A continuum theory of a plastic-elastic fibre-reinforced material

Int. J. Engng Sci. Vol. 7, pp. 129-I 52.

Pergamon Press 1969.

Printed in Great Britain

A CONTINUUM THEORY OF A PLASTIC-ELASTIC FIBRE-REINFORCED MATERIAL J. F. MULHERN Department of Mathematical Physics, University College, Galway, Ireland

T. G. ROGERS

and A. J. M. SPENCER

Department of Theoretical Mechanics, University of Nottingham, England Abstract-A

continuum model is proposed for describing the plastic-elastic behaviour of a composite material consisting of a plastic-elastic matrix reinforced by strong elastic fibres. The composite is transversely isotropic about the fibre direction, the local axis of transverse isotropy being convected with the material particles. The yield condition and constitutive equations are formulated for the material treated as a plasticelastic continuum. The theory is illustrated by its application to problems of uniform extension, and shear on planes containing parallel fibres in a direction oblique to the fibres. I. INTRODUCTION

A CONTINUUM model of an incompressible plastic-rigid material reinforced by inextensible fibres was proposed by us in [ 11. In that paper it was assumed that a plastic-rigid body was reinforced by a single family of fibres whose direction is defined by a unit vector d which in general varies from one point of the body to another. Although in a real fibre-reinforced composite material d exists only at points occupied by fibres, in formulating the continuum model it was assumed that d is a continuous vector field defined everywhere in the body. It was also assumed that d, like the fibres it describes, is convected with material line elements in the body. From these and other assumptions it follows that the material is locally transversely isotropic with its preferred direction defined by d; this direction, at a given position or particle, depends on the initial configuration and the subsequent deformation of the body. The material was assumed to be incompressible, non-hardening, plastic-rigid, with a plastic potential which coincided with the yield function, and the general theory of stress and deformation in such a material was constructed. The cases of plane strain and plane stress were analysed in some detail, and a few illustrative problems were solved. One of these made possible a limited, but very satisfactory, comparison of the theory with experiment. In a subsequent paper Mulhern[2] gave further solutions to some problems with cylindrical symmetry. The purpose of this paper is to construct, along similar lines, a continuum theory of a plastic-elastic fibre-reinforced material. This is a problem of considerable practical interest since a substantial effort is being devoted to the fabrication and design of composite materials, and in particular to composites in which metals are reinforced by strong fibres. If these composites are to be effectively used in engineering applications it will be necessary to develop methods of stress and strain analysis for them, and this paper is intended as a contribution towards this problem. We do not now assume that the material is incompressible, nor that the fibres cannot extend. We do, however, require (as in conventional plastic-elastic theories) that the density changes are essentially elastic, and also, since we have in mind composites in which ductile materials are 129

130

J. F. MULHERN,

T. G. ROGERS

and A. J. M. SPENCER

reinforced by brittle elastic fibres, that only elastic deformation occurs in the fibre direction. Kinematics are discussed in section 2. In addition to the usual kinematic variables we again introduce the fibre direction vector d in a manner similar to that used in [I]. We also define a continuous scalar function (T which is a measure of the fibre density at a point. The kinematic theory is formulated without any restriction on the magnitude of the strain components or of the dilatation. However, since the most useful applications of the theory are likely to be to materials in which a metal matrix is reinforced by elastic fibres with high elastic moduli, it may happen in practice that the dilatation and the strain component corresponding to the fibre direction are small, although other strain components may be large. Some implications of these restrictions are discussed in this and later sections; among other things they mean that certain yield stresses must be small compared to certain elastic moduli. The plastic-elastic theory is developed in sections 3 and 4. As far as possible we follow conventional plastic-elastic theory for an anisotropic material. Since our aim is to construct a theory which is sufficiently simple to enable the solution of some problems, we have not sought to obtain the maximum generality. Thus, for simplicity, we assume that the material is non-hardening, and that a plastic potential exists and is identical to the yield function. These restrictions are not essential and there should be no difficulty in constructing more general theories. The yield condition is discussed in section 3 and the constitutive equations for the elastic and plastic strain-rate components are formulated in section 4. The requirements of transverse isotropy about the axis defined by the fibre direction, and that the dilatation and fibre extension arise only from the elastic strain are incorporated in these equations. The final two sections are concerned with applications. Uniform extensions are considered in section 5. In section 6 we solve the problem of simple shear on planes containing parallel fibres, the direction of shear being in general oblique to the fibre direction.

2. KINEMATICS

The particles of the body are referred to a fixed rectangular Cartesian system Xi(i = I, 2, 3). The deformation is described by the dependence of the coordinates xi of a typical particle at time f on its coordinates X&4 = 1, 2, 3) at time t = 0 in a reference configuration, and on t. Thus Xi =

Xi(X,J t). 9

(2.1)

As is usual in plasticity theory, if quasi-static deformations are considered, time may be replaced by any convenient parameter which determines the sequence of events. The dependence of xi on X,q is assumed to be continuous and to possess continuous derivatives. Where no confusion is likely to arise, the argument t will be omitted. Components of particle velocity in the xi coordinates are denoted by tIi(xi, t) (i = 1.2,3). The density of the material at the particle XA at time t is denoted p(X,,t), and we also denote

PWA,0) = POW,).

(2.2)

A continuum theory of a plastic-elastic fibre-reinforced material

131

The continuity equation may be expressed in the equivalent forms (2.3) p +paV*lxi = 0,

or

where the superposed dot denotes the operation material particle, so that for any function g(xi, t)

(2.4) of time differentiation

following a

the paper the suffix summation convention is employed. Following the procedure used in Mulhern, Rogers and Spencer [ l] we define a unit vector d(X,, t). In a real fibre-reinforced composite d determines the fibre direction and is defined only on the axis of a fibre; in our continuum model we assume d to be defined everywhere and to be a continuous vector function of X,. We also denote

and here and throughout

WA, 0) =

(2.5)

doWA).

The components of d in the xi system are denoted by dj and those of 4 by dA(0).In view of (2.1), dj, and also p, may when convenient be regarded as functions of xi and t. The components di are determined by dAco)and the deformation in the following way. Consider a line element which at time t = 0 has length 63, and direction defined by 4. It is assumed that at time I the line element formed from the same particles has length 6s and direction defined by d (in physical terms, we assume that particles which initially lie on the axis of a fibre continue to do so throughout a deformation, although the fibre may extend or contract). The condition for this is d.t 6s = -

dXi

axA

Since di are components

dAco)6s w

(2.6)

of a unit vector

(6s)’=

axi dxi ax, ax, dAco)d,(O)@so)2,

(2.7)

and it follows from (2.6) and (2.7) that in the limit as, + 0 ds = dso

l/2

-/!?$!$ A

dA(O) dB(O) B

,

(2.8)

I

and

di =

(2.9)

132

J. F. MULHERN,

By taking the convected

T. G. ROGERS

and A. J. M. SPENCER

derivative of (2.9) we obtain, after some ma~pulation,

It may be verified that this equation is independent of superposed rigid rotations. We now define a quantity o, which we call the jibre density, as follows. In a real composite u(XA, t) will be defined to be the number of fibres which intersect unit area of surface normal to d at the particle X, at time t. We also denote

a(&, 0) = crO(XAh The initial and current fibre densities are related by the following considerations. Consider an elementary surface ES,, in the initial configuration bounded by a parallelogram with vertices X,, X, + 8XAn), XA+ aXA@),X, + 8XAc1)+ 8XA(2).The number of fibres intersecting &SOis o&MO, where 6A, = e.4BCdA’o’GX~“‘GX c (2)

(2.11)

is

the area of the projection of 6S0 onto the plane normal to 4 at X,, and eABcis the third order alternating tensor. In the deformed configuration, the particles which initially formed &So form a surface SS bounded by a parallelogram with vertices xi, Xi+&I(l), xi+ 8xi(2),xi+ 6~i(l)-tBxi(~) where axi

=

..!e$gxy=m, A

and the number of fibres which intersect SS is u&f, where (2.12) is the area of the projection of 6s onto the plane normal to d at XA and t. However 6S, and 6s are formed by the same particles and therefore intersected by the same number of fibres, so that (2.13)

crl$A, = a6A. Now, from (2.8), (2.9) and (2.12)

axi axj axk ds -l

6A = ejjkd!,f’ ax

ax

A

ax

B

ds

C (>

&!@‘~Xc,z)

0

(2.14)

A continuum theory of a plastic-elastic fibre-reinforced

material

133

It follows from (2.13) and (2.14) that

By taking

the convected

derivative of this relation, and using (2.4), we obtain its Euler-

ian form b+U(a

ij-

d*dj)aUJaXj

=

0.

(2.16)

To formulate the continuum mode1 we postulate the existence of functions a&CA) and u(XA, t), which are continuous functions of X,, related by (2.15), and such that uO(XA) is a measure of the density of fibres intersecting the surfaces normal to d,, in the reference configuration. In many cases we shall be concerned with composites in which during deformation the change in density is small compared to the initial density, and the extension ratio in the fibre direction has a value nearly equal to one. This is a valid approximation for many real composites, and in fact in the simpler mode1 considered by Mulhern, Rogers and Spencer [ I] the stronger assumptions that the material is incompressible and inextensible in the fibre direction were made. It will therefore frequently be assumed that -P = 1+ O(E). d$ = 1+ O(E)

PO

(2.17)

0

where E is a small constant, which in later sections we will identify as the ratio of a plastic yield stress to an elastic modulus. It follows from (2.15) that, if (2.17) hold, then also u

-

= 1+0(E).

(2.18)

CO

The relations (2.17) and (2.18) do not of course imply that all the displacement ients are small. 3. THE

YIELD

grad-

CONDITION

Genera1 theories of the elastic-plastic continuum have been formulated by Hill [3-51 and by Green and Naghdi[6]. The theory to be developed in this section and section 4 may be regarded as a special case of these theories. Our objective is to develop a theory of plastic-elastic fibre-reinforced materials which as far as possible follows the conventional theory of plasticity, but which also incorporates the kinematic properties of fibre-reinforced materials described in the previous section. In particular, we have to take account of the anisotropy introduced by the presence of the fibres. We therefore begin by postulating the existence of a yield function, which we assume depends on the current stress, the density, the current fibre direction, and the fibre density. For simplicity it is assumed that the material is non-hardening, so that the yield function does not depend on the history of strain. Thus f =

f(ofj, p, 4, a)

(3.1)

where oii are the components of stress measured as force per unit current area. Alternatively we could replace uti in (3.1) by the components Tij of Kirchoff stress (that is,

134

J. F. MULHERN.

T. G. ROGERS

and A. J. M. SPENCER

stress measured as force per unit area in the reference configuration),

where

PO TG = - u*j.

(3.2)

P

If dependence off on p is allowed, it is immaterial whether f is expressed in terms of oij or rW The yield function f has to be invariant under rigid body rotations of the body, and also under the transformation d = - d, since the sense of.d has no significance. It follows by well known results in the theory of invariants (Spencer and Rivlin[7] Spencer [8]) thatfcan be expressed in the form

where J, = didjc+ijy 52 = qiiy J, = (+uuij,

J, = didjuikujk

J, = (+ijujkuki.

Instead of the set of invariants J, to J5, we employ an equivalent set I, to I, and take (3.3) where I, = J,,

Zz = Jz-J1,

13 = J,-J;,

I, = Jd-2J:,-+JJ+JJ$,+jJJ, Z5 = +JJ5-&JJ,J, ++J4Jl +$JsJz -$J:sJ,

+QJI-+J;J,+$JJ2J;

+tJ:.

(3.4)

This choice of invariants is made largely for consistency with Mulhern, Rogers and Spencer[l]. In that paper it was assumed that the material is transversely isotropic with respect to the direction defined by d, that f is a function of the oij in a coordinate system in which the xQ axis coincided with the direction of d. It was then shown that f can be expressed as a function of I, to I, in a general coordinate system (the invariants denoted Z; and ZJ in the earlier paper are here denoted I, and Z5,and I, is here expressed in a different form from the one given previously). Moreover, it was shown in the earlier paper that Z3, Z4 and Z5 have the property, necessary in formulating the flow rule, that certain combinations of their derivatives with respect to the stress components vanish. In addition, they take quite simple forms when expressed in components referred to a coordinate system in which one of the axes coincides with the direction of d. The constitutive assumption (3.1) is equivalent to the assumption that the material is locally transversely isotropic about the direction defined by d; the present approach avoids the need to introduce a special coordinate system. If the relations (2.17) and (2.18) are assumed, and terms of order E are neglected, then p and u in (3.3) may be replaced by p,, and u. respectively. If in addition the material is homogeneous in its reference configuration, so that p. and o. are independent of X,, then to this order of approximation the composite remains homogeneous during

A continuum theory of a plastic-elastic fibre-reinforced

material

135

a deformation and for a composite with given fibre concentration p,, and u,, may be omitted from the yield function. Also in this case the distinction between TV and oii (but not necessarily between their time derivatives) is immaterial to this order of approximation. It can be argued that the spatial derivatives acr/aX, of o, and possibly higher derivatives also, should be included in the arguments off. However it seems inconsistent to do this without also including @/8X,. If these derivatives are included, the material can no longer be regarded as transversely isotropic with respect to a deformed configuration, and the form off is much more complicated. In any event, if the material is initially homogeneous, au/ax, is of order E and will be neglected. In applications it is usually necessary to adopt a special form for the yield function. Following the previous paper (Mulhern, Rogers and Spencer [ 11) we will in such cases use the quadratic yield function f=

(1,/k?) +3(1,/k%) - 1,

(3.5)

where k, and k2 are constants which may be related to the yield stress of the composite in shear on surfaces containing the fibres in directions along and perpendicular to the fibres. Further restrictions on the form off will be developed in the next section. The form (3.5) is consistent with these additional restrictions. 4. STRESS-STRAIN

RELATIONS

We make the usual assumption of plasticity theory that the strain-rate eij can be expressed as the sum of two parts, called the elastic and plastic strain-rate and denoted e; and ec respectively. Thus

Only stress states for whichfs _Oare admissible. Iff= 0 andf= 0 the material is in a plastic state, iff< 0 orf= 0 andfc 0 it is in an elastic state. Note that in evaluating fany dependence offon p and (T should be taken into account.

Plastic strain-rate

It is assumed that e$ can be derived from a plastic potential, and that this plastic potential is identical to the yield function. Hence e;=

0 if f<

0

or f=

f= 0

0

and

and

f<

f=O,

0, (4.2)

where A is a factor of proportionality. It is customary in metal plasticity to make the assumption, based on experimental observation, that density changes arise only from the elastic part of the strain-rate.

J. F. MULHERN,

136

T. G. ROGERS

and A. J. M. SPENCER

Since we have in mind application to composites in which the plastic constituent metal, we make the same assumption here, and assume that ek = 0.

is a

(4.3)

We also assume that the fibre is elastic, and that only elastic deformation occurs in the fibre direction. Thus it follows that the component of e$ which corresponds to the fibre direction is zero; that is did& = 0.

(4.4)

In a manner similar to that used in [l] it follows that in order for (4.2) to satisfy (4.3) and (4.4) f must be independent of I, and It, so that, omitting the arguments p and cr

f=fU3,14,15) *

(4.5)

Hence from (3.4), (4.2) and (4.5) e{=O

if f<

0

or

f=O

and

J‘< 0,

(4.6)

if f=O

and

f=

0.

If the yield function (3.5) is adopted, the second of (4.6) becomes e$ = h{-~Z~k,26~ + [- 2Z,kT2+ (&Zz+ II) kF2]didj + kg2cu + ( kT2- kT2) (did#jk +djd/Pik))v iff = 0 andf= 0. Elastic strain-rate As a constitutive equation for the elastic strain-rate e;, we postulate that e; are linear functions of the components of a suitable measure of the stress-rate, with coefficients which depend only on diy and that this relation is form-invariant under rotations about the direction defined by d. We also require that for sufficiently small strains relative to a reference configuration in which the stress components are all zero, the equations reduce to the equations of linear elasticity for transversely isotropic materials with preferred direction defined by d. These equations are given by Lekhnitskii [9] and may be expressed in the form yij

=

Ciikm~km

(4.8)

A continuum theory of a plastic-elastic fibre-reinforced

where yU denote components

material

137

of infinitesimal strain, and

1

-= Gz

2(1 +V) E’

E, E’, G1, G, are constant elastic moduli, and V, V’ are also constant and may be interpreted as Poisson’s ratios. Stress-rate

measures

Since large deformations are allowed, the stress-rate measure must be independent of superposed rotations. Of the infinite number of measures with this property, among those which appear most frequently in the literature are the ‘Oldroyd’ derivative of oij

-4e.i = dot

&fj-(Tikvj,k

(4.11)

-ujkvi,]c

which measures the rate of change of the oij when these are referred to convected coordinates and measured per current (variable) unit area: the derivative employed by Truesdell

dm = 41

bij-

ui]cvj,k-

ujkvi,k+

ufjvk,k

(4.12)

which measures rate of change of oti referred to convected coordinates and measured per unit area in the current configuration regarded as reference configuration (equivalently (4.12) may be regarded as the rate of change of the Kirchoff stress components, referred to convected coordinates and with the current configuration chosen as reference configuration); and the ‘Jaumann’ derivative Of oij (4.13) (where Wij=

$(Vi,j-Vj,*)

(4.14)

is the rate of rotation tensor) which measures rate of change of uii referred to coordinates which do not deform but are convected with and rotate rigidly with the particles of the body. Suitable time derivatives of the Kirchoff stress tensor may also be constructed. Any one of the following constitutive equations (as well as infinitely many others) (4.15)

J. F. MULHERN.

138

T. G. ROGERS

and A. J. M. SPENCER

et = Cijkmd~kmld~tr

(4.16)

e5 = cijt&~krnldJf,

(4.17)

has the necessary invariance properties under rigid rotations and can be integrated to give (4.8) if the strain is infinitesimal and the stress is initially zero. There seems to be no compelling reason to choose one of these relations in preference to the others, so we will employ whichever is most convenient in each given case. It is possible to give plausible arguments which suggest that if

which is often the case in plasticity problems, then it makes little difference which of (4.15)-(4.17) is used in a given problem: we do not however attempt a proof of this conjecture here. The complete system of equations, in Eulerian form, consists of the continuity equation (2.4), the relations (2.10) and (2.16) for di and (+, the yield condition (4.5), the constitutive equations (4.6) and one of (4.15)-(4.17) for the plastic and elastic strainrate components respectively, and the equations of motion which in the absence of body forces are (4.18) In (4.18) t of course represents the real time. It is convenient to record the form the equations take when they are referred to a coordinate system in which one of the coordinate axes, say the x3 axis, has the same direction as d, so that di = (0, 0, 1). The stress invariants (3.4) then reduce to 12 =

CT11 +

I,

=

u33r

I,

=

it-(crll-u22)*+2cT122,

cr22.

I3

=

uf3

+

I5

=

3(u,,-a,,)

ug3,

(uf3-u223)

+

~2~13f712~

(4.19)

so that the quadratic yield function (3.5) becomes f’=

u:3+

CT;3 + kT

The constitutive

cull

--22)“+4a122 4k

-1.

(4.20)

%

equations (4.6) for the plastic strain-rate then become

(4.2 1)

eb = 0,

A continuum theory of a plastic-elastic fibre-reinforced

material

139

or, if the special yield function (3.5) is used

e:/j= hkp{- $(ull+ u&s&+ eL = hk;2u,3,

The constitutive

ef&= 0,

U&o>

(4.22:

@,/I = 1,2).

equations (4.9) and (4.15) for the elastic strain-rate are now

l+V&,$ -ezp = E

dot

)

(a,/3 = 1,2).

(4.23)

If (4.16) or (4.17) is used instead of (4.15), then in (4.23) dooii/dot must be replaced by dTrij/dTIor d,Jcij/dJt.

In sections 3 and 4 we have not attempted to formulate the most general theory of plastic-elastic behaviour of fibre-reinforced composites. Our objective rather has been to formulate a model of the behaviour of such materials which is sufficiently realistic to give an adequate description at least for some purposes but retains sufficient simplicity to enable solutions to be obtained to some problems. The theory is clearly capable of generalisation in many directions. As far as formulation of the theory goes, for example, it is not necessary to replace p and u in the yield function (3.3) by p,, and u. respectively. Neither is it necessary to restrict f to be a function of the arguments. indicated in (3.3): we could, for example, admit dependence on du/dXA. It should also be straightforward to drop the assumption of ideal plasticity and permit f to depend on the strain history. In formulating constitutive equations for the plastic strain-rate the existence of a plastic potential is not essential, and if such existence is assumed, it is not necessary to identify the plastic potential with the yield function. For the elastic strain-rate, more general relations than (4.15) or (4.16) could be used; for example one could assume a relation of the form dgkmldot = h,,(di, uij, eFj)

(4.24)

with hkm linear in e& Finally, there should be no difficulty in extending the present purely mechanical theory to a thermodynamical theory of the kind described by Green and Naghdi [6]. We do not, however, attempt these generalisations at the present stage. It may be noted that the constitutive equation (4.16) for the elastic strain-rate somewhat resembles the equation proposed by Truesdell [ 101 for a ‘hypoelastic material of grade zero’ and correspondingly (4.24) resembles the constitutive equation for a general hypoelastic material. The correspondence is not exact because we allow dependence on the vector d (a hypoelastic body in which stress and strain both vanish together is necessarily isotropic). Nevertheless, experience with the theory of hypoelasticity is of value in formulating and using the present theory.

5. UNIFORM

EXTENSIONS

We consider the fibres to lie parallel to the x,-axis, so that d has components

(O,O,l)

J. F. MULHERN,

140

T. G. ROGERS

and suppose the material undergoes so that at time t xI=X,(l+elt),

and A. J. M. SPENCER

uniform extensions

x1=X2(1+e2f),

in the directions of the axes,

x3=X3(1+e3f)

(5.1)

and elX1

v1 =

elxl/ ( 1 + elt) , etc.,

=

(5.2)

where e,, e2, e3 are constants. If r represents real time, then the acceleration is zero; if t is some other ordering parameter we consider the motion to be sufficiently slow for the acceleration terms in (4.18) to be neglected, so that in either case (4.18) reduce to the equilibrium equations &r,,/ax, = 0. We seek solutions for which uii depend only on t, in which case these equilibrium equations are satisfied. We also seek solutions in which uLi = O(i # j) and assume that oij = 0 when t = 0. Since the rotations are zero in this deformation, it makes for algebraic simplicity to use the Jaumann definition (4.13) of stress-rate and the constitutive equation (4.17) in the form (4.23) with dgij/dot replaced by dJou/dJt for the elastic strain-rate. With the above assumptions we have d.1 (+ij --_=d,t

au ij at

For sufficiently small values of t such that f < 0 the elastic-strain-rate strain-rate, and is given by (4.23) which become

is the total

(5.3) =----

e3

1 +e,f

E’

at

On integrating and making use of the initial condition these give

U 11 =

E{(E’-E~v’~)In(l+e,t)+(E’v+Ev’~)ln(l+e,~)+E’v’(l+v)ln(l+e,tjj (l+z~){E’(l-~)-2Ev’~} E{(E’-Evf2)

In (1+ezt)+(E’v+Evr2)

U 22 =

(1

E’{Ev’ln U33

=

+v){E’(

(l+ert)(l+e,t)+E’(l--v) E’(l

-v)

In (l+elt)+E’v’(l+v) 1 -v) -2Evf2} In (l+est)}

-~Ev’~

From (4.19), we have I3 = I, = 0, and 14= Hull -(+22)“.

3

In (l+e3t)}

(5.4)

A continuum theory of a plastic-elastic fibre-reinforced

Without loss of generality put in the form

141

material

we may assume oll 2 uz2, and the yield condition

u11-u22

=

(5.5)

Y

where Y is a constant. If the yield function (4.20) is adopted, follows that yield first occurs when c = f,,, where Y(l+v) E If Y/E Q 1, then (5.6) may be approximated (e,-e2)t,=

can be

Y = 2k,. From (5.4) it

(5.6)

’

by Y(l+v)/E.

(5.7)

We denote the values of the stress components when yield first occurs, given by (5.4) (O).In subsequent plastic deformation with t > to, the non-zero with t = to, by u’,“:, u$?, u33 plastic strain-rate components are, from (4.2 1) e$ = - ef2 = A’ , Hence in plastic deformation form el

the total stress-strain

e2

+-

l+e,t+m=

(5.8)

w

relations can be expressed

au22 ---2d au,, at E’ at ’

(+11-u22 = y,

(5.9)

from which, with the condition that uI1, u22 and u33 are continuous that

Ull +u22 = u11-(+22

in the

EE’{ln (l+e,t)(l+e,t)+2v’ln E’(1-~)-2Ev’~

at t = to, it follows

(l-test)} 3

= Y,

(733=

~‘{Ev’ln

(l+e,t)(l+e,t)+E’(l-v)(ln E’(1 -v) -2Ei2

(l+M)) (5.10)

Alsofrom(5.10) In ((1+e,t)(1+e21)(l+e3~)}

=

(-!-$!-$)(ull+u22)+yu33,

(5.11)

142

J. F. MULHERN,

T. G. ROGERS

and A. J. M. SPENCER

We now use (5.11) to consider some of the implications of the assumption, suggested in section 2, that the extension in the fibre direction and the dilatation are of the order of a small constant E. In the present problem this means that In (1 + e,T) (1 + ezt)( 1 + eat) and In (1 + e,t) are of order E. It follows from (5.11) that oS3 is restricted to be of order rE’ and or1 + uz2 of order min (EE, EE’). These are not serious restrictions in practice, since most fibres likely to be used in real composites may be expected to fracture under tensile stresses which are only a small fraction of their elastic moduli, and in most applications will not be subjected to very large hydrostatic pressures. If we now consider deformation under a uniaxial tensile stress (+rl = Y, uz2 = oS3 = 0, it follows again from (5.11) that Y/E and Y/E’ must be of order E, so that now E may be identified with one of these ratios, as suggested in section 2. If the yield function (4.20) is used, Y = 2k2, so that k,/E and k,/E’ are of order E. This example imposes no restrictions on the ratios of k, to the elastic moduli, or of k, to G,, but in specific composites it may happen that the elastic moduli are of similar magnitude, and that k, and k, are of the same order of magnitude. To consider unloading from a plastic state we suppose that loading ceases at a time t, > lo and that for t > t1 the deformation is given by x,=X,[l+e,tl-ee:(t-rt,)],

X2=X2[1+e2fl-e~(t--t,)],

X3=X3[1+e3tl-eA(t-fi)].

(5.12)

The non-trivial equations in (4.23) then become (with eyj = 0) -e;

1 +eltl-eeI(t-_t,) -ei

l+ezt,--ei(t-t,) - ej

l+e,tl-eA(t-_t,)

=E’

at

(5.13)

These are similar to (5.3), and can be solved in an analogous way, with the conditions that cru, uz2 and us3 must be continuous with the values given by (5.10) when t = t,. The calculation is straightforward and we omit details. The constants ei and ei must be chosen so that (if ull > uz2) a(u,, -uz2)/at < 0 at I = tr. Provided v < 1. this means that eb el (5.14) ---=->o. 1 +e,t, 1 + e,t, If the deformation (5.12) is continued for sufficiently large time, reverse plastic yield will in general eventually occur. It is of interest to compare the behaviour of this model with that of real fibre composites in uniaxial tension in the fibre direction, that is when url = uz2 = 0, us3 # 0. With our model the behaviour is entirely elastic, the relation between us3 and the extension being u

33= E’ In (1 + e,t) = E’e,t.

(5.15)

A continuum theory of a plastic-elastic fibre-reinforced

material

143

Hill [ 1 l] and Mulhern, Rogers and Spencer [ 141 have investigated the behaviour under uniaxial stress of a composite element consisting of a circular cylindrical elastic fibre in a concentric cylinder of idea1 plastic-elastic material. To a good approximation for small strains the behaviour of this composite element under cyclic loading may be described by the stress-strain curve illustrated in Fig. 1. If Young’s Moduli for the fibre and matrix are El, E, respectively, c is the volume concentration of the fibre. and Y, is the tensile yield stress of the matrix, then the slope of OA, FB and DC is cE,+ (I - c)E,, the slope of AC and FD is cE,, and Y0 = Y, [ (1 -c) + c&E;~]. Behaviour of real composites which is in qualitative agreement with this theory has been reported by Baker and Cratchley [ 121. Our mode1 does not predict the hysteresis loop illustrated in Fig. 1. There are, however, at least three cases in which the mode1 can adequately describe the behaviour of plastic-elastic composites in uniaxial tension along the fibres: (a) us3 < 2Y,, in which case deformation takes place on the section FB of the stressstrain curve. In this case E’ may be interpreted as cEl + (1 - c) E2. (b) 033 * 2Y,, in which case deformation takes place effectively on the section BC or DF of the curve, these curves lying closely together. In this case E’ may be interpreted as cE,.

AXIAL

STRAIN

e,t

Fig. 1. ldealised behaviour of cylindrical composite element under uniaxial cyclic loading. OA, FB, DC have slope cE, + (1 - c)E,; AC, FD have slope cE,.

lJESVoL7Flo

2-B.

J. F. MULHERN.

144

T. G. ROGERS

and A. J. M. SPENCER

(c) c& s (1 - c)E,, in which case the discontinuities in slope are small. It is worth noting that a large ratio of El/E2 is desirable in real composites for effective transfer of load from the matrix to the fibre. We may also note that this case of uniaxial tension is probably the most stringent test of our model in that it directly tests our assumption that yielding is not affected by the normal stress component in the fibre direction, and we expect the model to be more effective when the other stress components are not insignificant. 6. SIMPLE

SHEAR

We again consider the fibres to lie parallel to the x8 axis, so that d has components (0, 0, I), and suppose that the deformation consists of simple shear on parallel planes containing the fibres, the direction of shear being, in general, oblique to the fibre direction. Thus we have Xl =

x,,

x2 + s,X1t,

x2 =

v, = 0,

v2

=

x3

&X,,

(6.1)

x3 + S3Xlfr

=

v3 =

S.Gx,,

(6.2)

where s2 and s3 are constants. Since in this deformation vk,li = 0, the Truesdell derivative (4.12) reduces to the Oldroyd derivative (4.11). It is convenient to use this derivative in this problem. From (4.11) and (6.2) we obtain dO(+ll

d,ju2.,

da,,

d,t=-$

d OU?3

-

=

dor

d

duw

or33 - dot

77==7”-2S2u’2J

6k.r~~

--,y2cJ*3

-

at

s3(T12,

d oc13 dot

-=--s3cr,1,

=

dU33 - at

-

2S,Cl,,

aal3

at

(6.3) Also from (6.2) e,, =

&ss,,

e13 =

is3,

e,,

=

e22 =

e33 =

e2:~ =

0.

(6.4)

As in section 5. we seek solutions in which uij are functions only oft. We note that the acceleration components are all zero, so that the equations of motion are satisfied if t represents real time. If t is some other ordering parameter the motion is regarded as quasi-static and the equations of equilibrium are identically satisfied. If all stress components are zero at c = 0, then for sufficiently small values of r the material behaves elastically. The elastic strain-rate is then the total strain-rate and is given by (4.23) which, with (6.3) and (6.4), become e

,1 =

o

_

e22=o=__-.

1 doa,, E

dot

1, douz

LJ’dou,,,

E

E'

dot

I A--.-doup., v Au,, E dot -t i? dot

d,,t v’ d,,u:jy E d,,t ’

A continuum theory of a plastic-elastic fibre-reinforced

145

material

(6.5) From the first three of these equations it follows that d,pJdot = dg2Jdot = dc33/ dot = 0, and it may then readily be shown that the solution of (6.9, with dou,/dot given by (6.3), andaij = 0 when t = 0, is u

11 -

~23

0,

u22

j(G1-t

=

=

G&Y,

G2)~2~3f~,

CT33 =

G 13ST

~13 =

G1~3l, o12 = G2~2f.

9

(6.6)

We note that the solution given by linear elasticity theory is recovered if the squares and products of s2t and s3t are neglected. The stress components are given by (6.6) for values of t such that the yield condition (4.20) is not violated, that is for t < to where t,, is the positive real root of {G:+~(Gl+G,)2s~t~}~ft~+{1+~~%t~}G%sft~=

1 (6.7)

k;

k:

If we neglect k,/G,, k,/G, and k,lG, compared to one, (6.7) may be approximated 1.

{(g+(-i)3ti=

(6.8)

For values of t > to, the strain-rate is the sum of the plastic strain-rate (4.22). and the elastic strain-rate given by (4.23). Hence for t > to e ,* =

0

=

_u22),

E dot

1 doa,, cz3 = 0 = --+2G, dot 1 2

-s3

given by

Id,cr,_udO(TPZ_Pd0033+h(u11 E dot

E’ dot

Y’ dous v doer,, g,, = 0 = ___+ldou22 ------((+11-u&, E dot E dot E’ dot

e 13 =

by

=

2k;

h 2k;

A kfuz3’

1 do,3 A -+j-$k. , d,,r

2G,

(6.9) and, in addition, vii satisfy the yield condition continuous at t = to.

(4.20) and the condition

that Vij are

146

J. F. MULHERN,

T. G. ROGERS

and A. J. M. SPENCER

The first three of (6.9) may be replaced by the equivalent relations d Ofl22 -+d dot T=” off11

(6.10)

ddot

or33 ---~

0

(6.11)

’

and oS3 now appears only in (6.11). It is convenient to introduce auxiliary variables w, $I. +, x defined by o11=k~(w--sin9sin$), (+IZ= k, sin + cos 4,

~~~=k~(~+sin$sin+), crx3= k, cos + sin X.

o13 = k, cos + cos x,

(6.12)

The yield condition is then identically satisfied. At t = to, w, 4. I,!I, x have the values wo. $o, IJ~.x0 respectively, where, from (6.6) and (6.12) G 2s2t2 20

tan C#J~ =

~s2t0.

00=2k2’ G&o

sin $. = - k2

(G,

[ 1 + QEtf] 1’2,tan x0 = -

+

Gzhto

2G

(6.13)

.

1

The first of (6.10) may now be written (6.14)

$-s,sin$cos+=O. Now eliminate A from (6.9), and (6. 1O)2and make the substitutions

4

sin $-+

dt

s2

I

(6.12). This gives (6.15)

=O.

Next eliminate A between (6.9), and (6.9&, to obtain cos~~-s,cosJ,cos2x+~

1[( kz) o+&

sinx-sinJ,cos(+-XX)

1

=O.

(6.16)

Again, by eliminating A from (6.9), and (6.1 O),, we obtain -cos$sin$z

4

1

-sin~sinx~+cos+cosx(~--~2)]--s3k,sin+cosg), (6.17) Equations (6.14), (6.15), (6.16) and (6.17) are four first-order differential equations

A continuum theory of a plastic-elastic fibre-reinforced

147

material

which, with the initial conditions (6.13), determine w. $, $ and x. The stress component oS3 is then given by (6.11). We have not found an exact solution of these equations in the general case, but later in this section we will consider their approximate solution for the case in which k,/G,, k,lG,, k,/G, and k,lG, are small. We note also that the equations are in a convenient form for numerical integration. Exact solution of the equations is possible in the special cases sq = 0 and sS = 0, that is, when the direction of shear is either parallel or perpendicular to the fibres, and we consider these cases first. If sz = 0, so that the shear is in the direction parallel to the fibres then, from (6.7) (6.18)

s,t, = k,lG, and, from (6.13) wg =

c#ao = $0 =

x0

It is easily verified that the solution of (6.14)-(6.17) cjJ=

The corresponding

c#l=

l/J=

which satisfies these conditions is

values of the stress components u11=

(~22 =

~12 =

crz3

0,

=

(6.20)

(t 2 to).

0

x=

(6.19)

0.

=

are, from (6.12)

u13

=

k,. (t a to).

(6.21)

It remains to determine crS3,which is given by (6.11) which, with (6.2 l), reduces to dum dt

--

2s,k, = 0.

From (6.6) and (6.18) we have (+33= k:/G, when t = to and hence, with (6.18) crss=$+2s3kl(t-t,,)

=2s,k,t-5

1

1

which completes the solution for the case s2 = 0. In the case ss = 0, the shear direction is perpendicular now gives

(t 2 to),

(6.22)

to the fibres. Equation (6.7)

(6.23) The values of oO, r#~,,,+0 are given by (6.13); in particular we now have sin+,=

1,

&=J7r.

(6.24)

For sQ= 0 and t 2 to we seek solutions of (6.14)-(6.17) in which + = +rr (that is, rr,3 = CT 23= 0). With these values (6.16) and (6.17) are identically satisfied and (6.14) and (6.15) become

148

J. F. MULHERN,

T. G. ROGERS do -&-S’COScj

and A. J. M. SPENCER

(6.25)

-0.

djj!+s,[(2+o)sinli,-

I] = 0.

(6.26)

From these we have d (sin+) (6.27)

dw This equation has the integrating factor exp (G&k,++&) 4 = & when w = w0 leads to the solution

which with the condition

sin+= d2F{(?+,),d2}

(6.28) where F(x) = exp (-x2) 1 exp p dt

(6.29)

0

is Dawson’s Integral, tables of which are available (Abromowitz and Stegun, [I 31 p. 3 19). With C#J given in terms of w by (6.28). w is then determined as a function oft by (6.25), which gives

sw

do -= cos cp w,,

S*(f--fO)

(t

z=

t”),

(6.30)

and 4 may be expressed in terms of t by substituting for w from (6.30) into (6.28). The stress components ~~~~CT**.o12. o13, oz3 are given in terms of w, C#I and $ = &I-by (6.12). To determine the sixth stress component ox3 we observe that, when ss = 0, we have from (6.6) fls3 = 0 when r G to, and (6.11) reduces to aa,,/& = 0. Hence ~33 =

0

(f 2 ?“)

in the case ss = 0. The above solution, though exact, is in a rather inconvenient form from which numerical values of the stress components can only be obtained by carrying out numerical integrations. We now seek approximate solutions for the case G,/k, s 1, which is the case of greatest practical interest. For this we neglect kZ/Gp in comparison with one, and we also assume that o and s,t are of order one or smaller. To this order of approximation we have from (6.23).

continuum theory of a plastic-elastic fibre-reinforced

material

szto = k,lG,

149

(6.31)

and from (6.13) (with s3 = 0) ikk,lG,,

w. =

Dawson’s integral F(x) has the property (Abramowitz

and Stegun[ 131 p. 298)

as x + a.

= &+0(P)

xF(.x)

(6.32)

$0 = +kk,lG,.

(6.33)

With (6.32) and (6.33), equation (6.28) becomes, approximately (6.34)

sinf$=$[l--Jexp(+-y)] and (6.30) gives simply w-oO=

s2(t-to),

or

w = srt-+~

2

(6.35)

Hence, from (6.34) (6.36) The expressions

for the stress components

~12

=

kp,

ITI3

=

now become

uz3

=

us3

=

0

(t 2 to).

(6.37)

We note that (6.34) and the subsequent results may be obtained more directly by neglecting the term o sin 4 in (6.27). The solution of (6.27) thus modified, subject to the initial conditions (6.32), is equation (6.34). Since an exact solution is possible in this case, it seems preferable to defer approximation to a later stage, as we have done, rather than introduce approximations into the differential equations before solving them. We now return to the general case in which s2 # 0, ss # 0. Here it seems to be necessary to introduce approximations at an earlier stage, and we will from the outset neglect k/G compared to one, where k/G represents any of the ratios k,/G, (a. p = 1,2), so that to is given by (6.8). The initial conditions (6.13) then become

W

” = y,

_

&

=

&to,

x0

=

G;gG2&fo 1

“

G2WO cosqJo=- G,Sst,,

sin I/J~= -

k,

’

k,

’

(6.38)

150

J. F. MULHERN,

T. G. ROGERS

Also, in view of these initial conditions assume that ~$41,

and the results of the special cases above, we weGlk.

x~l,

By introducing these approximations

and A. J. M. SPENCER

(6.39)

into (6.14), (6.15), (6.16) and (6.17), we obtain

do z-szsinJI=O,

(6.40)

3 d4 >

sin+ x--s2

cos$

(

dx z-s”

sin $[ k, [-x

= $6

>

s2Gp

+TC#I

(6.41)

= 0,

s&r & + - k, x-k,sin$=O,

sin $2

+ cos +(z

(6.42)

- sz)] - s3k2 sin +}.

(6.43)

Now substitute for d+/dr -sz and d>c/dt -s2 from (6.41) and (6.42) into (6.43). After rearrangement, (6.43) then becomes d$ _ kBstcos $ - kls, sin $I drkf ~cosz$+$sin2J, 2

(6.44)

1

We now denote tan+,

=*_ KI%

(6.45)

Then, with (6.38) and (6.45), equation (6.44) can be written in the form

$!L ‘dr

sin 2$0 sin (I& - $I) 2(cos$osin~Icos2++sin~ocos+,sin2+)

With some further rearrangement

*

this becomes

1 --ldr = sin 2$, rod+ and hence, on integrating and inserting the condition at r = r.

tani(dh - 4~) tan *(JiI - tj~~)= exp

-t--l (

1

>

sin2~,-2sin(~,-~,,)[c0s($,+~)-c0s($,+~,)] sin 2$, cos(JI, - tie) (r 2 to)

1. (6.46)

A continuum theory of a plastic-elastic fibre-reinforced

151

material

It follows from (6.46) that I,/J-+ I,!J~as t + ~0. Moreover, since s&,, S& 4 1, $ approaches this limiting value rapidly for t > t,,. It is worth noting that in simple shear of a plastic-rigid solid with the yield condition (4.20) and associated flow rule, JI has the constant value +I. With $ given as a function of t by (6.46), w, 4 and x are then determined by (6.40)(6.42) as follows

(6.47) s,+Ftan$

exp {g(t)}dr

1

1 ,

where h(t)

+j-t

2 S&I ST(t) = 7

to t

1 I to

cosec $(t)dr,

set

$(t)dt.

(6.48)

The limiting values of 4 and x as t + 00are

x+x1=

-$$ 3

COs+I + $ 1

sin $1. 1

These values are consistent with the assumptions that 4 G 1, x G 1. The stress components are given by (6.11) and (6.12), but in view of the complexity of the expressions for them we refrain from stating them explicitly. The problem of unloading in simple shear may be considered in a manner similar to that discussed in section 5 for uniform extensions, but except possibly in the cases sz = 0 or sg = 0 the algebra is involved and we will omit it. In the problem of uniform extensions considered in section 5 it was possible to make certain deductions about the magnitudes of the ratios of k, to the elastic moduli in terms of the dilatation and the extension in the fibre direction. No similar deductions can be made from the results of this section, because the deformation (6.1) is isochoric and involves no extension in the fibre direction. Acknowledgment-One of the authors (J.F.M.) is indebted to the National University award of a travelling studentship. REFERENCES [II J. F. MULHERN,T. G. ROGERS and A. J. M. SPENCER, [21 J. F. MULHERN, Q. JI. Marh. uppl. Mech. In press. [3] R. HILL,J. Mech. Phys. Solids 7,209 (1959).

of Ireland for the

Proc. R. Sot. A301,473 (1967).

152

J. F MULHERN.

T. G. ROGERS

and A. J. M. SPENCER

141 R. HILL,J.

Me&. Phys. Solids 10. 1 ( 1962). [51 R. HILL,J. Mech. Phys. Solids 10. 185 (I 962). [61A. E. GREEN and P. M. NAGHDI,Archsrution. Mech.Analysis 18,251 (1965). F'lA. J. M. SPENCER and R. S. RlVLIN,Archs ration. Mech. Analysis 9.45 (1962). 181 A. J. M. SPENCER, Archs rution. Mech. Analysis 18,5 1 (1965). 191 S. G. LEKHNITSKII, Theory ofE/asticity ofan Anisorropic Elastic Body. Holden-Day (1963). [lOI C. TRUESDELL,J. rution. Mech. Annlysis 4.83 (1955). [l I] R. HILL,J. Mech. Phys. Solids 12.213 (1964). [ 121 A. A. BAKER and D. CRATCHLEY, Appl. Muter. Res. 5.92 ( 1966). [ 131 M. ABRAMOWITZ and I. A. STEGUN. Handbook ofMuthematicul Functions. Dover Press (I 965). [I41 J. F. MULHERN, T. Cl. ROGERS and A. J. M. SPENCER,J. Inst. Math. & Applic. 3.2 I (1967). (Received

13 September

1968)

RCsumk- Les auteurs proposent un modele de milieu continu pour d&ire le comportement elasto-plastique d’un mat&au composite forme dune matrice tlasto-plastique renforcee par des tibres Clastiques de grande resistance. Le composite est isotrope dans les directions perpendiculaires aux fibres, I’axe local d’isotropie transversale Ctant orient6 suivant les grains du matbriau. Les conditions de fluage et les equations d’etat sont don&es en considerant le mattriau comme un milieu continu tlasto-plastique. La thtorie proposee est ensuite appliquee a des problbmes d’allongement uniforme et de cisaillement dans des plans contenant des fibres paralleles disposees obliquement par rapport a la direction de I’effort. Zusammenfassung-Zur Beschreibung des plastisch-elastischen Verhaltens eines Kunststoffes wird ein Kontinuumsmodell vorgeschlagen. Der Stoff besteht aus einer durch starke, elastische Fasern verstlrkten, plastisch-elastischen Matrix. In der Umgebung der Fasern ist der Stoff transversal isotrop; die lokale Axe der transversalen Isotropie und die Stoffpartikel werden durch voneinander abhangige Vektoren beschrieben. Es werden die Fliessbedingungen und die Konstitutivgleichungen formuliert. wobei der Stoff als plastischelastisches Kontinuum behandelt wird. Die Erlauterung der Theorie erfolgt durch ihre Anwendung auf Probleme der gleichmassigen Streckung, sowie auf das Problem der Scherung von Ebenen, die parallele Fasern enthalten. wobei die Scherbeanspruchung schrag zu Richtung der Fasern auftritt. Sommario- Si propone un modello continua per descrivere il comportamento plastico-elastic0 di un materiale compost0 consistente in una matrice plastico-elastica rinforzata conrobuste fibre elastiche. I1 materiale compost0 b isotropic0 in sense trasversale lungo la direzione della fibra e I’asse locale dell’isotropia trasversale e convettato con le particelle del materiale. Le equazioni costitutive e delle condizioni di snervamento son0 formulate come se il materiale fosse un continua plastico-elastico. La teoria i illustrata con l’applicazione al problemi dell’estensione uniforme e del taglio sui piani aventi fibre parallele in sense obliquo alle fibre. MODeJIb KOHTUHyyMa AJIB on.ucaHHB nnacrHHecKo-ynpyroro noBefleHHa A6e+pavr-DpeAnaraercB xOM6HHHpOBaHHOrO MaTepHana, COCTOflmerOH3 tlnaCTHYeCKO-ynpyrO8OCHOBbI apMHpOBaIiHOtiKpeIlKHMH y”pyrHMH BOIIOKHBMH. KOM6HHHpOBaHHblH MaTepHan RBJlfleTCIl H30TpOnHbIM B HaIlpaBneHUH nOnew'IHOM K HanpaBneHHto B~II~KOH, npltreM MecTnaR ocb nonepewfol5i3oTponHocTH nepeMewaeTca B~~~TB~TcTBHB @OpMy,l‘,pylOTCS, yC,tOBHR nOAaTJIHBOCTl4 W KOHCTIiTyTUBHble ypU,HeHU,, C YaCTHuaMHMaTepHana. nna MaTepeana paccMaTpaBaeMor0 KaK nnacTarecKo-ynpyrefi KOHTUHYYM. Teopmx LiJIn~pHpyeTc% npHMeHeHHeMK 3anaYaM n0 paBHOMepHOMypaCTsrHBaHHKJP rI0 Cpe3bIBaHHK)B ITBOCKOCTBX COAepXCaILHfX napannenbffble BonoKHa, wdelourebfy Hanpaenewie HaKJIOHHOe K BOBOKHBM.