Craze Initiation in Glassy Polymer Systems

Craze initiation in glassy polymer systems

MT02.002 O.F.J.T Bressers

Master report

Supervisors: dr. ir. J.M.J. den Toonder Philips Research ir. H.G.H. van Melick TU/e dr. ir. L.E. Govaert TU/e prof. dr. ir. H.E.H. Meijer TU/e Summary

This report deals with the numerical simulation of craze initiation in amorphous polymer systems. The approach is based on the view that the development of a craze is preceded 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 will take place leading to local development of voids. The voids grow, coalesce and the ligaments between the voids are subsequently super-drawn leading to the typical structure of 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 polymers in a well-defined experiment. A micro indenter with a 150 m sapphire sphere produces reproducible indents, which are later examined with an optical microscope. These observations lead to a critical force where crazes are initiated in polystyrene (PS). Combination of these experiments with a numerical study using the compressible Leonov-model showed that the loading part of the indentation can be accurately predicted. A critical hydrostatic stress of 39MPa is extracted from the numerical model by analysis of the local stress field at the moment the indentation force reaches the experimentally determined force level at which crazes were found to initiate. This criterion is validated by application of the model to indentations on samples with different thermal histories, and at various loading rates. Varying the network density of the polymer showed that the incline of hydrostatic stress during indentation is not influenced. At higher network density, the samples started to craze at higher forces and the corresponding hydrostatic stress is higher.

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

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

ii iii Contents

General introduction 1

Part 1 : Cavitation initiation of amorphous polymers 3 Abstract 3 Introduction 4 Experimental 7 Materials 7 Experimental set-up 7 Numerical methods 8 Material model 8 FEM model 10 Material characterisation 11 Identification of a cavitation initiation criterion 14 Annealed reference sample 14 Thermal history 17 Influence of the loading rate 18 Network density 19 Conclusions 20 References 21

Part 2 : Numerical prediction of a temperature-induced brittle-to- ductile transition in polystyrene 23 Abstract 23 Introduction 24 Experimental and numerical methods 25 Results 28 Uniaxial tensile tests 28 Large strain deformation of a RVE 29 Conclusion 32 References 33

Part 3 : Indentation on thin films 35 Introduction 35 Experimental and numerical modelling 36 Experimental 36 Numerical modelling 38 Results 39 Conclusion 41 References 41

Appendix A 43

Appendix B 45

iv v General introduction

This report deals with the numerical simulation of craze initiation in amorphous polymer systems. The approach is based on the view that the development of a craze is preceded 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 will take place leading to local development of voids. The voids grow, coalesce and the ligaments between the voids are subsequently super-drawn leading to the typical structure of a craze, a crack-like defect bridged by highly drawn filaments.

It is well known that the initiation and development of a localised plastic zone is dominated 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 the initiation of strain localisation, and 2) strain hardening, which stabilises the growth of the localised plastic zone. Subtle variations in the amount of strain softening or strain hardening can lead to extreme changes in the macroscopic deformation behaviour, changing the failure mode from tough to brittle. Over the past 15 years several constitutive models were developed that are able to describe this complex post-yield behaviour, enabling us to numerically simulate localisation phenomena in glassy polymers. Up till now, however, two factors hamper the adequate analysis of the stability of the deformation zone: 1) the absence of a validated criterion that can be used to detect incipient cavitation, and 2) the limited knowledge of the influence of ligament size on the intrinsic material properties. It are exactly these problems that are addressed in the present work.

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

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

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 is different from bulk material. The layers are spin-coated on glass and then indented with a micro indenter and a nano indenter. Experimental results are compared to results from numerical models.

1 2 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 glassy polymers. The approach is based on the view that the development of a craze is preceded 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 will take place, leading to local development of voids. The initiation of a localised plastic zone is numerically simulated using a constitutive model that incorporates an accurate description of the post-yield behaviour with the important phenomena of strain softening and strain hardening. Subsequent cavitation of the deformation zone is detected using a hydrostatic stress criterion. This criterion is identified and validated by confronting numerical simulations to experimental results of indentation experiments on polystyrene. A micro-indenter with a sapphire sphere of 150 m radius is used to produce indents that are later examined with an optical microscope. These observations lead to a critical force where crazes are initiated in polystyrene (PS). Combination of these experiments with a numerical study using the numerical model shows that the loading part of the indentation can be accurately predicted. A critical hydrostatic stress of 39MPa is extracted from the numerical model by analysis of the local stress field at the moment the indentation force reaches the experimentally determined force level at which crazes were found to initiate. This criterion is validated by application of the model to indentations on samples with different thermal histories, and at various loading rates. The influence of network on the value of the hydrostatic stress criterion is investigated by 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.

3 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 a microscopic level, they display large plastic deformations. For this reason, crazes are the most important source of fracture toughness in brittle glassy polymers, even though the volume fraction crazes during fracture is generally low. It is, therefore, not surprising that a vast amount of research has been done on all aspects of crazing: craze nucleation, growth and failure, the micro-structure of crazes, the influence of molecular parameters, etc., and a number of excellent reviews are available [1-3].

Figure 1 depicts some of the microscopic events that are likely to be involved in craze nucleation [1]. First, plastic deformation starts at a local stress concentration. The non-linear nature of the yield process and the strain softening character of polymer glasses will result in a localisation of deformation as the plastic strain increases. Since the material at some distance 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 can stabilise the strain-localisation process and the micro-shear band will spread out. Second, it has been shown [4] that the hydrostatic stress required to plastically expand a micro-porous region is greatly reduced if the material is in a state of flow. Therefore, if the lateral stresses become high enough, the material in the deformation zone will cavitate, and craze nucleation has been accomplished.

Figure 1: schematic drawing of microscopic events involved in craze nucleation: a) formation of a localised surface plastic zone and build-up of lateral stresses, b) cavitation of the plastic zone and c) deformation of the polymer 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 failure behaviour of a glassy polymer is determined by two factors: 1) the intrinsic post-yield behaviour of the material, and 2) its resistance against cavitation.

4 The post-yield behaviour of a glassy polymer is typically characterised by occurrence of strain softening and strain hardening [5]. Although the exact molecular origin of strain softening is still unknown, it is well documented that it is strongly influenced by the thermal and mechanical history of the material [5,6-12]. Slow-cooling rates tend to increase strain softening, 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 of physical ageing effects by plastic deformation: mechanical rejuvenation. The origin of strain hardening 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 by a subtle interplay between strain hardening and strain softening. The initiation of localised deformation zones is governed by strain softening, which allows the zone to grow with decreasing stress. The amount of strain hardening determines whether or not the deformation zone is stabilised. Strain softening, however, appears to be the key factor in these localisation phenomena. Upon removal, or strong reduction, of strain softening through thermal (quenching) or mechanical (plastic pre-deformation) methods, the occurrence of strain localisation 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 a mechanical 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 as result of physical ageing. In the past 15 years, considerable effort was directed towards the numerical simulation of strain localisation phenomena. The development of 3D constitutive models that were able to capture 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 der Giessen [18-19] and our group [10,11,20-23]. As a result of these activities the numerical simulation of plastic localisation in various loading geometries is now well established. An interesting application of these techniques is the field of polymer blends. It is well known that blend morphology has a pronounced influence on the macroscopic toughness of the material. Numerical studies on the evolution of deformation in microstructures of various compositions are now well within reach (see for instance the work of Smit et al. [23-26]), and offer new opportunities for fast material evaluation and development. However, in order to evaluate the toughness of such systems numerically, an additional criterion is required to detect whether or not a craze will initiate within the deformation zone.

Craze-initiation criterion that have been introduced so far generally involve a deviatoric stress component and a hydrostatic stress component [1,27,28]. Following the mechanism of craze initiation proposed by Kramer [1] (Figure 1), the formation of a local plastic deformation zone is a prerequisite. As mentioned above, the formation of such a strain localisation is determined by the post-yield behaviour of the material, which is completely determined by the deviatoric state of stress. Moreover, the development of such localised deformation 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 the deformation zone will cavitate, and craze nucleation has been accomplished. In previous work [29], it was investigated whether craze initiation could be envisaged as a plastic 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 rate dependence, described by a single Eyring process. This result strongly indicates that the

5 occurrence of small plastic deformation zones is the rate-determining step in craze initiation and, therefore, that the on-set of cavitation may be described by a local, rate-independent criterion. 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 critical hydrostatic stress h,c might be adequate.

The objective of this paper is to identify a local cavitation criterion, based on the assumption of a critical level of the hydrostatic stress. For this an experiment is used in which the 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 accurate experiment to generate crazes in a well-controlled and reproducible way. At a certain load during indentation, pile-up of material occurs next to the contact area between indenter and polymer. Beside plastic deformation, positive hydrostatic stresses evolve in this pile-up which results in small crazes just below the surface of the polymer. To identify the local stress distribution within the pile-up, finite element simulations of the indentation 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 quite well. The capability of this model to describe the indentation of amorphous polymers was already shown by Melick [30]. The hydrostatic stress criterion is obtained by a combination of numerical simulations and experimental observations. From the experiments the force is recorded at which the crazes are initiated. Using a numerical simulation the critical hydrostatic stress is identified as the maximum hydrostatic stress in the pile-up zone at this specific indentation force. The applicability of the obtained criterion is subsequently investigated by comparing numerical predictions of craze initiation with experimental observations in various loading rates (i.e. indentation force rates) and on samples with different thermal histories (quenched and annealed PS). Using the same approach the influence of network density on the critical hydrostatic stress is studied by indentation of blends of polystyrene and poly(2,6-dimethyl- 1,4-phenylene-oxide) (PPO).

6 Experimental

Materials The 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 is compression moulded into plates of 100x100x3mm in a stepwise manner. First the material is pre-heated in the mould at 90LC 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 to 300kN) 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 order to eliminate residual stresses and thermal history effects the platelets are heat-treated for 30 minutes at 15LC above Tg. Subsequently the platelets are given two thermal treatments known as annealing and quenching. During annealing the samples are held at 20LC below Tg for three days and then slowly cooled in one day in the oven to room temperature. During quenching the samples are cooled rapidly in ice-water from 15LC above Tg.

Experimental setup Indentations are performed with a micro indenter; a custom-designed built apparatus at Philips Research Laboratories in Eindhoven. The forces that 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 means of coils at the bottom of the indenter column. The spherical indenter used is a sapphire sphere, with a radius of 150 m, glued onto a brass holder. The compliance of the apparatus is determined by a reference measurement on silica glass. The elastic indentation depth-force curve is predicted by Hertz’ theory. From the deviation between the theoretical and experimental curve, the compliance is determined to be 6·10-2 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 the indenter towards the sample until the surface is contacted with a pre-load of 5mN. Next the platelet is loaded in force control up to a predefined maximum force at force rates ranging from 10mN/sec up to 1N/sec. When a predefined maximum force is reached the indenter is retracted in position control. The force required and the displacement of the indenter are recorded during indentation. The experiments are carried out in a sequence of increasing force steps. At each step of 0.5N, the indentations are repeated at least 10 times. After the experiments, the indents are examined using an optical microscope (Leica DM/RM) to check whether crazes are initiated by the applied force. This microscope uses 2 polarizers for the interference contrast to visualise the crazes. The critical force for cavitation initiation is identified by the indentation at which crazes occur first.

To obtain the material parameters, required for the numerical simulations, uniaxial compression tests are performed on a servo hydraulic MTS Elastomer Testing System 810. Cylindrical specimens are compressed at room temperature, at a constant logarithmic strain rate between two parallel, flat steel plates. The friction between the sample and the steel plates is 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

7 bulging or buckling of the sample is observed, indicating that the friction is sufficiently reduced. 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 data acquisition at an appropriate sample-frequency (depending on strain rate). A constant true logarithmic 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 carried out using a finite element model of the indentation experiment. Information that can not be extracted from the experiments is derived from the numerical model, such as stresses and strains in the indented material. By comparing the computed forces and displacements with the experimental results, the numerical model can be validated to the experiment. From the experiment the critical force is determined and from the simulation the accompanying quantitative cavitation criterion is obtained.

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

Material model In previous work, an elasto-viscoplastic constitutive equation for polymer glasses was introduced, the so-called compressible Leonov-model [20,21]. To include strain hardening and strain softening [10], the Cauchy stress tensor σ is composed of two contributions: The driving stress tensor s and the hardening stress tensor r respectively:

σ s r (1)

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

~ d s K(J 1)I GB e (2)

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

J tr(D) (3) o ~ d d ~ ~ d d Be (D Dp ) Be Be (D Dp ) (4)

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

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

~ d r GR B (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 strain hardening 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 total deformation. In fact, Equation (1) implies that strain softening and strain hardening are regarded to act simultaneously. The constitutive description is completed as the plastic deformation rate is expressed in the extra stress tensor by a generalised non-Newtonian flow rule:

s d D (6) p 2 ( eq , D, p) 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 the model. As suggested by Hasan et al. [15] a history variable D is specified, the softening parameter, which influences the viscosity η. During plastic deformation D evolves to a saturation level DA, which is independent of the strain history. The result for η reads:

/ ( , D, p) A (D, p) eq 0 (7) eq m 0 sinh( eq / 0 ) where the equivalent stress τeq is defined by:

1 d d eq 2 tr(s s ) (8) and : p . A (D, p) Aexp D/ (9) m / 0 0 1 1 p 3 tr(σ) 3 tr(s) (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:

Ω (11) V

9 The evolution of the softening parameter D is specified according to Hasan et al. [15]: . D / D h 1 / p (12) DA 0 with initial condition D = 0; h is a material constant describing the relative softening rate and p is the equivalent plastic strain rate , according to:

p tr(Dp Dp ) (13)

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

FEM model The 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 assumed to be frictionless, the influence is examined by varying the friction in the model and is found to be negligible on the results. The model is modelled as an axi-symmetric problem (see Figure 2). The dimensions of the PS should be large enough so that the indented region has no influence on the edges of the model, i.e. 10 times the indented region. The indenter is modelled as a half sphere.

y-direction

Force x-direction

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 to provide axi-symmetry. The PS is fixed to the world at the right side. The indenter is prevented to move in x-direction when no load is applied using a very weak spring with a modulus of 1 N/m. The force is applied to the left side of the indenter. The nodes of this side are linked to each 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 is assumed to be a linear elastic material with a Young’s modulus of 304GPa and a Poisson’s ratio of 0.234, Simmons [32]. The parameters of the PS are extracted from uniaxial compression tests and are represented in the results.

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

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

Material characterisation

Bulk properties The material parameters required for the numerical simulations are extracted from uniaxial compression tests at room temperature. In this loading geometry localised deformation is inhibited. 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 from 3·10-4 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 softening and strain hardening. The solid lines in Figure 3 are predictions using the compressible Leonov-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 is slightly underpredicted. As, in reality, the pre yield behaviour is (non linear) visco elastic, the model will be less suitable for predicting the behaviour in the unloading part of the indentation. The rate dependence of the yield point is captured quite well by a single Eyring process. Especially for large strains, the softening is underpredicted for large strain rates and for low strain rates overpredicted. However, during indentations the maximum strain is only slightly larger than the strain at the yield stress. Up to values of 0.2, the strain is described well including the softening, the differences between experimental and numerical results do not play a role at that moment.

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

11 Thermal history The 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 sample influences the material behaviour of a polymer. In Figure 4 the results are given of an uniaxial compression 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 the quenched sample. The strain hardening is however unchanged. From a true strain of 0.13 and more the two results coincide, the thermal history does no longer influence the behaviour upon further straining. The material is, so called, mechanical rejuvenated, this effect is described before by Hasan and Boyce [7]. Since the material behaviour is different, the thermal history also influences the material parameters; the A0 is increased for the annealed sample since the yield point is increased. The h and D also increase since the drop in stress is higher 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 of the samples. Slight differences between the thermal history of the samples of the indentation and the compression test are compensated with the A0. In Table 1 the material parameters are given for the samples used for the indentations.

12 Network density The 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 network density 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 remains largely unaffected by an increase of the network density. The post-yield behaviour is strongly influenced, the strain softening decreases with increasing network density. This must be ascribed to the fact that the stabilising contribution of the entangled polymer network has already a large effect at smaller strains and hence the true stress can not drop as much as in a looser 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 a good description of the uniaxial compression tests.

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

Type E GR A0 0 H Μ DA h [MPa] [-] [MPa] [s] [MPa] [MJ/mol] [-] [-] [-] Quenched PS 3300 0.37 13 2.62108 2.6 0.1723 0.14 7 40 Annealed 3300 0.37 13 11012 2.6 0.1723 0.14 11 75 reference PS Ref. + 20% 3000 0.37 25 11 2.63 0.157 0.14 13 65 PPO 1 10 Ref. + 40% 2700 0.37 50 10 2.63 0.156 0.14 20 68 PPO 1.4 10

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

13 Identification of a cavitation initiation criterion

Annealed reference sample In 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 indentations are compared. The experiments result in a critical force and with the numerical model the cavitation criterion is examined.

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

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

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

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

a b

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

The solid drawn line is the average of all crazes per maximum indentation force, the errorbars are drawn at a probability of 95% with a normal distribution. The dotted lines are used to determine what force is needed to initiate crazes. For the annealed sample occur at low 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 the 1.35N. From this point the number of crazes increases almost linearly. The dotted lines indicate 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 parameters for the reference annealed platelet, as mentioned in Table 1. In Figure 6 both the outcome of the numerical simulation and the experimental data are given.

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

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

39 MPA polymer

indent

0MPa local hydrostatic stress

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 highest hydrostatic stress is followed during the simulation of the previous experiment and is represented as a function of the force in Figure 9.

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

The loading force increases from 0N to 4N, the resulting hydrostatic stress also has a positive incline. The highest value of the hydrostatic stress varies in place and magnitude during the simulation. This sometimes results in a staggered path of the hydrostatic stress, a fit of the hydrostatic stress is also included (the solid line). In the experiment the platelet starts crazing at the force of 1.35N, the corresponding maximum hydrostatic stress in the numerical model is then 39MPa, based on the simulation. In Figure 9 also the upper and lower boundaries of the critical force are given, the horizontal lines represent the corresponding hydrostatic stress. The critical hydrostatic stress varies then from 38 to 42.5MPa.

16 The conclusion is drawn that the reference PS starts crazing at edge between the indenter and the sample. The numerical model describes the experiments well for the loading part, the crazes are initiated during this part. During the simulation the material has a local plastic deformation zone which is formed under the influence of the deviatoric stress, so based on the results is tentatively concluded that a critical hydrostatic stress is adequate to describe the cavitation 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 by Narisawa [36].

Thermal history To 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 section compared with a quenched platelet. The material parameters of the quenched sample are presented in Table 1. In Figure 4 it was already shown that the yield stress increases when the sample is annealed. When the indentation curves are observed this is also found since the indentation depth decreases when the sample is annealed (see Appendix B). A look at the crazes shows that the crazes of the quenched platelet occur at higher forces and are very small and thin compared to the reference platelet. The crazes of the quenched platelet are often very hard to see and can only be visualised by using polarizers. The force to initiate crazes for the quenched platelet is 3.5N (-0.9/+0.6N), the errorbars are longer compared to the reference annealed sample (see Figure 7).

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

The incline of the hydrostatic stress for the annealed and the quenched sample is the same up to 0.7N. When the hydrostatic stress is approximately 30MPa, the paths of the annealed platelet and the quenched platelet begin to diverge. At that point the hydrostatic stress of the annealed platelet is larger, the cavitation criterion of 39MPa is reached at 1.35N for annealed PS. The hydrostatic stress of the quenched platelet is then 36MPa. The cavitation criterion of 39MPa 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 lines the conclusion is drawn that the quenched sample can initiate crazes from 2.05N to 2.9N. The cavitation criterion holds within experimental error since in the experiments was shown that the 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 in experiments and in the simulations is shown that the crazes occur at a higher force, so the trend is predicted well. The increase of the hydrostatic stress is different for distinct thermal treatments. The exact force where crazes occur is underpredicted by the numerical model but within the experimental error, when the platelet is quenched. A possible explanation is that the value of the cavitation criterion is dependent of the thermal history, this was stated earlier by Ishikawa [37]. Another explanation is that because the crazes are hard to see when the platelet is quenched, the mean value of the crazing force is too high in the experiments. The bad visibility also makes the error in the estimated force larger.

17 Influence of the loading rate Another 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 in Figure 3. The PS in this experiment is annealed and therefore the numerical model uses the parameters of the annealed material of Table 1. The loading rate of the loading force is varied from 0.01 to 1N/sec.

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

Figure 10 shows the force at which crazes in the experiment are observed as a function of the loading rate. The errorbars presented are calculated in the same way as described in the section of the annealed reference sample. Also the corresponding critical force predicted by the simulation when the platelets should craze is given for different loading rates. This is the force 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 Z , experiments

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

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

18 Network density In 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 section the strain hardening is changed by adding PPO to the PS (see Figure 5). This influences the network density of the polymer and it may affect the value of the cavitation criterion. The samples are all annealed because this results in clearly visible crazes.

The network density can influence the point where polymers start to craze. PPO is an amorphous 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 elastic modulus decreases slightly when PPO is added. In this experiment the PPO content is varied in two steps: 20% and 40% PPO. The sample with 0% PPO is the reference sample. PPO has a 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 the PS increases. As the yield stress and the E do not vary much for the materials, the loading curves 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 used material parameters are given in Table 1. The corresponding hydrostatic stress for PS and 2 blends is given in Figure 11.

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

The increases of the hydrostatic stress for the different blends follow the same path. All curves have a positive incline with an increasing indentation force. The critical force for PS without PPO is reached at 1.35N (as shown before), this corresponds with a critical hydrostatic stress of 39MPa (-1/+2.5MPa). According to the simulations, the critical hydrostatic stress for the 20% PPO is reached at a force of 3.0N and is at that moment 50MPa

19 (±2MPa). For the 40% PPO the critical indentation force is reached at 5.1N and the corresponding 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 critical hydrostatic 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. The observed trend is in full agreement with Sauer [39].

Conclusions Uniaxial compression tests provide material properties that in combination with a compressible Leonov-model describe the experiments well. Indentations of PS with a micro indenter lead to crazes and with the use of a numerical model the hypothesis of Kramer is checked. Kramer stated that crazing of polymers begins with plastic deformation. Due to the nonlinear nature of the yield process the strain softening of the material results in a localisation of deformation as the plastic strain increases while the hydrostatic stress also is building up at that moment. The hydrostatic stress is the only parameter that initiates cavitation after the material shows local deformation and softening.

Crazes occur at the loading curve of an indentation and can be visualised by means of a microscope. By counting the numbers of crazes, a critical force was determined. The crazes start at the edge between indenter and platelets and occur in radial direction. The critical hydrostatic stress is determined to be 39MPa (-1/+2.5MPa) for an annealed PS sample and occur just outside the contact zone between PS and indenter. The criterion is validated by varying the thermal history and the loading rate of the indentation. Quenched PS shows very tiny crazes at a higher force, the trend depicted by the numerical simulation is good and the prediction by the cavitation criterion agrees with the observations. 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 and experiments resulted in the same force. The post yield behaviour is changed by adding PPO to the PS as the network density is increased. An increase of the network density result in a

20 higher value of the critical cavitation criterion, 50MPa (M2MPa) for the blend with 20% PPO and 55MPa (-2/+1MPa) for the blend with 40% PPO.

It is concluded that the combination of a numerical model, covering the rate dependent intrinsic yield behaviour, and a local, rate independent, cavitation criterion can be used effectively 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.

21 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.

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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.

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

22 Part 2

Numerical prediction of a temperature-induced brittle-to-ductile transition 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, under influence of temperature, is studied by means of finite element simulations. As the macroscopic deformation behaviour of amorphous polymers is determined by their post-yield behaviour, influencing this behaviour can results in major changes. Besides treatments like cross-linking and mechanical pre-conditioning, also the testing conditions proved to play a crucial role. Recently a failure criterion was reported which identifies a critical hydrostatic stress as the criterion for craze initiation, provided this is preceded by plastic deformation. Combining this criterion with the adequate 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 a brittle or ductile manner. Therefore in this paper, a heterogeneous structure is deformed in a tensile test. At a single strain rate and various temperatures simulations are performed to study the macroscopic response of voided polystyrene. At a temperature 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 in these simulations. These results correlate well which experimental observations.

23 Introduction

The macroscopic deformation behaviour of amorphous polymers is determined by their post-yield behaviour, i.e. strain softening and strain hardening [1,2,3]. This strain softening, a characteristic feature of most amorphous polymers, induces a localisation of strain in small material volumes. Depending on the amount of strain softening and contribution of the strain hardening, these strain localisations might be stabilised. In case of polystyrene, exhibiting a pronounced strain softening and a weak strain hardening, these strain localisations evolve to extremes. The mechanism of craze initiation, as proposed by Kramer [4], induces a built-up of high hydrostatic stresses in these plastic zones and as a result, voids can nucleate. With ongoing strain these voids coalescence and form a void network, also known as a craze. The break-up of these crazes ultimately results in macroscopic brittle fracture. By influencing the strain hardening and strain softening, the macroscopic response of a polymer can be tailored. Henkee and Kramer [5] showed that by cross-linking of polymethylmethacrylate (PMMA) the strain hardening can be enhanced, resulting in macroscopic 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 more homogeneous and ductile deformation behaviour. But this can not only be achieved by special treatments, also a careful choice of testing conditions, like strain rate pressure and temperature, can result in a remarkable increase of ductility. Already in the early sixties it was known that a brittle-to-ductile transition (BDT) in an uniaxial tensile test can be achieved in polystyrene by elevating the test temperature to some 80-90 K above room temperature [9]. Since the early Haward and Thackray model a lot of research effort has been put into the numerical description of the deformation behaviour of amorphous polymers. The models proposed 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 or ductile manner was still not possible as a failure criterion was still lacking. Recently a failure criterion was proposed by Bressers et al. [20]. They demonstrated initiation of crazes in polystyrene at a critical hydrostatic stress of 39MPa, provided that in this region cavitation was preceded by plastic deformation. This sequence of events was consistent with the mechanism of craze initiation as proposed by Kramer [4].

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

24 Experimental and numerical methods

Experimental method The 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 Elastomer Testing 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 strain rate of 10-3 s-1 and a temperatures ranging from 20oC up to 100oC The temperature in the oven could be controlled at an accuracy of M0.5oC. Prior to testing the tensile bars were mounted in the 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 an appropriate sample frequency. To obtain the material parameters, required for the numerical simulations, uniaxial compression tests were performed on the same servo hydraulic system. Cylindrical specimens were made and next compressed at different temperatures and logarithmic strain rates between two parallel, flat steel plates. The friction between the sample and the steel plates was 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 bulging or buckling of the sample is observed, indicating that the friction is sufficiently reduced. 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 data acquisition at an appropriate sample-frequency (depending on strain rate). A constant true logarithmic 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 different temperatures. In Figure 2.1 are the results of these tests for a constant true logarithmic strain rate of 0.001 s-1, the solid lines are the corresponding simulations using the material parameters of Table 2.1. The temperature strongly influences the yield stress: the magnitude of the yield stress decrease with increasing temperature. The modulus of the polystyrene and the strain hardening also decrease with increasing temperature.

Figure 2.1: uniaxial compression test with different temperatures

25 The corresponding material parameters of PS for different temperatures are given in Table 2.1.

Type E GR A0 0 H µ DA h [MPa] [-] [MPa] [s] [MPa] [MJ/mol] [-] [-] [-] PS, 293K 3000 0.37 13 1.38·109 2.6 0.1723 0.14 6.8 60 PS, 313K 2890 0.37 9 1.50·108 2.6 0.1723 0.14 6.0 60 PS, 333K 2730 0.37 6 3.64·107 2.6 0.1723 0.14 4.8 70 PS, 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 was introduced, the so-called compressible Leonov-model [16,17]. To include strain hardening and strain softening [8], the Cauchy stress tensor σ is composed of two contributions: The driving stress tensor s and the hardening stress tensor r respectively:

σ s r (14)

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

~ d s K(J 1)I GB e (15)

J tr(D) (16) o ~ d d ~ ~ d d Be (D Dp ) Be Be (D Dp ) (17)

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

~ d r GR B (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 strain hardening behaviour of polycarbonate in uniaxial compression, uniaxial extension and shear (torsion) [1].

26 It should be noted here that the strain hardening builds up gradually over the total deformation. In fact, Equation (1) implies that strain softening and strain hardening are regarded to act simultaneously. The constitutive description is completed as the plastic deformation rate is expressed in the extra stress tensor by a generalised non-Newtonian flow rule:

s d D (19) p 2 ( eq , D, p) 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 the model. As suggested by Hasan et al. [13] a history variable D is specified, the softening parameter, which influences the viscosity η. During plastic deformation D evolves to a saturation level DA, which is independent of the strain history. The result for η reads:

/ ( , D, p) A (D, p) eq 0 (20) eq m 0 sinh( eq / 0 ) where the equivalent stress τeq is defined by:

1 d d eq 2 tr(s s ) (21) and : p . A (D, p) Aexp D/ (22) m / 0 0 1 1 p 3 tr(σ) 3 tr(s) (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:

Ω (24) V

The evolution of the softening parameter D is specified according to Hasan et al. [13]: . D / D h 1 / p (25) DA 0 with initial condition D = 0; h is a material constant describing the relative softening rate and p is the equivalent plastic strain rate , according to:

p tr(Dp Dp ) (26)

27 Most of the used parameters can be extracted from uniaxial compression tests with varying loading 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 craze initiation, the heterogeneous structure is used here. The numerical model used consists of a Representative Volume Element (RVE) [21-23], a voided polymer matrix. The RVE is used in multi-level finite element simulations (MLFEM) [21-23] to couple the deformation of an heterogeneous structure on micro-level to the deformation of a structure on macro-level. The procedure to generate an RVE is described in Smit [21-23] and starts with the filling of unit box with 20% randomly placed mono-sized spheres (radius equals 3% of the dimension of the box). Next an arbitrary cross-section is taken from this box and this geometry is meshed by the auto-mesher of Mentat (MSC Software) which is represented in Figure 2.2. The structure consists of 5470 8-noded, second-order elements. To prevent rigid-body movement, the lower left corner of the mesh is fixed. Of the upper left corner the displacement in x-direction is inhibited. The lower right corner of the mesh a predefined linear strain rate of 10-3 s-1 is applied to deform the structure. Furthermore periodic boundary conditions are assumed at the edges of the RVE.

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

Results

Uniaxial tensile tests Specimens 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 to 373K the macroscopic response is determined. At low temperatures crazing is observed in all tensile bars prior to macroscopic failure. The strain to break decreases slightly with increasing temperature from 3% to 1.5% of macroscopic strain. More striking was the change in craze density with temperature, at low temperatures many crazes throughout the specimen were observed which covered the major part of the cross-sectional area. At a temperature of 353K only a few crazes were observed which covered much less of the cross-sectional area.

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

373 K 368 K 365 K 363 K 353 K 333 K 313 K 293 K

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 as the deformation mechanism shifts from crazing to shear yielding. These observations are consistent 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 RVE In 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 Figure 2.4 for increasing strain.

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

29 Temperature 293 K Equivalent strain 1.5

strain 1% strain 5% strain 10% strain 15% Temperature 353 K

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

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 is extracted 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 the strain softening is reduced, the deformation could be transferred to other regions at increasing macroscopic strain. This results in a more uniform divided deformation at higher temperatures.

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

39 MPA

293 K 1% 313 K 1.2%

0MPa

333 K 1.2% 353 K 10%

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 the critical hydrostatic stress at 1-1.2% macroscopic strains for temperatures between 293K and 333K. For 353K the model can be stretched much further up to 10% strain before the critical stress is reached. As mentioned before, at high temperatures a reduced strain softening leads to 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 the high macroscopic strain where the critical hydrostatic stress is reached at 353K.

31 The estimated failure strain can be extracted from Figure 2.6, this is the strain where the hydrostatic stress reaches the 39MPa. These estimated failure strains are given as a function of 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 333K the failure strain is 1.2%. For a temperature of 353K however the estimated failure strain is 10%. So between the 333K and the 353K the RVE can be suddenly stretch at larger strains before 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 useful to take more temperatures in to account and to extract material properties on the used tensile bars. When the tensile bars in the experiments are replaced with core shell bars with 20 volume % air the experiments and the numerical simulations can be compared.

Conclusion In 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 increasing temperature. At this temperature the strain to break increases from 3% up to 40%. Material parameters are extracted from uniaxial compression tests and together with the found cavitation criterion of PS, stated in Part 1, the transition is numerically described. Due to the intrinsic strain softening at high temperatures the localisation of strain softening is less extreme in the local plastic zones. This results in a slower built-up of positive hydrostatic stress, and moreover due to the lowered intrinsic yield stress of the material the maximum hydrostatic 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% macroscopic strain. It is concluded that the estimated transition temperature is between the 333K and the 353K. 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 the tensile bars can be replaced with core shell PS in further research.

32 References

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

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33 34 Part 3

Indentation on thin films

Introduction In 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 components approaches the nanometer scale. In both metals and polymers examples are reported that the properties of these materials at such a length scale can be quite different from the large scale properties. For instance glass transition temperature (Tg) of polystyrene to which the mechanical properties are strongly related, decreases dramatically when the thickness of a polymer 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 reproduced with permission, from Forrest [1]

Recent more examples are reported in this respect, which show that the properties of small polymeric structures and polymeric material near a free surface can deviate considerably from the bulk properties [2,3,4]. Generally the phenomenon of a reduced Tg in thin polymer films is assigned to an enhanced segmental mobility near the free surfaces of the polymer. It was shown by Melick et al. [5] by nano-indentation that the mechanical properties near a free surface 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 is studied by a combination of indentation experiments and a finite element modelling. This indentation technique, already successfully applied in ceramics and metals may be the solution for characterising the mechanical properties of polymers at nanometer scale. Thin polymeric layers, ranging from 20nm to 28 m, are spin coated onto thin glass substrates. After careful heat treatments, to allow evaporation of the solvent, the samples are indented by nano-indentation [5,6]. Afterwards the indentation procedure is simulated by means of a finite model. The aim is to estimate the mechanical properties at this length scale by comparison of the experimental and numerical data.

35 Experimental and numerical modelling

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

The material used here is polystyrene (Styron 634) supplied by Dow Chemical. The polymer dissolved in toluene and stirred, with a magnetic stirrer, to obtain good mixing. As a substrate 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 to facilitate breaking up the glass in the desired dimensions. Next the glass plates were thoroughly 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 a treatment in a UV-ozone photoreactor, The plate is exposed to UV-radiation in a ozone filled environment. The spinning of the polymer films was performed on a spin coat machine (Karl Suss CT 62). As the thickness of the spin coated layer is determined by the viscosity of the solution, spinning speed and spinning time, these quantities were adjusted to obtain the desired range of film thicknesses. In Table 3.1 both the parameters of the spin coat process and the obtained film thickness are given, the applied acceleration was always 3000 rad/sec2.

Volume % Proces Time Rotation speed Final thickness PS in toluene [sec] [rad/sec] 0.95 Dosing 3 400 50 nm Spinning 4 2000 Vaporisation 10 300 20 Dosing 5 300 28 m Spinning 4 300 Vaporisation 20 300

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 is necessary to cut the PS layer, otherwise the layer is pulled off the glass while breaking. The polymer 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 125LC for 3 days. This treatment, more essential for thick layers must be performed above the Tg of the material as in the glassy state diffusion of the solvent is inhibited. Any remainders of the solvent can be traced using a Differential Scanning Calorimeter (DSC) as it would lower the Tg of the material. The glass temperature of the polystyrene was measured by DSC after the heat treatment and proved to be equal to bulk Tg reported in literature.

36 Due to differences in the coefficient of thermal expansion of the polystyrene and the glass substrate, stresses were induced by the cooling of the films. This effect is, of course, related to the 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 delamination can be increased by slowly cooling after heat-treating. Another solution is to break the plates when 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 is promoted by poor adhesion between the glass substrate and the polystyrene, a way to prevent this delamination is applying a thin layer of a coupling agent. Therefore the glass substrate is coated with a mono-layer of phenyl-tri-methoxy-silaan. For thin layers this coating is not necessary and may even influence the measurements. Although the coating does not prevent that the layer come loose of the glass, it elongates the time.

The thickness of the spin coated layers varies from 20nm up to 28 m. In the experiments three layers are used; 50nm, 1.3 m and 28 m. The thickness of the layers is measured by an alpha stepper, Atomic Force Microscopy (AFM) and an optical profile meter (UBM). UBM uses optics and is therefore a non-contact device. AFM can also be used to examine the roughness of the top layer. The top layer is very smooth; the Ra roughness value is below 1nm.

Indenting Indentations 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 means of coils at the bottom of the indenter column. The spherical indenter used is a sapphire sphere, of 150 m radius, glued onto a brass holder. The compliance of the apparatus is determined by a reference measurement on silica glass. The elastic indentation depth-force curve is predicted by Hertz’ theory. From the deviation between the theoretical and experimental curve, the compliance is proved to be 6·10-2 µm/N, and the corresponding stiffness of the measuring system 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 an accuracy of 10µN. Indentation depths can be recorded up to 10 m with a theoretical accuracy of 0.04nm. The practical accuracy is 3nm. The indenters used have a radius of 2.2 m and 150 m. The apparatus is only force controlled and hence the load can only be applied at a certain loading rate.

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

37 Numerical modelling The 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 the polystyrene is assumed to be frictionless. The influence was examined by varying the friction coefficient in the model and the influence on the result proved to be negligible. The adhesion between the polystyrene and the glass is considered to be perfect. The model is modelled as an axi-symmetric problem (see Figure 3.2). The dimensions of the glass should be large enough so that the indented region has no influence on the edges of the model, i.e. 10 times the indented region. The indenter is modelled as a half sphere. PS glass indenter

y-direction

Force x-direction

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 the y-direction to provide axi-symmetric conditions. Furthermore rigid-body movements of the glass substrate are prevented. The indenter is prevented to move in x-direction when no load is applied using a very weak spring with a modulus of 1 N/m. At the start of the indentation a pre-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. The sapphire and the glass are assumed to be linear elastic materials with a Young’s modulus of 304GPa 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 parameters of the polystyrene are extracted from uniaxial compression tests on bulk polystyrene and are represented in Table 3.2.

Type E GR A0 0 H µ DA h [Mpa] [-] [Mpa] [s] [MPa] [MJ/mol] [-] [-] [-] Annealed 3300 0.37 13 1·1012 2.6 0.1723 0.14 11 75 reference PS

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]) and uses the material parameters described in Table 3.1. To analyse the model the options of large displacements, constant dilatation (to prevent locking) and updated Lagrange are used. To exclude any influence of mesh size a stepwise element size reduction is performed until the solution converged to a steady, mesh independent, result. In these simulations especially the PS layer is refined, in this way the shear behaviour of the polystyrene at the interface with the glass can be accurately described. In order to prevent excessive computation time the mesh refinement is restricted to areas of interest. A typical mesh is given is Appendix A.

38 Results

A 1.3 m layer is indented with both the nano and micro indenters with an indenter of 150 m 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 1.3 m layer

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

The measurement of the thick layer is done on the micro indenter with a sphere of radius 150 m. The experiment and the simulation are given in Figure 3.4.

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

39 The numerical simulation provides a good fit for the experiment. The withdrawal of the indenter is not simulated since this is not modelled well in the used material model (see Part 1) and to avoid large calculation times. The thin layer of 50nm is indented on the nano indenter with a sphere of 2.2 m radius. The experiment 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 0 N. The experiments show that first the layer is easily penetrated by the indenter, later the influence of the glass becomes more important and the resulting stiffness increases. The standard simulation begins with no pre load and is too stiff compared to the experiment, it does not show the characteristic slope of the first tens of nanometers as seen in the experiment. The incline of the simulation describes the experiments when the beginning is not noticed. The experiments and the simulation coincide when the simulation should be shifted 17nm. The stiffness of the curve of the numerical model must be ascribed 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 linear temperature field takes care of the drop in Tg at the surface (see Figure 3.1). The temperature is constructed in a way that the Tg at the surface is lowered to 20LC and at a depth of 50nm the Tg is normal, 106LC. The temperature field increases the indentation depth but the simulation can not describe the experiments (see Figure 3.5). Another attempt is made by simulating the contact between the glass and the layer to be friction less. The first part of the experiments is described in a proper way, but at larger indentation depths the simulations stops because of converge errors. So the beginning is described well and therefore it would indicate that the PS film does not adhere to the glass at all. To simulate this thin layer well the conditions between the glass and the polymer are extremely important. In further work these conditions must be examined and used in numerical models.

40 Conclusion It 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 the glass and the polystyrene holds can be elongated by adding a mono-layer of PTMS to the glass or by braking the glass before the polystyrene is dried. For thick layers it is possible to simulate the indentation behaviour with a compressible Leonov-model. The simulation of the thin layers is more difficult. Near the surface another material behaviour is considered that result in a lower modulus of indentation at the first nanometers. The numerical model under-predictes the indentation depth with the applied material parameters that are extracted from bulk polystyrene. An attempt to influence the material behaviour near the surface increases the indentation depth, but only a little. The characteristic measured indentation curve is not achieved. When, however, the thin layer was modelled to be non-adhesive to the glass, the slope at the beginning of the experiments was modelled well, so the beginning is described well and therefore it would indicate that the PS film 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 are extremely important. In further research a combination of a temperature field and a partial non-adhesive contact between the glass and the layer PS is recommended to describe the experiments with the simulations. Another point of research is to use the modulation option of the nano indenter, in this way the thin layer can be indented repetitive at the same position.

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41 42 APPENDIX A: numerical models

Numerical model for indentations on bulk PS

43 Numerical model for indentations on thin layers PS on glass

sapphire PS glass

44 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