Evolution of microstructure in semi-crystalline polymers under large plastic deformation

Materials Science and Engineering, A175 (1994) 183-199

183

Evolution of microstructure in semi-crystalline polymers under large plastic deformation C. G'sell a n d A . D a h o u n Laboratoire de MOtallurgie Physique &Science des MatOriaux (URA CNRS 155), Ecole des Mines, Parc de Saurupt, 54042 Nancy (France)

Abstract The plastic properties of semi-crystalline polymers up to large strains are reviewed in terms of the particular mechanisms activated in these materials. Also, a specific point is made on the transformations affecting the microstructure under the effect of the deformation (density, viscoelastic response and macromolecular orientation). The case of orthorhombic polyethylene and poly(ether ether ketone) are investigated for the illustration of the general principles emphasizing crystalline defects. The constitutive plastic behaviour is determined by means of a novel technique. The microstructural evolution is modelled in terms of glide in the crystalline lamellae, the development of damage and the modelling of the amorphous chains in the interlamellar regions.

1. Introduction Semi-crystalline polymers are the most developed thermoplastics for commodity and technical applications. Their microstructure is basically composed of lamellar crystallites (typically 20% to 80% by volume) embedded in a matrix of amorphous macromolecules which are partly bonded to the crystallites and act as tie molecules. Although the mechanical properties of these polymers have been the subject of a number of technical papers, it is only during the last 25 years that their plastic properties have been correctly described in terms of constitutive stress-strain relations representing the response of the intrinsic material [1-2] and that the elementary deformation mechanisms have been listed and analysed in the context of operational use, and not only for curiosity [3-4]. This international effort for assessing the behaviour and characterising the processes has opened the way toward realistic modelling of the phenomena, taking into consideration not only the original microstructure, but also its continuous transformation in the course of the plastic deformation. Modelling such a complex evolution demands a very critical selection of the salient processes in order to maintain the necessary simplicity. The pioneering models of Keller [5], Stein [6], Peterlin [7] and Schultz [8], although qualitative or schematic in nature, initiated this approach. They were widely applied to a variety of polymeric species, a number of papers being devoted to the detailed characterisation of the crystalline structure at various steps in the deformation. A remarkable example is provided by 0921-5093/94/$7.00 SSD1 0921-5093(93)03526-G

the series of papers on the deformation of polyethylene by Argon, Cohen and co-workers [9-11]. However, although these models help the interpretation of the structural observations under simple loading paths, they cannot predict quantitatively the constitutive stress-strain curves in complex loading conditions nor the microstructural transformations induced by the deformation. This is why it is important to develop a new generation of models taking consistently into account the conditions of triaxial deformation, the microscopic constraints of crystalline plasticity and the participation of the interlamellar amorphous chains. This modelling is currently being studied and has already been applied to several cases [12-17]. The object of this paper is threefold: i. to review some elementary results and mechanisms proposed in the literature, ii. to illustrate them in the particular cases of polyethylene (PE) and poly(ether ether ketone)(PEEK), and iii. to examine how a simplified self-consistent model may reproduce the mechanical and microstructural evolution under uniaxial tension and simple shear.

2. Principal ingredients of the deformation models for semi-crystalline polymers Among the abundant literature on the structure of crystallised polymers, we will review in this section a variety of basic processes which play a major role in © 1994 - Elsevier Sequoia. All rights reserved

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Microstructurein semi-crystallinepolymers

plastic deformation. This should not be considered as an exhaustive review, but rather as a deliberately simplified selection. The illustrations will also be restricted to the two cases of PE and PEEK, whose properties will be presented and discussed in a later section. 2.1. Crystal structure In a variety of polymers the macromolecules are capable of adopting two basic states: the amorphous state, in which the chains are statistically ravelled in interpenetrating coils, and the crystalline state, in which the chains adopt an elongated shape and arrange together in regular rows. In some instances several allotropic forms are encountered whose relative occurrence depends on the crystallisation conditions. In PE there are two crystalline forms, orthorhombic and monoclinic, but the former is the dominant one in samples normally cooled from the melt. In PEEK the only form is orthorhombic. Data concerning these structures are displayed in Table 1. It is interesting to note that the ideal density of the polymer single crystals is not more than about 14% higher than the relative density of the amorphous phase (da = 0.854 for PE [18] and da = 1.250 for PEEK [19]). The shape of the chains in the unit crystallographic cell is represented for both polymers in Fig. 1. There are two chains per cell, the central one being rotated to improve compactness. Additionally in PEEK, since the groups between the benzene rings are of two kinds, the central chain is translated along its axis. Thanks to their equilibrated atomic interactions, the single crystals keep a good elastic rigidity up to the melting point (about 300 GPa along the chains and 5 GPa across the chains for PE [20]). In the case of the PEEK, the fusion temperature being exceptionally high (335°C), the material is a good candidate for extreme service conditions. However, the actual behaviour of a real polymer grade is limited by the properties of the amorphous fraction, which suffers a sharp modulus drop at the glass transition temperature Tg (recorded at about - 2 0 ° C for PE and 143°C for PEEK, but which depends somewhat on the polymer microstructure as well). Above that temperature the amorphous phase is rubbery, so that the polymer may be considered as a

composite material with a soft matrix reinforced with the rigid crystals. For this work, two industrial grades (Table 2) were processed in the shape of thick plates cooled slowly from the melt after being extruded (or compressionmoulded), and then subjected to annealing under vacuum for 24 h at 120 °C (PE) or 180 °C (PEEK). The degree of crystallinity (deduced from density or calorimetric experiments) is about 73wt.% for the PE and 38wt.% for the PEEK, giving an apparent Young's modulus of the order of 1 GPa between Tg and Tm. 2.2. Lamellae and spherulites The microstructure of PE and P E E K samples solidified from the melt is polycrystalline. The crystallites have the shape of lamellae or ribbons radiating from crystallisation nuclei within well-defined spherulites [21,22] about 10/am in diameter. The thickness of the crystallites is assessed directly by means of electron microscope observation, or indirectly from small-angle X-ray scattering or differential scanning calorimetric data. It is of the order of 10 to 20 nm for PE [17,23], but only 2 to 6 nm for PEEK [17,22]. Locally, the lamellae are packed parallel to one another, separated with a layer of amorphous chains. The precise crystalline orientation within an individual lamella is still the object of controversial discussions, but there is a general agreement on the following basis, at least in PE. It was shown [23] that the growth direction (spherulite radius) is along the b = [010] axis, as illustrated schematically in Fig. 2. Furthermore, there is a strong indication that the main facets of the lamellae are close to the (201) plane, so that the direction of the chains (c = [001] axis) makes an angle of about 35 ° with the normal of the lamellae [21,24]. It is therefore evident that a given chain cannot fit in its extended shape within the thickness of a single lamella. In order to avoid this contradiction, several assumptions have been proposed. In the folded-chain model, the macromolecules form a fold at the lamella surface and re-enter immediately in the same crystallite, the fold plane being the (110) or the (100) plane, according to Keller [25]. In melt-crystallised PE, the latter habit plane is more likely. However, it appears from some

TABLE 1. Structural features of the PE and PEEK [18-19] Polymer

Lattice

Group

a (nm)

b (nm)

c (nm)

Helix

Tm (°C)

dc

AHm° (J g-')

PE PEEK

Orthorhombic Orthorhombic

Pnma P222

0.741 0.775

0.494 0.586

0.254 1.000

1,2/1 1.2/3

135 335

0.995 1.400

292 130

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185

).--.-o

b ,H

+ )..-.....o a

tt

u \

v

(a)

{b)

Fig. 1. Configuration of the chains in the crystalline cells of (a) PE and (b) PEEK.

experimental results [26] that the regular chain-folding model is a theoretical idealisation, and that reality is closer to the switchboard model, with the chains reentering through loose folds at non-adjacent sites or even forming tie-chains with a neighbouring lamella. The latter situation occurs very favourably in polymers of high molecular weight and provides a higher toughness to the material since it links the lamellae together. Although the amount of experimental characterisation of the crystallite structure is far smaller for PEEK, the

general features of the PE lamellae still hold in a first approximation [27]. However, in this polymer the chains are less flexible than in PE and more defects are expected in the lamellae. Also, because of the relatively small thickness of the crystallites, only four to ten phenyl rings are aligned at one time between chain folds [22]. An important point is the orientation distribution of the crystallites. At the scale of the spherulite substructure, the crystalline lamellae are arranged more or less

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TABLE 2. Actualdata for the materialsused in this work Polymer

Origin

Mn (g mole 1)

Mw (g mole-l)

Mw/Mn

d at 23 °C

PE PEEK

R6chling D 500,000 IC1450 G

37 200 41 400

172 000 110 000

4.6 2.65

0.956 1.306

Fig. 2. Schematic configuration of lamellae in spherulites (after refs. 65 and 66).

radially, and that plays some role in the deformation mechanisms. However, at a macroscopic scale, the crystalline texture can be regarded as isotropic provided that no crystallisation constraints (resulting from high shear rates in the melt during the sample processing) induce a preferential crystalline orientation. In the materials taken as examples in this work, the initial isotropy of the specimens was assessed by X-ray diffraction goniometry with a precision of a few degrees [17]. The evolution of this initial microstructure under the effect of an applied deformation will be a valuable indicator for assessing the active macromolecular mechanisms. 2.3. Deformation mechanisms in the crystalline phase The microscopic processes which control the mechanical response of semi-crystalline polymers have been largely assigned to crystallographic slip in the lamellae, and secondarily to twinning or phase transformation. The original pieces of work were the subject of several review papers (e.g. refs. 4 and 23). Plastic slip in orthorhombic polymer crystals is based on the formation and glide of dislocations. Because of the unavailability of large single crystals and the difficulty of direct TEM observation of the defects, the precise modelling of dislocation geometry and dynamics is not

Hm

(j g-l) 211 47

yet fully resolved. However, some general properties have been deduced by considering the particular features of the polymer crystals. Because of the high strength of the covalent bonds, it is evident, for example, that the dislocations should not provoke a kink in the chain axis, so that the glide plane is of the {hk0} type, for which only the Van der Waals bonds are distorted. Among the possible choices, several pieces of evidence [3,9,10,28,29,30] including direct microscopic observation promote firstly the "chain slip" (Fig. 3(a)) along the (100)[001] and (010)[001] slip systems, and secondarily the "transverse slip" (Fig. 3(b)) along the (100)[010] and (010)[100] systems. The former undoubtedly involve the glide of screw dislocations of Burgers vector c, while the precise mechanism for the latter are more questionable, owing to the possible splitting of the edge dislocations with Burgers vectors b or a (with the line along [001]) in the [110] planes [28,31]. In addition to the crystallographic considerations, the very small thickness of the lamellae plays a considerable role. On the one hand, in such a medium, the stress field induced by screw dislocations (such as those intervening in the chain slip) are partly relaxed at the surface [8,32] and thus the long-range interaction between dislocations gliding in parallel planes is weaker than in bulk crystals. On the other hand, the probability of forming sessile dislocation junctions is very small. It follows that the kinetics of dislocation multiplication in PE and PEEK crystals is controlled by the thermal nucleation of dislocations rather than by the activation of dislocation sources [28,33]. A complementary ingredient in the discussion of glide processes is the influence of the chain folds. In contrast to the case of metallic single crystals, the movement of a dislocation in a polymeric lamella may leave more than an elementary step at the lamella surface if the glide plane is not parallel to the fold plane. In such a case, the chain fold is subjected to a kind of "reptation" movement involving several coordinated conformation changes. The situation is easier if both the fold plane and the glide plane coincide with the dense (100) plane. In addition to the above processes, several authors have proved the occurrence, albeit of marginal importance, of twinning and phase transformations. Because of the covalent bonds along the chains, the twin planes must be of the {hk0} type [4,34]. Although the {110}

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G'sell, A. Dahoun

/ Microstructure in semi-co'stalline polymers

(a) (b) Fig. 3. "Chain slip" and "transverse slip" in a polymericcrystal.

twin plane is dominant, the {310} plane was identified as well in heavily deformed PE [9]. As for the straininduced transformation from orthorhombic to monoclinic, it was evoked several times in PE [4,23,34,35], with the following coincidence: (020)o r-- (210)Mo and [110]o r - [110]Mo. However, its importance in the overall deformation kinetics has not been ascertained. In conclusion to this .preliminary discussion, it appears that the variety of interacting phenomena during the plastic deformation of polymeric crystals, together with the unfavourable experimental conditions, leaves some uncertainty about basic data such as the critical resolved shear stress (CRSS) and the resolved strain hardening for the active glide systems. From direct measurements obtained with highly textured specimens, or from an adequate modelling of stress-strain data obtained with isotropic samples, several values of the CRSS for the chain slip systems were determined at room temperature in the range from nearly 20 MPa in earlier papers [36] to 7.2 MPa in recent studies [10]. Among these data, the choice is not easy, particularly because the effect of the strain rate is rarely taken into account. 2. 4. Deformation mechanisms in the amorphous phase A significant contribution to the overall behaviour of semi-crystalline polymers is provided by the amorphous phase, which represents in some cases the major volume fraction of the material (e.g. in the PEEK investigated in this work). As stated in a previous section, the mechanical response of amorphous polymers is highly dependent on temperature. For T> Tg, the Van der Waals interactions are overcome and the

187

amorphous polymers exhibit a rubber-like behaviour with an initial modulus of the order of only 1 MPa. The stress is of entropic nature, being controlled by the thermodynamic tendency of the stretched chains to return to their coiled geometry. Therefore there is no yield stress, and the tangent modulus increases more and more as the chains are elongated [37]. For T< Tg, the "Van der Waals interactions are effective. The polymer is then in a glassy state, the elastic modulus being equal to several gigaPascals. If no early damage occurs, the glassy polymers can exhibit plastic deformation. A transient yield point is observed for a typical stress of 50 to 100 MPa, followed by a sharp stress drop and then by a long strain-hardening stage [38-40]. The apparent similarity of the stress-strain curves at large deformation above and below ~ is due to the similar chain orientation mechanism which controls the mechanical response under both conditions. Various models were proposed to fit the hardening behaviour of amorphous polymers, based on a statistical analysis of the chain configurations under stress. In the approaches of Treloar [37], Boyce [41] and Van der Giessen [42], the stress tensor is computed from the combination of the elementary forces experienced by the individual sub-chains between cross-links (constituted by physical entanglements or by the chemical junctions), as depicted in Fig. 4. All the models are nearly equivalent at small strains, the tensile modulus being equal to n k T where n is the density of active subchains (the loose ends do not participate to the tensile force), k is Boltzmann's constant and T is the absolute temperature. Conversely, at large strains, the rise of the stress-strain curve is very dependent on the precise description of the chain structure (valence angles, bond conformation, excluded volume, entanglement sliding, etc.) and on the modelling of the representative material element (three chains [37], eight chains [41], random distribution [42]). The number of elementary links in the subchains, N, plays a dominant role at large strains, the shortest ones reaching their limit extension ratio earlier. However, the models available to date still involve many approximations, among them the perfect flexibility of the bond joints and the free interpenetrability of the chains. Despite these limitations, the model of Van der Giessen [42] seems preferable because the initial topology of the network is totally isotropic and independent of the reference loading axes. In the case of semi-crystalline polymers, it is not clear how much the behaviour of the amorphous chains confined between crystalline lamellae differs from the response of the same material in a totally amorphous form. This is because: i. the configurations of the tie molecules are presumably influenced by the growth conditions of the lamellae [43,44], and

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Microstructurein semi-crystallinepolymers

4)

Fig. 4. Random distribution of initial end-to-end vectors and affinely deforming network of cross-links for the modelling of rubber-like elasticity.

b)

ii. the constraints imposed by the confinement between the rigid crystalline blocks may influence the deformation of the amorphous chains [45,46]. Although it is not unrealistic, in a first approximation, to analyse the behaviour of a semi-crystalline material on the basis of a composite structure consisting of the crystalline phase and of the amorphous fraction [47,48], more specific models are now developed which take into consideration the micromechanical interactions between the amorphous zones and the lamellae [15]. In particular, the amorphous phase plays a major role in the processes of "lamella sliding" and "lamella separation" which take place during the spherulite deformation (Fig. 5). Additionally, much care should be taken in modelling the mechanical behaviour below Tg in polymers subject to extensive moisture absorption, as in the case of polyamides. In these materials, the yield behaviour of the glassy phase is very sensitive to the amount of absorbed water, and a strong softening effect is noted [38], especially in the range of moderate strains. Therefore the samples should be maintained under dry conditions in order to study the intrinsic polymer response.

3. Experimental investigation of PE and PEEK under uniaxial tension and simple shear The PE and PEEK industrial grades, whose structure has been characterised in the preceding section, were chosen to illustrate the mechanical behaviour and the induced structural evolution of two semi-crystalline polymers with the same crystallographic structure but different degrees of crystallinity and chain rigidity. Also, in order to provide comparisons, the mechanical

Fig. 5. Deformation of the amorphous phase during (a) lamella sliding and (b) lamellaseparation processes.

tests were performed at nearly corresponding temperatures between the glass transition temperature of the amorphous phase and the melting temperature of the crystallites, that is at 25 °C for PE and 180°C for PEEK. 3.1. Effective stress-strain behaviour It is well known after a number of studies in mechanical metallurgy that it is essential to characterise the plastic response of materials under various loading paths in order to interpret this response correctly in terms of microscopic mechanisms. In particular, it was shown that tests which do not prescribe the principal axes of stress or strain are very discriminating as to what is concerned with the hardening processes and the deformation textures. For this work on PE and PEEK, we focused our attention on uniaxial tension and simple shear, corresponding to very contrasting strain fields. In uniaxial tension, the major principal strain axis stays aligned with the tensile axis (Fig. 6(a)), while in simple shear it rotates continuously toward the shear direction as the applied shear increases (Fig. 6(b)). For the present investigation, we used a special video-controlled materials testing method developed in this laboratory (called "VideoTraction" or "Video-

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Microstructure in semi-co'smlline polymers

189

F

Re D

\\

\

~1=~;

el=lOg(y + Y2~/~~ )

82=-c/2 83= -~:/2

82=_log(y+ y2fy~+4)

83=0 ct =1/2 tan-l(2/y)

(a)

F

(a)

ib)

Fig. 7. Samples for (a) uniaxial tension test and (b) simple shear test.

(b)

Fig. 6. Principal strains in (a) uniaxial tension and (b) simple

shear.

Shear", according to the loading mode). It makes it possible to measure in real time the true strain and the true stress in a given material element while its true strain rate is maintained at a constant value during the test. Although the tension and shear variants were presented elsewhere before [49,50], we will recall briefly their main features below. The system is based on a servo-hydraulic testing machine equipped with a temperature-controlled environmental chamber. The deformation stage is designed specifically for tension or shear. In the case of tension, the samples have an hour-glass shape (Fig. 7(a)), which makes it possible [2] to monitor the true strain within the median plane from the reduction of the local diameter D during the test through the relation e = 2 log(D0/D) where D 0 represents the initial diameter. Also, the local true stress is assessed in the same zone by the relation

o =4F/JrD 2 x Fr where 4F/~D 2 is the Kirchoff stress and FT(Rc,D ) the Bridgman triaxiality factor, which corrects the perturbation due to the local radius of curvature, R c, of the specimen profile. The mechanical state in the centre of the specimen is thus characterised by the simultaneous measurement of F, D and Re, even after the necking phenomenon has appeared (usually at yield in polymers). The above equations are strictly valid for an isochoric deformation, but it was shown [2] that minor density changes in the material have negligible effects on the recorded stress-strain curves. In the case of the simple shear, the

specimens were designed in such a way that the deformation localises in a single shear band located at the thinnest cross-section of the round-based groove (Fig. 7(b)). The deformation is assessed within that band from the local distortion of a flexible ink marker printed on the flat surface prior to the test, the shear strain being defined by the relation y = tan(0max), where 0max is the maximum deflection angle of the marker at the root of the notch. As for the shear stress, it is simply deduced from the relation r = F/(Lto), where L and t 0 are respectively the length and thickness at the minimum cross-section. In order to control the tests in tension as well as in shear, the system utilises a special videometric device interfaced with a fast microcomputer equipped with an image-digitising board (Fig. 8). The computer analyses in real time the applied force and the geometric features acquired by the video camera (in a frame of 512 x 512 pixels), and displays immediately the current point of the stress-strain curve on a special monitor. Another function of the system is the control of the hydraulic ram velocity (through a digital regulation program) in such a way that the true strain rate (g = de/dt or ~ = dy/dt) is automatically maintained at a constant value during the whole test, even if a plastic instability develops in the specimen. In all cases, strain rates slower than about 1 0 - : s- l are chosen in order to avoid adiabatic heating. The curves in Fig. 9 represent the stress-strain response of PE and P E E K under uniaxial tension and simple shear. In order to compare the two loading modes more clearly, the shear strain and shear stress variables were transformed into the so-called "equivalent" strain and stress by using the von Mises relations: Gq = y/,]3 and O'eq= r.]3. In tension, these expressions simply reduce to the true stress and strain : eeq = e and

190

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Microstructure in semi-crystalline polymers

ANALOG-DIGITAL L O A D C E L L INTERFACE

fl

A

DISPLAY MONITORS

1 T E N S I L E TESTING MACHINE

DIGITAL-ANALOG RAMP GENERATOR

CURVE PLOTTER

Fig. 8. General diagram of the video-controlled testing system (illustrated here in the case of the tensile tests).

200

i

i

i

PEEK

n

150

C,~=

I¢1

¢n

I-Z LU ._1

5 o

~PEEK SHEAR

5. I0 "4 s-I

W F-

PE

T:o,ON. /

/

I00

/

/

180"C

/

50

UJ

~

0 0

25"(: I

I

|

0.5

1.0

1.5

2.0

EOUIVALENT STRAIN

Fig. 9. Equivalent stress-strain behaviour of polyethylene at 25 °C and poly(ether etherketone) at 180 °C under uniaxial tension and simple shear. The equivalent strain rate is eeq = 5 × 10 -4 s- 1 in both modes.

O'eq = O. It should be noted here that the choice of the above formulation for the equivalent variables is not obvious, since it has been shown several times in the literature (e.g. ref. 51 ) that the von Mises flow criterion is not strictly verified for polymers, the yield stress increasing somehow with the hydrostatic pressure. However, this influence is small, and it is visible in Fig. 9 that, for a given polymer, the "equivalent" stress-strain curves in tension and in shear are nearly superimposed up to the yield point. Conversely, at large strains the two loading modes are very contrasted, as we shall see below. In uniaxial tension, both materials are characterised by a long plastic range with increasing strain-harden-

ing. One notes that PEEK hardens much more than PE, the flow stress for the former being multiplied by a factor of about five between the elastic limit and the rupture occurring at e = 1, while in the latter polymer the five-fold increase is achieved at e = 1.8 only, the rupture occurring at a strain of about two. This gradual hardening in tension is a very general property of semicrystalline polymers, but this feature has been acknowledged only recently [1], since the first authors working on the stretching of plastics were often misled by the dramatic force drop recorded at yield in standard tests performed at a constant elongation rate, while the sample experiences an abrupt necking, and by the long force plateau while the neck propagates along the calibrated length. The absence of yield drop in the tensile true stress-true strain curves of Fig. 9 proves that the apparent yield softening displayed by the nominal force-elongation curves is not an intrinsic phenomenon but is merely due to the reduction of the crosssection at the neck. Under simple shear, the situation is very different. For PE, the post-yield behaviour is characterised by a weak stress drop whose amplitude is about 7 MPa. Obviously, this stress decrease cannot be caused by a geometric thinning, since in shear the specimen thickness is constant. Nor can the stress drop be entirely due to the effect of the normal stresses. Data published in an earlier work [52] showed that, although the stress drop would be reduced if the normal stresses were included in the expression of Oeq,it would remain significant, indicating that it is related to an intrinsic structural evolution of the material. When the equivalent strain reaches eeq = 1.73, the softening saturates and the stress is observed to rise again slowly. While in this work we stopped the tests at e~q = 2.02 (V = 3.5), tests driven up to eeq = 4.62 with PE in the earlier study [52] showed that the softening stage is actually followed by a marked hardening before rupture. In contrast, PEEK exhibits monotonic hardening after the yield point, and no stress drop is noted. The ultimate stress at rupture (for eeq= 1.44) is nearly three times larger than the yield stress. One advantage of the video-controlled material testing system is its capacity to maintain the strain rate at a constant value. We took advantage of this feature to analyse the influence of the strain rate on the flow stress. As in several other polymers [51] it was found here that a power law of the type Oeq=f(,f.eq)X(~eq/~.~eq)m is a good approximation, in which m is the "strain-rate sensitivity coefficient". For PE in both loading modes, we obtain a common value m = 0.11. The effect of strain rate is lesser in PEEK, with m~-0.025. It can thus be considered that the semi-crystalline polymers behave as viscoplastic materials.

C. G'sell, A. Dahoun

/

3.2. Strain-induced crystalfine textures In order to assess the structural evolution, tests were run up to specific values of the plastic deformation, at which the samples were quenched to room temperature and unloaded. Tiny cylinders 2 mm in diameter were then carefully machined within the precise zone to which the stress-strain behaviour was ascribed, with a view to X-ray diffraction goniometry with Cu K a j incident radiation. An advantage of this axi-symmetric shape is to allow the determination of complete pole figures with minimum absorption corrections, by rotating the specimens around the axis and around a direction normal to the axis. The most intense peaks used

~ DD

191

Microstructure in semi-co,stalline polyrners

for this study were (110), (011) and (200) for PE, and (110), (111) and (200) for PEEK. It is verified that the initial chain orientation is negligible. The normalised pole figures obtained with the (200) reflection are displayed in Fig. 10 for both materials at an equivalent plastic strain of one (i.e. for e = 1 in tension and V= 1.73 in shear). It is clear that the distribution of crystallite orientations is quite different according to the loading mode and secondarily according to the material. In tension, an axi-symmetric texture is observed, with the a vector oriented evenly in the transverse plane (TP) perpendicular to the drawing direction

~

PE TENSION c:l

PEEK TENSION E:=I

DD

~i~i¸:~

i~i!ii~I TP

I-'-I 0.88 E]

0.5e

r-7 1.08 r-7 1.~ z.08 z.se mm 3.00 mm 3.S8 4.08 )4

SD

/

/

/

/

DD

$D

/

:

O

i

¸

®

TD

-

--

SD

PE

SHEAR = 1.75 ( E ~ =11

~ ~ SD

PEEK SHEAR =1.73 ( E ~ =1)

Fig. 10. Crystalline (200) pole figures of PE and PEEK after deformation in tension and under shear for an equivalent strain

geq =

l.

192

C. G'sell, A. Dahoun

/

Microstructure in semi-crystalline polymers

(DD). In PE the concentration of poles near this plane is not yet very marked at e = 1, but it is much stronger in the pole figure recorded at e = 1.9, displayed elsewhere [17]. This preferential orientation, typical of the "fibre texture" early identified by many authors by means of flat-film Debye-Scherrer experiments [53, 62], is clearly associated with the progressive rotation of the c axis toward DD, the (a,b) diad being randomly distributed in the plane TE The same evolution is noted in the case of PEEK, with an even more pronounced texture for the same applied strain. No pole figure is shown for e > 1 because the samples fractured soon after this strain. The results displayed qualitatively in the stereographic projections of Fig. 10 were analysed quantitatively in terms of the Hermans orientation factors [54]. Originally introduced for textile fibres, these factors were lately generalised to the characterisation of the average orientation of a crystallographic axis u with respect to a macroscopic direction d through the relation Fu/d=(3(cosZ(a,/d))l)/2, where a,ea represents the angle between u and d For example, in tension, the Hermans factor of the c axis vs. the drawing direction, Fc/o~ is equal to 0.44 at e = 1 for PE and equal to 0.50 for PEEK at the same strain (note: F,/a equals 0 for a random orientation of u vs. d 1 for a perfect alignment along d and - 0 . 5 for perfect orthogonality). Also, the texture evolution was analysed in terms of the average directions of the crystallographic axes a, b and c computed from the three recorded X-ray pole figures. For clarity, the evolution of these average directions was displayed in a triangular graph ("Hermans Triangle") as in Fig. 11, where the corners of the triangle correspond to the macroscopic reference axes. Again the "fibre texture" is evident in this representation. Under shear, no symmetry is prescribed a priori, and thus it is interesting to analyse the rotation of the (200) normals in the experimental pole figures. In the case of PE, Fig. 10 shows that the a axis rotates toward the plane containing the shear direction SD and the normal direction ND perpendicular to the shear plane. In the pole figures (not shown here) for moderate strains, e.g. eeq = 0.5, the concentration of the normals is weak and the distribution is rather regular between SD and ND. Here, for eeq ~ 1, a bimodal distribution is noted, with the stronger population of the a axes near ND. At larger strains, e.g. eeq = 1.9, the rotation of the principal a component towards ND becomes more and more evident. Also, the secondary component of the bimodal distribution tends to vanish while the concentration of a becomes more regularly distributed in the plane containing ND and TD (TD is the transverse direction perpendicular to SD in the shear plane). The quantitative evolution of the average a, b and c axes, displayed in Fig. 11, shows unambiguously:

DD

DD

TE....~

TP~ '~ SD

ND

SO

TD ND

TD

Fig. 11. Hermans triangle representation of the evolution of the average a, b and c directions during tension and shear tests for PE and PEEK.

i. the rotation of c toward the shear direction, and ii. the slightly preferential orientation of a near the direction normal to the shear plane. The texture development of PEEK under shear is similar to the case of PE, with a progressive rotation of c toward SD. On the average, the rotation is a little more pronounced than in PE, with a Hermans factor Fc/sD equal to 0.27 at Eeq = 1, to be compared to 0.09 in PE for the same strain. Concerning the a axis, it turns toward ND as for PE, whereas a broad distribution is shown by the (200) pole figures rather than a well resolved bimodal distribution.

3.3. Structural damage caused by the deformation Plastic deformation not only provokes an orientation of the crystalline species, but also it induces various forms of structural perturbation, such as chain unravelling, lamella fragmentation and crazing. Although all these phenomena were already identified in semi-crystalline polymers [5,9-11,55-58, 62], a direct relation with the intrinsic mechanical behaviour was rarely established for loading modes other than tension. The basic characterisation performed in this work was the microdensitometric analysis of small fragments of polymer (about 8 mm 3) carefully cut from plastically deformed specimens. The density d was deduced from the apparent weight of the specimen in methyl alcohol, compared with the weight in air at 23 °C. The curves in Fig. 12 show the relative variation, did o with reference

C G kell, A. Dahoun

/

Microstructure in semi-co'stalline polymers 1.0

PE SHEAR

LO =

193

25 "C 0.6

0.9 0.8

0.2

PEEK 180 "C

SHEAR

1.0

1.0

TENSION

"0

-~

PEEK180 "C

0.6

SHEAR

0.9 0.8 0

i

i

05

1.0

0.2

1.5

2.0

0

EOUIVALENT STRAIN

Fig. 12. Relative variations of density for pre-deformed specimens.

to the density d 0 of the undeformed specimen. It appears that while the density of PE is reduced of about 10% by a tensile strain of e = 2, an equivalent strain in shear does not significantly affect the density. Although the density drift is progressive, one can assess the onset of associated damage at a tensile strain of the order of one ( i.e. for a stretching ratio ~, = 2.7). In the case of PEEK, although the analysis concerns a relatively limited strain range, it appears that i. in tension the density reduction is about half that of PE (at e = 1 ), and ii. in shear the damage is still negligible. Our last structural indicator is deduced from the transient viscoelastic behaviour of specimens unloaded at different prestrains, for which the test is resumed after a 300 s waiting time. As previously demonstrated in detail elsewhere [59], the transition from elastic to plastic becomes more and more gradual in semi-crystalline polymers as the prestrain, ep, is increased. This "viscoelastic" stage is fitted by the equation Oeq = Oy{l - - e x p [ - - w ( ~ - ep)]}, where Oy is the yield stress and w is a coefficient which increases with the viscoelastic compliance. The relative variations of w are displayed in Fig. 13 with reference to the undeformed state. Here again it is evident that the structural perturbation is less for shear than for tension, but now the effect is very pronounced for both materials, even more for PEEK. The particular evolutions of the densitometric and viscoelastic parameters shown above indicate clearly that some structural damage may occur in both polymers at large strains. Even if crazing is not very active under constant strain-rate conditions, some amount of cavitation should nevertheless be invoked, at least in PE under tension, since the final density is smaller than

-,, TENSION ,

,

0.5

1.0

,

1.5

2.0

EQUIVALENT STRAIN

Fig. 13. Relative variations of the "viscoelastic" coefficient for

pre-deformed specimens.

the amorphous density. However, the effects illustrated here should be mainly interpreted in terms of i. the unravelling of chains in the amorphous phase, and ii. the progressive fragmentation of the crystallites, inducing the transfer of chains from the lamellae to the rubber-like amorphous fraction. It is evident from the experimental results that these mechanisms are more active in tension, which is a highly dilatant loading mode, while the structure is less perturbed under shear, where the hydrostatic pressure is negligible.

4. Discussion

Although information on the elementary deformation mechanisms is still incomplete or, to some extent, qualitative in nature, knowledge about the structure of crystalline lamellae and the statistical configurations of amorphous zones is now sufficiently reliable to propose a tentative modelling of their respective behaviour under various loading paths. In this section, we shall briefly recall the bases of theories applicable to the two phases and will combine them to simulate the effective behaviour of both PE and PEEK.

4.1. Crystal deformation model Basically, a crystalline polymer can be regarded as a polycrystalline aggregate of randomly distributed crystallites which plastically deform in a co-operative manner. This representation of the microstructure is

194

C G'sell, A. Dahoun

I

Microstructure in semi-crystalline polymers

grossly oversimplified since it ignores the mesoscopic distribution of the lamellae within the spherulites, but nevertheless it constitutes a valuable starting point for a correct model, provided that factors such as the shape factor of the lamellae, the crystallographic structure, the active glide systems, the resolved slip kinetics and the crystal-to-crystal interactions are consistently taken into consideration. Also it is clear that the development of deformation textures under the effect of plastic deformation must be taken into account, since Fig. 9 shows unambiguously that the stress-strain curves diverge as soon as the strain increases above eeq -- 0.5. On these grounds, we applied the viscoplastic selfconsistent (VPSC) model introduced by Molinari et al. [60] and further developed by Ahzi et al. [61]. This model predicts the global stress-strain behaviour and the texture development for a polycrystalline aggregate subjected to large plastic strains. The version we consider here corresponds to the "one-site" scheme, in which each crystallite is supposed to interact with its neighbours like a plastic inclusion with a surrounding "equivalent homogeneous medium" whose properties are those of the macroscopic material. Within each lamella, the plastic deformation results from the combination of slip systems s, whose plastic shear rate is related to the corresponding resolved shear stress ps by the following non-linear viscous law: =

(1)

where r~ and 2~ are the reference stress and the reference shear strain, and m is the strain rate sensitivity coefficient. The strain-hardening of each slip system is potentially taken into account through the r~(~,s) function. The interaction of any crystalline lamella with the surrounding medium is predicted by the following relation o-

0 = ( r - ' + A ° ) ( t - i)

(2)

where i = (L + L v )/2 is the overall applied strain rate, 0 is the overall deviatoric Cauchy stress, o is the deviatoric stress in the lamella, t is the local strain rate tensor in the lamella and r an interaction tensor taking account of the effective tangent modulus A° and of the shape of the lamella (considered as a flat ellipsoid, to a first approximation). In the present state of the model, one takes for the homogeneous equivalent medium an isotropic tangent modulus /2 which is an unknown p_arameter depending on the applied strain rate tensor ~. With this simplification, the relation (3) is written as o - O=/~K(g- ~)

(3)

where the tensor K = F - 1 / / u +I depends only on the morphology of the crystallite. After the macroscopic strain rate tensor ~ and the morphological tensor K are fixed, the microscopic

stress and strain tensors are determined in such a way that the following self-consistency conditions are satisfied: (g) = ~;

(a.t~) = O.i

(4)

where the bracketed expressions are averaged in volume. This makes it possible to compute the resolved shear rates ~s and the induced rotation of the lamellae, through a double iterative scheme on the variables/, and 6. The above model is thus capable of predicting not only the macroscopic stress-strain behaviour from the microscopic response of the crystallites, but also the strain-induced texture of the polycrystalline aggregate. Originally the VPSC model of Molinari et al. [60] was designed for polycrystalline metals and geological structures. In order to use it for the case of crystallized polymers, we start neglecting for simplicity the contribution of the amorphous phase to the mechanical response of the material. This approximation is legitimate for PE, in which the amorphous fraction is less than 30%, whereas it is not well adapted to the modelling of PEEK, whose crystalline content is in minority. A given sample was considered as an aggregate of orthorhombic lamellae initially oriented at random. The shape factor of each lamella was fixed at a value of 20 in order to reproduce their fiat shape as closely as possible without numerical instabilities. The number of lamellae was fixed at 450. Since the large plastic strains were of principal interest in this work, the elastic deformation of the lamellae was neglected. As for the active glide systems, preliminary simulations taking into account only the chain slip systems led to the unrealistic result that the equivalent flow stress in shear is larger than in tension. This confirms the deductions drawn from experimental studies [9,10] that the activation of transverse slip plays a significant role in the plasticity of polymer crystals. However, lacking irrefutable data on the respective resolved shear stresses of the different systems, we chose the same CRSS for all the slip systems taken into consideration: (100)[001], (010)[001], {110}[001], (100)[010], (010)[100] and {110}(110). This is obviously an oversimplification, since transverse slip systems should be more difficult than chain slip systems. However, this first approximation has the merit of minimising the number of adjustable parameters. Furthermore, considering the argument developed in Section 2 on the weakness of the dislocation interactions, the resolved strain hardening was fixed at zero for all the systems. Finally, the resolved shear strain rate sensitivity coefficient was uniformly set equal to the macroscopically determined coefficient m. In the case of PE, the self-consistent model gives very encouraging results. Concerning the stress-strain

C. G'sell, A. Dahoun

/

Microstructurein semi-c~stalline polymers

curves, the graphs of Fig. 14 show that the computed results reproduce quite correctly the experimental results, i.e. the stress increases in uniaxial tension and the true stress decreases in simple shear. Furthermore, the CRSS (which is here the only adjustable parameter) is found at a value of about 11 MPa, which fits quite well in the middle range of the CRSS proposed by Argon and coworkers [10] for the active chain and transverse slip systems. However, it appears that the modelled strain hardening in tension is too large in comparison to the experiment, while the strain decrease in shear is a little too small. As for the simulated (200) pole figures (Fig. 15), they follow the general trends in respect of the rotation of c toward the drawing direction (DD) and the shear direction (SD), respectively. However, in simple shear the distribution of a does not seem to be quite correct, since one notes in the simulated figure a pole concentration near the transverse direction (TD), while the experimental distribution was rather depleted in this area. It is difficult to determine the exact cause of the residual discrepancies between the simulated and experimental results for PE. However, in regard to the problem of the a axis distribution, it may be envisaged that the uniform value of the CRSS for the different chain-slip systems could be questionable. Further runs should be tested with a relatively smaller CRSS for the (100)[001] system, in order to favour the glide on this plane and thus to make it turn preferentially toward an orientation close to the shear plane, i.e. its a normal near ND. As concerns the lower hardening in the experimental tensile curve w+. the computed curve, it is probable that it is due to the strain-induced damage

200

i

PE o o_

~- ~ = 5 . 1 0 " * s '4 8 Systems CRSS = II MPa

150

u~ u) W e~

I--.I00 z

W _J

>

50

O hi

SHEAR

0

0

I

I

I

0.5

1.0

1.5

2.0

E Q U I V A L E N T STRAIN

Fig. 14. Macroscopic stress-strain curves deduced from the Visco-Plastic Self-Consistent model of Molinari et al. [60] for PE in tension and under s h e a r (O'eq--~'eq).The experimental curves are displayedin dotted lines for comparison.

195

which does occur in the real samples (as indicated by Figs. 12 and 15), whereas it is neglected in the model. It is conceivable according to the literature [7,9] that the crystallites begin to undergo fragmentation and unfolding at strains between 0.5 and 1.0. This process undoubtedly increases the material compliance (as shown by the decrease of w in tension, Fig. 13) and is likely to reconcile the simulated and experimental curves if introduced in the model. One indication which corroborates this assumption is the improved fit of the predicted shear curve, because under this loading mode the strain-induced damage is reduced. In the case of PEEK, the VPSC model is not capable of reproducing the experimental stress-strain relation. In particular, whereas the computation continues to predict a strain softening in shear, the actual tests show a monotonic hardening after the yield point (Fig. 9). This is due to the large amorphous fraction in this polymer (62vo1.%), and we will examine now the specific behaviour of this phase. 4.2. Introduction of the amorphous phase As we explained in Section 2, the presence of rubber-like amorphous chains between the crystalline lamellae is of great importance, not only in the interpretation of the viscoelastic response prior to the initiation of crystal glide [56], but also at large strains during the crystallite fragmentation process [5,7,44]. Efforts are currently being developed to integrate fully the amorphous phase contribution directly within the selfconsistent code presented above, but this is a rather difficult task: not only is the combination of a viscoplastic constitutive equation (crystalline lamellae) with a hyperelastic equation (amorphous phase) difficult, but also the micromechanics of the confined deformation for thin elastomeric layers between the otherwise rigid lamellae are likely to give numerical instabilities. This is the reason why, in this preliminary approach, we chose to introduce the response of the amorphous phase through a simplified composite scheme. The constitutive behaviour of the rubber-like material was derived according to the model of Van der Giessen and collaborators [42]. In this approach, the polymeric chains are supposed to be highly ravelled, and the subchains between the entanglement (whose density is n per unit volume) are randomly distributed, each of them containing on average N chain segments capable of free rotations. One major assumption of the model is that the network of entanglements deforms homothetically with respect to the macroscopic strain tensor ("affine" theory). The rubber-like behaviour of an individual subchain, controlled by the maximum entropy criterion [63], is characterised by a very weak load increase at small elongation, followed by a rapidly increasing hardening which tends asymptotically to

C G'sell,A. Dahoun / Microstructureinsemi-crystallinepolymers

196

L DD

PE TENSION

t

PEEK TENSION

~ E = I

E--I

T==~.

iii~

~

/

/

I-'-I g,8g E ~ g.58

DD

I-"1 1.8g r-I

t.S8

2,88 2.58 mm 3,8g 3,58 mm 4.8g

SD

SO

),!

/

I i~ ~ L i

/

/

\

NO,

~

/ i!ii

/

SD

Nx

/'

~/

":

"

PE SHEAR eg

\,,.

/

"

SD~

%

/

PEEK SHEAR = 1.75 (Eoq

= I)

Fig. 15. Crystalline(200) pole figurespredicted by the VPSC model for PE and PEEK after deformation in tension and under shear for an equivalentstrain e~q= 1.

infinity as the stretching ratio tends to ~'max = ,iN. For a random network of entangled chains, Van der Giessen has shown that the constitutive equation in uniaxial tension is given by: Jr2;r

nkT,/N t"t" 0 0

/

_ l I 2 c ,,v

cosines of the chain end-to-end vector with the tensile and transverse axes, and 2 c the chain extension ratio, with ~,c2 = m l 2 exp(2e) + (m22 + m3 2) exp( - e). This model is also applicable to simple shear provided that special precautions are taken with the principal axis rotation. One thus gets the shear stress-strain relation:

/

ml exp(2e)-m22 e x p ( - e ) gin(0)d0d¢ x k 2c

(5)

where k is Boltzmann's constant, -~(x)= coth(x)- 1/x is the Langevin function, (ml, m2, m3) the directing

r=sin(tan-'(~))

nkT'/N?2i8~~)!d -~(~NN)

(m12exp(2el)-m22exp(2e2))sin(O)dOd~ X

•

~c

(6)

(7. G'sell, A. l ) a h m m

/

197

Microsmlcture ir~ semi-crystalline polymers

where e~ and e2 are the principal strains (Fig. 6) and 2~ the chain extension, with ,~c2=ml2exp(2e~)+ m22 exp(2e2) in this case. A systematic investigation of these laws shows that the equivalent stress-strain curves are systematically higher in tension than in simple shear for the same values of the two parameters n and N (Fig. 16). The strain-hardening at large strain increases for decreasing values of N. Prior tests in tension and shear on model amorphous polymers above T, have shown that eqns.

(5) and (6) predict the experimental results fairly well [64]. The point which is of importance here is that the rubber-like model predicts a monotonic strain-hardening in both loading modes, so that a combination of the viscoplastic crystalline law with the rubber-like amorphous behaviour can presumably fit the P E E K behaviour. In order to operate this combination, the parallel composite model of Voigt was applied in a first approximation [47,48] in which the two phases are supposed to undergo the same strain e~L,--,feq and the overall stress is given by the simple law of mixture: tl

20

6 = A'~o c UNIAXlAL TENSION

==

50

N = I0 I 2 0 /

~z =,

io

,,,

5

0 0.5

0

SIMPLE

,

,

,

,

2

LO

1.5

2.0

2.5

30

5.5

EQUIVALENT STRAIN

Fig. 16. Influence of the number of segments in the subchains, N, on the equivalent stress-strain curves under uniaxial tension and simple shear. (For all the curves, the density of subchains, n, is normalized to n k T = 1.)

200

i

TENSION o n

150

•

PEEK /" = 5.10 "4 s 4 T = 180 "C

03 Lt} LU n,"

F(n

I00

t-

SHEAR

Z hi ,J

> :2)

50

0h i

0

0

I

I

I

0.5

1.0

L5

2.0

EQUIVALENT STRAIN

Fig. 17. Modelling of the stress-strain behaviour of PEEK by means of the VPSC model combined with rubber-like elasticity for the amorphous fraction. The experimental curves are displayed in dotted lines for comparison.

+(1 - X ~ ) o ~'

(7)

where X c is the crystalline volume fraction. From this approximation, the parameters of the generalised model (CRSS, n and N) could be easily adjusted with the experimental curves (Fig. 17). An acceptable fit is found with a single set of parameters for the two loading modes: CRSS = 49 MPa, n = 3.46 x 1027 m 3 and N = 30. Although satisfactory for practical use, the fit of this simplified composite approach should not be considered as a proof of the model's pertinence in all its aspects, since it is based on too many approximations to be rigorous. Among them, the most serious ones are probably: i. the parallel combination, which overestimates the crystalline contribution, ii. the absence of micromechanical interaction between the crystallites and the confined amorphous layers, which is particularly sensitive in the lamella separation process, iii. the freely-jointed assumption for the amorphous chains, which overestimates their compliance, and iv. the ideal cross-link network, which neglects the possible sliding of entanglements. Also, in the combination of the crystalline and amorphous phases, the strain-induced damage of the crystalline lamellae was not taken into consideration. This is a difficult task because the chains unravelled by the crystallite fragmentation increase the amorphous fraction [44]. Therefore, taking this process into consideration would necessitate a dynamic redefinition of the structural model during the course of the simulation. Also, the development of cavities and crazes increases the compliance of the material in a way which is not fully modelled to date. Nevertheless, it is encouraging to note that the introduction of the amorphous phase does improve the original capabilities of the polycrystal deformation model, and it thus can be regarded as a step toward the correct modelling of semi-crystalline polymers.

198

C. G'sell, A. Dahoun

/

Microstructure in semi-crystalline polymers

5. Conclusions Samples of polyethylene and poly(ether ether ketone) were subjected to uniaxial tension and simple shear tests between Tg and Tm at constant true strain rate. The results show that after the yield point, which is correctly predicted by a simple isotropic von Mises criterion, the curves in tension and in shear diverge, with a much lower strain-hardening in shear, and even a true strain-softening in the case of PE. The analysis of the crystalline texture of deformed samples was performed by X-ray diffraction goniometry. It showed in both materials that the c axis of the orthorhombic crystalline cell rotates progressively the plastic deformation toward the stretching axis (in tension) and towards the shear direction (under simple shear). In the case of shear, the (100) plane tends to orient parallel to the shear plane. By means of densitometric and viscoelastic measurements, it appears that the tensile deformation induces significant damage into the structure. By contrast, in shear the cavitation process seems inactive but the conservative processes, such as crystallite fragmentation and chain unravelling, remain active at large strains. Based on a cross-review of the deformation mechanisms, the one-site, viscoplastic self-consistent model of Molinari et al. was applied to the case of PE and PEEK. It involves the activation of eight glide systems in the crystallites, among the chain slip systems and the transverse slip systems. Simulations run with only one adjustable parameter successfully predict the divergence of the stress-strain curves for the two loading modes and the main features of the strain-induced textures. However, it does not take correctly into account the monotonic strain-hardening observed in shear in the case of PEEK. In order to take the amorphous-phase contribution into consideration, the specific Van der Giessen model for rubber-like elasticity was applied to tension and shear strains and combined with the above crystalline model. It involves two parameters: the density, n, of flexible subchains limited by entanglements or by crystallites, and the number of rotating covalent bonds, N, within each subchain. Although this combination was treated in an oversimplified way (Voigt model), the introduction of the amorphous response improved significantly the fit with the experimental curves in both loading modes.

Acknowledgments The authors are very much indebted to Professor A. Molinari (University of Metz) and Dr. G. R. Canova

(CNRS, Grenoble) for adapting the simulation code to the case of polymeric structures and contributing to this study. The authors also express their gratitude to Professors M.J. Philippe and C. Esling (Metz University), who provided the experimental facilities for the X-ray diffraction goniometry.

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52 C. G'sell, S. Boni and S. Shrivastava, J. Mater. Sci., 18(1983) 903. 53 R.A. Horslex and H.A. Nancarrow, Brit. J. Appl. Phys., 2 (1951)345. 54 P.H. Hermans, in (kmtribution to the Physics of Cellulose l:ibetw, Elsevier, Amsterdam, Netherlands, 1946. 55 G. Meincl, N. Morosoff and A. Peterlin, .1. Pol. Sci., A 2 - 8 (1970) 1723. 56 P. Allan and M. Bevis, Philos. Mag., A 41 ( 1980)555. 57 F.J. Baha-Calleja and A. Peterlin, J. Macromol. Sci., B4 (1970)519. 58 B.Z. Jang, D.R. Uhlmann and J.B. Van der Sande, Polym. Eng. &i., 25(1985) 98. 59 C G'sell. N.A. Aly-Helal, L.S. Semiatin and J.J. Jonas, Polymer, 35' (1992) 1244. 60 A. Molinari. G.R. Canova and S. Ahzi, Acta Metall., 35 (1987) 2283. 61 S. Ahzi, A. Molinari and G.R. Canova, in J.P. Boehler (ed.), l'roc. EGF5, Mech. Eng. Publ.. London, UK, 1990. 62 E Wcynant, J.M. Haudin and C. G'sell..I. Mater. Sci., 15 (198(t) 2677. 63 M.C. Wang and E.J. Guth, J. ('hem. Phys., 20 (1952) 1144. 64 A. Souahi, Ph.D. Thesis, lnstitut National Polytechnique de Lorraine, Nancy, France, 1992. 65 C W. Bunn, in R. Hill (ed.), Fibres from ,~vnthetic Polymers, Elsevier, Amsterdam, Netherlands, 1953. 66 R J. Abraham and I.S. Haworth, Polymer, 32 ( 1991 ) 121.