Numerical simulation of fatiguefailure in polymer glassesD. de KanterMT04.28March 2004

TU/e Internship ReportMarch 2004Coaches:ir. R.P.M. Janssendr.ir. L.E. GovaertEindhoven University of TechnologyDepartment of Mechanical EngineeringMaterial Technology group

Abstract

This study focuses on the ability of the altered Leonov-model as suggested byKlompen [1] to predict time to failure for creep and fatigue. The model thatwas altered to describe the e�ects of softening and aging, will not be used toits full potential. The assumption is made that the e�ects of aging are to beneglected at the timescales investigated. The model is tested on two, di�erentnumerical methods and for two types of polycarbonate.The tests performed were tests on creep and fatigue. The fatigue was appliedby a sawtooth shaped dynamic stress signal on the polycarbonate. The time tofailure was de�ned as the moment of necking or brittle rupture of the material.The amplitude of the dynamic stress signal was varied to determine in uenceof the amplitude on the time of failure.The second part of the study introduces an acceleration factor based on thesimpli�cations of the model as suggested in this study. The acceleration factoris 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 basedon the time of failure for creep.Correspondence between the numerical simulations and experiments was achievedfor creep. The fatigue predictions did not appear as accurate as the predictionsfor creep. The acceleration factor proved to accurately predict the numericallysimulated times of failure for fatigue. The e�ects of aging are not to be assumednegligible in case of fatigue. Aging plays an important role in the explinationfor the di�erences between the experiments and the numerical simulations.

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Contents

Introduction 11 Materials and methods 31.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.1 Specimen standards . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1 Tensile tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 Creep and fatigue tests . . . . . . . . . . . . . . . . . . . . . . . 42 Numerical simulation 62.1 MARC n Mentat element simulation . . . . . . . . . . . . . . . . . . . . 62.2 Matlab simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Acceleration factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Results 93.1 Results of the experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Results of the simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Experiments and simulations . . . . . . . . . . . . . . . . . . . . . . . . 11Bibliography 17A Tensile test results and Da-values 18B Determination of the plastic strain rate formulation 19C Analytical solution accumulative plasticity 20

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Introduction

The polymers that are used in productional processes have di�erent thermal histories.The thermal history a�ects the macroscopic behavior of a polymer [1] and to predictthe macroscopic behavior of a speci�c polymer this history must be know. For the pre-diction 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 macroscopicbehavior. The correspondence between the numerical methods and tensile, compres-sion and creep tests has been proven by Klompen [1].Products are made for a variety of purposes. Many of those purposes are applicationsin which the product endures stresses of statical, but also dynamical form. Dynamicstress can lead to fatigue. During a course project (Appendix ??) the altered modelproved to be able to predict time to failure for creep in polycarbonate. In this studyattention is paid to the use of this model under fatigue conditions. The model is used topredict the failure of polycarbonate that is subjected to a static stress combined witha sawtooth shaped dynamic stress. The in uence of the amplitude of the dynamicstress on the time of failure compared to creep, is discussed. A simpli�ed method ofpredicting fatigue rupture is introduced. The simulations are veri�ed with experiments.The Leonov modelIn the Leonov model as suggested by Klompen[1] a state parameter to describe thethermal history is introduced. The state parameter is the di�erence between the presentstate and the rejuvenated state as is shown in Figure 1. In the used model aging is nottaken into account.The numerical simulations performed in this study are on base of the de�nition ofthe plastic strain rate. By de�ning an accumulative plastic strain the in uence of theloading over the time can be calculated. The assumption of a maximal allowable strainmakes it possible to determine a point of failure. Prior simulations and experimentsproved the model to be su�cient to predict creep rupture under these failure assump-tions (Appendix ??). The derivative of the plastic strain rate is de�ned by (1). Thisequation is derived as is depicted in Appendix B.

_ pl = 12AT exp �u(p3+�)3�0 �Da�(1+(s0�exp( pl))s1 ) s2�s1s1 !

(1)In the formulation pl is the accumulated plastic strain at a moment in time. AT ; s0; s1and s2 are �tting parameters suggested by Engels [2]. Da is the state parameter as

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CONTENTS 2

Figure 1: Visualization of the state parameter. The state parameter values thedi�erence 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 pa-rameter (Da) can be determined by the data from a tensile test following the methodof calculation shown in Appendix ??. For the numerical simulations the value of thestate parameters were determined from the experimental data.

PolycarbonateA0 1012 [s]s0 0,965 [�]s1 50 [�]s3 -5 [�]Table 1: Material properties for polycarbonate for numerical simulations. Ob-tained from the internal report of Engels [2].

Chapter 1

Materials and methods

The validation of numerical simulations was done on base of comparison with exper-imental data. The experimental data for this study was obtained from tensile, creepand fatigue testing. The data obtained from the tensile tests was used for determiningthe parameters needed for numerical simulation as will be discussed in Chapter 2. Thedata obtained from the creep and fatigue tests was used for validation of the numericalsimulation results. The experimental obtained data is given in Chapter 3. The choicesfor the used material, specimen shape and the experimental setups are discussed inthis chapter.1.1 MaterialsFor creep and fatigue rupture experiments use was made of two types of polycarbon-ate. Polycarbonate was chosen as testing material on the base of prior experiments(Appendix ??). The polycarbonate showed in prior experiments a stable necking orbrittle rupture behavior which formed a good indication point for a time of failure dueto creep or fatigue.Use was made of Bayer Makrolon and G.E. Lexan 101. The Makrolon specimenswere obtained from a provided extruded sheet. The choice for an extruded sheet wasmade to obtain specimens with only little orientation. The material is assumed tobe homogeneous during the experiments. Orientated samples can show di�erences inbehavior compared to not orientated materials. Due to a undersize in Makrolon duringthe experiments a shift had to be made to Lexan 101 that was available. The Lexan101 specimens were injection molded.1.1.1 Specimen standardsBoth types of material specimens were shaped following the ISO 527 standard. Toobtain this standard, the Makrolon specimens were machined by a milling procedure.The ISO 527 standard showed problems during prior experiments. Investigation ofthe material with polarized light showed two local concentrations of residual stress inthe ISO-norm specimens and the specimens used in prior testing. Local stress areasproved to have been formed during the mechanical interventions on the material thatwere done for acquiring the ISO-norm shape. The combination of milling and thestrong round o� in the material standard could be pointed out as the cause of the

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CHAPTER 1. MATERIALS AND METHODS 4development of these concentrations of stress. The stresses in the material would notlead to a shift in the data, but only to an induction of placed failure in the broader partof the specimen. Specimens that were obtained after the internship and were formedfollowing the ASTM D638M norm proved to be free of residual stress concentrations.1.1.2 Material propertiesThe material properties needed for the experiments were obtained from the internalreport of Engels [2] and the report of Klompen [1] and are given in Table 1.1. The ma-terial properties that indicate the present state of the material were obtained by tensiletesting. The determination of the state parameter (Da) is depicted in the appendicesof the course-report in Appendix ??. Apart from the state parameter the rest of thematerial properties are similar for both types of polycarbonate.

PolycarbonateE 2000 [MPa]� 0,03 [�]� 0,37 [�]�0 0,7 [MPa]Gr 26 [MPa]Table 1.1: Material properties of polycarbonate

1.2 Experimental setup1.2.1 Tensile testsThe tensile tests were performed on the servo-hydraulic MTS Elastomer Testing System810 and 831.10, both with a controlled temperature chamber. The specimens weretested on constant linear strain rates of 10�4, 10�3 and 10�2s�1. The specimens weretested on the temperature of 23; 0o Celsius (�0; 1o Celsius). To prevent temperaturedi�erences between the temperature chamber and the specimens, the specimens wereacclimatized for 15 minutes in the chamber before testing. The results of the tensiletests have been placed in Appendix A.1.2.2 Creep and fatigue testsCreep and fatigue rupture tests were performed on both types of materials discussed inthe section Materials. Both materials were subjected to a static load and in case of thefatigue measurements the static load was complemented with a dynamic contribution.The tests were performed on the servo-hydraulic MTS Elastomer Testing System 810and 831.10, both with a controlled temperature chamber. Failure of the polycarbonatedue to it is loading was de�ned as the moment in time on which necking or brittlefracture occurred in the polymer. The fatigue and creep-rupture measurements weredone at the same temperature (23o Celsius) as the tensile tests and under the same

CHAPTER 1. MATERIALS AND METHODS 5specimen conditions.The fatigue was formed by a sawtooth shaped stress signal with prede�ned amplitudeand frequency as shown in Figure 1.1. The sawtooth shape was chosen because of theanalytical possibilities that will be discussed in Chapter 2. The amplitude was variedfrom 0 MPa (creep) up to 10 MPa. Makrolon was measured at the amplitudes of 4and 6 MPa. Due to an undersize the series for Makrolon up to 10 MPa in steps of 2MPa were aborted after 6 MPa. The Lexan 101 was tested at the amplitudes of 5 and10 MPa.

0 0.5 1 1.540

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65Reproduction of fatigue stress signal

Time [s]

Ap

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[M

Pa] σ

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σm

Figure 1.1: Visualization of the fatigue signal. The state parameter is a pa-rameter indicating the parent state of the aged polymer (solid line) comparedto the rejuvenated polymer (dotted line).All measurements have been done with the maximal stress under the value of the yieldstress measured by tensile tests on the strain rate of 10�4s�1. The frequency was inall experiments 1,0 Hz. This refers to a maximum strain rate at the largest amplitudein the order of 10�2s�1. Simulations, with adiabatic conditions, showed that the as-sumption of no heating up of the specimen was reasonable.

Chapter 2

Numerical simulation

This chapter will discuss the numerical simulations performed and the simpli�cationsthat were made.2.1 MARC n Mentat element simulationFor numeric simulations the Leonov model is used in combination with the �nite el-ement program Marc n Mentat. By use of the parameters from table 1.1 and 1 asimulation is performed for a tensile bar to compare the numerically estimated param-eter ,Da, with the experimental determined Da. Due to simpli�cation of the reality inthe model, the measured value of Da has to be adjusted in the numerical simulations topredict the material behavior correctly. From the simulation of a tensile test the valueof the state parameter Da for numerical simulations can be determined. For Makrolonthe value of Da for numerical simulations was determined at 35,7. For Lexan 101 theDa value was determined at 31,4. Both di�erent from the experimental values as shownin Appendix A.Prior numerical simulations were performed with the mesh of a quarter of the specimenbar using its symmetry for modelling the whole specimen. To predict the failure of amaterial it is not necessary to make a mesh of the full specimen. The failure starts in allelements at the same time in case of a homogenous material with constant dimensionsof the modulated specimen. Because failure is based on the accumulative plastic strainin every element apart and in this survey was the only point of interest, the mesh wassimpli�ed. Considering the equality of elements, only one element has to be calculatedto predict the failure of the whole specimen. The mesh was reduced to a single elementwith axial, rotational symmetry. Simulations were performed for Lexan 101 for thesame conditions as the experiment considered in Chapter1.2.2 Matlab simulationInterested in the moment of failure the model still knows opportunities for further sim-pli�cation. Knowing the model to calculate the accumulative plastic strain rate in eachnodal point of the single element, reducing the element to a single point should lead to

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CHAPTER 2. NUMERICAL SIMULATION 7same results. Reduction of the mesh to a single point makes a �nite element programis no longer needed for the calculations. The calculations can now be taken over by amathematical program like Matlab that was used in this survey. In the Matlab sim-ulation attention is paid to the accumulative plastic strain and the elastic part of thematerial behavior is neglected. This should lead to a small decrease in the predictionsof the time to failure.2.3 Acceleration factorNot all parts of the used de�nition play a signi�cant role. The formulation can beparted into two contributions. The �rst part describing the in uence of the stress.The second part describing the evolution of the state parameter. A constant stateparameter in the order of 30 would only have a small in uence on the evolution of _ plfor the value of exp�Da approaches to 1. For the determination of the accelerationfactor the state parameter is neglected.

_ pl = 12AT exp��u(p3+�)3�0 � � exp� Da�(1+(s0�exp( pl))s1 ) s2�s1s1 !(2.1)

For the further simpli�cation the assumption is made that aging does not occurs on thetime scales investigated. The formulation will therefor reduce to a more workable form._ pl = 12AT exp��u(p3+�)3�0 � (2.2)

Considering the formulation (2.2) a new parting can be suggested. By parting theapplied stress (�u) into a static (�m) and a dynamic contribution (�d), as shown in(2.3), the de�nition of (2.4) is obtained. This de�nition shows the contribution of thedynamic 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) (2.3)

_ pl = 12AT exp��m(p3+�)3�0 � � exp��d�g(f;t)(p3+�)3�0 � (2.4)The introduction of an acceleration factor makes it possible to analytically predict thefatigue measurements after only measuring or numerically simulating creep. The accel-eration factor will be formulated as the rate in accumulative plastic strain contributionin a speci�c time period for fatigue and creep.

az(�m; �d; f) = R _ pl(�u) dtcycleR _ 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 amplitudelarger than zero the factor will increase. The combination of (2.4) and (2.5) can yield

CHAPTER 2. NUMERICAL SIMULATION 8

0 2 4 6 8 10 1210

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Amplitude of dynamic stress [MPa]

Acc

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Acceleration factors for different signals

SawtoothBlockSinus

Figure 2.1: The evolution of the acceleration factors for the di�erent signals.a de�nition that describes analytically the contribution of a sawtooth shaped stresssignal with de�ned frequency. The derivation of (2.6) has been depicted in AppendixC.

az = � 3�02�d(p3 + �)� � sinh��d(p3+�)3�0 � (2.6)

The acceleration factor for a sawtooth shaped signal is now only a function of the pres-sure dependency �, the model parameter � and the dynamic stress amplitude �d. Theformulation of the acceleration factor gives the opportunity to view the developmentof the acceleration factor by increasing amplitude of the sawtooth shaped stress signal.A similar analytical formulation can be derived for di�erent types of dynamic stresssignals. A de�nition for the contribution of a block-shaped dynamic stress signal isgiven in Appendix C. Mathematical problems arise for the calculation of an analyticalsolution for a sinus shaped stress signal. Gaining a good analytical solution for thesinus shape could lead to expressions for every type of dynamic stress by performing aFourier transformation on the signal and summarize the di�erent constitutional sinusesinto a acceleration factor.

Chapter 3

Results

This chapter covers the results obtained from the experiments discussed in Chapter 1and the numerical simulations in Chapter 2.3.1 Results of the experimentsThe Makrolon and Lexan 101 specimens were tested on MTS Elastomer Testing Sys-tems. Makrolon was tested at the amplitudes of 0, 4 and 6 MPa. Lexan 101 wastested at the amplitudes of 0, 5 and 10 MPa. In the Figures 3.1 and 3.2 the time tofailure is plotted on a logarithmic scale as function of the applied static stress. Thethree relations to be seen in both �gures correspond to the least square solutions forcreep, shifted in the time to �nd the best correspondence with the results. In Figure3.1 Makrolon shows almost parallel relations for creep and fatigue. An increase of thedynamic amplitude proves to result in a decrease in time to failure. In Figure 3.2 forLexan 101 the same type of shift to decreased time to failure for higher amplitudes. Forhigher amplitudes the correspondence of the measurements to the least square relationbecomes less. Possible e�ects like aging due to long experimental times and heating ofthe sample due to high stresses cause the change in slope in the �rst order least squaresolutions.3.2 Results of the simulationsAs 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 simulatedin MARC n Mentat as shown in Figure 3.3. Again, one can recognize a decrease oftime of failure for higher amplitude.In Chapter 2 the assumption was made that the mesh could be decreased to a singlepoint. The comparison of the simulations for one point done in Matlab and the sim-ulations done for a single element in MARC n Mentat show a small but constant gapbetween 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 factorof the order 2. An explanation can be found in the elasticity of the material that hasnot been taken into account in the Matlab simulation. A second reason that could haveled to this di�erence is the method of programming of the model in MARC n Mentat.

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CHAPTER 3. RESULTS 10

101 102 103 104 10540

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Time to failure [s]

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Fatigue measurements on polycarbonate (Makrolon)

CreepFatigue 4 MPaFatigue 6 MPa

Figure 3.1: Measurements on Makrolon. The least square method is applied toobtain a �rst order relation (dotted line)

101 102 103 104 10540

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Fatigue measurements on polycarbonate (Lexan 101)

CreepFatigue 5 MPaFatigue 10 MPa

Figure 3.2: Measurements on Lexan 101. The least square method is appliedto obtain a �rst order relation (dotted line)

CHAPTER 3. RESULTS 11

101 102 103 104 10540

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Fatigue simulations in MARC \ Mentat (Lexan 101)

CreepFatigue 5 MPaFatigue 10 MPa

Figure 3.3: Fatigue simulations of Lexan 101 in MARC n Mentat. The align-ment of the results is shown with the dotted line.MARC n Mentat makes use of a minimal plastic strain value to take it into account asplastic strain. On small time steps this can lead to the ignorance of the plastic strainbecause of a to small value. Matlab on the other hand takes every value of the plasticstrain into account and will therefor �nd an earlier time of failure.Despite the small di�erence, Matlab is showing the same acceleration factors as theMARC n Mentat simulations. This is shown in Figure 3.5. For determination of theacceleration factors Matlab is as adequate in the solutions for predicting fatigue asMARC n Mentat.A last simpli�cation was made by retrieving an analytically derived acceleration factorfor the prediction of the contribution of the dynamic stress signal on creep simulations.Comparing the analytical derived acceleration factors with the acceleration factorsthat can be derived from the numerical simulations (Figure 3.5 and Figure 3.6), theanalytical acceleration factors match the numerically found acceleration factors. Theanalytical solution is only just below the factors obtained from the numerical simula-tions. The di�erence can be explained with the neglecting of the strain hardening inthe analytical result.3.3 Experiments and simulationsMaking a comparison between the experimental and numerical simulation data asshown from Figure 3.7 till Figure 3.9, a conclusion can be drawn that the two donot yet correspond correctly. It's to be seen that with this model the creep ruptureis predicted correctly, but the acceleration factors calculated are lower than the ac-celeration factors from numerical simulations and the analytical prediction. A remarkmust be made regarding di�erence in evolution of the acceleration factor for Makrolon

CHAPTER 3. RESULTS 12

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Time to failure [s]

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Fatigue analytically predicted and numerically simulated (Lexan 101)

CreepFatigue 5 MPaFatigue 10 MPa

Figure 3.4: Fatigue simulations of Lexan 101 in Matlab (points) and MARC nMentat (dotted line)compared to Lexan 101. A possible explanation for the di�erence is the too simplis-tical model. Discarding aging leads to a decrease in time to failure as is also the casebetween the numerical simulations and the experimental measurements.

CHAPTER 3. RESULTS 13

101 102 103 104 10540

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Fatigue analytically predicted and numerically simulated (Lexan 101)

CreepFatigue 5 MPaFatigue 10 MPa

Figure 3.5: Comparison of the analytical solution for the acceleration factors(dashed line) and the factors obtained from obtained from MARC n Mentatsimulations (symbols).

0 2 4 6 8 10 1210

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Acceleration factors from simulations

AnalyticalMakrolonLexan 101

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. RESULTS 14

101 102 103 104 10540

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Fatigue experiments and simulations (Lexan 101)

CreepFatigue 5 MPaFatigue 10 MPa

Figure 3.7: The results of the experiments (symbols) and the numerical sim-ulations for Lexan 101 (dashed lines). The simulations for creep performed inMatlab and the fatigue predicted with the analytical acceleration factors.

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Fatigue experiments and simulations (Makrolon)

CreepFatigue 4 MPaFatigue 6 MPa

Figure 3.8: The results of the experiments (symbols) and the numerical sim-ulations for Makrolon (dashed lines). The simulations for creep performed inMatlab and the fatigue predicted with the analytical acceleration factors.

CHAPTER 3. RESULTS 15

0 2 4 6 8 10 1210

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Acceleration factors for a sawtooth signal on polycarbonate

AnalyticalMakrolonLexan 101

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 predictingtime of failure for fatigue type of loading. As Klompen [1] already showed, failure dueto creep loading proved to be predicted accurately in this study.The comparison between creep and fatigue loading led to good correspondence betweenthe two numerical methods. The results of Matlab have the advantage of been calcu-lated in less time due to a more simpli�ed representation of the material by single point.The analytical acceleration factor that was derived, showed the same predictions forfatigue loading on base of the creep predictions, as both the numerical simulation meth-ods. The advantages of the acceleration factor is the short time of determination andthe possibility to de�ne an acceleration factor for every type of fatigue loading. A sec-ond advantage is that fatigue simulations are no longer necessary as the accelerationfactor 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 accurateas for creep loading. A too simplistical model can be a reason to declare the di�erencebetween simulation and experiment. E�ects like aging can not be neglected withoutfurther determination of its in uence on fatigue.

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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:Tobe submitted. Eindhoven.[2] Engels, T.A.P. 2003. An investigation into the predictability of long-term ductile failureof glassy polymers. In: Internal Report (MT03.18) Eindhoven.

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Appendix A

Tensile test results andDa-values

Material Strain rate [s�1] Da Experimental [-] Da Numerical [-]Makrolon 10�2 26.5 31.4Makrolon 10�3 26.5 31.4Makrolon 10�4 26.5 31.4Lexan 101 10�2 31 35.7Lexan 101 10�3 31 35.7Lexan 101 10�4 31 35.7Table A.1: Experimentally and numerically derived state parameters

Material Amplitude of dynamic stress Acceleration factorMakrolon 0 1Makrolon 4 2.6Makrolon 6 7.4Lexan 101 0 1Lexan 101 5 2.7Lexan 101 10 30.3Table A.2: Experimentally derived acceleration factors

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Appendix BDetermination of the plasticstrain rate formulation

In his report, Klompen [1] writes the viscosity as a function of the equivalent stress (�), thestate parameter (Da) and the pressure (p).�(�� ;D; p) = A0�0 exp(�HRT ) exp��p�0 +D� ���0sinh� ���0 � (B.1)

The plastic strain rate is de�ned by: _ pl = ��2� (B.2)In the value of A0 depends on the temperature and is therefor rewritten to the value AT . AT istaken constant in the simulations.

A0;ref = A0 exp� �HRTref � (B.3)AT = A0;ref exp��HR �

� 1T � 1Tref �� (B.4)Combining all equations into equation B.1 leads to:_ pl = 12AT exp���p�0 �D( pl)� sinh� ���0� (B.5)In this function the pressure (p) is adjusted to uniaxial strain as was used in this study.

p = 13�u (B.6)For simpli�cation of the equation use was made of a mathematical simpli�cation followingshown rule. sinh� ���0� = 12 exp� ���0� (B.7)The softening as suggested by Klompen [1] can be added to the equation.

D( pl) = Da � (1 + (s0 � exp( pl))s1) s2�s1s1 (B.8)Inserting all equations in B.5 leads to the de�nition of the plastic strain rate as it will be usedin this study._ pl = 12AT exp �u(p3+�)3�0 �Da�(1+(s0�exp( pl))s1) s2�s1s1 ! (B.9)

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Appendix C

Analytical solutionaccumulative plasticity

This appendix discusses the derivation of the acceleration factor for a sawtooth shaped stresssignal. 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 strainrate. The softening is considered negligible.

_ pl = 12AT exp��u(p3 + �)3�0 � (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 afunction of time and frequency. Inserting C.2 into C.1 leads to:

_ pl = 12AT exp (�m + �d � g(f; t)) �p3 + ��3�0! (C.3)

The static stress contribution is invariant of time and can therefor be taken outside the integral. pl = 12AT exp �m(p3+�)3�0 ! t2Z

t1 exp��p3+�3�0 ��d�g(f;t)� dt (C.4)In the equation a notation is introduced that describes the contribution of the static stress,_ pl(�m).

_ pl(�m) = 12AT exp �m(p3+�)3�0 ! (C.5)20

APPENDIX C. ANALYTICAL SOLUTION ACCUMULATIVE PLASTICITY21Equation C.4 can now be rewritten.

pl = _ pl(�m) � t2Zt1 exp��p3+�3�0 ��d�g(f;t)� dt (C.6)

The new equation clearly shows the contribution of the dynamic stress as a factor on the staticcontribution. This division makes it possible to de�ne a acceleration factor. A accelerationfactor (az) can be de�ned as equation C.7.az(�m; �d; f) = R _ pl(�u) dtcycleR _ 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, theperiod is divided in three time parts. The integral over the whole period will be the same asthe sum of the three parts.Time interval De�nition of the stress signal (�u)0 � t < 14f �u = �m + 4f�dt14f � t < 34f �u = �m + 2�d � 4f�dt34f � t < 1f �u = �m � 4�d + 4f�dt

For every time interval a contribution can now be calculated. For the time interval 0 � t < 14fthe next equations can be obtained. pl = _ pl(�m) � 14fZ

0 exp��p3+�3�0 �4f�dt� dt (C.8)Inserting the time boarders.

pl = _ pl(�m) � 3�0�p3 + �� 4f�d!� exp��p3+�3�0 ��d�� exp0! (C.9)

In the same way a de�nition can be written for 14f � t < 34f . pl = _ pl(�m)�exp�2�d�p3+�3�0 �� 3�0�p3 + �� 4f�d

!� exp���d�p3+�3�0 ��

� exp��3�d�p3+�3�0 ��!(C.10)

To �nish with the last time period 34f � t < 1f . pl = _ pl(�m) �exp��4�d�p3+�3�0 �� 3�0�p3 + �� 4f�d

!� exp�4�d�p3+�3�0 ��

� exp�3�d�p3+�3�0 ��!(C.11)

APPENDIX C. ANALYTICAL SOLUTION ACCUMULATIVE PLASTICITY22The three formulations can be summarized to one equation.

pl = _ pl(�m) � � 3�02�d(p3 + �)� � exp��d(p3+�)3�0 �

�1! (C.12)The de�nition C.7 is used to de�ne a acceleration factor for the sawtooth shaped stress signal.The analytical solution for the acceleration factor is now only a function of the pressuredependency value (�), a model parameter (�) and the amplitude of the dynamic stress (�d).

az = � 3�02�d(p3 + �)� � exp��d(p3+�)3�0 �

� exp���d(p3+�)3�0 �! (C.13)This equation can be simpli�ed by using a mathematical approximation of the exponentialparts.

az = � 3�02�d(p3 + �)� � sinh��d(p3+�)3�0 � (C.14)

The same procedure can be followed to obtain a acceleration factor for a block-shape. The ana-lytical solution of the acceleration factor for a block shaped stress signal can then be written as:az = 12 � exp��d(p3+�)3�0 �+exp���d(p3+�)3�0 �! (C.15)

The solution for a de�nition of a sinus shaped stress signal is only to be determined bya numerical approximation. Di�erences in the evolution of the acceleration factors for thedi�erent signals is shown in Figure 2.1.