Plastic potential and yield function of porous materials with aligned and randomly oriented spheroidal voids

International Journal of Plasticity, Vol. 9, pp. 271-290, 1993

0749-6419/93 $6.00 + .00 Copyright © 1993PergamonPress Ltd.

Printed in the USA

PLASTIC POTENTIAL AND YIELD FUNCTION OF POROUS MATERIALS WITH ALIGNED AND RANDOMLY ORIENTED SPHEROIDAL VOIDS

Y. P. QIu* and G. J. WENt** *Sonoco Products C o m p a n y and **Rutgers University

Abstract-Based on an energy approach, the plastic potential and yield function of a porous material containing either aligned or randomly oriented spheroidal voids are developed at a given porosity a n d pore shape. The theory is applicable to both elastically compressible and incompressible matrix and, it is proved that, in the incompressible case, the theory with spherical and aligned spheroidal voids also coincides with Ponte Castaneda's bounds of the Hashin-Shtrikman and Willis types, respectively. Comparison is also m a d e between the present theory and those of Gurson and Tvergaard, with a result giving strong overall support of this new development. For the influence of pore shape, the yield function and therefore the stress-strain curve of the isotropic porous material are f o u n d to be stiffest when the voids are spherical, and those associated with other pore shapes all fall below these values, the weakest one being caused by the disc-shaped voids. The transversely isotropic nature of the yield function and stress-strain curves o f a porous material containing aligned pores are also demonstrated as a function of porosity and pore shape, and it is further substantiated with a comparison with an exact, local analysis when the void shape becomes cylindrical.

I. I N T R O D U C T I O N

This paper is concerned with the establishment of plastic potential and yield function of porous materials that contain either aligned or randomly oriented spheroidal voids. When such voids of identical shape are homogeneously dispersed in the ductile matrix, the resulting media are transversely isotropic and isotropic, respectively, with an overall property strongly dependent upon the porosity and pore shape. The choice of spheroidal shape is versatile enough to cover a wide range of void shapes, ranging from disc to sphere, and all the way to needle. These two types of porous media are schematically depicted in Fig. l(a) and (b). The principle to be developed is based upon an energy approach recently proposed by Q ~ and WENO [1992], where Von Mises' effective stress of a heterogeneously deformed matrix is defined from its distortional energy, instead of from its average stress alone. This new definition is capable of accounting for the influence of local yielding to certain extent and, as demonstrated by a comparison with the exact solution for spherical pores and inclusions under hydrostatic tension, can yield very accurate results. Furthermore, when the matrix is elastically incompressible it also delivers a rather desirable theoretical feature: The predicted stress-strain curve of the porous material with spherical voids always coincides with that derived from PONTE CASXA_m~DA'S [1991] lower bound of strain potential of the H A s r n ~ - S ~ [1963] type, and that with any other pore shapes the stress-strain relations always fall below this curve. The quantitative accuracy and the bound consistency as displayed are indicative of its theoretical strength, and for the isotropic case the theory can also deliver the additional information on the influence of pore shape, an important factor that both the exact analysis and the bound are not capable of doing at present. 271

272

Y.P. QIU and G..I. WENG

OI lOgO ! O ! O ! °

J

,

(a)

(b)

Fig. 1. Schematicdiagrams of porous materials with (a) aligned spheriodal voids and (b) randomly oriented spheroidal voids.

The paper in QIu and WENG [1992], however, was primarily concerned with the introduction of this new concept and its implications, with a strong reference on the plastic expansion under hydrostatic tension. The subjects of plastic potential and yield function, which have proved to be so useful for the constitutive descriptions, have not been addressed. This will be the focus of the present study. In addition, the case of aligned spheroidal voids, which also has not been studied there, will also be taken up. The problem of elongated or deflated voids is likely to have some connections with the description of void growth and collapse processes. Before we set out to develop the theory, it should be borne in mind that the nonlinear analysis of a heterogeneous problem is not a simple one, and that the proposed approach, which though to some extent could account for the heterogeneity of plastic deformation in the matrix, is largely a mean-field one. A limitation of such an approach is that it does not provide information on the initial, local yielding near the void surface, and, therefore, its prediction of the initial yield point is not very accurate. The accuracy will improve progressively as the matrix enters the fully plastic range, and, in view of its simplicity, the theory should serve a useful purpose. Due to its mean-field nature, however, its application must be confined to the low concentration range.

II. CONSTITUTIVE EQUATIONS OF THE DUCTILE MATRIX The porous medium can be regarded as a two-phase material in which the ductile matrix will be referred to as phase 0 and the vacuous voids as phase 1. The elastic bulk and shear moduli of the matrix are denoted as ro and #o, respectively, and the volume fraction of the r-th phase as c, (cl + Co = 1). Von Mises' effective stress oe and effective strain ee of the ductile matrix beyond the yield point can be characterized by the distortional part o f the strain potential f ( a e ) , such that

of(o~) %

--

Oa~

-- f ' ( a e ) .

(1)

Plastic potential and yield function of porous materials

273

In particular, when the stress and plastic-strain relation can be described by the modified Ludwik's equation Oe = try + h. (eep)n,

(2)

this potential will take the form 1

2

f ( a e ) = 6#00"e

n

1

+ n+l-- h I/'~ (o e

Oy)(n+l)/n - -

(3)

.

The constants oy, h, and n are the tensile yield stress, strength coefficient, and stainhardening exponent, respectively. This potential then yields the elastic and plastic components of the effective strain = e

-

oe

+

(4)

3#0 as desired. Due to plastic incompressibility the dilatational part of the potential is simply (1/2ro)a 2, in terms of the mean tension am(Ore = o k J 3 ) . Together, they yield the total potential 1

(5)

Oo(oe, Om) = f ( a e ) + ~ - - 0 2 , zKo

for the ductile matrix. By confining our analysis to the proportionally increasing combined stress, it is sensible to adopt the simple deformation theory of plasticity, so that the "secant" shear modulus at a given stage of deformation is given by O'e

/.,t6 --

3f,(Oe) •

(6)

In particular, when the Ludwik equation (2) is adopted, the secant Young's modulus is simply

E~ =

1

- 1- + E0

eep

,

(7)

% + h. (eeP)"

where E0 is the ordinary Young's modulus. The secant Young's and shear moduli, the secant Poisson ratio, and the plane-strain secant bulk modulus k~, are related to each other through the isotropic relations

tr~ #~)= 2 ( l + v g ) '

1 (~

vg= ~ -

)E~

- V o Eoo' k g = x o +

1

~t~).

(8)

274

Y . P . QIU and G. J. WENCJ

are

(AR-

n = 0.455.

(9)

For a 6061-T6 aluminum, its elastoplastic properties at room temperature [1984]; NmH 8, CHELLMAN [1984])

SENAULT

Eo = 68.3GPa,

Vo = 0.33,

Oy = 250MPa,

h = 173MPa

and

These values will be used in our subsequent calculations. It may be noted that, for most ductile materials, the nonlinear potential f(Oe), and therefore Uo(oe,om), are stronger than quadratic as the norm of the stress becomes large. For a linear comparison material whose elastic moduli are equal to the secant moduli of the nonlinear matrix, its strain potential is quadratic, given by 1

2

t

2

g ~ ( a e ' O m ) = 6#~ Oe + ~oKo am

1

1

2

2 a e f ' ( a e ) + 2K0 am"

(lO)

This potential is greater than Uo, with the difference AU 0 = Ug-

U 0 = lvOef'(ae) -- f ( a ~ ) .

(11)

Thus, in order to obtain the strain potential of the nonlinear material from that of the linear comparison material, this difference must be subtracted from the latter. This point will prove useful when we later wish to convert the quadratic strain potential of the linear comparison composite to the strain potential of the nonlinear porous medium.

I!1. P L A S T I C P O T E N T I A L A N D YIELD F U N C T I O N OF I S O T R O P I C P O R O U S M A T E R I A L S C O N T A I N I N G S P H E R I C A L OR R A N D O M L Y O R I E N T E D S P H E R O I D A L VOIDS

The energy approach proposed by QIu and WENC [1992] was based in part on the theoretical framework developed by TANDON and WEN6 [1988] for particle-reinforced solids, and WENC [1990a] for dual-phase metals. This approach makes use of a linear, homogeneous comparison material whose elastic moduli are taken to be equal to the secant moduli L~, of the ductile matrix at a given stage of deformation. The homogeneous comparison material is then filled with an eigenstrain e* in the regions occupied by the inclusions (voids here), its magnitude being determined so that the average stress in the inclusion regions of the comparison material is set equal to that in the original inclusions (or voids) of the two-phase medium (the stress vanishes in voids here). The adoption of the linear comparison material then allows one to use ESHELBY'S [1957] solution of an ellipsoidal inclusion and his equivalent inclusion principle, as well as MORI and TANAKA'S [1973] method of treating the inclusion-inclusion interactions. The average strains in the matrix and the inclusion regions of the e*-filled comparison material are then taken to represent the average strains o f a linear comparison composite, whose microgeometry is identical to that of the original two-phase system and whose constituent moduli are equal to L1 and L~. Though not fully nonlinear, the approach is analytically tractable and, without which, the problem may have to be solved numerically. As shown by WEN6 [1990b], this approach bears an identical theoretical structure as the HASHIN-SHTRIKMAN [1963] and WALPOLE [1966] bounds in the elastic case. With spherical voids, it provides the secant bulk and shear moduli o f the porous material (WENG [1984])

Plastic potential and yield function of porous materials

x~

1

Xo

1 -'l- CI 3Xo + 4#~) Co 4#8

,

#~

1

#~)

1 "l" C1 5(3Xo + 4#~)) Co 9Xo + 8#~)

275

,

(12)

which coincide with HASHIN-SHTRIKMAN'S [1963] upper bounds for the elastic solid. The strain potential of the linear comparison composite under 6 o is equal to 1 02 +

1

62,

(13)

which, upon differentiation, yields the desired overall effective and volume strains,

ge-

OUs 1 aUs 1 -- - - Oe, ~'kk-am. 06 e 3#~ Oam Ks

(14)

Thus at a given 6 o, if tt~ is known, the overall response of the porous medium can be determined. Alternatively, one can say that, if at a given/~) the overall stress a could be found, then the overall behavior of the porous material could also be determined. This latter view has prompted us to construct the yield function of the porous medium. Since the strain energy (13) is contributed solely by the matrix region, it follows that

1 (o~O)(x)e~O)(x) dV, a Vo

(15)

for a unit overall volume, where the local field in the matrix can be written as the sum of its mean values 0~o) and ~-~o), and the locally perturbed fields aiPt(°)(x) and efftt°)(x). Since the volume average o f the perturbed fields must vanish in the matrix, the energy density can be decomposed into the distortional and dilatational parts, as 1

1 r -(o)2

ab°)(x)e~°)(x) = 2~o [Ob(°)6~)(°) + abPt(°)(x)aoe'(°)(x)] + 9Xo ta** + o~,~(°)2(x)]. (16) Now following QIO and WENG [1992], the effective stress oe of the nonuniformly deforming matrix is defined from its distortional energy

O"ez = ~1 f v o 3~ (6/J°)6/~ (°~ +

obptt°)(x)abpt(°)(x))dV.

(17)

The inclusion of the perturbed field allows it to account for the effect of local yielding to a certain extent, and it recovers to its traditional f o r m when the field is uniform. It follows f r o m (13) and (16) that 1 -2

1 02m

6as o~ + ~

Co[ 1

2

1 , - (0)2

6t~[ O'e -[" 2r0 ['am

"+ (O'mPt(0)2(X)))],

(18)

276

Y.P. QIu and G. J. WENG

where the brackets ( . ) represent the v o l u m e average of the said q u a n t i t y over the matrix region. This e q u a t i o n can be cast in a normalized form to serve as the yield function

o =

-\oe/

+ --K, \ oe /

--Co - - + - /~g Ko

+

~-~

2 oe

=0,

(19)

where the relation Om = C00-(m °) has been used. The equality to zero defines the yield condition. The strain potential (13) and yield function (19) still hold with r a n d o m l y oriented spheroidal voids. T h e effective secant bulk a n d shear m o d u l i , however, are given by TANDON a n d WENC [ 1986]:

rs Ko

1 1 + clp'

#s ~o

1 1 + clq'

(20)

where

P = p2/Pl,

q ----q2/ql,

(21)

and, after m a k i n g use o f the p r o p e r t y that voids have v a n i s h i n g elastic m o d u l i , Pl : ql : Co,

1-v~ P2-

18o12-4(l+v~)(tx2-1)-312(1-2v~))(a2-1)+3(2t~2+l)]g

6(1 - 2 v ~ ) 8(1

-

2c~Z+ [(1 - 4 c ~ 2) + (1 + vg)(c~ 2 - 1 ) g ] g

pg)(o~

q2 --

5

2 -

1) [ [

'

1 4o~ 2 + [(1 - 2vg)(ot 2 - 1) - 3 ( ~ 2 + l ) ] g (22)

1

+

2 _ 4(1 - vg)(o~ 2 - 1) + [2(1 - 2vg)((x 2 - 1) - ~21g

1

1

]

+ 6 2o~ 2 + [(1 -- 4 a 2) + (1 + v~)(c~ 2 -- 1 ) g ] g

"

T h e shape p a r a m e t e r g is given by

1

needle void ((X 2 - - 1) 3/2 [Or( ( 2 2 -

g =

1) 1/2 - -

cosh-l(ct)]

2

(1 - - Or2) 3/2 [ C O S - I OL - - Or(1 - -

0

prolate void shape

sphere 0/2)1/2]

oblate void shape circular cracks

(23)

Plastic potential and yield function of porous materials

277

where ot is the aspect ratio (the length-to-diameter ratio) of the voids. As pointed out by TANDON and WENG [1986], this set of moduli always lie on or below HashinShtrikman's upper bounds. At the existing tre, the plastic potential U of the nonlinear porous material can be found from the quadratic Us of the linear comparison composite, by appealing to (11) (24)

U = Us - Co[~%f'(Oe) - f(Oe)],

as the difference now comes only from the matrix part. This difference still holds with the aligned pores to be discussed in Section 6, but with anisotropic U and Us.

IV. ELASTICALLY INCOMPRESSIBLE MATRIX AND COMPARISON W I T H GURSON'S AND TVERGAARD'S MODELS AND PONTE CASTANEDA'S BOUND

When the matrix is elastically incompressible (Xo - , ~), the yield function (19) reduces to the simple form (O~ee)2 + -3[~s - (Om12 -- - C o -~s - =0. Ks \ (~e/ [ZSO

t~ :

(25)

Furthermore, with spherical voids, the secant moduli in (12), can be simplified to Xs _ 4Co g~ 3cl '

P-s g~

Co 1 + 2~Cl '

(26)

such that the yield function becomes

\Oe/

4(1 + 2Cl)

a---~/

1 + 2Cl --O,

(27)

noting that 6,k = 36m. The effective stress ae represents the current flow stress of the matrix; the yield function is therefore applicable to a porous material with a workhardening matrix. When the matrix is ideally plastic it simply reduces to tre= Oy.

IV. 1. Comparison with Gurson's and Tvergaard's theories This yield function bears certain similarity to the widely used GURSON'S [1977] and TVERGAARD'S [1981] models. Gurson's original derivation makes use of a superposition o f two compatible displacement fields, one exact from the dilatational loading with an ideally plastic matrix and the other simply from a constant deviatoric (or shear) strain field. It is therefore an upper bound approach, with

\ Oy]

\ 20y /

(28)

278

Y . P . QIu and G . J . WENO

Tvergaard's modification of Gurson's model was based on his numerical calculations, with

~Tvergaard( 0_.)2~ =

+ 2 q l c a c o s h (q26k, t - (1 + q 3 c 2) : 0 . \ 2dy /

(29)

When the parameters q~, q2, and q3 are all set to 1, it returns to Gurson's model. For the numerical comparison we take the values of q~ = 1.5, q2 = 1 and q3 = q2 = 2.25, a possibility suggested by him. The corresponding yield surfaces at the porosities of ca = 0.2 and 0.3 are plotted in Fig. 2(a) and (b), respectively, where the results by the present theory are marked as "Qiu and Weng." Also marked by "x" on the abscissa is the exact asymptotic solution under a pure hydrostatic loading 6kk, through which Gurson's theory passes (as it was so derived). A quick glance indicates that, under 6kk alone, the present theory overestimates the yield stress whereas Tvergaard's theory significantly underestimates it. Under a pure deviatoric, or shear loading, Gurson's assumption of constant strain invariably leads to a higher yield stress than the other two, with the present theory lying in between. It is interesting to note that, under 6kk alone, the accuracy of the present theory actually improves with increasing porosity; more specifically 6,k

Present theory:

2Co

1

6 k___{

Gurson (exact):

2 In--',

=

O-y

Tvergaard:

C1

6,_~, = 2In oy

1

= 2In 2-~- with q~ = 1.5 and q3 = q2.

q~ cl

(30)

3cl '

Such dependencies are depicted in Fig. 3(a). It shows that, at ca = 0.3, the overestimate by the present theory is about 6°70, and at ca = 0.5, it reduces to only 2°70. The underestimates by Tvergaard's theory are 34°70 and 68%, respectively, with a deteriorating

--e/O y 10

. . . .

, . . . . . . . . .

,

. . . .

10

~e/d y

C~ =20Z

C, =307.

0 8 0 6

Qiu a n d Weng

e

r

g

o

a

r

0.8

- •

d

Qiu ond Weng Ourson Tvergaard

0.6

04

\

0.4

O.2

0.2

OC

L, 1

2 (a)

, 3

5 kk/ O:,,

0.0

1

2

3

ok~/oy

(b)

Fig. 2. Comparison of the yield functions of Gurson, Tvergaard, and the present theory at the porosities of (a) Cl = 2007o and (b) cl = 3007o.

Plastic potential and yield function of porous materials

"O kk/

10

O y

i

. . . .

i

. . . .

i

. . . .

--e/O ,

,

,

,

~.0

Qiu end Weng Exact & Gurson

8

,

y ,

,

,

f

. . . .

x 0.8

6

0.6

4

0.4

J

. . . .

- ~

-

\x~

0 0.00

0.25

0.50

0.75

\

\

. .. \

0.0 0.00

1.00

. . . .

Ourson Tvergaard

----

0.2 cI

l

Qiu and Weng

\\ 2

279

\\

"

.

i \\\

cl

0.25

0.50

(a)

0.75

1.00

(b)

Fig. 3. Comparison of the yield stresses of Gurson, Tvergaard, and the present theory under (a) hydrostatic and (b) deviatoric loadings.

accuracy. It drops to zero at the porosity cl = 1/q~ = 0.67, unless the value of ql is made Cl-dependent. Under a pure deviatoric loading, the corresponding yield stresses are: Present

theory:

tle/O'y ---- CO/%/'l + 2C1,

Gurson:

6e/Oy = Co,

Tvergaard:

(Te//O'y

---- X]I

--

2qlc~ + q3 c2 = 1 - q i c l , with q3

----

q2.

(31)

The simple rule-of-mixture relation in Gurson's theory is a reflection of his constantstrain assumption, and is always an overestimate. The porosity dependence in Tvergaard's relation is also linear, and again drops to zero at Cl = 1/ql = 0.67, if ql = 1.5. The present theory does not suffer from such a limitation. These porosity dependencies are plotted in Fig. 3(b). IV.2. C o m p a r i s o n with P o n t e Castaneda's b o u n d The elastic, or complementary energy given by (13) is that of the linear comparison composite, such that its total stress and strain relation is taken to be equal to that of the porous material at a given internal state/~. To explore the possible connection with Ponte Castaneda's bound, we note that, with r0 ~ 0%

1

+ 2 "~ Cl

= 6Wo

c----T-

9C l ~2m1 +

•

(32)

On the other hand, the yield condition (27) provides the effective stress in the ductile matrix at this stage o f deformation

1[( 1 +

ae = --

¢o

]

cl

+

9 cl

= s, say.

(33)

280

Y.P. QIu and G. J. WZN6

With this, the quadratic potential Us can be rewritten as Co

Us = ~

2

1

1

(34)

Oe = ~ Cot~ef'(Oe) = ~ CoSf'(s),

where the relation ae = 3 / ~ f ' ( a e ) has been used. At this oe the corresponding potential o f the nonlinear p o r o u s medium can be f o u n d f r o m (24), as U = cof(s),

(35)

which is exactly Ponte Castaneda's lower b o u n d o f the H a s h i n - S h t r i k m a n type (lower b o u n d for the strain potential would translate into a higher flow stress). IV.3. Randomly oriented spheroidal voids With elastically incompressible matrix (v~ = vo = 2!), the secant moduli (20) simplify to rs

2Co 2or 2 -

/x~

Ct

(o~ 2 -

1)g

2ot2+ 1

(36) '

/~)

1 + Cl --

Co

' q2

where q2 - -

4(ct 2 5

1) (

1

2

~ -4c~ 2 + 3(c¢ 2 + 1)g + 4 - 2c~ 2 - 3g (37) +

.

3(2 - 3g) [2c~ 2 - (o~ 2 - 1)g]

The yield function (25) becomes

= \Oe/ --

+ - -2

(Co + Cl q 2 ) [ - ~ ~--~ (c~ 2

1)g]

\ 0e ]

Co + clq2

which recovers to (27) for spherical voids (q2 = ~)" Using the properties o f 6061 a l u m i n u m given in (9), but setting Vo = ~, the dependence o f the yield function on the shape, or aspect ratio, o f the voids is displayed in Fig. 4(a), at the porosity o f Cl = 0.3. It is evident that, consistent with the lower-bound status o f the spherical voids, the yield surfaces associated with other void shapes all lie inside it. The disc-shaped voids are seen to weaken the p o r o u s medium m o r e severely than the needle-shaped voids. The sensitivity o f void shape by reducing the aspect ratio f r o m ~ = 0.1 to 0.01 is also greater t h a n by increasing it f r o m 10 to o¢. V. ELASTICALLY COMPRESSIBLE MATRIX

W h e n the matrix is elastically compressible, the contribution by the dilatational part o f the energy in the matrix, headed by 3t~s/~o in the yield function (19), is not negligi-

Plastic potential and yield function of porous materials

0.8

- -

e/O e

0.8

~e/O e

. . . . . . . . . E s

v °. ! 2 c,-0.3 a-i

281

cl-O,3

V o-0.33

yo

"0.8

& PC/HS bound

0.6

0.6 - " ~ ~ : ] . .

0.4

0.4

0.2

0.2 '

>\x

- ~ \

o':,, °_;, o. ~

a~

,!o, I 0.0 0.0

0.5

I

0.0

-6,./a e

0.0

.0

i

0.5 (b)

(a)

"Oral 0 e

.0

Fig. 4. The pore-shape dependence of the yield surface of an isotropic porous medium with (a) elastically incompressible and (b) elastically compressible matrix.

ble. While #~o) = (1/Co)6m, the mean of the perturbed t e r m in general is not known. Its evaluation requires a specific microgeometry, and can be done only numerically. Qiu and WENC'S [1992] analysis, however, suggests that this term is usually negligible, and, as an approximation, it will be neglected here. With spherical voids the yield function (19) then becomes 1 ~ispherical

\ ore /

4-

(~k__~k]2 --

6(tO + 2#~) \ Ore ] 1 +c I 9to + 81z~

C2

6(to + 2#~) 1 +c~ 9to + 8#~

= 0.

(39)

This function, as in the elastically incompressible case, provides a constant yield stress under the hydrostatic loading

6kk ore

2Co

- --. 4~

(40)

Under a pure deviatoric loading, however, the yield stress is not constant even with a perfectly plastic matrix (ae = try), such that ~'e

oe

CO

/

4

;

(41)

2 3(4 - 5~,~) 1 + ~cl 7-5u~

it increases from an initial value by setting v~ = % and reaches the asymptotic state

Co/41 + 2Cl/3, as in (31) by setting ~ = 1.

282

Y . P . QIU and G. J. WENG

No such explicit form can be written with randomly oriented spheroidal voids, where =

(Oe/2

\Oe/

+

3 C l ( 1 +p)lxSO

Co~o(1+ C l q )

(Om12 -\Oe/

CO

1 +clq

- O.

(42)

Again using the properties of 6061 aluminum, but this time with its real Poisson ratio, the void-shape dependence of the yield surface is shown in Fig. 4(b), also at the porosity of c~ = 0.3. The yield surface of the porous material with spherical voids is seen to contain all the others again, although no such lower-bound status can be claimed in this case. Regardless of whether the matrix is elastically compressible or incompressible, it is of considerable interest to derive the stress-strain relation of the porous material at a given porosity c~ and pore shape c~. This can be done by using the strain potential ( 1 3 ) - w i t h rs and #s given by ( 2 0 ) - a n d the yield function (19). Consider the proportionally increasing combined loading 6~i = o~j 6 ( t ),

(43)

where aij are the desired proportional ratios, and t is a time-like parameter indicating the increase of ft. We may start from a given e l , which provides E~, v~, and #8 by (7) and (8), and Oe by (2) or (6). The corresponding Ks and #s then follow from (20). To find the corresponding 5 ( t ) , we introduce OLe = ( 3_ t t 1/2 , 20lijOlij)

(44)

1 and olm = ~Olkk , in parallel to ae and am. The corresponding 6 ( t ) can be computed from (42), as ~(.f)

(re

__

C0

1 +clq

1/2 °re2 +

3Cl(l +_p)/x_~ ~x2 c--~r0(-1 + c~q)

(45)

The magnitudes 6ij are then known. With the computed xs and Izs, the strain potential ( 1 3 ) - o r simply ( 1 4 ) - a l s o provides the corresponding strain components. By increasing the value of eep and repeating the same process, the entire stress-strain curve can be determined. With this procedure and again using aluminum as the matrix, the effective stressstrain curves of the porous materials containing various shapes of randomly oriented spheroidal voids are plotted in Fig. 5(b). Similar curves are also given in Fig. 5(a), but here the Poisson ratio is artificially set to be v0 = 1/2 (elastically incompressible), so that in the case of spherical voids (a = 1), the result coincides with the curve derived from Ponte Castaneda's lower bound (or upper value in the sense of flow stress). Since all other curves lie below this one, the theory is seen to be consistent with the PC bound. With the elastically compressible matrix the curve with spherical voids also lies on the top. Disc-shaped voids, or penny-shaped cracks (with o~ = 0.01 here), are seen to cause the severest weakening effect for the porous material. VI. A L I G N E D S P H E R O I D A L VOIDS

With aligned spheroidal voids, such as the one depicted in Fig. l(a), the five effective secant moduli of the transversely isotropic material can be written as (TA~DO~ &

WENG [1984])

Plastic potential and yield function of porous materials

283

I[.......... -6 a (MPa)

-6 e (MPa)

.1 100 0 r 0

.

.

.

.

.

.

.

I ~ 0 (%) 2

0

~0 (%) 0

1

(a)

2

(b)

Fig. 5. The pore-shape dependence of the effective stress-strain relation of an isotropic porous medium with (a) elastically incompressible and (b) elastically compressible matrix.

1 -

EJ! = E~

2~-~g(4~

v~2 =

c , ( 1 - u~) 2Co

1 - vg - 2vgv~2 +

CI(1 -- v~) 0/2(1 + v~2)(2 -- 3g) - (ol 2 - 1)(1 - v~2 --2v~v~2)g 20/2 - - g ( 4 0 / 2 -

Co

CO[O2 1

2(1

v~)

2(0/T-

Cl

1) +

#j_22 = 1 1 -

2(1

•2

vl--L2 +

Ell

- 1 - 2v~ u~)

1 - 2vg -

+

+ 1 a 2-

1

1 ) + (1 + p 1 ~ ) ( 0 / 2 1)g2

,)]

( cl

Co

/~

E~

1)g 2

(l+v~)(1-2v~)

k~

-

)

[4v~(g -- 1) -- 2] (0/2 __ 1) "1- (2 -- 3 g ) 20/2 -- g(4ot 2 -- 1) + (1 + vg)(0/2 -- 1)g 2

x~ =

1

~l--~gT(~'-2 "~ " 1 ~ 2

20/2 -- g(40/2 -- 1) + (1 + v~)(0/2 -

2Co v~ 1+

1) + ( i

~--I,

0/2(2 -- 3g) + (1 + 2v~)(0/2 -- 1)g

c, (1 - v~)

1

v~

[4v~(g - 1) - 2] (0/2 __ 1) + (2 - 3 g )

v~

1 + c, - 2Co

4(0/2-

,(

l - 2v~

1)

g

0/2-S i

g

], (46)

4

w h e r e the a l i g n e d d i r e c t i o n is t a k e n t o b e d i r e c t i o n 1, a n d p l a n e 2-3 i s o t r o p i c . This set o f m o d u l i have been recently p r o v e d b y WENG [1992] to coincide with t h o s e derived f r o m Wn~us' [1977] u p p e r b o u n d o f strain energy, which in t u r n generates H a s h i n - S h t r i k m a n ' s

284

Y . P . QIu and G. ,I. WEN~

b o u n d s when c~ = 1, HILL'S [1964] a n d HASHIN'S [1965] b o u n d s when c~ --, o% and WALPOLE'S [1969] exact m o d u l i when a ~ 0. It must be noted that, a m o n g these moduli, only K23, s #~2 a n d #~3 q u a l i f y as u p p e r b o u n d s and the other three are merely m o d u l i derived f r o m the b o u n d (see WENG [1992]). U n d e r a general 6 a , the strain energy o f the linear c o m p a r i s o n c o m p o s i t e then can be written as 1 [ 1s

Us ~ 2

~ 1 1 ~21 --

2vi~2

Ef~

_ 1

1

O11(6.22 + 6.33) -'l- EJ---2(6.22 + 6.33) 2

1

k~3 6.226.33 + /~3

,

6.2 + _ _

~2

]

(6.22 + 6.123) .

(47)

This energy is a g a i n c o n t r i b u t e d by the m a t r i x p h a s e alone, given b y the right side o f (18). T h e effective stress o f the m a t r i x , after m a k i n g use o f 6.~0) = (1/Co)6.m,is given by

2

6"O[u s

(Ie =

1 l (6.2+Cg(OPmt(O)2(X)))l. 2 CoKo

C0

(48)

Connection with Ponte Castaneda's bound of Will&' type for the elastically incompressible matrix

VI. 1.

To m a k e this c o n n e c t i o n , we n o t e t h a t for the elastically i n c o m p r e s s i b l e m a t r i x (Vo = vd = 1/2), the effective secant m o d u l i can be s i m p l i f i e d to El1 _ 3#t~

1 = - - , 1 say 2 ( g - 2)(or 2 - 1) + (2 - 3g) ml D 4Co 2ix 2 - g(4ot 2 - 1) + 3 ( 0 / 2 - 1)g 2 C1

1+

cl

ix2(2 - 3g) + 2g(ot 2 - 1)

2Co 2o~ ~ - g ( 4 a

1

2-

1)+ 3(0/2-

1)g 2

CI 2 ( g - 2)(or 2 - 1) + (2 - 3g) l + m 4Co 2or 2 - g(4ot 2 - 1) + 3(c~2 - 1)g 2

1

K~3

3/z~

1

t~3 /z~

--

1+

2u12

+

-C0

1 C1 4(or 2 -

E~2 3M~

l+--

1)

1 = --, m4

c~

2(ct 2 -

say

1 = --,

1)

say

m5

CO (0/2 "{" 1)(3g - 2)

1 --

say

m3

[ 0~2 -- -g (iX22 -- 1)] (2 -- 3g)

1

/z~

1)(1 - 2v12)

Co (2or 2 -- 3g)

/x~2

- -

1 = --,

C 1 tX2(1 + 1'12)(2 -- 3g) - g ( o t 2 -

S2 m l P12 "{- (ms + 3 m 4 ) / 4

1 = --, m2

say.

(49)

Plastic potential and yield function of porous materials

285

It is important to note that, unlike (46), the right s i d e s - o r constants m l , m 2 , ' ' ", m 5 are all independent o f / ~ . The incompressibility, PiE = 1/2, holds only for cylindrical voids (tx --, oo). Ponte Castaneda's lower bounds or estimates of Willis' type for the strain potential under the present condition can be written as (his equation (3.6)) (..7_(6) = sup{ (~(6) - V(/2o)], in his notations f'o

= sup{ Us(6) - Co[ lf'(ae)Oe

-- f(tre)]},

in the present notations

(50) = Co sup - - S2 ~ 6tz~)

f'(s)s - f(s)

= cof(s),

where Us(6) = Co[1/(61zSo)]S 2, with s2

=

1

{m1021

_

2v~2mlOll (0.22 + 033) + m2(~r22 + 033) 2

Co

(51) + 3[--m4622633 + m4623 + m5(622 + 623)11.

The simple result in the last step of (50) can be accomplished because s in (51) is independent o f / ~ (a similar result can not be achieved if v0 *: 21) • This s is exactly equal to our oe in (48), when the matrix is elastically incompressible (do ~ oo). Then, as discussed in Section IV.2 for the isotropic, spherical case, the corresponding strain potential of the nonlinear porous medium calculated by way of (24) from Us in (47) is

6~

~e -

Co

~ef'(~e)

-- f ( O e )

= Cof(S)

= 0_.

(52)

The strain potential and, therefore, the strain-stress relations calculated from the present theory again coincide with Ponte Castaneda's lower-bound approach. Since only rs23, #~3 and/~2 qualify as bounds, only the stress-strain curves under the plane-strain biaxial loading, the transverse shear, and the axial shear can be said to be derived from the upper bounds. The axial (except for ot ~ 00) and transverse tensile stress-strain curves are merely estimates. No such connection can be established between the present theory and the PC bound (or estimate) when the matrix is elastically compressible. The optimization process involved in the second step o f (50) leads to a rather complicated result for the effective stress in the PC bound, and it is unlikely that such a complicated expression will coincide with the simple (48) of the present one (particularly with the contribution from the a~tt°)(x)-field neglected).

286

Y.P. Q[u and G..l. W'~N(;

VI.2. Yield f u n c t i o n The yield function of the porous medium can be constructed by equating (47) to the right side of (18), as ~(Oij ) ~

(011) 2 __ 2/,'f2(011t (022 \ °e

-]- 033 I

\ ae I \ °e

_[_

Oe ]

__Ell( 022_ ~- --033t2 E~2

ae

Oe l

Ells 0220332 --1---Ell (023)__ q- --EiSl[( __022) -{- (023/]_ /L23 Oe ~3 \ Oe #~2 [_\ O-e \ Oe / J - Co 3tzf,

C2oK{~

~

+ c2 <"m 0,2

/

(53)

: O,

where the term <~,(o)2(,)>can be neglected for simplicity. With elastically incompressible matrix, it reduces to + - -m2 (022 --+-\ a~l

\ a e l \ a,.

022033 ml Oe

3m4

d-

--

G,

t

m 1

(54)

-~- - -

m 1 \ ae /

03312 Oe /

O-e

--

ml

-~-

\ Oe /

--

=0.

(Te /J ml

Of particular interest is the axisymmetric loading, under Oil and 622 = 633. The voidshape dependence of such an anisotropic function is shown in Fig. 6(a) with u0 -- 1/2, and in Fig. 6(b) with u0 = 0.33 at E U E o = 0.8, again using the properties of 6061 aluminum for the other constants. The present theory and the Ponte Castaneda/Willis approach coincide in Fig. 6(a) for each c~ (which also defines the spheroidal symmetry in Willis' correlation function). The general trends of the c~-dependence as displayed in

1.0

N~/o e-~~/o

0,5

0.0 0.0

~_

1.0

322/0 ~-'~3/o

3_ v o - ~ ci-0.3 &PC~Willis

v~-0.33

(a)

Eo

0.5 0.01

~ 1 1 / o ~ 0.0

0.6

--E~=0,8

/

/I/

o.o

c~-0.3

1.2

0.0

i

i ii

0.i

l

l

0.6

r

~11/Oe

.2

(b)

Fig. 6. The pore-shape dependence of the yield surface of a transversely isotropic porous medium with (a) elastically incompressible and (b) elastically compressible matrix under axisymmetric loading.

Plastic potential and yield function of porous materials

287

Figs. 6(a) and (b) are seen to be identical, regardless of the elastic compressibility of the ductile matrix. VI.3. S t r e s s - s t r a i n c u r v e s The stress-strain curves of the transversely isotropic porous medium can be constructed from the strain potential (47) and yield function (53) under any proportionally increasing combined stress, as in (43). The procedure is entirely analogous to the one described in Section V for the isotropic case. Thus, one m a y start with a given e l , which gives rise to the secant moduli E~, ~ , ~ , and k~ from (7) and (8). The effective secant moduli then follow from (46), providing

au~

(55)

E u - 88ij'

at a given Oii. On the other hand at this assumed eep, the corresponding oe is given by (2), or more generally by (6). At this oe the magnitude of O(t) in (43) can be evaluated from the yield condition (53), as

oC

-

\3#~o/

[Eft a2,

E~--~u , , ( - 2 2

-

+ a,3) + ~2 (56)

_ 1

1 a223+

1 (a22+0~23)

+

1 ot2] - ' / 2 -

co ---o

j

'

where the contribution from Om pit°) has been neglected. This, in return, provides the magnitudes of 5/1 f r o m (43). At this a u the corresponding gu are given by (55). Then, by increasing the value of eep and repeating the same process, the entire stress-strain curves under any proportional combined loading can be determined.

-6,

(o22:°33)

MPa

1 000 . . . .

800 600

i=30

400

=0.I

200 i

0.0

i

,

,

I

0.5

,

~

,

i

I

1.0

,

h

i

,

I

1.5

, E0

i

,

2.0

Fig. 7. Comparison between the present theory and the exact, local analysis of a porous material with cylindrical pores in an elastically incompressiblematrix.

288

Y . P . QIu and G. J. WEN¢;

In order to check the quantitative accuracy of this approach, we first applied it to the condition of long circular voids (a --, 0o) under the axisymmetric, plane strain condition, where exact, local analysis can also be performed when the matrix is elastically incompressible. Under #22 = #33 ( = #, say) and ~-lJ = (1/ELI)[#l! - .v~2(#22 + #33)] = 0 , the tensile stress #11 = v~2(#22 + #3~) must be applied continuously to maintain the plane strain condition. In this comparison, the matrix phase is taken to have a bilinear stressstrain curve, whose tangent modulus in the plastic state is characterized by E6~. The ratio E~/Eotherefore defines the degree of work hardening, with the extreme values 1 and 0 representing an elastic and ideally plastic state. At the porosity of q = 30%, the biaxial stress-strain curves are shown in Fig. 7, where the solid lines represent the present theory and the dashed ones are the exact results o f the local analysis (see the Appendix). With the exception o f initial yielding, the present theory is seen to be generally ac-

011, 250

MPa

022

5

_i "Ioo ........

//#/'-

200

~ ~

.............

so

1 O0

,I/ ,"

0

,

i , 0.5

,

,

,

,

. . . .

. . . .

,

,

LO'91

-[

/i ~

2-_<__

,

=:_[

1oo

150

1 O0

, , 1.5

1.0

,

200

0 1 o~_

/",.,

. . . . ~=1io:2.

CI=20%

0.0

,

_

///,/,/

50

MPa

,

. . . .

25O

. . . . . . . . . . . . . . . . . . .

,

,

ca=20%

5O

e-~ (%)

,

,

o

,

,

O0

2.0

,

r

,

,

,

0.5

,

i

,

,

,

I .0

,

i

,

,

,

I5

,

~22

(%)

2.0

(b)

(a) (oa==~a~)

O, M P a 400

500

0

~

>S .................. c o.o! i i,.......

200 100

. . . .

0

,

0.0

. . . .

0.5

,

. . . . . . . . .

1.0

.5

"~

(%)

20

(c) 023

o12, MPa 1 20

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

MPa

150

.... . . . . . . . .

/ 0 O0

,

......... 0 : 0 !

,

0.2

0.01

= =_.~

__< ~ ~___

100

c~=20%

/'

,'

40

./-

/~/,,

100

80

a=l

,

,

, . . . . 0.5

i . . . . 10

(d)

, 15

,

,

50

~2

, 2.0

~#

CI=20%

(%) 0 0.0

0.5

1 .0

15

2.0

(e)

Fig. 8. The pore-shape dependence of the transversely isotropic behavior of a porous material containing aligned spheroidal voids under five respective loading conditions.

Plastic potential and yield function of porous materials

289

curate, and, consistent with the concept of a linear comparison material, the comparison gets progressively better as the work-hardening ratio becomes stronger. The void-shape dependence of the stress-strain relations for the transversely isotropic porous medium is displayed in Fig. 8 under five different loading conditions, at the porosity of cl = 20°70. Here the matrix is again taken to be 6061 aluminum. Under axial tension #11, prolate voids (~ = 5,100) are seen to give less weakening effect as compared to oblate, or disc-shaped voids (a = O.2,0.01). A reversed trend is observed under the transverse tension 622 and transverse shear #23. Under the plane strain biaxial loading, voids a r e - a s evidenced from the higher stress-strain curves-generally less damaging. The sensitivity of void shape under axial shear #t2 is seen to be weak when a __. 1, and becomes quite pronounced as the shape approaches that of a circular crack (~ = 0.01). Acknowledgement-This work was supported by the National Science Foundation, Mechanics and Materials Program, under Grants MSS-8918235 and MSS-9114745.

REFERENCES

1950 1957 1963

1964 1965 1966 1969 1973 1977 1977 1981 1984 1984 1984 1984 1986 1988 1990 1990a 1990b 1991

HILL, R., The Mathematical Theory of Plasticity, Clarendon Press, Oxford. ESHEtaY, J.D., "The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems," Proc. R. Soc. Lond., A241, 376. HASHIN, Z., and SHTRIKMAN,S., "m Variational Approach to the Theory of the Elastic Behavior of Multiphase Materials," J. Mech. Phys. Solids, 11, 127. HILL, R., "Theory of Mechanical Properties of Fiber-Strengthened Materials, I. Elastic Behavior," J. Mech. Phys. Solids, 12, 199. HASrtIN, Z., "On Elastic Behavior of Fiber Reinforced Materials," J. Mech. Phys. Solids, 13, 119. WALPOLE, L.J., "On Bounds for the Overall Elastic Moduli of Inhomogeneous System-I," J. Mech. Phys. Solids, 14, 151. WALPOLE, L.J., "On the Overall Elastic Moduli of Composite Materials," J. Mech. Phys. Solids, 17, 235. Moal, T., and TANAKA,K., "Average Stress in Matrix and Average Elastic Energy of Materials With Misfitting Inclusions," Acta Metall., 21, 571. GURSON, A.L., "Continuum Theory of Ductile Rupture by Void Nucleation and Growth: Part I Yield Criteria and Flow Rules for Porous Ductile Media," ASME J. Eng. Mat. Tech., Trans. ASME, 99, 2. WrLLXS,J.R., "Bounds and Self-Consistent Estimates for the Overall Properties of Anisotropic Composites," J. Mech. Phys. Solids, 25, 185. TVERGAARO, V., "Influence of Voids on Shear Band Instability Under Plane Strain Condition," Int. J. Fracture, 17, 389. ARSENAULT, R.J., "The Strengthening of Aluminum Alloy 6061 by Fiber and Platelet Silicon Carbides," Mat. Sci. Eng., 64, 117. NIEn, T.G., and CrtELLMAN, D.J., "Modulus Measurements in Discontinuous Reinforced Aluminum Composites," Script. Metall., 8, 925. TANDON, G.P., and WENG, G.J., "The Effect of Aspect Ratio of Inclusions on the Elastic Properties of Unidirectionally Aligned Composites," Polymer Composites, 5, 327. WENG, G.J., "Some Elastic Properties of Reinforced Solids, With Special Reference to lsotropic Ones Containing Spherical Inclusions," Int. J. Eng. Sci., 22, 845. TANDON, G.P., and WENG, G.J., "Average Stress in the Matrix and Effective Moduli of Randomly Oriented Composites," Comp. Sci. Tech., 27, 111. TANDON, G.P., and WENG, G.J., "A Theory of Particle-Reinforced Plasticity," J. Appl. Mech., 55, 126. Qiu, Y.P., Theory of Inclusions in the Elastic and Elastic-Plastic Deformation of Composite Materials, Ph.D. Dissertation, Rutgers University. WENG, G.J., "The Overall Elastoplastic Stress-Strain Relations of Dual-Phase Metals," J. Mech. Phys. Solids, 38, 419. WEN6, G.J., "The Theoretical Connection Between Mori-Tanaka's Theory and the Hashin-ShtrikmanWalpole Bounds," Int. J. Eng. Sci., 28, 1111. PONTE CASTANEDA,P., "The Effective Mechanical Properties of Nonlinear Isotropic Composites," J. Mech. Phys. Solids, 39, 45.

290

1992 1992

Y.P. QIu and G. J. ~'ENG

QIo, Y.P., and WENC, G.J., "A Theory of Plasticity for Porous Materials and Particle-Reinforced Composites," J. Appl. Mech., 59, 261. WENG,G.J., "Explicit Evaluation of Willis' Bounds With Ellipsoidal Inclusions," Int. J. Eng. Sci., 30, 83.

Sonoco Products Company 555 Science Drive, Suite B University of Wisconsin Research Park Madison, WI 53711, USA Department of Mechanical and Aerospace Engineering Rutgers University New Brunswick, NJ 08903, USA

(Received 2 February 1992; in final revised form 15 August 1992) APPENDIX: EXACT SOLUTION FOR THE OVERALL STRESS-STRAIN RELATION UNDER PLANE-STRAIN, BIAXIAL LOADING

Within the small strain range (without considering the void growth or collapse) and assuming a bilinear stress-strain relation (n = 1) for the elastically incompressible matrix, the overall stress-strain curve of a porous material with unidirectionally aligned cylindrical pores can be constructed in three steps: (i) Onset of yielding occurs at c0 6- = 6-22 = 6-33 = ~ oy,

(A.I) x/3Cl = 622 = ~33 --

2Eo

flY"

(ii) During partial yielding in the matrix with r = R denoting the elastic/plastic boundary: oy 6= ~ g-

1 - R- 2- + - 2Eo In-- + a2 Eo + h 1:11 ~

-1 (A.2)

X/3 R 2

2Eo a22 Oy, where h is the work hardening modulus, and is related to the elastic modulus Eo and tangent modulus E~' as h = 1/[(1/E~) - (l/E0)], and al and a2 are the inner and outer radii of the composite cylinder model, respectively. (iii) Fully plastic state: With the initial value x/3Oy/2Eo, the biaxial strain g increases gradually. At a given value ~ (~ > x/3Oy/2Eo), the stress is -

o-

-

Eo+h

[ ,nl

--

cl

+

-

T

1

t] ~

(A.3)

"

The procedure leading to these results was elementary but lengthy; it basically followed Hill's [1950] analysis but incorporated the work-hardening behavior of the matrix (Qiu [1990]).