Finite deformations of anisotropic polymers

Mechanics of Materials 15 (1993) 3-20 Elsevier

3

Finite deformations of anisotropic polymers J.J. Pereda, N. Aravas and J.L. Bassani Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia; PA 19104, USA

Received 24 February 1992; revised version received 7 October 1992

A model for the thermomechanical behavior of anisotropic polymers under finite deformations below the glass-transition temperature is developed. The anisotropy is deformation-induced and the principal axes of the plastic stretch tensor are used to define the local axes of orthotropy at each material point of the deformingpolymer. The formulation is based on a multiplicative decomposition of the deformation gradient tensor into elastic and plastic parts. An expression for the plastic spin, which is the average spin of the continuum as seen by an observer spinning with the axes of orthotropy, is derived. The numerical implementation of the model in a finite-element program is discussed and an algorithm for the numerical integration of the elastoplastic equations is developed. The problems of plane strain drawing and extrusion of polymeric sheets are solved using the finite-element method.

1. Introduction Polymeric materials develop anisotropies when deformed to large strains. The anisotropy is deformation-induced and associated with the stretch and alignment of the molecular chains (e.g., see Brown and Windle, 1984). Parks et al. (1984) were the first to develop a finite strain constitutive model for the elasticplastic behavior of glassy polymers. Their model was further developed by Boyce et al. (1988; 1989; 1991) who considered the effects of thermomechanical coupling. In these works, the anisotropy associated with the orientation of the molecular chains is accounted for by introducing a 'back stress' in the plasticity model and the overall continuum formulation is of the kinematic hardening type. Bassani and Batterman (1987) presented a model for the mechanical behavior of anisotropic polymers using Hill's (1950) orthotropic yield criterion, where at every instant of the deformation, the principal axes of orthotropy Correspondence to: Prof. N. Aravas, Department of Mechanical Engineering and Applied Science, 297 Towne Building, 220 South 33rd Street, University of Pennsylvania, Philadelphia, PA 19104-6315, USA.

are aligned with the principal axes of stretch. Dafalias (1991a,b) has also discussed recently the constitutive modeling of polymeric materials at large deformations and emphasized the importance of the plastic spin, which is the average spin of the continuum relative to the substructure (e.g., axes of anisotropy). A model for the thermomechanical behavior of anisotropic polymers under finite deformations below the glass-transition temperature is developed in this paper. The anisotropy is deformation-induced and the principal axes of the plastic stretch tensor are used to define the local axes of orthotropy at each material point of the deforming polymer. Hill's orthotropic yield criterion is used to describe the plastic behavior of the material. The formulation is based on a multiplicative decomposition of the deformation gradient into elastic and plastic parts. An expression for the plastic spin is derived. The coupled thermomechanical problems of plane strain drawing and extrusion of polymeric sheets are solved using the finite-element method. Standard notation is used throughout. Boldface symbols denote tensors, the order of which is indicated by the context. The summation convention is used for repeated indices unless otherwise

0167-6636/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

4

J.Z Pereda et al. / Anisotropic polymers

indicated. The prefices tr and det indicate the trace and the determinant, respectively, a superscript T the transpose, a prime, the deviatoric part of a second-order tensor, subscripts s and a the symmetric and antisymmetric parts of a second-order tensor, and a superposed dot the material time derivative. All tensor components are written with respect to fixed Cartesian coordinate system and the following products are used in the text: ( A . B ) i j = A i k B k i , (C: A)ij = CijklAkl , (A: C)ij =AktCktij, A: B =AijBij, (a -A)i = aiAji, (A. a)i = Aija j, and (ab )ii = aib j.

The deformation rate D and the spin W, defined as the symmetric and antisymmetric parts of L are be written as D=De+D

p,

and

W=W*+W

p,

(3)

where De = (/~e .Fe-1)s

'

W* = ( / + e ' F e - ~ ) a ,

(4)

D p = ( F e ./TP . F p - I . F p - l ) s , W P = ( F e " ITP " F p - I " F e - 1 ) a .

(5)

In the isoclinic configuration we also define 2. Kinematics L i =FP

The kinematics of finite elastoplastic deformation is described in terms of the multiplicative decomposition of the deformation gradient F: F = F e "F p

(1)

formally introduced in continuum mechanics by Lee and Liu (1967) and Lee (1969). The intermediate unstressed configuration ~'~, that is the configuration of the continuum after removal of F e, is not uniquely defined, since an arbitrary rigid body rotation can be superimposed on it and still leave the configuration unstressed. Motivated by the kinematics of the single crystal and in order to remove the aforementioned ambiguity Mandel (1971, 1973) introduced a triad of 'director vectors', say d i, embedded in the material substructure, and used the orientation of the di's with respect to a fixed Cartesian coordinate system to define the orientation of the intermediate configuration ~q~i.We choose to work in terms of the so-called 'isoclinic configuration' ~'i which is defined in such a way that the director vectors at ~'i have, at all times, a fixed orientation in reference to a global coordinate system (Mandel, 1971; 1973). The definition of the director vectors of our model is given in Section 3.2 below. In the current configuration .~', the velocity gradient L can be written as L = / + ' F -l =aCe . F e - 1 + F e . / ? p . F p-1 . F e - 1 .

(2)

.F p-1 '

DP = (/TP.Fp-1)s

,

and Wi p = (/~P " F P - l ) a

.

(6)

In anisotropic materials, the definitions of D e, W*, D p, and W p given above are appropriate when the intermediate configuration is isoclinic; when ~'i is not isoclinic, the material time derivatives in the aforementioned definitions must be replaced by derivatives co-rotational with the substructure that defines the material anisotropy (Dafalias, 1987).

3. Constitutive equations We define the material symmetries and write the constitutive equations in the isoclinic configuration. Isotropic hyperelasticity is used to describe the elastic response of the material. The plastic behavior of the polymer is assumed to be orthotropic in the isoclinic configuration; the plastic anisotropy is deformation-induced and the axes of orthotropy are aligned with the principal plastic stretches. 3.1. Elastic equations The thermo-elastic behavior of the polymer is assumed to be isotropic, and the elastic constitu-

J.J. Pereda et al. / Anisotropic polymers

tive equations are written in terms of the potential • as

5

so that Se = C:E e- (7"- ro)M ,

(17)

~}qD(ge, r ) s e=p0

,

(7)

where se = F e - 1 . "r" F e - T is the elastic second Piola-Kirchhoff stress tensor defined at ~'i, r = Jo" is the Kirchhoff stress tensor, J = det (F), tr is the true or Cauchy stress tensor, E e = 1 ( F e T . F e - - I ) is the elastic Green strain tensor, I is the second-order identity tensor, T is temperature, P0 is the mass density in the undeformed configuration ,~0, and ~P is an isotropic function of its arguments. Differentiating the above equation with respect to time, we find that Se = C : E e - M T ,

vE %jkt---- (1 + v)(1 -- 2v) OijOkl E ¢, ¢, + c o x. + 2(1 + u) 1[OikOjl OilOjk)' aE M = 1 - 2~, I '

(18)

(19)

where E is Young's modulus, v is Poisson's ratio, c is the specific heat, a is the coefficient of thermal expansion, TO is an initial reference temperature, and 8ij is the Kronecker delta.

(8) 3.2. The isoclinic configuration and yield condition

where a2cp C = P0 OE e a E e

and 02cp M = - P o i}E e 0 r

(9)

After some lengthy but straightforward calculations, it is found that the rate equation (8) can be written in the current configuration ~ as ~.__~e : D e _ M e T

(10)

where ¢:¢ ='? + ' r "

~Vge

-W

~g

-'r,

(11)

"~iijek l = FimFjnF~pFlqCmnpq, e e e e

(12)

. ~ f = ~ e + T,

(13)

Tijkl = ½(rikt~jl + "ril~jk + 8ik'rjl + ~ilrjk),

(14)

M e = F e " M ' F eT.

(15)

Assuming linear thermo-elasticity, we write p0

with

,(E e, r ) = =El e : C : g e -- ( r -

P°c "T ~ 0 ( - T°)2,

r0)M: g e

(16)

The yield condition and the plastic flow rule are written in the isoclinic configuration ~ i in terms of the plastic work conjugate quantities ] ; = R e T ' ¢ - g e and D~, where R e is the rotational part of F e (Freund, 1970). Following Batterman and Bassani (1987; 1990) we assume that the principal directions of the plastic stretch correlate with the molecular orientation or 'texture' of the polymer, and, therefore, the principal axes of plastic stretch are taken as the principal (orthogonal) axes of orthotropy. Under arbitrary histories of deformation, each material point of the deforming polymer will possess orthotropic symmetries about the principal axes of the plastic stretch B p = F p T ' F p in the intermediate configuration. As will be seen in Section 3.4 below, this physical assumption regarding the texture of the polymer completely determines the plastic spin Wip, which differentiates the average spin of the continuum from that of the microstructure. In Mandel's terminology, the eigenvectors of B p in the intermediate configuration play the role of the director vectors at each material point. We denote by Ap (A p >/Ap > h~) the eigenvalues of ( B P ) 1/2 and by m i the corresponding eigenvectors. Plastic incompressibility is assumed, so that ~ p1"'2"'3 l p : t p--- 1. At every instant, the arbitrary rota-

6

J.J. Pereda et al.

/ Anisotropic polymers

tion of the intermediate configuration at each point is chosen in such a way that the orientation of the mi's at that point does not vary with time and equals the orientation of the corresponding mi's when the material point yield for the first time. A simple example of the definition of the isoclinic configuration is given in Section 6.1 where the case of simple shear is analyzed in detail. In the isoclinic configuration the yield condition is expressed as f ( ~ , A,, A 2, /~P, Ep, T) = 0,

(20)

where A 1 = mlm 1, A 2 = BPl21'f/2, Ep is the equivalent plastic strain that controls the size of the yield surface, and f is an isotropic function of its arguments. Bassani and Batterman (1987; 1990) suggest, using Hill's (1960) orthotropic yield criterion, F( "~22 -- ~33) 2 + G( ~32 -- "a~ll) 2 + H( Z,I - X22) 2 + 2LZ~3 + 2MX21 + 2NZ22 - Cr~l(ap, T) = 0,

(21)

where the coordinate axes are along the axes of orthotropy (i.e., the principal plastic stretch axes), %1 is the yield stress in direction 1 and the parameters (F, G, H, L, M, N) are defined as

Oryl)2

1

--

+

1]

,22,

where the o-y's and ry's are the yield stresses in tension and shear, respectively. The evolution of the yield stresses with plastic deformation is discussed in Section 4 and is such that when two of the plastic stretches are equal at a material point, say h~ > h~ = h p, then

H=G,

N=M,

and

L=2F+G

(28)

and Eq. (21) reduces to the transversely isotropic version of Hill's criterion, with x 1 being the axis of rotational symmetry. When all three plastic stretches are equal at a material point, in which case hip = A~ = h~ = 1 because of plastic incompressibility, we have that F = G = H = 1/2 and L = M = N = 3/2

(29)

and the expression (21) reduces to von Mises' yield criterion, so that the materrial is instantaneously isotropic at that point. Dafalias and Rashid (1989) have shown that Hill's yield condition can be written in invariant form in terms of the orientation tensors A~ and A 2 and the variables and parameters in Eq. (21)

as ( L +M-N)

tr(17.2')

+ 2 ( U - L) tr(~;' "~;' "AI) + 2 ( U - M ) t r ( ~ ; ' - ~ ' "A2) +(4G +F+H-

1

G=~

1

O'y3 (~O.i---~)2-- (~yY~) 2

H=~ L =

M=

N=

t 2'

+( H + G + 4 F - 2L) t r / ( X " A 2 )

+11

(24)

~ry31!

1 ( OryI )2, -2 \ ry12

+ 2(2G + 2 F - H + N - M - L ) × tr(1;' .A,) tr(~;' .A,) - ~rZl(ep, T) = 0,

(30) (25)

2 \ ry23 !

1t ,1t 2

2 M ) tr2(X ' ' A , )

where a prime ' denotes the deviatoric part of a second-order tensor. (26)

3.3. The plastic flow rule (27)

The constitutive equations for D p and Wip, and the evolution equation for the equivalent

J.J. Pereda et al. / Anisotropic polymers plastic strain t ° are written in the isoclinic configuration as follows D p = ( A ) N i P ( ~ , A1, A2, Ap, ~P, T),

(31)

t~ip~- a~'(~, A,, A~, ~ , ,~, T)

(32)

7

where

sn:(.~: r =

+ T) (39)

X ( + N n : (Sa~ + f ) : N p' Of/OT - N n : M e

e=

~ f + Nn : (.~,ae + T) : N p

,

(40)

and

~p = ~(x, A,, A~, ~', ,P, r),

(33)

where Ni p, ~'~P, and g are isotropic functions of their arguments, ~{ is a loading parameter, and () are the Maccauley brackets. It is mentioned in .passing that the evolution of the plastic stretches Ap can be written as

AP = (A)AP(.~7, A1, A2, Ap, CP, r ) ,

(34)

)r = -

(0f_ OfXpl ' 0E----ps+ 0,~ ]

N " = F e-~" O f ' F e T , Of.r

(42)

Tt'jkl = "i'ik~jl - Til~jk"

(43)

Summarizing, we write the plastic rate of deformation tensor D p and the plastic spin tensor W v (36) in the current configuration as D p = h(J~)(NPr:

where AP is an isotropic function of its arguments; the exact form of AP is given in the Appendix. The corresponding expressions for D p and W p in the current configuration ~ ' are D p = ( A ) N p and

W p = ( A ) / ~ p,

(35)

D + eNPT)

and W p = h ( A ) ( / ~ P r : D + eg'~PT),

0f Nip = ~-~ = ~bll + ~b2~' + (haA'l

NP = [ Fe " ( NiP + a P ) " Fe-1]s

+ q~4a ~ + 6 5 ( ~ ' . a l + a 1 .,,~7') + ¢~6(,~' "h 2 + a 2 ",~'), (45)

and (36)

Taking into account that the orientation tensors A 1 and A 2 a r e kept fixed in the isoclinic configuration ~'i, we can write the consistency condition f = 0 as

where,

t~l = - { [ ( N - L )

tr(,~'-A1)

+(N-M)

(37)

tr(,~' .A2)],

(46)

62 = 2 ( L + M - N ) , ~b3 = 2 ( F + 4G + H -

Of Of ~p + ~p~p + 0fT= 0. /= ~ : £ + 0V 0r

(44)

where h(/i) is the Heaviside step function. In our model, the yield function is taken as the plastic potential (normality) in the isoclinic configuration, so that from (30)

where

a P = [F e • (NiP + / ~ p) -Fe-1 lJ a "

(41)

(47) 2M) tr(2'-A1)

+ 2(2G + 2 F - H + N - M - L ) X tr(~' .A2),

(48)

Using the consistency condition together with Eqs. (10), (31), (32) and (34), we find that the loading parameter can be written as

~b4 = ( 2 F + 2 G - H - L - M + N )

~, = 2 ( N - L),

(50)

A = r : D + eT,

~b6 = 2 ( N - M ) .

(51)

(38)

+2(4F+G+H-2L)

tr(~'.A1) tr(~,'.A2) ,

(49)

8

J.J. Pereda et al. / Anisotropic polymers

Following Hill (1950), we define the equivalent stress t~ in the isoclinic configuration as K : [(L + M - N )

The eigenvectors m i of B ° play the role of director vectors at each material point and define the local axes of orthotropy in the isoclinic configuration ~'~. The rate of change of the mi's can be written as

tr(~'.lT')

+ 2 ( U - L) t r ( ~ ' . ~ ' .A,) +2(N-M)

t r ( ~ ' .,~' .A2)

+(4G +F+H-

2 M ) tr2(l; '.AI)

+(H+G+4F-2L)

+2(2G + 2F-H+N-M-L) (52)

'/2

(63)

mi = WE "mi,

tr2(2;'.A2)

×tr(~' 'al)tr(X"al)]

3.4. The plastic spin

It can be readily shown that the corresponding work conjugate equivalent plastic strain rate ~P is

where the spin W E is defined from the expression (Mehrabadi and Nemat-Nasser, 1987) B p . D p . B p - I _ B p- 1. D p . B p = 2 ( W E - Wi p) - B P . ( W

E - WiP) . e p-1

- - B P - 1 " ( W E - t~iP) " B p,

~P= [C 1 tr(D p2) +C 2 tr(D~2"A1) +c 3 tr (DP, Z . A 2 ) + c 4 tr(D~'.a,)

which, with respect to the eigenvectors m i of B p, has the following component representation (Biot, 1965; Hill, 1970)

× t r ( D P ' A 2 ) + c 5 trZ(D{"Al) +c 6 tr2( DiP'A2)] 1/2,

(64)

(53)

WE -= 0,

(no sum over k),

where

(65)

,~p2 .}_ /~p2

1

1

1

C~=L+M

(54)

N'

= (wip)

,

APk4: Ar/,

( 1 1 1 c2=2 L +N M +

C3=

(55)

'

(56)

, H

1

c4=2 FG+GH+HF c5=

FG + GH + H F

c6=

FG+GH+HF

1

L + M

1 )

(57)

N ' (58)

L

'

(59) (60)

k -~ l ,

(no sum over k, l).

(66)

When the intermediate configuration is isoclinic rhi = 0, which requires that W E = 0. Equation (64) now reduces to B p .O p . B p - I _ B p - 1 .D p . B p = _ 2WiP + Bp . ~ i p . B p - 1 + B p - 1 . ~/ip . Bp '

(67) or, equivalently, (WiP),k = 0,

(no sum over k),

(68)

~.~2 + ~p2

so that W p = ~ : o p =6~p.

(WiO)~t (61)

Using the above equation, we can readily show that the function g that enters the evolution equation of Ep (see Eq. (33)) can be written as 1 = = , ~ " Ni p. or

(62)

A~_A 2 (DP),,, k4:l,

A~4:Af,

(no sum on k , l ) .

(69)

When two of the plastic stretches are equal, say A~ = A~, the material becomes locally transversely isotropic about the ml-direction in the isoclinic configuration. In such a case, in view of the local rotational symmetry, a spin about the ml-axis is

J.J. Pereda et al. / Anisotropic polymers

inconsequential and can be set arbitrarily to zero. Therefore, we write A~=A p,

k4=l.

(70)

Finally, when all three plastic stretches are equal (h p = h~ = h p = 1), the material is locally isotropic and (Dafalias, 1983; 1984; Loret, 1983) WiP = 0.

(71)

Zbib and Aifantis (1988) proposed expressions similar to (69) for the relative spin Ws/o, which is the "spin of stress" relative to the "spin of stretching". In the following, we show that the expression given in (69) for the plastic spin is consistent with the general form assumed for WiP in Eq. (32) of Section 3.3. Using Eq. (31) together with the respresentation theorems of isotropic functions, we arrive at a general expression that defines D p in terms of ~, A1, A2, Ap, eP and T. When this general expression for D p is substituted into Eqs. (68) and (69), the resulting expression for the plastic spin Wip is indeed an anti-symmetric isotropic function of the arguments that appear on the right-hand side of Eq. (32) (Pereda, 1992). Put in other words, Eqs. (68) and (69) above are a special case of the general form assumed for the plastic spin in Eq. (32) of Section 3.3. In a recent publication, Nemat-Nasser (1990) presented a detailed discussion of certain issues in finite deformation continuum plasticity. Using the polar decomposition theorem, he writes F p = R p" U p, where U pT= U p and R pT = R p - l , and defines the plastic deformation rate and the plastic spin as b p = (Up. Up-1)s

(72)

in the intermediate configuration ~'p defined by U p only. He then shows that ifP is completely defined by /)P, U p and its basic invariants. The relationship between (/gP, i f P) and the quantities (D p, Wip) used in the present paper is D p = R p • b p . g pT

(73)

where the differences are due to the rigid rotation R p that distinguishes ~'p from the isoclinic configuration ~'i- One way to interpret the second part of (73) is as follows. When WiP,/~v and U p (and thus ifP) are known, the second part of (73) defines the evolution of RP; when Wip is defined by Eq. (69), the corresponding R p is such that the intermediate configuration is isoclinic. We conclude this section with a general discussion on the definition of the plastic spin. Dafalias (1987) suggested the use of the plastic spin as a convenient means of differentiating the continuum spin from that of the substructure that defines the material symmetries. With respect to an arbitrary intermediate configuration ~'u defined by F p, Dafalias introduces the definitions D p = (iI~ p . F P - ' ) s = (FP . F P - ' ) s and Wup = ( F P " F P - 1 ) a = (/~P " F P - 1 ) a -

¢.Ou,

(74)

where cou is the substructural spin in the (arbitrary) intermediate configuration ~-~u, and /~P is the corrotational rate of F p with respect to the substructure, i.e.,

(75)

F P = F P - ~Ou" FP.

When the intermediate configuration is isoclinic ('-~u = ~ i ,

¢'Ou= 0),

DPu=Dp and

WuP=Wi p.

(76)

On the other hand, when ~'u = ~ p Nasser, 1990), we have that

and ~"P = ( U P " U p - 1 ) a ,

and t~i p = J~P" e pT + e p" i f P " e pT,

(WiP) k / ~---(WiP)lk = O,

when

9

Dup=/gp

and

Wf = i f p - ~ o

u.

(Nemat(77)

Note that ifP is completely defined by /)P and U p (Nemat-Nasser, 1990). The second part of Eq. (77) shows that Wup (together with /)P and U p) defines the substructural spin oJu in the intermediate configuration where the material symmetries are defined, i.e., Wu° provides information about the evolution of texture in the continuum.

10

J.J. Pereda et al. /

Anisotropic polymers

[(

3.5. Rate form of the elastoplastic equations

~'y23=0.yl 20"Yl]Z+2(0.Yl The rate form of the elasticity equation (10) can be written as

+ = ~ e : ( O _ O o) _MET.

(78)

Ty31

Using the plastic flow rule (44), we can rewrite the above equation as

4"=.~ e : ( J - h ( A ) N P r )

0"y2]

[ t O'y3

=0..L2 ,_2j +2

(79)

where J is the fourth-order identity tensor. Finally, introducing the Jaumann derivative ¢, ~ we find ~.= [..~e: ( J - h ( A ) N P r )

+h(A)

×(,.ap-ap.,)r]: D -[Me+h(A)e(.~ e : SP-,r.a

p

+a,.,-)]t.

(80)

4. Evolution of the yield stresses

It is well known that as the polymer is stretched, and the molecular chains become oriented, the polymer is stiffer in the direction of the maximum principal stretch than in transverse directions (Bassani and Batterman, 1987). We propose the following expressions for the evolution of the yield stresses (see also Batterman and Bassani, 1990)

1 + -eo -

l/n

,

/3 exp(MEp2), 0.ya

1 + Cl(h ~ - 1) "l

0.y2

1 + C1(A ~ - 1) "1'

°'J--~z= 1 - C2 tanh( ~-~3- 1), %3

,

(84)

0",_20"y2-1

,

(86)

~ 0.yl ]

where (Ap - 1) "1 - sign (h p - 1) l A/p - 11 .t, A~' is the maximum plastic stretch, e0, n, el, Cl, C 2 and a l are material constants, /3 and M are determined by requiring continuity of the 0"yl-e p behavior as well as continuity of slope at e p = e~, and 0"0(T) is a temperature-dependent reference stress to be defined in the following. Using the yield condition (21) together with the definitions (52) and (53) we can readily show that, in uniaxial tension,

~=%1=0. and e ° = e~n,

l(ffP)

2--1

~ 0.y3

:D

- ( M e +h(A)e.~¢ : NP)7",

0.yl= 0.0(T)

) 1 -lj2

if if

e p ~
(81)

e p >/e l,

(82) (83)

(87)

where or is the applied stress, and e~', is the corresponding logarithmic plastic strain.Therefore, the constants e0, n, /3, M and el that appear in the evolution equation of 0.yl can be determined from the uniaxial stress-strain curve of the polymer. The values e 0 = 0.05, n = 5, /3 = 1.41985, M = 0.5556 and e, = 0.4 are used in the calculations that are presented in the following. The remaining material parameters C1, C z and a l can be determined from plane-strain tension data and the observed directional dependence of the yield stresses relative to the initial draw direction (Brown et al., 1968; Shinozaki and Groves, 1973; Caddell et al., 1973) as discussed in the following. First, consider the case of plane-strain tension and ignore the effects of elasticity. Let 1 be the direction of stretching and 2 the constrained direction ( D z / = 0). It can be readily shown that (Bassani and Batterman, 1987) 0"22= O'llH/( F + H).

(88)

J.J. Pereda et aL / Anisotropic polymers

11

2 el

2'

=0.

c~=o.5=a. ,~,,/oo

3

1#

2

J

C1 = 0. 0"22/0" 0

1

C, = C, = 0

,

0

,

,

,

I

,

,

i

i

.25

Iii

i

i

i

.5

I

,

.75

,

,

,

)

0.5 1.

I

1

Fig. 2. Orientation of test specimen with respect to the drawing direction.

Ell

Fig. 1. Variation of the normal stresses in plane-strain tension with the parameter C 1.

With trn and o'22 required to have the same sign, H must be positive (note that F + H is always positive). Figure 1 shows the predicted variation of o'11 and 022 with axial logarithmic strain for different values of C 1 with C 2 = 0.05 and a l = 2. In order to make connection with the experimental results of Brown et al. (1968), we discuss the predictions of the proposed model for the case where tension or shear are applied at an angle to the initial draw direction. In the work of Brown et al. (1968), tensile specimens cut at an angle 0 to be the initial draw direction (IDD) were pulled at room temperature; shear specimens were also cut with the shear direction making angles, O, from 0° to 180° with the IDD. Similar experiments, including compression after drawing, were also carried out by Shinozaki and Groves (1973). In our calculations we assume that the specimen is first pulled to a stretch ratio A = L/L o = 3 and then subject to tension o'rr or shear o'r2' as shown in Fig. 2. It can be readily shown that the predicted yield stresses in tension (o-ti,) and shear ('/'yl,2,)

The predicted variation of o.tr with 0 is shown in Fig. 3 for C 1 --1, C 2 = 0.05 and a l - - 2 . The results shown in Fig. 3 agree qualitatively with the experimental data of Brown et al. (1968), Shinozaki and Groves (1973), and Caddell et al. (1973). Also, the prediction that ryr2, is independent of 0 is in agreement with the results of Brown et al. (1968) who observed a very small variation of "£y1'2'with 0. The yield stress of polymers depends strongly on temperature. The reference stress o'0(T) introduced above is assumed to vary with temperature as follows (Fager and Bassani, 1990)

O.o(T)=.~o( -TmTI ~, --~m ]

(91)

where ~0 is the yield strength of the material at zero absolute temperature, Tm is the glass transi3

2

are o'tl, = o . y l [ F sin40 + G COS40

+ H cosE20 + ½N sinE20]-t/2,

1

(89)

t7 t

Yl'/ao(T)

-'j2 0

7"yl'2'=O'yl[2+ (o.y2] j

= independent of 0.

I I I l ' ' ' l l k ) ' ' ~ ' ' J ' l ' ' ' ' ' l l i l l

0

(90)

30

60 9O 8 Fig. 3. Yield stress of the oriented polymer.

12

J.J. Pereda et al. / Anisotropic polymers

tion temperature of the polymer, and 6 is a material constant.

4.1. The strain induced heating We consider full thermomechanical coupling and a fraction of the rate of plastic dissipation appears as a heat source term in the energy balance equation. The following version of the energy equation is assumed in the deformed configuration

An outline of the integration scheme used in the computations is given in the following• The starting point is Eq. (6) which with (35) can be written as i~iP = L p " F i p = fit( Ni p + a i P ) • F p .

(93)

The direction of plastic flow is assumed to be constant over the increment and equal to (Ni p + O.P,)n=-X~. Integration of the above equation yields FnP+I1 = Vnp-I" exp( - AAX.)

k V2T+ f f W P = p c T ,

(92) =Fn p-I

where k is the thermal conductivity, c is the specific heat, p is the mass density in the deformed configuration ~', and ff is the fraction of plastic work that is converted into heat. Adams and Farris (1988) made deformation calorimetric measurements during uniaxial cold drawing of poly(bisphenol A carbonate) (PC) and found that ~" takes values in the range of 0.5-0.8. Higher values have been reported for polyethelene (see Fager and Bassani, 1990). Adams and Farris also report that the fraction of work dissipated varies with strain. The elastoplastic constitutive equations presented in Section 3 are rate-independent; however, in view of the thermomechanical coupling (see Eq. (92) above), the solution of thermomechanical boundary value problems does depend, in general, on the rate of the applied loading.

•

[I-AAXn+

~'

AA2X

2

+ O(ha3)] (94)

which is truncated to F n P- + , I - = F n P - I ' ( I - A A X n + T A A1 X n ) 2. 2

The evolution equation for the equivalent plastic strain is integrated using a forward Euler technique. A summary of the algorithm for the integration of the constitutive equations is given in the following 1 2 2 Fe+,=Ftriat'(l-AAXn+~AAXn),

(96)

e -Cn+l - E neT + l • f ne+ l ,

(97)

Een+l ~-- l ( c n e + l - - I ) ,

(98)

S~+1 = Po

,

In a finite-element environment the solution of the elastoplastic problem is developed incrementally and the constitutive equations are integrated at the element Gauss points. In a displacement based finite-element formulation the solution is deformation driven. At a material point, the solution (~n, ePn,Ff, Fn, T~) at time tn as well as the deformation gradient Fn+ 1 and the temperature T,+~ at time tn+ 1 = t , + At are supposed to be known and one has to determine the solution ( ' ~ n + l ' •P+I' FP+I )"

(99)

n+l

~n+l = s e + l ' C ,¢+ l ,

5. Numerical integration of the elastoplastic equations

(95)

=

°,

(101)

e,,P+, = F,, +1" r e + ? ,

(102)

BP+ 1 =

+ AA

(100)

FP+ 1 • F~+ pT 1 =,. (AP)n+I,

f('~'n+l,

(103)

A,, A 2, (AP),,+,, C + , , T.,+,)= 0, (104)

Fteial ~-- F n + 1" Fnp - 1. W e c h o o s e AA a s t h e primary unknown, treating the yield condition (104) as the basic equation for its determination. The solution is obtained using Newton's method. When the constitutive equations are properly normalized, AA can be interpreted as the magni-

where

J.J. Pereda et al. / Anisotropic polymers tude of the plastic strain increment. In such a case, it can be readily shown that

13

X3

F

Xa

f D

det(FP+l) = 1 + O(AA2),

(105) 1

i.e., the above algorithm preserves plastic incompressibility to within terms of O(AA 2) in comparison to unity (Aravas, 1992).

> A

B Xl

xI

C

6. Applications The constitutive model presented in the previous sections is implemented in the ABAOUS general purpose finite-element program (Hibbitt, 1984). This code provides a general interface so that a particular constitutive model can be introduced as a 'user subroutine'. The integration of the elastoplastic equations is carried out using the algorithm presented in Section 5. The finiteelement formulation is based on the weak form of the momentum balance and the energy equation, and the discretized non-linear equations are solved using Newton's method. In our calculations, we approximate the Jacobian of the Newton scheme by the tangent stiffness matrix derived from Eq. (80). Such an approximation is first-order accurate as At ~ 0; it should be emphasized, however, that the aforementioned approximation influences the rate of convergence only and not the accuracy of the results. The material constants used in the calculations are E = 45 MPa, v = 0.3, a = 10 -6 K -1, E0 = 0.05, n = 5, /3 = 1.41985, M = 0.5556, ~j = 0.4, C 1 = 1, C z = 0.05, a l = 2, "~0 = 1.8 × 108 MPa, Tm = 400 K, 6 = 1.5, k = 0.3 W / ( m 2 K), c = 1850 J / ( k g K), and P = 950 k g / m 3. The value of the fraction of work dissipated ~ is assumed to be constant and equal to 0.55. In all cases analyzed, the elastic properties are assumed to be temperature independent and a uniform initial temperature of To = 298 K is used.

\ Fig. 4. Simple shear.

vious section a single element is subjected to plane-strain simple shear transformation 3' along the x 1 direction as shown in Fig. 4. The material is assumed to be rigid-plastic obeying the plasticity equations described in Sections 3.2 and 3.3. This problem has been also analyzed by Dafalias and Rashid (1989) using a different plastic spin. The deformation gradient F is written as (106)

F = I + Y e l e 3.

The eigenvalues of ( F T. F ) 1/2 are 1

A1,3=[I+~T

2

1

2\1/2]

_-_+~/(l+zy )

]

1/2

,

A2=l (107)

and the corresponding eigenvectors in the deformed configuration .~' are M 1 = cos 0 e 1 + sin 0 e3, M 2 = e 2, M 3 = - s i n 0 e I + cos/9 e3,

(108)

where 6.1. Plane-strain shear

In order to check the consistency of the finiteelement formulation and the numerical implementation of the algorithm described in the pre-

0 = ~l a r c t a n ( 2 / y ) .

(109)

Since the material is rigid-plastic, the elastic part of the deformation gradient is a rigid body rotation, i.e., F e = RL Therefore, the plastic stretches

J.J. Pereda et al. / Anisotropic polymers

14

Ap are the same as the eigenvalues of ( F . F T ) 1/2 given in (107), and the eigenvectors m i of B p = F O T . F p in the intermediate configuration are given by m i = R eT "M i. At first yield, i.e., as y -~ 0 +, F, R e and F p all approach the second-order identity tensor I. Therefore, in that limit, the AP's and mi's behave as follows A~---)I +,

A~= 1,

A~I-

readily show that At O'11

^¢ 0"33

^ 0"13

C11

C33

el3

=

× [ F ( C , t + 2C33) 2 + G(C33 - C,,) 2

(110)

and

r)}

+ H ( 2 C 1 , + C33) 2 + 2MC23]

1/2~115)

where 1

m I ~ ~-(e

1 2F+H C n = 1--2 F G + G H + H F sin 20,

I +e3),

B)I 2 = e 2 , C33 =

1 m3 ~ - - ~ - ( - e l + e 3 ) . V2

F+2H 2F+HCll,

1

C13

4MCOS 20.

(116)

(111)

The rigid rotation F e - R e is chosen so that the orientation of the mi's at oqffi remains fixed for all times, i.e., ~'i is isoclinic. Therefore, referring to Fig. 4i we can write

The definition of the equivalent plastic strain rate (Eq. (53)) also implies that de p dy

1 I[ I~ F +H 2 G + G H + H F sine20 2

R e = cos 4) ( e l e I + e3e3) + sin 4) ( e l e 3 - e 3 e l )

+ e2e 2

+ ~cos220)

~ 1/2

.

(117)

(112) Assuming that there are no heat losses through the boundary of the specimen, we can write the heat-transfer equation as

and F p = cos 4) ele I + ( y cos 4) - sin 4))ele 3

dT + sin 4) e3e 1 + (',/sin4) + cos 4))e3e 3

where 4) = ~r/4 - 0. The non-zero components of the rate of deformation are = 7y 1 • sin 20,

pep

(118)

(113)

+ e2e2,

D = l- D 3 3l

de p

(~ - if--.

bl 3

= 7Y 1 • cos 20, (114)

where a caret indicates components with respect to the 21 - 2 3 coordinate system as shown in Fig. 4. It should be noted that, because of incompressibility, the stresses can only be determined to within an arbitrary pressure. Using the yield condition, the plastic flow rule, and the plane strain condition D22 = 0, we can

The last two equations arc integrated numerically via a forward Euler scheme, and the stress components are calculated from Eq. (115). Finally, the stress components with respect to the global coordinate system x~-x 3 are obtained through a simple Mohr circle transformation. In Figs. 5 and 6 the deviatoric stress components and the temperature are plotted versus 3'. In both figures, the solid lines are the results of the rigid-plastic calculations and the open symbols indicate the results of the elastic-plastic finiteelement analysis. The elastic moduli used in the finite-element calculations are three orders of magnitude larger than 0-0(T0) so that the role of

J.J. Pereda et al. / Anisotropic polymers

15

A specimen of initial length to width ratio

1.50

l/w = 2 is considered. Because of symmetry, only

o,3/oy(To)

1.00

o./oy(To) 0.500

0.000

tr33/Cry(Zo) 0.

0.500

1.00

1.50

2.{30

"7

Fig. 5. Evolution of stresses in simple shear.

elasticity becomes secondary. The results of the finite-element calculations agree well with the analytical solution.

6.2. Plane-strain necking Next we consider the plane-strain neck propagation in polymeric sheets. This problem has been also analyzed in detail using analytical and numerical techniques by Tu~cu and Neale (1987), and Fager and Bassani (1986; 1990).

320.

r(~:) 310.

300.

290.

280.

i

,

,

,

i

0.500

,

,

,

,

I

1.00

,

,

,

,

I

,

,

1.50

7 Fig. 6. E v o l u t i o n of t e m p e r a t u r e in s i m p l e shear.

i

,

I

2.00

one quarter of the specimen is analyzed. A small geometric imperfection Aw = 0.01w is introduced at the center of the specimen in order to initiate necking. Coupled temperature-displacement analyses are carried out using four-node isoparametric elements with 2 × 2 Gauss integration and an independent interpolation for the dilation rate in order to avoid artificial constraints on incompressible modes (Nagtegaal et al., 1974). Three different cases are studied. The m a t e rial parameters given in the beginning of Section 6 are used in the computations. First, we analyze isothermal necking in which all thermal effects are ignored. In the other two cases the coupled thermomechanical problem is solved for two different values of the nominal strain rate, namely ~0 = 0.005 s-1 and 0.5 s-1. We assume that there is no heat transfer between the specimen and the grips. There is natural convection heat transfer between the wall and the environmental air at 298 K. Assuming that the test is conducted in a vertical machine, we estimate the film coefficient of heat transfe? using the formula (Incropera and De Witt, 1981)

hi 4 (_G~)l/4 Nul

ka

3

g(Pr),

(119)

where Nu t is the average of the Nusselt number on the wall, h is the film coefficient, k a is the thermal conductivity of the air, Gr~ is the Grashof number, and g(Pr) is a known function of the Prandlt number. Using the last equation, we estimate the film coefficient to be h --- 10 W / ( m K). The development of the plastic zone is shown in Fig. 7 for the isothermal case. Initially, the whole specimen deforms plastically. When the neck forms, the active plastic zone decreases due to elastic unloading and is limited to the necking region. Finally, when the neck starts to propagate, the plastic zone spreads again toward the ends of the specimen. A similar behavior is observed in the other two cases analyzed. Figure 8 shows the load-deflection curves for the three cases analyzed. The letters A - B - C - D - E on the

J.J. Pereda et at /Anisotropic polymers.

16

Temperature (K) A~ = 0.125 l0

1 2 3 4 5 6

300 302 304 306 308 311

A~ = 0250 I()

Al

=

0.375 Ternperature (K)

~t = 0.500 1(i

1 2 3 4 5 6 7 8 9

300 302 304 306 308 311 313 315 317

S t r a i n R a t e d = 0.005s -1

_xj = 0.625 lO

Fig. 7. D e f o r m e d finite-ele~nent m e s h e s a n d active plastic zones ( i s o t h e r m a l case).

S t r a i n R a t e d = 0.Ss - l Fig. 9. C o n t o u r s of t e m p e r a t u r e at a n o m i n a l strain A I / I o = 0.625.

isothermal curve correspond to the five stages of deformation shown in Fig. 7. Figure 9 shows temperature contours for the two different nominal strain rates analyzed. The maximum increase in temperature is 15 K and occurs in the neck region of the specimen corresponding to the larger nominal strain rate.

1.5

I A

F Woao(To)

B

C

...... ....

Isothermal 0.005 s l 0.5 s "1

Another set of calculations was carried out using a nominal strain rate ~0 = 5 s - t . The solution is almost identical to that of ~0 = 0.5 s-1. It appears, therefore, that, for the geometry and material analyzed, when the nominal strain rate is of the order of 10 -~ or larger, the effects of heat conduction are minimal and the solution is practically adiabatic as analyzed by Fager and Bassani (1990). 6.3. Plane-strain extrusion

We consider plane-strain extrusion of a polymeric sheet. The height reduction is A h / h o = 1 / 4 and the length of the reduction region is L = 2h 0 (see Fig. 10), where h 0 is the height at the start of

1

.5

0

0

.125

.25

.375

LXUIo Fig. 8. L o a d - d e f l e c t i o n curve.

.5

.625 Fig. 10. T h e d e f o r m e d f i n i t e - e l e m e n t m e s h a n d the plastic zone.

J.J. Pereda et aL / Anisotropic polymers

the reduction region. The reduction area of the die is shaped in the form of a fifth-order polynomial with zero slope and curvature at both ends. A rigid smooth piston pressing against the rear face of the billet provides the driving force. The coefficient of friction along the die-polymer interface is assumed to vanish and any heat loss to the die is neglected. The coupled thermomechanical problem is solved and all walls are assumed to be adiabatic. The material properties used in the calculations are those listed in the beginning of Section 6. Four-node elements, similar to those used in the polymer drawing problem, are used in the computations. Figure 10 shows the deformed finite-element mesh and the extent of the active plastic elements at the end of the calculation. Notice the so-called 'die-swell' at the exit of the die where elastic unloading occurs; this is due to the recovery of the elastic strains which are of the order of 5%. Figure 11 shows the variation of the longitudinal stress Crxx along lines passing through different rows of elements in the billet; in that figure, the open triangles, circles, and squares correspond to the lower, fifth, and top row of elements, respectively. Figure 12 shows the variation of the residual longitudinal stress across the section of the billet after the exit from the die. It is interesting to note that the longitudinal residual

17

0.8

o6

h/ho

0.4

0.2

-0.4

-0.2

0.0

0.2

0.4

O'xx/Cr 0

Fig. 12. Residual-stress distribution across the section of the billet.

stress is compressive near the outer surface of the billet; a similar residual stress pattern has been found by Aravas (1991; 1992) who analyzed the plane strain extrusion of anisotropic metals. This contrasts with the results obtained for isotropic materials where the residual stress is found to be tensile near the free surface (Lee et al., 1977). Several calculations were carried out using different piston velocities. In al cases, the maximum calculated temperature rise is 3 K. It appears, therefore, that, for most practical cases, the temperature effects can be neglected in cold extrusion of polymeric sheets.

1.00

7. Closure 0.600

0.200

O'xx/ O'O -0.200

-0.600

-1.00 . -2.00

.

.

. 0.

. 2,00

4.00

6.00

x/xo Fig. 11. Distribution of the longitudinal stress qxx.

8.00

The orienting of long molecular chains during finite-strain deformation has rather significant implications on the mechanical behavior of polymers. These include the dramatic stiffening at large strains, which has been widely recognized and characterized, whereas the concomitant evolution of mechanical anisotropy and internal (frictional) heating has been less fully appreciated. For amorphous and semi-crystalline polymers, a phenomenological constitutive theory has been developed where the principal axes of anisotropy are aligned with the plastic stretch so that the anisotropy is generally orthotropic. Consistent

18

J.J. Pereda et al. / Anisotropic polymers

with experiments, the polymer is taken to be relatively stiffer in the direction of the maximum stretch and softer in transverse directions. Consequently, in a natural drawing process, for example, the resistance to any lateral constrains is less than it would be in a truly isotropic setting, so that relative to an isotropic prediction, the evolving anisotropy leads to a smaller drawing load and a greater draw ratio (Batterman and Bassani, 1990). Relative to isothermal deformations, internal heating tends to also decrease the drawing load and increase the draw ratio (Fager and Bassani, 1990). in this paper, a rigorous finite-strain constitutive model has been derived where the anisotropy is expressed in the intermediate (isoclinic) configuration with respect to the orientation of the principal axes of plastic stretch. The use of the isoclinic configuration has the advantage of yielding tractable rate forms of the anisotropic constitutive equations in terms of co-rotational derivatives formed on the spinning plastic stretch triad. These equations are readily incorporated into standard finite-strain finite-element algorithms. In closing, we note that use of the isoclinic configuration for defining anisotropy means that the principal axes of orthotropy are not precisely aligned with the principal axes of the total stretch in the current configuration; in fact, the difference is of the order of elastic strains and disappears in the limit of rigid plasticity. However, when elastic strains are finite, which is the case in rubbery materials and possibly in moderately cross-linked polymers, this idealization needs further investigation.

Acknowledgements JJP and NA acknowledge the support of NSF (Grant No. MSM-8657860) and Alcoa (matching funds). JLB acknowledges the support of the NSF MRL program at the University of Pennsylvania under Grant No. DMR-9120668. The ABAOUS general purpose finite-element program was made available under academic license from Hibbitt, Karlsson and Sorensen, Inc., Providence, RI.

References Adams, G.W. and R.J. Farris (1988), Latent energy of deformation of bisphenol A polycarbonate, J. Polyme Sci.: Part B: Polym. Phys. 26, 433. Aravas, N. (1991), Finite strain anisotropic plasticity: Constitutive equations and computational issues, in: N. Chandra and J.N. Reddy, eds., Advances in Finite Deformation Problems in Materials Processing and Structures, AMD-Vol. 125, ASME, New York, p. 25. Aravas, N. (1992), Finite elastoplastic transformations of transversely isotropic metals, Int. J. Solids Struct. 29, 2137. Bassani, J.L. and S.D. Batterman (1987), Deformation induced anisotropy and neck propagation in polymers, in: V.K. Stokes and D. Krajcinovic, eds., Constitutive Modeling for Nontraditional Materials, AMD-Vol. 85, ASME, New York, p. 33. Batterman, S.D. and J.L. Bassani (1990), Yielding, anisotropy, and deformation processing of polymers, Polym. Eng. Sci. 30, 1281. Biot, M.A. (1965), Mechanics of Incremental Deformations, Wiley, New York. Boyce, M.C., E.L. Montagut, and A.S. Argon (1991), The effects of thermomechanical coupling on the cold drawing process of glassy polymers, in: V.J. Stokes, ed., Plastics and Plastic Composites: Material Properties, Part Performance, and Process Simulation, MD-Vol. 29, ASME, New York, p. 47. Boyce, M.C., D.M. Parks and A.S. Argon (1988), Large inelastic deformation of glassy polymers, Part I: Rate-dependent constitutive model, Mech. Mater. 7, 15. Boyce, M.C., D.M. Parks and A.S. Argon (1989), Plastic flow in oriented glassy polymers, Int. J. Plast. 5, 593. Brown, D.J. and A.H. Windle (1984), Stress-orientationstrain relationship in non-crystalline polymers, J. Mater Sci. 19, 1997. Brown, N., R.A. Ducken and I.M. Ward (1968), The yield behavior of oriented polyethylene terephthalate, Philos. Mag. 18, 483. Caddell, R.M., R.s. Raghava and A.G. Atkins (1973), A yield criterion for anisotropic and pressure dependent solids such as oriented polymers, J. Mater. Sci. 8, 1641. Dafalias, Y.F. (1983), On the evolution of structure variables in anisotropic yield criteria at large plastic transformations, in: J.P. Boehler, ed., (Crit~res de rupture des matdriaux a structure interne orient~e, Colloque International du CNRS), No. 351, Villard-de-Lans, France, June, 1983, Editions du CNRS, Paris, in press. Dafalias, Y.F. (1984), The plastic spin concept and a simple illustration of its role in finite plastic transformations, Mech. Mater. 3, 223. Dafalias, Y.F. (1987), Issues in constitutive formulation at large elastoplastic deformation, Part 1: Kinematics, Acta Mech. 69, 119. Dafalias, Y.F. (1991a), Constitutive model for large viscoelastic deformations of elastomeric materials, Mech. Res. Commun. 18, 61.

J.J. Pereda et al. / Anisotropic polymers Dafalias, Y.F. (1991b), Constitutive modeling of polymeric materials at large deformations, in: C.S. Desai, E. Krempl, G. Frantziskonis and H. Saadatmanesh, eds., Constitutive laws for Engineering Materials, (Proc. of the Third Int. Conf. on Constitutive Laws for Engineering Materials: Theory and Applications, January, 1991, Tucson, Arizona, USA), ASME Press, New York, p. 587. Dafalias, Y.F. and M.M. Rashid (1989), The effect of plastic spin on anisotropic material behavior, Int. J. Plast. 5, 227. Fager, L.O. and J.L. Bassani (1986), Plane strain neck propagation, Int. J. Solids Struct. 22, 1243. Fager, L.O. and J.L. Bassani (1990), Neck propagation in polymers with adiabatic heat generation, Mech. Mater. 9, 183. Freund, L.B. (1970), Constitutive equations for elasti-plastic materials at finite strain, Int. J. Solids Struct. 6, 1193. Hibbitt, H.D. (1984), A B A Q U S / E P G E N -- A general purpose finite element code with emphasis on nonlinear applications, Nucl. Eng. Des. 77, 271. Hill, R. (1950), The Mathematical Theory of Plasticity, Oxford University Press. Hill, R. (1970), Constitutive inequalities for isotropic elastic solids under finite strain, Proe. R. Soe., Set. A 314, 457. Incropera, F.P. and D.P. De Witt, (1981), Fundamentals of Heat Transfer, Wiley, New York. Lee, E.H. (1969), Elastic-plastic deformations at finite strains, J. Appl. Mech. 36, 1. Lee, E.H. and D.T. Liu (1967), Finite-strain elastic-plastic theory with application to plane wave analysis, J. Appl. Phys. 38, 19. Lee, E.H., R.L. Mallett and W.H. Yang (1977), Stress and deformation analysis of the metal extrusion process, Cornput. Meth. Appl. Mech. Eng. 10, 339. Loret, B. (1983), On the effects of plastic rotation in the finite deformation of anisotropic elastoplastic materials, Mech. Mater. 2, 287. Mandel, J. (1971), Plasticitd Classique et V'tseoplasticitd, Courses and Lectures, No. 97, International Center for Mechanical Sciences, Udine, Springer-Verlag. Mandel, J. (1973), Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques, Int. J. Solids Struct. 9, 725. Mehrabadi, M.M. and S. Nemat-Nasser (1987), Some basic kinematical relations for finite deformations of continua, Mech. Mater. 6, 127. Nagtegaal, J.D., D.M. Parks and J.R. Rice (1974), On numerically accurate finite element solutions in the fully plastic range, Comput. Meth. Appl. Mech. Eng. 4, 153. Nemat-Nasser, S. (1990), Certain basic issues in finite-deformation continuum plasticity, Meccanica 25, 223. Parks, D.M., A.S. Argon and B. Bagepalli (1984), Large elastic-plastic deformation of glassy polymers, Part I: Constitutive modeling, MIT Program in Polymer Science Report, Massachusetts Institute of Technology. Pereda, J.J. (1992), Finite thermomechanical deformations of anisotropic polymers, Master's Thesis, University of Pennsylvania, Philadelphia, PA.

19

Shinozaki, D. and G.V. Groves (1973), The plastic deformation of oriented polypropylene Tensile and compressive yield criteria, J. Mater. Sci. 8, 71. Tu~cu, P. and K.W. Neale (1987), Analysis of plane-strain neck propagation in viscoplastic polymeric films, lnt. J. Mech. Sci. 29, 793. Zbib, H.M. and E.C. Aifantis (1988), On the concept of relative and plastic spin and its implications to large deformation theories. Part I: Hypoelasticity and vertex-type plasticity, Acta Mech. 75, 15.

Appendix The rate of change of the principal plastic s t r e t c h e s is d i s c u s s e d b r i e f l y in t h e f o l l o w i n g . Using the polar decomposition theorem, we can write

FP=R

p • UP=VP'R

3 p = Y'~ a y m j M y , j=l

(120)

where R p-1 = R pT, u p T = u p, vPT= V p, ay are the principal plastic stretches, and (mi, Mj) are the eigenvectors of (V p, UP). Differentiating (120) with respect to time, we find 3 ~'P = E (~.ymjMy + a y f a y M j + APmjMy). j=l

(121)

Using the above equation and taking into account that m j . thj = M j . ~,lj = O, (no sum over j),

(122)

we conclude that •~P

my" I ~p " M j,

=

(no sum over j).

(123)

T h e r a t e o f c h a n g e o f F p c a n b e w r i t t e n as

Pp =

+ w?).

= (fi)(NiPq-~'~iP) "

APmjMy J

.

(124)

20

ZZ Pereda et aL / Anisotropic polymers

Finally, substituting /fP into (123) we find that

where

Ap" = (~t)~ p,

-P-APmj'(NiO+OY)'mj,Aj-

(~25)

(no sum over j ) .

(126)