finite element simulation of deformation behavior of semi-crystalline polymers with...

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International Journal of Mechanical Sciences 52 (2010) 158–167

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International Journal of Mechanical Sciences

0020-74

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/ijmecsci

Finite element simulation of deformation behavior of semi-crystallinepolymers with multi-spherulitic mesostructure

Makoto Uchida a,�, Tsuruyo Tokuda b, Naoya Tada b

a Graduate School of Natural Science and Technology, Okayama University, 3-1-1, Tsushimanaka, Kita-ku, Okayama-City, Okayama 700-8530, Japan.b Daihatsu-Techner Co. Ltd., 9-37-2, Kitaitami, Itami-City, Hyogo 664-0831, Japan.

a r t i c l e i n f o

Article history:

Received 31 July 2009

Received in revised form

28 August 2009

Accepted 1 September 2009Available online 8 September 2009

Keywords:

Spherulite

Semi-crystalline polymer

Mesostructure

Multi-scale

Homogenization

FEM

03/$ - see front matter & 2009 Elsevier Ltd. A

016/j.ijmecsci.2009.09.002

esponding author. Tel./fax: +8186 2518031.

ail address: [email protected] (M

a b s t r a c t

Elasto-viscoplastic deformation behavior of semi-crystalline polymer with multi-spherulitic mesos-

tructure was investigated using large deformation finite element homogenization method. Multi-

spherulitic mesostructure was modeled in representation volume element (RVE) by Voronoi polygons,

and the deformation behavior during tension in different directions was numerically investigated.

Mesoscopic stress strain relations were almost similar for different mesoscopic structures except for a

quarter spherulite mesoscopic models, and its anisotropy decreased with increase in the number of

spherulites in RVE. Although anisotropy in mesoscopic stress strain relation disappeared for enough

number of spherulites, highly heterogeneous distribution of local strain and stress was observed in each

spherulite. Distribution of lamellar orientation in the spherulite caused the heterogeneity and played an

important role in the local deformation field.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Understanding the mechanical behavior of semi-crystallinepolymers is important to expand their use as structural materials.Multi-scale mechanical modeling based on structures in differenthierarchical scales is necessary because semi-crystalline polymershave complex hierarchical structures. The microscopic scale ofsemi-crystalline polymers consists of crystalline lamellae andamorphous phases. Molecular chains orient in a specific directionin the crystalline phase but they are distributed in randomdirections in the amorphous phase. Furthermore, spherulite isformed by radial arrangement of crystalline lamellae in meso-scopic scale of semi-crystalline polymer [1].

Similar to anisotropic deformation characteristic of metalsdepending on crystal orientation, deformation in spherulite ofsemi-crystalline polymer strongly depends on the lamellarorientation. Radial arrangement of lamellae in spherulite bringsabout non-uniform deformation behavior even under uniaxialloading condition, which is experimentally evaluated by a lot ofresearches [2–5]. Anisotropic plastic deformation is caused by theslip deformation of crystalline phase along the molecular chainand the deformation of amorphous phase between crystallinelamellae. These mechanisms need to be introduced into the

ll rights reserved.

. Uchida).

mechanical model of semi-crystalline polymers in order to predictnot only the macroscopic mechanical response but also theevolution of microscopic structural change with increase in plasticdeformation.

Micromechanical models of semi-crystalline polymers weredeveloped by Nikolov et al. [6] and van Dommelen et al. [7] basedon the inclusion model [8]. Effect of deformation rate andcrystallinity on the macroscopic mechanical response waspredicted by computational simulation. In these studies, Taylor,Sachs or their hybrid models were used to represent themechanical response of aggregation of randomly oriented crystal-line lamellae and amorphous layer. However, these models did notconsider the geometry of spherulite.

Deformation behavior of single spherulite was numericallyinvestigated by using microscopic to mesoscopic mechanical modelbased on the large deformation finite element homogenizationmodel [9] proposed by Tomita and Uchida [10,11]. In these studies,microstructural unit cells consisting of crystalline and amorphousphases were given for gauss integration point on finite elementin mesostructure with specific direction according to the loca-tion in spherulite. Although highly heterogeneous deformationbehavior in spherulite is evaluated by these models, the effects ofshape and size distribution of spherulite were not discussedbecause the model did not have boundaries of spherulites. Sincemesoscopic morphology of spherulite can be controlled by specialcrystallization process or heat treatment [12,13], it is desirable toknow the effect of the shape and size of spherulites.

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M. Uchida et al. / International Journal of Mechanical Sciences 52 (2010) 158–167 159

In this paper, the effect of distribution of spherulites on thedeformation behavior of semi-crystalline polymer is investigatedby using multi-spherulitic mesostructure represented by Voronoipolygons. Mesoscopic response of semi-crystalline polymer andstrain/stress fields in spherulite is evaluated by numericalsimulation of uniaxial tensile deformation. Then, the effect ofspherulite geometry on local deformation is discussed consideringthe distribution of lamellar orientation.

2. Constitutive equation

In order to describe the deformation behavior of semi-crystal-line polymer, the crystalline plasticity theory with the penaltymethod [10,11] and the nonaffine molecular chain network theory[14] were employed for crystalline and amorphous phases,respectively. Hereafter, the indices ‘‘C’’ and ‘‘A’’ are used torepresent the quantities for crystalline and amorphous phases,respectively.

The plastic strain rate dpij in the crystalline phase is modeled

using the crystalline plasticity theory [15] with shear strain rate_gðaÞpC on the ath slip system expressed by a power law [16], as

dpij ¼

XðaÞ

PðaÞij_gðaÞpC ; ð1Þ

_gðaÞpC ¼_g0C

tðaÞ

gðaÞtðaÞ

gðaÞ

��ð1=mC Þ�1

; ð2Þ

where _g0C is the reference strain rate in the crystalline phase, mC

the strain rate sensitivity exponent, g(a) the resistance to slip,t(a)=Pij

(a)sij the resolved shear stress, sij the Cauchy stress,PðaÞij ¼

�sðaÞi mðaÞj þmðaÞi sðaÞj

�=2 the Schmidt tensor, and sðaÞi and mðaÞi

are unit vectors along the slip direction and the slip plane normal,respectively. Here, the penalty method is employed to approxi-mately satisfy the inextensibility of the chain direction. Thecorresponding constitutive equation of the crystalline phase isexpressed as [10,11]

Sr

ij ¼ ðDCijklþl0cicjckclÞdkl �

XðaÞ

RðaÞij_gðaÞpC ; ð3Þ

RðaÞij ¼DCijkl PðaÞkl þðW

ðaÞik skjþsik W ðaÞ

kj Þ_gðaÞpC ; ð4Þ

W ðaÞij ¼

1

2ðsðaÞi mðaÞj �mðaÞi sðaÞj Þ; ð5Þ

where Sr

ij is the Jaumann rate of Kirchhoff stress, DCijkl the

anisotropic elastic modulus tensor, dij the total strain rate tensor,l0 the penalty constant, which has a large value, and physically, it

RepresentativeVolume Element

Center ofspherulite

(x2

x3

Fig. 1. Computational model of FE homogenization method. (a) Periodic

represents the chain directional stiffness, and ci the unit vector ofchain direction.

Subsequently, the constitutive equation for the amorphousphase [14] with plastic shear strain rate _gpA [17] is expressed as

Sr

ij ¼DAijkldkl � _gpA

2Gffiffiffi2p t�ðs0ij � BijÞ; ð6Þ

_gpA ¼ _g0A exp �A~s

T1�

t�~s

� �5=6( )" #

; ð7Þ

where DAijkl is isotropic elastic modulus tensor, _g0A and A are

constant, ~s ¼ s0þapP, s0 is the initial shear strength [18], ap

pressure coefficient, p the pressure, T the absolute temperature,and t* is the applied shear stress. Furthermore, Bij in Eq. (6) is theback-stress tensor and the principal components bi are expressedby employing the eight-chain model [18] as

bi ¼1

3CR

ffiffiffiffiNp V2

i � l2

lL�1 lffiffiffiffi

Np

� �; ð8Þ

where l2¼ ðV2

1 þV22 þV2

3 Þ=3, Vi is the principal plastic stretch, N

the average number of segments in a single chain, CR=nkBT aconstant, n the number of chains per unit volume, kB Boltzmann’sconstant, and LðxÞ ¼ cothðxÞ � 1=x the Langevin function. In thenonaffine eight-chain model [14], the number of entangled pointsN, in other words, the average number of segments, may changedepending on the distortion x, which represents the localdeformation of a polymeric material. The simplest expression ofthe number of entangled points is N=N0 exp{c(1�x)} with x=1 inthe reference state, and N0 is the number of segments in a singlechain in the reference state and c a material constant.

3. Computational model

Fig. 1 shows the microscopic to mesoscopic computationalmodel of multi-spherulitic structure of semi-crystalline polymerfor finite element homogenization method. The material wascomposed of representative volume element (RVE) ofmesostructure shown in Fig. 1(a) where mesoscopic coordinatesystem xi was taken to be parallel to the side of RVE at themesoscopic scale as shown in Fig. 1(b). Several numbers ofspherulites were given in RVE with straight boundaries to the nextspherulites.

A microstructure consisting of crystalline and amorphousphases with microscopic coordinate system yi was given in allGauss integration points of finite elements in mesoscopic RVE asshown in Fig. 1(c). Since the effect of heterogeneous deformationin each phase is not considered in the present study, number ofelement for each phase was one. Periodical allocation of micro-

Lamellar growthb-axis) direction

x1

y3y1 y2

Molecularchain

Crystallinephase Amorphous

phase

ab

c

al structure. (b) Mesostructure in RVE. (c) Microstructural unit cell.

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M. Uchida et al. / International Journal of Mechanical Sciences 52 (2010) 158–167160

structural unit cell forms microscopic laminated structure ofcrystalline and amorphous phases where y1 and y2 axes are ontheir interface plane and y3-axis is normal to the interface. On theother hand, a, b, and c are the bases of crystalline lattice incrystalline phase. b-axis corresponds to the growth directionof crystalline lamella and is parallel to y2-axis. However, directionof c-axis, which represents molecular chain direction, is not

Model A

Model DModel C

Model B

Fig. 2. Finite element mesh

Table 1Slip systems and normalized resistances of HDPE [8].

Type Chain slip

Indices (10 0)[0 0 1] (0 10)[0 0 1] {10 0}[0 0 1

g(a)/g0 1.0 2.5 2.5

Fig. 3. Relationships between mesoscopic true stress and strain for tensions in

parallel to the normal of interface. The angle between themis 301 [19].

Six kinds of RVE with different distribution of spherulitesshown in Fig. 2 were used. The number of spherulites in RVE forModels A to E were 1/4, 3, 4, 5, and 11, respectively. Model A hasno spherulite boundary inside and four neighboring RVEs make awhole spherulite. Spherulites in Models B to E are Voronoi

Model E

Model BP

division for each model.

Transverse slip

] (10 0)[0 10] (0 10)[10 0] {110}/11̄0S1.66 2.5 2.2

x1 and x2 directions. (a) Tension in x1 direction (b) Tension in x2 direction.

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polygons whose boundaries are perpendicular bisectors of the lineconnecting centers of neighboring spherulites. In order to evaluatethe effect of roughness or jaggy of spherulite boundary on thedeformation behavior, Model BP was also used. Lamellarorientation in each element was determined based on therelative location of center of element with respect to the centerof spherulite, and is indicated by short lines in Figs. 1 and 2.Uniaxial tensile deformation in the x1 or x2 direction was assignedwith macroscopic strain rate of _E0 ¼ 10�5. In order to representthe periodic deformation of RVE, uniform displacement in normal

Fig. 4. Relationships between mesoscopic true stress and strain f

Fig. 5. Relationships between spherulite-averaged equivalent stress a

direction to the load, which is obtained by additional boundarycondition

Pf ðiÞt ¼ 0, was applied. Here, f ðiÞt means nodal force in

normal direction to the load for the ith nodal point on the rightend boundary of RVE. This boundary condition indicates that foursymmetric RVEs correspond to the unit composition in periodicmesostrucutre.

The material parameters at T=256 K were specified fromreferences [7,8,14]. For the amorphous phase, EA/s0=23.7, As0/T=3.64, ap=0.01, _g0A =1�10–3, s0=4.4 MPa, CR/s0=0.36,

ffiffiffiffiffiffiN0

p¼ 7,

and c=0.1, and for the crystalline phase, _g0A =1�10–3, g0=8.0 MPa,

or each model. (a) Model B and BP (b) Model D (c) Model E.

nd strain. (a) Tension in x1 direction (b) Tension in x2 direction.

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and 1/mC=9. As for the components of anisotropic elastic modulusDC

aaaa=g0 =0.88�103, DCcccc=g0 =10.13�103, DC

aabb=g0 =0.48�103,DC

aacc=g0 =0.59�103, DCabab=g0 =0.19�103, and DC

acac=g0 =0.20�103,where the subscripts a, b, and c for anisotropic elastic moduluscorrespond to the axes of the HDPE crystalline lattice, respectively.The slip system and normalized resistance of HDPE were shown inTable 1. Furthermore, to reproduce sufficient inextensibility and tocarry out stable calculation, a penalty constant l0=106 wasemployed.

For the representation of stress state and deformation,equivalent stress, seq, and equivalent strain, eeq, were introducedfor each finite element as

s2eq ¼

3

2s0ij s0ij; ð9Þ

e2eq ¼

2

3

Zdij dij dt; ð10Þ

Fig. 6. Equivalent strain distribution in tension in x1 direct

where s0ij=(sij�skkdij/3) is deviatoric part of Cauchy stress and t

the time.

4. Result and discussion

Fig. 3 shows the mesoscopic responses under uniaxial tensionin the (a) x1 and (b) x2 directions. In this paper, the relationshipbetween true stress and strain for RVE is called a mesoscopicresponse. Mesoscopic response showed nonlinear behaviordepending on the deformation behavior of amorphous phaseand rotation of lamellae. Only the response in Model A, which hadno spherulite boundary inside the RVE showed different curvesfrom those in the other models in the earlier stage of deformation.This result indicates that a quarter spherulite RVE model is notenough to capture the generalized mesoscopic response of thematerial.

ion. (a) Model A (b) Model B (c) Model D (d) Model E.

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In order to investigate the isotropy of semi-crystalline polymeron the mesoscopic scale, true stress and strain relations of RVEwere compared between tension in the x1 and x2 directions. Theresults are shown in Fig. 4. The difference of mesoscopic responsesbetween two orthogonal directions was smaller for the RVE withincrease in the number of spherulites. Eleven spherulites wereenough to show isotropic mesoscopic response in this case. On the

Fig. 7. Equivalent strain distribution in tension in x2 direct

other hand, comparing the results for Models B and BP shown inFig. 4(a), the difference of mesoscopic responses was small. Thisfact suggests that the effect of roughness of spherulite boundaryline on the mesoscopic response is small.

In the theoretical modeling of mechanical behavior of poly-crystalline metal, Taylor and Sachs models are often used. Here,the mechanical response for each spherulite was evaluated to


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investigate which model is appropriate for the semi-crystallinepolymer. Fig. 5 shows relationships between spherulite-averagedequivalent stress ~seq and strain ~eeq for Models B to E in tension in(a) x1 and (b) x2 directions. Here, ~seq and ~eeq were area-averagedequivalent stress and strain for each spherulite. In order toevaluate the effect of spherulite shape on the spherulite response,~seq � ~eeq relationship is shown by different range of aspect ratio ofspherulite which is defined as the ratio of the maximum length intensile direction to that in transverse direction to tensile direction.The spherulite-averaged mechanical response deviated dependingon the spherulite in the RVE. The deviation tended to be largerwhen spherulites with different range of the aspect ratio existedin the RVE. Especially, spherulite with larger aspect ratio exhibitedhigher equivalent stress. A standard deviation at the end of thedeformation for Model E was about 2.22 MPa for equivalent stressand 0.057 for equivalent strain, which corresponded to 21.8% and

Fig. 8. Equivalent stress distribution in tension in x1 direct

22.8% of RVE-averaged values of those, respectively. From thisresult of similar degree of deviations in both of spherulite-averaged equivalent stress and strain, it is concluded thataccuracies of mechanical response predicted by Taylor and Sachsmodels are almost similar for the case of semi-crystallinepolymer.

Figs. 6 and 7 show the equivalent strain distribution in tensionin x1 and x2 directions, respectively. At the beginning of the tensiledeformation, localized strain caused at the center of spherulitewhere lamellae in various directions are concentrated. Then,locally deformed region expanded along the oblique directions tothe loading axis. At large strain, the deformed region in spheruliteexpanded to connect with each other across the boundaries.

Figs. 8 and 9 show the equivalent stress distribution in tensionin x1 and x2 directions. Equivalent stress locally concentratedat the central region of spherulite as similar to the strain


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distribution. Different point was that stress concentrating regiontended to expand to the parallel and orthogonal directions to theloading axis. With further deformation, region parallel to theloading axis in spherulite tended to show larger stress. It isconcluded from a series of results of equivalent strain and stressdistributions shown in Figs. 6–9 that heterogeneous deformationin spherulite depends on the radial distribution of lamellae andneighboring spherulites. Although no results are shown for ModelBP, it was confirmed that the roughness or jaggy of spherulite

Fig. 9. Equivalent stress distribution in tension in x2 direct

boundary had a very small effect on the distributions of equivalentstress and strain.

Stress/strain fields were qualitatively characterized from Figs. 6–9. However, it is difficult to capture the quantitative informa-tion from them. As mentioned above, distribution of lamellarorientation in spherulite is a key parameter to characterize thelocal deformation in spherulite and it varies depending on theshape of spherulite. Figs. 10 and 11 show the polar graphs ofdistribution of area-weighted equivalent strain eyeqAy=A and (b)


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stress syeqAy=A against the lamellar direction yl in Model E at

different degree of tensile deformation in x1 and x2 direction,respectively. eyeq and sy

eq (are the average values of eeq and seq insectorial area with lamellar direction yl. yl was calculated frominner product of lamellar growth direction b-axis and unit vectorin loading direction. Ay is the area of each sectorial area andthe central angle of each area, Dyl, is 10 deg. Furthermore, eyeqAy

and syeqAy were normalized by total area of RVE, A, which enables

the comparison of magnitude of strain and stress in allspherulites.

Distribution of area-weighted equivalent strain and stress foreach lamellar direction was quite different depending on thespherulite. Although spherulites near the boundary of RVE had acommon straight spherulite boundary, there was a tendency inthe distribution of equivalent strain that the area-weighted strainshows high values in the longest direction of spherulite. Withincrease in the strain E, the angle showing higher strain rotated tothe tensile direction with the rotation of lamellae. Area-weightedequivalent stress also tended to show higher value in thelongest direction of spherulite. However, when the longestdirection of spherulite was close to the loading direction,degree of stress became considerably large compared to otherspherulite.

Series of computational results revealed that local deformationin each spherulite depends on its shape and interacted by thesurrounding spherulites. In general, concentration of stressand strain or fracture of material is strongly affected by theheterogeneity in local deformation. The present simulationwith multi-spherulitic mesostructure is useful for thatpurpose.

Fig. 10. Area weighted equivalent strain and stress for each lamellar direction in tensio

equivalent stress.

5. Conclusion

Elasto-viscoplastic deformation behavior of semi-crystallinepolymer with multi-spherulitic mesostructure was investigatedby using large deformation finite element homogenizationmethod. Five different patterns of spherulitic structure in RVEwere configured based on the voronoi polygons and tensiledeformation behaviors in two different direction were numeri-cally investigated.

Mesoscopic responses of different patterns of multi-spheruliticmesostructures except for the case with periodic structure ofsquare spherulite showed almost similar behaviors independenton the number of spherulites in the RVE. Although slightdifference in the degree of stress levels between tensions indifferent directions were confirmed, it decreased with theincrease in the number of spherulites in RVE. The spherulite-averaged mechanical response deviated depending on the spher-ulite and similar degrees of deviation of stress and strain indicatedthat neither Taylor nor Sachs models are appropriate to predictthe mechanical response of semi-crystalline polymer.

At the beginning of the deformation, localized strain and stresscaused at the center of the spherulite. Then, locally deformedregion expanded along the oblique directions to the loading axiswhile region parallel and orthogonal direction to the load tends toshow larger stress. The strain and stress distributions were clearlycharacterized by distribution of lamellar orientation in thespherulite. Since distribution of lamellar orientation in spherulitecorresponds to the shape of spherulite, spherulite boundary shapeis useful information to predict the local deformation in thespherulite of semi-crystalline polymer.

n in x1 direction of Model E. (a) Area-weighted equivalent strain (b) Area-weighted

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Fig. 11. Area-weighted equivalent strain and stress for each lamellar direction in tension in x2 direction of Model E. (a) Area-weighted equivalent strain (b) Area-weighted

equivalent stress.


References

[1] Lin L, Argon AS. Structure and plastic deformation of polyethylene. J Mater Sci1994;29(2):294–323.

[2] Boni S, G’Sell C, Weynant E, Haudin JM. Microscopic in situ observation of theplastic deformation of polybutene-1 films under simple shear. Polym Test1982;3(1):3–24.

[3] Aboulfaraj M, G’Sell C, Ulrich B, Dahoun A. In situ observation of the plasticdeformation of polypropylene spherulites under uniaxial tension and simpleshear in the scanning electron microscope. Polymer 1995;36(4):731–42.

[4] Butler MF, Donald AM. Deformation of spherulitic polyethylene thin films. JMater Sci 1997;32(14):3675–85.

[5] Lee SY, Bassett DC, Olley OH. Lamellar deformation and its variation in drawnisolated polyethylene spherulites. Polymer 2003;44(19):5961–7.

[6] Nikolov S, Doghri I, Pierard O, Zealouk L, Goldberg A. Multi-scale constitutivemodeling of the small deformations of semi-crystalline polymers. J Mech PhysSolids 2002;50(11):2275–302.

[7] van Dommelen JAW, Parks DM, Boyce MC, Brekelmans WAM, Baaijens FPT.Micromechanical modeling of the elasto-viscoplastic behavior of semi-crystalline polymers. J Mech Phys Solids 2003;51(3):519–41.

[8] Lee BJ, Parks DM, Ahzi S. Micromechanical modeling of large plasticdeformation and texture evolution in semi-crystalline polymers. J Mech PhysSolids 1993;41(10):1651–87.

[9] Higa Y, Tomita Y. Computational prediction of mechanical properties ofnickel-based superalloy with gamma prime phase precipitates. In: Progress inmechanical behaviour of materials: ICM8, vol. III. Fleming Printing Ltd.; 1999.p. 1095–9.

[10] Tomita Y, Uchida M. Computational characterization of micro- to mesoscopicdeformation behavior of semicrystalline polymers. Int J Mech Sci 2005;47(4–5):687–700.

[11] Tomita Y, Uchida M. Computational characterization and evaluation ofdeformation behavior of spherulite of high density polyethylene in mesoscaledomain. CMES 2005;10(3):239–48.

[12] Chai CK, Auzoux Q, Randrianatoandro H, Navard P, Haudin JM. Influence ofpre-shearing on the crystallisation of conventional and metallocene poly-ethylenes. Polymer 2003;44(3):773–82.

[13] Jia J, Mao W, Raabe D. Changes of crystallinity and spherulite morphology inisotactic polypropylene after rolling and heat treatment. J Univ Sci TechnolBeijing 2008;15(6):514–20.

[14] Tomita Y, Adachi T, Tanaka S. Modeling and application of constitutiveequation for glassy polymer based on nonaffine network theory. Eur J Mech A/Solids 1997;16(5):745–55.

[15] Peirce D, Asaro RJ, Needleman A. Material rate dependence andlocalized deformation in crystalline solids. Acta Metall 1983;31(12):1951–1976.

[16] Hutchinson JW. Bounds and self-consistent estimates for creep of polycrystal-line materials. Proc R Soc London A 1976;348(1652):101–27.

[17] Argon AS. A theory for low-temperature plastic deformation of glassypolymers. Philos Mag 1973;28(4):839–65.

[18] Boyce MC, Parks DM, Argon AS. Large inelastic deformation of glassypolymers. Part I: rate dependent constitutive model. Mech Mater1988;7(1):15–33.

[19] Bassett DC, Hodge AM. On the morphology of melt-crystallizedpolyethylene, I. Lamellar profiles. Proc R Soc London A 1981;377(1768):25–37.

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