Craze initiation in glassy polymer systems

MT02.002

O.F.J.T Bressers

Master report

Supervisors:dr. ir. J.M.J. den Toonder Philips Researchir. H.G.H. van Melick TU/edr. ir. L.E. Govaert TU/eprof. dr. ir. H.E.H. Meijer TU/e

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Summary

This report deals with the numerical simulation of craze initiation in amorphouspolymer systems. The approach is based on the view that the development of a craze ispreceded by de formation of a localised plastic deformation zone. As this zone develops,the hydrostatic stress increases, and, when exceeding a critical stress level, cavitation willtake place leading to local development of voids. The voids grow, coalesce and theligaments between the voids are subsequently super-drawn leading to the typical structureof a craze: a crack-like defect bridged by highly drawn filaments.

In Part 1 a critical hydrostatic stress is examined as a cavitation criterion for polymersin a well-defined experiment. A micro indenter with a 150m sapphire sphere producesreproducible indents, which are later examined with an optical microscope. Theseobservations lead to a critical force where crazes are initiated in polystyrene (PS).Combination of these experiments with a numerical study using the compressibleLeonov-model showed that the loading part of the indentation can be accuratelypredicted. A critical hydrostatic stress of 39MPa is extracted from the numerical modelby analysis of the local stress field at the moment the indentation force reaches theexperimentally determined force level at which crazes were found to initiate. Thiscriterion is validated by application of the model to indentations on samples withdifferent thermal histories, and at various loading rates.

Varying the network density of the polymer showed that the incline of hydrostaticstress during indentation is not influenced. At higher network density, the samples startedto craze at higher forces and the corresponding hydrostatic stress is higher.

The cavitation criterion, validated in Part 1, is subsequently applied in Part 2 toanalyse the temperature dependence of the deformation behaviour of a heterogeneousPolystyrene/void structure. The material parameters are extracted from compression testsat different temperatures and then applied to the numerical model, the RVE. Firstly, it isshown that at low temperatures the deformation of the RVE is more local whereas at hightemperatures the deformation is more global. This is rationalised by the decrease of thestrain softening with increasing temperature. Secondly, it is shown that application of thecavitation criterion yields a macroscopic brittle-to-ductile transition at a temperaturebetween the 333K and 353K.

The indentation on thin PS films is described in Part 3. Thin films are made on a glassplate using a spin coat device. The layer thickness varies from 20nm up to 28m, thesolvent of the thick films is eliminated in three days in the oven. The films are indentedwith a micro indenter and a nano indenter. Small differences occurred between bothindenters. The indentations are also simulated with a numerical model. The thick filmsare simulated well. The experiments on the small films however showed a difference withthe simulations. Especially the first tenth of nanometers of the indentation deviatestrongly. The deviation appears to be related to the boundary conditions on thepolystyrene/glass interface.

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Contents

General introduction 1

Part 1 : Cavitation initiation of amorphous polymers 3Abstract 3Introduction 4Experimental 7

Materials 7Experimental set-up 7

Numerical methods 8Material model 8FEM model 10

Material characterisation 11Identification of a cavitation initiation criterion 14

Annealed reference sample 14Thermal history 17Influence of the loading rate 18Network density 19

Conclusions 20References 21

Part 2 : Numerical prediction of a temperature-induced brittle-to-ductile transition in polystyrene 23

Abstract 23Introduction 24Experimental and numerical methods 25Results 28

Uniaxial tensile tests 28Large strain deformation of a RVE 29

Conclusion 32References 33

Part 3 : Indentation on thin films 35Introduction 35Experimental and numerical modelling 36

Experimental 36Numerical modelling 38

Results 39Conclusion 41References 41

Appendix A 43

Appendix B 45

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1

General introduction

This report deals with the numerical simulation of craze initiation in amorphous polymersystems. The approach is based on the view that the development of a craze is preceded by deformation of a localised plastic deformation zone. As this zone develops, the hydrostatic stressincreases, and, when exceeding a critical stress level, cavitation will take place leading tolocal development of voids. The voids grow, coalesce and the ligaments between the voids aresubsequently super-drawn leading to the typical structure of a craze, a crack-like defectbridged by highly drawn filaments.

It is well known that the initiation and development of a localised plastic zone isdominated by the post-yield characteristics of the material. In the case of amorphous glasses,the post-yield behaviour is governed by two phenomena: 1) strain softening, leading to theinitiation of strain localisation, and 2) strain hardening, which stabilises the growth of thelocalised plastic zone. Subtle variations in the amount of strain softening or strain hardeningcan lead to extreme changes in the macroscopic deformation behaviour, changing the failuremode from tough to brittle. Over the past 15 years several constitutive models were developedthat are able to describe this complex post-yield behaviour, enabling us to numericallysimulate localisation phenomena in glassy polymers. Up till now, however, two factorshamper the adequate analysis of the stability of the deformation zone: 1) the absence of avalidated criterion that can be used to detect incipient cavitation, and 2) the limited knowledgeof the influence of ligament size on the intrinsic material properties. It are exactly theseproblems that are addressed in the present work.

The study consists of three parts. In Part 1 an approach with indentation is used to find, andvalidate, a cavitation criterion for polystyrene. Subsequently the influence of network densityon the resistance against cavitation is investigated.

The cavitation criterion, validated in Part 1, is subsequently applied in Part 2 to analyse thetemperature dependence of the deformation behaviour of a heterogeneous polystyrene/voidstructure.

In addition, indentation is used to investigate the mechanical behaviour of thin PS films.The results are presented in Part 3. At low layer thickness of films the material behaviour isdifferent from bulk material. The layers are spin-coated on glass and then indented with amicro indenter and a nano indenter. Experimental results are compared to results fromnumerical models.

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Part 1

Craze initiation in glassy polymers:Influence of thermal history, loading rate and network density

O.F.J.T. Bressers 1,2, H.G.H van Melick 1, L.E. Govaert 1, J.M.J. den Toonder 2, H.E.H. Meijer 1

1 Dutch Polymer Institute (DPI), Materials Technology (MaTe), Eindhoven University of Technology,P.O. Box 513, 5600 MB Eindhoven, The Netherlands

2 Philips Research Laboratories Eindhoven, Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands

Abstract

In this work a method is presented that can be used to predict craze initiation in glassypolymers. The approach is based on the view that the development of a craze ispreceded by de formation of a localised plastic deformation zone. As this zonedevelops, the hydrostatic stress increases, and, when exceeding a critical stress level,cavitation will take place, leading to local development of voids.The initiation of a localised plastic zone is numerically simulated using a constitutivemodel that incorporates an accurate description of the post-yield behaviour with theimportant phenomena of strain softening and strain hardening. Subsequent cavitationof the deformation zone is detected using a hydrostatic stress criterion. This criterionis identified and validated by confronting numerical simulations to experimentalresults of indentation experiments on polystyrene. A micro-indenter with a sapphiresphere of 150m radius is used to produce indents that are later examined with anoptical microscope. These observations lead to a critical force where crazes areinitiated in polystyrene (PS). Combination of these experiments with a numericalstudy using the numerical model shows that the loading part of the indentation can beaccurately predicted. A critical hydrostatic stress of 39MPa is extracted from thenumerical model by analysis of the local stress field at the moment the indentationforce reaches the experimentally determined force level at which crazes were found toinitiate. This criterion is validated by application of the model to indentations onsamples with different thermal histories, and at various loading rates. The influence of network on the value of the hydrostatic stress criterion is investigatedby indentation of blends of polystyrene and poly(2,6-dimethyl-1,4-phenylene-oxide).It is shown that the critical hydrostatic stress increases with network density.

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Introduction Macroscopic brittle fracture of glassy polymers is normally preceded by the formation of

crazes, small crack-like defects, bridged by super-drawn fibrils. As a result of these fibrils,crazes have, unlike real cracks, a considerable load-bearing capacity and when viewed on amicroscopic level, they display large plastic deformations. For this reason, crazes are the mostimportant source of fracture toughness in brittle glassy polymers, even though the volumefraction crazes during fracture is generally low. It is, therefore, not surprising that a vastamount of research has been done on all aspects of crazing: craze nucleation, growth andfailure, the micro-structure of crazes, the influence of molecular parameters, etc., and anumber of excellent reviews are available [1-3].

Figure 1 depicts some of the microscopic events that are likely to be involved in crazenucleation [1]. First, plastic deformation starts at a local stress concentration. The non-linearnature of the yield process and the strain softening character of polymer glasses will result ina localisation of deformation as the plastic strain increases. Since the material at somedistance of the local deformation zone is relatively undeformed, lateral stresses will develop.At this stage two things can happen. First, the strain-hardening response of the material canstabilise the strain-localisation process and the micro-shear band will spread out. Second, ithas been shown [4] that the hydrostatic stress required to plastically expand a micro-porousregion is greatly reduced if the material is in a state of flow. Therefore, if the lateral stressesbecome high enough, the material in the deformation zone will cavitate, and craze nucleationhas been accomplished.

Figure 1: schematic drawing of microscopic events involved in craze nucleation: a) formation of a localisedsurface plastic zone and build-up of lateral stresses, b) cavitation of the plastic zone and c) deformation of thepolymer ligaments between voids and coalescence of individual voids to form a void network (after Kramer [1]).

From the sequence of events mentioned above, it is clear that the macroscopic failurebehaviour of a glassy polymer is determined by two factors: 1) the intrinsic post-yieldbehaviour of the material, and 2) its resistance against cavitation.

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The post-yield behaviour of a glassy polymer is typically characterised by occurrence ofstrain softening and strain hardening [5]. Although the exact molecular origin of strainsoftening is still unknown, it is well documented that it is strongly influenced by the thermaland mechanical history of the material [5,6-12]. Slow-cooling rates tend to increase strainsoftening, whereas a quenching leads to moderate or, in the case of polyvinylchloride (PVC),even negligible softening [6]. The phenomenon appears to be related to the erasure ofphysical ageing effects by plastic deformation: mechanical rejuvenation. The origin of strainhardening is well established: contribution of the entanglement network as a result of stress-induced segmental mobility [5,11,13].

The extent in which the polymer will be susceptible to strain localisation is determined bya subtle interplay between strain hardening and strain softening. The initiation of localiseddeformation zones is governed by strain softening, which allows the zone to grow withdecreasing stress. The amount of strain hardening determines whether or not the deformationzone is stabilised. Strain softening, however, appears to be the key factor in these localisationphenomena. Upon removal, or strong reduction, of strain softening through thermal(quenching) or mechanical (plastic pre-deformation) methods, the occurrence of strainlocalisation is inhibited. Prime examples are the homogeneous deformation of quenched PVC[6] and the remarkable transition from crazing to macroscopic plastic flow in PS after amechanical treatment consisting of a thickness reduction of 30% by means of rolling [12].These effects are, however, of a temporary nature, as strain softening tends to return in time asresult of physical ageing.

In the past 15 years, considerable effort was directed towards the numerical simulation ofstrain localisation phenomena. The development of 3D constitutive models that were able tocapture the post-yield behaviour of glassy polymers started of with the work of Boyce and co-workers at MIT [14-17]. This work was later followed by studies of the group of van derGiessen [18-19] and our group [10,11,20-23]. As a result of these activities the numericalsimulation of plastic localisation in various loading geometries is now well established. Aninteresting application of these techniques is the field of polymer blends. It is well known thatblend morphology has a pronounced influence on the macroscopic toughness of the material.Numerical studies on the evolution of deformation in microstructures of various compositionsare now well within reach (see for instance the work of Smit et al. [23-26]), and offer newopportunities for fast material evaluation and development. However, in order to evaluate thetoughness of such systems numerically, an additional criterion is required to detect whether ornot a craze will initiate within the deformation zone.

Craze-initiation criterion that have been introduced so far generally involve a deviatoricstress component and a hydrostatic stress component [1,27,28]. Following the mechanism ofcraze initiation proposed by Kramer [1] (Figure 1), the formation of a local plasticdeformation zone is a prerequisite. As mentioned above, the formation of such a strainlocalisation is determined by the post-yield behaviour of the material, which is completelydetermined by the deviatoric state of stress. Moreover, the development of such localiseddeformation zones can be simulated effectively using numerical techniques.

Since the material at some distance of the local deformation zone is relatively undeformed,lateral stresses will develop. If the lateral stresses become high enough, the material in thedeformation zone will cavitate, and craze nucleation has been accomplished.

In previous work [29], it was investigated whether craze initiation could be envisaged as aplastic localisation process, followed by (rate-independent) cavitation of the deformed zones.It was shown that both craze initiation fracture and yield have the same strain ratedependence, described by a single Eyring process. This result strongly indicates that the

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occurrence of small plastic deformation zones is the rate-determining step in craze initiationand, therefore, that the on-set of cavitation may be described by a local, rate-independentcriterion. As the initiation of the plastic zone is already covered by the post-yield behaviour,this criterion does not necessarily involve a deviatoric stress component and hence a criticalhydrostatic stress h,c might be adequate.

The objective of this paper is to identify a local cavitation criterion, based on theassumption of a critical level of the hydrostatic stress. For this an experiment is used in whichthe defect-sensitivity of polystyrene is circumvented. As a spin-off of a previous project [30],indentation with spherical indenters on flat polymeric surfaces has proven to be an accurateexperiment to generate crazes in a well-controlled and reproducible way. At a certain loadduring indentation, pile-up of material occurs next to the contact area between indenter andpolymer. Beside plastic deformation, positive hydrostatic stresses evolve in this pile-up whichresults in small crazes just below the surface of the polymer.

To identify the local stress distribution within the pile-up, finite element simulations of theindentation experiment are performed. As a material model, the compressible Leonov-model[10,20,21] is used, which can capture the complex yield behaviour of glassy polymers quitewell. The capability of this model to describe the indentation of amorphous polymers wasalready shown by Melick [30].

The hydrostatic stress criterion is obtained by a combination of numerical simulations andexperimental observations. From the experiments the force is recorded at which the crazes areinitiated. Using a numerical simulation the critical hydrostatic stress is identified as themaximum hydrostatic stress in the pile-up zone at this specific indentation force.

The applicability of the obtained criterion is subsequently investigated by comparingnumerical predictions of craze initiation with experimental observations in various loadingrates (i.e. indentation force rates) and on samples with different thermal histories (quenchedand annealed PS). Using the same approach the influence of network density on the criticalhydrostatic stress is studied by indentation of blends of polystyrene and poly(2,6-dimethyl-1,4-phenylene-oxide) (PPO).

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Experimental

MaterialsThe base material used is a commercial grade polystyrene (Styron 634, Dow Chemical)

and two blends of polystyrene (PS) and poly(2,6-dimethyl-1,4-phenylene-oxide) (PPO 803,General Electric Plastic), that contain respectively 20 and 40% PPO. The granular material iscompression moulded into plates of 100x100x3mm in a stepwise manner. First the material ispre-heated in the mould at 90C above its glass-transition temperature (Tg) for 15 minutes.Next the material is compressed, at the same temperature, in 5 steps of increasing force (up to300kN) during 5 minutes. In between these steps, the pressure is released to allow degassing.Next the mould is placed into a cold press and cooled to room temperature at a moderate force(100kN). From the centre of these plates 9 small square platelets are cut (20x20mm). In orderto eliminate residual stresses and thermal history effects the platelets are heat-treated for 30minutes at 15C above Tg. Subsequently the platelets are given two thermal treatments knownas annealing and quenching. During annealing the samples are held at 20C below Tg for threedays and then slowly cooled in one day in the oven to room temperature. During quenchingthe samples are cooled rapidly in ice-water from 15C above Tg.

Experimental setupIndentations are performed with a micro indenter; a custom-designed built apparatus at

Philips Research Laboratories in Eindhoven. The forces that can be measured, range from20mN up to 20N with an accuracy of 2mN. The accuracy of the displacement is 20nm. Forcesand displacements are measured by means of coils at the bottom of the indenter column. Thespherical indenter used is a sapphire sphere, with a radius of 150m, glued onto a brassholder. The compliance of the apparatus is determined by a reference measurement on silicaglass. The elastic indentation depth-force curve is predicted by Hertz’ theory. From thedeviation between the theoretical and experimental curve, the compliance is determined to be6·10-2 µm/N, and the corresponding stiffness of the measuring system is 1.67·107 N/m.

A typical indentation procedure begins with a position-controlled movement of theindenter towards the sample until the surface is contacted with a pre-load of 5mN. Next theplatelet is loaded in force control up to a predefined maximum force at force rates rangingfrom 10mN/sec up to 1N/sec. When a predefined maximum force is reached the indenter isretracted in position control. The force required and the displacement of the indenter arerecorded during indentation. The experiments are carried out in a sequence of increasing forcesteps. At each step of 0.5N, the indentations are repeated at least 10 times. After theexperiments, the indents are examined using an optical microscope (Leica DM/RM) to checkwhether crazes are initiated by the applied force. This microscope uses 2 polarizers for theinterference contrast to visualise the crazes. The critical force for cavitation initiation isidentified by the indentation at which crazes occur first.

To obtain the material parameters, required for the numerical simulations, uniaxialcompression tests are performed on a servo hydraulic MTS Elastomer Testing System 810.Cylindrical specimens are compressed at room temperature, at a constant logarithmic strainrate between two parallel, flat steel plates. The friction between the sample and the steel platesis reduced using PTFE tape (3M 5480, PTFE skived film tape) onto the sample and a soap-water mixture on the surface between the steel and the tape. During the compression test no

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bulging or buckling of the sample is observed, indicating that the friction is sufficientlyreduced. The relative displacement of the steel plates is recorded by an Instron extensometer(Instron 2630-111). The displacements of the extensometer and force are recorded by dataacquisition at an appropriate sample-frequency (depending on strain rate). A constant truelogarithmic strain rate varying from 110-4 up to 110-2 s-1 is achieved in strain control.

Numerical methods

To obtain a quantitative relation for cavitation initiation, numerical simulations are carriedout using a finite element model of the indentation experiment. Information that can not beextracted from the experiments is derived from the numerical model, such as stresses andstrains in the indented material. By comparing the computed forces and displacements withthe experimental results, the numerical model can be validated to the experiment. From theexperiment the critical force is determined and from the simulation the accompanyingquantitative cavitation criterion is obtained.

The numerical simulations are only useful if the material behaviour of the experiment isrepresented well. The material model used will be described first including the materialparameters needed.

Material modelIn previous work, an elasto-viscoplastic constitutive equation for polymer glasses was

introduced, the so-called compressible Leonov-model [20,21]. To include strain hardeningand strain softening [10], the Cauchy stress tensor σ is composed of two contributions: Thedriving stress tensor s and the hardening stress tensor r respectively:

rsσ (1)

The expression for s is derived from the compressible Leonov-model [20]:

de1 B~I)(s GJK (2)

In this equation I is the unit tensor, the superscript d denotes the deviatoric part, and K andG are the bulk modulus and the shear modulus respectively. The relative volume change J andthe isochoric elastic left Cauchy Green deformation tensor eB~ are implicitly given by [20]:

)D(trJ (3)

)DD(B~B~)DD(B~ dp

dee

dp

de

o

(4)

The left-hand side of this equation represents the (objective) Jaumann derivative of theisochoric elastic left Cauchy Green tensor. The tensor D denotes the deformation rate tensor,Dp the plastic deformation rate tensor.

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The hardening behaviour of the material is described with a neo-Hookean relation for thehardening stress tensor r:

dR B~r G (5)

where GR is the strain hardening modulus (assumed temperature independent). The neo-Hookean approach shown in Equation (6) proved to be very successful in describing the strainhardening behaviour of polycarbonate in uniaxial compression, uniaxial extension and shear(torsion) [11].

It should be noted here that the strain hardening builds up gradually over the totaldeformation. In fact, Equation (1) implies that strain softening and strain hardening areregarded to act simultaneously. The constitutive description is completed as the plasticdeformation rate is expressed in the extra stress tensor by a generalised non-Newtonian flowrule:

),,(sD

pDeq

d

p 2 (6)

where eq, D and p are state variables to be defined in the following.

Particularly the driving stress tensor s is relevant for the incorporation of softening in themodel. As suggested by Hasan et al. [15] a history variable D is specified, the softeningparameter, which influences the viscosity η. During plastic deformation D evolves to asaturation level D, which is independent of the strain history. The result for η reads:

)/(/

),(),,(0eq

0eq0meq sinh

pDApD (7)

where the equivalent stress τeq is defined by:

)ss( dd21

eq tr (8)

and :

DpApDA

0m exp

),( (9)

)s()σ( trtr 31

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p (10)

where p is the pressure (positive in compression). The parameter is a pressure coefficient,related to the shear activation volume V and the pressure activation volume according to:

VΩ

(11)

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The evolution of the softening parameter D is specified according to Hasan et al. [15]:

p1

DDhD (12)

with initial condition D = 0; h is a material constant describing the relative softening rate andp is the equivalent plastic strain rate , according to:

)DD( ppp tr (13)

Most of the used parameters can be extracted from uniaxial compression tests with varyingloading rates and temperature. The model, presented above, was implemented in the MARC(MSC Software) finite element program using a subroutine [31].

FEM modelThe numerical model that describes the indentation consists of two deformable bodies: the

indenter and the examined material. The contact between the indenter and the PS is assumedto be frictionless, the influence is examined by varying the friction in the model and is foundto be negligible on the results. The model is modelled as an axi-symmetric problem (seeFigure 2). The dimensions of the PS should be large enough so that the indented region has noinfluence on the edges of the model, i.e. 10 times the indented region. The indenter ismodelled as a half sphere.

Figure 2: numerical model of the indentation

At the centre line of the model, the indenter and the PS are fixed in the y-direction toprovide axi-symmetry. The PS is fixed to the world at the right side. The indenter is preventedto move in x-direction when no load is applied using a very weak spring with a modulus of 1N/m. The force is applied to the left side of the indenter. The nodes of this side are linked toeach other and therefore the left side can only move uniformly in x-direction.

Two materials are used in the numerical simulations: sapphire and PS. The sapphire isassumed to be a linear elastic material with a Young’s modulus of 304GPa and a Poisson’sratio of 0.234, Simmons [32]. The parameters of the PS are extracted from uniaxialcompression tests and are represented in the results.

To analyse the model the options of large displacements, constant dilatation (to preventlocking) and updated Lagrange are used. To exclude any influence of mesh size a stepwiseelement size reduction is performed until the solution converges to a steady, mesh

y-direction

x-directionForce

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independent, result. In order to prevent excessive computation time the mesh refinement isrestricted to areas of interest. An example of a mesh is given in Appendix A.

Material characterisation

Bulk propertiesThe material parameters required for the numerical simulations are extracted from uniaxial

compression tests at room temperature. In this loading geometry localised deformation isinhibited. Therefore the true stress/true strain behaviour can be obtained up to large(compressive) strains. Figure 3 shows the rate dependence of PS at strain rates ranging from3·10-4 up to 3·10-3 s-1, which is representative for the strain rate in the PS during indentation.

Figure 3: rate dependence of polystyrene in an uniaxial compression test. It is, in Figure 3, observed that the strain rate predominantly affects the level of the (initial)

yield stress and has a less pronounced influence on the post yield behaviour: strain softeningand strain hardening. The solid lines in Figure 3 are predictions using the compressibleLeonov-model with the parameters indicated in Table 1. The model, described above,assumes linear elastic behaviour up to the yield point, and as a result the strain at yield isslightly underpredicted. As, in reality, the pre yield behaviour is (non linear) visco elastic, themodel will be less suitable for predicting the behaviour in the unloading part of theindentation. The rate dependence of the yield point is captured quite well by a single Eyringprocess. Especially for large strains, the softening is underpredicted for large strain rates andfor low strain rates overpredicted. However, during indentations the maximum strain is onlyslightly larger than the strain at the yield stress. Up to values of 0.2, the strain is describedwell including the softening, the differences between experimental and numerical results donot play a role at that moment.

The tests are performed with different strain rates to describe the A0, 0 and H. Othermaterial parameters that can be extracted are h, D and Gr. As mentioned before, the softeningis in fact rate dependent, however this is not modelled in the Leonov-model and therefore thesoftening is described using an average softening. Since the E and can not properly bedescribed they are extracted from a tensile test. is found in literature [33].

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Thermal historyThe effect of thermal ageing in glassy polymers is known for some time. It appears to be

first observed by Horsley [34] in 1958 on PVC. For PS it is described by Marshall and Petrie[35] and by Hasan and Boyce [7]. They showed that the thermal history of the sampleinfluences the material behaviour of a polymer. In Figure 4 the results are given of an uniaxialcompression test done on a quenched and an annealed sample.

Figure 4: dependence of polystyrene on thermal history in an uniaxial compression test.

The magnitude of the yield stress for the annealed sample is higher compared to thequenched sample. The strain hardening is however unchanged. From a true strain of 0.13 andmore the two results coincide, the thermal history does no longer influence the behaviourupon further straining. The material is, so called, mechanical rejuvenated, this effect isdescribed before by Hasan and Boyce [7]. Since the material behaviour is different, thethermal history also influences the material parameters; the A0 is increased for the annealedsample since the yield point is increased. The h and D also increase since the drop in stress ishigher for the annealed sample.

The indentation samples are first heated in an oven to eliminate stress and thermal history,next the samples are cooled. The cooling rate of the samples influences the thermal history ofthe samples. Slight differences between the thermal history of the samples of the indentationand the compression test are compensated with the A0. In Table 1 the material parameters aregiven for the samples used for the indentations.

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Network densityThe post-yield behaviour is strongly influenced by the network density. To investigate the

post-yield behaviour PS is compared with two PS/PPO blends containing 20% and 40% PPO.The network density increases when PPO is added to the PS. The influence of the networkdensity in an uniaxial compression test is given in Figure 5.

Figure 5: network density dependence of polystyrene in an uniaxial compression test.

From the Figure 5 it follows that when PPO is added to the PS, the yield stress remainslargely unaffected by an increase of the network density. The post-yield behaviour is stronglyinfluenced, the strain softening decreases with increasing network density. This must beascribed to the fact that the stabilising contribution of the entangled polymer network hasalready a large effect at smaller strains and hence the true stress can not drop as much as in alooser network. The solid lines in Figure 5 are predictions using the compressible Leonov-model with the parameters indicated in Table 1. The compressible Leonov-model provides agood description of the uniaxial compression tests.

The thermal history affects the modelling since the samples that are used for thecompression are not identical to the platelets used for the indentations and therefore themagnitude of the yield stress can vary. By varying the A0, the thermal history of the plateletsis taken care of. The A0 presented is the A0 used for the numerical indentations. The values ofA0 are obtained by fitting the data of the experimental indentations to numerical results. InTable 1 the material parameters are given for the platelets used in the indentations.

Type E[MPa]

[-]GR[MPa]

A0[s]

0[MPa]

H[MJ/mol]

Μ[-]

D [-]

h [-]

Quenched PS 3300 0.37 13 2.62108 2.6 0.1723 0.14 7 40

Annealedreference PS 3300 0.37 13 11012 2.6 0.1723 0.14 11 75

Ref. + 20%PPO 3000 0.37 25 11011 2.63 0.157 0.14 13 65

Ref. + 40%PPO 2700 0.37 50 1.4 1010 2.63 0.156 0.14 20 68

Table 1: material parameter sets for PS and PS blends with PPO

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Identification of a cavitation initiation criterion

Annealed reference sampleIn this section an indentation of a reference platelet is described. The reproducibility of the

indentations is discussed and the results of the experimental and the numerical indentationsare compared. The experiments result in a critical force and with the numerical model thecavitation criterion is examined.

The annealed reference platelet is indented more than a hundred times with differentforces. Typical load-displacements curves with varying maximum loads are presented inFigure 6.

Figure 6: indentation curves with an indenter (radius 150m) on PS, numerical and experimental results

The area between the loading and the unloading curve is the hysteresis work. This is thework done to deform the material plastically. The curves have the shape that is characteristicfor polymer indentation. The transition of loading-unloading is not sharp, but somewhatsmoothed. This is caused by the time dependent behaviour of the material. The remainingplastic depth is obviously larger for indentions with larger maximum loads. All the curves ofthis figure follow the same loading curve and therefore the reproducibility is good.

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At large indentation force crazes occur. The crazes begin at the edge of the contact areabetween the indenter and the sample and they are orientated in radial direction (see Figure7b). By using a special set-up, that is used to observe the initiation of crazing in situ, it isfound that crazes appear during the increasing load part of the indentation. Under amicroscope the crazes are counted. The number of crazes are presented as a function of themaximum applied force in Figure 7a.

a b

Figure 7: a, counted numbers of crazes at corresponding maximal indentation forcesb, typical photo of crazes around an indent.

The solid drawn line is the average of all crazes per maximum indentation force, theerrorbars are drawn at a probability of 95% with a normal distribution. The dotted lines areused to determine what force is needed to initiate crazes. For the annealed sample occur atlow loading forces no crazes and when the force is 1.5N, for the first time, crazes are seen.When the average line is extrapolated it is concluded that the craze initiation occur around the1.35N. From this point the number of crazes increases almost linearly. The dotted linesindicate that this value can vary from 1.05N to 1.65N.

The conclusion is drawn that pure annealed polystyrene crazes at a loading force of 1.35N,with an error of 0.3N, and a loading rate of 0.1N/sec.

The experiment is simulated with a numerical model with the correct material parametersfor the reference annealed platelet, as mentioned in Table 1. In Figure 6 both the outcome ofthe numerical simulation and the experimental data are given.

The loading part of the experiment is described well by the numerical model. Theunloading part, however has a different slope than the measured curves. This results in anoverprediction of the remaining plastic depth of the indent by the numerical model This ismost probably caused by the fact that the relaxation of the material is described by a singerelaxation mode [21]. A plastic deformation remains after the indentation, this is arequirement for the hypothesis of Kramer. The general conclusion is that the loading part ofthe indentation is described well and so the numerical model can be used to find a cavitationcriterion since the crazes initiates during loading.

16

At a certain load during indentation, pile-up of material occurs next to the contact areabetween indenter and polymer. Besides plastic deformation also a positive hydrostatic stressevolve in this pile-up. The hydrostatic stress is presented in Figure 8.

Figure 8: the numerical mesh with a closer look to the area before the contact zone at an indentation force of 1.35N

This is the same position where the crazes occur in the experiments. The highesthydrostatic stress is followed during the simulation of the previous experiment and isrepresented as a function of the force in Figure 9.

Figure 9: hydrostatic stress as a function of the maximal indentation forcefor an annealed and a quenched sample

The loading force increases from 0N to 4N, the resulting hydrostatic stress also has apositive incline. The highest value of the hydrostatic stress varies in place and magnitudeduring the simulation. This sometimes results in a staggered path of the hydrostatic stress, afit of the hydrostatic stress is also included (the solid line). In the experiment the platelet startscrazing at the force of 1.35N, the corresponding maximum hydrostatic stress in the numericalmodel is then 39MPa, based on the simulation. In Figure 9 also the upper and lowerboundaries of the critical force are given, the horizontal lines represent the correspondinghydrostatic stress. The critical hydrostatic stress varies then from 38 to 42.5MPa.

local hydrostatic stress

39 MPA

0 MPa

indent

polymer

17

The conclusion is drawn that the reference PS starts crazing at edge between the indenterand the sample. The numerical model describes the experiments well for the loading part, thecrazes are initiated during this part. During the simulation the material has a local plasticdeformation zone which is formed under the influence of the deviatoric stress, so based on theresults is tentatively concluded that a critical hydrostatic stress is adequate to describe thecavitation initiation. For the reference annealed PS the critical hydrostatic stress is 39MPa,with an error of -1MPa and +3.5MPa, which compares well to the value of 40MPa found byNarisawa [36].

Thermal historyTo validate the conclusion that the hydrostatic stress is appropriate to describe cavitation

initiation, the thermal history is varied. The annealed reference platelet is in this sectioncompared with a quenched platelet. The material parameters of the quenched sample arepresented in Table 1.

In Figure 4 it was already shown that the yield stress increases when the sample isannealed. When the indentation curves are observed this is also found since the indentationdepth decreases when the sample is annealed (see Appendix B). A look at the crazes showsthat the crazes of the quenched platelet occur at higher forces and are very small and thincompared to the reference platelet. The crazes of the quenched platelet are often very hard tosee and can only be visualised by using polarizers. The force to initiate crazes for thequenched platelet is 3.5N (-0.9/+0.6N), the errorbars are longer compared to the referenceannealed sample (see Figure 7).

The conclusion from the experiments is that the force needed to initiate the crazesincreases when the sample is quenched and that the crazes are smaller and hence moredifficult to observe. The simulated maximal hydrostatic stresses for the quenched and thereference platelet are given in Figure 9.

The incline of the hydrostatic stress for the annealed and the quenched sample is the sameup to 0.7N. When the hydrostatic stress is approximately 30MPa, the paths of the annealedplatelet and the quenched platelet begin to diverge. At that point the hydrostatic stress of theannealed platelet is larger, the cavitation criterion of 39MPa is reached at 1.35N for annealedPS. The hydrostatic stress of the quenched platelet is then 36MPa. The cavitation criterion of39MPa is for the quenched platelet reached when the indentation force is approximately 2.2N.The horizontal lines indicated the boundaries of the cavitation criterion, based on these linesthe conclusion is drawn that the quenched sample can initiate crazes from 2.05N to 2.9N. Thecavitation criterion holds within experimental error since in the experiments was shown thatthe 95% reliability ranges from 2.6N to 4.1N for the quenched sample.

The conclusion is drawn that a thermal treatment influences the crazing behaviour. Both inexperiments and in the simulations is shown that the crazes occur at a higher force, so thetrend is predicted well. The increase of the hydrostatic stress is different for distinct thermaltreatments. The exact force where crazes occur is underpredicted by the numerical model butwithin the experimental error, when the platelet is quenched. A possible explanation is thatthe value of the cavitation criterion is dependent of the thermal history, this was stated earlierby Ishikawa [37]. Another explanation is that because the crazes are hard to see when theplatelet is quenched, the mean value of the crazing force is too high in the experiments. Thebad visibility also makes the error in the estimated force larger.

18

Influence of the loading rateAnother validation of the failure criterion is done by varying the loading rate during

indentation. The stress-strain behaviour of PS is influenced by the loading rate as shown inFigure 3. The PS in this experiment is annealed and therefore the numerical model uses theparameters of the annealed material of Table 1. The loading rate of the loading force is variedfrom 0.01 to 1N/sec.

For low loading rates the indentation depth at the maximum force is smaller an also theremaining plastic depth is smaller (see appendix B). This difference in indentation depths areattributed to the visco-elasticity of the material. At low loading rates the time of theindentation is longer and the sapphire sphere can penetrate the PS more and the effect of thevisco-elasticity is present more strongly.

Figure 10 shows the force at which crazes in the experiment are observed as a function ofthe loading rate. The errorbars presented are calculated in the same way as described in thesection of the annealed reference sample. Also the corresponding critical force predicted bythe simulation when the platelets should craze is given for different loading rates. This is theforce where the hydrostatic stress becomes higher than the critical 39MPa found before.

Figure 10: influence of the loading rate at the force where crazes initiate, simulations , experiments

The critical force in the experiments where crazes initiate does not differentiate much, theyall start around the same force. The errorbars are all of the same length. In the numericalmodel the incline of the hydrostatic pressure is the same for all loading rates and therefore thecavitation criterion is reached at the same force for different loading rates. The predictedcritical force satisfies the experimental forces including the errorbars.

The conclusion of varying the loading rates is that the cavitation criterion is loading rateindependent. This is found both by the experiments and the numerical model.

19

Network densityIn the previous two sections the magnitude of the yield stress was changed and it was

shown that the crazes appearing could be explained by the cavitation criterion. In this sectionthe strain hardening is changed by adding PPO to the PS (see Figure 5). This influences thenetwork density of the polymer and it may affect the value of the cavitation criterion. Thesamples are all annealed because this results in clearly visible crazes.

The network density can influence the point where polymers start to craze. PPO is anamorphous polymer that can be mixed with PS to increase the stiffness of the network [38].The network density does not affect the magnitude of the yield stress much. The elasticmodulus decreases slightly when PPO is added. In this experiment the PPO content is variedin two steps: 20% and 40% PPO. The sample with 0% PPO is the reference sample. PPO hasa higher Tg than PS and the resulting Tg of the blend increases when PPO is mixed in the PS.

The depth of the remaining plastic indent decreases when the percentage PPO added to thePS increases. As the yield stress and the E do not vary much for the materials, the loadingcurves do not differentiate much (see Appendix B).

The force needed to initiate crazes is higher for an increasing PPO content in the sample.For the blend with 20% PPO the force is 3N (±0.2N), for 40% PPO this is 5N (±0.2N)according to the measurements. From the experiments the conclusion is drawn that the force,needed to initiate crazes, increases with the network density.

The experiments of the three blends are simulated with a numerical model. The usedmaterial parameters are given in Table 1. The corresponding hydrostatic stress for PS and 2blends is given in Figure 11.

Figure 11: hydrostatic stress as a function of the indentationforce for PS and 2 blends PS/PPO

The increases of the hydrostatic stress for the different blends follow the same path. Allcurves have a positive incline with an increasing indentation force. The critical force for PSwithout PPO is reached at 1.35N (as shown before), this corresponds with a criticalhydrostatic stress of 39MPa (-1/+2.5MPa). According to the simulations, the criticalhydrostatic stress for the 20% PPO is reached at a force of 3.0N and is at that moment 50MPa

20

(±2MPa). For the 40% PPO the critical indentation force is reached at 5.1N and thecorresponding hydrostatic stress is then 55MPa (-2/+1MPa).

PS has a network density of 3.0•1025 chains/m3, the blend with 20% PPO 4.9•1025

chains/m3 and the blend with 40% PPO 7.9•1025 chains/m3, Melick [38]. The criticalhydrostatic stress is given as a function of the network density in Figure 12.

Figure 12: critical hydrostatic stress as a function of the network density

The critical hydrostatic stress increases when the network density is increased. Theobserved trend is in full agreement with Sauer [39].

ConclusionsUniaxial compression tests provide material properties that in combination with a

compressible Leonov-model describe the experiments well. Indentations of PS with a microindenter lead to crazes and with the use of a numerical model the hypothesis of Kramer ischecked. Kramer stated that crazing of polymers begins with plastic deformation. Due to thenonlinear nature of the yield process the strain softening of the material results in alocalisation of deformation as the plastic strain increases while the hydrostatic stress also isbuilding up at that moment. The hydrostatic stress is the only parameter that initiatescavitation after the material shows local deformation and softening.

Crazes occur at the loading curve of an indentation and can be visualised by means of amicroscope. By counting the numbers of crazes, a critical force was determined. The crazesstart at the edge between indenter and platelets and occur in radial direction. The criticalhydrostatic stress is determined to be 39MPa (-1/+2.5MPa) for an annealed PS sample andoccur just outside the contact zone between PS and indenter.

The criterion is validated by varying the thermal history and the loading rate of theindentation. Quenched PS shows very tiny crazes at a higher force, the trend depicted by thenumerical simulation is good and the prediction by the cavitation criterion agrees with theobservations. The exact value of the crazing force for the quenched sample is hard to predict.Varying the loading rate does not influence the crazing force, both the numerical model andexperiments resulted in the same force. The post yield behaviour is changed by adding PPO tothe PS as the network density is increased. An increase of the network density result in a

21

higher value of the critical cavitation criterion, 50MPa (2MPa) for the blend with 20% PPOand 55MPa (-2/+1MPa) for the blend with 40% PPO.

It is concluded that the combination of a numerical model, covering the rate dependentintrinsic yield behaviour, and a local, rate independent, cavitation criterion can be usedeffectively to predict craze initiation in glassy polymers.

References

1. Kramer EJ, Adv. Pol. Sci., 1983;52/53:1-56.

2. Kramer EJ, Berger LL, Adv. Pol. Sci., 1990;91/92:1-68.

3. Kinloch AJ, Young RJ, Fracture behaviour of polymers, 2nd ed., Elseviers Appl. Sci. Publ.Ltd, London and New York, 1985.

4. Argon AS, Hannoosch JG, Phil. Mag., 1995;36(5):1195.

5. Haward RN, Young RJ, The Physics of Glassy Polymers, 2nd ed., Chapman & Hall, London,1997.

6. Cross A, Haward RN, Polymer, 1978;19:677-82.

7. Hasan OA, Boyce MC, Polymer, 1993; 34: 5085-92.

8. Bauwens JC, J. Mater. Sci., 1978;13:1443-8.

9. G’Sell C, In: Queen HJ, ed., Strength of metals and alloys. Oxford: Pergamon Press,1986:1943-82.

10. Govaert LE, Timmermans PHM, Brekelmans WAM, J. Eng. Mat. Tech. 2000; 122:177-185.

11. Tervoort TA, Govaert LE, J. Rheol. 2000; 4: 1263-77.

12. Govaert LE, Melick van HGH, Meijer HEH, Polymer, 2001;42:1271-1274.

13. Melick van HGH, Govaert LE, Meijer HEH, Polymer, submitted.

14. Boyce MC, Parks DM, Argon AS, Mech. Mater., 1988;7:15-33.

15. Hasan OA, Boyce MC, Li XS, Berko S, J. Polym. Sci., Part B: Polym. Phys., 1993;31:185-197.

16. Arruda EM, Boyce MC, Int. J. Plast.,1993;9:697-720.

17. Boyce MC, Arruda EM, Jayachandran R, Polym. Eng. Sci., 1994;34:716-725.

18. Wu PD, Giessen van der E, J. Mech. Phys. Solids, 1993;41:427-456.

19. Wu PD, Giessen van der E, Int. J. Plast., 1995;11:211-235.

20. Tervoort TA, Smit RJM, Brekelmans WAM, Govaert LE, Mech. Time-depend. Mat.,1998;1:269-291.

21. Tervoort TA, Klompen ETJ, Govaert LE, J. Rheol., 1996;40:779-797.

22. Smit RJM, Brekelmans WAM, Meijer HEH, Comp. Meth. Appl. Mech. Eng. 1998;155:181-192.

22

23. Smit RJM, Brekelmans WAM, Meijer HEH, J. Mech. Phys. Sol., 1999;47:201-221.

24. Smit RJM, Brekelmans WAM, Meijer HEH, J. Mat. Sci., 2000;35:part1 2855-2867.

25. Smit RJM, Brekelmans WAM, Meijer HEH, J. Mat. Sci., 2000;35:part2 2869-2879.

26. Smit RJM, Brekelmans WAM, Meijer HEH, J. Mat. Sci., 2000;35:part3 2881-2893.

27. Kambour RP, J. Polymer Sci: Macromolecular Reviews, 1973;7:1-154.

28. Kausch HH, Polymer Fracture, Polymers/Properties and Applications, 1978.

29. Govaert LE, Tervoort TA, Meijer HEH, Polymer, submitted.

30. Melick van HGH, Dijken van AR, Toonder JMJ, Govaert LE, Meijer HEH, Phil. Mag. A,2002, accepted.

31. Aa van der MAH, Schreurs PJG, Baaijens FPT, Mech. of Mat., 2001;33:555-572.

32. Simons G, Wang H, Single Crystal Elastic Constants and Calculated Aggregate Properties: AHandbook, 2nd ed., 1971.

33. Melick van HGH, PhD thesis, Eindhoven University of Technology, Department of MaterialsTechnology, 2002.

34. Horsley RA, Plastic progress, Iliffe ans Sons, London and New York, 1958:77.

35. Marshall AS, Petrie SEB, J.Appl. Phys., 1975: 46: 4223.

36. Narisawa, Yee, in E.L.Thomas: Mat. Science and technology, 1993, vol.12.

37. Ishikawa, Narisawa, Ogawa, J. Pol. Sci, Pol. Phys., 1977; 15:1791.

38. Melick van HGH, Govaert LE , Meijer HEH, Macromolecules, submitted.

39. Sauer JA, Hara M, Advances in Polymer Science 91/92, pag 82.

23

Part 2

Numerical prediction of a temperature-induced brittle-to-ductiletransition in polystyrene

O.F.J.T. Bressers , H.G.H van Melick , L.E. Govaert , H.E.H. Meijer

Dutch Polymer Institute (DPI), Materials Technology (MaTe), Eindhoven University of Technology,P.O. Box 513, 5600 MB Eindhoven, The Netherlands

Abstract

In this paper the transition from brittle to ductile deformation behaviour, underinfluence of temperature, is studied by means of finite element simulations. As themacroscopic deformation behaviour of amorphous polymers is determined by theirpost-yield behaviour, influencing this behaviour can results in major changes. Besidestreatments like cross-linking and mechanical pre-conditioning, also the testingconditions proved to play a crucial role. Recently a failure criterion was reportedwhich identifies a critical hydrostatic stress as the criterion for craze initiation,provided this is preceded by plastic deformation. Combining this criterion with theadequate description of the deformation behaviour by the compressible Leonov-model, a tool is found which is able to predict whether a structure will deform in abrittle or ductile manner. Therefore in this paper, a heterogeneous structure isdeformed in a tensile test. At a single strain rate and various temperatures simulationsare performed to study the macroscopic response of voided polystyrene. At atemperature of 353K a critical hydrostatic stress of 39MPa is exceeded, at 10%macroscopic strain, in this structure and a transition is found from brittle to ductile inthese simulations. These results correlate well which experimental observations.

24

Introduction

The macroscopic deformation behaviour of amorphous polymers is determined by theirpost-yield behaviour, i.e. strain softening and strain hardening [1,2,3]. This strain softening, acharacteristic feature of most amorphous polymers, induces a localisation of strain in smallmaterial volumes. Depending on the amount of strain softening and contribution of the strainhardening, these strain localisations might be stabilised. In case of polystyrene, exhibiting apronounced strain softening and a weak strain hardening, these strain localisations evolve toextremes. The mechanism of craze initiation, as proposed by Kramer [4], induces a built-up ofhigh hydrostatic stresses in these plastic zones and as a result, voids can nucleate. Withongoing strain these voids coalescence and form a void network, also known as a craze. Thebreak-up of these crazes ultimately results in macroscopic brittle fracture.

By influencing the strain hardening and strain softening, the macroscopic response of apolymer can be tailored. Henkee and Kramer [5] showed that by cross-linking ofpolymethylmethacrylate (PMMA) the strain hardening can be enhanced, resulting inmacroscopic ductile behaviour. On the other hand it was shown by many studies that by pre-conditioning of polymers the strain softening can be strongly reduced or even eliminated[1,6,7,8]. In a subsequent tensile test, localisation of strain was inhibited resulting in morehomogeneous and ductile deformation behaviour.

But this can not only be achieved by special treatments, also a careful choice of testingconditions, like strain rate pressure and temperature, can result in a remarkable increase ofductility. Already in the early sixties it was known that a brittle-to-ductile transition (BDT) inan uniaxial tensile test can be achieved in polystyrene by elevating the test temperature tosome 80-90 K above room temperature [9].

Since the early Haward and Thackray model a lot of research effort has been put into thenumerical description of the deformation behaviour of amorphous polymers. The modelsproposed by Wu and van der Giesen [10,11], Boyce et al. [12-15] and Govaert et al. [1,2,16-19] proved that they can adequately describe this complex deformation behaviour.

Despite these research efforts, a prediction whether a material would deform in a brittle orductile manner was still not possible as a failure criterion was still lacking.

Recently a failure criterion was proposed by Bressers et al. [20]. They demonstratedinitiation of crazes in polystyrene at a critical hydrostatic stress of 39MPa, provided that inthis region cavitation was preceded by plastic deformation. This sequence of events wasconsistent with the mechanism of craze initiation as proposed by Kramer [4].

The numerical description of the intrinsic deformation behaviour by the compressibleLeonov-model, as proposed by Govaert et al. [8,16,17] and the failure criterion proposed byBressers et al. [20] are combined here to predict the macroscopic deformation behaviour ofpolystyrene. As inhomogeneous deformation is a prerequisite to study such a transition, arepresentative volume element (RVE) [19,21-23] is deformed during these simulations. Thematerial parameters of polystyrene used in these simulations are provided by uniaxialcompression tests.

25

Experimental and numerical methods

Experimental methodThe experimental verification of a brittle-to-ductile transition was done by uniaxial tensile

tests at various temperatures. The tests were performed on a servo hydraulic MTS ElastomerTesting System 810, equipped with a temperature chamber. Dumbbell shaped (ASTM D538)tensile bars of PS (N5000, Shell) were subjected to an uniaxial tensile test at a linear strainrate of 10-3 s-1 and a temperatures ranging from 20oC up to 100oC The temperature in the ovencould be controlled at an accuracy of 0.5oC. Prior to testing the tensile bars were mounted inthe clamps and left for 15 minutes at the testing temperature to regain thermal equilibrium.The displacements and forces were recorded during the test by the control unit at anappropriate sample frequency.

To obtain the material parameters, required for the numerical simulations, uniaxialcompression tests were performed on the same servo hydraulic system. Cylindrical specimenswere made and next compressed at different temperatures and logarithmic strain ratesbetween two parallel, flat steel plates. The friction between the sample and the steel plateswas reduced using PTFE tape (3M 5480, PTFE skived film tape) onto the sample and a soap-water mixture on the surface between the steel and the tape. During the compression test nobulging or buckling of the sample is observed, indicating that the friction is sufficientlyreduced. The relative displacement of the steel plates is recorded by an Instron extensometer(Instron 2630-111). The displacements of the extensometer and force were recorded by dataacquisition at an appropriate sample-frequency (depending on strain rate). A constant truelogarithmic strain rate varying from 110-4 up to 110-2 s-1 is achieved in strain control.

The material parameters of PS are extracted from uniaxial compression tests at differenttemperatures. In Figure 2.1 are the results of these tests for a constant true logarithmic strainrate of 0.001 s-1, the solid lines are the corresponding simulations using the materialparameters of Table 2.1. The temperature strongly influences the yield stress: the magnitudeof the yield stress decrease with increasing temperature. The modulus of the polystyrene andthe strain hardening also decrease with increasing temperature.

Figure 2.1: uniaxial compression test with different temperatures

26

The corresponding material parameters of PS for different temperatures are given in Table2.1.

Type E[MPa]

[-]GR[MPa]

A0[s]

0[MPa]

H[MJ/mol]

µ[-]

D [-]

h[-]

PS, 293K 3000 0.37 13 1.38·109 2.6 0.1723 0.14 6.8 60PS, 313K 2890 0.37 9 1.50·108 2.6 0.1723 0.14 6.0 60PS, 333K 2730 0.37 6 3.64·107 2.6 0.1723 0.14 4.8 70PS, 353K 2570 0.37 3 5.75·105 2.6 0.1723 0.14 3.4 60

Table 2.1: material parameters for PS at different temperatures

Numerical method

In previous work, an elasto-viscoplastic constitutive equation for polymer glasses wasintroduced, the so-called compressible Leonov-model [16,17]. To include strain hardeningand strain softening [8], the Cauchy stress tensor σ is composed of two contributions: Thedriving stress tensor s and the hardening stress tensor r respectively:

rsσ (14)

The expression for s is derived from the compressible Leonov-model [16]:

de1 B~I)(s GJK (15)

In this equation I is the unit tensor, the superscript d denotes the deviatoric part, and K andG are the bulk modulus and the shear modulus respectively. The relative volume change J andthe isochoric elastic left Cauchy Green deformation tensor eB~ are implicitly given by [16]:

)D(trJ (16)

)DD(B~B~)DD(B~ dp

dee

dp

de

o

(17)

The left-hand side of this equation represents the (objective) Jaumann derivative of theisochoric elastic left Cauchy Green tensor. The tensor D denotes the deformation rate tensor,Dp the plastic deformation rate tensor.

The hardening behaviour of the material is described with a neo-Hookean relation for thehardening stress tensor r:

dR B~r G (18)

where GR is the strain hardening modulus (assumed temperature independent). The neo-Hookean approach shown in Equation (6) proved to be very successful in describing the strainhardening behaviour of polycarbonate in uniaxial compression, uniaxial extension and shear(torsion) [1].

27

It should be noted here that the strain hardening builds up gradually over the totaldeformation. In fact, Equation (1) implies that strain softening and strain hardening areregarded to act simultaneously. The constitutive description is completed as the plasticdeformation rate is expressed in the extra stress tensor by a generalised non-Newtonian flowrule:

),,(sD

pDeq

d

p 2 (19)

where eq, D and p are state variables to be defined in the following.

Particularly the driving stress tensor s is relevant for the incorporation of softening in themodel. As suggested by Hasan et al. [13] a history variable D is specified, the softeningparameter, which influences the viscosity η. During plastic deformation D evolves to asaturation level D, which is independent of the strain history. The result for η reads:

)/(/

),(),,(0eq

0eq0meq sinh

pDApD (20)

where the equivalent stress τeq is defined by:

)ss( dd21

eq tr (21)

and :

DpApDA

0m exp

),( (22)

)s()σ( trtr 31

31

p (23)

where p is the pressure (positive in compression). The parameter is a pressure coefficient,related to the shear activation volume V and the pressure activation volume according to:

VΩ

(24)

The evolution of the softening parameter D is specified according to Hasan et al. [13]:

p1

DDhD (25)

with initial condition D = 0; h is a material constant describing the relative softening rate andp is the equivalent plastic strain rate , according to:

)(tr ppp DD (26)

28

Most of the used parameters can be extracted from uniaxial compression tests with varyingloading rates and temperature. The model, presented above, was implemented in the MARC(MSC Software) finite element program [24].

As inhomogeneous deformation is a prerequisite to study localisation of strain and crazeinitiation, the heterogeneous structure is used here. The numerical model used consists of aRepresentative Volume Element (RVE) [21-23], a voided polymer matrix. The RVE is usedin multi-level finite element simulations (MLFEM) [21-23] to couple the deformation of anheterogeneous structure on micro-level to the deformation of a structure on macro-level. Theprocedure to generate an RVE is described in Smit [21-23] and starts with the filling of unitbox with 20% randomly placed mono-sized spheres (radius equals 3% of the dimension of thebox). Next an arbitrary cross-section is taken from this box and this geometry is meshed bythe auto-mesher of Mentat (MSC Software) which is represented in Figure 2.2. The structureconsists of 5470 8-noded, second-order elements. To prevent rigid-body movement, the lowerleft corner of the mesh is fixed. Of the upper left corner the displacement in x-direction isinhibited. The lower right corner of the mesh a predefined linear strain rate of 10-3 s-1 isapplied to deform the structure. Furthermore periodic boundary conditions are assumed at theedges of the RVE.

Figure 2.2: numerical model (RVE) used in the simulations

Results

Uniaxial tensile testsSpecimens of aged polystyrene (N5000) are subjected to uniaxial tensile tests at various

temperatures. At a single (linear) strain rate of 10-2 s-1 and temperatures ranging from 293K to373K the macroscopic response is determined. At low temperatures crazing is observed in alltensile bars prior to macroscopic failure. The strain to break decreases slightly with increasingtemperature from 3% to 1.5% of macroscopic strain. More striking was the change in crazedensity with temperature, at low temperatures many crazes throughout the specimen wereobserved which covered the major part of the cross-sectional area. At a temperature of 353Konly a few crazes were observed which covered much less of the cross-sectional area.

29

As can be seen in Figure 2.3, at 368K a sharp increase in ductility is observed as the strainto break increases from approximately 1.5% to more than 40%. With increasing temperaturethis trend continues and failure is no longer observed due to the limited span of the tensiletester.

Figure 2.3: experimental brittle-to-ductile transition of PS and a photo of the used bars

The photo incorporated in Figure 2.3 clearly demonstrates the brittle to ductile transition asthe deformation mechanism shifts from crazing to shear yielding. These observations areconsistent with previously reported temperature induced brittle to ductile transitions [9],although it must be added that the exact transition temperature is influenced by the thermo-mechanical history of the material [3].

Large strain deformation of a RVEIn the numerical simulations the RVE is elongated at a strain rate of 0.001 s-1 and the

equivalent strain is followed. The deformed meshes at two temperatures are given in Figure2.4 for increasing strain.

In these simulations is the cavitation criterion not applied and therefore the macroscopicstrains are presented up to 15%. The distribution of the equivalent strain gives an indication ofthe degree of localisation induced by the deformation. In the upper part of Figure 2.4 the RVEat 293K is plotted. It can be seen that the deformation is strongly concentrated in a fewligaments. The holes adjacent to these highly stretched ligaments form a sort of localisationpath. In the lower part of Figure 2.4 the RVE at 353K is plotted. The deformation is muchmore spread over the RVE and although the ligaments of the previously mentionedlocalisation path are still fairly stretched, the deformation is transferred to other regions of theRVE.

373 K368 K365 K363 K353 K333 K313 K293 K

30

Figure 2.4: deformed meshes at 293K and 353K for increasing strains

The differences in degree of localisation in Figure 2.4 can be explained when the stress-strain curve is looked at, for these temperatures. This curve is given in Figure 2.5, the data isextracted from numerical simulations.

Figure 2.5: stress-strain curves at 293K and 353K

Figure 2.5 shows that the strain softening decreases with increasing temperature. When thestrain softening is reduced, the deformation could be transferred to other regions at increasingmacroscopic strain. This results in a more uniform divided deformation at highertemperatures.

1.5

0

Temperature 293 K

strain 1% strain 5% strain 10% strain 15%

Temperature 353 K

Equivalentstrain

strain 1% strain 5% strain 10% strain 15%

31

Due to the intrinsic strain softening at high temperatures the localisation of strain softeningis less extreme in the local plastic zones. It is likely that this results in a slower built-up ofpositive hydrostatic stress, and moreover due to the lowered intrinsic yield stress of thematerial the maximum hydrostatic stress which are reached will be lower. In Part 1 ahydrostatic stress of 39MPa is presented as a critical cavitation stress. So at a criticalhydrostatic stress of 39MPa the mesh is considered to be broken. At that point the meshes arerepresented in Figure 2.6, the estimated failure strain and the temperature are also given. Thequantity plotted is the hydrostatic stress.

Figure 2.6 : estimated failure strains of RVE’s, at different temperatures,at a critical hydrostatic stress

Figure 2.6 shows the RVE at four different temperatures. The numerical model reaches thecritical hydrostatic stress at 1-1.2% macroscopic strains for temperatures between 293K and333K. For 353K the model can be stretched much further up to 10% strain before the criticalstress is reached. As mentioned before, at high temperatures a reduced strain softening leadsto a more uniform deformation and this leads to a lower built-up of hydrostatic stress.Combining this with a lower yield stress, leading to lower hydrostatic stresses, explain thehigh macroscopic strain where the critical hydrostatic stress is reached at 353K.

293 K 1% 313 K 1.2%

353 K 10%333 K 1.2%

39 MPA

0 MPa

32

The estimated failure strain can be extracted from Figure 2.6, this is the strain where thehydrostatic stress reaches the 39MPa. These estimated failure strains are given as a functionof the temperature in Figure 2.7.

Figure 2.7: estimated failure strain as function of the temperature

For a temperature of 293K the estimated failure strain is 1% and for both 313K and 333Kthe failure strain is 1.2%. For a temperature of 353K however the estimated failure strain is10%. So between the 333K and the 353K the RVE can be suddenly stretch at larger strainsbefore the critical hydrostatic stress is reached. This incline is the onset of the brittle-to-ductile transition as seen before in experiments.

The numerical simulations are performed at 4 temperatures, in further research it is usefulto take more temperatures in to account and to extract material properties on the used tensilebars. When the tensile bars in the experiments are replaced with core shell bars with 20volume % air the experiments and the numerical simulations can be compared.

ConclusionIn experiments it is found that annealed PS has a brittle-to-ductile transition temperature

around the 368K. Below this temperature the number of crazes decreased with increasingtemperature. At this temperature the strain to break increases from 3% up to 40%. Materialparameters are extracted from uniaxial compression tests and together with the foundcavitation criterion of PS, stated in Part 1, the transition is numerically described. Due to theintrinsic strain softening at high temperatures the localisation of strain softening is lessextreme in the local plastic zones. This results in a slower built-up of positive hydrostaticstress, and moreover due to the lowered intrinsic yield stress of the material the maximumhydrostatic stress which are reached will be lower.

The critical hydrostatic stress for 293K up to 333K is reached at values below the 2%macroscopic strain. At 353K the critical hydrostatic stress is reached at 10% macroscopicstrain. It is concluded that the estimated transition temperature is between the 333K and the353K. The numerical simulations, using the cavitation initiation criterion proposed in Part 1,indeed predict the observed brittle-to-ductile transition well. For a better comparison thetensile bars can be replaced with core shell PS in further research.

33

References

1. Tervoort TA, Govaert LE, J. Rheol. 2000; 4: 1263-77.

2. Govaert LE, Melick van HGH, Meijer HEH, Polymer, 2001;42:1271-1274.

3. Melick van HGH, Govaert LE, Meijer HEH, Polymer, submitted.

4. Kramer EJ, Adv. Pol. Sci., 1983;52/53:1-56.

5. Henkee CS, Kramer EJ, J. Polymer Sci: Polymer Physics Edition, 1984;22:721-737.

6. Bauwens JC, J. Mater. Sci., 1978;13:1443-8.

7. G’Sell C, In: Queen HJ, ed., Strength of metals and alloys. Oxford: Pergamon Press,1986:1943-82.

8. Govaert LE, Timmermans PHM, Brekelmans WAM, J. Eng. Mat. Tech. 2000; 122:177-185.

9. Vincent PI, Polymer 1960;1:1

10. Wu PD, Giessen van der E, J. Mech. Phys. Solids, 1993;41:427-456.

11. Wu PD, Giessen van der E, Int. J. Plast., 1995;11:211-235.

12. Boyce MC, Parks DM, Argon AS, Mech. Mater., 1988;7:15-33.

13. Hasan OA, Boyce MC, Li XS, Berko S, J. Polym. Sci., Part B: Polym. Phys., 1993;31:185-197.

14. Arruda EM, Boyce MC, Int. J. Plast.,1993;9:697-720.

15. Boyce MC, Arruda EM, Jayachandran R, Polym. Eng. Sci., 1994;34:716-725.

16. Tervoort TA, Smit RJM, Brekelmans WAM, Govaert LE, Mech. Time-depend. Mat.,1998;1:269-291.

17. Tervoort TA, Klompen ETJ, Govaert LE, J. Rheol., 1996;40:779-797.

18. Smit RJM, Brekelmans WAM, Meijer HEH, Comp. Meth. Appl. Mech. Eng. 1998;155:181-192.

19. Smit RJM, Brekelmans WAM, Meijer HEH, J. Mech. Phys. Sol., 1999;47:201-221.

20. Bressers OFJT, Melick van HGH, Govaert LE, Toonder JMJ, Meijer HEH, Craze initiation inglassy polymers, Polymers, 2002 submitted.

21. Smit RJM, Brekelmans WAM, Meijer HEH, J. Mat. Sci., 2000;35:part1 2855-2867.

22. Smit RJM, Brekelmans WAM, Meijer HEH, J. Mat. Sci., 2000;35:part2 2869-2879.

23. Smit RJM, Brekelmans WAM, Meijer HEH, J. Mat. Sci., 2000;35:part3 2881-2893.

24. Aa van der MAH, Schreurs PJG, Baaijens FPT, Mech. of Mat., 2001;33:555-572.

34

35

Part 3

Indentation on thin films

IntroductionIn different industrial areas there is an ongoing trend in miniaturisation of products. In

particular in the electronics and chip industry the typical length scale in componentsapproaches the nanometer scale. In both metals and polymers examples are reported that theproperties of these materials at such a length scale can be quite different from the large scaleproperties. For instance glass transition temperature (Tg) of polystyrene to which themechanical properties are strongly related, decreases dramatically when the thickness of apolymer film is reduced to less than 70nm [1], see Figure 3.1.

Figure 3.1: dependence of the Tg as function of the film thickness, partly reproducedwith permission, from Forrest [1]

Recent more examples are reported in this respect, which show that the properties of smallpolymeric structures and polymeric material near a free surface can deviate considerably fromthe bulk properties [2,3,4]. Generally the phenomenon of a reduced Tg in thin polymer films isassigned to an enhanced segmental mobility near the free surfaces of the polymer. It wasshown by Melick et al. [5] by nano-indentation that the mechanical properties near a freesurface on this length scale might also deviate from the bulk properties.

In this part the mechanical behaviour of thin polystyrene (PS) films on glass substrates isstudied by a combination of indentation experiments and a finite element modelling. Thisindentation technique, already successfully applied in ceramics and metals may be thesolution for characterising the mechanical properties of polymers at nanometer scale. Thinpolymeric layers, ranging from 20nm to 28m, are spin coated onto thin glass substrates.After careful heat treatments, to allow evaporation of the solvent, the samples are indented bynano-indentation [5,6]. Afterwards the indentation procedure is simulated by means of a finitemodel. The aim is to estimate the mechanical properties at this length scale by comparison ofthe experimental and numerical data.

36

Experimental and numerical modelling

ExperimentalSpin coating

The thin polymeric films were made by spin coating. A solution, containing a certainamount of polymer, is dosed on a spinning plate. By the rapid spinning of the plate, thesolution is homogeneously distributed over the plate, while the solvent evaporates. Thepolymer remains as a thin homogeneous, well-defined film on the substrate. The thickness ofthe residual layers depends mainly on 3 factors: the viscosity of the solution, the speed of theplate and the time the plate spins. The theoretical background is given in Emslie [7] andMeyerhofer [8].

The material used here is polystyrene (Styron 634) supplied by Dow Chemical. Thepolymer dissolved in toluene and stirred, with a magnetic stirrer, to obtain good mixing. As asubstrate thin glass plates (150x150x1mm) were used. As for the indentations specimens of(20x20mm) are required, the plates were pre-scratched before cleaning and spin coating tofacilitate breaking up the glass in the desired dimensions. Next the glass plates werethoroughly cleaned; first the plates are washed with hot-water and soap (10% Extran MA 02)and dried with compressed air. Then the plates are washed in three steps: with demi-water,ethanol and heptane, between all steps the plate is dried with compressed air. The last step is atreatment in a UV-ozone photoreactor, The plate is exposed to UV-radiation in a ozone filledenvironment. The spinning of the polymer films was performed on a spin coat machine (KarlSuss CT 62). As the thickness of the spin coated layer is determined by the viscosity of thesolution, spinning speed and spinning time, these quantities were adjusted to obtain thedesired range of film thicknesses. In Table 3.1 both the parameters of the spin coat processand the obtained film thickness are given, the applied acceleration was always 3000 rad/sec2.

Volume %PS in toluene

Proces Time[sec]

Rotation speed[rad/sec]

Final thickness

Dosing 3 400Spinning 4 2000

0.95

Vaporisation 10 300

50 nm

Dosing 5 300Spinning 4 300

20

Vaporisation 20 300

28 m

Table 3.1: spin coating of thin polymeric layers

Next the glass was broken in small platelets (20x20mm). For layers thicker than 6 m it isnecessary to cut the PS layer, otherwise the layer is pulled off the glass while breaking. Thepolymer films were heat-treated above their Tg to allow evaporation of any remaining solvent.During this treatment the plates were kept in an oven at 125C for 3 days. This treatment,more essential for thick layers must be performed above the Tg of the material as in the glassystate diffusion of the solvent is inhibited. Any remainders of the solvent can be traced using aDifferential Scanning Calorimeter (DSC) as it would lower the Tg of the material. The glasstemperature of the polystyrene was measured by DSC after the heat treatment and proved tobe equal to bulk Tg reported in literature.

37

Due to differences in the coefficient of thermal expansion of the polystyrene and the glasssubstrate, stresses were induced by the cooling of the films. This effect is, of course, related tothe thickness of the film and hence, thin layers were not noticeable influenced. Thick layers,however, delaminates from the glass after approximately 3 days. This time to delaminationcan be increased by slowly cooling after heat-treating. Another solution is to break the plateswhen the layer is still wet. In this way the solution covers also the small sides of the platelets.Of course residual stresses remain in the layer after the heat-treatment. As the delamination ispromoted by poor adhesion between the glass substrate and the polystyrene, a way to preventthis delamination is applying a thin layer of a coupling agent. Therefore the glass substrate iscoated with a mono-layer of phenyl-tri-methoxy-silaan. For thin layers this coating is notnecessary and may even influence the measurements. Although the coating does not preventthat the layer come loose of the glass, it elongates the time.

The thickness of the spin coated layers varies from 20nm up to 28m. In the experimentsthree layers are used; 50nm, 1.3m and 28m. The thickness of the layers is measured by analpha stepper, Atomic Force Microscopy (AFM) and an optical profile meter (UBM). UBMuses optics and is therefore a non-contact device. AFM can also be used to examine theroughness of the top layer. The top layer is very smooth; the Ra roughness value is below1nm.

IndentingIndentations were performed with a micro indenter and a nano indenter. The micro

indenter is a custom-designed built apparatus at Philips Research Laboratories in Eindhoven.The forces, which can be measured, range from 20mN up to 20N with an accuracy of 2mN.The accuracy of the displacement is 20nm. Forces and displacements are measured by meansof coils at the bottom of the indenter column. The spherical indenter used is a sapphire sphere,of 150m radius, glued onto a brass holder. The compliance of the apparatus is determined bya reference measurement on silica glass. The elastic indentation depth-force curve is predictedby Hertz’ theory. From the deviation between the theoretical and experimental curve, thecompliance is proved to be 6·10-2 µm/N, and the corresponding stiffness of the measuringsystem is 1.67·107 N/m.

The nano indenter is a commercial apparatus (Nano-Test 600) manufactured by Micro-materials Ltd (Wrexham, UK). This apparatus can measure forces up to 500mN with anaccuracy of 10µN. Indentation depths can be recorded up to 10m with a theoretical accuracyof 0.04nm. The practical accuracy is 3nm. The indenters used have a radius of 2.2m and150m. The apparatus is only force controlled and hence the load can only be applied at acertain loading rate.

A typical indentation procedure begins with a position-controlled movement of theindenter towards the sample until the surface is contacted with a pre-load of 5mN (microindenter) or 0mN (nano indenter). Next the platelet is loaded in force control up to apredefined maximum force at force rates ranging from 10mN/sec up to 1N/sec. When apredefined maximum force is reached the indenter is retracted out in position control (microindenter) or force control (nano indenter). The force required and the displacement of theindenter are recorded during indentation.

38

Numerical modellingThe numerical model that describes the indentation consists of two deformable bodies: the

indenter and the polystyrene on the glass substrate. The contact between the indenter and thepolystyrene is assumed to be frictionless. The influence was examined by varying the frictioncoefficient in the model and the influence on the result proved to be negligible. The adhesionbetween the polystyrene and the glass is considered to be perfect. The model is modelled asan axi-symmetric problem (see Figure 3.2). The dimensions of the glass should be largeenough so that the indented region has no influence on the edges of the model, i.e. 10 timesthe indented region. The indenter is modelled as a half sphere.

Figure 3.2: numerical model of the indentation

At the centre line of the model, the indenter, the polystyrene and the glass are fixed in they-direction to provide axi-symmetric conditions. Furthermore rigid-body movements of theglass substrate are prevented. The indenter is prevented to move in x-direction when no loadis applied using a very weak spring with a modulus of 1 N/m. At the start of the indentation apre-defined force was applied at the flat side of the indenter which moves uniformly in x-direction.

Three materials are used in the numerical simulations: sapphire, polystyrene and glass. Thesapphire and the glass are assumed to be linear elastic materials with a Young’s modulus of304GPa respectively 72GPa and a Poisson’s ratio of 0.234 respectively 0.23, Simmons [9].The simulations confirmed the validity of the assumption of linear elasticity. The parametersof the polystyrene are extracted from uniaxial compression tests on bulk polystyrene and arerepresented in Table 3.2.

Type E[Mpa]

[-]GR[Mpa]

A0[s]

0[MPa]

H[MJ/mol]

µ[-]

D [-]

h[-]

Annealedreference PS 3300 0.37 13 1·1012 2.6 0.1723 0.14 11 75

Table 3.2: material set used in the numerical simulations

The material model used is the compressible Leonov-model (see Part 1 and [10,11,12]) anduses the material parameters described in Table 3.1. To analyse the model the options of largedisplacements, constant dilatation (to prevent locking) and updated Lagrange are used. Toexclude any influence of mesh size a stepwise element size reduction is performed until thesolution converged to a steady, mesh independent, result. In these simulations especially thePS layer is refined, in this way the shear behaviour of the polystyrene at the interface with theglass can be accurately described. In order to prevent excessive computation time the meshrefinement is restricted to areas of interest. A typical mesh is given is Appendix A.

y-direction

x-directionForce

glassPSindenter

39

Results

A 1.3m layer is indented with both the nano and micro indenters with an indenter of150m radius. In Figure 3.3 the resulting curves are given.

Figure 3.3: comparison of the micro indenter and the nano indenter,indentations performed on a 1.3m layer

Figure 3.3 shows two almost identical indentation curves. The loading and unloading curvefollow the same path except for a slight difference around the transition from loading tounloading. This difference occurred as the micro indenter can not perform this transition asfast as the nano indenter can and hence the maximum force was held for a short period. Thenano indenter, on the other hand can be retracted immediately. The final plastic indentationdepths are identical.

The measurement of the thick layer is done on the micro indenter with a sphere of radius150m. The experiment and the simulation are given in Figure 3.4.

Figure 3.4: numerical and experimental indentation of a 28 m layer

40

The numerical simulation provides a good fit for the experiment. The withdrawal of theindenter is not simulated since this is not modelled well in the used material model (seePart 1) and to avoid large calculation times.

The thin layer of 50nm is indented on the nano indenter with a sphere of 2.2m radius. Theexperiment and the simulations are given in Figure 3.5.

Figure 3.5: numerical and experimental results on a thin layer, a The standard simulation, b Simulation with a temperature field, c Simulation with frictionless contact between layer and glass

In Figure 3.5 the maximal force is at its maximum 0.3mN, the loading rate is 0.02mN/sec.The experiment begins with a pre-load of 0N. The experiments show that first the layer iseasily penetrated by the indenter, later the influence of the glass becomes more important andthe resulting stiffness increases. The standard simulation begins with no pre load and is toostiff compared to the experiment, it does not show the characteristic slope of the first tens ofnanometers as seen in the experiment. The incline of the simulation describes the experimentswhen the beginning is not noticed. The experiments and the simulation coincide when thesimulation should be shifted 17nm. The stiffness of the curve of the numerical model must beascribed to the fact that the influence of the glass is direct noticed.

In order to describe the experiments a temperature field is applied to the model. A lineartemperature field takes care of the drop in Tg at the surface (see Figure 3.1). The temperatureis constructed in a way that the Tg at the surface is lowered to 20C and at a depth of 50nmthe Tg is normal, 106C. The temperature field increases the indentation depth but thesimulation can not describe the experiments (see Figure 3.5).

Another attempt is made by simulating the contact between the glass and the layer to befriction less. The first part of the experiments is described in a proper way, but at largerindentation depths the simulations stops because of converge errors. So the beginning isdescribed well and therefore it would indicate that the PS film does not adhere to the glass atall. To simulate this thin layer well the conditions between the glass and the polymer areextremely important. In further work these conditions must be examined and used innumerical models.

41

ConclusionIt is shown that with spin coating a range of PS layers can be made. The thickness of the

layers can be controlled well and the surface is smooth. The time that the bond between theglass and the polystyrene holds can be elongated by adding a mono-layer of PTMS to theglass or by braking the glass before the polystyrene is dried.

For thick layers it is possible to simulate the indentation behaviour with a compressibleLeonov-model. The simulation of the thin layers is more difficult. Near the surface anothermaterial behaviour is considered that result in a lower modulus of indentation at the firstnanometers. The numerical model under-predictes the indentation depth with the appliedmaterial parameters that are extracted from bulk polystyrene. An attempt to influence thematerial behaviour near the surface increases the indentation depth, but only a little. Thecharacteristic measured indentation curve is not achieved. When, however, the thin layer wasmodelled to be non-adhesive to the glass, the slope at the beginning of the experiments wasmodelled well, so the beginning is described well and therefore it would indicate that the PSfilm does not adhere to the glass at all. This simulation did not converge at 33nm and stopped.To simulate this thin layer well the conditions between the glass and the polymer areextremely important.

In further research a combination of a temperature field and a partial non-adhesive contactbetween the glass and the layer PS is recommended to describe the experiments with thesimulations. Another point of research is to use the modulation option of the nano indenter, inthis way the thin layer can be indented repetitive at the same position.

References1. Forrest JA, Mattsson J, Phys. Rev. E., 2000; 61, R53

2. Keddie JL, Jones RAL, Cory RA, Europhys. Lett., 1994;27:59

3. Keddie JL, Jones RAL, Dynamics in Small Confining Systems II, Material Research SocietySymposium Proceedings, 1995;366:183-188

4. Zanten van JH, Wallace WE, Wu W-L, Phys. Rev. E., 1996; 53, R2035

5. Melick van HGH, Dijken van AR, Toonder JMJ, Govaert LE, Meijer HEH, Near-surfacemechanical properties of amorphous polymers, 2001, submitted

6. Strojny A, Xia X, Tsou A, Gerberich WW, J. Adhesion Sci. Tech., 1998;12:1299-1321

7. Emslie AG, Bonner FT, Peck LG, J. Appl. Phys., 1958; 29:5.

8. Meyerhofer D, J. Appl. Phys., 1978;49:7.

9. Simons G, Wang H, Single Crystal Elastic Constants and Calculated Aggregate Properties: AHandbook, 2nd ed., 1971.

10. Tervoort TA, Smit RJM, Brekelmans WAM, Govaert LE, Mech. Time-depend. Mat. 1998; 1:269-291.

11. Tervoort TA, Klompen ETJ, Govaert LE, J. Rheol. 1996; 40:779-797.

12. Aa van der MAH, Schreurs PJG, Baaijens FPT, Mech. of Mat., 2001;33:555-572.

42

43

APPENDIX A: numerical models

Numerical model for indentations on bulk PS

44

Numerical model for indentations on thin layers PS on glass

sapphire PS glass

45

APPENDIX B: Indentations (experimental and numerical)

Indentation on PS with distinct thermal histories

Indentation on PS with distinct loading rates

Indentation on PS blends with PPO added