Constitutive relationships for anisotropic high-temperature alloys

389

Nuclear Engineering and Design 83 (1984) 389-396 North-Holland, Amsterdam

CONSTITUTIVE

RELATIONSHIPS

FOR ANISOTROPIC

HIGH-TEMPERATURE

ALLOYS* D.N. ROBINSON

University of Akron, Akron, OH, USA Received April 1984

A constitutive theory is presented for representing the anisotropic viscoplastic behavior of high-temperature alloys that possess directional properties resulting from controlled grain growth or solidification. The theory is an extension of a viscoplastic model that has been applied in structural analyses involving isotropic metals. Anisotropy is introduced through the definition of a vector field that identifies a preferential (solidification) direction at each material point. Following the development of a full multiaxial theory, application is made to homogeneously stressed elements in pure shear and to a uniaxially stressed rectangular block in plane stress with the stress direction oriented at an arbitrary angle with the material direction. It is shown that an additional material parameter introduced to characterize the degree of anisotropy can be determined on the basis of simple creep tests.

1. Introduction The need for greater efficiency in aircraft engines places increasing demands on the high-temperature structural alloys used for engine components. As higher operating temperatures are sought, advanced materials are being developed to meet these increased demands. Good examples are the single crystal (SC) and directionally solidified (DS) polycrystalline materials finding application as turbine airfoil components. An advantage of these materials over conventionally cast alloys is their increased strength (e.g., creep and creep-rupture strength, yield strength, etc.) in a preferential (grain growth) direction, which in the case of a turbine blade can be advantageously oriented radially (centrifugally). Improved creep and creep-fatigue properties result as well as reduced susceptibility to grain boundary corrosion and oxidation. The directional properties of SC or DS metals render them highly anisotropic relative to conventional alloys. This introduces additional complexity in understanding and mathematically representing their mechanical behavior over and above the already * Research sponsored by the National Aeronautics and Space Administration, Lewis Research Center, Cleveland, OH, 44135, USA.

enormous complexities associated with elevated temperature. Here, the unifiecl constitutive model of Robinson [1,2] that has found application in representing important behavioral features of high-temperature isotropic metals is extended to account for the effects of anisotropy. Each material point is taken to have a uniquely identifiable direction designated by a vector. A n extended material body is thus treated as being locally transversely isotropic although the preferential direction may vary from point to point as represented by a vector field. It is believed that this relatively simple model captures the essence of anisotropy as induced by directional grain growth and solidification without undue complication. The isotropic, isothermal form of the Robinson viscoplastic theory is first discussed with emphasis on its derivability from a potential function. Full isotropy is treated by taking the applied and internal stress dependence of the potential function in terms of the principal invariants of the stress tensors. The extension to anisotropy is made by replacing the principal invariants with another set of stress invariants that reflect the appropriate material symmetry. Following the development of the full multiaxial theory, application is made to simple states of shear stress oriented transverse to and along the preferential material direction. A final application is made to a

0029-5493/84/$03.00 O Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

390

D . N . Robinson / Constitutive relationships for high-temperature alloys

uniaxially stressed rectangular block of material in plane stress with a uniformly oriented material direction taken at an arbitrary angle with the direction of stress.

2. The isotropic viscoplastic model The flow and evolutionary equations in the Robinson model are taken to be derivable from a potential function g] of the applied and internal stress. The components of these stress tensors are denoted by ~q and a,i, respectively. Thus, we have O = O(cr,, ~,,).

(1)

and inelastic strain rate over a volume of material that is large compared to the crystal size. The derivative in eq. (6), however, as applied to an individual slip system, requires constant local stress, whereas in the present context it implies constant average stress. As pointed out in ref. [4], it is not generally possible to assume that constant average stress implies constant local stress and, consequently, that eq. (6) remains exactly true for a polycrystalline material. Nevertheless, constitutive relationships have been derived form eqs. (5) and (6) that are consistent generalizations of well established classical equations (refs. [3,4]) and that are capable of accurately representing important features of the high temperature behavior of metals including rate-dependent plasticity, creep, recovery and their interactions (refs. [1,2]). The function ~ in the Robinson model can be written as

For the sake of simplicity, the present development is restricted to isothermal conditions. Extension to nonisothermal conditions follows the development presented in refs. [1,2]. As moderate hydrostatic stress is known to have essentially no effect on inelastic behavior, the stress dependence is taken in terms of the deviatoric components of the applied stress

where the stress dependence enters through the scalar functions

Sq = o'q - ~ trkk6,i,

F(,~j) and G(a,,)

(2)

and of the internal stress aq = aq - ~-akkS,j,

(3)

where the symbol 6q denotes the usual Kronecker delta. We further identify ~,q = S o - a o

(4)

as the effective stress. The potential nature of ~ is expressed by d,j = Og][Oo'#

1

R

dG)

(8)

taken as depending on the effective and internal stress, respectively. The functions f and g and the material parameters K, /x, R, and H are assumed known for present purposes; they are determined as described in earlier writings (refs. [1,2]). Under full isotropy, the functions F and G can be taken to depend on the principal stress invariants A = ~ x,j~j,, L -~-1-~,j-~jk-~k, ,

(9)

(5) and

and - fio/h = O.O/Octq,

2

(6)

where i~i represents the inelastic strain rate and h is a scalar function of the internal stress. Eq. (5) is termed the flow law; eq. (6) is termed the evolutionary law. The appropriateness of eqs. (5) and (6) has been discussed on physical and thermodynamical grounds by several authors including Rice [3], Ponter and Leckie [4], Valanis [5], and Robinson [6]. Eqs. (5) and (6) are shown in ref. [4] to hold exactly for an individual slip system in a polycrystalline metal deforming at high temperature where aq is interpreted as the local internal flow stress on the slip plane. Eq. (5) remains valid fief. [3]) for a polycrystalline metal where o'~j and ~0 are interpreted as the average stress

~2 - 1 - ~ aqa~, , ~3 = ~ aqa~kak, .

(10)

In the spirit of von Mises, we retain only -/2 and ~2, quadratic in stress, and take F = ~--~- I

(ll)

and G = •2/K 2.

(12)

Eq. (11) plays the role of a (Bingham) yield condition with K denoting the magnitude of the threshold shear stress below which inelastic deformation does

D.N. Robinson / Constitutive relationships [or high-temperature alloys

not occur - inelastic strain occurs only for F > 0. For our purposes, we treat K as a constant; more generally it is considered a scalar state variable. The flow and evolutionary equations are determined directly from eqs. (5) and (6) making use of eqs. (7) to (12) and taking h = High(G)

(13)

in eq. (6). The details of this development are given in Appendix A. Here we state just the result, i.e., the flow law 2 ue,, = [(F)Z,~

(14)

H diJ = gh(G) 4ij -- Rgr(G) a#,

(15)

in which g,(G) = g ( G ) / g h ( G )

3. E x t e n s i o n t o a n i s o t r o p y

The direction of grain growth or solidification at each point in a SC or DS solid can be characterized by a field of unit vectors di(xk). The mechanical behavior at each point must then depend not only on the stress and deformation history at the point but also on the local preferential direction. This requires that dependence on d~ be included in F and G in eq. (8). However, as the sense of di is immaterial, the dependence is properly taken in terms of the product didj. Thus, we replace eq. (8) with F(Z~j, d~d) and G ( % , d,d) .

and the evolutionary law

(16)

This is essentially the form of the Robinson model for a fully isotropic material and for isothermal conditions. Some important features of the model, such as the accompanying inequalities (refs. [1,2]), are not expressed or discussed here as they do not pertain directly to the extension to anisotropy. The evolutionary law [eq. (15)] is of the widely accepted Bailey-Orowan type, which presumes that high-temperature deformation occurs under the action of two simultaneously competing mechanisms, a hardening process proceeding with accumulated deformation [characterized by the first term in eq. (15)] and a recovery term proceeding with time (characterized by the second term). Steady state then corresponds to the condition where the two competing mechanisms balance and d 0 = 0. In most applications of the theory to date, the function f has been chosen as

391

(21)

As indicated in Appendix B, the theory of tensorial invariants (refs. [8-12]) requires that for form-invariance under arbitrary rigid-body rotations F and G must be expressible in terms of the principal invariants of their respective tensorial arguments and invariants involving various products of these tensors. Here, as argued in Appendix B, we take the functions F and G as depending on the subset of these invariants, It --- ~t ZijZji

,

(22)

12 = d,d,Z,k,Y,k, , / 3_ --- :1 a , a , g ~

for F, and ~l = ½% % ,

(23)

•¢2 = d, dja, kak,, & = ½d, g a , i ,

for G. Thus, we take I1 1 1 F = ~5 + (~-~)(I2~,

1

1

~)- 1 ,

(24)

(25)

f( F) = F o ,

or

(17)

[ ( F ) = (sinh F ) " ,

and, as suggested by the experimental results of Mitra and McLean [7], g(G) = G m ,

(18)

gh(G) = 1 / G ¢~ ,

(19)

so that gr(G) = G " 0, where n, m, and /3 are constants.

(20)

As in the fully isotropic development we have sought generalizations of a v o n Mises type theory and have, therefore, restricted our choice of invariant expressions to those quadratic in stress. Analogous to the earlier development, K denotes the threshold (Bingham) shear stress transverse to the preferential material direction and Kd denotes the same for shear along the material direction (fig. 1). For K = Kd, indicating no difference in shear strength across and along the direction d~, eqs. (24) and (25) reduce to their isotropic counterparts [eqs. (11) and (12)]. As before, the flow and evolutionary equations are

D.N. Robinson / Constitutive relationshipsfor high-temperature alloys

392

obtained from eqs. (5) and (6), this time by making use of eqs. (7), (13), and (22) to (25). Again, the details are reserved for Appendix A. The resulting flow law is given by

' dtdk~kt(t~ii

-4- d~dj)]]

.

(26)

(28)

constant,

and 0/22 ~

0/33 =

0/12 ~

0/13 ~

0 ,

(29)

From eq. (24) we obtain for F

( ~ - s)2 F

K2

gtq= g~-'~ E#- Rgr( G)[ a# + (-~d-1)[ dflkaki + dkd~ajk -- ~dfdkakt( 6,j + d,dj)]

T ~

0/23 = s # 0.

and the evolutionary law by H

0"23 ~

0/I 1 ~

21z~# = f(F)[ Zo + (~a--1)[dflk~k~ + dkd~Zjk --2

~ l t = 0"22 = 0"33 = o"12 = 0 - t 3 = 0 ,

.

(27)

=

K2

1 ,

which depends only on the parameter K. From eq. (26), the flow law for the shear rate component 5'23=-5',r is

• = ((~2s) 2

Note that ~, = 0 and d, = 0 , the former indicating incompressibility of the inelastic deformation and the latter confirming the deviatoric nature of %. As before, when K = Ka eqs. (26) and (27) reduce to those of the isotropic case [eqs. (14) and (15)]. Recall that the functions f, g and gh and the material constants K, /~, R, and H are determined just as in the isotropic case. Determination of Ko or alternatively the ratio K/Ka is discussed in the following section.

(30)

/xy,r

\

K2

)"

1

(r-s),

(31)

where we have made use of the first of eqs. (17). Now for creep under longitudinal shear [fig. l(b)], we write 0"11 =

0"22 ~

0"12 ~

"/" ~

0-33 ~

O"13 ~

0-23 ~

0 ,

(32)

constant,

and 0/I I =

0/22 =

0/33 =

O/13 =

0/23 =

0 ,

(33) a,2 = g# 0. This time F from eq. (24) is

4. Applications

(~._ g)2 F-

We first consider applications of the foregoing theory to the cases of homogeneously stressed elements in pure shear, transverse [fig. l(a)] and longitudinal [fig. l(b)] to the preferential material direction. In each case, the x, coordinate direction is aligned with the material direction, i.e., d = (1, 0, 0). For creep in transverse shear [fig. l(a)], we have

Xl

X2 (al Transverse shear state.

X3

T T0 X2 (b) Longitudinal shear state.

Fig. 1. Homogeneously stressed elements in pure shear.

1

(34)

which now depends only on Ka. The flow law for "Y12~ "i'~ois likewise obtained from eq. (26) and is ((~._ g)2 ~ " " = \ K~a

1)" K2

(-~)('r-g).

(35)

With F ~>0 and s > g > 0 (corresponding to the initial stage of a creep test) we obtain by dividing eq. (35) by eq. (31) and solving for K2/K~o,

K2_

X1

T 0, 00,

K~-

K2

X3

\~',/

(36)

As the parameter n is assumed known, the ratio K/K~ is defined by eq. (36) in terms of the ratio of creep strain rates along and transverse to the preferential material direction• Simple shear tests of this type can, in principle, be used to determine the ratio K/Kd; however, a more practical method on the basis of uniaxial tests will be suggested in the following paragraphs. Next consider an application of the theory to that

D.N. Robinson / Constitutiverelationshipsfor high-temperature alloys 1 = ~

f(v)(o-

~)

~

393

-

x (4 cos 2 9 + 3 cos 2 9 sin 2 9 + sin 2 9 ) ] , _

×2

&

3H

1

1 +~

2gh(G) ~ - R g ' ( G ) a

(41)

-1

x (4 cos 2 9 + 3 cos 2 9 sin 2 9 + sin2 9)] •

(42)

As expected, the in-plane shear strain rate 4/= 4/~2 is not generally zero and is given by:

= ~ f ( F ) ( o - a) Fig. 2. Rectangular uniaxial element in plane stress. Material direction is oriented at an angle q~ with the x~ axis.

of uniaxial plane stress as depicted in fig. 2 T h e plane stress element lies in the plane x3 = 0 with the tensile stress applied along x~. T h e preferential material direction is constant throughout the body and, in this case, makes an angle 9 with the x~ axis (fig. 2). T h e stress components are 022 ~

033

:

012 ~

O13 ~

- 1 sin 2 9 sin 2 9 .

T h e shear strain rate is zero when the preferential material direction is 9 = 0 ° or 9 = 90 ° and when the material is isotropic, i.e., K/Kd = 1. With 9 = 0 ° and f ( F ) = F", eqs. (39) and (41) become G,-

( 0 " - 0l) 2 3 ~

1,

(44)

F~,(K~) ( o - a) -

(45)

023 ~ 0

~,, = ~ with

(43)

(37)

o1~ = o ~ 0 .

With 9 = 90°, eqs. (39) and (41) give

T h e unit vector d denoting the material direction is

Fgo ( o - a) 2 [ -- ~ 3 K

d = (cos 9, sin 9, 0),

1 [K2 l+~/~-l)l-1,

•

1

~

1

K2 (47)

U n d e r constant stress creep conditions with F >>0, the ratio of initial creep rates corresponding to 9 = 0 ° and 9 = 90 ° is given by

F-(°-'~)~[I+I(K2~£_ t 3\~d-- 1)[4COS~9+sin: 9 - ~ (2 cos 2 9 - sin 2 9)2]] - 1

(39)

and

=

G =~5

[1 + ~1( ~ K-' -

(46)

(38)

F and G from eqs. (24) and (25) become

O~2

\]

i-/--K-7-

\-'~.

(48)

1)[4cos 2 9+sin 2 9 solving for K2/K~d we g e t

- } ( 2 cos 2 9 - s i n 2 9)2] 1 ,

(40)

where a = e ~ is the uniaxial c o m p o n e n t of the internal stress %. The governing equations for the extensional strain rate 4 = G~ are given by eqs. (26) and (27) as:

K:_( Koz

\ 4 - (4./~0)'/'"+n] •

(49)

As the material parameter n is presumed known, eq. (49) allows the determination of the ratio K/Ka

394

D.N. Robinson I Constitutive relationships for high-temperature alloys

from uniaxial tests with stress directed along and transverse to the grain growth or solidification direction. Uniaxial tests conducted with the applied stress at arbitrary angles to the preferential material direction will provide information on which assessments of the present theory can be made. For a complete elastic-viscoplastic theory, needed for structural analysis, a compatible anisotropic elasticity formulation must be coupled with the present model. This will not be dealt with here but will constitute a topic of subsequent research.

Appendix A. Derivation of flow and evolutionary equations A . 1. lsotropic case

We first present the derivation of the flow and evolutionary laws for the fully isotropic case, i.e., leading to eqs. (14) and (15). From eq. (5) we write: 011

e0 -

Oo'~i

-

0.(2 d F OJ20Skl

OF d J2 OSkt 0o-, "

( I A)

Making use of eqs. (7) to (12), we have Og] 5. Summary and conclusions

OF

K2 -

dF/d~

A constitutive theory has been presented for representing the anisotropic viscoplastic behavior of high-temperature alloys that have directional properties resulting from controlled grain growth or solidification. The theory is constructed by defining a vector field that identifies the preferential direction at each material point. This results in a locally transversely isotropic model with allowance for spatially varying directional properties. The anisotropic theory is based on the isotropic viscoplastic model of Robinson that has already been successfully applied in elevated temperature structural analysis. Application of the anisotropic theory is made to homogeneously stressed elements in pure shear with the shear direction taken transverse to and along the preferential material direction. These simple applications help to illustrate the physical origin of the pertinent material parameters K and Kd. Application is also made to a uniaxially stressed rectangular block in a state of plane stress with the spatially constant material direction making an arbitrary angle with the stress direction. As expected, the results indicate that shear strain generally develops in the absence of shear stress. In other words, the principal axes of stress and strain are not in alignment as is generally true under conditions of anisotropy. It is shown that the critical material parameter Kd (or alternatively the ratio K / K , , ) can be determined on the basis of uniaxial creep tests with the uniaxial stress direction along and transverse to the preferential material direction (grain growth or solidification direction). For a complete elasto-viscoplastic model, an appropriate elasticity formulation must be coupled with the present model.

2~

f(F),

= 1/K 2 ,

O~/OS~j = Z , , ,

(2A)

(3A) (4A)

and OSk,/Oo-,, = 6~,6,j - ', 6#6k,.

(5A)

Substitution of eqs. (2A) to (5A) into eq. (1A) leads directly to eq. (14), 2 ~ , , = f(F)~,,

(6A)

Next, from eq. (6) we write: 0~ d~j = - h - - = 0 o%

h ( O g ~ d F OJ2 O a d G OJ2~ Oak, \ O F d J . Oa~ + OG d ~ Oakfl OOtq '

(7A)

where in addition to eqs. (2A) to (5A), we have Og~/OG = K2( R / H ) g ( G) ,

(8A)

dG/d~2 = 1/K 2 ,

(9A)

OJ2/Oa# = - X,j,

(10A)

O¢2/Oaij = a 0 ,

( 1 1A)

and O a J O % = 8ki6o - ~6,,6k~ .

(12A)

Combining these and using eq. (13) gives eq. (15), d, i = ~

A.2.

H

eq - R g r ( G ) % .

(13A)

Anisotropic case

The derivation of eqs. (26) and (27) from eqs. (5) and (6) will now be outlined. From eq. (5) we write:

D.N. Robinson ~ acr,i

395

Constitutive relationships for high-temperature alloys

O0 OF

OG_

--~

d,4aj,,

(31A)

05~3

x ( O F OIt +OF OI2 +OF OI~OSkl

\J, o&l oi2 a&,

~

ask,!

(14A)

o~,,

Using eq. (7) together with eqs. (22) to (25) we have [without repeating terms already included in eqs. (2A) to (121))

05~3/0a, = ½did ` .

(32A)

Combining the appropriate terms in eq. (23A) and again making use of eq. (13) we get eq. (27), d, -

~ii - R & ( G )

% +

- 1

OFIM, = 1/K 2 ,

(15A)

g.(O)

Oil/c] & ~- ~,q,

(16A)

× [didkak~ + dkd, ask- ½d, dkak,(6 o + d~di)]] . (33A)

c]F

(17A)

1 -K-~ ,

J

o/2 Appendix

c]I2/OS,i = djdk,~k, + dkd~Z~ ,

(18A)

c]F_

(19A)

1--~

d,4N~, '

c]I31c]S,i = ½did,, c]&dOtro _--

~k,60 --

(20A) ~i 6#~kt

•

K~ 2 ~ o = f ( F ) I Z , + (--~d --1) [ dflk.~k, + d~d,.~jk

4g)]] •

of invariants

It follows from the theory of algebraic invariants (refs. [8] to [10]) that the scalar function F, specified in eq. (21) as a function of the two symmetric tensors .~ = [Z,j],

(1B)

(21A)

Substituting the appropriate terms from eqs. (2A) to (12A) and eqs. (15A) to (21A) into eq. (14A) we get eq. (26),

- ½d, dkZk,(~, +

B. Basis for selection

and D = [d, dj],

(2B)

is form-invariant under arbitrary rigid body rotations if expressed in terms of the invariants tr D = tr O z = t r D 3--- 1 ,

(22A)

tr 2; = 0, tr ,~2, tr ~ 3 ,

(3B)

tr D X = tr D2.~, tr O,~ z .

Finally, we write from eq. (6)

A set of nontrivial invariants extracted from eq. (3B) is

d,i= _hOg~(c]F c]I1 +OF C]Iz +OF c]I3] Oak, O-F \0I~ c]a~ 012 c]ak~ c]I30a~d aa o On IC]G 05~i + C]G a5~2+ aO 05~3~Oak, - h ~ ~05~1 Oak, C]~-2c]akl C]Y3c]akt/ aotq"

tr ~2, tr .~3, tr D Z and tr D.~2.

(23A)

(4B)

Seeking generalizations of a yon Mises type, we limit the dependence of F on combinations of the invariants in eqs. (4B) that are quadratic in stress, i.e.,

The terms in eq. (23A) not previously stated are

011/3a o = - • ,

(24A)

3Iz/3a o = --(4dkXk, + dkd~Xjk),

(25A)

11 -_ ~1 tr X 2 - ~1 Z,,g, 12 = tr D.~ 2 = d~ds.~jkXk,. = (½tr O Z ) 2 = (~d, I cliZ;,) 2 .

Oak,/Oa,j = 6k,60 - ~ 8,~8k~,

(26A)

In particular, we take

OG/05~l = I l K 2 ,

(27A)

Od~JOa o = a o ,

(28A)

OG

~2-

1

(~-~-i)

1

,

o~doa,, = gdka~, - dkd, ajk ,

(291) (30A)

F=~

Ii + ( 1 1 \ \ K - - - zK - ~~/, ( I 2 - g ) - ~ 1

(5B)

(6B)

as expressed in eq. (24). A parallel argument applies to the function G. According the eqs. (21) G depends on the symmetric tensors

396

D . N . Robinson / Constitutive relationships [or high-temperature alloys

A = [ao],

(7B)

and

o = [d,a~],

(8B)

and we are led, using arguments similar to the above, to express G in terms of the invariants o~1 -- - ~ 1 tr

A 2 = ~! aqaj~,

~z = tr D A z = d~dja~kak~ ,

(9B)

5~z = (~ tr D A ) z = (½ d, djaj,) 2 .

c~ /3 j, 6~j ~ g2 ~r ~i ~~#

Specifically, we write t~ G=_~2+/ 1 1 \ z ~-~d2 -- ~--~) (~¢: -- S3), K as given by eq. (25).

Nomenclature a~i d~ F f G g gh gr H h Ii #~ J~ ~ K Kd m n R s g S0 x,

components of deviatoric internal stress components of unit vector scalar function of stress material function scalar function of stress material function hardening function r e c o v e r y function material constant hardening function invariants of effective stress invariants of internal stress principal invariants of effective stress principal invariants of internal stress threshold transverse shear stress threshold longitudinal shear stress material constant material constant material constant internal stress in transverse shear internal stress in longitudinal shear c o m p o n e n t s of applied deviatoric stress coordinate directions

uniaxial internal stress material constant shear strain rate K r o n e c k e r delta uniaxial inelastic strain rate components of inelastic strain rate potential function uniaxial normal stress components of applied stress shear stress components of effective stress angle between x~ axis and stress direction material constant

(10B)

Reterences [1] D.N. Robinson, A unified creep--plasticity model for structural metals at high temperature, ORNL/TM 5969 (1978). [2] D.N. Robinson and R.W. Swindeman, Unified creep-plasticity constitutive equations for 241CR-1 Mo steel at elevated temperature, ORNL/TM-8444 (1982). [3] J.R. Rice, J. Appl. Mech. 37, no. 3 (1970) 728. [4] A.R.S. Ponter and F.A. Leckie, J. Eng. Mater. Technol. 98 no. 1 (1976) 47. [5] K.C. Valanis, The viscoelastic potential and its thermodynamic foundations, Iowa State University, Eng. Res. Inst. Rep. 52 (1966). [6] D.N. Robinson, On the concept of a flow potential and the stress-strain relations of reactor systems metals, ORNL/TM 5571 (1976). [7] S.K. Mitra and D. McLean, Proc. Roy. Soc. (London) 295(A), no. 1442 (1966) 288. [8] A.J.M. Spencer, Continuum Physics, Vol. 1, ed. A.C. Eringen (Academic Press, London, 1971) p. 240. [9] A.J.M. Spencer, Deformation of Fibre-reinforced Materials (Clarendon Press, Oxford, 1972). [10] A.J.M. Spencer and R.S. Rivlin, Arch. Ration. Mech. Anal. 9 (1962) 45. [11] J.P. Boehler and A. Sawczuk, Acta Mechanica 27 (1977) 185. [12] J.P. Boehler and J. Raclin, A unified theoretical and experimental study of anisotropic hardening, Proc. 6th Intern. Conf. on Structural Mechanics in Reactor Technology, Paris, France (1981) L3/2.