Deformation and Failure in Semi-Crystalline Polymer Systems · Deformation and Failure in...

Deformation and Failure inSemi-Crystalline Polymer Systems

Roel JanssenMaster Thesis, Eindhoven University of Technology, MT 02.14

Prof. Dr. Ir. H.E.H. MeijerDr. Ir. L.E. GovaertIr. R. SchellekensIr. B.A.G. Schrauwen

Eindhoven, October 2002

Abstract

This study deals with the relation between the microstructure of semi-crystalline polymersand their intrinsic deformation and fracture behaviour. For this purpose, the deformationand failure of well-characterised PET, PE and PP samples have been investigated thoroughly.Compression tests on amorphous and cold crystallised PET showed that the strain hardeningremained unchanged, despite the development of a crystalline phase by cold crystallisation.This suggests a direct relation between entanglements and strain hardening, similar to amor-phous polymers, and the slip of crystallites after yielding. True stress-strain curves of annealedPE confirm that strain hardening is not affected by the degree of crystallinity. However, meltcrystallisation and a lower molecular weight grade for PE and PP resulted in a distinctly lowerstrain hardening modulus. This observation confirms that entanglements affect the strain hard-ening behaviour, since melt crystallisation and a reduction in molecular weight promotes a lossof entanglements during lamellar folding.

Tensile tests were performed to examine the failure behaviour. The melt crystallisation pro-cess has proved to be of influence on failure, as its lower strain hardening appeared not to beable to stabilise strain localisation. Moreover, an increase in crystallinity due to an increasein yield stress results in a larger localisation of deformation. Since a higher molecular weightpolymer grade hampers the formation of a crystalline phase, localisation after yielding will beless. This and a larger strain hardening modulus enlarge the chance of ductile behaviour intension for polymers with a larger chain length.

Finally, an attempt was made to predict the intrinsic deformation behaviour of PP and PE.The compressible Leonov model appeared to be capable of generating an appropriate descrip-tion of the intrinsic deformation behaviour over a range of strain rates. For PE, even the pecu-liar double yield behaviour was modelled adequately by the compressible Leonov model.

Keywords: Semi-Crystalline Polymers, Morphology, Intrinsic Deformation, Strain Hardening,Failure, Entanglements, Constitutive Modelling

Contents

Abstract i

1 Introduction 3

I Deformation and Failure in Semi-Crystalline Polymer Systems 4

2 Theoretical Background 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Microstructure vs. Intrinsic Deformation . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Macroscopic Deformation & Failure . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Materials & Methods 153.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Characterisation of the Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Mechanical Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Intrinsic Deformation 194.1 Strain Hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Yield Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Macroscopic Deformation & Failure 285.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Bibliography I 31

II Modeling of Intrinsic Deformation in Semi-Crystalline Polymer Systems 34

6 Introduction 356.1 History of the Leonov Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.2 The Leonov Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

7 Results & Discussion 387.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Bibliography II 40

8 Conclusions 43

CONTENTS 2

A Young’s Dislocation Model 44

B X-Ray Diffraction 47B.1 Wide Angle X-ray Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47B.2 Small Angle X-ray Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

C Microstructural Uniform Samples 50

Chapter 1

Introduction

In the past decades, polymers have become more and more recognised as an appropriate ma-terial for many diverging applications. Examples of polymer products can be found in pack-aging, automotive industry, house-holding as well as in many engineering applications. In thepackaging industry, some examples of the use of polymers are plastic bags, food storage boxesand beverage bottles, respectively made of PE, PP and PET. Producing lighter cars for environ-mental reasons, is partially achieved by assembling light-weight thermoplastic components,such as bumpers and many interior applications. Nowadays, the know-how of polymers andtheir mechanical background has developed to the extent that they can be used in mechani-cally high-demanding applications, such as bullet resistant vests and cordage in the shippingindustry. However, in many of those applications the polymer material and their dimensionsrequired to fulfill the mechanical requirements are selected through experience during proto-typing or experience from applications of the same kind. Thus, the development of productsis a time-consuming and costly business. A thorough understanding of the microstructuralprocesses during deformation, and the ability of modelling the deformation in complex ge-ometrical structures might accelerate this process and reduce the costs of the developmentalstage.

Nowadays, the deformation and failure of amorphous polymers is well-established [1].Apart from a well-known relationship between the deformation and failure and the microstruc-ture, it was demonstrated that the deformation up to large strains of amorphous polymerscan be accurately captured by constitutive modelling [1, 2]. However, due to its complex mi-crostructure, various and sometimes contradicting views exist with respect to the deforma-tional background in semi-crystalline polymers. Therefore, a comprehensive study has beenperformed in an attempt to relate the deformation and failure behaviour of semi-crystallinepolymers to the microstructure. After an initial literature review, an attempt has been made tounderstand the deformation and failure by performing experiments, such as compression test-ing and tensile testing. Therefore, a set of samples in a wide range of crystallinities has beencreated by quenching, annealing and melt crystallisation of three semi-crystalline polymers(PET, PP and PE) of which the microstructural aspects of the crystalline phase are characterisedby means of small angle and wide angle X-ray scattering. This investigation is divided in twoparts. In the first part, the relation between microstructural processes and intrinsic deforma-tion behaviour is discussed by true stress-strain curves obtained by compression testing. Afterthat, the results and discussion of the failure in semi-crystalline polymers, as a result of thetensile testing, are presented. In the second part, an attempt is made to predict the intrinsicdeformation behaviour of PE and PP by means of the compressible Leonov model.

Part I

Deformation and Failure inSemi-Crystalline Polymer Systems

Chapter 2

Theoretical Background

2.1 Introduction

In this chapter, a review is presented on the views as they currently exist on the relationshipbetween microstructure and deformation and failure of amorphous and semi-crystalline poly-mers. In the case of amorphous polymers, the correlation between the polymer microstructureand the intrinsic deformation behaviour is well-established. However, assigning the intrin-sic deformation mechanisms, such as yield stress and strain hardening, to the microstructureof semi-crystalline polymers is much more complicated. Whereas amorphous polymers onlyconsist of an entangled polymer network of coils, semi-crystalline polymers are composed ofamorphous and crystalline regions. The amorphous structure is mainly characterised by en-tanglements, which are physical nodal points between molecular chains. Semi-crystalline poly-mers are characterised beside entanglements in the amorphous fraction, by so-called crystallinelamellae. The thickness of these highly ordered folded chains is expected to affect the intrinsicdeformation of semi-crystalline polymers. A schematic representation of the amorphous andsemi-crystalline structure is depicted in figure 2.1. Since semi-crystalline polymers also pos-sess an amorphous phase, for which deformation mechanisms are well-characterised, firstly,the background of intrinsic deformation in amorphous polymers is presented in order of thesuccessive stages of deformation. Thereafter, a brief overview is presented of the microstruc-tural background of deformation in semi-crystalline polymers. However, due to its complexstructure, different, sometimes contradicting, views exist on the background of the post-yieldbehaviour of semi-crystalline polymers. A number of views are discussed in order of the suc-cessive intrinsic deformation stages. At this point, it is worth mentioning that most studiesonly focus on the crystalline properties, such as crystalline content and lamellar thickness, andoften ignore the influence of crystallisation procedures and the molecular chain configuration,which strongly affect the amorphous properties.

Entanglement

a

amorphous phase

lamellar thickness

crystalline phase

b

Figure 2.1: a) Randomly folding of molecular chains in amorphous polymers. b) Semi-crystalline polymers are composed ofhighly ordered folded crystalline regions and amorphous regions.

Theoretical Background 6

2.2 Microstructure vs. Intrinsic Deformation

Amorphous Polymers

The intrinsic behaviour of polymers can be studied using video-controlled tensile testing [3] oruniaxial compression testing [4,5], since in these loading conditions no localisation phenomenaare present. The true stress-strain curves of glassy polymers, obtained by these experimentaltechniques, exhibit several characteristic stages of deformation. These typical intrinsic defor-mation stages of glassy polymers are schematically depicted in figure 2.2. Initially the polymershows a reversible (visco)-elastic deformation. At a certain amount of stress, deformation be-comes of an irreversible nature, which is recognisable by a yield point in the true stress-straincurves. After the yield point, a decrease in stress is observed, which is usually referred to asstrain softening. With proceeding deformation, a strain hardening behaviour is observed byan increase in stress. In the past decades, the microstructural background of this post-yieldbehaviour has become well-established.

Figure 2.2: The intrinsic deformation of an amorphous polymer.

It is well-known that the amount of strain softening depends on the thermo-mechanical his-tory of the material. It has been shown that strain softening can be reduced or even completelyremoved by a thermal or mechanical treatment [6–9], resulting in an uniform deformation ofPC [8] and a more ductile failure of PS during tensile testing [9]. These observations indicatethat strain softening is closely related to the process of physical ageing. During processing,the polymer is rapidly brought from the rubbery state into the glassy state, accompanied by aloss in segmental mobility of the polymer chains. Due to this instantaneous vitrification, themolecules can no longer maintain their thermo-dynamical equilibrium and, therefore, their oc-cupied volume is larger than the equilibrium volume. However, as the thermal mobility hasnot completely disappeared, the polymer molecules attempt to achieve their thermo-dynamicequilibrium in the course of time. During this process of physical ageing, volume relaxationtakes place, which is coupled with an increase in yield stress and a simultaneous increase ofthe strain softening. This is demonstrated by figure 2.3, where the physical ageing process inPC is accelerated through annealing at 140ÆC for 2 weeks. The observed increase in modulusand yield stress are indicative for a denser molecular composition, in which the interactionsbetween molecules are stronger. Strikingly, the large strain deformation is identical for boththe annealed and quenched PC. This implies that the effect of thermal history of the material isremoved upon plastic deformation.


0 −0.25 −0.5 −0.750

25

50

75

100

true strain [−]

com

pr. t

rue

stre

ss [M

Pa]

annealed

quenched

Figure 2.3: An increase in yield stress and consequently strain softening behaviour is observed for PC, which is annealed for 2weeks at 140ÆC. [1]

The relation between the strain hardening behaviour and the amorphous microstructurehas been investigated more extensively. It has been shown that plastic deformation in glassypolymers can be completely recovered by an annealing treatment above the glass-transitiontemperature [10–13]. This reversibility of plastic deformation suggests the entangled polymernetwork to be the origin of the large strain behaviour. This hypothesis is supported by the factthat the strain hardening behaviour of amorphous polymers is accurately described by a neo-Hookean relation [14,15]. A review of experimental results of various publications satisfied thefollowing neo-Hookean relation up to large strains

�true = �yield +Gr(�2 � �

�1) (2.1)

Where, � is the draw ratio, Gr is the strain hardening modulus and �yield is the yield stress.Moreover, Govaert and Tervoort found a neo-Hookean relation in strain hardening by mechan-ically rejuvenated PC tensile bars [16]. The neo-Hookean description of strain hardening in theso-called compressible Leonov model fitted very well onto experimental results [17]. Also, van

Figure 2.4: The strain hardening modulus of PS increases for an increasing entanglement density obtained by blending withPPO. [1]


Melick was also able to predict the strain hardening behaviour of PS-PPO blends adequatelyby a neo-Hookean relation, implemented in the compressible Leonov model [1]. Furthermore,he provided experimental support, which confirmed the relation between entanglements andstrain hardening behaviour in amorphous polymers. By varying the network density of PSby blending with PPO or cross-linking during polymerisation, Van Melick showed that strainhardening is proportional to the entanglement density, see figure 2.4.

Semi-Crystalline Polymers

The origin of yielding in semi-crystalline polymers is still subject of much debate. A raise inyield stress is noticed for polymers with increasing degree of crystallinity [18–23]. In these stud-ies, mainly the yielding behaviour of PE has been examined. At room temperature, the amor-phous phase of PE is in the rubbery state. Therefore, it does not contribute to the yield stress.These conditions facilitate the investigation of the yielding behaviour of the crystalline phase.Several theories have been developed to attribute the yielding to the crystalline fraction. Besidea melting and recrystallisation theory [24], the nucleation of dislocation-defects [25–27] is an in-terpretation of the yielding behaviour on a microscopic level. However, evidence against thefirst theory is provided by SAXS-studies [28]. The latter model is the more established one, andwas firstly proposed by Young [25, 26]. The background and some more recent developmentsaround Young’s model are presented in appendix A. It is based on the concept that at yield theenergy necessary to nucleate a screw dislocation of Burgers vector b lying distance d within thecrystalline lamellae is reached, which is described by

�y =Kb

r0�exp

��2��Gc

lcKb2+ 1

�(2.2)

Where r0 is the core radius of the dislocation, K is a function of the shear modulus of the crys-tal, lc is the crystal thickness and �Gc is the Gibbs free energy associated with the change inenergy during the formation of a dislocation. This shows that the model relates the yield stress�y directly to temperature (through �Gc) and lamellar thickness lc. Experimental observationsthat demonstrate that the yield stress is proportional to lamellar thickness [20, 21, 29] seem tojustify Young’s model. Moreover, the model has been successfully compared to experimentalresults of PE [20, 23, 30–35] and PP [21, 29]. However, the prediction of the yield stress variedfrom fair [23, 30, 31] to a close match of the experimental observations [20, 21, 29, 32–35], whichcan probably be accounted to an inconsistent use of several model parameters.

The initial deformation of semi-crystalline polymers can be explained by a scheme of inter-and intra-lamellar shear of the lamellae, see figure 2.5. Firstly, the deformation is (visco-)elastic,which is caused by straining the inter-lamellar amorphous domains [36], as is visualised byfigures 2.5b, 2.5c and 2.5d. At yield, two lamellar deformation mechanisms occur, whichare schematically represented in figures 2.5e and 2.5f. The first mechanism is the chain slipthrough the lamellae, where after the second process of coarse slip of lamellae occurs. Thisintra-crystalline shear results in a fragmentation of the lamellae [37]. Subsequently, the lamel-lae disintegrate and their remains will be transformed into a fibrillar phase coupled with theappearance of voids in the amorphous layers. Upon further deformation, the fibrils stretch,causing the chains to align and causing a lateral coalescation of voids, eventually resulting in amacroscopic crack due to breaking fibrils [36]. As already mentioned, the yield stress is foundto be proportional to lamellar thickness [20, 21, 29]. It was proposed that thinner lamellae en-courage the fine chain slip mechanism, since a lower chain length more easily slides throughthe crystallites [18].


a b c

d e f

Figure 2.5: a) Lamella before deformation. b) Elastic deformation: Inter-lamellar separation. c) Elastic deformation: Inter-lamellarrotation. d) Elastic deformation: Inter-lamellar shear e) Plastic deformation: Fine chain slip f) Plastic deformation:Coarse slip.

The intrinsic strain softening is argued to be a property of the glassy phase only. Since thepolymer molecules are fixed in the crystallites, they do not attribute to the physical ageing pro-cess, as described for amorphous polymers. Therefore, with increasing crystalline content andthus a decreasing amorphous fraction, the strain softening is expected to diminish to very lowvalues for highly crystalline polymers. Furthermore, strain softening is only present in semi-crystalline polymers of which the amorphous phase is in its glassy state.

Haward [15] was able to find a neo-Hookean relation for the strain hardening behaviourof semi-crystalline polymers, similar to amorphous polymers. This neo-Hookean relation isbased on the entanglement density in the amorphous phase, and thus the findings of Hawardshow that this phase attributes in the strain hardening behaviour of semi-crystalline polymers.This is confirmed by a study by Bastiaansen [38] et al. of the influence of processing and crys-tallisation of UHMW-PE. They were able to draw solution crystallised samples up to very highstrains, whereas melt crystallised samples showed a much larger strain hardening and subse-quently fracture of the sample is observed. This difference in large strain behaviour can beexplained by the entanglement density. For the solution crystallised samples, a high degreeof disentangling of chains is expected, which enables the polymer coils to become stretched ina fibre configuration, whereas for the melt crystallised samples the highly entangled networkcauses fracture of chains. Smith and Lemstra [39] also noticed a difference in strain hardeningfor melt and solution crystallised samples. They obtained true stress-strain curves by drawingUHMW-PE at 120ÆC at which the polymer deforms homogeneous. In figure 2.6, it can be seenthat the melt crystallised sample exhibits a larger strain hardening modulus than the solutioncrystallised sample. This strongly suggests that strain hardening is affected by the number ofentanglements. A different view is presented by Hiss et al. [19]. They examined the influence ofcrystalline content to strain hardening behaviour by means of video-controlled tensile testingof several poly(ethylene)-grades and related copolymers. They found that the strain hardeningmodulus is larger for grades with a higher crystalline content. As a result, they assume thatthe raise in modulus yields from the crystallites acting as fillers and as cross-links. Fillers resultin a larger entangled network density, assuming that the entanglements in the melt are shifted


Figure 2.6: The difference in large strain deformation behaviour of melt and solution crystallised UHMW-PE is attributed to theentanglement density. Tested at 120oC. [39]

into the amorphous regions during crystallisation. The view that anchored tie molecules incrystallites can be conceived as entanglements, leads to an even higher entanglement densityfor higher crystalline grades, since more anchored molecules are expected to be present. Thelatter concept, however, is inconsistent with the idea of the slip of fragmented crystallites afteryielding, which shows, once more, that the relationship between microstructure and deforma-tion in semi-crystalline polymers is not fully understood yet.

The observations presented, all suggest that the entangled polymer network at least par-tially contributes to the strain hardening behaviour. This means that crystallisation procedures,which affect the entanglement distribution in polymers, also play a role in the strain hardeningbehaviour of semi-crystalline polymers. Basically, two procedures can be distinguished whenthe polymers are crystallised by a thermal treatment. The first process is called melt crystallisa-tion, where the ordered regions originate from the disordered entangled melt. Slowly coolingfrom the melting temperature gives the polymer chains time to crystallise. The crystallites areformed by the reeling in of polymer chains from the entangled melt accompanied by a lossof the entanglements [40–42], see figure 2.7a. This crystallisation process is called adjacentre-entry and results in the formation of highly ordered lamellae. In addition to melt crystallisa-tion, solid semi-crystalline polymers can be crystallised by annealing at elevated temperatures.Then, the frozen-in disordered coils gain mobility and will crystallise. At these annealing tem-peratures, however, mobility of the chains is only moderate, and only crystallites of a very lowdegree of perfection emerge. Now, there is no reeling in of amorphous chains and thus thenumber of entanglements are preserved, see figure 2.7b. This crystallisation process is calledcold crystallisation for initially glassy polymers or annealing for semi-crystalline polymers.

a b

Figure 2.7: a) Melt crystallisation: adjacent re-entry, reeling in of chains results in a loss of entanglements. b) Cold crystallisation:incidental and random folding without loss of entanglements.


Figure 2.8: The difference in fracture behaviour of melt and solution crystallised LDPE is attributed to the entanglements. Testedat 75ÆC. [38]

Moreover, crystallisation procedures are also expected to contribute to the failure of semi-crystalline polymers, as is shown by the following two examples. The melt crystallised LDPEin figure 2.8 shows ductile behaviour, whereas the solution crystallised LDPE fails brittle [38].The authors attribute this difference to entanglements, since in solution crystallised samplesfewer entanglements are present. An investigation by Plummer, also suggests that entangle-ments, and thus the amorphous phase, strongly affects the failure behaviour in semi-crystallinepolymers [43]. By means of impact testing, they demonstrated that the fracture toughness ofPOM decreases with increasing crystallisation temperature, as can be seen in figure 2.9. In theright part of the graph, experimental data show a decrease in fracture toughness for increasingcrystallisation temperature. This is explained in terms of an increased loss in entanglementswith increasing crystallisation temperature, as the chain mobility increases. Moreover, a lowertoughness for lower molecular weight grades confirms the idea of entanglement loss duringlamellar folding at high crystallisation temperatures. A more detailed view on the failure insemi-crystalline polymers is presented in the following paragraph.

Figure 2.9: A decrease in fracture toughness of POM is explained by a loss in entanglements. [43]


2.3 Macroscopic Deformation & Failure

The deformation behaviour of the polymers investigated can be analysed by an approach de-rived by Haward [44]. In order to predict whether deformation is inhomogeneous or homo-geneous, Haward applied Considere’s construction, assuming there is no strain softening andneo-Hookean strain hardening. Inhomogeneous deformation occurs due to the presence of ge-ometric softening. Geometric softening is the decrease in stiffness due to a decrease in areawhich is subjected to a load. This construction is schematically explained by figure 2.10. Infigure 2.10a, four true stress-strain curves with increasing yield stress, but identical post-yieldbehaviour are depicted. Here, the relation between true stress and true strain is

�true = �yield +Gr(�2 �

1

�) (2.3)

When transformed to engineering stress-strain curves, see figure 2.10b, it can be seen that thepost-yield behaviour for each of the curves changes, according to

�eng =�true

�=

�yield

�+Gr(��

1

�2) (2.4)

0 1 2 3 40

10

20

30

λ2 − λ−1 [−]

True

Stre

ss [M

Pa]

a 1 3 5 7

0

10

20

30

λ [−]

Eng

inee

ring

Stre

ss [M

Pa]

b

Figure 2.10: a) Schematic true stress-strain curves with increasing yield stress and identical post-yield behaviour. b) The truestress-strain curves transformed to engineering stress-strain curves. The post-yield behaviour is no longer identical.

In this example, the strain hardening modulus is 3 MPa and the yield stress is 4, 9, 14 and 19MPa respectively. The occurence of geometric softening can be recognised in the engineeringstress-strain curves, which are depicted in figure 2.10b. The curve with a yield stress of 4 MPashows an increase in stress after yielding, which indicates that deformation is homogeneous.The following curve, however, will only just be prevented from necking, whereas the curveswith a yield stress of 14 MPa and 19 MPa both exhibit geometric softening, recognised by the adrop in stress after yielding. Now, Considere’s condition for necking requires that d�eng/d� �0. If subsequently the assumption is made that yielding of polymers occurs at very low strains( � � 1 ), then it can be concluded from equation 2.4 that inhomogenous deformation occurs if

�yield

Gr

> 3 if � � 1 (2.5)

Thus, an increase in yield stress or a decrease in strain hardening enhances the localisation ofdeformation.


The stability of inhomogenous deformation cannot be predicted by this criterion. Neckingis stable if the load transferred by the neck equals the load required to induce yielding in theundeformed zone. Now, the question arises whether this situation can be achieved. Therefore,a force equilibrium for a tensile bar exhibiting necking during deformation is formulated byHaward to derive a stability criterium. This leads to the following equations.�

�yield +Gr

��2

n �1

�n

��A0

�n= �yieldA0 (2.6)

Removing A0 on both sides, and formulating equation 2.6 again in terms of �yield/Gr, leads to

�yield

Gr

=�2n �

1

�n

�n � 1(2.7)

The �n represents the draw ratio of the neck for which necking is stable. This relation is plottedin figure 2.11a, in which the macroscopic deformation behaviour is illustrated by tensile bars.Again, it can be seen that necking takes place for �yield/Gr larger than three. The solid line rep-resents the neck draw ratio for which deformation is stable. Now the question arises, whetherneck growth is stable and when fracture is about to occur. Supposing that the tensile strengthis a factor � larger than the yield stress, then

��yield = �yield +Gr(�2

n �1

�n) (2.8)

and

�yield

Gr

=

�2n �

1

�n

(�� 1)(2.9)

The maximum tensile strength is represented by the dashed line in figure 2.11b. The solidline represents the neck draw ratio for which deformation is stable until the tensile strengthis reached. The intersection of those lines gives the draw ratio in the neck for which fractureoccurs. For very large �yield/Gr-values, large strain localisation occurs at such a small scale thatmacroscopic brittle failure is observed, without the presence of necking.

Figure 2.11: a) Neck stability analysis: the solid line represents the draw ratio for which stable necking occurs b) The dashed linerepresents the tensile strength of the neck.


The approach presented above is only valid for polymers which do not display strain soft-ening. This is the case for semi-crystalline polymers of which the amorphous phase is situatedin the rubbery state. However, if the amorphous phase is in its glassy state, strain softening canbe observed during the deformation of semi-crystalline polymers. Strain softening has its owncontribution to the macroscopic deformation and failure in polymers. This can be analysed bythe approach of van Melick [1]. Van Melick showed that the extent of strain softening affectsthe deformation and failure behaviour. The extent of strain softening is defined as the differ-ence between the yield stress and the stress of the strain softening plateau. Supposing that theyield stress is a factor ky larger than the stress in the strain softening plateau �yield;s, and thusbecomes ky�yield;s. The force equilibrium in equation 2.6 becomes�

�yield;s +Gr

��2

n �1

�n

��A0

�n= ky�yield;sA0 (2.10)

and as a consequence

�yield;s

Gr

=�2n �

1

�n

ky�n � 1(2.11)

This relation is graphically displayed in figure 2.12 by the solid lines. It shows that for a largerstrain softening the tensile strength is reached at a lower draw ratio in the neck. Furthermore,it can be seen that in the case of strain softening, the equilibrium line starts at �yield/Gr equal to0, and as a result inhomogeneous deformation is always present.

0 2 4 6 81

2

3

4

5

σy / G

r [−]

λ n [−]

larger strainsoftening

fracture fracture

Figure 2.12: Neck stability analysis in which strain softening is included: A larger strain softening enhances brittle failure. Thedashed line represents the tensile strength of the neck.

Chapter 3

Materials & Methods

3.1 Materials

In order to investigate the influence of crystallinity on the (post-)yield behaviour, various semi-crystalline polymers (PET, PP and PE) were investigated. Since testing took place at roomtemperature, a distinction of the materials can be made in terms of glass-transition tempera-ture. The glass-transition temperature of PE, PP and PET is situated, respectively, far below,around and far above the testing temperature.

Poly(ethylene terephthalate), PET, was selected for its slow crystallisation rate. The materialused was a non-commercial PET-grade (Mn = 36.2 kg/mol) supplied by DSM, The Netherlands.This PET was post-condensated in order to increase the molar mass and contains isophatalic-acid. Both result in a decrease of crystallisation rate facilitating the production of low crys-talline samples, of which afterwards crystallinity can be raised quite accurately by means ofcold crystallisation. It is well known that the presence of moisture may induce hydrolysisand thus might alter the molecular weight distribution during processing. Therefore, prior toprocessing, the PET-pellets were dried in a vacuum oven for at least 10 hours at 110 ÆC. ThePET pellets were compression-moulded into 5mm plates at 280 ÆC and stepwise the force wasraised. Since there is a reasonable chance of chain degradation occurring at this temperature,the press was not released during moulding. This was done in order to prevent contact of sur-rounding air (oxygen) to the polymer melt, which causes the degradation. Next, the polymermelt was quenched in a cold press of 15 ÆC. This cooling took place on a time-scale in which nocrystalline phase was developed.

Also, samples were created by placing polymer melts into a hot press of, respectively, 200ÆC, 150 ÆC and 135 ÆC. Finally, slowly cooled samples were created by leaving the melt in thehot press after switching it off; the so-called melt crystallised samples. To get an indication ofdifferences in cooling rate, a thermocouple was inserted into the mould to registrate the tem-perature profile during cooling. It showed that the quenched plates were cooled at 22.5 ÆC/sand the cooling of the melt crystallised samples occurred at 0.01 ÆC/s. In order to obtain a setof samples containing a wide range of crystallinities, the quenched samples of PET were coldcrystallised in a silicone oil bath at 100 ÆC for 4, 24 and 72 hours and at 110 ÆC for 15 minutesand 4 and 54 hours. The samples were wrapped up in aluminum foil and teflon tape to avoidpossible alterations in material properties due to contact with the silicon oil. Prior to the test-ing, the cylindrical PET samples were conditioned in RH 0% for 72 hours.

Materials & Methods 16

Polymer-Grade Used Name Mw Mn

commercial name [kg/mol] [kg/mol]non-commercial PET PET ? 36.2StamylanHD9089 PE LM 70 11StamylanHD8621 PE HM 215 7StamylanP512 PP LM 205 41StamylanP15m10 PP HM 354 64

Table 3.1: Molecular weight of the PET, PE and PP-grades.

Two grades of poly(ethylene), PE, having different molecular weights are used in this in-vestigation; StamylanHD9089, Mw = 70 kg/mol, and StamylanHD8621, Mw = 215 kg/mol.Both grades were compression moulded to 5mm plates at 200ÆC with a stepwise raise in forceand subsequently quenched or melt crystallised. The quenched plates were cooled at 12ÆC/sand the melt crystallised at 0.01ÆC/s. Only quenched PE LM samples were annealed at 100ÆC,110ÆC, 120ÆC and 125ÆC for 8, 24, 48 or 72 hours. Poly(propylene), PP, was also available attwo molecular weights; StamylanP15M10, Mw = 354 kg/mol, and StamylanP512MN10, Mw =205 kg/mol. PP was compression moulded at 220ÆC and only quenched and melt crystallisedsamples were produced.

3.2 Characterisation of the Microstructure

The properties of the crystalline phase, such as degree of crystallinity and lamellar thickness,can be determined by means of wide angle and small angle X-ray scattering methods (WAXSand SAXS), respectively. The WAXS measurements were performed at the ID13 beamline (mi-crofocus) of the European Synchrotron Radiation Facility (ESRF) in Grenoble, France, whereasthe SAXS measurements were executed at the CRG BM 26 beamline (Dubble) of the ESRF. Thedetermination of degree of crystallinity and lamellar thickness by these scattering methods andtheir background is expounded briefly in appendix B. The outcome of these measurements arepresented in table 3.2.

As expected, the degree of crystallinity proves to be strongly dependent on crystallisationprocess, crystallisation time and crystallisation temperature. For PET, the prerequisite of ob-taining an amorphous sample and a wide set of crystallinities is met. It becomes amorphouswhen quenched and develops a crystalline content of 42% when melt crystallised. Cold crys-tallisation of the quenched PET resulted in intermediate levels of crystallinity up to 30%. How-ever, the crystallite thickness alters simultaneously with degree of crystallinity, which hampersdrawing conclusions of mechanical aspects with respect to the degree of crystallinity and lamel-lar thickness. For PE, on the other hand, a broad set of samples was created, including sampleshaving a similar degree of crystallinity, but differ in lamellar thickness. See for example, themelt crystallised, Tcc=125Æ for 48 hours and Tcc=120Æ for 72 hours samples of PE LM, of whichcrystallinity is about 76.5% and crystallite thickness respectively diminishes from 28.1 nm and27.5 nm to 25 nm.


Material & Treatment Xc (%) L (nm) lc (nm)PETquenched 0 - -Tcc=100Æ, 4h 0 - -Tcc=100Æ, 24h 11.6 - -Tcc=100Æ, 72h 27.1 10.5 2.9Tcc=110Æ, 15min 21.7 - -Tcc=110Æ, 4h 26.4 11.9 3.1Tcc=110Æ, 54h 29.1 10.2 3.0Tmc=135Æ 31.4 13.4 4.2Tmc=150Æ 32.2 13.5 4.4Tmc=200Æ 34.7 16.8 5.8melt crystallised 42.6 16.1 6.9

PE LMquenched 68.4 26.7 18.3Tcc=100Æ, 8h 69.9 27.4 19.2Tcc=100Æ, 24h 71.0 27.3 19.4Tcc=100Æ, 48h 73.5 29.4 21.6Tcc=100Æ, 72h 73.8 28.6 21.1Tcc=110Æ, 8h 72.3 29.0 21.0Tcc=110Æ, 48h 73.3 29.5 21.6Tcc=120Æ, 8h 74.1 31.4 23.3Tcc=120Æ, 24h 75.6 32.4 24.5Tcc=120Æ, 72h 76.5 32.7 25.0Tcc=125Æ, 8h 75.6 34.7 26.2Tcc=125Æ, 48h 76.6 35.9 27.5melt crystallised 76.7 36.7 28.1

PE HMquenched 68.4 29.9 20.5Tcc=120Æ, 72h 72.9 34.9 25.4melt crystallised 73.8 40.5 29.9

PP HMquenched 65.6 14.6 9.6melt crystallised 66.7 22.8 15.2

PP LMquenched 64.5 x xmelt crystallised 69.5 x x

Table 3.2: Degree of crystallinity Xc, long period L and lamellar thickness lc of differently treated PET, PE and PP-grades.


3.3 Mechanical Testing

For compression testing, the compression moulded plates were sawn into bars of 5x5mm, andsubsequently machined to cylindrical 4x4mm samples on a milling cutter. It was chosen to pro-duce samples as small as possible in order to obtain microstructural homogeneous samples. Inappendix C, it is confirmed for polystyrene, PS, that this sample size ensures valid and repro-ducible results. The compression tests were performed on a servo-hydraulic MTS ElastomerTesting System 810 at a constant logarithmic strain rate, prescribed by the following equation:

L(t) = L0e_�t (3.1)

Where, L(t) is the momentary length, Lo is the initial length, _� is the strain rate and t is time.Compression took place at a strain rate of - 3�10�3 s�1 between two flat steel plates and con-trolled by a MTS extensometer. Prior to testing, the actual dimensions of the samples wereaccurately determined by a micrometer, leading to true stress-strain curves carefully derivedfrom the measured force and extensometer displacement. In order to minimise friction, a thinfilm of teflon tape (3M 5480, PTFE skived film tape) was attached to the top and bottom of thesamples. Also, both contact areas between plates and sample were lubricated by a soap-watermixture. Both measures prevented the cylindrical samples to barrel or buckle, ensuring uni-form deformation during the compression test. All compression tests were performed at roomtemperature.

stamps

stamps

polymer

displacement

extensometer

extensometercontrolledsample

Figure 3.1: Schematic representation of the compression test set up.

The tensile test samples are also obtained from the 5mm plates in order to approximate mi-crostructural properties similar to the compression test samples. For PE LM and PE HM, thequenched, melt crystallised and the samples annealed at 120ÆC for 72 hours were examined.For PP, only the quenched and melt crystallised samples were investigated of the PP HM. Thequenched 5mm PET plates, however, visually appeared to have an inhomogeneous structure.Therefore, plates were produced in a 1mm mould. This resulted in visually clear and homoge-neous samples. A part of the obtained amorphous PET samples were placed in a ventilated hotair oven at 110 ÆC and were cold crystallised for 114 hours. The melt crystallised PET sampleswere produced in a 5mm mould, as the PE and PP. Hereafter, the PE and PP were milled toa 2mm plate, of which dumb-bell shaped tensile samples of 80x5x2mm were machined by acomputer-controlled milling machine. Since the thickness of the amorphous PET samples is1mm, the 5mm melt crystallised PET plate was reduced to 1mm. Thus for PET, tensile sampleswere created of 80x5x1mm. Tensile tests were performed on a Zwick Z010 tensile test machine,at a speed of 5mm/min. The gap between the clamps was 55mm, which equals an initial strainrate of 1.5�10�3 s�1. All tensile tests were performed at room temperature.

Chapter 4

Intrinsic Deformation

4.1 Strain Hardening

The results of the crystallinity measurements (see table 3.2) showed that the quenched PETsamples are completely amorphous. Since the intrinsic deformation behaviour of amorphouspolymers is known, firstly the behaviour of amorphous PET is considered. The amorphousPET constitutes a starting point for the investigation of the intrinsic deformation behaviour ofsemi-crystalline polymers.

The true stress-strain curve, obtained by the uniaxial compression, of an amorphous sampleis shown in figure 4.1a (red curve). Here, the true stress is plotted according a neo-Hookeandescription against �2��1, because of the expected proportionality of strain hardening modu-lus at large strains. This was already demonstrated for various amorphous and semi-crystallinepolymers [14, 15]. The obtained true stress-strain curve is typical for an amorphous polymer.After the initial elastic response, the curve exhibits yielding and strain softening behaviour,which is followed by a linear strain hardening. Subsequently, figure 4.1a also shows the truestress-strain curves of two cold crystallised samples with 21% (blue line) and 29% (black line)crystalline content. It is obvious that due to the crystallisation yield stress increases, whereasthe strain softening becomes smaller for an increasing degree of crystallinity. Since the increa-

0 1 2 3 4 50

25

50

75

100

125

| λ2 − λ−1 | [−]

Com

pres

sive

Tru

e S

tress

[MP

a]

amorphous and cold crystallised PET

Xc = 0%

Xc = 21.7%

Xc = 29.1%

a 0 1 2 3 4 5

0

25

50

75

100

125

| λ2 − λ−1 | [−]

Com

pres

sive

Tru

e S

tress

[MP

a]

cold crystallised

melt crystallised

b

cold crystallised and melt crystallised PET

Figure 4.1: a) True stress-strain curves of amorphous PET: Xc=0% (red line), cold crystallised PET: Xc=21.7% (blue line) andXc=29.1% (black line). b) True stress-strain curves of cold crystallised PET, Xc=29.1% (black line) and melt crystallisedPET, Xc=42.6% (blue line).

Intrinsic Deformation 20

sing degree of crystallinity leads to a decrease of the amorphous fraction, the latter observationconfirms that strain softening is a feature only controlled by the amorphous phase. Strikingly,however, the true stress-strain curves of the cold crystallised PET show the same strain hard-ening behaviour as observed in the amorphous PET. By way of illustration, the strain hard-ening part of the curves is extrapolated (dashed lines), elucidating that the strain hardeningbehaviour is the same. In all three cases, it is demonstrated that cold crystallisation of amor-phous PET, despite a large increase in crystallinity (approx. 30%), does not affect the strainhardening behaviour. The observation that the large strain deformation of the semi-crystallinePET shows an identical behaviour to amorphous PET, suggests that the crystalline phase hasno contribution to the deformation behaviour after yield. In contrast to cold crystallisation,crystalline samples of PET were created by crystallisation from the melt, which have a largercrystalline content. In figure 4.1b, the true stress-strain curves of a cold crystallised (black line)and a melt crystallised (blue line) sample are displayed. Once again, the strain hardening partof the curve is extrapolated to compare the moduli of the cold and melt crystallised samples.It is obvious that the higher crystalline melt crystallised sample shows a distinct lower strainhardening modulus. Whereas in the range of 0-30% crystallinity no changes in strain hardeningwere observed, the strain hardening modulus of the higher crystalline melt crystallised sampleis clearly smaller. This might suggest a transition point where the crystalline phase becomesmore dominant in its contribution to strain hardening.

In order to examine this possible transition point, the strain hardening moduli of all ex-amined amorphous, cold crystallised and melt crystallised PET samples are plotted againstcrystallinity in figure 4.2. Again, it can be seen that the strain hardening modulus of the coldcrystallised samples closely match the strain hardening modulus of the amorphous sample.Since strain hardening of the amorphous and cold crystallised PET is similar, it is concludedthat the strain hardening behaviour is not affected by cold crystallisation, and thus the crys-talline phase. At about 30% crystallinity a rather abrupt decrease can be noticed for the strainhardening modulus. However, not only the crystalline content of the samples changes aroundthis transition, but also the crystallisation process. After the 30% crystallinity, the samples areobtained by melt crystallisation instead of cold crystallisation. The melt crystallised PET (cross-symbols) clearly has a distinctly smaller strain hardening modulus compared to the amorphousand cold crystallised samples (circles).

0 10 20 30 40 500

4

8

12

16

Crystallinity [%]

Stra

in H

arde

ning

Mod

ulus

[MP

a]

cold crystallisation melt crystallisation

Figure 4.2: Strain hardening modulus vs. Crystallinity. The strain hardening modulus of the cold crystallised PET is similar to thestrain hardening modulus of the amorphous PET, whereas the melt crystallised PET shows a distinctly smaller strainhardening modulus.


Remarkably, all these observations assign the strain hardening to entanglements. Dur-ing melt crystallisation, highly perfect crystallites are formed by the reeling in of molecularchains from the amorphous phase. This is known to be accompanied by a loss of entangle-ments [40–42]. Since strain hardening only alters by different crystallisation procedures, it canbe concluded that the strain hardening behaviour is not affected by degree of crystallinity, butby the processing conditions. This observation means that, similar to amorphous polymers,the strain hardening of semi-crystalline polymers should be directly related to the entangle-ment density. This contradicts the findings of Hiss et al. [19], who found an increase in strainhardening proportional to the degree of crystallinity for PE. An explanation for this discrep-ancy may be the method by which a wide range of crystallinities is created. Hiss et al. usedvarious PE-grades, varying from several molecular weights of HDPE to LLDPE and PEVA-grades, whereas in this study only one polymer-grade is taken into consideration. Despite thewell-characterised crystalline phase, differences in chain configuration, branching and linear-ity, of the various PE-grades by Hiss may affect amorphous microstructural properties, suchas entanglement density. Thus, Hiss only focused on the crystalline phase and neglected theamorphous phase by his approach of obtaining a wide range of crystallinities.

To investigate the findings of Hiss et al., a similar investigation is performed on PE LM.Contrary to Hiss et al., a range in crystallinities is acquired by annealing of PE LM samplesquenched from the melt. In figure 4.3a, the true stress-strain curves of a quenched PE LM sam-ple together with two annealed PE LM samples are presented. Similar to the cold crystallisedPET, a rise in yield stress can be recognised with an increasing degree of crystallinity, whereasstrain softening appears to be hardly present in the true stress-strain curves of PE LM. At roomtemperature, PE is situated above its glass-transition temperature, which means that the amor-phous phase does not display strain softening. It is clear from figure 4.3a, that the degree ofcrystallinity does not affect the strain hardening behaviour of PE in contradiction to the find-ings of Hiss et al. For PE, also the influence of chain length is examined. In figure 4.3b, thetrue stress-strain curves of quenched and melt crystallised PE LM (blue) and PE HM (black)are displayed.

0 1 2 3 4 50

10

20

30

40

50

| λ2 − λ−1 | [−]

com

pres

sive

true

stre

ss [M

Pa]

quenched and annealed PE LM

Xc = 68.4%

Xc = 76.6%

Xc = 72.3%

a 0 1 2 3 4 5

0

10

20

30

40

50

| λ2 − λ−1 | [−]

Com

pres

sive

Tru

e S

tress

[MP

a]

quenched

melt crystallised

PE LM PE HM

b

Figure 4.3: a) True stress-strain curves of quenched PE LM: Xc=68.4% (red line) and annealed PE LM: Xc=72.3% (blue line) andXc=76.6% (black line). b) True stress-strain curves of quenched PE LM: Xc=68.4% (lower blue line) and melt crystallisedPE LM: Xc=76.7% (upper blue line) and quenched PE HM: Xc=68.4% (lower black line) and melt crystallised PE HM:Xc=73.8% (upper black line)


Once more, a decrease in strain hardening is observed for the melt crystallised samples.Here, it is emphasised that the annealed PE LM sample of 76.6% crystallinity, in figure 4.3a,equals the melt crystallised PE LM samples in degree of crystallinity. Thus, despite a corre-sponding degree of crystallinity a distinct difference in strain hardening is observed betweenthe annealed PE LM and melt crystallised PE LM. The black curves are the true stress-straincurves of a higher molecular weight PE-grade, PE HM. It is obvious, that for both the meltcrystallised and annealed PE, a higher molecular weight results in a higher strain hardeningmodulus. These observations are explained more clearly by means of figure 4.4, where thestrain hardening modulus is plotted against crystallinity. The quenched and annealed PE LM(blue circles), are clearly positioned within the range of a modulus of 3-4.5 MPa, whereas themelt crystallised PE LM has a modulus of approximately 1.5 MPa. For PE HM, which has ahigher modulus, a similar conclusion for the modulus of the quenched and melt crystallisedsamples can be drawn. Beside an unaffected strain hardening by cold crystallisation and thelower strain hardening by melt crystallisation, the dependency on molecular weight also indi-cates that the entangled polymer network is the origin of strain hardening in semi-crystallinepolymers. The larger chains of a higher molecular weight grade result in a larger number ofentanglements, which in its turn enlarge the stress contribution of the network at large defor-mations.

65 70 75 800

2

4

6

Crystallinity [%]

Stra

in H

arde

ning

Mod

ulus

[MP

a]

PE LM quenched and annealedPE LM melt crystallisedPE HM quenchedPE HM melt crystallised

quenched

meltcrystallisation

quenched

meltcrystallisation

Figure 4.4: Strain hardening modulus vs. Crystallinity. The quenched and cold crystallised PE HM and PE LM show a distinctlylarger strain hardening modulus than the melt crystallised PE HM and PE LM, and PE HM has a higher modulus thanPE LM.

In figure 4.5a and 4.5b, similar results can be seen for, respectively, PP HM and PP LM;the quenched samples clearly possess a larger strain hardening modulus compared to the meltcrystallised samples and for the higher molecular weight grade the decrease in modulus formelt crystallised samples is smaller. In table 4.1, the derived strain hardening moduli of allPET, PE and PP samples are presented. Here, the strain hardening modulus is derived byfitting a linear curve on the data between j�2-��1j = 3 and j�2-��1j = 4. The obtained values forquenched PP match the values derived by Haward [15] very closely and the strain hardeningmodulus of PE is in the same order of magnitude. For PET, a corresponding strain hardeningmodulus of approximately 10 MPa is derived from the true stress-strain curves presented byZaroulis and Boyce [45].


0 1 2 3 4 50

10

20

30

40

50

| λ2 − λ−1 | [−]

Com

pres

sive

Tru

e S

tress

[MP

a]

melt crystallised

quenched

PP LM

a 0 1 2 3 4 5

0

10

20

30

40

50

| λ2 − λ−1 | [−]

Com

pres

sive

Tru

e S

tress

[MP

a]

melt crystallised

quenched

PP HM

b

Figure 4.5: a) True stress-strain curves of quenched PP LM: Xc=64.5% (red line) and melt crystallised PP LM: Xc=69.5% (blackline). b) True stress-strain curves of quenched PP HM: Xc=65.6% (red line) and melt crystallised PP HM: Xc=66.7%(black line).

The following conclusions can be drawn. In this study, three kinds of semi-crystalline poly-mers have been taken into consideration with respect to glass-transition temperature. At roomtemperature, at which testing took place, PET is situated in its glassy state, whereas PE is inits rubbery state. The glass-transition temperature of PP is situated close to the testing tem-perature. Since similar results are obtained for the strain hardening behaviour of PET, PE andPP, the following conclusions with respect to strain hardening of semi-crystalline polymers ingeneral can be drawn:

1. Crystallites do not affect the strain hardening behaviour, which can be rationalised by theslip of crystallites after yielding.

2. Similar to amorphous polymers, only the entangled polymer network, which depends onprocessing conditions and chain length, is the origin of the strain hardening behaviour.


Material Treatment Xc (%) L (nm) lc (nm) Gr (MPa)PET Tcc=100ÆC, 4h 0 - - 11.2 �0:5

Tcc=100ÆC, 24h 11.6 - - 11.1 �1:8

Tcc=100ÆC, 72h 27.1 10.5 2.9 11.3 �1:8

Tcc=110ÆC, 15min 21.7 - - 12.7 �1:2

Tcc=110ÆC, 4h 26.4 11.9 3.1 11.4 �2:2

Tcc=110ÆC, 54h 29.1 10.2 3.0 11.6 �3:3

melt crystallised 42.6 16.1 6.9 7.9 �2:4

Tmc=135ÆC 31.4 13.4 4.2 8.2 �2:8

Tmc=150ÆC 32.2 13.5 4.4 10.0 �1:9

Tmc=200ÆC 34.7 16.8 5.8 6.9 �2:0

PE LM quenched 68.4 26.7 18.3 3.3 �0:5

Tcc=100ÆC, 8h 69.9 27.4 19.2 3.3 �0:8

Tcc=100ÆC, 24h 71.0 27.3 19.4 3.1 �0:2

Tcc=100ÆC, 48h 73.5 29.4 21.6 3.3 �0:6

Tcc=100ÆC, 72h 73.8 28.6 21.1 2.5 �0:4

Tcc=110ÆC, 8h 72.3 29.0 21.0 3.6 �0:3

Tcc=110ÆC, 48h 73.3 29.5 21.6 3.2 �0:4

Tcc=120ÆC, 8h 74.1 31.4 23.3 3.5 �0:9

Tcc=120ÆC, 24h 75.6 32.4 24.5 2.5 �0:4

Tcc=120ÆC, 72h 76.5 32.7 25.0 3.1 �0:4

Tcc=125ÆC, 8h 75.6 34.7 26.2 3.2 �0:5

Tcc=125ÆC, 48h 76.6 35.9 27.5 3.5 �0:3


PE HM quenched 68.4 29.9 20.5 4.6 �0:3


PP HM quenched 65.6 14.6 9.6 4.0 �1:1


PP LM quenched 64.5 x x 3.9 �0:5

melt crystallised 69.5 x x 2.8 �0:4

Table 4.1: Strain hardening modulus of differently processed PET, PE and PP-grades. The strain hardening modulus is derived byfitting a linear curve on the data between j�2-��1

j = 3 and j�2-��1j = 4. A distinctly lower strain hardening is observed

for the melt crystallised samples compared to quenched and annealed samples.


4.2 Yield Stress

Yield stress appeared to be the only deformation stage controlled by the crystalline phase.Therefore, a light is thrown on the relation between yield stress and the crystalline phase ofPE. At room temperature PET is in the glassy state, which means that the yield stress is com-posed of an amorphous and a crystalline contribution. In order to accurately determine thecontribution of the crystalline phase to the yield stress, a coupling must be made between theamorphous and crystalline yield stress. For that reason, only the yield stress of PE is taken intoconsideration. For PP, insufficient data are available to take the yield stress into consideration.

For PE, however, a striking feature is observed during deformation. PE appears to have twoseparate yield points, as can be seen in figure 4.6a. So far, the occurrence of double yieldingin PE has mainly been reported for engineering stress-strain curves [18, 19, 46–49]. Since aninvestigation into the origin of double yielding is beyond the aims of this survey, the approachof Butler et al. and Brooks et al. is followed. They suggest that the first yield point originatesfrom fine chain slip mechanism and the second yield point can be attributed to coarse slipresulting in lamellar fragmentation [18, 46, 47].

0 0.5 1 1.50

10

20

30

40

Compressive True Strain [−]

Com

pres

sive

Tru

e S

tress

[MP

a]

1st Yield Point

2nd Yield Point

a

68 71 74 7720

26

32

38

44

Crystallinity [%]

Yie

ld S

tres

s [M

Pa]

2nd Yield Stress

1st Yield Stress

b

Figure 4.6: a) Double yield behaviour of PE. b) First and second yield stress exhibit a similar increase with degree of crystallinity.

Furthermore, in figure 4.6b, the first and second yield stress are plotted against crystallinity.Here, it can be seen that the first and second yield stress show a corresponding increase asa function of crystallinity. Since the second yield stress alters almost analogously to the firstyield stress, conclusions drawn for the change in yield stress with respect to lamellar thick-ness and crystallinity are assumed to be similar for both the yield stresses. Therefore, only thefirst yield stress is taken into consideration, which is in accordance with results published inliterature [32]. A three-dimensional plot of the yield stress vs. crystallinity and lamellar thick-ness is depicted in figure 4.7. It is abundantly clear that yield stress increases as crystallinityand lamellar thickness increase, which are undoubtedly related to each other. However, thisgraph does not clarify the individual contribution of lamellar thickness or crystallinity to theyield stress. For that reason, the yield stress with a standard deviation of four measurements isplotted against crystallinity in figure 4.8a. In this figure, samples with a closely correspondinglamellar thickness are represented by similar symbols. Again, an upward trend in yield stresscan be seen for increasing lamellar thickness and crystallinity. An eye-catching feature is thatthe yield stress for samples of equal lamellar thickness is not affected by an increase in crys-


6870

7274

7678

1820

2224

2628

3022

26

30

34

Crystallinity [%]Lamellar Thickness [nm]

Yie

ld S

tres

s [M

Pa]

Figure 4.7: 3D-plot of the yield stress vs. crystallinity and lamellar thickness of PE LM.

tallinity. This suggests that lamellar thickness contributes more strongly to the yield stress thanthe degree of crystallinity, which is in agreement with findings of Brooks et al. [20]. To examinethe influence of lamellar thickness on the yield stress, the yield stress is plotted against lamel-lar thickness, see figure 4.8b. It can be seen that the yield stress clearly shows a linear increasewith increasing lamellar thickness. Since the yield stress is said to originate from slip of chainsthrough the lamellae, thinner lamellae will result in a lower yield stress, which is in accordancewith results published by various authors [18, 20, 21, 29]. A smaller length which the chainshave to slide through facilitates the fine chain slip mechanism, and as a result a lower yieldstress is observed.

Mainly the nucleation of dislocations theory is followed to predict the yield behaviour forsemi-crystalline polymers [20, 23, 29–35]. Since this model is based on the lamellar thickness, itcan be used as a useful tool to examine the experimentally derived relation between yield stressand lamellar thickness more clearly. The background of this model is discussed in appendix A.Here, the approach of Brooks and Mukhtar of Young’s model is followed [50]. They derivedthe following equation

65 70 75 8020

25

30

35

Crystallinity [%]

Yie

ld S

tres

s [M

Pa]

lc = 18.97 ± 0.59 nm

lc = 24.75 ± 1.20 nm

a 15 20 25 30

20

25

30

35

Lamellar Thickness [nm]

Yie

ld S

tres

s [M

Pa]

b

Figure 4.8: a) Yield Stress vs. Crystallinity PE LM for lc = 18.97 � 0.59 nm (Æ) and lc = 24.75 � 1.20 nm (x). b) Yield Stress vs.Lamellar Thickness of PE LM.


�y =K�(T )

�exp

��2��Gc

lcKb2+ 1

�(4.1)

where

�(T ) =b

r0exp

�2�E0

Klb2

�(4.2)

A relation for �(T ) was found by fitting experimental results. In this study, the validity ofthis relation is verified. Only now the value of K is derived by linearly interpolating the theo-retical values of Karasawa et al. [51] to a temperature of 293K. In figure 4.9 the energy range inwhich the activation of nucleation of dislocations takes place 40-80kT [27], is represented by thedashed lines. The measured yield stresses in this study prove to be within the range of 40-80kT.Thus, in addition to the results of Brooks and Mukhtar, the model adequately predicts the yieldstresses obtained in this study, which confirms the strong relation between lamellar thicknessand yield stress. However, since Brooks and Mukhtar fitted the parameter �(T ) on experimen-tal results, without actually verifying its physical value, the outcome of this approach has to beinterpreted carefully.

0 10 20 300

10

20

30

40

Lamellar Thickness [nm]

Yie

ld S

tress

[MP

a] 40kT

80kT

Figure 4.9: Yield stress vs. Lamellar thickness of PE LM, the obtained yield stresses are well-predicted by the Brooks and Mukhtarapproach of Young’s dislocation model.

A brief examination of the yielding of the semi-crystalline polymer PE resulted in the fol-lowing conclusions. Lamellar thickness is found to attribute stronger to yield stress than thedegree of crystallinity. A linearly increasing yield stress for increasing lamellar thickness sug-gests that the fine chain slip and coarse slip of lamellae mechanisms are valid, since thinnerlamellae facilitate the slip of chains through lamellae. An attempt to model the yield stress byBrooks and Mukhtar’s approach of Young’s nucleation-controlled dislocations model resultedin a proper description. No further conclusions are drawn for the yield stress and crystallinity,since an investigation of the strain hardening behaviour in semi-crystalline polymers is themajor aim of this study.

Chapter 5

Macroscopic Deformation & Failure

5.1 Results

The red curve in figure 5.1, is the engineering stress-strain curve of an amorphous PET sam-ple obtained by tensile testing. After yielding, the curve exhibits a peculiar decrease in stress,which can be attributed to the presence of shear banding, where after a stable neck developsrecognised by a constant force in tension. Thus, a ductile behaviour of the amorphous PETis observed. The melt crystallised PET, however, shows brittle fracture occurring even beforemacroscopic yielding.

1 1.10

25

50

75

Draw Ratio [−]

Eng

inee

ring

Stre

ss [M

Pa]

quenched

meltcrystallised

f

sb n

a

brittlefracture

(f)

(sb)

stable necking(n)

b

shear banding

Figure 5.1: a) Failure behaviour of amorphous (Xc = 0%) and melt crystallised (Xc = 42.7%) PET in tension. b) Schematic repre-sentation of deformed tensile bars: Brittle failure for the melt crystallised PET, the shear bending shortly after yieldingand stable necking for the quenched PET.

For PE two molecular weights are investigated. Firstly, the low molecular grade PE LM isdiscussed, see figure 5.2a. The quenched PE LM, which has a crystallinity of 68.4%, exhibitsductile behaviour, and the constant stress for large deformation indicates that a stable neck hasdeveloped. The quenched PE LM, was subsequently annealed to a degree of crystallinity of76.5%. As can be observed in figure 5.2a, the annealed sample also exhibits ductile behaviour.The melt crystallised PE LM has a similar crystalline content, but shows brittle behaviour. In fi-gure 5.2b, the results of the tensile tests on the high molecular weight PE, PE HM, are presented.Here, the quenched, annealed as well as melt crystallised samples show ductile behaviour.

Macroscopic Deformation & Failure 29

1 1.2 1.4 1.60

10

20

30

Draw Ratio [−]

Eng

inee

ring

Stre

ss [M

Pa]

annealed

melt crystallised tensile testing PE LM

quenched

a 1 1.2 1.4 1.6

0

10

20

30

Draw Ratio [−]

Eng

inee

ring

Str

ess

[MP

a]

tensile testing PE HM

melt crystallised

quenched

annealed

b

Figure 5.2: a) Failure behaviour of quenched (Xc = 68.4%), annealed (Xc = 76.5%) and melt crystallised (Xc = 76.7%) PE LM intension. b) Failure behaviour of quenched (Xc =68.4%), annealed (Xc = 72.9%) and melt crystallised (Xc = 73.8%) PEHM in tension.

In figure 5.3, similar results can be seen for PP HM; the melt crystallised sample clearlyexhibits brittle fracture and the quenched sample shows a ductile behaviour, despite a nearlyidentical degree of crystallinity.

1 1.1 1.2 1.30

10

20

30

40

Draw Ratio [−]

Eng

inee

ring

Stre

ss [M

Pa]

melt crystallised

quenched

tensile testing PP

Figure 5.3: Failure behaviour of quenched (Xc = 65.6%) and melt crystallised (Xc = 66.7%) PP HM in tension.

5.2 Discussion

Now the results of the tensile tests are discussed in relation to the neck stability analysis, whichwas presented in paragraph 2.3. By means of figure 5.5a, which is derived from the stabilityanalysis without strain softening, the observations of PE LM and PE HM in tension are madeclear. According to this analysis all samples should develop a neck, since their �y �Gr ratio islarger than three. It was already shown that the annealed PE LM, with a �y � Gr ratio of 9.9,indeed exhibits necking. The melt crystallised PE LM, has a larger �y �Gr ratio (18.3), due toa higher yield stress and a smaller strain hardening. Therefore, a larger draw ratio in the neckis expected. However, fracture is observed before necking is able to set in. This indicates that


the tensile strength is reached in the localised zone of the melt crystallised PE LM. Since the an-nealed PE LM shows ductile behaviour, it can be stated that at no further specified draw ratioin the neck, the tensile strength line intersects the equilibrium line of stable necking betweenthe predicted neck draw ratios of the annealed and melt crystallised PE LM. In figure 5.5a, thetensile strength is represented by the dashed line. Thus, melt crystallisation enhances the brit-tle fracture in semi-crystalline polymers.

Furthermore, it can be seen that in the higher molecular weight PE, the degree of strainlocalisation is smaller. And even the melt crystallised PE HM exhibits stable necking, as op-posed to the melt crystallised PE LM. This is remarkable, since both the melt crystallised PE LMand PE HM display a nearly identical yield stress. For that reason, the degree of localisationis expected to be similar. Yet, not only the yield stress, but also strain hardening plays a roleand in chapter 4 it was demonstrated that the strain hardening modulus is larger for a highermolecular weight. This implies that in a higher molecular weight strain localisation is moreeasily stabilised by a larger strain hardening and thus enhances ductile behaviour.

0 10 20 301

10

20

30

σy / G

r [−]

λ n [−]

stability analysis PE LM and PE HM

fracture PE LM

annealed PE HM

annealed PE LM

melt crystallised PE HM

melt crystallisedPE LM

a 0 5 10 15

1

4

7

10

13

16

σy / G

r [−]

λ n [−]

stability analysis PP HM

melt crystallised

fracture PP HM

quenched

b

Figure 5.4: a) Neck stability analysis of PE LM and PE HM. b) Neck stability analysis of PP HM.

In figure 5.5b, the results of PP in tension are analysed. The results are discussed by thestability analysis, which includes strain softening, since the amorphous phase of PP is in itsglassy state at room temperature. The observations made for PP are similar to those of PE LM.The melt crystallised PP HM fails brittle, but the stabilising effect of the larger strain hardeningin the quenched PP HM prevents that its tensile strength will be reached. Thus, for PP HM, thetensile strength intersects the equilibrium line of PP HM between the predicted draw ratios ofthe quenched and the melt crystallised PP HM. Again, it is mentioned that the exact positionof this line cannot be specified by the methods performed in this study.

PET has also to be discussed by the analysis which includes strain softening. The strain soft-ening in the melt crystallised PET is expected to be smaller, since it has a smaller amorphouscontent than the quenched (fully amorphous) PET. This is confirmed by the position of theequilibrium lines. A larger strain softening shifts the equilibrium line to the left and will thusintersect the tensile strength line earlier. This implies that a larger strain softening enhancesbrittle failure. However, it is observed that the quenched PET exhibits stable necking, whereasthe melt crystallised PET fails brittle. This means that the tensile strength line must intersectthe equilibrium lines between the predicted equilibrium draw ratios of the quenched and the


melt crystallised PET. Once more, it is shown that melt crystallisation enlarges the chance ofbrittle failure. This is assigned to a larger localisation due to a rise in yield stress, which cannotbe stabilised by its lower strain hardening. Remarkably, the melt crystallised samples fractureeven before the expected macroscopic yield point is reached, see figure 5.1a. Presumably, plas-tic deformation already occurs at a local level, where the low strain hardening behaviour is notable to stabilise the local deformation.

0 2 4 6 8 101

3

5

7

9

σy / G

r [−]

λ n [−]

stability analysis PET

quenched

melt crystallised

fracture

Figure 5.5: Neck stability analysis of PET.

In summary, it is shown for all examined polymer-grades that the failure behaviour of semi-crystalline polymers depends on a combination of yield stress, strain hardening and tensilestrength. The brittler failure due to an increase in yield stress induced by crystallisation can becompared to the increase in yield stress by ageing in glassy polymers. Due to a rise in yieldstress, the localisation of deformation increases. Moreover, similar to amorphous polymers,the entanglement density plays an important role in the failure behaviour of semi-crystallinepolymers, which was already suggested by Bastiaansen [38] and Plummer [43]. The entangle-ment density affects the strain hardening behaviour of the polymers, which in its turn affectsthe amount of stabilising of deformation in strain localisation zones. The brittler failure of themelt crystallised samples and the more ductile response of a higher molecular weight indicate astrong influence of entanglements, and thus strain hardening, in their attribution to the failurebehaviour.

Bibliography

[1] Melick, H.G.H. van, Deformation and Failure of Polymer Glasses, PhD thesis, Eindhoven Uni-versity of Technology (2002)

[2] Goessens, P.J.C., and Janssen, R.P.M., Characterisation and modeling of strain rate dependentsoftening of PMMA and PC., Internship Report no. MT 01.008, Eindhoven University ofTechnology (2001)

[3] G’Sell, C., Hiver, J.M., Dahouin A., Souahi, A., J. Mat. Sci., 27, p.5031 (1992)

[4] Arruda, E.M. and Boyce, M.C., Int. J. Plast., 9, p.697 (1993)

[5] Boyce, M.C., Arruda, E.M., Jayachandran, R., Pol. Eng. Sci., 34, p.716 (1994)

[6] G’Sell, C., Plastic deformation of glassy polymers: constitutive equations and macromolecularmechanisms., In: H.J. Queens et al., Eds. Strength of metals and alloys, Pergamon Press,Oxford, p. 1943-1982 (1986)

[7] Bauwens, J.C., J. Mat.Sci., 13, p.1443 (1978)

[8] Govaert, L.E., Timmermans, P.H.M., Brekelmans,W.A.M., J. Eng. Mat. Tech., 122, p.177(2000)

[9] Govaert, L.E., Melick, H.G.H. van, Meijer, H.E.H., Polymer, 42, p.1271 (2001)

[10] Gurevich, G., and Kobeko, P., Rubber Chem. Tech., 13, p.904 (1940)

[11] Haward, R.N., Trans. Faraday Soc., 38, p.394 (1942)

[12] Hoff, E.A.W., J. Appl. Chem., 2, p.441 (1952)

[13] Haward, R.N., Murphy, B.M., White, E.F.T., J. Pol. Sci., Part A2, 9, p.801 (1971)

[14] Haward, R.N., J. Pol. Sci., Part B: Pol.Phys., 33, p.1481 (1994)

[15] Haward, R.N., Macromolecules, 26, p.5860 (1993)

[16] Tervoort, T.A. and Govaert, L.E., J. Rheo., 44, p.1263 (2000)

[17] Tervoort, T.A., Smit, R.J.M., Brekelmans, W.A.M., Govaert, L.E., Mech. Time-Dep. Mat., 1,p.269 (1998)

[18] Butler, M.F., Donald, A.M., Ryan, A.J., Polymer, 38, p.5521 (1997)

[19] Hiss, R., Hobeika, S., Lynn, C., Strobl, G., Macromolecules, 32, p.4390 (1999)

[20] Brooks, N.W., Ghazali, M., Duckett, R.A., Unwin, A.P., Ward, I.M., Polymer, 40, p.821(1998)

BIBLIOGRAPHY 33

[21] O’Kane, W.J., Young, R.J., Ryan, A.J., J. Macromol. Sci.-Phys., B34, p.427 (1995)

[22] Capaccio, G. and Ward, I.M., J. Polym. Sci., Polym. Phys. Edn., 22, p.475 (1984)

[23] Kennedy, M.A., Peacock, A.J., Failla, M.D., Lucas, J.C., Mandelkern, L., Macromolecules,28, p.1407 (1995)

[24] Flory, P.J., and Yoon, D.Y., Nature, 272, p.226 (1978)

[25] Young, R.J., Mater. Forum, 11, p.210 (1988)

[26] Young, R.J., Phil. Mag., 30, p.85 (1974)

[27] Shadrake, L.G. and Guiu, F., Phil. Mag., 34, p.565 (1976)

[28] Ryan, A.J., Bras, W., Mant, G.R., Derbyshire, G.E., Polymer, 35, p.4357 (1994)

[29] O’Kane, W.J., Young, R.J., J. Mater. Sci. Lett., 14, p.433 (1995)

[30] Kennedy, M.A., Peacock, A.J., Mandelkern, L., Macromolecules, 27, p.5297 (1994)

[31] Crist, B., Fisher, C.J., Howard, P., Macromolecules, 22, p.1709 (1989)

[32] Sirotkin, R.O., Brooks, N.W., Polymer, 42, p.3791 (2001)

[33] Darras, O. and Seguela, R., J. Polym. SC. Part B, 31, p.759 (1993)

[34] Graham, J.T., Alamo, R.G., Mandelkern, L.,J. Pol. Sci., Part B: Pol.Phys, 35, p.213 (1997)

[35] Seguela, R., J. Pol. Sci., Part B: Pol.Phys., 40, p.593 (2002)

[36] Peterlin, A., J. Mater. Sci., 6, p.490 (1971)

[37] Bowden, P.B., and Young, R.J., J. Mater. Sci., 9, p.2034 (1974)

[38] Bastiaansen, C.W.M., Meijer, H.E.H., Lemstra, P.J., Polymer, 31, p.1435 (1990)

[39] Smith, P. and Lemstra, P.J., Colloid Polymer Sci., 258, p.898

[40] DiMarzio, E.A., Guttman, C.M., Hoffman, J.D.,Farad. Disc., 68, p.210 (1979)

[41] Hoffman, J.D., Polymer, 23, p.656 (1982)

[42] Hoffman, J.D., Polymer, 24, p.3 (1983)

[43] Plummer, C.J.G., and Kausch, H.-H., J. Macromol. Sci.-Phys., B35(34), p.637 (1996)

[44] Haward, R.N., Polymer, 28, p.1485 (1987)

[45] Zaroulis, J.S. and Boyce, M.C., Polymer, 38, p.1307 (1997)

[46] Brooks, N.W., Ducket, R.A., Ward, I.M., Polymer, 33, p.1872 (1992)

[47] Brooks, N.W., Unwin, A.P., Ducket, R.A., Ward, I.M., J. Macromol. Sci.-Phys., B34, p.29(1995)

[48] Seguela, R., and Rietsch, F.J., J. Mater. Sci. Lett., 9, p.46, (1990)

[49] Seguela, R., and Darras, O., J. Mater. Sci., 29, p.5342, (1994)

[50] Brooks, N.W.J. and Mukhtar, M., Polymer, 41, p.1475 (2000)

[51] Karasawa, N., Dasgupta, S., Goddard, W., J. Phys. Chem., 95, p.2260 (1991)

Part II

Modeling of Intrinsic Deformation inSemi-Crystalline Polymer Systems

Chapter 6

Introduction

6.1 History of the Leonov Model

The compressible Leonov model has successfully been used in order to predict the large straindeformation behaviour of amorphous polymers [1, 2]. The finite deformations of polymersare classically described using yield criteria, followed by hardening, which is sometimes pre-ceded by intrinsic strain softening [3]. The eventual description of the model has a long history.Describing this behaviour using a stress-dependent relaxation time dates back to Tobolsky andEyring [4]. This model was later refined by adding the Langevin-spring [5] and Boyce et al. [6,7]modified the model into a 3-D finite strain formulation using the Argon theory and the three-chain model [8]. Arruda [9] and Wu and van der Giessen [10] further modified the model withrespect to the strain hardening response, followed by Hasan et al. [11], who included the ef-fects of ageing and rejuvenation. More recently, the modelling of the (non-)linear visco-elasticdeformation was improved by using a spectrum of relaxation times instead of one [12], andTervoort et al. [13] showed that the model is able to describe the plastic deformation behaviour.The model resulting from the latest modification is commonly called the compressible Leonovmodel. At this point, the model predicts a yield point, which appears to be proportional to thestrain rate and a post-yield behaviour which is independent of strain rate. However, this lastobservation is shown to be incorrect for PMMA and PS [14]. Moreover, despite many stud-ies indicating that for PC, there is only one molecular process contributing to the stress at lowstrain rates and room temperature [15–19], other observations exhibit a change in proportion-ality in yield stress at a certain strain rate and a strain rate dependence of the softening [20].This change in proportionality and strain rate dependency of the post-yield behaviour was alsoreported for PMMA [16, 18, 21–24], for PS [16, 18], for PET [24] and for PEMA [23]. This meansthat the compressible Leonov model is not able to describe the post-yield behaviour adequately.However, it has been shown by Klompen [25] that the compressible Leonov model can be mod-ified to describe the effect of the presence of two molecular processes on the behaviour of thematerial, the �-process and the �-process, both with their own individual softening evolution.This model has proved to be valid for PMMA [2].

In this survey, an attempt is made to obtain the compressible Leonov model parameters forthe semi-crystalline polymers PE and PP. By means of compression testing at various strainrates, the model parameters are determined by which an effort was made to describe the ex-perimentally obtained true stress-strain curves.

Introduction 36

6.2 The Leonov Model

The compressible Leonov model is composed of a part that describes the (visco-)elastic prop-erties at small deformations and yielding, and a part describing the large strain behaviour, as aresult of the polymer network. The total Chauchy stress � is decomposed in the driving stresss and the hardening stress r, according to

� = s + r (6.1)

The driving stress consists of a hydrostatic part and a deviatoric part.

s = K(Je � 1)I +G~Bd

e (6.2)

With G and K the shear modulus and the bulk modulus respectively. The moduli can be de-duced from the elastic parameters E and � using the following equations

K =E

3(1 � 2�)G =

E

2(1 + �)(6.3)

The elastic volume change factor Je and the volume invariant left Chauchy Green deformationtensor ~Be are given by

_Je = Jetr(D�Dp) (6.4)

Æ

~Be = (Dd �Dp) � ~Be + ~Be � (Dd �Dp) (6.5)

withÆ

~Be the Jaumann rate of ~Be. The plastic rate of deformation Dp is expressed in the driv-ing stress s by a non-Newtonian Eyring equation, including a pressure dependence and theinfluence of regions with elevated values of local free volume

Dp =sd

2�(�eq;D; p)(6.6)

It is chosen not to include parameters responsible for the pressure- and temperature depen-dence of � in the equations. Since the effects are present, however, they are included in theviscosity function.

� = Aps�0

ssd : sd

2�20

sinh

ssd : sd

2�20

(6.7)

Aps = A0exp[�D] (6.8)

with A0 a constant pre-exponential factor involving the fundamental vibration frequency and�0 a characteristic stress. The evolution of the softening parameter D is specified by

_D = h

�1�

D

D1

�_ eq (6.9)

Initially D equals 0. Assuming that after yielding, the plastic strain rate is equal to the equiva-lent strain rate, and assuming that the strain rate is constant, this equation becomes

D = D1

�1� exp

��h � eqD1

��(6.10)

Introduction 37

Here, h is a material constant describing the relative softening rate and _ eq is the equivalentplastic strain rate, according to

_ eq =

qtr(Dp�Dp) =

�eq

2=

1

A0

sinh

��eq

�0

�exp(�D) (6.11)

The hardening stress is given by an incompressible neo-Hookean relation based on Gaussianstatistics

r = Gr~Bd ~B = J

�2

3

e F � Fc (6.12)

where Gr is the hardening modulus, ~B the left Chauchy Green deformation tensor and F the de-formation gradient tensor. A schematic representation of this model for a 1-D situation is givenin figure 6.1a. In order to enable the description of thermo-rheologically complex materials the

G r

ηG

r

s

G r

ηGs

ηG

β

α α

β

β

α

r

Figure 6.1: a) Schematic representation of the compressible 1-D Leonov model for thermo-rheological simple materials. b)Schematic representation of the compressible 2-D Leonov model for thermo-rheological complex materials.

model is extended mainly by modifying equation 6.2

s = K(Je � 1)I +G�~Bd

e;� +G�~Bd

e;� (6.13)

The expression for the viscosity � now reads

� = Am;�(Dk; p)�0;��eq;�=�0;�

sinh(�eq;�=�0;�)+Am;�(Dk; p)�0;�

�eq;�=�0;�

sinh(�eq;�=�0;�)(6.14)

where

Am;k(Dk; p) = Akexp(�Dk) k = �; � (6.15)

Dk = D1;k

�1� exp

��h � eqD1;k

��(6.16)

The proper way to describe the individual contribution of each molecular process is by a par-allel configuration of non-linear flow elements, such as Maxwell elements, which is shown infigure 6.1b.

Chapter 7

Results & Discussion

7.1 Results

In this chapter, an attempt is made to simulate the (post-)yield deformation behaviour of thesemi-crystalline polymers PP HM and PE LM by means of the extended compressible Leonovmodel. The model parameters were derived from experimental true stress-strain curves, whichwere obtained from compression tests performed at various strain rates. The post-yield be-haviour appeared to be nearly similar for all strain rates, except _� = 10�2 s�1. This is explainedby the occurrence of thermal softening caused by the internal heat generation due to plasticwork at this high strain rate. At high strain rates, the process is not able to emit the generatedheat to the environment, making the process adiabatic. However, since thermal softening ishardly present at strains where yield starts to set in, this strain rate is still useful to obtain theyield model parameters, �0 and A0. These parameters can be determined by means of equa-tion 6.11. At yield, the softening contribution can be removed from this equation and sincestrain hardening is constant, it will not affect the slope in the �eq - _ eq graph. Under theseconditions, equation 6.11 becomes

�eq = �0asinh(A0 _ eq) = �0ln(A0 _ eq +

q1 + (A0 _ eq)2) (7.1)

where,

�eq =1

3

p3� ; _ eq =

p3 _ (7.2)

10−5

10−4

10−3

10−2

10−1

0

10

20

30

40

|equivalent strain rate| [s−1]

|equ

ival

ent y

ield

stre

ss| [

MP

a]

a

10−5

10−4

10−3

10−2

10−1

0

10

20

30

40

|equivalent strainrate| [s−1]

|equ

ival

ent y

ield

stre

ss| [

MP

a]

b

Figure 7.1: a) Fitting of the yield parameters �0 and A0 on the experimental results of quenched PP HM (Æ) and melt crystallisedPP HM (�). b) Fitting of the yield parameters �0 and A0 on the experimental results of PE LM.

Results & Discussion 39

MATERIAL E � A0 �0 h D1 Gr

[MPa] [-] [s] [MPa] [-] [-] [MPa]PP HM quenched 400 0.4 19�109 1.33 8.5 8 4.0PP HM melt crystallised 400 0.4 7�109 1.32 10 9.5 3.8PE LM �� process 500 0.4 4�106 1.32 0 0 3.3PE LM � � process 150 0.4 1�1010 0.15 22 30 3.3

Table 7.1: Leonov model parameters for quenched and melt crystallised PP HM (both single process) and quenched PE LM (dou-ble process due to double yielding).

Now, for every strain rate, the equivalent yield stress can be plotted against the equivalentstrain rate on a logarithmic scale. In this graph, �0 is represented by the linear relationship be-tween �eq and _ eq and A0 determines the vertical position of the linear curve. In figure 7.1, theexperimentally determined yield stresses of the quenched and melt crystallised PP HM and thequenched PE LM are plotted vs. strain rate, together with the determined fits. Since PE LM ex-hibits double yielding, both the yield points are plotted and it appeared to be possible to createa reasonable fit for both the yield points with a corresponding slope. Now the yield parametersare known, the remaining parameters can be determined by a Fortran-routine. The routine,of which the parameters can easily be altered in an input-file, solves differential equations, de-duced from the constitutive Leonov-model presented in chapter 6.2, assuming a constant strainrate and homogeneous deformation. For PP HM, firstly, the yield points of the modelling re-sults are positioned on the experimental obtained yield points by altering Young’s modulus Eand adjusting yield parameter A0. Subsequently, the strain hardening parameter Gr, directlyacquired from the experimental results, see table 4.1, is taken into account, after which the soft-ening is considered. The softening is governed by the parameters h and D1, of which h is ameasure for the softening slope and D1 is responsible for the saturation level of the amountof softening. This fitting procedure yielded the material parameters presented in table 7.1. Fi-gures 7.2a and 7.2b demonstrates that for, both the quenched and melt crystallised PP HM, theobtained model parameters accurately predicts the yield and large deformations behaviour incompression until a true strain of 1.5 for strain rates in the range of 3�10�5 to 3�10�3 s�1. Due tothe thermal softening occurring after yielding for a strain rate of 10�2 s�1, only the yield pointis predicted well by the model, in which thermal softening is not implemented.

Since PE LM exhibits the peculiar double yield behaviour, it cannot be modelled by simplyone process. In order to fit accurate simulation results onto the experimental true stress-straincurves, two processes, � and �, are required. The double yield behaviour is preceded by a(visco-)elastic response and subsequently characterised by the first yield point, whereupon anearly linear rise in stress follows until the second yield point is reached. Then, the well-known characteristic features during deformation of polymers, such as strain softening andstrain hardening are displayed. Now, the �-process is used to position the first yield point andthe �-process positions the second yield point and, thereafter, describes the strain softeningand hardening behaviour. Again, the in figure 7.1b derived �0;� and A0;�, together with theYoung’s modulus, E�, prescribe the first yield point. Then, the region in between both theyield stresses, is linearly described by the strain hardening of the �-process, and the elastic, E� ,and yield, �0;� and A0;�, contribution of the �-process. Despite the corresponding inclinationsof both the yield data in figure 7.1b, a low value of �0;� appeared to be necessary to simulatethe second yield point adequately. Thereafter, it proved to be feasible to describe the post-yield behaviour by the experimentally determined strain hardening parameter, Gr, and the

Results & Discussion 40

0 0.5 1 1.50

10

20

30

40

50

| ln(λ) | [−]

Com

pres

sive

Tru

e S

tress

[MP

a]

0 0.5 1 1.50

10

20

30

40

50

| ln(λ) | [−]

Com

pres

sive

Tru

e S

tres

s [M

Pa]

b

Figure 7.2: The predicted (compressible Leonov) true stress-strain curves in compression (solid line) compared to the experimentaltrue stress-strain curves (o) at the strain rates 3�10�5, 3�10�4 and 3�10�3 for a) the quenched PP HM and b) the meltcrystallised PP HM.

fitted softening parameters, D1;� and h� , properly. Figure 7.3 proves that, despite the unusualdouble yield behaviour, the compressible Leonov model is also capable of providing a gooddescription of the yield and post-yield behaviour of PE LM until a true strain of 1.5 for strainrates in the range of 3�10�5 to 3�10�3 s�1. Thermal softening appearing at the strain rate of 10�2

s�1, prevented the model to accurately describe the deformation behaviour at this strain rate.

0 0.5 1 1.50

10

20

30

40

50

| ln(λ) | [−]

Com

pres

sive

Tru

e S

tres

s [M

Pa]

Figure 7.3: The predicted (compressible Leonov) true stress-strain curves in compression (solid line) compared to the experimentaltrue stress-strain curves (o) for quenched PE LM at the strain rates 3�10�5 , 10�4 , 3�10�4 , 10�3 and 3�10�3 .

In conclusion it can be stated that the compressible Leonov model is capable of predictingthe (post-)yield behaviour of the semi-crystalline polymers PE and PP. At room temperatureand within the set of strain rates (3�10�5 s�1 - 3�10�3 s�1), the yield and post-yield behaviourof PP can be described by a single molecular process. Since PE exhibits the double yield be-haviour, the extended compressible Leonov model is used to model the deformation behaviourof PE. The �-process is used to describe the first yield point, whereas the �-process capturedthe second yield point and the following strain softening and strain hardening, for which, bothby PE and PP, the experimentally derived strain hardening moduli are implemented.

Bibliography

[1] Goessens, P.J.C., and Janssen, R.P.M., Characterisation and modelling of strain rate dependentsoftening of PMMA and PC., Internship Report no. MT 01.008, Eindhoven University ofTechnology (2001)

[2] Melick, H.G.H. van, Deformation and Failure of Polymer Glasses, PhD thesis, Eindhoven Uni-versity of Technology (2002)

[3] Ward, I.M., Mechanical Properties of Solid Polymers., 2nd Ed., John Wiley & Sons, Chichester

[4] Tobolsky, A.V., and Eyring, H., J. Chem. Phys., 11, p.125 (1943)

[5] Haward, R.N. and Thackray, G., Proc. Roy. Soc. London A, 302, p.453 (1967)

[6] Boyce, M.C., Parks, D.M., Argon, A.S., Mech. Mat., 7, p.15 (1988)

[7] Boyce, M.C., and Arruda, E.M., Pol. Eng. Sci., 30, p.1288 (1990)

[8] James,H.M., and Guth, E., J. Chem. Phys., 11, p.455 (1943)

[9] Arruda, E.M. and Boyce, M.C., J. Mech. Phys. Solids, 41, p.389 (1993)

[10] Wu, P.D. and van der Giessen, E.J., J. Mech. Phys. Solids, 41, p.427 (1993)

[11] Hasan, O.A., Boyce, M.C., Li, X.S., Berko, S., J. Polym. Sci. Polym. Phys. Ed., 31(2), p.185(1993)

[12] Tervoort, T.A., Klompen E.T.J., Govaert, L.E., J. Rheo., 40, p.779 (1996)

[13] Tervoort, T.A., Smit, R.J.M., Brekelmans, W.A.M., Govaert, L.E., Mech. Time-Dep. Mat., 1,p.269 (1998)

[14] Meijer, H.E.H., Govaert, L.E., Smit, R.J.M., A Multi-Level Finite Element Method for Mod-eling Rubber-Toughened Amorphous Polymers., In: R. Pearson, Ed., Toughening of Plastics,American Chemical Society, Boston, p.50-70 (1999)

[15] G’Sell, C., Hiver, J.M., Dahouin A., Souahi, A., J. Mat. Sci., 27, p.5031 (1992)

[16] Lohr, J.J., Appl. Pol. Symposia, 1, p.55 (1965)

[17] Bauwens-Crowet, C., and Homes, G., C.R. Acad., Sci. Paris, 259, p.3434 (1964)

[18] Brady, T.E., and Yeh, G.S.Y., J. Appl. Phys., 42, p.4622 (1971)

[19] Bauwens-Crowet, C., Bauwens. J.-C., Homes, G., J. Pol. Sci. Part A-2, 7, p.735 (1969)

[20] Ekvall, R.A., and Low, J.R., J. Appl., Pol. Sci., 8, p.1677 (1964)

BIBLIOGRAPHY 42

[21] Bauwens-Crowet, C., Bauwens. J.-C., Homes, G., J. Mat. Sci., 7, p.176 (1972)

[22] Roetling, J.A., Polymer, 6, p.311 (1965)

[23] Roetling, J.A., Appl. Pol. Symposia, 5, p.161 (1967)

[24] Ducket, R.A., Rabinowitz, S., Ward, I.M., J. Mat. Sci., 5, p.909 (1970)

[25] Klompen, E.T.J., and Govaert, L.E., Mech. Time-depend. Mat., 3, p.49 (1999)

Chapter 8

Conclusions

The intrinsic deformation behaviour of semi-crystalline polymers was examined by compres-sion testing of PET, PE and PP. By means of cold crystallisation, a crystalline phase was intro-duced in the initially amorphous PET, and the crystalline content of PE is raised by annealing.This crystalline phase, however, appeared to have no influence on strain hardening. For thatreason, the strain hardening behaviour of semi-crystalline polymers is attributed solely to theentangled amorphous network. This is confirmed by a lower strain hardening for melt crys-tallised samples and for a lower molecular weight grade. The presence of a high chain mobilityduring melt crystallisation enables the polymer network to disentangle, whereas a lower chainlength simply facilitates the disentangling process. Furthermore, the identical strain harden-ing for cold crystallised PET and annealed PE confirms the slip of crystallites after yielding.The yield stress, on the other hand, proved to be affected by the crystalline phase. However,it is found that the lamellar thickness contributes more strongly to the yield stress than thecrystalline content. Thinner lamellae exhibit a lower yield stress, which seems to confirm thefine chain slip and lamellar coarse slip mechanisms during yielding. Moreover, Young’s modelbased on the nucleation of dislocations during yielding properly describes the yield stress fora given lamellar thickness. However, results have to be interpreted carefully, since dissentingviews exist concerning exact model parameters.

The macroscopic deformation and failure in semi-crystalline polymers depends on a com-bination of yield stress, strain hardening and tensile strength. A larger yield stress (due to alarger crystalline content) promotes strain localisation. The stress in the localised zones is largerdue to a reduction in loaded area. As a result, macroscopic brittle failure can be noticed whenthe localised zones have reached their tensile strength. Only if strain hardening sufficientlystabilises the deformation in the localised zones, deformation shifts and a ductile behaviourmight be observed. Thus, macroscopic failure is enhanced by melt crystallisation and/or alower molecular weight, which lower the strain hardening.

An attempt was made to describe the intrinsic deformation of PP and PE by means of con-stitutive modelling. A fitting procedure of parameters of the compressible Leonov model ontrue stress-strain curves obtained by compression testing demonstrated that it is possible tocapture the deformation of PE and PP until true strains of 1.5 over a range of strain rates. How-ever, above a strain rate of 3�10�3 s�1, deformation is not properly predicted by the model dueto thermal softening occurring during compression testing, which is not implemented in themodel used. PP is modelled by a single Leonov mode, but the peculiar double yield behaviourof PE could only be captured by the 2-D Leonov model.

Appendix A

Young’s Dislocation Model

Based on the assumptions that yield behaviour originates from dislocations, a model basedon a nucleation-controlled dislocation process was first proposed by Young [26], and subse-quently refined by Shadrake and Guiu [27]. More recently, Brooks et al. [20, 32], Darras andSeguela [33, 35], Crist et al. [31] and Kennedy et al. [23, 30] compared experimentally obtainedyield stresses to this model. However, dissenting views existed with respect to the exactmodel parameters, leading to more and less successful modelling results. Therefore, Brooksand Mukhtar reconsidered the model first derived by Young [50]. They reintroduced a termwhich was considered negligible, namely the core energy E0, which is the elastic strain energyof the screw dislocation within the core radius. This leads to an extra term, �(T), including theunknown parameter E0, and the parameter r0, the core radius of the dislocation. Mainly, theapplied value of this latter parameter varied in the previous attempts to model the yield stressand probably caused the inconsistent results [20, 23, 30–35]. However, Brooks and Mukhtardetermined the newly introduced term �(T) by means of a fitting procedure on experimentallyacquired yield stresses. This resulted in a proper description of experimentally determinedyield stresses [50]. Here, Young’s model and the interpretation of Brooks and Mukhtar is re-peated briefly.

The model developed by Young assumes that the yield stress of a semi-crystalline poly-mer results from the energy required to nucleate a screw dislocation with the Burgers vectorof the dislocation parallel to the chain axis. A schematic representation of a screw dislocationof Burgers vector b lying distance d from the edge is given in figure A.1, where lc is the crystalthickness, � is the applied shear stress.

τ

d

τ

lc

Figure A.1: Schematic representation of a screw dislocation a lamella.

Young’s Dislocation Model 45

The change in energy �G during the formation of a dislocation is given by

�G =lcKb

2

2�ln

d

r0� lcd�b (A.1)

where r0 is the core radius of the dislocation and K is a function of the shear modulus of thecrystal. A dislocation will be activated when the critical value dc produces a maximum in �G,which is

dc =Kb

2��(A.2)

Rewritten to the applied shear stress � and implemented in equation A.1, the correspondingchange in energy for dislocation activation �Gc gives

�Gc =lcKb

2

2�

�ln

d

r0� 1

�(A.3)

Substituting equation A.2 in equation A.3 gives the shear stress � required to activate adislocation, which reads

� =Kb

2r0�exp

��2��Gc

lcKb2+ 1

�(A.4)

When applying the Tresca yield criterion, where

�y = 2� (A.5)

the following equation shows that the model relates the yield stress �y directly to tempera-ture (�G) and lamellar thickness lc.

�y =Kb

r0�exp

��

2�c

lcKb2+ 1

�(A.6)

Brooks and Mukhtar [50] reconsidered Young’s model, and reintroduced a term which wasconsidered negligible, namely the core energy E0. This is the elastic strain energy of the screwdislocation within the core radius. This changes equation A.6 to

�y =K

�

�b

r0exp

�2�E0

Klcb2

��exp

��2��Gc

lcKb2+ 1

�(A.7)

In this approach, the unknown temperature-dependent term between the braces is called�(T ). Then, equation A.7 becomes

�y =K�(T )

�exp

��2��Gc

lcKb2+ 1

�(A.8)

where

Young’s Dislocation Model 46

�(T ) =b

r0exp

�2�E0

Klb2

�(A.9)

The Gibbs free energy, �Gc, is assumed to be in the range 40kT-60kT [27], with k the Boltz-mann constant and T the absolute temperature. Brooks and Mukhtar fitted a third order poly-nomial on the theoretically derived values for K by Karasawa et al. [51], such that:

K(MPa) = 3740:806 � 8:106T + 0:01897T 2 � 0:000025T 3 (A.10)

With mentioned values and �Gc is 60kT, Brooks and Mukhtar [50] fitted the parameter�(T ) over a range of temperatures on their experimental results, which lead to

�(T ) = 0:4992 � 0:00119T (A.11)

In this way, they managed to obtain a model parameter by way of experimental results. Itis mainly the value of the core radius ro that varied in the many approaches of Young’s model.By fitting this parameter combined with other parameters over a range of temperatures, theyobtained a reliable parameter.

Appendix B

X-Ray Diffraction

Here, the background of WAXS and SAXS will be explained briefly. Both methods are basedon the superposition of waves by interference. The resulting amplitude of two interfering in-phase waves is the sum of the amplitudes of the individual waves and thus there will be anamplification of intensity when recorded. Scattering by layers of atoms or lamellae, as is thecase by respectively WAXS and SAXS, will be discussed in more detail.

2θ

sd

r

transmitted beam

crystalline sample

detector

diffract

ed beam

incident X−ray beam

a

dsin θ

diffracted

d

d

B

X−ray beam

θ

X−ray beam incoming

2θ

θθ

A C

atom

b

Figure B.1: a) Schematic representation of the geometry of wide angle X-ray scattering. b) Diffraction of X-ray waves from twosubsequent layers of atoms.

B.1 Wide Angle X-ray Scattering

Consider a lattice of regularly spaced atoms by distance d, as depicted in figure B.1b. When anincident X-ray beam hits an atom, it bounces off and a diffracted beam will travel on at the sameangle �. When another parallel beam hits the underlying atom of an adjacent layer separatedby a distance d, it also bounces off, but has to travel further by the path ABC. Now, the waveswill only end up in-phase if the difference in path equals a whole number of wavelengths n�.Basically, this describes Bragg’s law, which is given by:

n � � = 2d � sin2� (B.1)

In order to determine crystallinity and lamellar thickness, the intensity of the diffracted beamsis recorded by a detector, as is shown in figure B.1a. The sample-detector distance sd and theposition of the recorded diffraction pattern from the centre r, will give the diffraction angle 2�.

X-Ray Diffraction 48

a b

Figure B.2: a) Wide angle X-ray scattering detection of semi-crystalline PET-sample. b) Integrated intensity plotted against 2�of semi-crystalline PET-sample,the area below the dotted line represents the amorphous part, the area above, thecrystalline part.

Since the wave-length is monochrome and known, the obtained 2�-angle will give the distancebetween the atoms d. A semi-crystalline polymer consists of disordered amorphous and or-dered crystalline regions and, generally, an X-ray beam will cross crystalline and amorphousregions. The amorphous region will reflect the incoming waves in all directions, since thereare no ordered polymer chains in this phase. However, the ordered crystalline region can beregarded as in figure B.1b, and will thus yield the characteristic intensity rings. Consequently,the intensity measured by the detector is composed of the high intensity peaks from the crys-talline regions and a broad intensity scattering of the amorphous regions. An example of aWAXS detection for a semi-crystalline polymer is given in figure B.2a. Before analysing the re-sults, the intensity of the background scattering was subtracted from the WAXS measurements.Then, a two-dimensional plot can be made, where the intensity of the diffracted waves is plot-ted against the diffraction angle by radially integrating the intensity over 2�. A typical WAXScurve for a semi-crystalline polymer is displayed in figure B.2b, in which the peculiar crys-talline peaks and the amorphous broad hump are explicitly present. Now, the mass fraction ofthe crystals Xc can be determined by:

Xc =Ac

Aa +Ac

(B.2)

where Aa is the area under the amorphous curve and Ac is the remaining area under the crys-talline peaks. Ideally, the amorphous curve is obtained experimentally, by measuring a glassysample. However, for PE this is an almost unattainable task due to the high crystallisation rate.Then an estimation of the amorphous scattering based on experience with other amorphouspolymers can be made.

B.2 Small Angle X-ray Scattering

The lamellar stacks can also be considered as the layers in figure B.1b. This means that Bragg’slaw, repeated in equation B.3, also can be applied to determine the lamellar thickness.

n � � = 2d � sin 2� (B.3)

Here, d is the periodicity of the lamellar stacks. Since now the spacing is larger, comparedto the atoms by WAXS, the X-ray waves will be diffracted at a smaller angle, and hence, the

X-Ray Diffraction 49

lamellar thickness is determined by SAXS. Since the sample-detector distance and the positionof the diffracted waves on the detector are known, the corresponding 2� angle can be calcu-lated. After which, Bragg’s law is applied in order to determine the lamellar thickness. Theobtained lamellar spacing is called the long period L and consists of an amorphous and crys-talline length, la and lc, see figure B.3. Assuming an ideal two-phase model, the thickness of the

L

la

lc

Figure B.3: Schematic representation of an amorphous phase situated between two lamellae. L: long period, lc : crystalline lamellarthickness, la: amorphous lamellar thickness.

crystalline lamellae lc can be ascertained by means of the degree of crystallinity determined byWAXS. Then, the fraction of crystallinity in the polymer sample corresponds with the thicknessof the crystalline part of a lamellar stack, see equation B.4.

lc = Xc � L (B.4)

The intensity can be plotted against the long spacing L or the crystallite thickness lc. However,usually the intensity is plotted against the parameter q, which has the following relation withlc,

q =2�

lc(B.5)

In figure B.4, two different SAXS-profiles are shown. Two major distinctions can be observed.The position of the maximum of the curves differ from each other and there is a differencein width of the curves. The width gives information about the thickness distribution of thelamellae, whereas the position of the maximum of the peak is a measure for the average lamel-lar thickness. For example, the dashed curve exhibits a smaller width and a peak of whichthe maximum corresponds to a smaller q compared to the solid curve, which means that thedashed curve has a larger average lamellar thickness and a smaller thickness distribution.

0 0.2 0.4 0.6 0.80

2

4

6

8

10

q [nm−1]

Inte

nsity

[−]

Figure B.4: SAXS intensity vs. q plot of two differently treated PE samples.

Appendix C

Microstructural Uniform Samples

The cylindrical compression test samples were obtained by compression moulding. Since ther-mal conditions highly affect the microstructural and mechanical properties of semi-crystallinematerials, slight differences in cooling rate throughout the compression-moulded plate mightresult in a variance of microstructural characteristics. Therefore, the objective was to createsmall cylindrical samples of 3x3mm or 4x4mm, in order to obtain as good as uniform samples.Already a lot of compression tests have been performed on 6x6mm amorphous samples of PC,PS and PMMA with good result [1, 2].

0 0.2 0.4 0.6 0.8 10

25

50

75

100

compressive true strain [−]

com

pres

sive

true

stre

ss [M

Pa]

a 0 0.2 0.4 0.6 0.8 1

0

25

50

75

100


com

pres

sive

true

stre

ss [M

Pa]

b

0 0.2 0.4 0.6 0.8 10

25

50

75

100


com

pres

sive

true

stre

ss [M

Pa]

c 0 0.2 0.4 0.6 0.8 1

0

25

50

75

100


com

pres

sive

true

stre

ss [M

Pa]

d

Figure C.1: a) True stress-strain curves of 3x3mm cylindrical PS samples. b) True stress-strain curves of 4x4mm cylindrical PSsamples. c) True stress-strain curves of 6x6mm cylindrical PS samples. d) True stress-strain curves of 3x3mm (O),4x4mm (�) and 6x6mm (4) cylindrical PS samples.

Microstructural Uniform Samples 51

For this study, reproducible true stress-strain curves were obtained of polystyrene samplesof 3x3mm, 4x4mm and 6x6mm samples in compression, see figures C.1a, C.1b and C.1c. Infigure C.1d, it can be seen that the post-yield behaviour of both the 3x3mm and the 4x4mmsamples in compression is in good accordance with the results of the 6x6mm samples. Now, itcan be concluded that the dimensions of the compression samples have hardly any influenceon the post-yield behaviour. Nevertheless, a slight decrease in yield stress is noticed withdecreasing samples dimensions. The 3x3mm samples are expected to have a better uniformcomposition. However, handling of these samples was not ideal. Therefore, 4x4mm sampleswere produced for the compression tests.

Deformation and Failure in Semi-Crystalline Polymer Systems · Deformation and Failure in...

Documents

Transcript of Deformation and Failure in Semi-Crystalline Polymer Systems · Deformation and Failure in...