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Numerical simulation of flat-tip micro-indentation of glassy polymers: Influenceof loading speed and thermodynamic stateL. C. A. van Breemen a; T. A. P. Engels a; C. G. N. Pelletier a; L. E. Govaert; J.M.J. den Toonder a

a Department of Mechanical Engineering, Eindhoven University of Technology, 5600MB Eindhoven, TheNetherlands

Online Publication Date: 01 March 2009

To cite this Article van Breemen, L. C. A., Engels, T. A. P., Pelletier, C. G. N., Govaert, L. E. and den Toonder, J.M.J.(2009)'Numericalsimulation of flat-tip micro-indentation of glassy polymers: Influence of loading speed and thermodynamic state',PhilosophicalMagazine,89:8,677 — 696To link to this Article: DOI: 10.1080/14786430802441188URL: http://dx.doi.org/10.1080/14786430802441188

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Philosophical MagazineVol. 89, No. 8, 11 March 2009, 677–696

Numerical simulation of flat-tip micro-indentation of glassy polymers:Influence of loading speed and thermodynamic state

L.C.A. van Breemen, T.A.P. Engels, C.G.N. Pelletier,L.E. Govaert * and J.M.J. den Toonder

Department of Mechanical Engineering, Eindhoven University of Technology,P.O. Box 513, 5600MB Eindhoven, The Netherlands

(Received 8 May 2008; final version received 27 August 2008 )

Flat-tip micro-indentation tests were performed on quenched and annealedpolymer glasses at various loading speeds. The results were analyzed using anelasto-viscoplastic constitutive model that captures the intrinsic deformationcharacteristics of a polymer glass: a strain-rate dependent yield stress, strainsoftening and strain hardening. The advantage of this model is that changes inyield stress due to physical aging are captured in a single parameter. The twomaterials studied (polycarbonate (PC) and poly(methyl methacrylate) (PMMA))were both selected for the specific rate-dependence of the yield stress that theydisplay at room temperature. Within the range of strain rates experimentallycovered, the yield stress of PC increases linearly with the logarithm of strain rate,whereas, for PMMA, a characteristic change in slope can be observed at higherstrain rates. We demonstrate that, given the proper definition of the viscosityfunction, the flat-tip indentation response at different indentation speeds can bedescribed accurately for both materials. Moreover, it is shown that the modelcaptures the mechanical response on the microscopic scale (indentation) as well ason the macroscopic scale with the same parameter set. This offers promisingpossibilities of extracting mechanical properties of polymer glasses directly fromindentation experiments.

Keywords: instrumented indentation; polymer glasses; yield stress; mechanicalproperties

1. Introduction

Instrumented indentation is a versatile technique to probe local mechanical properties of films and/or bulk materials [1,2]. In principle, a well-defined body is pressed into thesurface of a material while measuring both the applied load and penetration depth. Theobtained data can subsequently be analyzed to determine the mechanical properties of theindented material. Regarding elastic modulus, in particular, quantitative analyticalanalysis methods are available [3,4]. With the aid of the elastic-viscoelastic correspondenceprinciple, these methods are also applicable to quantitatively assess viscoelastic properties[5–9]. With respect to large strain mechanical properties, the analysis of indentation data isless straightforward. Even for the determination of yield strength, a direct analyticalmethod of analysis is not available and an estimate can only be obtained using empirical

*Corresponding author. Email: [email protected]

ISSN 1478–6435 print/ISSN 1478–6443 onlineß 2009 Taylor & FrancisDOI: 10.1080/14786430802441188http://www.informaworld.com



scaling laws. Although these have been proven to be quite useful, the scaling factorbetween hardness and yield strength is not universal for all materials [10–12].

The development of FEM-based analysis methods has opened up new possibilities.Supported by the development of appropriate finite-strain constitutive relations, detailedanalysis of local deformation and stress fields is feasible. An excellent example is the workof Larsson’s group on Vickers [13] and Berkovich [14] indentation of elasto-plasticmaterials. In the case of polymeric materials, the analysis of such a contact problem iscomplicated due to the complex large strain behavior, characterized by a pronouncedstrain rate and pressure dependence of the yield stress and a post-yield response, which isgoverned by a combination of strain softening and strain hardening. In the case of amorphous polymer glasses, considerable effort has been directed towards the develop-ment of 3D constitutive models capable of capturing the experimentally observed intrinsicbehavior, especially by, for example, the group of Mary Boyce at MIT [15–17], the groupof Paul Buckley in Oxford [18–20], and in our Eindhoven group [21–23]. Thesedevelopments have enabled a quantitative analysis of localization and failure in polymerglasses [22,24–29] and revealed the crucial role of the intrinsic post-yield characteristics onmacroscopic strain localization.

Van Melick et al. [28] were the first to apply such a constitutive model to spherical-tipindentation of polystyrene (PS) for the analysis of radial craze formation. Theydemonstrated that the load–penetration depth curves could be well reproduced fordifferent indentation speeds by numerical simulations, using the Eindhoven GlassyPolymer Model (EGPM) [21]. In a subsequent study, Swaddiwudhipong et al. [30] showedthat the same model was unable to describe the Berkovich indentation response of anotherglassy polymer; polycarbonate (PC). To reproduce the response at different indentationspeeds correctly, they required an additional strain gradient effect. It should be noted,

however, that they adopted the parameters for polycarbonate from Govaert et al. [21]without verifying that this set was appropriate for the thermodynamic state of their ownpolycarbonate samples. In a more recent study, Anand and Ames [31] presented anextension of the BPA model [16], which proved successful in describing the conical–tipindentation of PMMA, albeit at a single indentation speed.

In the present study, we demonstrate that the constitutive model (EGPM), developedby our group [22], is also capable of quantitatively describing the indentation response of PC and PMMA over a range of indentation speeds. A flat-ended cone was chosen as theindenter body, since this specific tip geometry results in a load–penetration depth curve, inwhich elastic and plastic ranges are clearly distinguishable. At low indentation depths, the

response is governed by elastic deformation, whereas, at large depths, plastic deformationsets in, leading to a marked change in slope and resulting in a characteristic knee-shapedload–displacement curve [11,32,33]. Moreover, we will show that this is accomplished byusing a parameter set, which also quantitatively describes the materials mechanicalresponse in macroscopic testing.

2. Finite strain deformation of glassy polymers

2.1. Phenomenology

To study the intrinsic stress–strain response of polymers, an experimental set-up is

required in which the sample can deform homogeneously up to large plastic deformations.

678 L.C.A. van Breemen et al.



Examples of such techniques are uniaxial compression tests [15,34] or video-controlledtensile tests [35]. An illustrative example of the intrinsic stress–strain response of a polymerglass is presented in Figure 1a. Typical features are strain softening, the decrease in truestress that is observed after passing the yield point, and strain hardening at largedeformations. Strain hardening is generally interpreted as the result of a stresscontribution of the orienting molecular network [15,34,36,37]. Strain softening is closelyrelated to the fact that polymer glasses are not in a state of thermodynamic equilibrium.Over time, the glass will strive towards equilibrium, a process usually referred to asphysical aging [38] and, as a result, its mechanical properties change. This is demonstratedin Figure 1a, which compares the intrinsic response of two samples with different

thermodynamic state. It is clear that physical aging results in an increase of both modulusand yield stress but, upon plastic deformation, the differences between the curvesdisappear and eventually they fully coincide at a strain of approximately 0.3. Apparently,all influence of thermal history is erased at that strain and both samples are transformed toa similar, mechanically ‘rejuvenated’ state. From Figure 1a, it is clear that an increase of yield stress, due to a thermal treatment, will directly imply an increase in strain softening.The influence of molecular weight on the intrinsic response is usually small and negligible[20,22], which makes thermal history the key factor in influencing the intrinsic propertiesof a specific polymer glass. The thermal history is also reflected in the long-term failurebehavior of polymer glasses. This was demonstrated for PC, where an annealingtreatment, leading to an increase in yield stress, improved the life-time under constantstress by orders of magnitude [39].

The intrinsic stress–strain response of glassy polymers also displays a pronounceddependence on the time-scale of the experiment. This is illustrated in Figure 1b, where thestrain-rate dependence of the compressive stress–strain response of poly(methylmethacry-late) (PMMA) is shown [40]. It is clear that, with increasing strain rate, the overall stresslevel in the yield and post-yield range increases. Also, the amount of strain softening andstrain hardening appears subject to change. At strain rates over 3 Â 10À 2 sÀ 1 , the materialheats up due to viscous dissipation and, as a result, strain hardening disappears [41].

The strain-rate dependence of the yield stress for PMMA and PC is shown inFigure 2a. For the latter, the yield stress increases linearly with the logarithm of strain rate,

which indicates that in this range of strain rates the deformation of PC is governed by a

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

20

40

60

80

100

120

comp. true strain [ −]

c o m p .

t r u e s

t r e s s

[ M P a

]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

20

40

60

80

100

120

140

comp. true strain [ − ]

viscous heating

c o m p .

t r u e s

t r e s s

[ M P a

]

(a) (b)

Figure 1. Stress–strain response of PMMA in uniaxial compression: (a) influence of thermal history;annealed sample (- -) and quenched sample (-); (b) influence of strain rate.

Philosophical Magazine 679



single molecular relaxation process [42,43], i.e. the amorphous -transition (main-chainsegmental motion). Although this paper only focuses on the isothermal response, it isrelevant to note that the -stress contribution displays an Arrhenius-type temperaturedependence, which leads to a horizontal shift of the yield stress characteristic along thelogarithmic strain rate axis. This type of behavior is generally referred to as thermo-rheologically simple behavior.

In the case of PMMA, the strainrate dependence of the yield stress displays a clearchange in slope, which was shown to be related to onset of a stress contribution of asecond molecular process – the -transition [44,45] – a secondary glass transition related toside-chain mobility. A successful description of such a yield response is obtained using a

Ree–Eyring approximation, where, as schematically represented in Figure 2b, it is assumedthat each process can be described with an Eyring flow rule, whereas the stresscontributions of both molecular mechanisms are additive [46]. In the case of PMMA, itshould be noted that each process possesses its own characteristic activation energy,implying that curves measured at different temperatures will no longer coincide withhorizontal shifting. A correct translation to another temperatures can only be achieved byapplication of rate-temperature superposition on each contribution separately: this isgenerally referred to as thermo-rheologically complex behavior.

2.2. Numerical model

In the present study, we employ a 3D elasto-viscoplastic constitutive model that accuratelycaptures the intrinsic deformation characteristics of polymer glasses [21,23,39,47]. Thebasis of this constitutive model is the decomposition of the total stress into two separatecontributions, as first proposed by Haward and Thackray [36]:

p ¼ p s þ p r : ð1Þ

Here, p r denotes the strain hardening contribution that is attributed to molecularorientation of the entangled network, described here with a simple Neo-Hookean elasticexpression [21,37]:

p r ¼ Gr ~ Bd

, ð2Þ

10 −5 10 −4 10 −3 10 −2 10 −1

60

90

120

150 PMMAPC

y i e l d s t r e

s s

[ M P a

]

strain rate [s −1]log (strain rate)

y i e l d s

t r e s s

α

β

α +β(a) (b)

Figure 2. (a) Yield stress of PMMA and PC in uniaxial compression as a function of strain rate;(b) decomposition of the strain rate dependence of the yield stress into two separate molecularcontributions.




where Gr is the strain hardening modulus and ~ Bd the deviatoric part of the isochoricleft-Cauchy–Green strain tensor.

The driving stress p s is attributed to intermolecular interactions on a segmental scale[23,39] and is split into a hydrostatic ( p h

s ) and a deviatoric part ( p d s ) [47,48]:

p s ¼ p hs þ p d

s ¼ K J À 1ð ÞI þ G ~ Bd e , ð3Þ

where K is the bulk modulus, J is the relative volume change, G is the shear modulus, and~ B

d e is the deviatoric part of the elastic isochoric left Cauchy–Green strain tensor. The

evolution of J and ~ Be are implicitly given by

_J ¼ J Átr Dð Þ ð4Þ

~ Bo

e ¼ D d À D d p

Á~ Be þ ~ Be Á D d À D d

p

, ð5Þ

where ~ Bo

e is the Jaumann derivative of ~ Be, D d is the deviatoric part of the rate of deformation tensor. The plastic part of the rate of deformation tensor D p is crucial foradequate evolution of the driving stress, which is related to the deviatoric driving stress bya non-Newtonian flow rule:

D p ¼p d

s

2 " , p, S að Þ: ð6Þ

A correct expression for the solid state viscosity is essential in obtaining an accuratedescription of the 3D stress–strain response. For glassy polymers, this choice mainly

depends on the number of molecular relaxation mechanisms that contribute to the stress.In the simplest case, only a single molecular mechanism is active, the - process, and anexpression for can be obtained by taking the pressure-modified Eyring flow equation[22,49,50] as a starting point. In the isothermal case, this leads to

_" p " , p, S ð Þ ¼ _ 0, r Ásinh ð"= 0, Þ

|fflfflfflfflfflfflfflffl{zfflI

Áexp ðÀ p= 0, Þ

|fflfflfflfflfflfflfflfflII

Áexp ÀðS Þ

|fflfflfflfflffl{III

, ð7Þ

where _" p represents the equivalent plastic shear rate ( _" p ¼ ffiffiffiffiffi2tr ðD p ÁD pÞp ). The partmarked (I), where " is the equivalent shear stress ( " ¼

ffiffiffiffiffiffiffi1=2tr ðp d Áp d Þ

p ), represents the stress

dependence of the viscosity governed by the parameter 0 . Part (II), where p is thehydrostatic pressure ( p ¼ À 1=3tr ðp Þ), yields the pressure dependency governed byparameter . Part (III) represents the dependence of viscosity on the thermodynamicstate of the material expressed by the state parameter S . Finally, _ 0, r is a pre-exponentialfactor representative for the rejuvenated, unaged state, where the index refers to theidentity of the contributing molecular process.

Equation (7) leads to the following expression for the stress-dependence of the solidstate viscosity:

" , p, S ð Þ ¼"

_" p

" , p, S ð Þ

¼ 0, r Á"= 0,

sinh "= 0,

À Á" #Áexp p= 0,

À ÁÁexp S ð Þ, ð8Þ




where 0, r (¼ 0, = _ 0, r ) is the zero-viscosity for the rejuvenated state. The part betweenbrackets can be regarded as a stress-dependent shift factor, which equals one at equivalentshear stresses lower than 0, and decreases exponentially with increasing equivalent shearstress.

The dependence of the viscosity on physical aging and rejuvenation (strain softening) isincluded by defining [22]:

S ¼ S a tð Þ ÁR " pÀ Á: ð9Þ

Here, parameter S a can be regarded as a state parameter that uniquely determines thecurrent thermodynamic state of the material. Evolution of S a allows us to capturethe time-dependent change in mechanical properties as a result of physical aging [22]. Inthe present investigation, however, we will only consider materials with different initial S avalues (obtained by application of different thermal histories). The function R describesthe strain softening process, i.e. the erasure of thermal history with plastic deformation.

It is expressed as

R " pÀ Á¼1 þ r0 Áexp " pÀ ÁÀ Á

r1À Áðr2 À1Þ=r1

1 þ r r10À Á

r2À1=r1, ð10Þ

where r0 , r1 and r2 are fit parameters, and " p denotes the equivalent plastic strain. The initialvalue of R equals unity and, with increasing equivalent plastic strain, R decreases to zero.

The essence of the influences of physical aging and strain softening, modelled withinthe state parameter S (Equation (9)), is illustrated in Figure 3a, which shows the strainratedependence of the yield stress resulting from Equations (8) and (9). In the reference state,i.e. the fully rejuvenated state, parameter S a will be equal to zero. With physical aging, also

taking place during processing, the value of S a will increase, which leads to a shift in theyield stress versus strain rate characteristic along the logarithmic strain rate axis. At aconstant strain rate, the result will be an increase in yield stress compared to that of therejuvenated state. Upon deformation, the increasing equivalent plastic strain " p triggersstrain softening (Equation (10)) and the yield stress shifts back to that of the rejuvenatedstate. As a result, the yield stress drops with increasing strain and the intrinsic stress–straincurve evolves back to that of the rejuvenated state (see Figure 3b).

log (strain rate)

y i e l d s

t r e s s

aged

rejuvenatedS a

γ p

true strain

t r u e s

t r e s s

aged

rejuvenated

S a

(a) (b)

Figure 3. (a) Schematic representation of the influence of thermal history and strain softening on thestrain-rate dependence of the yield stress; (b) model prediction of the intrinsic stress–strain curveindicating the influence of physical aging.




It is important to note that, in the case of PC, the parameters in the model (see Table 1[39]) proved to be independent of the molecular weight distribution, and the keyparameter, required to adjust for differences in thermal history (illustrated in Figure 1), isthe initial value of the state parameter S : S a . This parameter can, in principle, bedetermined directly from the yield stress value of a single tensile test [22,39] or, if thethermal history during processing is known, be calculated directly [51,52].

Equations (7) and (8) both describe a linear increase in yield stress with the logarithmof the strain rate: a situation that can be observed for all glassy polymers, albeit over alimited range of temperature and strain rate. When studied over a sufficiently large rangeof temperature and strain rate, most glassy polymers reveal a change in slope that is relatedto the stress contribution of a secondary relaxation mechanism (generally referred to as the

-process) [43,45,53]. A successful description of such a yield response can be obtainedusing the Ree–Eyring model [46], where it is assumed that both molecular relaxationmechanisms act in parallel. This implies that the total stress can be additively decomposedinto their individual contributions:

tot ¼ þ : ð11Þ

This approach was successfully employed to describe the strainrate dependence of theyield stress of various amorphous and semi-crystalline polymers [44,45,53–56]. Anexpression for the individual stress contributions can be obtained by rewriting thepressure-modified Eyring flow expression (Equation (7)) in terms of equivalent shearstress:

" tot ¼ 0, Ásinh À1_" p

_ 0, r

Áexp p

0,

Áexp S að Þ

" #þ 0, Ásinh À1

_" p_ 0, e

Áexp p

0,

" #: ð12Þ

While the expression for the -contribution is equivalent to that of the -process, thereare two amendments. To a first approximation, we assume that the pressure dependence of the -contribution is identical to that of the -process. Moreover, since for the materialsunder investigation (PC and PMMA), the -contribution is already in its equilibriumstate [57] and does not change position during aging, there appears to be no use for a stateparameter S a, . The state is fixed by the equilibrium value of _" 0, ; _" 0, e .

Here, we follow an alternative route, where we capture the slope change in a singleviscosity expression. To accomplish this, we first approach the response of the material inthe þ -range, where both the - and the -process contribute to the stress, as a singleflow process:

_" pl " , pð Þ ¼ _ 0, þ Ásinh"

0, þ Áexp p

0, , ð13Þ

Table 1. Material parameters used for the numerical simulation of tensile and compression testson PC.

K [MPa] G [MPa] Gr [MPa] 0 , r[MPa/s] 0, [MPa] S a [–]

3750 321 26 2.1 Â 1011 0.7 0.08 À




where again it is assumed that the pressure dependence is identical to that of the -process.Changes in thermodynamic state are captured in the pre-exponential factor _ 0, þ .Equation (13) leads to an expression for the stress- and temperature-dependence of thesolid-state viscosity in the þ -range:

þ " , pð Þ ¼ " _" pl " , pð Þ

¼ 0, þ Á "= 0, þ

sinh "= 0, þÀ Á" #Áexp p= 0,À Á, ð14Þ

where

0, þ S að Þ ¼ 0, þ

_ 0, þ S að Þ; 0, þ ¼ 0, þ 0, , ð15Þ

and

_ 0, þ S að Þ ¼exp 0, ln _ 0, r

À ÁÀ S a

Â Ãþ 0, ln _ 0, e

À Á 0, þ

: ð16Þ

To obtain a single viscosity function that covers both the -range as well as theþ -range, we define the total viscosity as

tot " , pð Þ ¼ þ þ

¼ 0, r Á"= 0,

sinh "= 0,À Áþ 0, þ S að Þ

0, r Áexp S að ÞÁ

"= 0, þ

sinh "= 0, þÀ Á" #Áexp p= 0,À ÁÁexp S a ÁR " pÀ ÁÀ Á: ð17Þ

A schematic representation of this decomposition is given in Figure 4a. The expressionbetween brackets is again a stress-dependent shift factor. Its value equals one at shearstresses well below 0 , and decreases towards zero with increasing shear stress. An essentialconsequence of Equation (17) is that the - and the -contribution will both display identicalsoftening. As illustrated in Figure 4b, upon plastic deformation, the yield stress charac-teristic will shift horizontally along the logarithmic strain rate without any shape change.

α

βα +β

equivalent stress

l o g

( v i s c o s i t y

)

α

α +β

original rejuvenatedγ p

log (strain rate)

y i e l d s

t r e s s

(a) (b)

Figure 4. (a) Schematic representation of the decomposition of the stress-dependence of the viscosityof a thermo-rheologically complex material into two separate parts, the and viscosity functions;(b) influence of strain softening on the strain-rate dependence of the yield stress.




3. Experimental

3.1. Materials and sample preparation

3.1.1. Materials

The materials used in this study were polycarbonate (PC; Makrolon Õ , Bayer) obtained in

the form of extruded sheet of 3 mm thickness and poly(methylmetacrylate) (PMMA;Perspex Õ , ICI) obtained in the form of extruded rods of 6 mm diameter.

3.1.2. Sample preparation: PC For planar extension tests, rectangular samples with a dog-bone-shaped cross-section weremilled from the sheet [50]. The testing region has a thickness of 1.7mm over a length of 10 mm and a width of 50 mm. Owing to the large width-to-length ratio, contraction of thematerial is constrained, creating a plane strain condition [58]. For simple shear tests,samples similar to those used in the planar tests were used, now with a width of 100 mm

instead of 50 mm. With a gauge length of 10 mm, this results in an aspect ratio of 10.To avoid any influence of orientation effects due to the extrusion process, all samples weretaken from the same direction. For uniaxial tensile tests, samples according to ASTMD638 were milled from the extruded sheet. To avoid the influence of a processing-inducedyield stress distribution over the thickness of the samples [52], the tensile bars were milledto a thickness of 1.7 mm, i.e. identical to that of the test section of the samples for planarextension and shear.

To enable a direct comparison between the indentation tests and conventionalmacroscopic tests, indentation experiments were performed on a cross-section of a planarextension sample. A small specimen was cut from the gauge-section of the sample andsubsequently the cross-sectional surface was cryogenically cut, using a microtome, toobtain a smooth surface. Flat-tip indentation tests were performed in the middle of thesample area. For other indentation tests, samples of 10 Â 10mm were cut from theextruded PC sheet.

To change the thermodynamic state of the material, some of the samples were annealedat 120 C for 48 h in an air-circulated oven and subsequently slowly air-cooled to roomtemperature.

3.1.3. Sample preparation: PMMACylindrical samples of Ø 6 Â 6 mm were cut from the extruded rod. The end-faces of the

cylinders were machined to optical quality employing a precision-turning process with adiamond cutting tool. Indentation and uniaxial compression tests were performed on thesame samples. To vary their thermodynamic state, some samples were annealed at 95 Cfor five days in an air-circulated oven and subsequently slowly air-cooled to roomtemperature.

3.2. Techniques

Indentation experiments were performed using a nanoindenter XP (MTS Nano-Instruments, Oak Ridge, TN, USA) under displacement control. The geometry of the

tip was a flat-ended cone, chosen for the fact that the elastic and the plastic regions in the




load–displacement curve can be clearly distinguished. Unfortunately, this flat-tip geometryhas the drawback in that the force–displacement response is very sensitive to tip-samplemisalignment. This problem was solved by sample re-alignment using a specially designedalignment tool. Details on the alignment procedure can be found elsewhere [59]. Thegeometry of the tip was characterized using SEM and AFM and proved to have a tip-diameter of 10 mm (Figure 5a), a top angle of 72 (see Figure 5b), and an edge radius of 1 mm (Figure 5b).

Uniaxial compression tests were performed on a servo-hydraulic MTS ElastomerTesting System 810. The specimens were cylindrical-shaped and compressed under truestrain control, at constant true strain rates of 10 –4 to 10 –2 s –1 between two parallel, flat steelplates. Friction between samples and plates was reduced by an empirically optimizedmethod. Onto the sample ends, a thin film of PTFE tape (3M 5480, PTFE skived filmtape) was applied and the contact area between steel and tape was lubricated using a 1:1mixture of liquid soap and water. During the test, no bulging of the sample was observed,indicating that the friction was sufficiently reduced.

Uniaxial and planar tensile tests were performed on a Zwick Z010 tensile tester, atconstant linear strain rates of 10 –5 –10 –1 s –1 . Shear tests were performed on a Zwick 1475 atshear rates from 10 –5 to 10 –2 s –1 . Stress–strain curves were recorded and, whereappropriate, true stresses were calculated assuming incompressible deformation.

3.3. Numerical simulations

Axi-symmetric simulations were performed using MSC Marc/Mentat, a finite elementpackage. The constitutive model, as outlined in Section 2.2, was implemented in thispackage by means of the user-subroutine HYPELA2. The axi-symmetric mesh consists of 3303 linear quad4 elements, using full integration. The size of the mesh, which is0.05 Â 0.05 mm, is chosen such that the edges do not influence the stress distribution. Theindenter, a flat-ended cone with geometrical specifications as determined by SEM andAFM (see Figure 5), is modelled as an impenetrable body where no friction between

indenter and sample is taken into account.

0 2 4 6

2

4R=1 μm

h e

i g h t [ μ m

]

distance [ μm]

(b)(a)

Figure 5. Characterization of the tip: (a) top view SEM picture; (b) side view SEM picture with tipprofile obtained by AFM (lower picture).




The finite element mesh used for the simulation is shown in Figure 6. To exclude anymesh-dependence, a stepwise element refinement was performed until the solutionconverged to a steady, mesh-independent result. To prevent excessive computation times,the mesh refinement was restricted to areas of interest (Figure 6).

4. Results and discussion

4.1. Thermorheologically simple behavior: PC

4.1.1. Material characterizationIn the case of PC, only a single molecular process contributes to the yield stress, whichimplies that the viscosity function defined in Equation (8) can be applied. Besides theparameters in this expression ( 0, r , 0, , , r0 , r1 , r2), the model requires the determinationof the strain-hardening modulus Gr , the elastic shear modulus G and the bulk modulus K .Most of these parameters can be determined from fitting the results of uniaxial

compression tests at different strain rates. A proven strategy is to start by fitting theresponse of a rejuvenated material ( S a ¼ 0) on the strain-hardening regime of theexperimental curves, which yields the values for 0, , 0, r and Gr . Next, the softening canbe added and r0 , r1 , r2 and S a can be determined.

To enable these model simulations, we first need the values of the elastic bulk modulusK , the shear modulus G, and the pressure dependence . The value of K was calculatedfrom the values of the elastic modulus E and the Poisson ratio . The latter weredetermined in a uniaxial tensile test, yielding values of 2250 MPa for the elastic modulus E and a value of 0.4 for he Poisson ratio [47]. Using the interrelation

K ¼E

3 1 À 2ð Þ, ð18Þ

Figure 6. Mesh used to simulate indentation tests.




a value of 3750 MPa was found for the bulk modulus K . In the present, single modeapproach, the elastic shear modulus G is first chosen slightly too low such that thepredicted yield strain approximately equals that experimentally observed; this facilitatesthe characterization of the post-yield response. For polycarbonate, a value of G ¼ 320 MPaproved optimal.

An excellent method of obtaining pressure dependence is to perform experimentsdirectly under superimposed hydrostatic pressure [60–62]. Therefore, was determinedby numerically predicting the yield data obtained from compression tests at different true

strain rates and, finally, from the tensile tests under superimposed hydrostatic pressure, asreported by Christiansen et al. [60]. Figures 7a and b show that an excellent descriptionwas obtained using the material parameters given in Table 1 with an initial S a -value of 27.0for the compression and 34.0 for the yield experiments, representing the difference inthermal history between the two material sets used.

4.1.2. Macro-scale simulations

Since both rejuvenation and aging kinetics proved to be independent of the molecularweight of the polymer, the only unknown parameter in the model is the initial value of thestate parameter, S a , which can be directly determined from the yield stress, as measured ina simple tensile test at a single strain rate. This is demonstrated in Figure 8a, which showsthe results of uniaxial tensile tests at a strain rate of 10 –3 s –1 , for the as-received polymersheet as well as for a sample annealed for 48 h at 120 C. As a result of this thermaltreatment, the yield stress of the material increased substantially [22,63,64]. In both cases,the samples showed necking shortly after reaching the yield stress.

For both thermodynamic states, the S a -value is determined by matching theexperimental yield stress with the yield stress of a FEM-simulation using an axisymetricmodel of a tensile bar with a small imperfection in the middle (for details, see [65]). Thesimulations, shown in Figure 8a, yielded S a ¼ 31.7 for the as-received material and S a ¼ 39for the annealed material. With these values we have now obtained a complete parameter

set for both materials.

0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

true strain [ −]

t r u e s t r e

s s

[ M P a

]

0 250 500 750 10000

50

100

150

200

250

hydrostatic pressure [MPa]

t r u e y

i e l d s t r e s s

[ M P a

]

(a ) (b )

Figure 7. (a) Compression tests, experiments (open symbols) compared with numerical simulations(-) using material parameters as presented in Table 1 for three different true strain rates: 10 À 2 sÀ 1

(^ ), 10À 3sÀ 1 (œ ), 10 nms À 1 ( ); (b) yield stress versus superimposed hydrostatic pressure, modelprediction (-) compared to experimental results ( ) by Christiansen et al. [63] at a strain rate of 1.7Â10À 4sÀ 1 .




To demonstrate the capability of the model to describe the mechanical response indifferent loading conditions, we present, in Figure 8b, the strain rate dependence of theyield stress of the as-received material in planar extension, shear and uniaxial extension.The solid lines are predictions of the model, using the parameter set presented in Table 1with the S a value of 31.7 that was determined in Figure 8a. The predictions were obtainedby performing FEM-simulations on the actual sample geometries (see also [39]). It is clearthat an accurate, quantitative description is obtained. In the next section, we will

investigate the predictive capabilities of the model in micro-indentation.

4.1.3. Micro-scale simulationsBefore we apply the model to numerical simulations of micro-indentation tests, we firsthave to address an imperfection within the model. In the fitting procedure describedabove, we employed a shear modulus of 320MPa, which implies an elastic modulus of 900 MPa, considerably smaller than the 2250MPa observed in uniaxial extension. In thecase of indentation, where the elastic deformation will also significantly contribute atlarger depths, the low value of the modulus will lead to a drastic underestimation of thematerials’ resistance to deformation. The most elegant method to improve the descriptionof the pre-yield behavior is by means of a multimode extension of the present approach. Ina previous study, we demonstrated that the pre-yield response can be accurately capturedby a parallel arrangement of 18 modes [23]. Unfortunately, this solution wouldtremendously increase the required computational time. Instead, we chose to simplyincrease the shear modulus G until the initial slope of the compressive stress–strain curveswas properly described. This was achieved with G ¼ 784 MPa. To ensure that the yield andpost-yield response remain identical, this change of G subsequently requires an adaption of the rejuvenated zero viscosity 0, r . The resulting new data set is provided in Table 2.

The consequences of these changes are demonstrated in Figure 9. Figure 9a shows theinfluence of an increase in shear modulus G on the compressive stress–strain curve. The

initial modulus increases and the strain-at-yield decreases, leading to a shift in the yield

0 0.02 0.04 0.06 0.08 0.10

10

20

30

40

50

60

70

80

strain [ −]

s t r e s s

[ M

P a

]

10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 010

20

30

40

50

60

70

80

90

strain rate [s −1]

y i e l d s

t r e s s

[ M P a

]planar extension

uniaxial extension

simple shear

(a ) (b)

Figure 8. Experiments (open symbols) compared with the numerical simulation (-) on PC: (a) tensiletests at a strain rate of 10 À 3sÀ 1 for two different thermal histories with S a ¼ 31.7 for the as-received( ), and S

a¼ 39 (œ ) for the annealed material; (b) predicted yield stress at different strain rates,

Sa ¼ 31.7, for planar extension ( ^ ), uniaxial extension ( ) and shear ( œ ).




and post-yield response to lower strain values. As mentioned before, the value of the yieldstress as well as the shape of the post-yield behavior remains unaffected.The influence on the numerical simulation of a flat-tip indentation test is shown in

Figure 9b where experimental data of the as-received material is compared to numericalsimulations using the S a -value of 31.7 determined previously (Figure 8a). From Figure 9b,it becomes clear that a correct modulus leads to an accurate, quantitative prediction of theloading path of the indentation test, whereas the low modulus value leads to a largeunderestimation of the indentation resistance. Figure 9b also shows that the currentapproach is reasonably successful in capturing the unloading path. We expect that thisprediction could even be improved if a multimode approximation would be employed.

In Figure 10d, a characteristic loading curve of an indentation measurement is shown.In this figure, three points are marked: A, B and C. In Figures 10a–c, graphicalrepresentations of the development of the plastic deformation under the tip are given forthese three points. From the numerical evaluations, it is derived that the plasticdeformation starts at the edge of the indenter, as can be seen in Figure 10a, and grows inthe form of a hemisphere towards the symmetry axis. This is a result of the fact that stresslocalizes at the edge of the indenter. Around point B (Figure 10b), the plastic deformationzone concludes the formation of the hemisphere. From point C onwards (Figure 10c), thishemisphere then starts to expand in thickness. These results correspond well withexperimental observations made by others [11].

The strength of our approach is further demonstrated in Figure 11, which shows the

influence of thermodynamic state (Figure 11a) and strain rate (Figure 11b) on the

0 0.1 0.2 0.3 0.4 0.50

10

20

30

40

50

60

70

80

comp. true strain [ − ]

c o m p .

t r u e s

t r e s s

[ M P a

]

0 1 2 3 4 50

10

20

30

40

50

displacement [ μ m]

l o a

d [ m N ]

(a) (b)

Figure 9. Effect of the elastic modulus; (a) simulated compression tests for two different values of the elastic modulus: G = 321 MPa (- -) and G = 784 MPa (-) compared with the experiments ( œ );(b) the two different moduli, but now for the indentation tests.

Table 2. Material parameters with increased G, used for the numerical simulation of indentationtests on PC.

K [MPa] G [MPa] Gr [MPa] 0, r [MPa/s] 0, [MPa] S a [–]

3750 784 26 2.8 Â 1012 0.7 0.08 À




response in a flat-tip indentation test. In Figure 11a, the results of indentationexperiments (at a rate of 50 nm/s) on both the annealed and the as-received PC sheet arepresented. It is clear that the numerical simulations, employing the S a -values determinedin Figure 8a, quantitatively predict the load–displacement curves. The agreement is soprecise that the flat-tip indentation test can be used to determine the S a -value of a PCsample with an unknown prior thermal history. To achieve this, the load–displacementcurve has to be fitted with a numerical simulation. For PC, the S a -value can bedetermined in this way, in practice, with an accuracy of Æ1 (corresponding to a yieldstress inaccuracy of Æ1 MPa).

The feasibility of characterizing the S a -value from an indentation test is demonstratedin Figure 11b, which presents micro-indentation results at different indentation rates (5, 50and 200 nm/s). The material indented is a PC, which was annealed for a few hours at120 C. An S a -value of 34 was determined by fitting the numerical prediction to the load– displacement curve of 5 nm/s. The loading curves of the other indentation rates weresubsequently simulated with this value. The result is clear: the influence of strain rate is

also quantitatively predicted by our model.

(a) (b)

(d)(c)

0 1 2 3 4 50

10

20

30

40

50

displacement [ μm]

l o a

d [ m N ]

A

B

C

Figure 10. Simulation of the development of plastic deformation at different indentation depths forPC: (a) at 460 nm; (b) 965 nm; (c) 3 mm; (d) here, the points a, b and c indicate the load-displacementresponse for the different stages of plastic deformation.




4.2. Thermorheologically complex behavior: PMMA

4.2.1. Material characterization

In the case of PMMA, there are two molecular processes that contribute to the yield stress,which implies that the viscosity function defined in Equation (17) must be applied.This means that, besides the parameters already discussed in the previous section ( 0, r ,

0, , , r0 , r1 , r2 , Gr , G, and K ), we also have to determine the values of 0, þ and 0, þ .For the characterization, we make use of the fact that the -contribution is only present athigh strain rates and, therefore, the stress–strain response in the low strain rate range willbe determined by the -process only. As a consequence, we can use the samecharacterization strategy at low strain rates as employed in the previous section for PC.For the elastic properties of PMMA, we used a bulk modulus K of 3 GPa [66], theappropriate value of the shear modulus G was determined to be 760 MPa from the initialslope of the compressive stress–strain curves. To facilitate the fitting procedure, we initiallyadopted a lower value for G (630 MPa). After characterization of the -parameters, thevalues for 0, þ and 0, þ are subsequently determined by fitting numerical predictionsto the compressive stress–strain curves obtained at higher strain rates ( þ region).

To obtain the true value of the pressure-dependence parameter , we used a methodinspired by the work of Bardia and Narasimhan [67], who employed a sphericalindentation test to characterize the pressure sensitivity index of the Drucker–Pragerconstitutive model. Here, we follow a similar route. Since the compression tests and theindentation tests were performed on the same sample, the S a value is identical in bothcases. In the -range, the only unknown parameter is, therefore, the pressure dependence

. Similar to the approach for polycarbonate, we again generated different parameter setsby fitting the compression data for different values of . Here, it should be noted thateach set described the compressive stress–strain curves equally well (Figure 12a). Withthese data sets, we subsequently predicted the load–deformation curve for an indentationrate of 5 nm/s (see Figure 12b) and found that a value of ¼ 0.13 is in good agreement

with the experiment. The complete data sets, used for the predictions in Figure 12, are

0 0.5 1 1.5 2 2.50

5

10

15

20

25

30

l o a d

[ m N ]

displacement [ μm]0 0.5 1 1.5 2 2.5

0

5

10

15

20

25

30

l o a

d [ m N ]

displacement [ μm]

(a) (b)

Figure 11. Flat-tip indentation experiments (open symbols) compared with the numerical prediction(-): (a) as-received (S a ¼ 31.7) ( ) and for annealed (S a ¼ 39) (œ ) PC at an indentation speed of 50nms À 1 and (b) for speeds of 5 nm s À 1 ( ), 50nm s À 1 (œ ) and 200 nms À 1 (Á ) on the annealed PC(Sa ¼ 34).




tabulated in Table 3 (compression) and Table 4 (indentation). The corresponding S a valuewas determined to be 7.4.

Finally, indentation tests were performed at indentation rates of 5, 10, 20 and 40 nm/s.The results are compared to numerical predictions (employing the parameters listed inTable 4) in Figure 13a. It is clear that our numerical predictions are in excellent agreementwith the pronounced rate-dependence observed in the experimental force–displacementcurves. To demonstrate the presence of a -contribution in the indentation response, weperformed simulations of indentation tests at rates of 0.1 and 40 nm/s, with, as well aswithout, a -contribution (this implies 0, þ ¼ 0). The results are presented in Figure 13band show that, at low indentation rate (0.1 nm/s), the -contribution is negligible, whereas

at higher rates a significant contribution is visible.

0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

120

140

160

true strain [ −]

t r u e s

t r e s s

[ M P a

]

0 1 2 3 40

10

20

30

40

50

l o a

d [ m

N ]

displacement [ μm]

(a) (b)

Figure 12. Experiments (open symbols) compared to numerical simulation (-) for: (a) compressiontests ( ) performed on PMMA at a strain rate of 10 À 4sÀ 1 , 3Â 10À 4sÀ 1 , 10À 3sÀ 1 , 10À 2sÀ 1 ,3Â 10À 2sÀ 1and; (b) flat-tip indentation performed at 5 nm s À 1 (œ ).

Table 4. Parameter set with corrected G, used for the numerical simulation of indentation tests onPMMA.

K [MPa]

G[MPa]

Gr[MPa]

0,[MPa]

0, þ[MPa]

0 , r[MPa/s]

0 , þ

[MPa/s] S a ro R 1 r2

3000 728 26 2.71 7.05 9.27 Â 106 2.22 Â 105 0.13 7.8 0.96 30 À3.5

Table 3. Parameters used for the numerical simulation of compression tests on PMMA.

K [MPa]

G[MPa]

Gr[MPa]

0,[MPa]

0, þ[MPa]

0 , r[MPa/s]

0 , þ

[MPa/s] S a ro R 1 r2

3000 628 26 2.71 7.05 8.13 Â 106 2.12 Â 105 0.13 7.8 0.96 30 À3.5




5. Conclusion

In the plastic regime, glassy polymers possess a rather complex intrinsic behavior, with apronounced pressure- and rate-dependence of the yield stress as well as a post-yield regiondisplaying both strain softening and strain hardening. We employed a state-of-the-artconstitutive model, previously developed in our group, which describes this intrinsicbehavior, to numerically predict the indentation response. In the model, a singleparameter, the state parameter S a , is used to uniquely determine the initial yield stress of the material and capture all variations in its thermal history. We demonstrated that thismodel can capture the rate- and history-dependence of PC and PMMA on both themacroscopic and microscopic scale. The obtained accuracy of the description also createsthe possibility of extracting the state parameter S a directly from micro-indentationexperiments. This offers interesting possibilities with respect to quality control of load-bearing polymer products. Moreover, it was found that the pressure dependence of theyield stress can also be obtained by combining indentation and compression tests on thesame samples.

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Numerical Simulation of Flat-tip Micro-Indentation of Glassy Polymers-Holandeses