Characterisation of new Leonov parameters for PET

Gerard QuaakInternship MT04.23

Prof. Dr. Ir. M.G.D. GeersDr. Ir. P.J.G. SchreursIr. M.J. van den Bosch

January, 2005

Abstract

The implementation of a new version of the Leonov model was done by Lambert vanBreemen. The used implementation is one of the earlier versions of the new approach,from October 2004. This implementation uses some different parameters for describingsoftening and incorporating thermo-mechanical history of a material. These parametersare D0 instead of the earlier used D∞ and the parameters for the softening function: s0,s1 and s2. These new parameters are determined with the use of indentation tests andcompression tests. Also some problems encountered with the use of MSC. MARC for theindentation simulations are described. Some problems found where the instability of theLeonov model when dealing with distorted elements and the use of the internal MSC.MARC remeshing procedures.

CONTENTS 1

Table of contents

Contents

1 Introduction 2

2 The Leonov model 3

3 Methods 53.1 Validating old parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Determining the softening parameters . . . . . . . . . . . . . . . . . . . . 63.3 Determining D0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Results 84.1 Validated parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 Determining softening parameters . . . . . . . . . . . . . . . . . . . . . . 84.3 Determining D0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.4 Determining D0 with compression data . . . . . . . . . . . . . . . . . . . 134.5 The parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 MSC MARC problems 16

6 Conclusions & Recommendations 19

A The modified Carreau-Yassuda Function 21

B MSC MARC simulation settings 22

C FakeLeonov indentation simulation 23

1 INTRODUCTION 2

1 Introduction

At Corus IJmuiden polymer layered steel is being developed, which behaves differentfrom plain steel or plain polymers. Therefore it is necessary to determine behaviour ofthis combination to make simulations possible, which can later on speed up developmentof products with this material. A useful way to describe the behaviour of a glassy poly-mer, is the Leonov model, which has been used for this purpose before. Because of theimplementation of a new model - with the description of ageing and an easier way towork with thermo-mechanical history differences - to describe the behaviour of the poly-mer and because of the different behaviour of the thin layer from the bulk material, theparameters for the model have to be determined again.

Therefore we will first look at the Leonov model and how it was altered. This isfollowed by the description of the methods used to determine the parameters used inthe model. The experiments will be described, followed by the fitting that has to bedone with the help of simulations. Because of difficulties running certain simulations, adifferent approach will also be discussed here. This is followed by the obtained resultsand manipulating these to find the needed parameters.

The results section can be seen as a report of the steps to be taken to find theparameters, the methods chapter describes the theory behind the steps.

The first Leonov model was implemented by Robert Smit. The new Leonov model, asmentioned earlier, is being implemented in MSC. MARC by Lambert van Breemen andPiet Schreurs. When using the model, it is important to verify what version is being used,as the implementation is not yet ready. For this report a release of the implementationof Lambert van Breemen was used, to make use of the new approach.

2 THE LEONOV MODEL 3

2 The Leonov model

Modelling of a polymer layer can be done with the so-called Leonov model, developedby Tervoort et al. [8]. Later refinements with respect to strain hardening were alsodeveloped by Tervoort et al. [7]. Strain softening is based on the ideas of Hasan et al.[4] and implemented by Govaert et al. [3].

This model gives a fairly accurate description of the stress-relations, although ageingof the material was not yet included in the model. This was done by Klompen et al.[5]. With the implementation of ageing, also a new approach was used which makes itpossible to describe materials with a different thermo-mechanical history by changingonly one parameter.

A simple representation of the Leonov model is given in figure 1 (left). The two partsin the model correspond with the intermolecular forces and network forces, representedin figure 1 (right) which describes the behaviour of a glassy polymer. The Leonov modelhas strainrate dependence and pressure dependence, as drawn in figure 1.

Figure 1: Representation of the Leonov Model with a) Leonov model represented by a spring-damper system and b) effects of strainrate and pressure

The three dimensional non-Newtonian flow rule describes the rate of deformationtensor Dp, which in the end is part of the description of the total deformation. This rateof deformation tensor depends on the Eyring viscosity, which is given by:

η(τeq, p, T,D) = A0τ0 · exp

(

∆H

RT+

µp

τ0

− D

)

· τeq/τ0

sinh(τeq/τ0)(1)

with µ the pressure dependency, τeq and τ0 the equivalent stress and initial stress, A0 apre-exponential factor for fitting, ∆H the activation energy of the material.

Softening is introduced in the model via a parameter D, depending on D∞ which isthe saturated value for D, which is described by equation 2:

D(γp) = D∞

(

1 − exp

(

− h

D∞

γp

))

(2)

Here initially D(γp) = 0 (no softening), evolving to the saturation value D∞ forγp → ∞. This model describes the softening in the material, as is extensively describedin the report of Engels [2]. γp is defined as γp =

√3(ε − εy).

A shortcoming of the earlier mentioned model is the fact that ageing is not allowedto be incorporated. With looking at equation 1 it can by seen that the pre-exponentialfactor A0 determines the height of the yield-stress. Softening on the other hand gives

2 THE LEONOV MODEL 4

(in the end) a reduction of the yield-stress. This means that both A0 and D∞ should bemodified to incorporate ageing. The report of Engels [2] introduces a new way of definingD, whereby only D depends on the age of the material. Equation 2 now becomes:

D(γp) = D0 · gs(γp) (3)

With this new definition of D(γp) not only softening is defined, D acts as a stateparameter determining the height of the yield stress and describes softening behaviour.Therefore also a new value for A0 is determined, based on the rejuvenated state:

A0,rej = A0 · exp(−D∞) (4)

This means the new value of A0 can easily be determined from earlier obtained datafor the older Leonov model. (See Engels [2])

As stated above, D also describes the softening in the material. The report of Engels[2] gives a different approach for the softening function, as an alternative for the earlierused first order approximation. This approach uses the so called Yasuda model (see [6]),which gives a fit of the softening with three parameters. A brief explanation of thisfunction and the influence of the parameters can be found in appendix A.

3 METHODS 5

3 Methods

For obtaining the data to eventually determine D0, the report of Engels [2] was fol-lowed. This report describes in more detail why certain steps where taken and proof isgiven for the used assumptions. Also a short explanation is given in the previous sectionof this report. The actual fitting of the data is described in chapter 4.

3.1 Validating old parameters

As could be seen in the Eyring equation (equation 1), parameters for the pressuredependence (µ), initial stress (τ0), elasticity modulus (E), poison ratio (ν) and a pre-exponential factor (A0,rej) are to be determined.

Both µ and τ0 have the same definition as in the old model, therefore the same valuesthat where derived by Van der Aa [1] can be used. These values where derived andverified with data from the experiments from Leeds (1998) and Chrisfield (1971). Theseexperiments give values µ = 0.047 and τ0 = 0.9. These values will therefore be used inthe model from now on.

The elasticity modus is defined different in new model. This can be seen in figure 2.The difference is because of the new definition of the softening and the shift in the yieldpoint that was introduced because of that. The new value for E will be a higher becauseof this new definition.

Figure 2: Difference in the E-modulus in the old and the new model

The Poisson’s ratio is the same as for the old model, the value ν = 0.4 was taken fromVan der Aa [1]. The value for A0,rej is one of the main differences in the new approach.The report of Engels [2] gives both the old and the new approach for determining a valueA0, important is that A0,rej can be determined from the old data, using equation 4.

3 METHODS 6

3.2 Determining the softening parameters

To make determining the softening function possible, uniaxial compressive data is firststripped of the strain hardening component. The data from an uniaxial compression testgives a plot of the intrinsic behaviour as seen in figure 3. Here a plot is made of |λ2−λ−1|versus the true stress. This is done because the hardening stress is defined as:

σr = Gr(λ2 − λ−1) (5)

Here Gr is the hardening parameter and λ is the elongation, defined as λ = LL0

= 1 + ε.This implies that Gr[MPa] is the slope of the stress-strain curve during hardening. Thishas also been drawn in figure 3 (left), where Gr can be graphically determined.

With the hardening component removed, the resulting driving stress (σs) remains.The softening is now described from the yielding point (γ̄y) to the point with the pointwhere dσs

dε= 0. In figure 3 (right), the values of the yielding stress ∆σy and the rejuvenated

stress σrej can be determined. These values are used to scale the softening part of thestress behaviour by applying equation 6, where γ̄p =

√3(ε − εy).

g(γ̄p) =σs − σrej

∆σy

(6)

The definition for γ̄p follows from integration of the definition given for the equivalentplastic strain rate:

˙̄γp =√

2tr(Dp · Dp) (7)

Plastic deformation is only taken into account from the yield point (εy) and further,which implies Dp = D. In the case of uniaxial compression D is given by:

D =

ε̇ 0 00 −1

2ε̇ 0

0 0 −1

2ε̇

(8)

With these definitions ˙̄γp is defined as√

3|ε̇|. Now γ̄p can be determined by integratingequation 7 to time:

γ̄p =

∫ t

0

˙̄γpdt =√

3|ε̇|(t − ty) =√

3(ε − εy) (9)

The softening function as found by applying the earlier steps can now be approximatedusing a best fit based on the Yasuda model (Engels [2]):

gs(γ̄p) = (1 + (s0 · exp(γ̄p))s1)

s2−s1

s1 (10)

This function can then be fitted on the experimental data, with the help of a fitting routineor by hand. For graphicly determining the parameters s0, s1 and s2 it is necessary toknow the effect of the different parameters. These effects can be found in appendix A.

3.3 Determining D0

With all the parameters known except for the state parameter for the softening D0,it is possible to determine the value of this parameter by fitting simulations onto datafrom experiments. Because of thermo-mechanical history dependency, these experiments

3 METHODS 7

Figure 3: Determine softening and hardening parts: yield-point γ̄p and hardening parameterGr

have to be done with the material that needs to be described by the model and can notbe done with older or other data. The fit with the simulation can be done on a simpleuniaxial compression tests when we are working with bulk material, but because of thethin layer of polymer this is impossible. The thin layer can not be compared with thebulk material because of the difference material behaviour. The used samples were coatedsteel (240 [µm] with 30 [µm] layers of PET) of 10 x 30 [mm x mm], provided by Corus.No further pretreatment was performed.

The thin layer can be indented with a micro indentor, which gives a force-displacementdiagram. The simulations can be fitted with adjusting D0 and this way a value for D0 isfound. This indentation was performed at the Philips Natlab in Eindhoven on a microindentor. The used tip was a flat punch with a diameter of 10 [µm] and an indentationspeed of 1 [mN ] per second. These tests give the data to fit on with the MSC. MARCsimulations.

Another way to determine D0 is by fitting simulations of a uniaxial compression testonto the compression data. This is not possible for the thin layer, because it is notpossible to do compression test on the layer, but for bulk materials this is an usablesolution.

4 RESULTS 8

4 Results

This chapter only gives the steps as they where performed to manipulate the data andfind the parameters needed for the model. The first section describes the finding of thesoftening parameters, with which in the second section the simulations are performed.In this second section D0 should be found, but because of problems with the simulationsthis was not done. The next chapter describes some problems and probable solutions thework around those problems and find the value for D0. The second section does give avalue for D0, but this value was obtained by fitting the uniaxial compression tests of Vander Aa [1]. This value is therefore right for the experiments of Van der Aa, but not forthe PET layer, because of a different thermo-mechanical history.

4.1 Validated parameters

The parameters that are determined without the use of experimental data, or weredetermined from the old parameters, are given here.

The values µ = 0.047 [-] and τ0 = 0.9 [-] where already known and determined by Vander Aa [1]. These values will be used again. The value for the Elasticity modulus E wasgraphicly determined, as shown in the previous chapter. This gives a value E = 1100[MPa]. The value for ν was not changed and remained ν = 0.4 [-]. With the data fromVan der Aa and the use of equation 4, the old value of A0 = 1.23 · 1025 [-] and D∞ = 27.3[-] give a value of Arej = 1.72 · 1013 [-].

4.2 Determining softening parameters

0 0.5 1 1.5 20

10

20

30

40

50

60

70

80

90

| λ2 − λ−1 | [−]

Tru

e st

ress

[MP

a]

ε = 0.0025 s−1

ε = 0.025 s−1

ε = 0.25 s−1

Gr = 15 [MPa]

Figure 4: Determine hardening parameter: Gr (Data Van der Aa [1])

Following the method as described in the previous chapter, the uniaxial compressiondata of Van der Aa [1] was taken and the hardening component was removed by graphicly

4 RESULTS 9

determining the slope of the hardening part. This is shown in figure 4. Important to noticeis the fact that the hardening is not linear and therefore the parameter Gr depends on theposition where the slope is measured. For small deformations it is better to determinethe slope more to the left, for large deformations a position more to the right is to betaken. Here a position in the middle was taken, because this was the position where thevalue for Gr is the same for different strainrates. One of the reasons for the non-linearbehaviour of hardening, could be the friction in the compression test. With the chosenhardening parameter this friction is neglected.

From this figure also the yield point and the values for ∆σy and σrej can be determined,as described in the methods section. εy = 0.078 [-] is the value that was found and willbe used. These value are then used to determine the values of γ̄p which are used todetermine the softening function, by using equation 6. The values for ∆σy and σrej aregiven in table 1.

Strain rate [s−1] σref [MPa] ∆σy [MPa]0.25 38 35.00.025 36 33.00.0025 32 30.5

Table 1: Values of σrej and ∆σy for different strain rates

With only the softening part drawn in figure 5, the yasuda function can be fitted ontothe data. This was done graphicly in Matlab. It is vital to understand the influence ofthe different parameters, which is explained in appendix A.

0 0.1 0.2 0.3 0.4 0.5 0.6

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

γp,gem

[−]

g [−

]

Yasuda approximation

Figure 5: Fit of the Yasuda function with s0 = 0.992, s1 = 230 and s2 = −7.0 (Data Van derAa [1])

In figure 6 the first part of the fit, mainly determined by s0 and s1 is shown. Thedetermined parameters are s0 = 0.992 [-], s1 = 230 [-] and s2 = −7.0 [-]. For theseparameters it should also be noted that they are a compromise: as can be seen in figure

4 RESULTS 10

5 the fit follows closely in the first part and the last part, but in the middle the fit is notso good. The main reason for this is the fact that the data does not perfectly follow theexponential function used to fit, which can be seen clearly when the data is plotted on alogarithmic scale for the stress. The straight line of the fit is not followed, because of thefact that there is a bend in the data, see figure 7.

4.3 Determining D0

For the indentation at the Philips Natlab a flat indentor as shown in figure 8 was used.These indentation tests gave an force-displacement curve as can be seen in figure 9. Fromthe test it can be concluded that the indentation of the thin layer is fairly reproducible,although only one indentation speed was used: 1 [mN/s].

On the results of the indentation tests, a fit can be made with simulation data obtainedfrom MSC. MARC. The settings for the simulation can be found in appendix B. Becauseall the parameters are known with exception of D0, this last parameter can be determinedby fitting the simulation data on experimental data.

With the simulations some problems were encountered. These will be shortly ex-plained here.

The first problem encountered was element distortion. The Leonov model appears tobe very sensitive for the distortion of elements, and therefore the indentation simulationcould not run because no convergence was reached. The best elements to work with aresquare ones. A solution for the problem is the use of a remesher. In this study only theinternal MSC. MARC remesh procedures were used, because of a limited amount of time.

When using remesh, there are two general options. The first is the use of a localremeshing criterium, the second is the use of a global remeshing criterium. Both havetheir advantages and disadvantages. The local remesh is gives the possibility to only doa remesh on a part of the mesh, the part where remeshing is needed most. The problemis that the local remesh only divides the existing elements, which is not a solution for thedistortion problem, because the elements are still far from square. The local remesh doesgive lots of different criteria, but it can not be used in a way useful for the simulationsand therefore was not looked further into.

The global remesh does give the opportunity to do a total remesh of a contact body.The big advantage is the fact that the remesh lays a totally new mesh over the selectedarea, giving almost square elements after every remesh. This is a solution for the distor-tion problem and therefore global remesh can be used as a solution for this problem. Abig disadvantage of the global remesh procedure is the fact that it can only be done on acontact body, which means that a large area is going to be remeshed and that the remeshis not refined at the right position. The possibility to refine the mesh on the edges issome sort of solution, but this still gives only a slightly better result, because the wholeedge is refined, not only the edge where the indentation is performed. A solution for thislast problem is the use of more contact bodies, but this solution is not perfect becausethe edges of the contact bodies behave different then for one whole body.

With the use of global remesh, distortion is not a problem anymore, but the use ofremeshing introduces some other problems. The most important problem that makes theuse of the internal MSC. MARC remesher useless, is the fact that on the edges of thecontact bodies separation occurs after several remeshes. This only happens when theLeonov model is used, probably because of the fact that not all the data is transferredcorrectly. A second problem, also because of the data transfer to the new nodal points, is

4 RESULTS 11

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

γp,gem

[−]

g [−

]

Figure 6: Close-up of the first part of the softening function (Data Van der Aa [1])

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

10−1

100

γp,gem

[−]

g [−

]

Figure 7: Logarithmic description of the softening (Data Van der Aa [1])

4 RESULTS 12

Figure 8: Closeup of the indentor side- and topview

0 2 4 6 8 10 120

10

20

30

40

50

60

70

80

90

100

Indentationdepth [µm]

For

ce [m

N]

data1data2data3data4data5data6data7

Figure 9: Results of the indentation tests on the polymer layer, indentation speed 1 [mN/s]

4 RESULTS 13

a convergence problem. As stated above, the Leonov model is very sensitive and developsproblems in reaching convergence very easy. This also happens with the remesh, whenthe new nodal points do not have exactly the same values as the old ones, therefore givingthe model difficulties in reaching convergence. This problem can clearly be seen in figure10, where there are clearly visible jumps in the force.

0 0.002 0.004 0.006 0.008 0.01 0.0120

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Displacement [mm]

For

ce [N

]

Figure 10: First part of the simulation with Leonov model, clearly showing jumps in the forcebecause of remesh

The mentioned problems made it impossible to determine D0 on the intended way,therefore another solution was tried. This solution is described in the next section andmakes use of the earlier obtained data of Van der Aa [1].

4.4 Determining D0 with compression data

Because of the problems mentioned above, a different approach was chosen to finda value for D0. The compression data from Van der Aa [1] was used to make a fiton and determine a value for D0. This value is not the right value for the PET layer,because of a different thermo-mechanical history, but it shows how D0 can be found witha compression test.

The simulations were done with a 1 x 6 [mm x mm] round bar, as used by Van der Aa[1]. For the fit, only the value of D0 has to be changed, until the simulation data followsthe experimental data as good as possible. Within a couple of simulations, a value forD0 is found, and this value can be tested with different strainrates. This give figure 11,where it can be seen that the different strainrates are described well.

The results as given in figure 11 were obtained with a value for A0 = 2000 [-], for thereal value of A0 a similar procedure can be followed. Because of the time needed to runthe simulations this was not done again. The D0 found for these parameters was D0 = 20[-], which then can be used for the different strain rates. The three simulations in figure11 were all with the same parameters for the Leonov model, with only a difference in thestrain rate at which compression was performed.

4 RESULTS 14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

10

20

30

40

50

60

70

80

True strain [−]

Tru

e st

ress

[MP

a]

Exp. Data; ε = 0.25Exp. Data; ε = 0.025Exp. Data; ε = 0.0025Sim. Data; ε = 0.25Sim. Data; ε = 0.025Sim. Data; ε = 0.0025

Figure 11: Fits of the simulations for D0 = 20 [-] (and with A0 = 2000 [-])

A important thing that can be seen in the results of the fitting, is that the fit in thiscase is very good for small strains, but gets worse for large strains. There is no possibilityto get a good fit for both small and large strains, because of the non-linear behaviour ofthe hardening in the experiments and the linear approach of hardening in the simulations.Therefore, a choice must be made were to make the best fit. In this case small strainswere chosen, mainly because this is the easiest place to make the fit and because withthe data and an other hardening approach, the data should stay the same.

4 RESULTS 15

4.5 The parameters

The parameters found are for the material data of Van der Aa [1]. To determine theparameters for the PET layer a way must be found to make the MSC. MARC simulationsrun.

Parameter ValueE 1100 [MPa]ν 0.4 [-]Gr 14.5 [MPa]D0 5-30 [-]A0 1.72e13 [-]s0 0.992 [-]s1 230 [-]s2 -7.0 [-]∆H 230000 [J/mol]τ0 0.90 [-]µ 0.047 [-]Value 12 1.0Value 13 14.96Value 14 0

Table 2: Parameters for the Leonov model in MARC

The three last parameters are parameters of the Leonov model itself and cannot bechanged. Because of a wrong value used for A0 in the simulations to fit the compressiondata of Van der Aa [1], the value of D0 is not given. For A0 is 2000, the wrong assumptionmade for the simulations, D0 was found to be 20.5 [-].

5 MSC MARC PROBLEMS 16

5 MSC MARC problems

Because of the problems with convergence and separation with the use of the Leonovmodel for the indentation experiments, a different route was chosen to find the value forD0. Therefore an easier model was used, consisting of a simple elasto-plastic materialmodel with a Leonov like plastic behaviour. This model, referred to as Fake Leonov,was used to simulate the indentation experiments. These simulations where then used tocheck out some of the problems encountered with the Leonov model and to verify wetherthese problems are related to the Leonov model implementation or to the MSC. MARCinternal remesh procedure.

The used model is drawn in figure 12. For the following test that where performed,it is important to look at the size of the remeshed contact body, which is the part in thelower left corner of the model. The choice for the size of the contact bodies was made asfollows: the first contact body was made as small as possible, without getting distortionsat the edges because of the contact bodies pushing into each other. The second contactbody was used for remesh, but this seems to be unnecessary because nothing of theseremeshes can be found back in the results, except for extra jumps in the force. The uppercontact body does virtually nothing but holding the rest in place and the green contactbody is the steel part, mainly used as a placeholder also. All the simulations where donewith a loadcase of 1 second, and with standard parameters of 2000 elements, 2000 stepsand 250 steps before remesh. To look at the influence of the different parameters, onlythe considered parameter was changed.

The influence of the remesh can be seen in figure 13. As can be seen, the number ofremeshes has no significant influence on the results of the simulation.

The influence of the number of time steps chosen for the loadcase can be seen in figure14. Here a loadcase with a duration of 1 second was used, with 1000, 4000, 8000 and16000 steps. The figure clearly shows there is almost no influence as the number of timesteps is increased, although the force becomes a little bit higher. This increase in forceseems to be small enough to be neglected.

The influence of the number of elements in the remesh area was investigated and theresults are shown in figure 15. From the figure it can be concluded that the numberof elements used also does not have a significant influence on the simulations. Only500 elements seems to differ a little from the other results, but even this difference isnegligible.

From the above results it can be concluded that the internal remesher from MSC.MARC works for a simple material model on the indentation experiments. It can alsobe concluded that the number of elements, the number of time steps and the number ofremeshings do not influence the simulation a lot. Apparently the remesh does not workfor the Leonov model, because of the way it transfers data. It seems that not all the datais transferred for the Leonov model, and the data that is transferred by interpolation isnot close enough to the real values, which can be seen in the figures as the jumping inthe force.

The problems with the Leonov model and remeshing seem to be related to the in-ternal remeshing procedures of MSC. MARC. Therefore it would be wise to develop abetter insight in the internal remeshing procedures or to develop an external remeshingprocedure. The latter also has an other advantage, which is the possibility to use localrefinements on the positions most needed for the simulation.

5 MSC MARC PROBLEMS 17

cbody1

cbody2

cbody3

cbody4

cbody5

X

Y

Z

1

Figure 12: Simulations with different remesh criteria (Fake Leonov model)

1 2 3 4 5 6 7 8 9 10

x 10−3

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Displacement X [mm]

Rea

ctio

n F

orce

[N]

100 increments250 increments500 increments750 increments

Figure 13: Simulations with different remesh criteria (Fake Leonov model)

5 MSC MARC PROBLEMS 18

1 2 3 4 5 6 7 8 9 10

x 10−3

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Displacement X [mm]

Rea

ctio

n F

orce

[N]

1000 steps 4000 steps 8000 steps16000 steps

Figure 14: Simulations with different timesteps (Fake Leonov model)

1 2 3 4 5 6 7 8 9 10

x 10−3

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Displacement X [mm]

Rea

ctio

n F

orce

[N]

500 elements 1000 elements 5000 elements10000 elements

Figure 15: Simulations with different number of elements (Fake Leonov model)

6 CONCLUSIONS & RECOMMENDATIONS 19

6 Conclusions & Recommendations

From this study we can conclude that the new Leonov implementation has potentialand describes the real material behaviour well. The only part that could use some extraattention is the hardening, because the linear approach might not always be valid.

Further it was shown that the model as it is now, can not describe the whole range ofstrains. It fits on small strains or large strains, but not both for materials as PET. Thisis because of the way hardening is implemented.

The new Leonov implementation is usable to describe a simple compression test, butfor a more complicated indentation test the model seems to be a little unstable. This ismainly because of the problems with distorted elements, that are not handled right bythe model. A solution for this problem is remeshing the deformation zone.

The remeshing needed to keep the indentation simulations going shows problems whenusing the internal MSC. MARC remesh procedures. These procedures do not give thedesired options and do not transfer the data correctly to the newly created nodal points.Because of this last problem, the indentation simulations can not run with the new Leonovmodel, therefore preventing the user from obtaining the desired data, a value for D0.

A last conclusion to be drawn, is that if possible, compression data test are a betterand easier alternative to do the fitting on. Of course this means using bulk material,which can not be done in all cases.

To make the use of the new Leonov model easier and better, it I recommend lookinginto to option of developing external remesh procedures. These have the advantage ofmore transparency for the user, better control over the transferring of data to the newnodal points and more options for refining the mesh on the positions where this is mostneeded.

A second recommendation is to look at the description of the hardening in the model,possible to incorporate non-linear hardening in the model in the future. This can makethe model a better description for both small strains and large strains.

REFERENCES 20

References

[1] M.A.H. van der Aa, Wall ironing of polymer coated sheet metal, Ph.D. thesis, Tech-nische Universiteit Eindhoven, 1999.

[2] T.A.P. Engels, An investigation into the predictability of long-term ductile failure of

glassy polymers, Tech. report, Techn. University of Eindhoven, 2003.

[3] L.E. Govaert, P.H.M. Timmermans, and W.A.M. Brekelmans, The influence of in-

trinsic strain softening on strain localization in polycarbonate: modelling and experi-

mental validation, J. Eng. Mat. Tech. 122 (2000).

[4] O.A. Hasan, M.C. Boyce, X.S. Li, and S.J. Berko, An investigation of the yield and

postyield behaviour and corresponding structure of polymethylmethacrylate, J. Pol.Sci.: Part B: Pol. Phys. 31 (1993).

[5] E.T.J. Klompen, T.A.P. Engels, L.E. Govaert, and H.E.H. Meijer, Extending elasto-

viscoplastic modelling of large strain deformation of glassy polymers to incorporate

ageing kinetics, (2003).

[6] C.W. Macosko, Rheology; principles, measurements, and applications, VCH Publish-ers, New York, 1994.

[7] T.A. Tervoort and L.E. Govaert, Strain-hardening behaviour of polycarbonate in the

glassy state, J. Rheol. 44 (2000).

[8] T.A. Tervoort, R.J.M. Smit, W.A.M. Brekelsmans, and L.E. Govaert, A constitutive

equation for te elasto-viscoplastic deformation of glassy polymers, Mech. Time-Dep.Mat. 1 (1998).

A THE MODIFIED CARREAU-YASSUDA FUNCTION 21

A The modified Carreau-Yassuda Function

The modified Carreau-Yassuda function [6]:

g(γ̄p) = (1 + (s0 · exp(γ̄p))s1)

s2−1

s1 (11)

The parameters s0, s1 and s2 determine the output of the function. In figure 16 this isgraphicly shown.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−2

10−1

100

γp [−]

g s [−]

s0 = 0.965; s

1 = 50; s

2 = −5

s0 = 0.800; s

1 = 50; s

2 = −5

s0 = 0.965; s

1 = 10; s

2 = −5

s0 = 0.965; s

1 = 50; s

2 = −6.5

s0

s1

s2

Figure 16: The Carreau-Yassuda function with different values of the parameters s0, s1 ands2

It can be seen that s0 determines the position of the bendingpoint in the function.Decreasing s0 moves the bendingpoint to the right.

The value of s1 determines the sharpness of the bend. A larger s1 gives a sharperbend.

The last parameter, s2 determines the steepness of the function. The bigger s2 theflatter the function.

B MSC MARC SIMULATION SETTINGS 22

B MSC MARC simulation settings

The model to be used in MSC MARC uses hypoelastic material behaviour, withthe user subroutine hypola2 selected to incorporate the leonov material model.

Some extra settings have to be activated or deactivated in order to make the Leonovmaterial behaviour work as needed. The selection of the hypoelastic behaviour wasalready described, another option that must be checked is the deformation gradient,which must be set to the rotation tensor.

The elements chosen can not be of elementtype 80, every other element can be used.For the simulations in this report element type 10 was used (axisymmetric solid).

Constant dilation under geometry must be deactivated for the simulations, becausethis assumption is not valid for the indentation experiments.

The convergence testing parameter has to be small, to prevent having a solutionthat is to far away from the next solution in the next increment, which can fail to convergethen. This is because of the Leonov model, where requesting better convergence can inthe end speed up the simulations.

The constant timestep under loadcase should be chosen small, to prevent takensteps bigger then the element size and to prevent large differences in the results whichcan give convergence problems. Automatic timestep cut back should be enabled, tomake sure the simulation continues as the increment fails to converge. Normally this isnot a good thing to do, but for the Leonov model the option should be enabled to keepthe simulations running when convergence is not reached for an increment.

In the jobs section under analyses options the options large displacement andlarge displacement updated lagrange under rubber must be activated. Also large

strain multiplicative under plasticity must be set and updated lagrange as onlyoption under advanced job options.

In job parameters the user subroutine USDATA must be selected and user data

memory allocation should be at a large number, for example 1.000.000. # state

variables must be set to 33 and shell beamlayers must be at 1, otherwise the simu-lations will not run, this is because of the implementation of the Leonov model.

C FAKELEONOV INDENTATION SIMULATION 23

C FakeLeonov indentation simulation

Because of the problems with the Leonov model for the indentation simulation tests,a different approach was tried to find the D0 for the PET layer. The idea behind itwas to use the FakeLeonov material model to fit the indentation tests and then fit acompression simulation test with the Leonov model on a compression simulation testwith the FakeLeonov model. This approach did not work, but the results are given hereto show what was done. This approach has one big disadvantage, which is neglectingthe strainrate dependency of the Leonov model, because this is not implemented in theFakeLeonov material model.

The fit of the indentor tests with the FakeLeonov model were done, and these gave aresult as can be seen in figure 17.

Figure 17: Fit on the indentation data with the FakeLeonov model

This FakeLeonov model was build as an isotropic elastic plastic material, with anelasticity modulus of 1100 [MPa] and a yieldstress of 80 [MPa]. The softening and harden-ing was inserted with a table, with so called hardening going from 1 down to about 0.5and then up to higher values to incorporate hardening.

The problem with the model was the fact that hardening has almost no effect on theforces in the indentation tests, but has a much larger effect on the stress-strain relationsin the compression model. This makes it almost impossible to make a reliable fit, becauseboth simulations behave very different with respect to the parameters. Because of theproblem with the hardening, this approach was not followed and the earlier mentionedfit of the compression data of Van der Aa [1] was performed.