Overall finite deformation of elastic and elastoplastic composites

Mechanics of Materials 5 (1986) 73-86 North-Holland

73

OVERALL FINITE DEFORMATION O F EI,ASTIC AND ELASTOPLASTIC C O M P O S I T E S Jacob ABOUDi Department of Solid Me('hanics. Materials and Stru('tures. Fa('ultv of Engineering. Tel-,4rir Unirerslty. Ramat-Arir. 60078. Israel

Received 2 May 1985: revised version received 4 October 1985

A set of constitutive relations are derived for the prediction of the average finite deformation of c,~mposite matenab, whose constituents are either nonlinearlv elastic or nonlinearly elastoplastic materials. The average respoase of the composite to a given type of loading is solely determined from the known mechanical properties of the phases aad their relati,,e volume,,. Particulate composites including porous materials, as well as short-fiber, long-fiber and periodically bilaminated composite,,. are obtained by a proper selection of some geometrical parameters. Comparisons with other approaches are made.

Introduction The average behavior of linearly elastic composite materials is characterized by their effective moduli. Methods for the prediction of these elastic constants from the properties of the constituents and their volume concentrations are well developed, see Christensen (1979), the survey by Hashin (1983) and references cited there. The determination of the overall behavior of elastoplastic composites is obviously more complicated even when the strain may still be considered to be small. In a recent publication (Aboudi, 1984). a continuum theory is proposed for the modeling of elastoplastic composites which consist of inelastic matrix and inclusions. The theory completely determines the average behavior of the two-phase composite from the elastic and inelastic properties and volume concentration of the constituents. Very little information is known about the behavior of composites with large deformation (associated with finite strains). For nonlinearly elastic composites, results for the overall bulk modulus are given by Ogden (1974), in the case of initially spherical inclusions, dilutely suspended in the matrix, in such a way that t.here is no mutual interaction between the spheres. A simplified method for the determination of the behavior of an elastic porous material subjected to finite uniaxial strains is given by Feng and Christensen (1982). In the present paper we propose a continuum model for the prediction of the average behavior of two-phase composites with finite deformation. The inclusions are represented by rectangular parallelepipeds which are imbedded regularly within the matrix. By an appropriate selection of some geometrical parameters, particulate (including porous materials), short-fiber, long-fiber and periodically bilaminated composites are obtained as special cases. Both matrix and inclusion constituents are taken to be either nonlinearly elastic or nonlinearly elastoplastic materials. When both phases are nonlinearly elastic, the composite effectively behaves as a nonliner elastic material as well, and its effective response ",redicted by the present theory is expressed in the form of average-stress-average-deformation gradient relations. Due to the nonlinearity of the governing equations, the response curves to a given type of loading are computed incrementally up to the desired value of deformation. Linearization of the governing equations for infinitesimal strains reduces the derived constitutive relations to those given previously by the author (Aboudi, 1983), for the determination of the 0167-6636/86/$3.50 '~ 1986, Elsevier Science Publishers B.V. (North-Holland)

J. Aboudi / Elastoplastic composites

74

effective moduli. Results are given for the finite deformation of a foam rubber subjected to several types of loading, and contrasted with the available measured data of Blatz and Ko (1962) for this porous material. An additional comparison of the predicted nonlinear response by the present theory is provided by the results of the homogenization theory for a nonlinear porous elastic material given by Abeyaratne and Triantafyllidis (1984). Next, consider two-phase composites which consist of an elastopiastic matrix containing elastoplastic inclusions. Both matrix and inclufions may undergo finite deformation. The overall behavior of the elastoplastic composite was estimated by Nema:-Nasser and lwakuma (1983; 1985) by using the self-consistent method. In this paper we consider two-phase composites in which the constituents are elastic-viscoplastic work-hardening materials. Each constituent is represented in the framework of a unified theory of plasticity without a yield function or loading or unloading conditions by Bodner's (1984) equations. This class of representation is characterized by the use of kinetic equations and internal variables with appropriate evolutionary equations for treating all aspects of inelastic deformation. Bodner's equations are generalized here to represent the large elastoplastic deformation of the constituents. Since these equations are of a rate type, we use the Jaumann co-rotational form for the stress rates appearing in these equations. The average finite deformation of the obtained elastoplastic composite is predicted by the continuum theory presented in this paper. The resulting inelastic constitutive relations for the composite with finite deformation are rate-dependent and an incremental procedure in time is required to obtain the average field quantities for the desired amount of deformation. In the special case of small strains the constitutive relations of the constituents, as well as those of the composite, reduce to those given by Aboudi (1984). The predicted inelastic average response is checked in the fol!evdag two situations: (a) Elagtoplastic matrix containing a doubly periodic array of cylindrical cavities, (for this two-d/mensional problem the finite elemet~t solution of Needleman (1972) is used for comparison with the predicted average field) and (b) elastoplastic porous material which is subjected to a hydrostatic tension. Here the results obtained from Gurson's (1977) continuum theory for ductile porous materials and its modification by Tvergaard (1982) are compared with the overall predicted response.

Basic equations Cartesian coordinates are used throughout. Let a = (a~, a2, a3) denote the position of a material point in the undeformed configuration at time t = 0. The location of this point in the deformed configuration is denoted by x = (x 1, x2~ x3). This current position is given by

.r=a+u(a, t)

(1)

where u = (u~, /'/2, 113) is the displacement vector. Consider a particulate composite in which the inclusions at the undeformed configuration are represented by a triply periodic array of identical rectangular parallelepipeds. The initial volume cf a parallelepiped is d l h l l ~ (see Fig. l(a)). The parameters dE, h E and !2 represent the initial spacing of the inclusions within the matrix in the al, a 2 and a3 directions~ respectively. By appropriately chosing the parameters d~, h~, l~, the cases of long-fiber, short-fiber and particulate composites are obtained. Porous materials are given in the spacial case of voided inclusions. Due to the assumed periodic distribution, it is sufficient to consider a representative cell of initial dimensions (d~ + d2) , (h 1 + hE) and (11 + 12), as shown in Fig. l(b). The cell is divided into eight subcells or, fl, V = 1, 2, and eight fixed local systems of coordinates (fi~"), ~t2#), ~t3v)) are introduced whose origins are located in the center of each subcell and are oriented parallel to the fixed system (a 1, a2, a3). Here

75

J. Aboudi / Elastoplastic composites

a:2,.G:l,r:2

a:2,~

/"i

:1 , r : ,

-,-";,~

~-

-m a I -(i

o=l,a=l .r=2

/'1

,,,'1 ~

Ol ~(1

I (1:2 • .8:2 I F : I

--

II I "~ll I"1 I I II L--~'ll L / I I . / i

1L.."I

I t,,"l

I G$

/

~ /q

~

I11" @

(z:l, B : 2 , r:-"

i hI --'~h2---~,.I I1

(b)

*~

(o)

=z

Fig. 1. (a) A composite with triply periodic array of rectangular parallelepiped inclusions. (b) A representative cell.

and in the sequel the subscripts or superscripts a, fl, 7, will indicate that quantities belong to one of the eight subcells. Repeated a, fl or 7 do not imply summation. The displacements in each subcell can be expressed in terms of the local coordinates in a first order expansion in the form (2) where w (~#v) is the displacement vector at the center point of the subcell and ~,~/~v) X(,,/~v~ ~/,(-,pa,) characterize the linear dependence of the displacements on the local coordinates witlfin the subcell. Equation (2) consists of the linear terms of a series expansion of the displacements in terms of the local coordinates. In the present paper this first order expansion is used in the derivation of constitutive relations for the determination of the overall behavior of composites with large deformation. By applying a procedure similar to that used by Aboudi (1982) for composites with continuous fibers, it can be shown that the continuity conditions of the displacements on the average bases along the interfaces of the subcells give w(IlI)

____

W(I12)

=

Hy(,2l)

...(122),r = .;.(211~.;. =

=

W~219~ . . . . ---: :;,,'221)

=

W (222) ~

W,

(3)

and (4a)

d]~ OBv) + d2q)(2By) = (d] + d2) a---~ i)i] w , 3 hlX(alv) _t..h2X(~,2v)=. (h I ..~-J ~ 2 ) ~ w •

~a 2

.(4b) '

a 1,¢ ('~') + 12¢ (~2) = (/1 + 12) ~--~a3"-

(40

The components of the displacement gradient tensor of the motion in the subcell are given by a.,,(~Y)

i, j = 1 2, 3

(5)

76

J. A b o u d i / Elastoplastic composites

where ~,, is the Kronecker delta, and Ol =

O/Oa~"~, ~2 = ~)/Oa~t~, and

[1

~)3= i)/Oa~ "~- Using (2), (5) provides

]

=

•

(6)

The average displacement gradient. F. in the composite is given by 2

1

7 5-"

(7)

where ~,,,t~v=d,,hal.~ is the volume of the subcell and v = ( d ~ +d.,)(h~ +h2)(! ~ + i z ) being the total volume of the representative cell. Using (3)-(4) it can be readily verified that

(8)

+

Let us denote by T ~'i~v~ the first Piola-Kirchhoff (not symmetric) stress tensor in the subcell. The average stress within the subce!l is given by S,,,~v,=

1

fa,,/2 fhM2 ftC2

T,,,/~v, d a l . , da~/~, da~V ,

(9)

12a~SV "~ - d , , / 2 " - hl~/2~' - Iv~2

The average stress ",,,;..thin the composite is obtained from

!U

E

(10)

c~.fl.y = 1

The concrete expressions of T ~'t~v~ are determined from the adopted constitutive relations for the material within the subcell (a/~,). Using (2) and (5) in conjunction with the relevant constitutive expressions, the components of S ~t~'~ can be derived from (9). For infinitesimal elasticity (where the various stress measures are practically identical), for example, T ~#v~ is related to F ~'°v~ according to the Hooke's law, and the resulting expressions of S ~'av~ are given by Aboudi (1983), equations (22). For elastoplastic materials with small strains, S ~"t~v~ are given by Aboudi (1984, equations (10)). The final set of basic equations are obtained by imposing the continuity of the tractions across the subcell interfaces on the average basis. This provides the relations SI]/3"/) = OliC (2/~Y) ,

°2iC(•IY ) = °2iC ( ~2~' ) ,

°3iC~"al ~=

"-'3ic~"a2~.

(11 a,b,c)

In the present paper we deal with the finite deformation of elastic and elastoplastic work-hardening materials and in the following sections the above basic relations will be used for the determination of the overall behavior of elastic and elastop!astic composites with finite deformation. The derivation given in the sequel will describe the average behavior of the composite in the form of relations between the average stress T and the average deformation gradient F.

Effective behavior of nonlinearly elastic composites Suppose that both matrix and inclusions behave as nonlinearly elastic materials whose internal energy functions per unit undeformed volume are W~"aV~(F t'/~vl). The stresses are derived from

7";'-"/~YI=,'i~W'"~;"/~Fi~ '#~v',

i, j = 1, 2, 3.

(12)

J. Aboudi / Elastoplastic composites

77

Using (6) and (12) in (9) provide the requested expressions for the average stresses S"q~") in the subcell in terms of ~,,~v, X~,,t~, a~d Q,~"~. Let us first assume that the loading is specified by the imposed average displacement gradient F. With (8), (4) and (11) form a system of 72 nonlinear algebraic equations in the 72 unknown microvariables qJ~'q~v), X~'t~v~ and ~(~t~v). The solution is used in (10) to yield the average stresses T so that the desired average-stress-average-displacement gradient relation is determined. If, on the other hand, the specified loading involves tractions, the system of algebraic equations should be supplemented with the additional imposed traction_ conditions. Forexample, in the case of a uniaxial stress loading in the ardirection (say) for which T~ 4:0 and all other T~j are zero, the following set of 8 equations should be added To=0,

i+j4:2.

(13)

The 8 components F~j (i + j 4: 2) appear this time as additional unknowns in the system of equations. For a given type of loading the syslem of nonlinear algebraic equations should be solved incrementally until the desired amount of deformation is achieved. Linearization of the above system of equations, in the case of compressible constituents and infinitesimal deformation, provides the corresponding relations derived previously by Aboudi (1983), for the determination of the effective moduli of short-fiber composite, in which the constituents behave according to the Hooke's law. The reliability of the prediction was examined (Aboudi. 1983: 1984) by comparison with various results and good agreement was obtained. In order to assess reliability of the prediction of the present theory with large deformation, we utilize the experimental study of Blatz and Ko (1962) on the finite deformation of polyurethane foam rubber. This porous rubberlike polymer consists of 47 percent volume of voids and exhibits pronounced dilatation effects. The matrix is a polyurethane rubber and the measurements of Blatz and Ko (1962) reveal the value o f / t = 0.23 MPa for the shear modulus of this isotropic incompressible material. We use the neo-Hookean incompressible law to represent the behavior of the continuum rubberlike matrix, i.e., 1 W = ~btt 11 - 3)

(14)

where 1~ = trace(FF r ) with F r being the transpose of the deformation gradient tensor. Using (12) we obtain the following constitutive relation for the matrix T = - p F -1 + ttF v

(15)

where p [s an indeterminate pressure. For a uniaxial stress condition in the a~-direction (say) we obtain from (15) the relation

r,, =

1/x )

(16)

with h~ = 1 + Ou,/0a, (no sum) being the stretches. The uniaxial lateral contraction ratios are related to the axial stretch in the form ~ _ = ~., = 1 / ~ ' ( 2

(17)

such that J = dot F = 1. The curves based on (16) and (17) (with /t = 0.23 MPa) are shown in Fig. 2. together with the measured values of Blatz and Ko (1962) and good agreement can be seen. E,aving established the representative equations of matrix (incompressible continuum rubber) we can proceed for the prediction of overall response of the foam rubber (47 percent porosity). By substituting (6) in (15) and using (9) we obtain = + ]T

J. Aboudi / Eiastoplastic composites

78 0.4

1.0

i ~2

T 0.8

T C 1.0

i 1.2

~ 1.4

I 1.6

I 1.8

I 2.0

0"6r ~

1.0

~

I

i

I

I

1.2

1.4

1.6

1.8

2.0

--'~X,

~k,

Fig. 2. Uniaxial stress and uniaxial lateral contraction ratio versus longitudinal extension ratio of the continuum incompressible rubber (matrix). The measured values are given by Blatz and Ko 0962).

together with j(~t~v) = 1. where for the porous material/t,/~v = # (the elastic constant of the matrix), and ot + fl + ~, :# 3. For 47 percent of voids we have d ! = h I = ! l, d 2 = h E = i 2 and d3/(dz + d 2 ) 3 = 0.47. Blatz and Ko (1962) measured the response of the foam rubber in the following three situations.

Uniaxial tension m

Here Tn ~ 0 and all other TU are zero. In Fig. 3 the predicted response of the porous material and the experimental data are shown in the form of average uniaxial stress-average longitudinal stretch and average lateral contraction ratio-average longitudinal stretch curves. The agreement is reasonable. The measure of the dilatation of the foam is given by J = det F = )~i~2

(19)

(since 7~2 =7~3) and it can be noted that the compressibility of the foam is significant.

Uniaxial tension under plane strain conditions Here the following loading is applied: T1~ =~0, T22=0, ~ 3 = 1 and all ~ j are zero for i ~ j . The resulting response of the foam is shown in Fig. 4 together with the measured values. The agreement is very

_°.21 |

..

. ; '..y"

N

I,¢ 0.8

1

. 2.6

1.0

;..

Fig. 3. Average uniaxial stress and average uniaxial lateral contraction versus average longitudinal extension ratio of the foam rubber. The measured valucs are given by Blatz and Ko (1962).

J. Aboudi / Elastoplasticcomposites

79

0.2 /

1.2

n

1.0

:Z v O.l

.

iJ

T

T 0.8

"

,

L8

2.0

0.6

%

,'.2 ,',

,'.s ,'8 £o

1.0

1.2

1.4

1.6

Fig. 4. Average stress and average contraction ratio versus longitudinal extension ratio of the foam rubber under uniaxial tension and plane strain loading conditions. The measured values are given by Blatz and Ko (1962).

good. In the present state of loading the dilatation is given by

(20)

J = ~1~2 indicating again the importance of this effect. Biaxial tension under plane stress conditions

!a the present situation the loading is described by A] =~2, T33 = 0 and all ~j are zero for i ~=j. In Fig. 5 the predicted and measured response is shown and the agreement is fairly good. The dilatation is given this time by

;=7~,2.

(21)

Simple shear

Consider a simple shear deformation in which F32 ~t=0, F23 = 0 such that T32 #=0, TEa 4= 0 and all other T~/are zero. When the homogeneous incompressible rubber matrix (which, described by the neo-Hookean constitutive relations (15)), is subjected to this type of loading, it can be easily verified that a pure shear

0.2

1.2 1.0

X

0.I

|

*"

,~ o.a " ~ " ' - - ~ " 0.6

0 i.O

1.2

' 1.4

'

I16

I

LO

I

1.2

I

I

1.4

I

I

1.6

Fig. 5. Average stress and average contraction ratio versus longitudinal extension ratio of the foam rubber under biaxial tension and plane stress loading conditions. The measured values are given by Blatz and Ko (1962).

J. Ahoudi / Elastophlstic composites

80 0.20 -

1.0

0.16 0.8 0.12 |'3

T I

0.6

0.04

0~ I

0

0.4:

I

I

O4

--b

I.,

l

L

0

0.8

I

1

I

0.4

I

0.8 I

F32

----~.F3z

(Q)

(b)

Fig. 6. (a) Average Cauchy stress versus average deformation gradient of the foam rubber under simple shear loading. (bJ Overall dilatation versus average deformation gradient.

deformation is obtained in which T23 = T32 =ttt F32.

F =

[i °° 1

62

0 1

(22)

and p =/.t. The deformation is obviously incompressible so that J = 1 for all values of F32. Let us apply this type of loading, to the foam rubber (47 per cent of voids), i.e., ff3z 4: 0, ffz3 = 0 and ~j = 0 for i + j 4: 5. The resulting response gives rise to T23 = T32 and the average stress component 523 of the Cauchy stress tensor 6 = FT/J

t'~a~

is shown in Fig. 6(a) against if3_,- The deviation from iinearity should be noted. Due to the compressibility of the porous material, the simple shear loading produces a considerable amount of dilatation. This is exhibited in Fig. 6(b) where J = det F is shown versus if.a2- This Kelvin effect (Jaunzemis, 1967) does not exist of course in an incompressible material. Biaxial tension under plane strain conditions

Another comparison with the prediction of the present theory is provided by the recent results of Abeyaratne and Triantafyllidis (1984). ]'hey applied the homogenization theory for periodic structures to predict the response of a nonlinearly elastic matrix containing a doubly periodic array of identical cylindrical cavities. The homogenized method utilizes a singular perturbation analysis based on two scales of asymptotic expansions, provided the period of the structure is small, and calculates the limiting material properties as the size of the microstructure tends to zero. The matrix material is taken by Abeyaratne and Triantafyllidis (1984) as an isotropic compressible material which is characterized by the plane strain internal ~nergy function W= ½~[(I:- 2)- 2(J-

1)+ (J-

1)2/(1 - 2v)].

(24)

J. Aboudi / Elastoplastic composites

81

2.° r

i

1.0

1.4

1.8

J

Fig. 7. Average Cauchy stress against average stretch of a nonlinearly elastic material with cylindrical cavities subjected to equibiaxial stretching in plane strain conditions. The solid and dashed lines show the respon~ as predicted by the present theory and the homogenization theory, respectively.

For small strains (24) provides "-e "-looke's law for a linearly elastic material whose shear modulus and Poisson's ratio are ~ and v, respectively. The finite element procedure is used by Abeyaratne and 'rriantafyllidis (1984) to solve the resulting field equations of the homogenization theory for this two-dimensional problem. The composite is subjected to a biaxial stretching with plane strain conditions. Assuming that the cylindrical cavities extend in the a3-direction, we have ~ = ~ 2 , h3 = 1 and all ~j are zero for i 4:j. The matrix material is characterized by 1, = 0.49 and contains 50 percent of voids. Applying our method on this two-dimensional case we have d t = hl, d2 = 12, 1~ --, ~ , !., is finite and d ~ / ( d l + d z ) 2 = 0.5. In Fig. 7 the average Cauchy stress Oll= O2z is shown against the average stretching ~ . Also shown in the figure are the results of the finite element solution of the homogenization field equations. The two curves were adjusted to coincide in the linear range of small strains. The agreement between the entirely two different methods is reasonable. It should be mentioned that in a particulate composite, for example, the resulting field equations of the homogenization theory, can be solved by the application of a three-dimensional finite element procedure which may involve extensive computer time and memory.

Effective behavior of elastoplastic composites with large deformations Let V t"t~v' denote the spatial gradient of the particles velocity field within the material in the subcell (aflT). It is given in terms of the displacemem gradient tensor by V(.~v, = F~/~T,[ F(.~v)]- ]

(25)

where the dol expresses lhe lime derivative at fixed a. that is at a fixed material particle. Expressing V ~/~v) as the sum of its symmetric and anti-symmetric parts V(,,13v) = D,,,/~v~ + ~2~,,/~v)

(26)

determines the rate of deformation t.ensor D ~"/~v~ and the spin f~,,av). The former expresses the rate of deformation at the current configuration x which is appropriate for plasticity analysis. For finite deformation elastoplastic theory the rate of deformation tensor is decomposed into elastic and plastic parts as follows D(aBv)= DEL(a/~V) + DPL(oBV).

(27)

82

J. Aboudi / Elastoplasticcomposites

In the most commonly used law, the elastic part is given in terms of the Cauchy stress o <"/]v) in the form

[

DELta/],) = [(1 + v./]v ) ~t-/]v, - p./]v

(28)

trace(~"/]V')l]/E,/]v.

Here E~/]v and ~,./]v are the elastic Young's modulus and Poisson's ratio of the material, I is the unit matrix, and ,,t~/]v) dt./]v )

Qt"/]v)ot"/]v) + ot~/]v)Q ~"/]v)

(29)

is the Jaumann rate of change of stress which is used to assure objectivity to superimposed spin of the constitutive law. The plastic deformation of the material is described by the unified elastoplasticity with isotropic work-hardening of Bodner and his coworkers (see Bodner, 1984). In the framework of this unified theory the material behavior is represented by a single set of constitutive relations without a yield criterion, nor loading or unloading conditions. Plastic deformation always exists, but it is negligibly small when the material behavior should be essentially elastic. It is assumed that the Prandtl-Reuss flow rule holds so that D P L ( aft, ) = At ./], )6 t-/iv )

(30)

where 0 t~'/]') = o t'/]') - ~I trace(o <"/]v)) is the deflator of the Cauchy stress tensor, and At'/]'~ is the flow rule function. Using (27) and (29), equations (28) can be inverted to yield d t'/]') = [ h.~, trace(D t~/]Y)) ! + 21t.avDt"/]v) - ot"/]v)f/t"/]Y: + Ut"/]v)o <'/]') ] - 2tt,.,/]vD l'L("/]') (31) where/.t./]v = 0.5E, t~,/(1 + ,,/iv), ~'-/]v = 2/~ ,,/],v.t_~/(1 - 2~,./]v) and the incompressibility property of the flow rule (30) has been used. When the linear expansion (2) of the displacement vector in the ubcell (aft),) is used, the resulting form of the displacement gradient is given by equati,~ (6), from which expressions for F t"/]v), D t'/]v) and f~t-/]v) are deduced (by using (25)-(26)). Definir:g the average Cauchy stress tensor over the subcell in terms of the average first Pioia-Kirchhoff stress tensor S t"/]Y) (given in (9)) by (32)

s t~/]') = F~"~v)St"/]v)/J ~"/]'~ ,

we obtain from (31) ~= IX./], trace(Dt"/]v))! + 2~,/]v D ("/]'i - s'"/]v)$~ ('/]') + l~t"/]V)s("/]')] - 2/t,./],L ('/]').

(33)

In (33), L ("/]') involves the inelastic effects in the subcell. Its evolution law is given by

(34)

L ~/]') = At"/]v)~ ("a')

where ~t-/]v) is the deviator of s t~/]'~. As in Aboudi (1982), the flow rule function is given by the framework of the unified theory by At.0v)__ Dot./],)expl

)

j(a/],) ~l/2, ,

n./]v + l i½(Zt./]v))2/jt2./]v)l,,,,a~ ' / ( 2

'",/],'~p

/'-'0

1'

~p("~v) = 2At"/]v)J2(~/]v)' j2t-/]'¢) = l~(-/]v)~l-/]v)2.ij ~o "

(35)

J. Aboudi / Elastoplastic composites

83

The five parameters Zo~#Y), Z~ ~#v~, m,~#v, n,~#v, and Do~'#v~ specify the material in the subcell in the inelastic range. The parameter Z0~#v) is related to the 'yield stress" of the uniaxial stress-strain curve of the material and Z[~#v~ is proportional to the ultimate stress. The rate of work hardening is controlled by the constant ma#v, and n~#v determines the rate sensitivity of the material. Finally, Do~a#Y~ is the limiting rate of deformation and usually it can be arbitrarily chosen as Do 1 = 10 -4 sec, serving as a reference measure of time. It should be noted that by selecting na/~v large enough (e.g., n,,#v = 10), the response of the material would be essentially rate insensitive for rate of deformations less than about 10/sec. Therefore, with n ~ v = i0, the ma:eriai is practically rate independent for an applied rate of deformation of 0.01/sec, chosen in this paper for all cas~.~s of loading. The computation of the various field quantities is carried out by an incremental procedure in time which involves the integration of J.~#v~ and " ' t o ~ given by (34)-(35). Using tentatively the values of the microstructure variables ~b~'~v ~, Xt ~#v~, 6t ~#v ~Vobtained at the previous time step, we can determine D ~'#v and flt~#v~, and the Cauchy stress s ~'#v~ can be estimated from the integration of (33) where L ~o#v~ at time t (obtained from the integration of (34)) is used. Since S i~#v~ can be determined from (32), it follows that for a given type of loading and in conjunction with (4) and (11), the nonlinear system of algebraic equations in the unknown microvariables can be solved. Its solution naturally involves an iterative process during which S ~#v~ is progressively corrected. When the final form of S ~a#v~ is known at time t the average stress T is readily obtained from (10). In the special case of infinitesimal strains, the derived equations for the overall response of the composite reduce to those given by Aboudi (1984). However, even in this latter case the incremental proceda,e in time is still necessary due to the evolution equations of plastic deformations. We choose to compare the prediction of the present theory with the two-dimensional (plane strain) finite element solution of Needleman (1972) for an elastoplastic work-hardening medium containing a doubly periodic square array of circular cylindrical cavities. The material (matrix) is described in Needleman (1972) by the conventional concepts of elastoplasticity with a yield surface, and a powerhardening relation is used in the representation of the uniaxial tensile stress-strain behavior. Assuming that the cylindrical cavities extend in the a3-direction, the loading conditions imposed by Needleman (1972) are: Tt! = 0, X2 is prescribed, ~ = 1 (plane strain) and all shearing stresses T,j (i ~ j ) are zero. In the framework of the unified theory (Bodner, 1984)) the elastoplastic matrix is characterized in Table 1 where all quantities are normalized with respect to the yield stress Y in simple tension. It sho~:!d be noted that with the value n = 10, the response of the material would be essentially rate independent for rate of deformation gradients less than 10 sec-t. As was mentioned before, in all cases given in this paper, the prescribed loading is applied at the rate of 0.01 sec- t. In Fig. 8 the average stress-average stretching (T_,2 - ~ 2 ) predicted by the present theory are shown for the homogeneous matrix (characterized in Table 1) and the voided matrix with 4.9~ and 19.6~ initial volume fraction of voids. In all cases the above plane strain loading conditions are applied together with d t = ht, d 2 = h2, I1 ~ ~ , ! 2 is finite such that [d~/(d~ + d2)] 2 = 0.049 and 0.196 for the present case of cylindrical cavities. The prediction of the continuum theory is compared in Fig. 8 with the finite element solution of Needleman (1972) and a fair agreement can be observed, except in the case of the higher porosity where different, tendencies are exhibited in the higher values of deformation. This deviation may be attributed to the smearing effect of the elastic and plastic zones in the present continuum approach. The

Table 1 Material properties of the elaqoplastic work-hardening matrix used by Needleman (1972). The parameters E. ~, and Y are its Young's modulus, Poisson's ratio and yield stress in simple tension. E~ Y 200

v 0.33

Do 1 (sec) 10-4

Zo/ Y 1.3

ZI / Y 1.9

at 20

10

84

J. Aboudi / Eiastoplastic composites mcztrix

matrix

,6

"19.6"/o

0 t_._.

IiO

.

. I.:~.

.

. 1.4.

i

!

I. I

1.0

i

i

1.02

~2 (a)

(b)

Fig. 8. Average stress against average stretch of an elastoplastic matrix with cylindrical cavities for two values of porosities: 4,9% and 19.6%. The composite is subjected to a uniaxial stressing in plane strain conditions. The solid ( ) and dashed lines (. . . . . . ) show the response as predicted by the present theory and the finite element solution of Needleman (1972), respectively.Also shown is the behavior of the homogeneous matrix under the same type of loading. In this latter case the solid ( ) line corresponds to the response based on the elastic and inelastic parameters of the matrix given in Table 1, and the dashed lines (. . . . . . ) correspond to its characterization by Needleman (1972). (a) T22 against ~2 for 1 ~<}~2~<1.5; (b) T22 against }~2 for 1 ~<}~2~<1.025. peaks in the stress-stretch relations are obtained because the nominal stresses are shown in the figure rather than the Caucl~y stresses. It is obvious that the presence of voids lowers the load required to achieve a given stretch and decreases the value of stretch at which the m a x i m u m load occurs. Another comparizon with the prediction of the present theory is provided by the c o n t i n u u m model for porous elastoplastic solids developed by Gurson (1977). In Gurson's analysis the void-matrix aggregate is idealized as a homogeneous isotropic spherical body with a concentric spherical cavity, the void volume fraction being identical to that of the aggregate. The behavior of this single shell is determined using certain simplyfying assumptions in a rigid-plastic limit analysis. In particular G u r s o n proposed an approximate yield surface which for zero volume fraction of voids reduces to the Mises yield function. Tvergaard (1981) modified Gurson's yield function by introducing three parameters q~, q2 and q3 such that for q] = q2 = q3 = 1 Gurson's function is recovered. Tvergaard proposed qt = 1.5, q2 = 1, q3 = q2. The plastic rate of deformc, tion of the void-matrix aggregate in the Gurson c o n t i n u u m model is expressed in terms of the proposed yield surface in accordance with the normality rule (in the absence of nucleation of new voids). Tvergaard (1982) computed the response of a porous ductile material subjected to a hydrostatic tension by using a spherically symmetric model (one-dimension), Gurson's model q~ = q2 = q3 = 1 and its modification (q~ = 1.5, q2 = 1, q3 = q2), The matrix material is elastoplastic, whose uniaxial stress-strain relation is described in the plastic region by a power-hardening law. The appropriate description of this material it, the framework of the unified Bodner equations is given in Table 2. The hydrostatic tension loading is obtained by prescribing ~ = ~ 2 = ~3 together with ~ j = 0 for i ~ j . In Fig. 9 we compare the results of the prediction of the present c o n t i n u u m theory with those provided Table 2 Material properties of the elastoplastic work-hardening matrix used by Tvergaard (1982). The parameters E, v and Y are its Young's modulus, Poisson's ratio and yield stress in simple tension. E~ Y

v

Do t (sec)

Zo/Y

Z1 / Y

m

n

250

0.33

10-4

1.3

1.9

10

10

J. Aboudi / Elastoplastic composites

85

I%

, .,,--'2 "" . . . . . . .

-

,.c'- - "

>.,

2

q): I

----'-'-

./.

.... -'2"-

t.

3%

3

-. ......

~.;-'-'-'--'.2

~

--" L - " - ' . :

q~=l.5

I, I

0.02

ql = I ".21.2

"._" -" L ' - - ' . . " - - ' - -

>"

I.

i

:.-.-

I

t

. . . . . .

__

I

I

0.04

0

'

O. 0 2

'

0 . 0'4

--h Fig. 9. Average stress against average stretch of a porous material subjected to a hydrostatic tension for two values of porosity: 1~ and 3%. The elastoplastic homogeneous matrix is characterized in Table 2. The curves describe the response provided by the present theory ( ), by Gurson's (1977) theory ( . . . . . . ) (ql = 1 ) and its modification by Tvcrgaard (1982) (qj = 1.5), and by the simple spherical model ( . . . . . . ).

by: (a) the spherical model (one-dimensional analysis of a spherically symmetric deformation around spherical cavity), (b) Gurson's model (ql = q2 = q3 = 1) and (c) its modification by Tvergaard (1982) (qt = 1.5, q2 = 1, q3 = qlz)- Two values of porosity are selected: 1% and 3% which correspond to d t = h t = ! t, dt = h 2 = 12, and d3/(dt + d2) 3 =0.01 and 0.03, respectively. The figure shows that the response to hydrostatic tension predicted by the present theory is in a better agreement with the response given by the modification of Tvergaard to Gurson's model. The agreement is better in the case of higher porosity. It should be noted that for the continuum matrix (i.e., at zero porosity) the inelastic effects are absent in the present case of hydrostatic loading.

Conclusions Constitutive equations are derived for the prediction of the overall finite deformation of nonlinearly elastic and ductile composites. The resulting average response to a given type of loading is determined solely from the properties of the individual constituents. The composite is modeled as parallelepiped inclusions imbedded periodically in the matrix and various types of composites are obtained as special cases. These include particulate, short-fiber, long-fiber and periodically bilaminated composites. The obtained predicted behavior is compared with various theoretical and experimental results. For infinitesimal deformation, the proposed constitutive relations reduce to those derived previously by the author. The latter were critically checked, compared and discussed in previous investigations (e.g., Aboudi, 1983; 1984).

Acknowledgement The a,thor is grateful to Prof. Y. Benveniste for fruitful discussions. This research was sponsored by the Air Force Office of Scientific Research/AFSC under Grant AFOSR-84-0042, through the European Office of Aerospace Research and Development (EOARD), U.S. Air Force.

References

Abeyaratne, R. and N. Triantafyllidis (1984), "An investigation of localization in a porous elastic material using homogenization theory", J. Appl. Mech. 51,481.

Aboudi. J. (1982). "A continuum theory for fiber-reinforced elastic-viscoplastic composites", lnternat. J. Engrg. Sci. 20, 605.

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J. Aboudi / Elastoplastic composites

Aboudi, J. (1983), "The effective moduli of short-fiber composites", Internal J. Solids and Structures 19, 693. Ahoudi, J. (1984), "Elastoplasticity theory for porous materials", Mechanics of Materials 3, 81. Blatz, P.J. and W.L. Ko (1962), "Application of finite elastic theory to the deformation of rubbery materials", Trans. Soc, Rheoiogv 6, 223. Bodner, S.R. (1986), "Review of a unified elastic-viscoplastic theory" in: A.K. Miller, ed,, Unified Constitutive Equations for Plastic De.formation and Creep of Engineering Alloys, Elsevier Applied Science Pub., Barking. Christensen, R.M. (1979), Mechanics of Co,nposite Materials, Wileyolnterscience, New York. Feng, W.W. and R.M, Christensen (1982), "Nonlinear deformation of elastomeric foams", Internal J. Non-Linear Mech. 17, 355. Gurson, A.L. (1977), "' Continuous theory of ductile rupture by void nucleation and growth: part I--yield criteria and flow rules for porous ductile media", J. Engrg. Mat. Tech. 99, 2. Hashin, Z. (1983), "Analysis of composite materials-a survey", J. Appi. Mech. 50, 481.

Jaunzemis, W. (1967), Continuum Mechanics, MacMillan, New York. Needleman, A. (1972), "'Void growth in an elastic-plastic medium", J. Appl. Mech. 39, 964. Nemat-Nasser, S. and T. lwakuma (1983), '" Finite elastic-plastic deformation of composites", in: Z. i-iashin and C.T. Herakovich, eds, Mechanics of Composite Materials: Recent Advances, Pergamon Press, Oxford, 47. Nemat-Nasser, S. and T. lwakuma (1985), "Elastic plastic composites at finite strains", Internat. J. Solids and Structures 21, 55. Ogden, R.W. (1974), "'On the overall moduli of non-linear elastic composite materials", J. Mech. Phys. Solids 22, 541. Tvergaard, V. (1981), "Influence of voids on shear band instabilities under plane strain conditions", Internat. J. Fracture 17, 389. Tvergaard, V. (1982), "'On localization in ductile materials containing spherical voids", lnternat. J. Fracture 18, 237.