Modeling of large plastic deformation in crystalline polymers

Journal of the Mechanics and Physics of Solids49 (2001) 2719–2736

www.elsevier.com/locate/jmps

Modeling of large plastic deformationin crystalline polymers

Wei Yanga ; ∗, Ming-Xiang Chenb

aDepartment of Engineering Mechanics, Tsinghua University, Beijing 100084,People’s Republic of China

bSchool of Civil Engineering, Wuhan University of Hydraulic and Electric Engineering, Wuhan 430072,People’s Republic of China

Abstract

When a crystalline polymer grain undergoes plastic deformation, four independent slip systemsmay operate with the chain direction inextensible. The inextensibility leads to texture evolutionfor a polycrystalline polymer under large plastic deformation. This paper develops a microme-chanics model for the three-dimensional large plastic deformation of crystalline polymers. A con-tinuous orientation distribution function is introduced to characterize the distribution of the chainaxes. Restrictions on the orientation distribution imposed by material symmetry are considered.A micro–macro interaction law for deformation accounting for molecular chain inextensibilityis proposed by employing tensor function representation theorems. By the equivalence betweenthe microscopic and macroscopic work rate, an expression for the macroscopic stress is ob-tained. The evolution for the orientation distribution is quanti6ed. The proposed model enablesus to simulate not only the rotating process of chain axes toward the direction of the maxi-mum stretch but also the texturing hardening under axi-symmetric straining and simple shear.? 2001 Elsevier Science Ltd. All rights reserved.

Keywords: A. Microstructure; B. Constitutive behavior; B. Polymeric material; B. Finite strain;Tensor representation

1. Introduction

The plastic deformation of semicrystalline polymers involves several deformationmechanisms of crystalline and amorphous phases (Lin and Argon, 1994). Most of thesemechanisms are crystallographic in nature, similar to those found in the plastic deforma-tion of metallic single crystals. This is particularly true for the semicrystalline polymers

∗ Corresponding author. Tel.: +86-10-6278-2642; fax: +86-10-6256-2768.E-mail address: [email protected] (W. Yang).

0022-5096/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved.PII: S0 0 2 2 -5 0 9 6 (01)00076 -X

2720 W. Yang, M.-X. Chen / J. Mech. Phys. Solids 49 (2001) 2719–2736

with high crystallinity, such as high-density polyethylene (HDPE) whose crystallinityis typically between 70% and 80%. Among these mechanisms, the crystallographic slipmodes both in the chain direction and transversal to the chain are the most important.This paper focuses on the plastic deformation mechanism of semicrystalline polymersdue to the crystallographic slip. Besides the crystallographic slip, deformation mecha-nisms acting in the amorphous layers between lamellae, such as interlamellar sliding,are involved. They play a role mainly in the early phase of the deformation process,Bartczak et al. (1992a). The role played by the amorphous phase will be taken intoaccount through its inEuence on the crystallographic slip.

Though slip may occur in the chain direction, the chain direction itself is rigid dueto the stiF covalent bonding. The crystallographic slip provides only four independentslip systems with the chain direction of the crystal inextensible. During large plasticdeformation, the crystallographic chain axes tend to align preferentially with the direc-tion of the maximum stretch. The preferential orientation induces high anisotropy of themacroscopic mechanical behavior in the materials. For instance, an upturn hardeningis observed in tension but not in simple shear of polyethylene (see G’Sell and Jonas,1979).

In simulating both the macroscopic constitutive behavior of polycrystalline aggregateand the evolution of crystallographic texture, the Taylor assumption, which imposes thesame deformation on each grain, is often employed. However, the Taylor assumption isclearly unsuitable for crystalline polymers, since the inextensible constraint in the chaindirection renders the crystal incapable of accommodating an arbitrary deformation. Oth-erwise, the estimate of aggregate stress generally becomes unbounded. Parks and Ahzi(1990) developed a physically based micro-mechanical model for large plastic defor-mation of crystalline polymers. They explicitly accounted for the kinematics due to theinextensibility of molecular chains by introducing an intermediate traceless deformationrate in the micro–macro interaction. In the spirit of Parks and Ahzi (1990), Chen et al.(1995) developed a model for planar crystalline polymers based on a continuous ori-entation distribution function (ODF). The predicted upturn in the stress–strain curveby texture rotation, however, is stiFer than the actual crystalline polymers. Such a de-6ciency was removed in another work of Chen et al. (1996) by accounting for theheterogeneous spin of crystals induced by the amorphous phase within the aggregate.In the above mentioned works of Chen et al., the material was treated as planar poly-crystalline in which the chain axes lie in the common plane of deformation and the slipsystem of each grain was represented by a single slip system aligned with the chainaxis. The planar polycrystalline assumption can be real in polymer 6lms, but cannotbe realized in bulk polymers. It is, therefore, necessary to develop a model to accountfor three-dimensional chain orientation distribution.

This paper extends the planar model to the three-dimensional case, beginning withthree-dimensional description of the constrained single crystals in the next section. InSection 3, de6nition of the orientation distribution of the crystals and the restriction onthe orientation distribution imposed by material symmetry is given. In Section 4, anaveraging scheme within the aggregate is given and the micro–macro transition laws fordeformation and stress are established. In Section 5, the evolution of ODF is quanti6ed.The model is used to predict the texture in Section 6 under axi-symmetric deformation

W. Yang, M.-X. Chen / J. Mech. Phys. Solids 49 (2001) 2719–2736 2721

and simple shear. Comparison between the predicted stress–strain curves and the ex-perimental data in axi-symmetric tension and simple shear supports the present theory.

The following tensor notations are used. The tensor product is indicated by ⊗. LetR be a tensor whose order is not lower than another tensor T . The complete con-traction R •T is de6ned by (R •T)�:::� = R�:::��:::�T�:::�. The symbol operations apply:(AB)ij =AikBkj. The pre6x tr indicates the trace, a superposed dot refers to the materialtime derivative or rate, and the superscript ′ denotes the irreducible part of a tensor.

2. Constrained single crystals

The polymeric crystals share a feature uncommon to the metal grains: the chaindirection is inextensional. This feature leads to the constrained deformation and thetexture development. We explore the basic characteristics of constrained single crystalsin this section, together with a brief list of the material parameters involved.

2.1. Kinematics

Consider the plastic deformation due to crystallographic slip. The crystal lattice ofcrystalline polymers such as HDPE is orthorhombic with the c-axis in the chain di-rection. Here we single out the kinematics feature associated with the c-axis, whilethe diFerence in the a-axis and the b-axis for HDPE is averaged out by transverserandomization. The slip planes are always parallel to the c-axis. A slip direction canbe either parallel to the c-axis (chain slip) or perpendicular to the c-axis (transverseslip). Details on the slip systems and their plastic resistance of crystalline polymerscan be found in Bartczak et al. (1992b). Chain slip and transverse slip provide fourindependent slip systems.

Consider a crystal possessing K slip systems. Let s� and n� be unit vectors repre-senting the slip direction and the normal to the slip plane of the �th slip system, and�� the shear rate of that slip system. The symbols D and W designate the symmetricand skew parts of the velocity gradient, respectively. At the large strain regime, theelastic deformation is negligible when compared to the plastic deformation. Then, thedeformation rate D generated by all slip systems is expressed by

D ≈ Dp =K∑�=1

12��(s� ⊗ n� + n� ⊗ s�) =

K∑�=1

��R�; (1)

where R� is the symmetric part of the Schmid tensor s� ⊗ n� associated with the slipsystem �.

The material spin W in the crystal can be additively decomposed as the sum of theplastic spin W p and the lattice spin W 1 (see Asaro and Rice, 1977)

W =W 1 +W p =W 1 +K∑�=1

��A�; (2)

where A� is the skew part of the Schmid tensor associated with the slip system �.


Five independent slip systems are needed to accomodate an arbitrary incompressibledeformation. However, only four independent slip systems are available in crystallinepolymers. Each slip system, with slip planes penciled by the chain direction, allowsno stretching in the chain direction. One may express the constraint condition on thelocal deformation as (Parks and Ahzi, 1990)

C ′ •R� = 0 or C ′ •D = 0; (3)

where C ′ = c ⊗ c − (1=3)1 is the deviatoric part of the dyadic c ⊗ c, with c being theunit vector in the chain direction, and 1 the identity tensor of the second rank.

2.2. Constitutive equations

The constitutive description of a single crystal consists of a relation between the shearrate �� and the resolved shear stress ��. They were usually assumed (see Hutchinson,1976; Parks and Ahzi, 1990) to be related through a power law of the form

�� = �0��

g�

∣∣∣∣ ��g�∣∣∣∣n−1

; (4)

where the material parameters n, �0 and g� are the rate exponent, the reference shearrate and the shear strength associated with the �th slip system, respectively.

The local Cauchy stress deviator S in the crystal can be decomposed uniquely intothe direct sum of a stress tensor S∗ normal to C ′ and a component SC aligned withC ′

S = S∗ + SCC ′; (5)

where SC = (3=2)tr SC ′. Using the constraint condition (3) and decomposition (5), onecan show that the resolved shear stress �� for each slip system is independent of thecomponent SC (Parks and Ahzi, 1990). Then the constitutive equation for a constrainedsingle crystal can be expressed as

D =

�0

K∑�=1

1g�

∣∣∣∣S∗ •R�

g�

∣∣∣∣n−1

R� ⊗ R�

•S∗: (6)

As a result of the constraint condition (3) and the normality Eow rule (4), the defor-mation rate D is independent of SC. The undetermined reaction stress component SC

induced by the inextensible constraint (3) should be determined from the equilibrium.

3. Orientation distribution

Consider a representative element of polycrystalline aggregate that contains many in-dividual grains whose mechanical behavior is characterized by the relations describedin Section 2. Among the microstructure of the aggregate such as grain shape, grainsize, grain boundaries, and the chain orientation within a grain, the latter has a dom-inant eFect on the macroscopic behavior of the polymeric material. Accordingly, the


Fig. 1. Chain orientation in a single crystal.

present work characterizes the microstructure by the orientation distribution of chainaxes within the aggregate.

The orientation of the chain axis of a single crystal is described by a unit vectorc=cos � sin�e1 +sin � sin�e2 +cos�e3 in the sphere as depicted in Fig. 1. Consider anin6nitesimal neighborhood N (c) of the orientation c in the unit sphere. Let dV (N (c))be the volume of the crystals whose chains axes lie in N (c) and V be the totalvolume of the aggregate. An orientation distribution function (ODF), �(c), is de6nedas follows:

�(c) dA = dV (N (c))=V; (7)

where dA denotes an area element on the surface of the unit sphere, i.e., dA =sin � d� d�: The chain axis of a crystal grain changes its orientation under large plasticdeformation. Consequently, the ODF evolves. In view of de6nition (7), the continuityand the conservation on the orientation distribution requires

�(c) dA = �0(c0) dA0; (8)

where the subscript 0 labels the reference state.As shown by Zheng and Collins (1998), also see Kanatani (1982), the ODF can be

expanded in a Fourier series with coeOcients as follows:

�(c) =1

4�

∞∑n=1

(4n + 1)!

4n[(2n)!]2 a′2n •c⊗2n; (9)

where c⊗2n is the abbreviation of the 2nth tensor product of c, e.g., c⊗2=c ⊗ c; c⊗4=c⊗ c ⊗ c ⊗ c, and so on. The symbol a′2n denotes the irreducible part of the momenttensor, and is de6ned by the orientation average of c⊗2n over the aggregate

a′2n =∮c⊗2n�(c) dA = 〈c⊗2n〉; (10)

where cusp brackets 〈〉 denote the average over orientations. The reader can refer toAppendix A for detailed representation of a′2n in terms of a2n.


Undeformed melt-crystallized semicrystalline polymers often show a spherulitic struc-ture that consists of a radial arrangement of broad thin crystalline lamellae separated byamorphous layers. All possible orientations of chain axes occur with equal frequency.The corresponding ODF is isotropic. The material thus behaves as globally isotropic.Under large deformation, molecular chain axes align preferentially with the directionof the maximum stretch, and the spherulitic region becomes textured. The orientedspherulites, as well as the corresponding orientation distribution, are assumed to pos-sess a global orthotropy. By the assumption, the ODF is an orthotropic scalar-valuedfunction of the c-vector. Let bi (i = 1; 2; 3) denote three unit vectors along the or-thotropic axis. The structure tensor which characterizes the orthotropy can be formedas (Zheng, 1994)

M = b1 ⊗ b1 − b2 ⊗ b2: (11)

Then, the ODF is an isotropic scalar-valued function of c and M by the isotropiciza-tion theorem (Boehler, 1978; Zheng, 1994). Furthermore, the moment tensors a′2n, asaverage quantities over c, are isotropic functions of M . As shown in Appendix A, thereare n + 1 independent components �(2n)0; �(2n)(2r) (r = 1; 2; : : : ; n) in the 2nth momenttensors a′2n. Then the general representation of the ODF is

� = �(C ′;M ; �20; �22; : : : ; �(2n)(2r); : : :): (12)

4. Micro–macro transition

4.1. Averaging scheme

Consider an aggregate described at the beginning of Section 2. Within it, the micro-scopic variables, such as deformation and stress, of each grain are generally non-uniform.In order to determine the macroscopic behavior of the aggregate, an averaging schememust be speci6ed to de6ne the macroscopic variables according to the microscopic vari-ables. On the other hand, a transition law must be formulated to relate the microscopicvariables to the macroscopically imposed variables.

In a volume-average polycrystalline assembly, the overall quantities are evaluatedas the volume averages of the corresponding grain quantities (Hill, 1972). Accordingto the description in Section 3, the microscopic variables mainly depend on the grainorientation and the orientation distribution of the aggregate. It is, therefore, assumedthat the grains with the same orientation have the same microscopic deformation andstress. Then, with de6nition (7), the volume average can be replaced by the averageweighted by the orientation distribution. The global compatibility within the aggregatecan be written in a self-consistent manner as

〈D〉 = SD and 〈W〉 = SW ; (13)

where SD and SW are the macroscopic deformation rate and macroscopic spin, respec-tively. It states that the orientation average of microscopic deformation should give themacroscopic one.


4.2. Transition law for the deformation rate

The microscopic deformation rate D of a single crystal within the aggregate de-pends on the macroscopic deformation SD and SW , the chain axis orientation C ′ of thecrystal and the orientation distribution �(c) of the aggregate. In view of (12), D is atensor-valued function of SD; SW ; C ′;M and the components �20; �22; �40; �42; : : : . TheNeuman principle (Neuman, 1885) imposes that this tensor-valued function must beisotropic if the material is initially isotropic in the macroscopic sense. Without loss ofgenerality, one can write D as follows:

D = D( SD; SW ;M ;C ′; �20; �22; �40; �42; : : :) + �C ′; (14)

where D is an isotropic tensor-valued function. Since D is traceless by incompress-ibility and C ′ is traceless by de6nition, one concludes that D is traceless by Eq. (14).Multiplying both sides of Eq. (14) by C ′ and using the constraint conditions (3), oneobtains the parameter � as −( 3

2 )trC ′D. Inserting this into Eq. (14) and rearranging,one arrives at

D = P • D; (15a)

where

P = I − 32C ′ ⊗ C ′ (15b)

is a fourth-order tensor which projects the symmetric deviatoric second-order tensorspace into its four-dimensional subspace orthogonal to C ′. In Eq. (15b), the fourth-orderidentity tensor I describes the identity transformation on second-order symmetric ten-sors.

The transition (15) possesses the same form as that proposed by Parks and Ahzi(1990) using an intermediate traceless deformation rate. The derivation here is free ofassumptions and, therefore, suitable for general situations. The material symmetry isnaturally incorporated into the model. As shown in the sequel, the analytical power ofrepresentation theorems for tensor functions helps the derivation of the explicit formof D.

For simplicity, we make the similar assumption to that proposed by Chen et al.(1996) for the planar crystalline polymer case, in the spirit of Parks and Ahzi (1990):the local deformation rate D is a quadratic function of the current orientation C ′.This imposes a quadratic expansion of C ′ for the non-uniform deformation rate withina grain. Under this assumption and the form of Eq. (15), D is necessarily independentof C ′ and, therefore, a macroscopic variable. Substituting Eq. (15a) into the globalcompatibility (13)1 results in

〈P〉• D = SD or D = 〈P〉−1 • SD: (16)

From Eqs. (15b) and (10), the macroscopic tensor 〈P〉 depends on the second andfourth moment tensors. Eq. (16) shows that the form of D is independent of themacroscopic spin SW , and linearly related to the macroscopic deformation rate SD.


Therefore, the micro–macro transition (15) for the deformation rate is rate independentand depends on the second and fourth moment tensors.

By representation theorems (Zheng, 1994) and the traceless nature of D, the com-plete and irreducible representation of D as an isotropic tensor-valued function is (seeAppendix B)

D= (�1trM 2 SD + �2trM SD)(M 2 − 2

3 1)

+ (�3trM 2 SD + �4trM SD)M

+ �5 SD + �6(M SD + SDM) + �7(M 2 SD + SDM 2); (17)

where �1; : : : ; �7 are the functions of the components �20; �22; �40; �42 and �44. Ap-pendix B shows the determination for the coeOcients �1; : : : ; �7, by means of Eqs. (17),(15b), (10), (A.2) and (A:6).

4.3. Transition law for the spin

In specifying the micro–macro transition for spin, many authors equated W withthe macroscopic spin SW (see Parks and Ahzi, 1990; Chen et al., 1995). The latterworks showed that a direct imposition of W = SW may lead to sharp texture in thefully crystallized polymers. The rotation of the chain axes is actually a process thatresults from a variety of crystallographic slips and interlamellar slidings. As pointedout by Galeski et al. (1992), the crystallites rotate by interlamellar sliding throughshear of the amorphous phase, and that rotation is in the direction to compensate thatdue to crystallographic slip. This implies that the amorphous network surrounding thecrystalline lamellae resists the rotation of the crystals, giving rise to the heterogeneousspins of the crystals throughout the aggregate in crystalline phase.

Similar to the microscopic deformation rate D, the microscopic spin W of a singlecrystal within the aggregate should be an isotropic tensor-valued function of SD; SW ;C ′;M and components �20; �22; �40; �42; : : : . Like the deformation rate, the micro–macrotransition for the spin is assumed to be rate independent and to depend on only thesecond and fourth moment tensors. The rate independence implies that the spin Wdepends linearly on the macroscopic velocity gradient SD+ SW . The heterogeneity of thespin W is assumed to depend linearly on the orientation C ′. Under these restrictions,the spin W can be constructed as follows:

W = W + (C ′D − DC ′); (18)

where W and D are the isotropic second-order skew-symmetric and symmetric tensor-valued functions respectively. They are independent of C ′. The 6rst term in Eq. (18)describes a homogeneous spin while the second term in the brackets describes itsheterogeneous correction. Because the spherical part of D has no eFect on the spinW ; D may be regarded as traceless. Following the same procedure as for D, one canderive a representation of D as

D= (�1trM 2 SD + �2trM SD)(M 2 − 1

31)

+ (�3trM 2 SD + �4trM SD)M

+ �5(trN SD) + �6(trN1 SD)N1 + �7(trN2 SD)N2; (19)


where �1; : : : ; �7 are the characteristic parameters of the material, N is given inEq. (A.5), and N1;N2 are given in Eq. (A.7). According to the assumption givenabove, they depend linearly on SD and are functions of the second and fourth momenttensors. Inserting Eq. (18) into Eq. (13)2 and using Eqs. (10) and (A.2), one easilyobtains

W = SW − (Da′2 − a′2D): (20)

The transition law (18) for the spin W has direct impact on the texture evolution.That inEuence will become transparent in the next section by Eq. (25). In the presentmodel, the amorphous phase has its role in the construction of the heterogeneous spin,and consequently has its inEuence on the texture evolution.

4.4. Stress equilibrium

The self-consistent conditions for global equilibrium within the aggregate requiresthat the orientation average of the local stress be equal to the macroscopic stress SS

〈S〉 = SS : (21)

The component S∗ of the local stress S (see Eq. (5)) can be determined in terms of themacroscopic deformation rate SD by combining the micro–macro transition (15) withthe constitutive equation for a single crystal (4). On the other hand, the component SC

is a reaction stress due to the inextensible constraint and should be determined fromthe equilibrium. Parks and Ahzi (1990) approximated this local stress component as thecorresponding projection of the macroscopic stress SS , namely, SC = ( 3

2 )tr SSC ′. Chenet al. (1996) employed representation theorems to give its complete and irreducibleforms. Here, we adopt a diFerent approach.

Consider a polycrystalline aggregate as described at the beginning of Section 2. If thecontribution from the non-equilibrated traction along the grain boundary is neglected,one can employ the equilibrium condition to derive

〈S •D〉 = SS • SD; (22)

where the incompressibility trD=tr SD=0 is used. Relation (22) states that the orientationaverage of microscopic work rate equals macroscopic work rate. Using Eqs. (5), (3),(15) and (16), one obtains

S •D = S∗ •�P • (〈P〉−1 • SD)�: (23)

Inserting Eq. (23) into Eq. (22) and using S∗ •P = S∗, one obtains

SS = 〈P〉−1 •〈S∗〉: (24)

The micro–macro transition law for the stress also depends on the second- and fourth-order moment tensors. From the discussion given above, the macroscopic tensor 〈P〉plays a dominant role in de6ning the micro–macro transition for both deformation andstress.


5. Texture evolution

If the deformation rate D and the spin W of a single grain within the aggregate areknown, one can write the rotation rate of its chain axis c as

c =Wc − [(c ⊗ c)D −D(c ⊗ c]c: (25)

In order to solve the texture evolution, it is necessary to represent the evolution equa-tion (25) of the chain orientation in terms of the macroscopic deformation. InsertingEqs. (15) and (18) into Eq. (25) and rearranging, one obtains

c = Wc − [(c ⊗ c)(D − D) − (D − D)(c ⊗ c)]c: (26)

Because W ; D and D are all macroscopic variables, Eq. (26) implies that the chainaxes rotate with the macroscopic spin W and the macroscopic deformation rate D− D.

Recall that the chain axis of a grain is associated with a c-vector in the unit sphere.Consider that the unit sphere undergoes the uniform deformation governed by the spinW and the deformation rate D−D. The c-vector in the unit sphere rotates in accordancewith Eq. (26). Take a solid angle dA = sin �0 d�0 d�0 in the initial state. Under theuniform deformation, the unit sphere becomes an ellipse and the solid angle changes.The conservation of the volume and the continuity of the orientation distribution requirethat

R3 sin � d� d� = 13 sin �0 d�0 d�0; (27)

where R denotes the stretch of the cone axis of the solid angle in the deformed state.Three principal axes of the deformed ellipse determine three privileged orthogonal axesbi of the orientation distribution. The stretches in the directions of three principal axes(i.e. the principal stretches) are given by

�i = exp[∫

bi • (D − D)bi dt]: (28)

In the frame associated with the principal axes, one can express R as

R =

[(cos � ∗ sin�∗

�1

)2

+(

sin � ∗ sin�∗

�2

)2

+(

cos�∗

�3

)2]−1=2

; (29)

where the superscript ∗ labels the polar angles of the c-vector with respect to theprincipal frame.

A comparison of Eq. (29) with the conservation equation (8) of the ODF results in

�(c) =�0(c0)R3

=1

4�

[(cos � ∗ sin�∗

�1

)2

+(

sin � ∗ sin�∗

�2

)2

+(

cos�∗

�3

)2]−3=2

; (30)

where the initial value of �0(c0) = 1=4� is used for normalization. According toEqs. (28), (16) and (19), the principal stretches �i are the functions of the com-ponents of the second and fourth moment tensors. Thus, one needs to solve for thesemoment tensors in order to obtain a closed form solution of the ODF.


The evolution equation for the second moment tensor can be obtained using thesame procedure as proposed in Chen et al. (1996), namely

a′2 = Wa2 − a2W + (D − D)a2 + a2(D − D) − 2a4 • (D − D): (31)

Eq. (31) depends on the fourth moment tensor and thus is not closed. As discussedin Chen et al. (1996), this equation can be closed by the co-axial assumption betweenODF and SB, namely the privileged orthogonal axes (namely, the principal axes of a′2)coincide with those of the macroscopic left deformation tensor SB. Once the privilegedorthogonal axes are known, one can express the second and fourth moment tensors,respectively, in terms of �i by employing Eqs. (30), (A.4a) and (A:6). Eliminating�i from these expressions, one can express the fourth moment tensor in terms of thesecond moment tensors. When combined with this result, Eq. (31) can be solved inclosed form to deliver the second-order moment tensor.

With the above assumption, the spin of the principal axes of a′2 has to be equal tothat of SB,

�a′2i′∗j∗ = � SB

i′∗j∗ (i∗ = j∗): (32)

This provides three equations to quantify three parameters among �1; : : : ; �7. Let i∗ andj∗ label the principal axes of SB. One has the spin of the principal axes of a′2

�a′i∗j∗ =

1

�a′2j∗ − �a

′2i∗

(a′2)i∗j′∗ (i∗ = j∗; no summation); (33)

where �a′2i∗ are the principal values of a′2 and may be represented in terms of �20 and

�22 from Eq. (A.6a). The spin of the principal axes of SB is

� SBi∗j∗ = SWi∗j∗ +

S�2j∗ + S�

2i∗

S�2j∗ − S�

2i∗

SDi∗j∗ (i∗ = j∗; no summation); (34)

where S�i are the principal values of SB.When Eq. (31) is represented in terms of the components in the principal frame by

means of Eqs. (A:2) and (A:6), one obtains three equations on (a′2)i∗j′∗ and

�20 =(− 3

2�− 3

7�20 +

47�40

)trM 2(D − D) − 4

7(3�40 + 10�42)trM(D − D);

�22 =3�22 + 10�42

7trM 2(D − D) +

(1

8�− �20

28− �40

126− 40�44

3

)trM(D − D):

(35)

Combining Eq. (35) with Eqs. (19), (28) and (29), one easily shows that �i depend on�1; : : : ; �4 but not on �5; �6 and �7. One can further show that �1; : : : ; �4 have eFectson the evolution of the principal values of the ODF (see Eq. (30)), while �5; �6 and�7 only on the spin of the privileged axes of ODF. Under the co-axial assumptionbetween ODF and SB, one only needs to identify �1; : : : ; �4 for the evolution of theODF (see Eq. (30)). The other three parameters �5; �6 and �7 can be chosen to satisfythe assumption (32).


6. Comparison with experiments

The present model contains three material parameters n, �0 and g� in the single crys-tal description and seven material parameters �1; : : : ; �7 in the micro–macro transitionfor spin. The parameters �1; : : : ; �7 reEect the resistance of the amorphous phase tothe rotation of the chain axes in the crystalline lamellae, and aFect the second-ordermoment by Eq. (31). As discussed above, only four parameters �1; : : : ; �4 need tobe identi6ed. If one can measure the second-order moment of the specimen at aspeci6c stage of texture evolution, he or she can check the experimental data withEq. (31) to identify the value of �1; : : : ; �4. Unfortunately, the experimental data forthe second-order moment of HDPE, to the knowledge of the authors, are not availablein the literature. As an alternative way, we identify the values of �1; : : : ; �4 by 6ttingthe experiment data of stress response. Here we choose �1 and �2 so that

bi • (D − D)bi = �∗bi • SDbi ; (36)

where �∗ is a characteristic constant of the material. The special selection of �∗ = 1refers to the case of texture evolution dictated by macroscopic deformation.

We now prescribe the material parameters suitable for HDPE. The strain-rate sensi-tivity exponent n is assigned the same value 9 as given by Lee et al. (1993). The strainhardening of a slip system is neglected by taking g� to be a constant. The normalizedresistance, g�=�0, of the slip system is chosen to be the same value as tabulated byLee et al. (1993), where �0 = g(1 0 0)[0 0 1] is the initial shear strength on the easiest slipsystem. The above constitutive relations will be used to explore the behavior of HDPEunder global uniaxial stretching and uniaxial compressing.

Let us 6rst consider the case of uniaxial stretching. By 6tting the experiment data ofthe stress–strain curve, we take �∗ to be a constant of 1.1. The orientation distributionversus � curves for diFerent values of logarithmic tensile strains are plotted in Fig. 2.

The ODF is concentrated to the values of � = 0 and � = �. As the aggregate ex-tends, the chain axes rotate gradually towards the direction of stretching until they areperfectly oriented in that direction and form a fully textured aggregate.

Fig. 4 shows the macroscopic equivalent stress, S eq =√

3=2 SS • SS , as a function of the

macroscopic equivalent strain, S! eq =∫ t

0

√2=3 SD•D dt. The calculated equivalent stress–

strain curve in stretching exhibits a strong textural hardening, in agreement with theexperiment data obtained by G’Sell and Jonas (1979). The strong textural hardeningis the consequence for the inextensible chain axes to rotate towards the direction ofstretching.

Next consider the case of simple shear. The same set of material constants as instretching are used. Shear proceeds in the direction of x1. The non-zero components ofdeformation rate and spin are

SD12 = SD21 = SW 12 = − SW 21 = 12 $: (37)

In contrast to the axisymmetric deformation, two assumptions are involved for the sim-ple shear case. First, the ODF tensor a2 is assumed to be macroscopically orthotropic.This assumption holds for the case of axisymmetric straining, but seems questionable


Fig. 2. Evolution for the orientation distribution function under global uniaxial tension. Four curves corre-spond to logarithmic strains of 0.0, 0.5, 0.75 and 1.0.

for the simple shear case. An approximation is made such that the ODF is macroscopi-cally orthotropic at any extent of shear, though the axes of the macroscopic orthotropyrotate as the specimen shears. Second, the principal axes of ODF co-rotate with theaxes of SB, namely the rotation of the orthotropic axes is driven and synchronized bythe macroscopic left deformation tensor during shear. These two assumptions will bejusti6ed by the simulation results.

Fig. 3 shows the orientation distribution versus � curves for diFerent values of shearamount $. As the shear proceeds, the texture evolves to align the c-axes of variousgrains in the direction of the largest stretch, which varies from �=4 under zero shearstrain to 0 under in6nitely large shear strain. The larger the shear strain, the strongerthe texture. The calculated equivalent stress–strain curve is also included in Fig. 4.That curve exhibits rather mild hardening, in agreement with the experimental resultsby G’Sell et al. (1983) under simple shear. The mild hardening for the simple shearcase can be attributed to the gradual rotation of the texture.

7. Concluding remarks

This paper presents a physically based three-dimensional mechanical model for tex-ture evolution and microscopic plastic responses in crystalline polymers. The presentwork clearly describes the key role that the ODF plays in inEuencing the macroscopicconstitutive responses of the material. The ODF reEects the evolution of the internalstructure of the material and links the microscopic and macroscopic scales. Therefore,


Fig. 3. Evolution for the orientation distribution function under simple shear. Four curves correspond toshear strains of 0.0, 0.5, 0.75 and 1.0.

Fig. 4. Calculated S eq=�0 vs. S! eq × !=�0 curves in uniaxial stretching and simple shear. Experimental datawere normalized by using �0 = 7:8 MPa (see G’Sell and Jonas, 1979; G’Sell et al., 1983).


such an approach provides the physical basis for the internal variable plasticity. Sym-metry constraint allows a simple representation of the ODF. Combined with symmetry,representation theory of tensor functions is employed as an important tool for thespeci6cation of the micro–macro transition. The role played by the amorphous phaseis taken into account through its inEuence on the crystallographic slip. A method toclose the evolution equations of the ODF is suggested. Closed solutions are obtained.In axi-symmetric stretch and simple shear, the predictions of the present model are ingood agreement with the experimental data.

In the previous analysis for the plane strain case (Chen et al., 1996), we compared theplane strain predictions for compression (identical to the prediction under plane straintension) and simple shear with the experimental data by Galeski et al. (1992) and byG’Sell et al. (1983), respectively. Though the predicted stress–strain curves roughlyagree with the experiments, the predicted textures are rather sharp. By accommodatingthree-dimensional deformation, the present model not only predicts accurate stress–strain curves, but also more attenuated texture evolutions.

Appendix A

Kanatani (1982) and Zheng and Collins (1998) showed that the irreducible parts ofcomplete symmetric tensors such as a2n can be expressed as

a′2n = a2n − %(2n)1�a2n−2 ⊗ 1� + %(2n)2�a2n−4 ⊗ 1⊗2� − · · ·

=n∑

r=0

(−1)r%(2n)r�a2n−2r ⊗ 1⊗r�

with %(2n)0 = 1; %(2n)(r−1) = [3 + 4n + 2(r − 3)(2r − 1)]%(2n)r (r = 1; 2; : : : ; n):(A.1)

The notation �A� is assigned to represent the sum of all possible diFerent permutationsof tensor A. For example,

�1⊗ 1� = e� ⊗ e� ⊗ e� ⊗ e� + e� ⊗ e� ⊗ e� ⊗ e� + e� ⊗ e� ⊗ e� ⊗ e�;where {e�} is an orthonormal frame of three-dimensional Euclidean space. The irre-ducible parts of the second and fourth moment tensors are

a′ij = aij − 13�ij; a′ijkl = aijkl − 1

7��ijakl� + 135��ij�kl�: (A.2)

Zheng and Collins (1998) showed that the general representations of the momenttensors under the restriction of global orthotropy are

a′2n = 2�4n[(2n)!]2

(4n + 1)!�(2n)0T(2n)0 + 4�

n∑r=1

4n−r(2n + 2r)!(2n)!(4n + 1)!

�(2n)(2r)T(2n)(2r);

(A.3)


where �(2n)0; �(2n)(2r) are scalars given by

�(2n)(2r) =2(4n + 1)

�(2n− 2r)!(2n + 2r)!

∫ �

0P(2n)(2r)(cos�)sin�

∫ �=2

0�(�; �)cos(2r�) d� d� (A.4a)

with P2n being Legendre functions. The symbols T(2n)0 and T(2n)(2r) in Eq. (A.3) areirreducible tensors given by

T(2n)0 = Ren∑

m=0

(−1

4

)m�(1−M 2)⊗n−m ⊗ (M 2 + iNM)⊗m�; (A.4b)

T(2n)(2r) = Ren−r∑m=0

(−1

4

)m�(1−M 2)⊗n−r−m ⊗ (M + iN)⊗r ⊗ (M 2 + iNM)⊗m�;

(A.4c)

where the pre6x Re indicates real part, and i =√−1 is the unit imaginary number.

Tensor M is de6ned by (11) and N is given by

N = b1 ⊗ b2 + b2 ⊗ b1: (A.5)

The irreducible parts of the second and fourth moment tensors, for example, are

a′2 =4�15

�20

(1− 3

2M 2

)+

8�5�22M ; (A.6a)

a′4 =16�315

�40[21⊗ 1− I − 52

(1⊗M 2 +M 2 ⊗ 1)

+154M 2 ⊗M 2 +

5�8

(M ⊗M +N ⊗N)]

+16�21

�42[1⊗M +M ⊗ 1− 32

(M ⊗M 2 +M 2 ⊗M)

+N1 ⊗N1 −N2 ⊗N2]

+32�

3�44(M ⊗M +N ⊗N); (A.6b)

where

N1 = b1 ⊗ b3 + b3 ⊗ b1; N2 = b2 ⊗ b3 + b3 ⊗ b2: (A.7)

It can be shown that the even-order tensor products N ⊗ N ; N1 ⊗ N1; N2 ⊗ N2 andtherefore a′2n are functions of M ; 1 ⊗ 1 and I . From Eq. (A.3), there are n + 1


independent components �(2n)0; �(2n)(2r) (r=1; 2; : : : ; n) for the 2nth moment tensors a′2n.Consequently, the orientation distribution function � can be expressed by an isotropicscalar-valued function of C ′;M , and the coeOcients �20; �22; : : : ; �(2n)(2r); : : : ;

� = �(C ′;M ; �20; �22; : : : ; �(2n)(2r); : : :): (A.8)

In the case of the transverse isotropy, �(2n)(2r) = 0 (r= 1; 2; : : : ; n), i.e. there is only oneindependent component �(2n)0 in the moment tensors of any order.

Appendix B

The second-order symmetric and traceless tensor SD can be uniquely decomposedinto

SD= 32 trM 2 SD

(M 2 − 2

31)

+ 12 [(trM SD)M + (trN SD)N + (trN1 SD)N1

+(trN2 SD)N2] (B.1)

In view of trD = 0 (see Eq. 14), inserting Eq. (B.1) into Eq. (17), yields

D= [h1trM 2 SD + h2trM SD](M 2 − 2

31)

+ [h3trM 2 SD + h4trM SD]M

+h5(trN SD)N + h6(trN1 SD)N1 + h7(trN2 SD)N2; (B.2)

where h1 = �1 + 32�5 + �7; h2 = �2 + �6; h3 = �3 + �6; h4 = �4 + �7; h5 = 1

2 (�5 + �7);h6 = 1

2 (�5 +�6 +�7); h7 = 12 (�5 +�6 −�7). With the help of Eqs. (15b), (10) and (A.2),

Eq. (16)1 can be rewritten as

45 D + 2

7 tr(a′2D)1− 37 (a′2D + Da′2) − 3

2a′4 • D − SD = 0: (B.3)

Inserting Eqs. (B.2), (B.1) and (A:6) into Eq. (17), after lengthy but straightforward

manipulation, one obtains

h1 =1h

[85

+�

105(24�20 − 4�40 − 6720�44)

];

h2 = h3 =�

35h(32�22 − 80�42);

h4 =1h

[8

15− 8�

105(�20 + �40)

];

h5 =12

[45

+�

105(12�20 − 2�40 + 3360�44)

]−1

;

h6 =12

[45− �

105(6�20 + 72�22 − 8�40 + 560�42)

]−1

;


and

h7 =12

[45− �

105(6�20 − 72�22 − 8�40 − 560�42)

]−1

; (B.4)

where

h = (h1h4 − h2h3)−1:

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Modeling of large plastic deformation in crystalline polymers