the displacement was to be put directly to where it is measured, a lot of work would be
saved. Therefore a geometry containing only the section between the measurement points in
Figure 7 was made. The smaller geometry was locked on one side of the specimen and the
exact displacement extracted from tests was given to the other side, this was considered to be
a more efficient work flow. In appendix Figure 35 shows that the difference between the
simplified and full geometry is small. The smaller geometry can be seen in Figure 8. The
difference in plastic strain between the simplified geometry and the full size was evaluated.
Figure 8: The smaller FFA geometry used so that boundary conditions were easier to apply
The final geometries used for simulations, the FFA, Optus and Central hole can be seen in
Figure 9.
18
Figure 9: The models used in simulations. The red line shows the symmetry plane used
3.2 Meshingofthespecimens
The mesh was done in a program called Hypermesh. Three different element sizes are used
denoted E1, E2 and E4 which means that the elements are cubes with the length
corresponding to one, two and four elements through the thickness of the model (half the
thickness of the specimen).
It has been previously shown at GKN that it is important to have a good aspect ratio on the
elements. High aspect ratio for the elements close to the fracture zone gives large errors and
fracture will not be predicted properly. The aspect ratio can be calculated with the longest
element length and the smallest element area, a perfect cube has the aspect ratio equal to 1.
The mesh was first created in 2D and solid elements were then extruded in the normal
direction of the 2D mesh. The specimen was divided into smaller areas to mesh, seen in
Figure 10; this made it possible to have elements closer to a cubic shape where fracture
occurred.
To save calculation time the elements were made larger far away from the notch, this is seen
in Figure 11.
3.2.1 Elementtype
The element type used in this thesis was a fully integrated solid intended for elements with
poor aspect ratio and has an efficient formulation.
19
Fracture
Fracture
Figure 10: Lines created to get a better mesh
Figure 11:Mesh for Optus geometry
The failure strain is highly dependent on mesh size; this is because the strain is calculated by
use of the element size. It is therefore very important that the elements near the fraction zone
are of the same size for all geometries.
3.2.2 Shearmesh
The area where fracture occurs lays in a varying geometry and the area had to be divided into
tiny pieces. These pieces were made with the width of one element to get the best possible
mesh. This is seen in Figure 12.
20
Figure 12: Shows the mesh for the shear specimen. Between the yellow dots help lines were made to get a better mesh.
3.3 Stress‐straincurve
3.3.1 QS
In the tensile tests it was possible to measure the force at a given displacement which is
needed to calculate the stress strain curve. The force-displacement data was then used with
equations (22) and (23) to calculate the engineering stress-strain curve up to necking. The
engineering stress-strain was used in equations (24) and (25) to find the true stress-strain. The
mean values from three different tensile tests were used.
Calculating the true stress and strain directly from experiments is only possible up to necking.
After necking, simulations need to be used due to the large area reduction. The smooth
specimens also break close to the necking point, thus giving no distinct localization or drop in
load. For larger strains the smooth specimen data has to be extrapolated or extended using a
notched specimen, where a larger strain can be achieved.
Simulations using the FFA geometry were conducted to determine the last part of the stressstrain curve. A graphical comparison was made between measurement data and simulation
results. Changes were then done to the last part of the stress-strain curve found from the
tensile test until a good match was found between the simulation and force-displacement data
from the test.
The only part of the stress-strain curve that is defined in LS-DYNA is the plastic, a point to
separate the plastic and the linear elastic region was chosen. Different methods can be used to
separate the plastic and elastic region. In this thesis the 0.2% offset method was used to find
the yield strength, which is set to be the point separating the elastic and plastic parts (William,
Callister (2003). This means that a line with the slope equal to the E-modulus where drawn at
0.2% strain. Then the stress where the two curves cross defines the yield limit. This method is
visualized in Figure 13.
21
σ
σy
0,2%
ε
Figure 13: Separation of the elastic and plastic region
3.3.2 HSRcurve
A stress strain curve for high strain rates was determined from data from tensile tests for a
smooth specimen done at a strain rate of 1000s-1. Tensile tests where the grip moves at 20 m/s
was used to reach such strain rate. Because of the high speed and that the deformation was
localized over a 20 mm long section; the higher strain rate had much lower accuracy than for
QS. The test data for QS and HSR can be seen in Figure 14. Figure 14 has a normalized
stress-axis, the axis is normalized to the yield strength according to
. The axis is
normalized because the data used in this thesis is not open to the public. Some parts of the
HSR tensile test results were not used, this is partly because of oscillations but also because
the slope of the elastic part differs greatly from the QS case. This difference should not be
there since the elastic slop is not dependent on strain rate.
Generally the QS results are scaled to a HSR curve with a scale factor. The scale factor is
calculated from the HSR test at typically 10-15% strain. The scale factor in this thesis is
calculated at 10% strain which is the same as has been done before at GKN. 10% strain is
chosen because the measured data in the plastic region has stabilized. It is also well before
necking. The scale factor was found with (26),
%
%
where
,
(26)
is the scale factor and
% is the value of the stress at 10% strain for HSR,
% is the stress for 10% strain in the QS case.
22
The scaling of the HSR curve
1,8
Stress σnorm [‐]
1,6
(0,10; 1,466)
1,4
(0,10; 1,328)
1,2
QS Tensile test
1
HSR tensile test
0,8
0,6
0,4
0
0,05
0,1
0,15
0,2
0,25
0,3
Strain [‐]
Figure 14. Tensile test results for QS and the HSR test at 1000 s-1. Points to calculate the scale factor is marked in the
figure.
If the values from Figure 14 are used in equation (26) the following scale factor is found,
%
%
,
,
1,104.
3.3.2.1 Adiabaticheating
Heat is produced when the material undergoes plastic deformation and with a higher
temperature the material has a different stress strain relationship. This difference was
compensated by increasing the stress with 1-2% for larger strains. There are different methods
to calculate the compensation but no deeper evaluation of this was done. The compensation
used is based on earlier work done by GKN in the same field.
3.3.3 ThecalculatedHSRcurveusedinasimulationandcomparedtotensiletest
results
The same comparison done with the QS stress-strain curve used in a simulation is done to the
found HSR-curve. However, the HSR curve compared is a test for Optus done at 1000 s-1
which means that the tensile tests data have a lot of disturbances. This was only seen as an
interesting comparison and no work on GKN has been done on this before, therefore no
editing of the found curve was done after the comparison.
3.4 FE‐calculationsfortheQS‐case
In
23
Table 1 it is seen that the geometries where HSR data were not available are the shear and
punch specimens. However, QS data with force-displacement up to fracture was available.
Simulations up to the displacement where fracture occurred were performed to get the stress
state and plastic strain history.
The QS simulations are done with an implicit method.
3.5 FE‐calculationsfortheHSRcase
While solving the QS-case the total time that it took to perform the test is not important as
each point is treated as at the static equilibrium and therefore not time dependent. However,
for a dynamic (e.g. HSR) test it is more important that the time of the test until fracture is the
same for calculation as in the measurements. To make the dynamic calculation for the slowest
strain rate takes time, this is because of the relatively long time of the tensile tests and the
short time steps used in calculations. However, it is possible to do mass scaling to shorten the
calculation time.
All HSR simulations are done with an explicit method.
3.5.1 Thedisplacementusedinsimulations
At the end of a tensile test necking gives a rapidly lowered force and a quickly increasing
displacement, this makes it hard to find at what displacement fracture occurs. An illustration
of this is found in Figure 15. It was therefore tested how big effect the uncertain area of the
force-displacement curve has on the plastic strain at fracture. Calculations were made from
the point where the force starts to drop to the point where the strain found with ARAMIS is
reached.
When correlating the analysis models to test results, it would never fit both the ARAMIS
strains and the force-displacement results (if to use the sudden drop in force as the point of
fracture). In Figure 15 the strain is found to be around 23% for this force drop and not the
31,6% found by ARAMIS. Here was a decision taken to proceed by only using the ARAMIS
results for correlation purpose.
Different measurement length used by the ARAMIS-system would give different plastic
strains. The optical measurement length used in the test was between E1 and E2, but because
of uncertainties around how this relate to element size it might also be between E2 and E4.
The element size E2 was therefore used. E2 was then used as a reference to find the
displacement at fracture. With a known displacement at fracture the fracture strain for the
other element could be calculated.
24
Strain result difference
20
18
16
force [kN]
14
12
Tensile test 1
10
Tensile test 2
8
Simulation
6
31,6% strain
4
27,4% strain
2
22,9% strain
0
0
0,5
1
1,5
2
Displacement [mm]
Figure 15: Plastic strain found in simulations for different displacements plotted on the simulated force displacement
curve. The simulated curve is run up to the fracture strain found with ARAMIS. The Data is for Optus at a strain rate
of 80 s-1. The simulation is for the element size E2.
3.5.2 Massscaling
To calculate the case with the strain rate of 1 s-1, which is close to QS, the end time in the
simulations were around a second until fracture which result in long calculation times for a
finer mesh. Calculation times of up to several days was found, this is because of the great
number of short time steps. How long the time steps are depends on the density given in the
FE-program. One way to reduce the total time of the calculations is therefore giving the
material a higher density, this may lead to unwanted effects and a slightly incorrect answer
but for only a small change the results are almost the same. The mass scaling was only used
for the strain rate of 1 s-1. The density used was set to 100 times higher than the given for the
material.
3.6 CalculationofMMCfractureparameters
The following parameters are needed to calculate strain at fracture with varying lode
parameter and triaxiality: , , , , . These parameters are not directly used by the FEprogram but are used to calculate failure strain for every lode and triaxiality, which is
implemented in the FE-program.
3.6.1 Aandn
The first two, A and n, were calculated by use of the stress- strain curve found. A best fit
function of the calculated stress strain curve is used to get the parameters as close as possible
to equation (27) in every point.
∙
(27)
25
3.6.2 Fracturelocus
A best fit of equation (17) for every geometry was used to calculate the fracture parameters.
The data needed for the equation is found in the history of the element with highest strain at
fracture displacement. All principal stresses are needed to calculate triaxiality and lode
parameter according to equation (5) and (7), the plastic strain in this element is also needed.
The fracture parameters were used to calculate the failure strain at different triaxialities and
for different lode-angle at that triaxiality. The lode-parameters used were -1 to 1 with the step
of 0.2, thus giving 11 columns. The lode parameter was then recalculated with (28) to fit LSDYNAs definition of lode. For every column the failure strain was calculated at every
triaxiality shown in (29), the result was a 22x11 matrix covering every failure strain for the
different lode and triaxialities.
cos 3
sin
,
(28)
where is the lode angle and ̅ is the lode parameter and L the lode definition used by LSDYNA.
The used triaxialities was,
[-2, -1, -0.667, -0.64, -0.62, -0.6, -0.58, -0.56, -0.54, -0.52, -0.5, -0.47, -0.42, -0.34, -0.33,
-0.2, -0.1, 0, 0.1, 0.2, 0.3, 0.33].
(29)
3.7 Elementsizeregularization
Since failure strain is mesh dependent a regularization was required, this will make it possible
to use the material description for a mesh with varying element size. It was done with one
element size as a reference. The compensation for the element sizes was by the ratio between
the failure strain of the searched element size and the reference. The value of the ratio varies
with stress state therefore was the compensation described in the plane stress so that it only
varies with element size and triaxiality.
3.8 Ballisticsimulations
The result from the ballistic testing reported in Sjöberg (2014) was used as a validation point
for the derived failure surface. The ballistic simulations were made by a premade model and a
material description produced from the stress strain curve at QS and HSR, failure locus
regularization, and a temperature dependent stress strain curve. The temperature dependent
curve was based on earlier curves found at GKN for Inconel 718 but with a different heat
treatment.
Simulations using different impact velocities were made to find when a first crack appears.
The damage parameter was also increased so that the elements didn’t erode, this so that the
velocity where the damage was equal to one could be found. When the damage is one the just
cracked velocity was found.
26
4 Result
4.1 Stressstraincurve
4.1.1 QS
In 3.5.1 it was described that simulations gave lower plastic strain at fracture using the forcedisplacement data compared to the strain found with the ARAMIS-system. The difference
between the results simulated from the two measurements methods were reduced by altering
the stress strain curve found from the smooth specimen tensile tests. In Table 2 the stressstrain curve found from tensile test is compared to the altered curve. The last two points,
which were not found with LTU experiments, were chosen to give a hardening effect even for
high strain values. The last of these two points were given a very high strain value, this is
done after the method developed at GKN.
Table 2: The new stress strain curve compared to the old one
Plastic strain 0 0,005 Altered Stress σnorm Stress σnorm Original 1
1
1,092
1,076
0,015
1,151
1,126
0,05
1,261
1,244
0,1
1,353
1,336
0,123
1,387
1,378
0,19 1,454
1,445
0,23 5
1,462 1,51
1,462 1,51
The two stress-strain curves were used in simulations and the results in force-displacement
data is compared with tensile tests in Figure 16 and Figure 17. In Figure 16 the altered stressstrain curve gives a result closer to the tensile test of central hole and Figure 17 shows that for
the FFA geometry and another strain rate the original gave a better match.
Force displacement curve
20
18
16
Force [kN]
14
12
10
Tensile test
8
Altered
6
Original
4
2
0
0
0,2
0,4
0,6
0,8
Displacement [mm]
1
1,2
27
Figure 16 : The new stress-strain curve for QS compared to tensile test and the old curve. The plot is for Central hole
at the strain rate 80 s-1
Force displacement comparison 60
Force [kN]
50
40
30
Tensile test
Altered
20
Original
10
0
0
0,5
1
1,5
2
Displacement [mm
Figure 17: New stress-strain curve compared to tensile test and old curve. The figure is for FFA at QS
4.1.2 Thecurveforastrainrateof1000s‐1
It is shown in Figure 18 how the HSR curve is created by scaling the QS curve. The scaling of
the HSR curve does not give a perfect match to test results. However, even if the scaled curve
is little to high it follows the trend of the HSR tensile test. For a better fit it is possible tune
the HSR curve.
28
Resulting stress strain curves for QS and HSR
1,8
1,6
Stress σnorm [‐]
1,4
1,2
QS curve
QS tensile test
1
HSR curve
0,8
HSR tensile test
0,6
0,4
0
0,1
Strain [‐]
0,2
0,3
Figure 18: Stress strain curve comparison between QS and HSR scaled from the QS curve. The tensile test for both
cases is included.
4.1.3 ThefoundHSRcurvecomparedtoatensiletest
Because the HSR stress-strain was scaled from the QS curve, it was of interest to test the
curve by simulating a different geometry. Figure 19 shows the correlation for the Optus
geometry at 1000 s-1. The simulation result gave a more wobbling response than the tensile
test. A good dynamic correlation was not expected due to the selected boundary conditions.
Also the force output from the test machine is filtered to reduce the noise, without the filter
the force from the tensile test result is assumed to wobble as well. Part from that the two
curves follow the same trend.
29
Simulation and test for Optus geometry at 1000 s‐1
25
Force [kN]
20
15
Tensile test
Simulation
10
5
0
0
0,5
1
1,5
2
Displacement [mm]
Figure 19: Comparison between results found from the tensile test and simulation of the Optus geometry at a strain
rate of 1000 s-1
4.2 Massscaling
The longest simulation for the element size E4 took more than 60 hours with considerable
computer power to complete, therefore mass scaling was applied. Time step length in LSDYNA is calculated from the wave propagation velocity, which depends on the density.
The density was scaled by a factor 100 to reduce calculation time. The difference in time and
results are seen in Table 3. Higher scaling on density might be possible but the resulting
calculation time was satisfactory.
30
Density Calculation Plastic
scale
time
strain [-]
factor
1
100
3h 4min
20sec
20min
17sec
First
Force before
fracture [kN]
principal
stress [kPa]
max
0.181625
max
1946.15
max
19.492
0.181625
1946.15
19.492
Table 3: Results for the scaling of density in the simulations for FFA geometry at a strain rate o 1s-1 with the element
size E1
The result in Table 3 are for the strain rate 1 s-1 where the tensile test for FFA took
approximately 400 ms until failure. For the tensile test at 80 s-1 the test time was significantly
lower with 5,7 ms. Because the time step do not depend on the end time a simulation at 1 s-1
takes 70 times longer time to run than at 80 s-1. Also it is not recommended to use mass
scaling at 80 s-1 because of the more dynamics at higher strain rates.
4.3 SimplifiedgeometryfortheHSRcase
Figure 20 shows the difficulty of having the entire geometry compared to only use the
simplified HSR geometry. In the full geometry the displacements from tests are put at the top
of the specimens. It was hard to implement the tensile test result there because the
displacement is measured elsewhere. The closes curve that could be found is seen in Figure
20.
If a displacement is to be put as movement on the top part the value that represent the test data
is not known and different displacement curves have to be tested until one is found close to
the curve given in tests. This would need to be done for every geometry and is very time
consuming.
Figure 20: Displacement comparison between the full geometry and the simplified
31
4.4 Fractureparameters
4.4.1 Aandn
With a best fit to the found stress-strain curve A and n were calculated (see 3.6.1), these
values give the best fit hardening curve in Figure 21. As seen in the figure there are some
small differences between the best fit curve and the curve used in simulations but the shape of
them is almost the same and the deviation was acceptable.
Figure 21: The best fit curve made with A and n
4.4.2 Calculations of , and Histories for plastic strain and principal stresses were extracted from simulations up to
and
in
fracture. These results were then used to best fit the fracture parameters ,
equation (17). Due to element size dependency new parameters were generated for every
element size.
4.4.3 Failurelocus
The found fracture parameters were used in equation (17) and the failure strain depending on
lode parameter and triaxiality was calculated. This is seen as a failure locus in Figure 22.
Because the failure strain
depends on both triaxiality and lode parameter the result is in
three dimensions.
32
Shear
Punch
Optus
FFA
Figure 22: 3D failure locus with the plane stress case plotted in the plane.
The failure locus is a best fit for stress state and therefore the results from all different
geometries will not fit to the surface perfectly. The error for all geometries used are shown in
Table 4, where a value above one is a failure strain history that passes through the surface and
values below are a history coming up short. The maximum error for every element size is seen
in Table 5.
Table 4: The table shows how close the failure strain is to the plane for all the test cases and every element size. The
tests furthermost are marked in bold.
Test
FFA
80 s-1
1 s-1
Optus
1 s-1
80 s-1
Central hole
80 s-1
1 s-1
Shear
QS
Punch
33
speed
E4
E2
E1
1,04192 0,97837 1,03346 0,95807 1,01093 1,04192 0,99785 1,01065
0,99147 0,92311 0,94663 0,92011 0,92199 0,95561 0,92016 1,07988
1,11264 1,02854 0,99504 1,005467 0,88583 0,923785 0,886489 1,114157
Table 5: The largest error for every element size
Element
Largest
size
error
E4
0,042
E2
0,080
E1
0,114
To make the result easier to understand the case of plane stress is used. In plane stress lode
relates to triaxiality according to equation (28) and a plot with failure strain against triaxiality
can be made, see Figure 23. In the literature this is a common way to plot the failure strain. In
Figure 22 the black line in the three dimensional fracture plane is the plane stress case.
In Figure 23 the failure strain is high for triaxialitys less than -0.1. In chapter 2.4.1.1 it can be
read that this is the case of compression.
Figure 23: The failure locus in the plane stress case. This is the two dimensional curve seen in the plane in Figure 22
4.5 Triaxialitysfordifferentgeometriesduringloading
It is seen in Figure 24 that the triaxiality is not constant for the geometries and varies up to the
failure strain. Tensile tests for the areas not covered in the figure are assumed to only give
only a slightly better failure locus, this because the punch test, which is the most important
test, and also because many important triaxialities are covered. Most of the geometries lie in
34
the area, or close by, of a tip or valley, this was known before and the geometries were picked
to give a good curve with few tensile tests.
In Figure 24 it is seen that the results for 1 and 80 s-1 lie close and that the failure strain is
approximately the same.
Figure 24: Triaxiality and failure strain history for the different geometries. The figure also shows how the geometries
lay in the plane stress case. The results are taken for the element size E4
4.6 Failurestrainfordifferentelementsizes
Failure strain depends on element size and in Figure 25 it is shown that larger elements gives
lower failure strain.
35
E4
E2
Figure 25: The OPTUS geometry with the same displacement in every figure above. The displacement used in the
figures is the one found from tensile tests. The same legend has been used in all three element sizes.
Figure 26 displays a higher failure strain for E2 than E4 for negative triaxiality. Similar
results have been seen previously at GKN. It may seem like convergence can be found, but
previous work at GKN shows that this is not the case.
36
Figure 26: The failure locus for different element sizes at plane stress.
The failure locus used in the material description is created for the element size E4. The
difference seen in Figure 26 is compensated with the regularization described in 3.7. This is
done so that the material description can be used in models with different element sizes. The
compensation was made dependent on triaxiality.
The time consumption difference for the different element sizes is eight times. Meaning that
an analysis with E1 takes eight time longer than E2 and E2 takes eight times E4.
37
4.7 Stressthroughthethickness
Figure 27 shows the difference in stress through the thickness of the specimen. The stress in
the figure is in the same direction as the displacement.
Stress difference in the pull direction through the thickness
1,12
Stress σnorm [‐]
1,1
1,08
Stress
1,06
1,04
1,02
1
0,98
0
0,2
0,4
0,6
0,8
1
1,2
Thickness
Figure 27: The stress in the direction of the displacement through the thickness of the specimen
4.8 Ballisticresults
Ballistic results reported in Sjöberg (2014) are used for verifying the calibrated material
model. In the ballistic test performed by Sjöberg (2014) a plate was fired at a stationary rod,
the schematic is seen in Figure 28. Simulations for the velocities 40, 50, 60 and 70 m/s were
used to calculate the just cracked case. The simulated force in the impact was also compared
to the ballistic tests.
Figure 28: The schematic of the ballistic test at LTU
4.8.1 Damagefromtheballisticimpact
The material model used in LS-DYNA includes erosion and fracture, the accumulated damage
is calculated in the material. When the damage in an element reaches one, the element is
removed.
38
Seen in Figure 29 a velocity of 60 m/s is not enough to get a crack although a damage value
above 0,9 is found. The figure also shows a simulation where the velocity was 70 m/s which
is seen to give a crack. The exact value of the damage for different velocities can be found in
Figure 32.
Figure 29: The damage for a plate fired at an initial velocity of 60 m/s
Figure 31: Damage levels for the figure
to the right, where the value 1 is when
an element is removed
Figure 30: The damage for a plate fired at an initial velocity of 70m/s
39
To find the just cracked velocity, erosion in the 70 m/s simulation was turned off and a
damage value above 1 could be found. The damage shown in Figure 32 is seen to be almost
linear to the velocity. From these results the just cracked case was found to be 63 m/s. In
Sjöberg (2014) a crack was found for 60 m/s. Beside what was reported, several shots were
performed at around 55 m/s where four out of six showed a small crack.
Simulated damage
1,4
(80; 1,284)
1,3
1,2
Damage [‐]
1,1
(63; 1)
1
Simulated damage
0,9
Just cracked
0,8
Earlier predicted velocity
0,7
0,6
0,5
35
45
55
65
75
85
Velocity [m/s]
Figure 32: The damage depending on velocity. The damage at the just cracked velocity predicted with the old failure
model is also shown.
4.8.2 Impactforce
The force during impact for a velocity of 50 m/s is presented in Figure 33. In the figure the
force is predicted with good accuracy for 50 m/s. The maximum value is predicted well. The
only part of the curve with less accuracy is between 140 and 170 µs, however this is after the
major impact and is not an area of grate importance. The prediction of maximum force is
more important. Overall a good similarity is found. The tests and simulations for 40 m/s show
the same trend, this results can be found in appendix.
40
Impact force for 50 m/s
18
16
14
Force [kN]
12
Shot 1
Shot 2
Shot 3
Shot 4
Shot 5
Simulation
10
8
6
4
2
0
0
50
‐2
100
150
200
250
Time [µs]
Figure 33. The force under the duration of impact. The simulated force is compared to impact test
Figure 34 show that when increasing to 60 m/s, the correlation of the force time history was
not as good, the calculated force was too high. The over prediction of the force may come
from the absence of a crack in the simulations.
Velocity of 60 m/s
22
Force [kN]
17
Shot 1
Shot 2
Shot 3
Shot 4
Shot 5
Simulation
12
7
2
0
‐3
50
100
150
200
Time [µs]
Figure 34. The force under the duration of impact. The initial velocity of the plate was 60 m/s
250
41
5 Discussion
5.1 Failureprediction
The results in this report show that it is possible to get a fairly good prediction at which
velocity fracture occurs. A failure criterion based on data from the same material used in the
ballistic tests gives a much better prediction, thus a new failure criterion is needed if changes
in heat treatment and grain size are made.
When the just crack velocity for ballistic test is approximately 55 m/s and the predicted 63
m/s, the error is around 15%. The calculated failure velocity is lower in this thesis compared
to earlier results at GKN because a new failure locus is used. A failure locus that is calculated
from the same material with the same heat treatment and grain size. This means that even if
the material is the same a new failure locus is needed if small changes are made to the
material treatment.
The force at 60 m/s is predicted to be much higher in the simulation than the ballistic test, this
might be because of the fracture that occurs in the ballistic test and not in the simulations. The
highest value of the force found in the ballistic test for 50 and 60 m/s is almost the same, this
might also be the case for simulation at higher velocities.
The prediction done in this report is made optical strain results. If these results can be used in
this thesis the results from optical measurements can probably be used in other cases. The
optical measurement can therefore be used in tests at the higher strain rates where forcedisplacement data becomes more unreliable.
The MMC failure criterion has shown to give a reliable prediction of failure. It is also a
criterion that can be used for many different stress states. This means that the failure locus can
be used for many different impact cases which is preferable when dealing with turbine blades,
this is necessary because at turbine blade failure the shape of the debris is varying.
5.2 The need for a new failure locus when the material properties
change
The large difference between the prediction in this thesis and earlier work is that the resulting
failure locus in this thesis can only be used for this material with these specific material
properties. If the material changes in any way, even small changes, it is an open question if
these changes will affect the failure locus. A new failure locus is needed to get reliable results.
5.3 Sourcesoferror
5.3.1 Stressstraincurves
The stress-strain curves both for the QS and HSR case would be different if they were finetuned compared to other geometries than the ones used. To eliminate this error different
curves for different triaxialitys are needed.
The curves are also corrected for strains hard to achieve in tensile tests. These corrections
may add small errors compared to the real material properties.
42
5.3.2 Numberoftensiletest
The data used in this thesis only contains one tensile test for every geometry with a few
exceptions. The small amount of tests lead to uncertainties if the result for every geometry
was representative. Tensile test at a higher strain rate sometimes differ a lot and only with
three or four tests can only a trend be found. For a more accurate failure locus the simulations
should be run on a mean value from several tensile tests.
5.4 TheuseofARAMISresults
When the force-displacement from testing used in a simulation didn’t give the same failure
strain that the ARAMIS system did, one of three choices had to be made. One was to trust the
ARAMIS results and scale the displacement data until fracture strain from ARAMIS was
reached in the simulations. Second was to only trust the force-displacement data and ignore
ARAMIS results. Earlier work done at GKN had been based on force-displacement because
no ARAMIS pictures were available. The third opinion was to change the material properties
used in simulations until the failure strain found in ARAMIS was the same as found by use of
force –displacement data.
The third alternative was tried but the difference in strain was too big and a similar result gave
large changes to the material properties. To completely change the material properties was
determined to be wrong and a combination of alternative 1 and 3 was used. The choice to
change the material description was an assumption necessary to continue the thesis when the
early simulations did not give the answer found with ARAMIS. The assumption may give
some errors but with ballistic test results the final failure locus can be tested and discarded if
the difference is too big.
6 Conclusion




MMC has predicted the outcome for the ballistic test with good accuracy. It has been
shown that this can be done with just a few material tests and simulations.
The result from the different initial velocities of the ballistic test is predicted well
when it comes to the force and the just cracked case. However, due to fracture for a
slightly higher velocity in the simulation, the force for 60 m/s is not predicted correct.
Earlier calculations at GKN with a failure locus for a different material treatment gave
a good result for the ballistic test, but that failure locus do not predict well for other
material treatments. Because the new failure locus was based on the same material
properties in the ballistic specimen as in the tensile tests used to determine the failure
locus, a good result is found. The results in this thesis confirm GKN:s work method
and show that a new failure locus is needed if changes are made of the material
properties.
In this case the failure locus based on ARAMIS results give a good prediction of
failure.
43
6.1 Answerstoquestionsinobjectdefinition
Is it possible to get a good prediction of an impact test by use of a method based on
only results from material testing?
The impact in this thesis is predicted fairly well and the velocity for just cracked is
almost within 10% compared to ballistic test, which means that it is possible to predict
impact fracture with data based on material testing.
Do the used failure criterion give a good prediction at every stress state?
It was found in the literature that MMC gives an excellent prediction for thin
specimen, but the accuracy is less for round bars. It is concluded that the prediction is
worse for cases where fracture depend on void growth.
If the failure strain is predicted badly for one geometry or some geometries do this
give a bad failure plane?
It has been shown that the failure locus is calculated as a mean result for several tests.
If a stress state gives a slightly wrong value and the rest of the results are good, then
the resulting failure locus is not affected much. However, if one of the geometries give
a very wrong failure strain the best fit function will give a result with the highest error
as small as possible which will move the whole locus in the direction of the bad test.
Is it necessary to calculate a new fracture criterion for the same material but different
material treatment and grain size?
A failure locus exists before this thesis at GKN, for the same material but with a
slightly different heat treatment and grain size. This locus did not predict the just
cracked case very well. However, the failure locus in this thesis is calculated by use of
material tests from the same material with the same heat treatment used in the ballistic
tests. Results show that the just cracked velocity is predicted fairly well, it is therefore
needed to calculate a new failure locus when changes are made to the material
treatment.
7 Futurework
Of interest would be to use a constant velocity as boundary condition and the whole geometry
when simulating the HSR case. The simulation would be run until the strain found with
ARAMIS is reached, this for the element corresponding to the optical measurement length.
In this thesis the results from ARAMIS are used as a way of finding when the specimen in the
simulations reached failure. If the strain found with ARAMIS and the force-displacement data
is used separately two failure loci could be calculated and a comparison would be possible.
The failure locus in this report is calculated for room temperature. With data from high
temperature tensile test a failure locus could be calculated for the temperature that the engine
operates at.
44
8 References
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46
Appendix
A.1:Simplifiedgeometry
Stress time comperison
1,8
1,6
Stress σnorm [‐]
1,4
1,2
1
0,8
Simplified
geometry
Full geometry
0,6
0,4
0,2
0
0
0,002
0,004
0,006
0,008
time [s]
Figure 35 The stress over time result from simulations with the full size geometry and the simplified geometry. The
same force has been applied to both specimens. The geometry used is FFA
A.2:Ballisticresult
Impact force for 40 m/s
14
12
10
Axelrubrik
Shot 1
8
Shot 2
Shot 3
6
Shot 4
4
Shot 5
Simulation
2
0
0
50
‐2
Figure 36: Impact force for 40 m/s
100
150
Axelrubrik
200
250