STEADY-STATE CREEP OF FIBER-REINFORCED COMPOSITES ...

hf. J. Solids SrmcturesVol. 32, No. 15, pp. 2219-2244, 1995 Copyright 0 1995 Elswier Science Ltd Printed in Great Britain. All rights reserved 002&7683/95 99.50 + .OO

Pergamon

0020-7683(94)00251-7

STEADY-STATE CREEP OF FIBER-REINFORCED COMPOSITES : CONSTITUTIVE EQUATIONS AND COMPUTATIONAL ISSUES N. ARAVAS, CA0 CHENG and P. PONTE CASTANEDA Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104, USA (Received 2 March 1994; in revisedform

22 September

1994)

AIrstract-The general form of the constitutive equations that describe steady-state creep of fibcrreinforced metal-matrix composites with transversely isotropic overall symmetry is developed. The physical meaning of the constitutive functions involved is discussed in detail. A method for the numerical integration of the constitutive equations is developed. The “linearization moduli” associated with the integration algorithm are computed, and the constitutive model is implemented in a general purpose finite element program. A constitutive model for steady-state creep of fiberreinforced composite that has been developed recently by deBotton and Ponte Castadeda (1993) is also considered. A number of “unit cell” problems with periodic boundary conditions, consistent with the requirements of homogenization theory, are solved by using the finite element method, and the results are compared with the predictions of the analytical model of deBotton and Ponte Castadeda.

1. INTRODUCTION

Fiber-reinforced metal-matrix composites are expected to play a key role in achieving the performance goals of the next generation of aircraft engines. Compared with traditional metal alloys, these materials have superior creep resistance at elevated temperatures, as well as a high strength-to-stiffness ratio. In view of their potential as high-temperature structural materials, metal-matrix composites have attracted increasing attention recently, and various attempts to develop constitutive models for the mechanical behavior of such materials have been made. Several anedimensional models that can be used to predict the creep behavior of fiber-reinforced composites under simple types of loading are already available in the literature ; we mention amongst these the work of Mileiko (1970), Kelly and Street (1972), McLean (1985, 1988), Goto and McLean (1991a,b), and McMeeking (1993a,b). Johnson (1977) appears to have been the first to propose a set of three-dimensional constitutive equations for creeping transversely isotropic materials ; he based his model for steady-state creep in directionallysolidified eutectic alloys on a generalization of the Bailey-Norton law, which connects the creep strain rate and applied stress by a power-law relation. More recently, deBotton and Ponte Castafieda (1993) have developed estimates as well as rigorous bounds for the dissipation functions of multiple-phase fiber composites, in which the constituent phases are non-linear isotropic materials. The work of deBotton and Ponte Castafieda was presented in the context of non-linear elastic materials, but it can be also used to describe the steadystate creep of fiber-reinforced transversely isotropic composites. A review of several models for the effect of fibers on the creep characteristics of unidirectional composites has been presented by McMeeking (1993a). It appears that, whereas some progress has been made in developing constitutive equations for creeping anisotropic composites, there have been few experimental studies on fiber-reinforced composite systems having practical utility at elevated temperatures. The creep behavior of metal-matrix composites reinforced with continuous fibers was studied recently by Weber et al. (1992). Their results show that, when both the matrix and the fibers creep, the composite exhibits steady-state behavior, following an initial transient ; however, when the fibers do not creep, transient creep of the composite is observed, with a creep strain limited by the elastic deformation of the fibers. 2219

N. Aravas

2220

et al.

In this paper, we develop the general three-dimensional form of the constitutive equations that describe steady-state creep of fiber-reinforced metal-matrix composites with transversely isotropic overall symmetry. The constitutive equations for the creep strain rate involve four scalar functions that depend on the five transversely isotropic invariants of the stress tensor ; the physical meaning of these constitutive functions is discussed in detail. The numerical implementation of the general form of the transversely isotropic constitutive equations in a finite element program is discussed, and a method for the numerical integration of such equations is presented. A specific constitutive model for steady-state creep of fiber-reinforced composites that has been developed recently by deBotton and Ponte Castafieda (1993) is examined. The predictions of the model are compared with the solutions of a number of “unit cell” problems; periodic boundary conditions, consistent with the requirements of homogenization theory, are imposed on the unit cell problems, and the solutions are obtained by using the finite element method. Finally, the model of deBotton and Ponte Castafieda (1993) is implemented in a general purpose finite element program, and the problem of a composite plate with a hole is solved. Standard notation is used throughout. Boldface symbols denote tensors, the order of which is indicated by the context. All tensor components are written with respect to a fixed Cartesian coordinate system, and the summation convention is used for repeated indices, unless otherwise indicated. The prefices tr and det indicate the trace and the determinant, respectively, a superscript T indicates the transpose of a second order tensor, a superposed dot denotes the material time derivative, and the subscripts s and a indicate the symmetric and anti-symmetric parts of a second order tensor. Let a and b be vectors, A and B secondorder tensors, and C and D fourth-order tensors ; the following products are used in the text : (ab), = a&j, (A a), = Ai+~j, (a. A)f = a/t,, (A. B), = AikBk,,A : B = A,B,, (8AjaB)iik. I-- aAij/aBk,, (AB),,w = A,B,,, (C : A)!, = Cij!Jk/, and (C : D)+, = CrlmnDrnnk,.

2. CONSTITUTIVE

EQUATIONS

We consider infinitesimal deformations and write the infinitesimal strain tensor E as the sum of the elastic and the creep strains, i.e. E =

&‘+Ec’.

(1)

The focus of this paper is the steady-state creep behavior of fiber-reinforced composites. However, for comparison purposes, we start with a brief discussion of some commonly used constitutive equations for creeping isotropic materials. 2.1 Creep of isotropic materials The constitutive equation for the steady-state creep strain rate is of the form .

e =

p23

-(I

K23, v12),

(14)

-2~,2K23r

where the x,-axis is in the direction of transverse isotropy. 2.2.2 Creep. The constitutive equation for the steady-state creep strain rate is of the form t? = f(a,s),

(15)

where f is a transversely isotropic function, and s is collection of material parameters such as the volume fraction of the fibers, the material constants that enter the constitutive equations of the matrix and the fibers, etc. Using the results of Liu (1982) together with the representation theorems for isotropic functions (Wang, 1970a,b; Smith, 1971), we can readily show that the most general form of the above constitutive equations is s”’= alI+2a,a+3u3az+u,a+u,(a~a+a~a)+u,(a2~a+a~a2)

= f(a,s),

(16)

where the uis are functions of s, and the following five transversely isotropic invariants : I, = tr(a),

Z2 = tr(a’),

Z3 = tr(a3),

Z4 = tr(a-a)

Z5 = tr(a2.a)

= n-u*n,

= n.6’.n. (17)

An alternative set of commonly used transversely isotropic invariants is given in Appendix A. If the creep response of the material is incompressible, then the following equation must be satisfied for all values of u : 3u, +2u,z,

+3u,z~+u,+2u~z,+2u,z,

= 0.

(18)

We consider composite materials in which both the fibers and the matrix are isotropic with constitutive equations such that ,F’(- a) = -P(u), and require that the constitutive equation of the composite, eqn (16), satisfies the condition ECr(-a) = -$‘(a) as well. Then, one can readily show that Y(O) = 0, and that the functions ui are such that ui(-zL,z2,

-I3*

-Z,,Zs) = -u,(Z~,Z~,Z~,Z~,Z~)

for

i= 1,3,4,6,

(19)

and ad--I,,z,,

-13,

-Zb,Zs) = ai(Zi,Z~,Z3,Z~,Z,)

for

i== 2,5,

(20)

which imply that

a, = u3 = u4 = u6 = 0

when

I, = Z, = Z4 = 0.

(21)

We assume next that the creep strain rate L” is derived from a creep potential Y = ‘Y(u, s). In view of the assumed transverse isotropy, Y must be of the form

Steady-state creep of fiber-reinforced composites

y =

2223

wz,,z*,z3,z4,z5,~).

(22)

Using the chain rule, we can readily show that

ay 5 ay azi i” = z = 2 - - = alI+2a2a+3a,a2 i=l aZi ab

+a,a+u,(a*a+a*a),

(23)

where now

ui=z.

ay (24)

Note that eqn (23) is a special case of the more general form, eqn (16), with u6 = 0. Consider now the case in which both the matrix material and fiber material creep according to the power-law equation, eqn (8). Since quadratic stress generators are absent in (8), we introduce the assumption that the corresponding creep equation for the composite does not involve quadratic stress generators either, i.e. u3 = aY/aZ, = 0. We emphasize, however, that this is meant to be only a reasonable approximation ; in general, such quadratic generators may appear in eqn (23) even though they are absent in eqn (8). Summarizing, we mention that the assumed constitutive equation for the creep strain rate of the fiber-reinforced material is 8” = u,I+2u,a+u,a+u,(a*a+a*a) 3. IDENTIFICATION

= f(a,S).

OF MATERIAL

FUNCTIONS

(25) a,

Let the coordinate axis x3 be along the direction of the fibers, so that n = e3, where e3 is the unit base vector along the x,-axis. Then the constitutive equation, eqn (25), can be written as eFJ = u,6ij+2U*aij+U,8i3~jj3

+U5(gj3dj3

+Bi3Oj3),

(26)

or, equivalently sf; = a, +2u*a,,,

(27)

ey* = a, +2u*0**,

(28)

273

=

U,

+2(u*

+a4

(29)

+u5)033,

E;; = 2u2aL2,

(30)

K3

(31)

=

VU2

+U5)a23,

SyI = (2~~ + u5)a3,.

(32)

The relevant invariants Zinow take the form II = n(a),

I2 = tr(a’) ,

Z4 = a33 ,

Z5 = ai1 +a:*

+az3.

(33)

Equations (27)-(32) show that : ??

u2 and a, relate to the response of the composite under shear, whereas a, and u4 refer to longitudinal and transverse tension ;

N. Aravas

2224

et al.

if a5 = d W/al, z 0, then the response of the material is identical under longitudinal (gZ3,03,) or transverse shear (g,J ; ?? in view of eqn (21), longitudinal shear loading (i.e. c,3 # 0 and/or uZ3# 0, other cij = 0) causes longitudinal shear strain rates only, i.e. eT; = SyZ= EyJ = ef; = 0 ; ?? in view of eqn (21), transverse shear loading (i.e. ei2 # 0, other flij = 0) causes transverse shear strain rates only, i.e. s;‘, = ~7~= E?~= of; = eys = 0. ??

The last two conclusions are true even when the quadratic stress generator is included in the constitutive equation for P, because the function a3 vanishes under longitudinal or transverse shear, in view of eqn (21).

4. FINITE

ELEMENT

IMPLEMENTATION

OF THE

CONSTITUTIVE

MODEL

In this section, we discuss the implementation of the general form of the constitutive model described in Section 2 in a finite element program. In a finite element environment, the solution of the creep 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 (u,, E,) at time t, as well as the strain E,, , at time t,+ , = t, + At is supposed to be known, and one has to determine the solution b,,, , . 4.1 Numerical integration of the constitutive equations We start with the elasticity equation, eqn (10) : btl+1 =

c:g+, = C:(s:,+As-AsC’)

=a’-C:As”‘,

(34)

= s:+, --ET are the total- and creep-strain increments, and where BE = E,+ , -E, and A.E~~ a’ = a,+C : As is the (known) elastic predictor. The creep constitutive equation, eqn (25), is integrated by using a generalized trapezoidal method : As=’= [af(u,+ I, s)+(l

-a)f(a,,s)]At

(0 < a < 1).

(35)

When a = 0, 1, or l/2, the integration scheme reduces to the forward Euler, backward Euler, or the trapezoidal method, respectively. Summarizing, we write F(AsC’) = AsCr- [af(u,+, (Ae”),s) + (1 -a)f(u,,

s)]At = 0,

(36)

where u,+ , (AE~‘)= d’- C : As”.

(37)

We choose AE”as the primary unknown and treat eqn (36) as the basic equation in which b,+ , is defined by eqn (37). The solution is obtained by using Newton’s method. The first estimate for A&’used to start the Newton loop is obtained by using a forward Euler scheme, i.e. (AE~‘)_~= f(u,, t,, s) At. An alternative and more accurate estimate for Aecr can be obtained by using the so-called “forward gradient” technique as described in the following. The function f(a,+ ,) is approximated by

: c :(AE -

AE”)

n

and substituted into eqn (36), which then yields the following estimate for de”:

(38)


2225

(39)

The Jacobian associated with the Newton loop in eqn (36) is given by

.ab,+l= J+uAt n+, . aA&“’

8F

-= aal?

a’ : c. 0 a0 ll+i

(40)

Once As” is found, eqn (37) defines the stress anfl, and this completes the integration procedure. In the following, we derive the general form of af/aa. Recalling the creep constitutive equation

kc’ =

2 = i E 3 = qa>, aZi

i-1

aa

(41)

we readily conclude that

(42)

where

az,

a*4

I

-=

aa

’

81, -=3a2,

y

ax

84

aa

’

--f&c

aa

-_=a

o

ab=

’

ab=

3(6ikaQ+ aiks,i),

kl

(45)

o

(46)

’ azz,

(47) We conclude this section with a brief discussion of the appropriate time increment used in the integration procedure. Let a,, be the maximum of the absolute values of the stress components (i.e. amax= max la& and define

CETOL = O.lF,

11

(48)

where E, 1 is the elastic modulus defined in Section 2.2.1. The time increment At is chosen so that the maximum difference in the creep strain increment calculated from the creep strain rate based on the conditions at the beginning and at the end of the increment is always less than CETOL, i.e.

N. Aravas el al.

2226 IAj(a.+

I) -fj(a,)

1At < CETOL

for all i, j.

(49)

4.2 Linearization moduli In an implicit finite element code, the overall discretized equilibrium equations are written at the end of the increment, resulting in a set of non-linear equations for the nodal unknowns. If a full Newton scheme is used to solve the global non-linear equations, one needs to calculate the so-called “linearization moduli” 8, given by

For simplicity, we drop the subscript (n+ 1) with the understanding that all quantities are evaluated at the end of the increment, unless otherwise indicated. Starting with the elasticity equation, eqn (34), we find da =

cd&-cafw.

(51)

The differential dA&’is evaluated from eqn (35) as follows :

(52) Substituting the last equation into eqn (51) and solving for au/&, we find

(53)

Note that ahj/aok, is symmetric with respect to the pair of indices (i, j) and (k, I). Hence, in view of eqn (53) and the usual symmetries of the elasticity tensor C, the Jacobian fijk, is also symmetric with respect to (i, j) and (k, I), which leads to a symmetric “stiffness matrix” in the finite element computations. 4.3 The case of plane stress In this section, we consider the case in which the fibers are all parallel to the x3 = 0 plane (i.e. n = rz,e, + n2e2)and the applied loads are such that c33 = (r3, = 032 = 0. The stress and strain tensors are now of the form a = fs‘aSe.eB and

E = &upe.eS+&3,e3e3,

(54)

where Greek subscripts range over the integers (1,2). In such problems, the strain increments A&,,,AE~>,and AE,~are known, but the out-ofplane component of the strain increment A\E~~ is not defined kinematically ; some modifications to the method described in Section 4.1 are therefore needed. The total- and creep-strain increments are written as As = AsZge.eB+ A&e,e,

and

AE” = AE$e,eg + Aey3e,e,.

(55)

The plane stress condition (rj3 = 0 requires that C33ij(Asij_As~) = 0 Summarizing, we write

or

AE,, =(C,~~/AEC,‘-CC~~.BAE,B)IC,,~~.

(56)


2221

F(A.$‘) 3 As=‘- [af(a(Ae”), S) + (1 - a)f(a,, s)]At = 0,

(57)

8; = (a,)ij + CijapAE,b.

(60)

where

We choose AeY as the primary unknown and treat (57) as the basic equation in which a(Ae”) is defined by eqns (58)-(60). The solution is obtained by using Newton’s method. The corresponding Jacobian is f3F

af

= J-&t--_:aAecr aa

a0

= .J+aAtE:(C-A),

aw

(61)

where -4, = Cij33C33kllC3333. 5. AN ANALYTICAL

MODEL FOR CREEPING

(62)

FIBER-REINFORCED

MATERIALS

deBotton and Ponte Castafieda (1993) have recently presented a constitutive model for non-linear composite materials reinforced by continuous fibers. They developed their model in the context of infinitesimal non-linear elasticity, but their results can be used to describe steady-state creep as well. The derivation of the model is based on a variational principle that enables the expression of the effective energy functions of non-linear composites in terms of optimization problems. For the special case of incompressible behavior in which the creep potential for both the matrix and the fibers is a function of the von Mises equivalent stress a,, their model can be summarized as follows. Let the creep potentials for the matrix and the fibers be of the form y(k) =y”k’(cr)P 9 k=

1, 2 3

(63)

where k = 1 refers to the matrix and k = 2 to the fibers. If the matrix is the “weaker” material (i.e. ‘@“(a,) > Y”‘(a,) for all rre), then the creep potential of the composite is estimated to be (deBotton and Ponte Castafieda, 1993) Y(Z,) Z,,ZJ =

qa,z, 0;) = ngl

where c1 and c2 are the volume fractions (c, +cZ = l), and a:“(a,o,q)

[C1Y'(')(a!'))+C2Y(2)(b~2))],

of the matrix and the fibers, respectively,

= {[(1+C,o)*+c*o’]as’(a)+(1+c,~)2a~(a))”2,

r7:2)(6,0,rf)= [(l-c,w)2af(a)+(l-c,v])2.:(,)J1’2, &4

= 3(2,2+$

= i(31, -I:)-$(&

a:(a)

= (an-a,)2

= i(3Z.4 -ZJ2,

0”

=

n’b’n,

(64)

(65) (66)

-34)2,

(67)

(69)

N. Aravas

2228

et al.

up = $7 : (I--n).

(70)

The quantities cr,, aP, r,, and zP are the alternative set of transversely isotropic invariants discussed in Appendix A. Note that, in eqn (64), the creep potential Y is independent of Is, which implies that the predicted response of the composite will be the same under longitudinal and transverse shear. Let fl and ti be the values of r~and o that minimize the right-hand side of eqn (64) for a given stress state b, i.e. Y =

c,Y(“(a,~,y”)+C,Y’2’(u,Ci),q).

(71)

In general, 4 and 0 are functions of 0, and the corresponding equation for the creep strain rate is

8”‘=c,-

av’ au

i-C2

F)$.

(72)

However, in view of the minimization in eqn (64), each of the terms in parentheses on the right-hand side of the above equation vanishes identically. Therefore, in computing P, one may regard Q and 9 as constants (deBotton and Castafieda, 1993). The corresponding constitutive equation for the creep strain rate can be now written as

ayr(l)(dlq ST=

c,

[

=

c-1

i&y +Q

5L’ i [(l+

awyd2))

e1

au,

c2h)2 +c,~2]oS$

0

=

1,

(73) l&q = lj

+(1 +clti)‘o,z ‘J I

(74) where (75) and (76)

0d2

= f(an-u,)(3njnj-6,). 1,

(77)

It should be noted that the above model is independent of the hydrostatic stress p = akk/3 and that the predicted creep response is volume preserving, i.e. irk = 0. If the fibers are rigid, then one can formally set Y (2)= 0. The minimization in eqn (64) then yields


1

1

2229

1

o=-l+cz

and

ri= -cz.

Then ai’) = a,/,/= and the estimated creep potential and the corresponding constitutive equations of the composite become Y(ZI

,z*,Z‘J

=

clY’(‘)(op),

(79)

Using the last equation, one can readily show that “.E”

‘II

=

(81)

0,

i.e. the composite is inextensible in the direction of the rigid fibers. If n = e3, then the constitutive equations (80) become

ST; =

-ST2

=

-f&2),

ts; = 0,

and

ST = Aav

for

i #j,

(82)

where (83)

6. COMPARISON

WITH RESULTS OF HOMOGENIZATION

THEORY

The predictions of the constitutive model described in the previous section are compared here with the results of periodic homogenization theory (Sanchez-Palencia, 1980 ; Bakhvalov and Panasenko, 1989). The homogenization techniques were originally developed in the context of linear elasticity, but they have been extended to infinitesimal non-linear elasticity (Suquet, 1982 ; Jansson, 1992). The comparisons are carried out for non-linear elastic materials, for which the model of deBotton and Ponte Castaiieda (1993) has been developed. The non-linear composite is assumed to be macroscopically homogeneous and we seek to determine an effective constitutive equation of the form X = g(E), where E and E are the macroscopic stress and strain respectively, and g is a tensor-valued constitutive function to be determined. The macroscopic field equations for a certain elasticity problem involving the composite then become

IL = g(E),

(85)

E_=l au,+!!2 ‘1

2 ( axj

ax, )7

(86)

where u is the displacement field, and b = b(x) is the body force. In the following, we summarize some of the results of homogenization theory as developed by Sanchez Palencia (1980) [see also Lene and Leguillon, 1982 ; Suquet, 1982, 1987 ; Lene, 1986 ; Jansson, 19921.

N. Aravas

2230

er al.

6.1 Homogenization theory

The composite is now modeled as an inhomogeneous continuum made of two different homogeneous non-linear elastic constituents. The microstructure is assumed to be periodic, i.e. the constituents of the composite are arranged in such a way that it can be constructed by the periodic repetition of self-similar elements. We define the “unit cell” as the smallest such repeatable element. The characteristic length I of the unit cell is assumed to be small compared with any characteristic dimension L of the body, i.e. 1

is=- YC2’(a,) for ce,/oo > (1 l/4)“’ = 1.16. The “deformation plasticity” model in ABAQUS has an additional “linear-elastic” term on the righthand side of eqn (113a) ; the elastic moduli used in the finite element computations are four orders of magnitude larger than co, so that the contribution of the additional elastic terms

f

Yi

2b

b-2&b

Fig. 1. Hexagonal array of fibers and the corresponding unit cell.

N. Aravas et al.

Fig. 2. Finite element mesh

becomes negligible. The volume fraction of the fibers is 39.5%, i.e. c2 = 0.395, c, = 0.605. Figure 2 shows the finite element mesh used in the problems that do not involve longitudinal shear (i.e. X,3 = 0) ; the dark and white regions in Fig. 2 represent the fibers and the matrix, respectively. The layout shown in Fig. 2 is repeated in the third direction to produce the layer of three-dimensional elements used for the solution of the problems that involve longitudinal shear. The type of elements and the exact boundary conditions used in the calculation are discussed in Appendix B. Figure 3 shows the calculated macroscopic stress-strain curves. In Fig. 3, the dotted line is the prediction of the model of deBotton and Ponte Castafieda, whereas the solid line indicates the results of the finite element calculations. Figure 3(a) shows the calculated longitudinal stress-strain curve. The predictions of the analytical model agree well with the solution of the unit cell problem. Figure 3(b) shows the calculated transverse stress-strain curve. At a transverse strain E,, = DOES,there is less than 2% difference between the prediction of the analytical model and the finite element solution. Figure 3(c) shows the transverse shear stress-strain response. At a transverse shear strain 2E12 = lOy, (y,, = $sO), there is a less than 8% difference between the prediction of the analytical model and the unit cell solution. The finite element solution of the problem of longitudinal shear produces a shear stress-strain curve identical to that shown in Fig. 3(c) for the transverse shear. This is consistent with the structure of the analytical model, which also predicts an identical response to longitudinal and transverse shear. Figure 4 shows the calculated macroscopic stress-strain response when equal amounts of normal and shear stresses are applied simultaneously ; Figs 4(a) and 4(b) correspond to combined transverse tension and shear and Figs 4(c) and 4(d) show that the corresponding results agree well with the predictions of the analytical model. Note that, for any given point (El,, C,,) on Fig. 4(a), the corresponding point on Fig. 4(b) is that for which Xl2 = C,,. A similar comment applies to Figs 4(c) and 4(d). Figures 3 and 4 show that, at a given strain level, the stress predicted by the analytical model is always higher than that of the finite element solution. This is consistent with the fact that the complementary elastic energy function Y developed by deBotton and Ponte Castaiieda is an upper bound to the actual complementary energy of the composite. It should also be noted that the model of deBotton and Ponte Castafieda is developed for a transversely isotropic composite with a random distribution of fibers, whereas the unit cell calculations refer to a composite with a given periodic microstructure (hexagonal array). 7. AN EXAMPLE:

A PLATE WITH A HOLE

The model of deBotton and Ponte Castafieda is implemented in the ABAQUS general purpose finite element program. This code provides a general interface so that a specific


0 -

0

0

2

4

2235

6

8

W%

10

10 6 8 2 4 Fig. 3. (a) Longitudinal stress-curve. (b) Transverse stress-strain curve. (c) Stress-strain curve for transverse shear (q, = ~,,/a, y,, = ,/&,).

constitutive model can be introduced as a “user subroutine.” The constitutive equations are integrated by using the method presented in Section 4 with 01= l/2 (trapezoidal method). Figure 5 shows a schematic representation of a plate with a hole ; the plate is reinforced by continuous aligned fibers as shown in the figure, and the ratio w/2a equals 6. Elasticity and creep are assumed to be the only possible mechanisms of deformation. The diameter of a typical fiber is assumed to be small compared with the size of the hole and the thickness of the plate, so that the continuum model described in Section 5 can be used in the calculations. As a model material, we consider a y-TiAl matrix reinforced by polycrystalline A1,03 continuous fibers. The fiber volume fraction is assumed to be 20%, i.e. c2 = 0.20. Typical values of the elastic constants are E = 200 GPa and v = 0.3 for y-TiAl, and E = 385 GPa and v = 0.33 for A1203, where E and v are Young’s modulus and Poisson’s ratio, respectively. Using the estimation procedure described by Christensen (1979), we find the following values for the elastic constants of the composite : E, 1 = 235 GPa, pr2 = 85 GPa, pZj = 85 GPa, KZ3 = 220 GPa, and v12= 0.31. The matrix and the fibers are assumed to creep according to a power-law equation of the form ) k=1,2.

(114)

The model of deBotton and Ponte Castafieda is used to describe the creep behavior of the composite. The following creep constants are used in the computations: n, = 2.6 n2 = 1 and and for the matrix, 6,,/e”,\ = 1.304 x lop9 MPa-“1 *s-l and ~,,/a”o; = 100~ MPa-“2 as-’ for the fibers. Plane stress conditions are assumed, and two types of loading are considered, in which a constant tensile stress of 70 MPa is applied (a) in the direction of the fibers, and (b) in

N. Aravas et al.

2236

0

-WE, 0 2

0 4

6

8

10

0

2

4

6

8

0 2W% 4 6 8. 10 0 2 8 4 6 2 Fig. 4. (a), (b) Stressstrain curves for the combined transverse tension and shear with ): , , = Z ,2 = C2I. (c), (d) Stress-strain curves for the combined longitudinal tension and shear with &g = z,, = 2x1. 0

I I W

Ll .ww

Fig. 5. Schematic representation of a plate with a hole. The fibers are in the x2 coordinate direction.


2237

the transverse direction. In both cases, the load is applied suddenly to the plate at the time t = 0. The instantaneous response of the material is elastic, and the elastic stress distribution provides the initial condition for the creep problem. Four-node isoparametric elements with 2 x 2 Gauss integrations are used in the calculations. The analysis is carried out incrementally, and the maximum size of the time increment is controlled by the formula in eqn (49). At every element integration point, the values of ci, and 9 that minimize the right-hand side of eqn (64) are found by using the values of the stress tensor a, at the beginning of each increment. Figures 6-9 show contour plots of several transversely isotropic invariants of the creep strain t?’at a time t = 1 h for both cases analyzed. The invariants plotted in these figures are (deBotton and Ponte Castafieda, 1993) : 8; = ;g-:g

(115)

= i(&Y* +&S;),

(116)

i : 5

coP?rouRs +0 .OOE-00 +2.00E-03 +4.00E-03 +6.00E-03 +E.OOE-03

&:

loading t=l hour

axial

CONTOURS

-7.OOE-03 -5.OOE-03 -3.OOE-03 -l.OOE-03

Fig. 6. Contours of creep strain invariant E,.

N. Aravas et al.

2238

coNnJJRs 1

2 3 4 5

-4.OOE-03 -2.753-03 -1.50E-03 -2.503-04 +l.OOE-03

21 3 4

CONTOURS +l.OOE-03 +2.00E-03 +3.00E-03 +4.00E-03

axial ioading t=l hour

Fig. 7. Contours of creep strain invariant eP.

2239


f r

CONTOURS

: 5

+l.OOE-03 +3.00E-03 +5.00E-03 +7.00E-03 +9.00E-03

axial liading t=l hour

coNTouFts

+S.OOE-02 +7.503-02 +l.OOE-01 +1.25E-01 +1.50E-01 +1.75E-01 +2.00E-01

Fig. 8. Contours of creep strain invariant y..

N. Aravas et al.

2240

CONTOURS 1 2 3 4

+I. .OOE-03 +2 .OOE-03 +3 .OOE-03 +4.00E-03

3 4

CONTOURS +O.OOE-00 +l .OOE-01 +2.00E-01 +3.00E-01 +4,00E-01 +5.00E-01 +6.00E-01

2 7

t=l hour

Fig. 9. Contours

of creep strain invariant y,,.

(yg)’= ~(e~r.B):(e~“.B)-a(e~r:B)* =(&)2+3&;; (y;)* = (C)* : a- (8” : a)’ = (ET;)*+ (.&)*,

_&q*,

(117)

(118)

where n = e,, /I = I - nn = e,e, + e,e,, and the Cartesian components refer to the coordinate system shown in Fig. 5. The strain in the direction of the fibers E, and the in-plane “volumetric” strain eP attain their maximum values at point A (see Fig. 5), which appears to be a possible site of fiber failure and debonding. Figures 9 and 10 show that the longitudinal (y,J and transverse (y,,) shear stresses reach their maximum values on the surface of the hole.


2241

n =?l

A

Fig. 10. Transversely isotropic stress invariants.

Acknowledgments-Fruitful discussions with Professors G. deBotton and P. M. Suquet are gratefully acknowledged. NA and CC acknowledge the support of the Office of Naval Research contract NOOO14-92-J-1808through sub-agreement KK3006 from the University of California, Santa Barbara. PPC acknowledges the support of the NSF MRL Program at the University of Pennsylvania under Grant No. DMR-9120668. The ABAQUS finite element code was made available under academic license from Hibbitt, Karlsson and Sorensen Inc., Providence, RI.

REFERENCES Aravas, N. (1992). Finite elastoplastic transformations of transversely isotropic metals. Znf. J. Solids Structures 29,2137-2157. Bakhvalov, N. and Panasenko, G. (1989). Homogenisation: Avera.qin.q _ . Processes in Periodic Media. Kluwer Academic Publishers, Dordrecht, The Netherlands. Christensen. R. M. (1979). Mechanics of Comoosite Materials. Wilev. New York. NY. USA. delotton, d. and Ponte Castafieda, P. (1993j. Elastoplastic constitutive relations for fiber-reinforced solids. In?. J. Solids Structures 30, 1865-1890. Goto, S. and McLean, M. (1991a). Role of interfaces in creep of fibre-reinforced metal-matrix composites-I. Continuous fibers. Acta Metall. Mater. 39, 153-164. Goto, S. and McLean, M. (1991b). Role of interfaces in creep of fibre-reinforced metal-matrix composites-II. Short fibers. Acta MetaN. Mater. 39, 165-l 77. Green, A. H. and Adkins, J. E. (1960). Large Elastic Deformafions. Clarendon Press, Oxford, UK. Hibbitt, H. D. (1984). ABAQUS/EPGEN-A general purpose finite element code with emphasis on nonlinear applications. Nucl. Engng Des. 77,271-297. Jansson, S. (1992). Homogenized nonlinear constitutive properties and local stress concentrations for composites with periodic internal structure. Znt. J. Solids Structures 29,2181-2200. Johnson, A. F. (1977). Creep characterization of transversely-isotropic metallic materials, J. Mech. Phys. Solids 25, 117-126. Kelly, A. and Street, N. Creep of discontinuous fibre composites II. Theory for the steady state. Proc. Roy. Sot. Lond. A 323,283-293. Lene, F. (1986). Damage constitutive relations for composite materials. Engng Pratt. Mech. 25,713-728. Lene, F. and Leguillon, D. (1982). Homogenized constitutive law for a partially cohesive composite. Znt. J. Solids Structures l&443458.

Liu, I.-S. (1982). On representations of anisotropic invariants. Znt. J. Engng Sci. 20, 1099-l 109. McLean, M. (1985). Creep deformation of metal-matrix composites. Compos. Sci. Technol. 23,37-52. McLean, M. (1988). Mechanisms and models of high temperature deformation of composites. Mater. Rex Symp. Proc. 120,67-79.

McMeeking, R. M. (1993a) Models for the creep of ceramic matrix composite materials, to appear. McMeeking, R. M. (1993b). Power law creep of a composite material containing discontinuous rigid aligned fibers, to appear. Mileiko, S. T. (1970). Steady state creep of a composite material with short fibres, J. Mater. Sci. 5,254-261. Nasrtenaal. J. D.. Parks. D. M. and Rice. J. R. (1974). On numerically accurate finite element solutions in the _ f&tplastic range. Comput. Meth. Apple Mech.‘Engng 4, 153-177. Sanchez Palencia, E. (1980). Non-homogeneous Media and Vibration Theory, Lecture Notes in Physics, Vol. 127. Springer-Verlag, Berlin, Germany. Smith, G. F. (1971). On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors. Znt. J. Engng Sci. 9,899-1916.

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Suquet, P. M. (1982). PlasticitC et Homogbksation, These de doctorat d’Etat, Universite Paris VI. Suquet, P. M. (1987). ‘Elements of homogenization for inelastic solids’, in Homogenization Techniques for Composite Media (Edited by E. Sanchez Palencia and A. Zaoui), Lecture Notes in Physics, Vol. 272, pp. 193278. Springer-Verlag, Berlin, Germany. Wang, C.-C. (1970a). A new representation theorem for isotropic functions : an answer to Professor G. F. Smith’s criticism of my papers on representations of isotropic functions. Part 1. Scalar-valued isotropic functions. Arch. Ration. Mech. Anal. 36, 166-197.

Wang, C.-C. (1970b). A new representation theorem for isotropic functions : an answer to Professor G. F. Smiths criticism of my papers on representations of isotropic functions. Part 2. Vector-valued isotropic functions, symmetric tensor-valued isotropic functions, and skew-symmetric tensor-valued isotropic functions. Arch. Ration. Mech. Anal. 36, 198-223.

Weber, C. H., Lofvander, J. P. A. and Evans, A. G. (1993). The creep behavior of CAS/Nicalon continuous-fiber composites. Acta Metall. Mater. 41,2681-2690.

APPENDIX A Transversely isotropic invariants

An alternative set of commonly used transversely isotropic invariants is J, =cr,=$r:/?,

(Al)

JZ = a. = o : a,

642)

.I, = T,’ = f[(u.8):(u.B)-l(a:B)*],

(A3)

J,=r.‘=a’:a-(t~:a)~,

(A4)

Js = det (a),

(A5)

where a = nn, and/l = I-a. Physically, (up, u n,r p, r n) correspond to the in-plane hydrostatic stress, the longitudinal stress, the maximum transverse shear stress, and the resolved longitudinal shear stress, respectively. A schematic representation of the above invariants is shown in Fig. 10. For convenience, we also define CT:=(u,,-0,)’

and

=(in*d*n)*

u: =

3(7,2+7,2),

646)

and note that u: =(u~-u,)*+3(r;+T,2)

=u.:+u:.

(A7)

If the fibers are aligned with the xg coordinate direction, i.e. n = e, and 1 = e,e, +e+,, then up = $r, I +o**h

(‘48)

0, = 033,

649)

7;

=

a:2 +gu,

t:

=

41

I -uz#,

+a:,,

and

-aFig. Al. Schematic representation of the two-dimensional unit cell.

(AlO) (Al 1)

Steady-state creep of fiber-reinforced composites u: = ($Q.

2243 (A12)

(T“3r I”Tn) : In the following, we state the relationships between (I,, I,, 14,I,) and (a .D’

0, = fu, -LA

W3)

0”= 4,

6414)

T;=;I,-I,+fI:-;(I,-I‘#,

@Is)

m; = I,-&

C416)

and I, = o;, +20,,

W7)

I, = u~+~u,++(T,+T~.~),

(‘418)

14 = e,,

(‘419)

I, =

6420)

0.‘+7.‘.

APPENDIX B The unit cellproblem

We start with the case in which the unit cell occupies the region 0 Q y, Q CI,0 Q y, Q b, -c < y, < c, and thenletchco. For simplicity, we consider first the case in which there is no axial shearing, i.e. E,, = E,* = 0 and Z,, = Z3, = 0. The geometry of the unit cell and the applied loads are now symmetric with respect to y, = 0. Let u(‘)(y) be the solution for the Y-periodic field of the unit cell problem and define g(y) by ti,(.!+JZ~Y,) = Q(Y,,Y2r-Y3)r

(Bl)

&(Y,,Y*,Y,) = -4”(Y,,Y,,---Y,),

(B2)

where Greek subscripts range over the integers (1,2). One can readily show that ii(y) satisfies the governing equations and the boundary conditions (98)-(102), i.e. ff is another solution of the unit cell problem. Since the solution is unique, this implies that B = u(‘I, i.e. II(‘)has the following symmetries :

4’(Y,,Y2r-Y3) = 4”(Y,,Y,,Y,),

U33)

~S”(Y,,.Y1,-Y,)

(B4)

= --uW,,Y,,Y,).

We therefore have

4”(Y,,Y,,O) = 0.

(W

When c + co, there is no way to distinguish the location of y3 = 0 ; hence, in view of the above symmetries, II(‘) must be independent of y,, i.e. a!‘) = 4”(YI ,Y*),

(B6)

ui” = 0.

(B7)

The solution of the unit cell problem can be now written as &(Y,,Y,) = J%,Ys+u!“(Y,,Y,), &(J+) = &Y,>

(B8) (B9)

which corresponds to a state of “generalized plane strain”. Referring to Fig. Al, we can arbitrarily set II = 0 at y = 0, which implies also that II(‘)= 0 at y = 0. Using eqn (B8) and taking into account the Y-periodicity of II(‘),we can readily show that

&(a,~~)-&(O,Y~ =&a

and %(~~,b)-ti.(~~,O) = EA.

@lo)

The components of the macroscopic strain tensor E play the role of the “applied loads” for the unit cell problem. In order to impose eqn (BlO) in a tinite element computation, we introduce two fictitious nodes, say, M and N, and define the corresponding nodal displacement components as

N Aravas et al.

2244

p 1 = 0 , and zP=E a 01 101)

tir = E,,b,

6: = 0.

(B11)

The conditions (BlO) can be now written as = 6,” and

Ua,h-k(O,h)

ti,(y,,b)-li.(y,,O)

= ~2:.

(BW

The corresponding macroscopic stresses Z,, can be calculated as

(B13) so that (E: E)(ab) = F”‘.tiw+FN .iI”,

(B14)

for all E, where FM and FN are the nodal forces work conjugate to tiMand fi” at nodes M and N. In a generalized plane strain calculation, the whole finite element mesh has one common degree of freedom in they, coordinate direction. We assume that the mesh is of unit thickness in the y, direction and let liX= E,, be that common degree of freedom. Then the corresponding macroscopic stress component & is given by

where F3 is the nodal force conjugate to iii. The conditions (B12) are enforced by using the so-called “multi-point constraints” in ABAQUS, which allows constraints to be imposed between different degrees of freedom in the model. Depending on the problem under consideration, we prescribe the relevant components of E and Z, or equivalently of FM, FN and II”‘,u”‘.For example, in the case of axial tension we let E,, = known,

and

Z,, = E122= C,, = 0,

(B16)

z& = E?j = known,

and

Fy = Fy = Fy = F: = 0,

(B17)

or, equivalently,

and determine E,, = F,/(ab), and

E,Z

=

&(Y,>b)-4(y,>O) , b

E

=

22

&(y,>b)-h(.v,>O) b

(B19)

The correctness of the numerical solution can be checked by verifying that E, I = E,, and E, 2 = E,, = 0. The generalized plane strain solutions are obtained by using four-node elements with 2 x 2 Gauss integration ; an independent interpolation for the dilatation rate is used in order to avoid artificial constraints on incompressible modes (Nagtegaal et al., 1974). In the cases where there is axial shearing (i.e. EiI # 0 and/or .&, # 0), the unit cell problem is formulated in a similar way and the calculations are carried out by using a layer of eight-node three-dimensional brick elements with 2 x 2 x 2 Gauss integration. The layout shown in Fig. 2 is repeated in the third direction to produce the aforementioned layer of three-dimensional elements.