Numerical simulation of fatigue
failure in polymer glasses
D. de Kanter
MT04.28
March 2004
TU/e Internship Report
March 2004
Coaches:
ir. R.P.M. Janssen
dr.ir. L.E. Govaert
Eindhoven University of Technology
Department of Mechanical Engineering
Material Technology group
Abstract
This study focuses on the ability of the altered Leonov-model as suggested by
Klompen [1] to predict time to failure for creep and fatigue. The model that
was altered to describe the eects of softening and aging, will not be used to
its full potential. The assumption is made that the eects of aging are to be
neglected at the timescales investigated. The model is tested on two, dierent
numerical methods and for two types of polycarbonate.
The tests performed were tests on creep and fatigue. The fatigue was applied
by a sawtooth shaped dynamic stress signal on the polycarbonate. The time to
failure was dened as the moment of necking or brittle rupture of the material.
The amplitude of the dynamic stress signal was varied to determine inuence
of the amplitude on the time of failure.
The second part of the study introduces an acceleration factor based on the
simplications of the model as suggested in this study. The acceleration factor
is the analytical contribution of the dynamic stress in the total stress signal.
The acceleration factor is used to predict the time of failure for fatigue based
on the time of failure for creep.
Correspondence between the numerical simulations and experiments was achieved
for creep. The fatigue predictions did not appear as accurate as the predictions
for creep. The acceleration factor proved to accurately predict the numerically
simulated times of failure for fatigue. The eects of aging are not to be assumed
negligible in case of fatigue. Aging plays an important role in the explination
for the dierences between the experiments and the numerical simulations.
i
Contents
Introduction
1
1
3
Materials and methods
1.1 Materials . . . . . . . . . . . .
1.1.1 Specimen standards . .
1.1.2 Material properties . . .
1.2 Experimental setup . . . . . . .
1.2.1 Tensile tests . . . . . . .
1.2.2 Creep and fatigue tests
2
3
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3
3
4
4
4
4
Numerical simulation
6
2.1 MARC n Mentat element simulation . . . . . . . . . . . . . . . . . . . .
2.2 Matlab simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Acceleration factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
6
7
Results
9
3.1 Results of the experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Results of the simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3 Experiments and simulations . . . . . . . . . . . . . . . . . . . . . . . . 11
Bibliography
A Tensile test results and
17
Da -values
18
B Determination of the plastic strain rate formulation
19
C Analytical solution accumulative plasticity
20
ii
Introduction
The polymers that are used in productional processes have dierent thermal histories.
The thermal history aects the macroscopic behavior of a polymer [1] and to predict
the macroscopic behavior of a specic polymer this history must be know. For the prediction of the macroscopic behavior tools have been developed. One tool that is used,
is the Leonov model [1] which is capable of relating the thermal history to macroscopic
behavior. The correspondence between the numerical methods and tensile, compression and creep tests has been proven by Klompen [1].
Products are made for a variety of purposes. Many of those purposes are applications
in which the product endures stresses of statical, but also dynamical form. Dynamic
stress can lead to fatigue. During a course project (Appendix ??) the altered model
proved to be able to predict time to failure for creep in polycarbonate. In this study
attention is paid to the use of this model under fatigue conditions. The model is used to
predict the failure of polycarbonate that is subjected to a static stress combined with
a sawtooth shaped dynamic stress. The inuence of the amplitude of the dynamic
stress on the time of failure compared to creep, is discussed. A simplied method of
predicting fatigue rupture is introduced. The simulations are veried with experiments.
The Leonov model
In the Leonov model as suggested by Klompen[1] a state parameter to describe the
thermal history is introduced. The state parameter is the dierence between the present
state and the rejuvenated state as is shown in Figure 1. In the used model aging is not
taken into account.
The numerical simulations performed in this study are on base of the denition of
the plastic strain rate. By dening an accumulative plastic strain the inuence of the
loading over the time can be calculated. The assumption of a maximal allowable strain
makes it possible to determine a point of failure. Prior simulations and experiments
proved the model to be sucient to predict creep rupture under these failure assumptions (Appendix ??). The derivative of the plastic strain rate is dened by (1). This
equation is derived as is depicted in Appendix B.
_ pl =
1
exp
2AT
s s !
u (p3+) D (1+(s exp( ))s1 ) 2s1 1
a
0
pl
30
(1)
In the formulation pl is the accumulated plastic strain at a moment in time. AT ; s0 ; s1
and s2 are tting parameters suggested by Engels [2]. Da is the state parameter as
1
2
CONTENTS
Figure 1: Visualization of the state parameter. The state parameter values the
dierence between the aged polymer (solid line) and the rejuvenated polymer
(dashed line).
shown in Appendix A. The strain rate dependency (0 ) and the pressure dependencies
() are listed in table 1.1. The other parameters are listed in table 1. The state parameter (Da ) can be determined by the data from a tensile test following the method
of calculation shown in Appendix ??. For the numerical simulations the value of the
state parameters were determined from the experimental data.
A0
s0
s1
s3
Polycarbonate
1012
[s ]
0,965
[ ]
50
[ ]
-5
[ ]
Table 1: Material properties for polycarbonate for numerical simulations. Obtained from the internal report of Engels [2].
Chapter 1
Materials and methods
The validation of numerical simulations was done on base of comparison with experimental data. The experimental data for this study was obtained from tensile, creep
and fatigue testing. The data obtained from the tensile tests was used for determining
the parameters needed for numerical simulation as will be discussed in Chapter 2. The
data obtained from the creep and fatigue tests was used for validation of the numerical
simulation results. The experimental obtained data is given in Chapter 3. The choices
for the used material, specimen shape and the experimental setups are discussed in
this chapter.
1.1
Materials
For creep and fatigue rupture experiments use was made of two types of polycarbonate. Polycarbonate was chosen as testing material on the base of prior experiments
(Appendix ??). The polycarbonate showed in prior experiments a stable necking or
brittle rupture behavior which formed a good indication point for a time of failure due
to creep or fatigue.
Use was made of Bayer Makrolon and G.E. Lexan 101. The Makrolon specimens
were obtained from a provided extruded sheet. The choice for an extruded sheet was
made to obtain specimens with only little orientation. The material is assumed to
be homogeneous during the experiments. Orientated samples can show dierences in
behavior compared to not orientated materials. Due to a undersize in Makrolon during
the experiments a shift had to be made to Lexan 101 that was available. The Lexan
101 specimens were injection molded.
1.1.1
Specimen standards
Both types of material specimens were shaped following the ISO 527 standard. To
obtain this standard, the Makrolon specimens were machined by a milling procedure.
The ISO 527 standard showed problems during prior experiments. Investigation of
the material with polarized light showed two local concentrations of residual stress in
the ISO-norm specimens and the specimens used in prior testing. Local stress areas
proved to have been formed during the mechanical interventions on the material that
were done for acquiring the ISO-norm shape. The combination of milling and the
strong round o in the material standard could be pointed out as the cause of the
3
CHAPTER 1.
MATERIALS AND METHODS
4
development of these concentrations of stress. The stresses in the material would not
lead to a shift in the data, but only to an induction of placed failure in the broader part
of the specimen. Specimens that were obtained after the internship and were formed
following the ASTM D638M norm proved to be free of residual stress concentrations.
1.1.2
Material properties
The material properties needed for the experiments were obtained from the internal
report of Engels [2] and the report of Klompen [1] and are given in Table 1.1. The material properties that indicate the present state of the material were obtained by tensile
testing. The determination of the state parameter (Da ) is depicted in the appendices
of the course-report in Appendix ??. Apart from the state parameter the rest of the
material properties are similar for both types of polycarbonate.
E
0
Gr
Polycarbonate
2000
[MP a]
0,03
[ ]
0,37
[ ]
0,7
[MP a]
26
[MP a]
Table 1.1: Material properties of polycarbonate
1.2
1.2.1
Experimental setup
Tensile tests
The tensile tests were performed on the servo-hydraulic MTS Elastomer Testing System
810 and 831.10, both with a controlled temperature chamber. The specimens were
tested on constant linear strain rates of 10 4 , 10 3 and 10 2 s 1 . The specimens were
tested on the temperature of 23; 0o Celsius (0; 1o Celsius). To prevent temperature
dierences between the temperature chamber and the specimens, the specimens were
acclimatized for 15 minutes in the chamber before testing. The results of the tensile
tests have been placed in Appendix A.
1.2.2
Creep and fatigue tests
Creep and fatigue rupture tests were performed on both types of materials discussed in
the section Materials. Both materials were subjected to a static load and in case of the
fatigue measurements the static load was complemented with a dynamic contribution.
The tests were performed on the servo-hydraulic MTS Elastomer Testing System 810
and 831.10, both with a controlled temperature chamber. Failure of the polycarbonate
due to it is loading was dened as the moment in time on which necking or brittle
fracture occurred in the polymer. The fatigue and creep-rupture measurements were
done at the same temperature (23o Celsius) as the tensile tests and under the same
CHAPTER 1.
5
MATERIALS AND METHODS
specimen conditions.
The fatigue was formed by a sawtooth shaped stress signal with predened amplitude
and frequency as shown in Figure 1.1. The sawtooth shape was chosen because of the
analytical possibilities that will be discussed in Chapter 2. The amplitude was varied
from 0 MPa (creep) up to 10 MPa. Makrolon was measured at the amplitudes of 4
and 6 MPa. Due to an undersize the series for Makrolon up to 10 MPa in steps of 2
MPa were aborted after 6 MPa. The Lexan 101 was tested at the amplitudes of 5 and
10 MPa.
Reproduction of fatigue stress signal
65
Applied static stress [MPa]
60
σd
σm
55
50
45
40
0
0.5
1
1.5
Time [s]
Figure 1.1: Visualization of the fatigue signal. The state parameter is a parameter indicating the parent state of the aged polymer (solid line) compared
to the rejuvenated polymer (dotted line).
All measurements have been done with the maximal stress under the value of the yield
stress measured by tensile tests on the strain rate of 10 4 s 1 . The frequency was in
all experiments 1,0 Hz. This refers to a maximum strain rate at the largest amplitude
in the order of 10 2 s 1 . Simulations, with adiabatic conditions, showed that the assumption of no heating up of the specimen was reasonable.
Chapter 2
Numerical simulation
This chapter will discuss the numerical simulations performed and the simplications
that were made.
2.1
MARC
n
Mentat element simulation
For numeric simulations the Leonov model is used in combination with the nite element program Marc n Mentat. By use of the parameters from table 1.1 and 1 a
simulation is performed for a tensile bar to compare the numerically estimated parameter ,Da , with the experimental determined Da . Due to simplication of the reality in
the model, the measured value of Da has to be adjusted in the numerical simulations to
predict the material behavior correctly. From the simulation of a tensile test the value
of the state parameter Da for numerical simulations can be determined. For Makrolon
the value of Da for numerical simulations was determined at 35,7. For Lexan 101 the
Da value was determined at 31,4. Both dierent from the experimental values as shown
in Appendix A.
Prior numerical simulations were performed with the mesh of a quarter of the specimen
bar using its symmetry for modelling the whole specimen. To predict the failure of a
material it is not necessary to make a mesh of the full specimen. The failure starts in all
elements at the same time in case of a homogenous material with constant dimensions
of the modulated specimen. Because failure is based on the accumulative plastic strain
in every element apart and in this survey was the only point of interest, the mesh was
simplied. Considering the equality of elements, only one element has to be calculated
to predict the failure of the whole specimen. The mesh was reduced to a single element
with axial, rotational symmetry. Simulations were performed for Lexan 101 for the
same conditions as the experiment considered in Chapter1.
2.2
Matlab simulation
Interested in the moment of failure the model still knows opportunities for further simplication. Knowing the model to calculate the accumulative plastic strain rate in each
nodal point of the single element, reducing the element to a single point should lead to
6
CHAPTER 2.
7
NUMERICAL SIMULATION
same results. Reduction of the mesh to a single point makes a nite element program
is no longer needed for the calculations. The calculations can now be taken over by a
mathematical program like Matlab that was used in this survey. In the Matlab simulation attention is paid to the accumulative plastic strain and the elastic part of the
material behavior is neglected. This should lead to a small decrease in the predictions
of the time to failure.
2.3
Acceleration factor
Not all parts of the used denition play a signicant role. The formulation can be
parted into two contributions. The rst part describing the inuence of the stress.
The second part describing the evolution of the state parameter. A constant state
parameter in the order of 30 would only have a small inuence on the evolution of _ p l
for the value of exp Da approaches to 1. For the determination of the acceleration
factor the state parameter is neglected.
_ pl =
1
2AT
exp
(p3+) u
30
s2 s1 !
Da (1+(s0 exp(pl ))s1 ) s1
exp
(2.1)
For the further simplication the assumption is made that aging does not occurs on the
time scales investigated. The formulation will therefor reduce to a more workable form.
_ pl =
1
2AT
exp
(p3+) u
30
(2.2)
Considering the formulation (2.2) a new parting can be suggested. By parting the
applied stress (u ) into a static (m ) and a dynamic contribution (d ), as shown in
(2.3), the denition of (2.4) is obtained. This denition shows the contribution of the
dynamic stress as an acceleration factor on the statical stress. In the formulation,
g (f; t) is the relation describing the fatigue signal as a function of frequency and time.
In this survey a sawtooth was used as is shown in Figure 1.1.
u = m + d g (f; t)
_ pl =
1
2AT
exp
(p3+) m
30
exp
(2.3)
g(f;t)(p3+) d
30
(2.4)
The introduction of an acceleration factor makes it possible to analytically predict the
fatigue measurements after only measuring or numerically simulating creep. The acceleration factor will be formulated as the rate in accumulative plastic strain contribution
in a specic time period for fatigue and creep.
R
_ pl (u ) dtcycle
az (m ; d ; f ) = R
_ pl (m ) dtcycle
=
Z
_ pl (d ) dtcycle
(2.5)
In case of creep the acceleration factor will be of the value of 1. In case of an amplitude
larger than zero the factor will increase. The combination of (2.4) and (2.5) can yield
CHAPTER 2.
8
NUMERICAL SIMULATION
Acceleration factors for different signals
3
10
Acceleration factor [−]
Sawtooth
Block
Sinus
2
10
1
10
0
10
0
2
4
6
8
10
12
Amplitude of dynamic stress [MPa]
Figure 2.1: The evolution of the acceleration factors for the dierent signals.
a denition that describes analytically the contribution of a sawtooth shaped stress
signal with dened frequency. The derivation of (2.6) has been depicted in Appendix
C.
az =
30
2d ( 3 + )
p
sinh
(p3+) d
30
(2.6)
The acceleration factor for a sawtooth shaped signal is now only a function of the pressure dependency , the model parameter and the dynamic stress amplitude d . The
formulation of the acceleration factor gives the opportunity to view the development
of the acceleration factor by increasing amplitude of the sawtooth shaped stress signal.
A similar analytical formulation can be derived for dierent types of dynamic stress
signals. A denition for the contribution of a block-shaped dynamic stress signal is
given in Appendix C. Mathematical problems arise for the calculation of an analytical
solution for a sinus shaped stress signal. Gaining a good analytical solution for the
sinus shape could lead to expressions for every type of dynamic stress by performing a
Fourier transformation on the signal and summarize the dierent constitutional sinuses
into a acceleration factor.
Chapter 3
Results
This chapter covers the results obtained from the experiments discussed in Chapter 1
and the numerical simulations in Chapter 2.
3.1
Results of the experiments
The Makrolon and Lexan 101 specimens were tested on MTS Elastomer Testing Systems. Makrolon was tested at the amplitudes of 0, 4 and 6 MPa. Lexan 101 was
tested at the amplitudes of 0, 5 and 10 MPa. In the Figures 3.1 and 3.2 the time to
failure is plotted on a logarithmic scale as function of the applied static stress. The
three relations to be seen in both gures correspond to the least square solutions for
creep, shifted in the time to nd the best correspondence with the results. In Figure
3.1 Makrolon shows almost parallel relations for creep and fatigue. An increase of the
dynamic amplitude proves to result in a decrease in time to failure. In Figure 3.2 for
Lexan 101 the same type of shift to decreased time to failure for higher amplitudes. For
higher amplitudes the correspondence of the measurements to the least square relation
becomes less. Possible eects like aging due to long experimental times and heating of
the sample due to high stresses cause the change in slope in the rst order least square
solutions.
3.2
Results of the simulations
As shown in Chapter 2 numerical simulations can be performed with the use of the
nite element program MARC n Mentat. To illustrate Lexan 101 has been simulated
in MARC n Mentat as shown in Figure 3.3. Again, one can recognize a decrease of
time of failure for higher amplitude.
In Chapter 2 the assumption was made that the mesh could be decreased to a single
point. The comparison of the simulations for one point done in Matlab and the simulations done for a single element in MARC n Mentat show a small but constant gap
between the results shown in Figure 3.4. The slopes of the relations equal each other.
The gap between the results of Matlab and MARC n Mentat is on every spot a factor
of the order 2. An explanation can be found in the elasticity of the material that has
not been taken into account in the Matlab simulation. A second reason that could have
led to this dierence is the method of programming of the model in MARC n Mentat.
9
CHAPTER 3.
10
RESULTS
65
Fatigue measurements on polycarbonate (Makrolon)
Applied static stress [MPa]
60
55
50
45
40
1
10
Creep
Fatigue 4 MPa
Fatigue 6 MPa
2
10
3
10
Time to failure [s]
4
10
5
10
Figure 3.1: Measurements on Makrolon. The least square method is applied to
obtain a rst order relation (dotted line)
65
Fatigue measurements on polycarbonate (Lexan 101)
Applied static stress [MPa]
60
55
50
45
40
1
10
Creep
Fatigue 5 MPa
Fatigue 10 MPa
2
10
3
10
Time to failure [s]
4
10
5
10
Figure 3.2: Measurements on Lexan 101. The least square method is applied
to obtain a rst order relation (dotted line)
CHAPTER 3.
11
RESULTS
65
Fatigue simulations in MARC \ Mentat (Lexan 101)
Applied static stress [MPa]
60
55
50
45
40
1
10
Creep
Fatigue 5 MPa
Fatigue 10 MPa
2
10
3
10
Time to failure [s]
4
10
Figure 3.3: Fatigue simulations of Lexan 101 in MARC
ment of the results is shown with the dotted line.
5
10
n Mentat.
The align-
MARC n Mentat makes use of a minimal plastic strain value to take it into account as
plastic strain. On small time steps this can lead to the ignorance of the plastic strain
because of a to small value. Matlab on the other hand takes every value of the plastic
strain into account and will therefor nd an earlier time of failure.
Despite the small dierence, Matlab is showing the same acceleration factors as the
MARC n Mentat simulations. This is shown in Figure 3.5. For determination of the
acceleration factors Matlab is as adequate in the solutions for predicting fatigue as
MARC n Mentat.
A last simplication was made by retrieving an analytically derived acceleration factor
for the prediction of the contribution of the dynamic stress signal on creep simulations.
Comparing the analytical derived acceleration factors with the acceleration factors
that can be derived from the numerical simulations (Figure 3.5 and Figure 3.6), the
analytical acceleration factors match the numerically found acceleration factors. The
analytical solution is only just below the factors obtained from the numerical simulations. The dierence can be explained with the neglecting of the strain hardening in
the analytical result.
3.3
Experiments and simulations
Making a comparison between the experimental and numerical simulation data as
shown from Figure 3.7 till Figure 3.9, a conclusion can be drawn that the two do
not yet correspond correctly. It's to be seen that with this model the creep rupture
is predicted correctly, but the acceleration factors calculated are lower than the acceleration factors from numerical simulations and the analytical prediction. A remark
must be made regarding dierence in evolution of the acceleration factor for Makrolon
CHAPTER 3.
12
RESULTS
Fatigue analytically predicted and numerically simulated (Lexan 101)
65
Creep
Fatigue 5 MPa
Fatigue 10 MPa
Applied static stress [MPa]
60
55
50
45
40
1
10
2
10
3
10
4
10
5
10
Time to failure [s]
Figure 3.4: Fatigue simulations of Lexan 101 in Matlab (points) and MARC n
Mentat (dotted line)
compared to Lexan 101. A possible explanation for the dierence is the too simplistical model. Discarding aging leads to a decrease in time to failure as is also the case
between the numerical simulations and the experimental measurements.
CHAPTER 3.
13
RESULTS
Fatigue analytically predicted and numerically simulated (Lexan 101)
65
Creep
Fatigue 5 MPa
Fatigue 10 MPa
Applied static stress [MPa]
60
55
50
45
40
1
10
2
3
10
4
10
5
10
Time to failure [s]
10
Figure 3.5: Comparison of the analytical solution for the acceleration factors
(dashed line) and the factors obtained from obtained from MARC n Mentat
simulations (symbols).
Acceleration factors from simulations
3
10
Acceleration factor [−]
Analytical
Makrolon
Lexan 101
2
10
1
10
0
10
0
2
4
6
8
10
12
Amplitude of dynamic stress [MPa]
Figure 3.6: Comparison of the analytical solution for the acceleration factors
(line) and the factors obtained from obtained from both numerical simulations
(symbols).
CHAPTER 3.
14
RESULTS
65
Fatigue experiments and simulations (Lexan 101)
Applied static stress [MPa]
60
55
50
45
40
1
10
Creep
Fatigue 5 MPa
Fatigue 10 MPa
2
10
3
10
Time to failure [s]
4
10
5
10
Figure 3.7: The results of the experiments (symbols) and the numerical simulations for Lexan 101 (dashed lines). The simulations for creep performed in
Matlab and the fatigue predicted with the analytical acceleration factors.
65
Fatigue experiments and simulations (Makrolon)
Applied static stress [MPa]
60
55
50
45
40
1
10
Creep
Fatigue 4 MPa
Fatigue 6 MPa
2
10
3
10
Time to failure [s]
4
10
5
10
Figure 3.8: The results of the experiments (symbols) and the numerical simulations for Makrolon (dashed lines). The simulations for creep performed in
Matlab and the fatigue predicted with the analytical acceleration factors.
CHAPTER 3.
15
RESULTS
Acceleration factors for a sawtooth signal on polycarbonate
3
10
Acceleration factor [−]
Analytical
Makrolon
Lexan 101
2
10
1
10
0
10
0
2
4
6
8
10
12
Amplitude of dynamic stress [MPa]
Figure 3.9: The weighted acceleration factors from experiments on Makrolon
(diamond) and Lexan 101 (circle) compared to the analytical prediction (line).
Conclusion
The study is performed to verify whether the Leonov-model was capable of predicting
time of failure for fatigue type of loading. As Klompen [1] already showed, failure due
to creep loading proved to be predicted accurately in this study.
The comparison between creep and fatigue loading led to good correspondence between
the two numerical methods. The results of Matlab have the advantage of been calculated in less time due to a more simplied representation of the material by single point.
The analytical acceleration factor that was derived, showed the same predictions for
fatigue loading on base of the creep predictions, as both the numerical simulation methods. The advantages of the acceleration factor is the short time of determination and
the possibility to dene an acceleration factor for every type of fatigue loading. A second advantage is that fatigue simulations are no longer necessary as the acceleration
factor can predict the results on base from the creep simulations.
The prediction on the other hand of time to failure for fatigue loading is not as accurate
as for creep loading. A too simplistical model can be a reason to declare the dierence
between simulation and experiment. Eects like aging can not be neglected without
further determination of its inuence on fatigue.
16
Bibliography
[1] Klompen, E.T.J., van Haag, C. and Engels, T.A.P. 2003. Elasto-viscoplastic modelling of
nite strain deformation behavior of glassy polymers; Incorporating aging kinetics. In:To
be submitted. Eindhoven.
[2] Engels, T.A.P. 2003. An investigation into the predictability of long-term ductile failure
of glassy polymers. In: Internal Report (MT03.18) Eindhoven.
17
Appendix A
Tensile test results and
Da-values
Material Strain rate [s 1] Da Experimental [-] Da Numerical [-]
Makrolon
10 2
26.5
31.4
Makrolon
10 3
26.5
31.4
4
Makrolon
10
26.5
31.4
Lexan 101
10 2
31
35.7
3
Lexan 101
10
31
35.7
Lexan 101
10 4
31
35.7
Table A.1: Experimentally and numerically derived state parameters
Material Amplitude of dynamic stress Acceleration factor
Makrolon
0
1
Makrolon
4
2.6
Makrolon
6
7.4
0
1
Lexan 101
Lexan 101
5
2.7
Lexan 101
10
30.3
Table A.2: Experimentally derived acceleration factors
18
Appendix B
Determination of the plastic
strain rate formulation
In his report, Klompen [1] writes the viscosity as a function of the equivalent stress ( ), the
state parameter (Da ) and the pressure (p).
p
H
+D
(
; D; p) = A0 0 exp( RT ) exp 0
The plastic strain rate is dened by:
0
sinh
0
(B.1)
(B.2)
2
In the value of A0 depends on the temperature and is therefor rewritten to the value AT . AT is
taken constant in the simulations.
H
RT
ref
(B.3)
A0;ref = A0 exp
RH T1 T 1
ref
AT = A0;ref exp
(B.4)
_ pl =
Combining all equations into equation B.1 leads to:
p
1
_ pl =
exp 0 D(p l) sinh
2 AT
0
(B.5)
In this function the pressure (p) is adjusted to uniaxial strain as was used in this study.
1
p = u
(B.6)
3
For simplication of the equation use was made of a mathematical simplication following
shown rule.
1
sinh
= exp
(B.7)
0
2
0
The softening as suggested by Klompen [1] can be added to the equation.
s2 s1
D(p l) = Da (1 + (s0 exp(pl ))s1 ) s1
(B.8)
Inserting all equations in B.5 leads to the denition of the plastic strain rate as it will be used
in this study.
s s !
u (p3+) Da (1+(s exp( ))s1 ) 2s1 1
0
pl
30
1
_ pl =
exp
(B.9)
2AT
19
Appendix C
Analytical solution
accumulative plasticity
This appendix discusses the derivation of the acceleration factor for a sawtooth shaped stress
signal. Only the steps that are crucial for this derivation are written down in this appendix.
The starting point of this derivation is the formulation for the derivative of the plastic strain
rate. The softening is considered negligible.
_ pl =
1
2 AT
exp
p
u ( 3 + )
30
(C.1)
The rst step is dividing the stress u in a static stress m and a dynamic stress contribution
d .
u = m + d g (f; t)
(C.2)
In this equation g(f; t) represents the function describing the dynamic stress signal and is a
function of time and frequency. Inserting C.2 into C.1 leads to:
(m + d g(f; t))
_ pl =
exp
2AT
30
1
p
3+
!
(C.3)
The static stress contribution is invariant of time and can therefor be taken outside the integral.
!
p
m (p3+) Zt2
3+
g(f;t)
30
1
d
3
0
exp
dt
(C.4)
pl =
exp
2 AT
t1
In the equation a notation is introduced that describes the contribution of the static stress,
_ pl (m ).
!
m (p3+)
30
1
_ pl (m ) =
exp
(C.5)
2 AT
20
APPENDIX C. ANALYTICAL SOLUTION ACCUMULATIVE PLASTICITY
Equation C.4 can now be rewritten.
pl = _ pl (m ) Zt2
p
33+
0
exp
d g(f;t)
(C.6)
dt
t1
The new equation clearly shows the contribution of the dynamic stress as a factor on the static
contribution. This division makes it possible to dene a acceleration factor. A acceleration
factor (az ) can be dened as equation C.7.
R
_ pl (u ) dtcycle
az (m ; d ; f ) = R
=
_ pl (m ) dtcycle
Z
_ pl (d ) dtcycle
(C.7)
To calculate the integral over the time of one period of the sawtooth shaped stress signal, the
period is divided in three time parts. The integral over the whole period will be the same as
the sum of the three parts.
Denition of the stress signal (u )
u = m + 4fd t
u = m + 2d 4fd t
u = m 4d + 4fd t
Time interval
0 t < 41f
3
1
4f t < 4f
3
1
4f t < f
For every time interval a contribution can now be calculated. For the time interval 0 t < 41f
the next equations can be obtained.
1
p
Z4f
33+
0 4fd t dt
pl = _ pl (m ) exp
(C.8)
0
Inserting the time boarders.
pl = _ pl (m ) p 30
3 + 4f d
!
p
33+
0
exp
d
!
exp
0
In the same way a denition can be written for 41f t < 43f .
!
p
p
d 33+
2d 33+
3
0
0
0
p pl = _ pl (m ) exp
exp
exp
3 + 4f d
To nish with the last time period
p
4d 33+
0
pl = _ pl (m ) exp
3
4f
t < f1 .
p 30
3 + 4f d
!
exp
4d
p
33+
0
(C.9)
3d
33+
0
!
(C.10)
exp
p
3d
p
33+
0
!
(C.11)
21
APPENDIX C. ANALYTICAL SOLUTION ACCUMULATIVE PLASTICITY
The three formulations can be summarized to one equation.
pl = _ pl (m ) p30
2 d ( 3 + )
exp
d (p3+)
30
!
1
(C.12)
The denition C.7 is used to dene a acceleration factor for the sawtooth shaped stress signal.
The analytical solution for the acceleration factor is now only a function of the pressure
dependency value (), a model parameter ( ) and the amplitude of the dynamic stress (d ).
p
!
d ( 3+)
d (p3+)
30
3
3
0
0
p
az =
exp
exp
(C.13)
2d ( 3 + )
This equation can be simplied by using a mathematical approximation of the exponential
parts.
p
d ( 3+)
30
3
0
p
az =
sinh
(C.14)
2 d ( 3 + )
The same procedure can be followed to obtain a acceleration factor for a block-shape. The analytical solution of the acceleration factor for a block shaped stress signal can then be written as:
p
!
d ( 3+)
d (p3+)
1
3
3
0
0
az = exp
+ exp
(C.15)
2
The solution for a denition of a sinus shaped stress signal is only to be determined by
a numerical approximation. Dierences in the evolution of the acceleration factors for the
dierent signals is shown in Figure 2.1.
22