The Size Effect on Void Growth in Ductile Metals

8/12/2019 The Size Effect on Void Growth in Ductile Metals

1/17

Journal of the Mechanics and Physics of Solids

51 (2003) 11711187

www.elsevier.com/locate/jmps

The size eect on void growthin ductile materials

B. Liua, X. Qiub, Y. Huanga ;, K.C. Hwangb, M. Lic, C. Liud

a

Department of Mechanical and Industrial Engineering, University of Illinois, 1206 W. Green Street,Urbana, IL 61801, USA

bDepartment of Engineering Mechanics, Tsinghua University, Beijing 100084, ChinacAlcoa Technical Center, Aluminum Company of America, Alcoa Center, PA 15069, USA

dMaterials Science and Technology Division, Los Alamos National Laboratory,

Los Alamos, NM 87545, USA

Received 1 March 2002; accepted 19 February 2003

Abstract

We have extended the RiceTracey model (J. Mech. Phys. Solids 17 (1969) 201) of voidgrowth to account for the void size eect based on the Taylor dislocation model, and have found

that small voids tend to grow slower than large voids. For a perfectly plastic solid, the void size

eect comes into play through the ratio l=R0, where l is the intrinsic material length on the

order of microns, the remote eective strain, and R0 the void size. For micron-sized voids and

small remote eective strain such that l=R06 0:02, the void size inuences the void growth rate

only at high stress triaxialities. However, for sub-micron-sized voids and relatively large eective

strain such that l=R0 0:2, the void size has a signicant eect on the void growth rate at all

levels of stress triaxiality. We have also obtained the asymptotic solutions of void growth rate at

high stress triaxialities accounting for the void size eect. For l=R0 0:2, the void growth rate

scales with the square of mean stress, rather than the exponential function in the RiceTracey

model (1969). The void size eect in a power-law hardening solid has also been studied.

? 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Voids; Size eect; Void growth rate; Strain gradient plasticity; Taylor dislocation model

1. Introduction

The micro-indentation hardness experiments have repeatedly shown that the indenta-

tion hardness of metallic materials displays strong dependence on the indentation depth

Corresponding author. Tel.: +1-217-265-5072; fax: +1-217-244-6534.

E-mail address: [email protected] (Y. Huang).

0022-5096/03/$ - see front matter? 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0022-5096(03)00037-1
mailto:[email protected]:[email protected]


2/17

1172 B. Liu et al. / J. Mech. Phys. Solids 51 (2003) 1171 1187

(e.g., Nix, 1989, 1997; Stelmashenko et al., 1993; Ma and Clarke, 1995; Poole et al.,

1996; McElhaney et al., 1998; Swadener et al., 2002). The indentation hardness may

increase by a factor of two or even three as the indentation depth decreases to mi-crons and sub-microns. Similar size dependence of plastic behavior of materials at the

micron scale has also been observed in micro-twist of thin copper wires (Fleck et al.,

1994), micro-bend of thin nickel foils (Stolken and Evans, 1998;Shrotriya et al., 2003)

and thin aluminum beams (Haque and Saif, 2003), in an aluminum matrix reinforced

by SiC particles (Lloyd, 1994; Xue et al., 2002a), and in a micro-electro-mechanical

system (MEMS) named Digital Micromirror Device (DMD) (Douglass, 1998; Xue

et al., 2002b). Direct dislocation simulations have also repeatedly shown strong size

dependence of metallic materials at the micron scale under various loading conditions

(Cleveringa et al., 1997, 1998, 1999a, b, 2000; Needleman, 2000).

Classical theories of plasticity fail to explain the observed size dependence of ma-terial behavior at the micron and sub-micron scales since their constitutive models

possess no internal material length. This has led to the development of strain gradient

plasticity theories (e.g., Fleck and Hutchinson, 1993, 1997, 2001; Fleck et al., 1994;

Nix and Gao, 1998; Gao et al., 1999a, b; Huang et al., 1999, 2000a, b; Acharya and

Bassani, 2000; Acharya and Beaudoin, 2000; Qiu et al., 2001; Gurtin, 2002; Hwang

et al., 2002, 2003) based on the concept of geometrically necessary dislocations in

dislocation mechanics (Nye, 1953; Cottrell, 1964; Ashby, 1970; Arsenlis and Parks,

1999; Gurtin, 2000). The Taylor dislocation model (Taylor, 1934, 1938), which links

the ow stress and the dislocation density, is the basis for some strain gradient plastic-

ity theories (e.g.,Gao et al., 1999b;Huang et al., 1999, 2000a, b;Qiu et al., 2001), andthese theories have shown excellent agreement with the micro-indentation experiments,

such as with McElhaney et al.s (1998)of copper hardness data (Huang et al., 2000b),

Stelmashenko et al.s tungsten data (Qiu et al., 2001), and Saha et al.s (2001) data

for an aluminum thin lm on a glass substrate. These theories also agree well with

other micron scale experiments, such as Fleck et al.s (1994) micro-twist and Stolken

and Evans (1998) and Haque and Saifs (2003) micro-bend experiments (Gao et al.,

1999a; Haque and Saif, 2003), with Lloyds (1994) particle-reinforced metal-matrix

composite materials (Xue et al., 2002a), as well as with Douglasss (1998) MEMS

named DMD (Xue et al., 2002b). Jiang et al. (2001)andShi et al. (2001)have shown

that these strain gradient plasticity theories based on the Taylor dislocation model canexplain cleavage fracture in ductile materials observed in experiments (Bagchi et al.,

1994; Elssner et al., 1994; Bagchi and Evans, 1996).

The ductile failure of metallic materials results from the nucleation, growth and

coalescence of microvoids. For a rigidperfectly plastic solid under high stress triax-

iality, the pioneer work of Rice and Tracey (1969) established that the void growth

rate increases exponentially with the mean stress surrounding the void. Attention was

limited to a spherical void in an innite rigidperfectly plastic solid characterized by

the Mises yield condition, e =Y, where the eective stress e was related to the

deviatoric stresses sij

= ij

1

3

kk

ij by

e= (3s

ijs

ij=2)1=2, and

Ywas the yield stress.

The solid was subjected to remote axisymmetric tension, 11 ,

22 =

11 , and

33 , sat-

isfying the Mises yield condition 33 11= Y. The non-vanishing strain rates in theremote eld were 33= 211= 222 , and the corresponding eective strain rate was


3/17

B. Liu et al. / J. Mech. Phys. Solids 51 (2003) 1171 1187 1173

= (2ij

ij =3)1=2 = 33 . Rice and Tracey (1969) gave the void growth rate as

D=

V

3V = 0:2833 exp3m

2Y

; (1)

where V was the void volume and m =

kk=3 was the mean stress in the remote

eld. The RiceTracey model (1969) has been widely used to study void growth in

ductile materials, and has been extended to viscoplastic materials (e.g., Budiansky

et al., 1982), to account for the eect of low stress triaxiality (e.g., Huang, 1991),

and to establish yield criteria for isotropic (e.g., Gurson, 1977; Tvergaard, 1990),

anisotropic (e.g., Chen et al., 2000) and viscoplastic solids containing microvoids

(Chen et al., 2003).

Recent experimental investigations (e.g., Schluter et al., 1996; Khraishi et al., 2001)

and numerical studies (e.g., Fond et al., 1996; Fleck and Hutchinson, 1997;Shu, 1998;Huo et al., 1999; Zhang et al., 1999; Zhang and Hsia, 2001) have shown that void

growth in ductile materials also depends strongly on the void size. The micron- and

sub-micron-sized voids tend to grow slower than large voids under the same stress level.

This size eect is consistent with the aforementioned experiments of micro-indentation,

micro-twist, micro-bend, particle-reinforced composite, and MEMS, i.e., smaller is

stronger. The RiceTracey model (1969) as well as other void growth models (e.g.,

Budiansky et al., 1982; Huang, 1991) cannot account for the void size eect since

the classical plasticity theories possess no intrinsic material lengths. Since the Taylor

dislocation model (Taylor, 1934, 1938) can characterize the plastic deformation rather

accurately at the micron and sub-micron scales, it is used in the present paper to extendthe RiceTracey model (1969) of the void growth rate accounting for the void size

eect. Section 2 provides a summary of the RiceTracey model of void growth, while

Section3gives the Taylor dislocation model, which relates the ow stress to the plastic

strain and plastic strain gradient. Section 4 studies the void size eect in a perfectly

plastic solid, while Section 5 gives the analytical expression of the void growth rate

for the high stress triaxiality limit. The void size eect in a power-law hardening solid

is studied in Section 6.

2. Variation principle for a rigidperfectly plastic solid

Rice and Tracey (1969) established a variation principle for an innite, rigid

perfectly plastic solid containing a single void. The variation principle, also adopted by

Budiansky et al. (1982) and Huang (1991) to study void growth in ductile materials,

involves the minimization of the following functional of velocity u,

(u) =

[sij() sij ]ijd ij

S

niujdS; (2)

where is the innite volume exterior to the void, S the void surface, ni the unitnormal on S pointing into the solid; u is the velocity eld satisfying the incompress-

ibility condition u k; k = 0, strain rates ij= (u i; j+ uj; i)=2; sij and s

ij are the deviatoric

stresses and their counterparts in the remote eld, respectively, and are given in terms


4/17


of the strain rates via plastic ow normality,

sij= Yij

( 32klkl)1=2; sij =

Y

ij

( 32

kl

kl)1=2: (3)

Here the Mises yield condition e =Y has been used, and Y is the tensile yield

stress.

For prescribed remote strain rates ij , an additional velocity eld u

i is dened such

that

ij=

ij +

ij;

ij= 12

(ui; j+ u

j; i): (4)

Among all admissible additional elds satisfying u =o(R3=2) as the distance to the

void centerR , the exact eld minimizes the functional (Rice and Tracey, 1969).Similar to Rice and Tracey (1969), Budiansky et al. (1982) and Huang (1991), we

decompose the admissible velocity eld u i into the following three parts,

(i) a uniform velocity eld ij xj associated with the uniform remote strain rates

ij ,

where xj are the Cartesian coordinates; this eld does not change the void volume

due to incompressibility of the remote eld, kk = 0;

(ii) a volume-changing, spherical symmetric additional velocity eld uDi correspond-

ing to a change in the void volume but no change of shape, i.e., uDR= (R0=R)2R0,

uD= uD= 0 in spherical coordinates (R; ; ), where R0 is the void radius and R

the distance to the center of spherical void; and

(iii) a shape-changing additional velocity eld, u

E

i, decaying at remote distances, whichchanges the void shape but not its volume, i.e., uEi(R ) = o(R3=2),S

uERdS= 0, where integration is over the void surface S.

Rice and Tracey (1969) showed that the above shape-changing additional velocity

eld (iii) has essentially no eect on the void growth rate at high stress triaxiality.

The velocity eld is then written as

u i=

ij xj+R20R3

R0xi=

ij xj+ DR30R3

xi; (5)

where = 2ij

ij

=3 is the eective strain rate in the remote eld,

D=R0

R0=

V

3V (6)

is the void growth rate consistent with Rice and Tracey (1969) given in Eq. (1), and

V is the void volume. The strain rates can be obtained from (5) as

ij=

ij + D Dij (7)

and the non-vanishing components of Dij are

DRR=

2 D=

2 D=

2

R30

R

3 (8)

in the spherical coordinates.

The parameter to be determined in the velocity eld (5) is the void growth rate D.

For a rigidperfectly plastic solid, the minimization of velocity functional in Eq.


5/17


(2) with respect to D gives the following non-linear integral equation forD (Rice and

Tracey, 1969)

[sij(D) sij ] Dijd=mR0

S

dS; (9)

where sij(D) are obtained from (3), (7), and the Mises yield condition e = Y; s

ij are

obtained similarly in terms of the remote strain rates ij ; the non-vanishing components

of Dij are given in Eq. (8), m =

kk=3 is the remote mean stress, and R0 the void

radius. Note that in computing derivatives for the minimization, we have used

@sij(D)

@D ij= 0 (10)

since the stress derivative is tangent to the yield surface and is therefore normal to ij.

Rice and Tracey (1969) obtained the solution of (9), which gives the void growth rate

in Eq. (1) under high stress triaxialities.

In the present study of the void size eect on the void growth rate D, we modify the

RiceTracey model to replace the tensile yield stress Y by the ow stress obtained

from the Taylor dislocation model, as discussed in the following sections.

3. The Taylor dislocation model

The Taylor dislocation model (Taylor, 1934, 1938) gives the shear ow stress interms of the dislocation density by

=b

=b

S+ G ; (11)

where is the shear modulus, b the Burgers vector, an empirical material constant

around 0.3 (e.g., Taylor, 1934, 1938; Wiedersich, 1964), S and G are densities of

statistically stored dislocations and geometrically necessary dislocations, respectively.

The above relation has been rewritten to give the tensile ow stress in terms of the

plastic strain and plastic strain gradient (Nix and Gao, 1998; Huang et al., 2000b).

The tensile ow stress is related to the shear ow stress in Eq. (11) by =M,

where M is the Taylor coecient; M= 3 for an isotropic solid, and M= 3:08 for aface-centered-cubic (FCC) crystal (Bishop and Hill, 1951a, b; Kocks, 1970).

The density of geometrically necessary dislocations is related to the gradient of

plastic deformation by (Ashby, 1970; Nix and Gao, 1998; Huang et al., 2000b)

G = rp

b; (12)

where b is the Burgers vector, p the eective plastic strain gradient to be given later,

r is the Nye factor (Arsenlis and Parks, 1999)to account for the eect of discrete slip

systems on the distribution of geometrically necessary dislocations, and r is around

1.9 for FCC crystals (Arsenlis and Parks, 1999). The density of statistically storeddislocations is determined from the relation between stress and plastic strain p in

uniaxial tension, = reffp(p), where ref is a reference stress (e.g., yield stress Y).

This is because the density of geometrically necessary dislocations vanishes in uniaxial


6/17


tension since there is no strain gradient [see (12)]. The Taylor dislocation model (11)

then gives

reffp(p)M

=bS; (13)where the tensile ow stress has been substituted by the uniaxial stressplastic strain

relation. The substitution of (12) and (13) into (11) gives the ow stress in terms of

eective plastic strain and eective plastic strain gradient,

=

2reff2

p(p) + 1822bp=ref

f2p(p) +lp; (14)

where p =

pdt is the eective plastic strain, p =

23

pij

pij the eective plastic strain

rate, and pij the plastic strain rate tensor; the eective plastic strain gradient p is given

later;

l= 182

ref

2b (15)

is the intrinsic material length associated with the strain gradient eect, and is on the

order of microns (Huang et al., 2000b). It should be pointed out that, even though

the intrinsic material length l depends on ref, the ow stress in Eq. (14) depends

only on the overall uniaxial stressstrain curve reffp(p) and strain gradient p, and

is independent of the choice of ref.

Gao et al. (1999b) developed dislocation models to calculate the density of geomet-

rically necessary dislocations in bending, torsion and void growth. Based on dislocation

models, they determined the eective plastic strain gradient p as

p=

1

4ijk

ijk;

ijk=pjk; i+

pik;j pij;k; (16)

where pij=

pijdt is the plastic strain tensor, and

pij is the corresponding plastic strain

rate tensor. It is noted that Fleck and Hutchinson (1997) also determined the eective

strain gradient in terms of the second order invariants of the strain gradient tensor, but

the coecients are dierent from 14

in Eq. (16).

In the following sections, we use the ow stress in Eq. ( 14) based on the Taylordislocation model to replace the tensile yield stress Y in Eq. (3). Therefore, such

an approach does not involve higher order stresses, contrary to some strain gradient

plasticity theories. Moreover, since the ow stress in Eq. (14) depends on the plastic

strain and strain gradient but not their increments, the variational principle (2) still

holds in the present study.

4. The size eect on void growth in a perfectly plastic solid

We investigate the void size eect on void growth rate in a perfectly plastic solidbased on the Taylor dislocation model. FollowRice and Tracey (1969), we neglect the

shape-changing additional velocity eld such that the strain rates ij and velocities u iare still given by (7) and (5), and are linearly dependent on the void growth rate D.


7/17


Similar to (3), normality of plastic ow gives the deviatoric stresses in terms of strain

rates as

sij(D) = eij(D)

[ 32

kl(D)kl(D)]1=2

; sij =Y

ij

( 32

kl

kl)1=2

; (17)

where the dependence on the void growth rate D is made explicit; e is the eective

stress, and the eective stress in the remote eld is the yield stress Y, as shown later.

The yield condition in the Taylor dislocation model gives the eective stress as

e= ; (18)

where is the ow stress in Eq. (14) accounting for the strain gradient eect. For a

perfectly plastic solid, reffp(p) =Y, the ow stress becomes

=

2Y+ 18

22bp=Y

1 +lp; (19)

which diers from the yield stress Y by the eective plastic strain gradient term, lp,

where l is the intrinsic material length in Eq. (15), and p is given in Eq. (16). Unlike

Rice and Tracey (1969), the ow stress in Eq. (19) depends on the displacement eld

through plastic strain gradient p.

If we assume proportional deformation, the displacement eld can be obtained from

the velocity eld (5) as

ui=

ij xj+R2

0R3 u

0xi=

ij xj+ DR3

0R3 x

i; (20)

where ij and =

2ij

ij =3 are the strains and eective strain in the remote eld,

respectively; u0 is the increase of average void radius, and D = u0=(R0) is identical to

the void growth rate in Eq. (6) under proportional deformation. Since elastic deforma-

tion is neglected (Rice and Tracey, 1969; Budiansky et al., 1982; Huang, 1991), the

eective plastic strain gradient p can be obtained from the above displacement eld

via (16),

p= 35

2

R20R4 u0= 3

52

R30R4 D: (21)

Its substitution into (19) gives the ow stress as

=Y

1 + 3

5

2

R20R4

u0l=Y

1 + 3

5

2

R0

R

4l

R0D: (22)

It is straightforward to verify that the above degenerates to yield stress Y in the remote

eld (R ).As discussed in Section 3, the variation principle (2) still holds after the strain

gradient eect is accounted for in the present study for a rigidperfectly plastic solid.For the velocity eld in Eq. (5), minimization of velocity functional (2) also leads

to (9), except that the deviatoric stresses sij are obtained from (17)(19) to account

for the strain gradient eect. By changing the integration variables to = R30=R3 and


8/17


=cos , we rewrite (9) to the following integral equation for the void growth rate D,

1

0

d

1

0

(1 + 4D 32

)

1 + 35

2

l

R0D4=3

1 + 2D+ 4D22 6D2 1 32

d= 3mY ; (23)where m =

kk=3 and are the remote mean stress and eective strain, respectively,

Y the yield stress, l the intrinsic material length in Eq. (15), and R0 the void radius.

The void size eect comes into play through the ratio l=R0. For void size much larger

than the intrinsic material length which is on the order of microns, R0l, the void

size eect disappears. Eq. (23) then degenerates to Rice and Traceys (1969) analysis

and gives the void growth rate in Eq. (1) under high stress triaxialities. It is observed

that, for vanishingly small remote eective strain ( 0), the void size eect alsodisappears since the ratio l=R0 0.

Eq. (23) can be simplied to the following integral equation after the integration

with respect to is carried out analytically,

10

1 + 3

5

2

l

R0D4=3

|1 2D| 1 2D 12D

226D

sin1 6D

1 + 2D+ 4D22 d

4D2 =

3m

Y: (24)

It can be shown that the above integrand is non-singular around = 0. The solution

of (24) takes the form

D=D

m

Y; l

R0

; (25)

which once again indicates that the void size eect comes into play through the

non-dimensional combination of l=R0.

Fig. 1 shows the void growth rate D versus normalized mean stress m=Y for

l=R0 =0, 0.004, 0.01 and 0.02. The limit ofl=R0 =0 corresponds toRice and Traceys

(1969) void growth rate in classical plasticity. The void growth rate decreases with

increasing l=R0, which suggests that, at a given remote eective strain and intrinsic

material length l, small voids tend to grow slower than large voids. This is consis-

tent with the aforementioned experimental observations (Schluter et al., 1996; Khraishi

et al., 2001) and numerical studies of void growth (Fond et al., 1996; Fleck and

Hutchinson, 1997; Shu, 1998; Huo et al., 1999; Zhang et al., 1999; Zhang and Hsia,

2001). It should be pointed out that the inuence of void size depends strongly on

the mean stress level. At relatively small mean stresses (e.g., m=Y 1) as in most

laboratory experiments, void size has little or essentially no eect on the void growth

rate. At the large mean stress m=Y= 4:5 of cavitation instability (Huang et al., 1991;Tvergaard et al., 1992) observed in highly constrained ductile materials (Ashby et al.,

1989), small voids grow signicantly slower than large voids. Even at an intermediate

mean stress m=Y = 2:6 which is a representative value for a mode-I crack tip in a


9/17


Fig. 1. The void growth rate D versus normalized mean stress m=Y for a perfectly plastic solid and

l=R0 = 0, 0.004, 0.01 and 0.02, where D is dened in Eq. (6), m and are the mean stress and eective

strain in the remote eld, respectively, Y the tensile yield stress, l the intrinsic material length in Eq. (15),

and R0 the void radius. The curve for l=R0 = 0 corresponds to the RiceTracey model (1969) of void

growth in classical plasticity.

perfectly plastic solid, the void growth rate forl=R0 = 0:02 is still more than 30% less

than the prediction of classical plasticity (l=R0= 0).

The eect of void size can be magnied by the remote eective strain since void

size comes into play through the ratio l=R0. We may take aluminum subjected to mean

stress m=Y = 2:6 as an example. The intrinsic material length l for aluminum is a

few microns (Xue et al., 2002a; Haque and Saif, 2003). For a relatively small eective

strain = 0:002, the void size eect is signicant (l=R0= 0:02) for a sub-micron-sized

void (l=R0 = 10), but is much less signicant (l=R0= 0:004) for a micron-sized void

(l=R0=2). However, at a larger eective strain=0:01, the size eect for a micron-sizedvoid (l=R0 = 2) becomes signicant (l=R0= 0:02).

5. High stress-triaxiality approximations

Follow Rice and Tracey (1969), we establish the asymptotic expression of void

growth rate under high stress triaxiality accounting for the void size eect in this

section. As shown in Fig. 1 [and also (1)], high stress triaxiality m=Y1 leads to

large void growth rate, D1. Under high stress-triaxiality limit the governing equation

(24) for void growth rate can be simplied to give analytic or semi-analytic solutions.We rst study the void growth rate for small ratio of l=R0 as in Fig. 1, l=R01.

This corresponds to relatively small remote eective strain ( up to a few percent) and

micron-sized voids (l=R0 less than or on the order of 1). It can be shown that, for


10/17


Fig. 2. Comparison of the void growth rate in a perfectly plastic solid with its asymptotic solution ( 26) for

high stress triaxialities m=Y 1 (D1) and small ratio l=R01. All parameters and notations are the

same as in Fig. 1. The asymptotic solution (26) involves no tting parameters.

l=R0

1 and D

1, the asymptotic analysis of (24) gives

D exp

9

10

16

l

R0D

= 0:2833 exp

3m

2Y

for

l

R01: (26)

Its right-hand side is Rice and Traceys (1969) void growth rate under high stress

triaxiality given in Eq. (1). Forl=R0 =0, (26) degenerates toRice and Traceys (1969)

solution in classical plasticity. The algebraic equation (26) forD is much simpler than

the integral equation (24) and involves no tting parameters. Its solution, marked as

the asymptotic solution in Fig. 2, has good agreement with the numerical results of

(24) for the same set of l=R0.

Eq. (26) has the following approximate but explicit solution for small ratio l=R0

D= 2DRT

1 +

1 + 8:506(l=R0)DRTfor

l

R01; (27)

where DRT = 0:2833 exp(3m=2Y) is Rice and Traceys (1969) void growth rate given

in Eq. (1). The right-hand side of (27) comes from the asymptotic analysis of (26)for

l=R01, but the coecient 8.506 inside the square root is determined by tting the

numerical solution of (24). As shown in Fig. 3, (27) also gives a good approximate

solution of the void growth rate to account for the void size eect for l=R01. This

approximate solution shows that, once the void size eect is accounted for, the void

growth rate is no longer an exponential function of mean stress as the RiceTraceymodel (1969).

It should be emphasized that the asymptotic solution (26)and approximate solution

(27) hold strictly for l=R01. For a sub-micron-sized void l=R0 = 10 and a large


11/17


Fig. 3. Comparison of the void growth rate in a perfectly plastic solid with the approximate solution ( 27)

for high stress triaxialities m=Y 1 (D1) and small ratio l=R01. All parameters and notations are

the same as in Fig. 1. The approximate solution (27) involves one tting parameter. The void growth rate

is no longer an exponential function of mean stress as the RiceTracey model (1969) after the void size

eect is accounted for.

remote eective strain = 0:1, the ratio l=R0 becomes 1 such that (26) and (27) are

not applicable anymore. We have also analyzed the asymptotic limit of (24) forD1

and l=R0 on the order of 1, and obtained the following asymptotic solution for the

void growth rate without any parameter tting

D= 4

45

l

R0

2

m

Y

2for

l

R0 0:2: (28)

This asymptotic solution suggests that, for suciently small voids and large eective

strain such that l=R0 0:2, the void growth rate D scales with the square of mean

stress instead of the exponential function in Eq. (1) at high stress triaxialities. Fig. 4

shows the void growth rate D versus the normalized mean stress m=Y for l=R0= 1.

Both numerical solution of (24) and asymptotic solution (28) are shown, along with

the void growth rate in classical plasticity (l=R0 = 0). For a sub-micron-sized void

l=R0 =10 and large eective strain = 0:1, the void size eect becomes signicant even

at relatively small mean stresses (e.g., m=Y 1), as seen from the large dierence

between the solid curve for l=R0 = 1 and that for classical plasticity l=R0= 0. The

asymptotic solution (28)is a good representation of the void growth rate only at largemean stresses. This is because, for l=R0 = 1, the void size eect has signicantly

reduced D to 5 even at a large mean stress m=Y= 4:5 for cavitation instability, and

such a value of 5 does not satisfy D1 required in the asymptotic analysis.


12/17


Fig. 4. The void growth rate D versus normalized mean stress m=Y for a perfectly plastic solid and a

large ratio l=R0 = 1. All notations are the same as in Fig. 1. The curve for l=R0 = 0 corresponds to the

RiceTracey model (1969) in classical plasticity. The asymptotic solution (28) for high stress triaxialities

m=Y 1 (D1) and large ratio l=R0 (on the order of 1) is also shown. The asymptotic solution ( 28),

involving no tting parameters, scales with the square of mean stress, (m=Y)2, instead of the exponential

function in the RiceTracey model (1969) for l=R0 on the order of 1.

6. Void growth in a power-law hardening solid

We investigate the void size eect on void growth rate in a power-law hardening

solid in this section. The relation between stress and plastic strain in uniaxial tension

can be written as reffp(p) =refNp , where ref is a reference stress and N ( 1) is

the plastic work hardening exponent. The strain rates ij, velocities u i, displacements

ui and eective plastic strain gradient p are still given by (7), (5), (20) and (21),

respectively. For monotonically increasing and proportional deformation, the eective

plastic strain p is obtained from the displacements in Eq. (20) as

p=

1 + 2(1 3cos2 )DR

30

R3 + 4D2

R60R6

; (29)

where is the eective strain in the remote eld, and is the spherical angle.

The deviatoric stresses are given by (17)except that the eective stress in the remote

eld is no longer the yield stress Y. The yield condition in the Taylor dislocation

model (18) still holds. The ow stress in Eq. (14) accounting for the strain gradient

eect becomes

=

[refNp]

2

+ 182

2

bp=ref

2N

p +lp; (30)

where l = 182(=ref)2b is the intrinsic material length for a power-law hardening

solid, and p and p are given in Eqs. (29) and (16), respectively.


13/17


For a strain hardening solid Rice and Tracey (1969) showed that, among all admis-

sible velocity elds, the exact eld minimizes the following functional of velocity

(u) =

[sij() sij ]ijd

[sij() sij ]ij d ij

S

niujdS:

(31)

The above functional degenerates to (2) for a perfectly plastic solid. For the velocity

eld in Eq. (5), minimization of velocity functional (31) give an integral equation

for the void growth rate D. By changing the integration variables to = R30=R3 and

= cos , this integral equation becomes

10

d 1

0

(1 + 4D

32)[1 + 2D(1

32) + 4D22]N + 352 12NlR0

D4=3

1 + 2D+ 4D22 6D2

d= 3mf

; (32)

where m =

kk=3 and are the mean stress and eective strain in the remote eld,

respectively, f=refN is the ow stress corresponding to the remote eective strain

, N the plastic work hardening exponent, l the intrinsic material length in strain

gradient plasticity, and R0 the void radius. The void size eect comes into play through

the ratio 12Nl=R0. This ratio 12Nl=R0 degenerates to l=R0 for a perfectly plastic

solid (N = 0). Therefore, the void size eect disappears for voids much larger thanthe intrinsic material length (R0l) or for vanishingly small remote eective strain

( 0).The void growth rate governed by (32) takes the form

D=D

m

f;12Nl

R0; N

=D

m

refN;

12Nl

R0; N

: (33)

Fig. 4 shows the void growth rate D versus normalized mean stress m=f for 12Nl=

R0= 0, 0.024, 0.048 and 0.12 and N= 0:2. The limit of 12Nl=R0= 0 corresponds to

classical plasticity. Similar to Fig. 1, small voids tend to grow slower than large voids,

which is consistent with the experimental and numerical studies of void growth (Fond

et al., 1996; Schluter et al., 1996; Fleck and Hutchinson, 1997; Huo et al., 1999;

Zhang et al., 1999; Khraishi et al., 2001; Zhang and Hsia, 2001). The void size has

essentially no eect on the void growth rate at small mean stresses (e.g., m=Y 1),

but at relatively large mean stresses (e.g., m=Y 2) the void size eect may become

signicant. The void size eect is also magnied at a large remote eective strain

because of the ratio 12Nl=R0 (Fig. 5).

7. Concluding remarks

We have extended the RiceTracey (1969) model of void growth to account for the

void size eect. The following objectives have been achieved in this paper.


14/17


Fig. 5. The void growth rate D versus normalized mean stress m=f for a power-law hardening solid and

12Nl=R0 = 0, 0.024, 0.048 and 0.12, where D is dened in Eq. (6), m and are the mean stress and

eective strain in the remote eld, respectively, f the ow stress in the remote eld corresponding to the

eective strain , N = 0 :2 the plastic work hardening exponent, l the intrinsic material length in Eq. (15),

and R0 the void radius. The curve for 12Nl=R0= 0 corresponds to the RiceTracey model (1969) of void

growth in classical plasticity.

(i) We have studied the size eect on void growth in ductile materials based on the

Taylor dislocation model. It is established that small voids tend to grow slower

than large voids. This is consistent with the prior experimental and numerical

studies of void growth, which cannot be explained by classical plasticity theories.

(ii) For a perfectly plastic solid, we have found that the void size eect comes into

play through the ratio l=R0, where l is the intrinsic material length given in Eq.

(15) and is on the order of microns, the remote eective strain, and R0 the

void size. This suggests that the void size eect can be magnied by the remote

eective strain. For micron-sized voids and small remote eective strain such that

l=R06 0:02, the void size inuences the void growth rate only at high stress

triaxialities. For sub-micron-sized voids and relatively large eective strain such

that l=R0 0:2, the void size has a signicant eect on the void growth rate at

all levels of stress triaxiality.

(iii) We have obtained analytically the asymptotic solutions of void growth rate at

high stress triaxialities accounting for the void size eect. Even for l=R06 0:02,

the void growth rate already deviates from the RiceTracey model (1969) of void

growth based on classical plasticity. For l=R0 0:2, the void growth rate scales

with the square of mean stress (m=Y)

2

, rather than the exponential function inthe RiceTracey model (1969), exp(3m=2Y).

(iv) For a power-law hardening solid, we have observed similar size eect on the void

growth rate, i.e., small voids grow slower than large voids. The void size comes


15/17


into play through the ratio 12Nl=R0, where N is the plastic work hardening

exponent.

Acknowledgements

YH acknowledges the support from NSF (grant CMS-0084980 and a supplemental

to grant CMS-9896285 from the NSF International Program). KCH acknowledges the

support from NSFC.

References

Acharya, A., Bassani, J.L., 2000. Lattice incompatibility and a gradient theory of crystal plasticity. J. Mech.Phys. Solids 48, 15651595.

Acharya, A., Beaudoin, A.J., 2000. Grain-size eect in viscoplastic polycrystals at moderate strains. J. Mech.

Phys. Solids 48, 22132230.

Arsenlis, A., Parks, D.M., 1999. Crystallographic aspects of geometrically necessary and statistically stored

dislocation density. Acta Mater. 47, 15971611.

Ashby, M.F., 1970. The deformation of plastically non-homogeneous alloys. Philos. Mag. 21, 399424.

Ashby, M.F., Blunt, F.J., Bannister, M., 1989. Flow characteristics of highly constrained metal wires. Acta

Metall. 37, 1857.

Bagchi, A., Evans, A.G., 1996. The mechanics and physics of thin lm decohesion and its measurement.

Interface Sci. 3, 169193.

Bagchi, A., Lucas, G.E., Suo, Z., Evans, A.G., 1994. A new procedure for measuring the decohesion energy

of thin ductile lms on substrates. J. Mater. Res. 9, 17341741.Bishop, J.F.W., Hill, R., 1951a. A theory of plastic distortion of a polycrystalline aggregate under combined

stresses. Philos. Mag. 42, 414427.

Bishop, J.F.W., Hill, R., 1951b. A theoretical derivation of the plastic properties of a polycrystalline

face-centered metal. Philos. Mag. 42, 12981307.

Budiansky, B., Hutchinson, J.W., Slutsky, S., 1982. Void growth and collapse in viscous solids. In: Hopkins,

H.G., Sewll, M.J. (Eds.), Mechanics of Solids, The Rodney Hill 60th Anniversary Volume. Pergamon

Press, Oxford, pp. 1345.

Chen, B., Wu, P.D., Xia, Z.C., MacEwen, S.R., Tang, S.C., Huang, Y., 2000. A dilatational plasticity theory

for aluminum sheets. In: Chuang, T.-J., Rudnicki, J.W. (Eds.), Multiscale Deformation and Fracture in

Materials and Structures, the James R. Rice 60th Anniversary Volume. Kluwer Academic Publishers,

Dordrecht, The Netherlands, pp. 1730.

Chen, B., Huang, Y., Liu, C., Wu, P.D., MacEwen, S.R., 2003. A dilatational plasticity theory for viscoplasticmaterials. Mech. Mater., in press.

Cleveringa, H.H.M., van der Giessen, E., Needleman, A., 1997. Comparison of discrete dislocation and

continuum plasticity predictions for a composite material. Acta Mater. 45, 31633179.

Cleveringa, H.H.M., van der Giessen, E., Needleman, A., 1998. Discrete dislocation simulations and size

dependent hardening in single slip. J. Phys. IV 8, 8392.

Cleveringa, H.H.M., van der Giessen, E., Needleman, A., 1999a. A discrete dislocation analysis of bending.

Int. J. Plasticity 15, 837868.

Cleveringa, H.H.M., van der Giessen, E., Needleman, A., 1999b. A discrete dislocation analysis of residual

stresses in a composite material. Philos. Mag. A 79, 893920.

Cleveringa, H.H.M., van der Giessen, E., Needleman, A., 2000. A discrete dislocation analysis of mode I

crack growth. J. Mech. Phys. Solids 48, 11331157.

Cottrell, A.H., 1964. The Mechanical Properties of Materials. Wiley, New York, p. 277.Douglass, M.R., 1998. Lifetime estimates and unique failure mechanisms of the digital micromirror device

(DMD). Annual ProceedingsReliability Physics Symposium (Sponsored by IEEE), March 31April 2,

1998, pp. 916; also http://www.dlp.com/dlp/resources/whitepapers/pdf/ieeeir.pdf.
http://www.dlp.com/dlp/resources/whitepapers/pdf/ieeeir.pdfhttp://www.dlp.com/dlp/resources/whitepapers/pdf/ieeeir.pdfhttp://www.dlp.com/dlp/resources/whitepapers/pdf/ieeeir.pdf


16/17


Elssner, G., Korn, D., Ruehle, M., 1994. The inuence of interface impurities on fracture energy of UHV

diusion bonded metal-ceramic bicrystals. Scripta Metall. Mater. 31, 10371042.

Fleck, N.A., Hutchinson, J.W., 1993. A phenomenological theory for strain gradient eects in plasticity. J.

Mech. Phys. Solids 42, 18251857.

Fleck, N.A., Hutchinson, J.W., 1997. Strain gradient plasticity. In: Hutchinson, J.W., Wu, T.Y. (Eds.),

Advances in Applied Mechanics, Vol. 33. Academic Press, New York, pp. 295361.

Fleck, N.A., Hutchinson, J.W., 2001. A reformulation of strain gradient plasticity. J. Mech. Phys. Solids 49,

22452271.

Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W., 1994. Strain gradient plasticity: theory and

experiment. Acta Metall. Mater. 42, 475487.

Fond, C., Lobbrecht, A., Schirrer, R., 1996. Polymers toughened with rubber microspheres: an analytical

solution for stresses and strains in the rubber particles at equilibrium and rupture. Int. J. Fract. 77, 141

159.

Gao, H., Huang, Y., Nix, W.D., 1999a. Modeling plasticity at the micrometer scale. Naturwissenschaften 86,

507515.Gao, H., Huang, Y., Nix, W.D., Hutchinson, J.W., 1999b. Mechanism-based strain gradient plasticityI.

Theory. J. Mech. Phys. Solids 47, 12391263.

Gurson, A.L., 1977. Continuum theory of ductile rupture by void nucleation and growth: part 1yield

criteria and ow rules for porous ductile media. ASME Trans. J. Eng. Mater. Tech. 99, 215.

Gurtin, M.E., 2000. On the plasticity of single crystals: free energy, microforces, plastic-strain gradients. J.

Mech. Phys. Solids 48, 9891036.

Gurtin, M.E., 2002. A gradient theory of single-crystal viscoplasticity that accounts for geometrically

necessary dislocations. J. Mech. Phys. Solids 50, 532.

Haque, A., Saif, M.T.A., 2003. Strain gradient eect in nanoscale thin lm. Acta Mater., in press.

Huang, Y., 1991. Accurate dilatation rate for spherical voids in triaxial stress elds. ASME Trans. J. Appl.

Mech. 58, 10841086.

Huang, Y., Hutchinson, J.W., Tvergaard, V., 1991. Cavitation instabilities in elasticplastic solids. J. Mech.Phys. Solids 39, 223241.

Huang, Y., Gao, H., Hwang, K.C., 1999. Strain-gradient plasticity at the micron scale. In: Ellyin, F., Provan,

J.W. (Eds.), Progress in Mechanical Behavior of Materials, Vol. III. pp. 10511056.

Huang, Y., Gao, H., Nix, W.D., Hutchinson, J.W., 2000a. Mechanism-based strain gradient plasticityII.

analysis. J. Mech. Phys. Solids 48, 99128.

Huang, Y., Xue, Z., Gao, H., Nix, W.D., Xia, Z.C., 2000b. A study of micro-indentation hardness tests by

mechanism-based strain gradient plasticity. J. Mater. Res. 15, 17861796.

Huo, B., Zheng, Q.-S., Huang, Y., 1999. A note on the eect of surface energy and void size to void growth.

Euro. J. Mech.A/Solids 18, 987994.

Hwang, K.C., Jiang, H., Huang, Y., Gao, H., Hu, N., 2002. A nite deformation theory of strain gradient

plasticity. J. Mech. Phys. Solids 50, 81 89.Hwang, K.C., Jiang, H., Huang, Y., Gao, H., 2003. Finite deformation analysis of mechanism-based strain

gradient plasticity: torsion and crack tip eld. Int. J. Plasticity 19, 235251.

Jiang, H., Huang, Y., Zhuang, Z., Hwang, K.C., 2001. Fracture in mechanism-based strain gradient plasticity.

J. Mech. Phys. Solids 49, 979993.

Khraishi, T.A., Khaleel, M.A., Zbib, H.M., 2001. Parametric-experimental study of void growth in

superplastic deformation. Int. J. Plasticity 17, 297315.

Kocks, U.F., 1970. The relation between polycrystal deformation and single crystal deformation. Metall.

Trans. 1, 11211144.

Lloyd, D.J., 1994. Particle reinforced aluminum and magnesium matrix composites. Int. Mater. Rev. 39, 1

23.

Ma, Q., Clarke, D.R., 1995. Size dependent hardness in silver single crystals. J. Mater. Res. 10, 853863.

McElhaney, K.W., Vlassak, J.J., Nix, W.D., 1998. Determination of indenter tip geometry and indentationcontact area for depth-sensing indentation experiments. J. Mater. Res. 13, 13001306.

Needleman, A., 2000. Computational mechanics at the mesoscale. Acta Mater. 48, 105124.

Nix, W.D., 1989. Mechanical properties of thin lms. Metall. Trans. 20A, 2217 2245.


17/17


Nix, W.D., 1997. Elastic and plastic properties of thin lms on substrates nanoindentation techniques. Mater.

Sci. Eng. A234236, 3744.

Nix, W.D., Gao, H., 1998. Indentation size eects in crystalline materials: a law for strain gradient plasticity.

J. Mech. Phys. Solids 46, 411425.Nye, J.F., 1953. Some geometrical relations in dislocated crystals. Acta Metall. Mater. 1, 153 162.

Poole, W.J., Ashby, M.F., Fleck, N.A., 1996. Micro-hardness of annealed and work-hardened copper

polycrystals. Scripta Metall. Mater. 34, 559 564.

Qiu, X., Huang, H., Nix, W.D., Hwang, K.C., Gao, H., 2001. Eect of intrinsic lattice resistance in strain

gradient plasticity. Acta Mater. 49, 39493958.

Rice, J.R., Tracey, D.M., 1969. On the ductile enlargement of voids in triaxial stress elds. J. Mech. Phys.

Solids 17, 201217.

Saha, R., Xue, Z., Huang, Y., Nix, W.D., 2001. Indentation of a soft metal lm on a hard substrate: strain

gradient hardening eects. J. Mech. Phys. Solids 49, 19972014.

Schluter, N., Grimpe, F., Bleck, W., Dahl, W., 1996. Modelling of the damage in ductile steels. Comp.

Mater. Sci. 7, 2733.

Shi, M., Huang, Y., Jiang, H., Hwang, K.C., Li, M., 2001. The boundary layer eect on the crack tip eld

in mechanism-based strain gradient plasticity. Int. J. Fract. 112, 2341.

Shrotriya, P., Allameh, S.M., Lou, J., Buchheit, T., Soboyejo, W.O., 2003. On the measurement of the

plasticity length scale parameter in LIGA nickel foils. Mech. Mater. 35, 233243.

Shu, J.Y., 1998. Scale-dependent deformation of porous single crystals. Int. J. Plasticity 14, 10851107.

Stelmashenko, N.A., Walls, M.G., Brown, L.M., Milman, Y.V., 1993. Microindentation on W and Mo

oriented single crystals: an STM study. Acta Metall. Mater. 41, 28552865.

Stolken, J.S., Evans, A.G., 1998. A microbend test method for measuring the plasticity length scale. Acta

Mater. 46, 51095115.

Swadener, J.G., George, E.P., Pharr, G.M., 2002. The correlation of the indentation size eect measured

with indenters of various shapes. J. Mech. Phys. Solids 50, 681694.

Taylor, G.I., 1934. The mechanism of plastic deformation of crystals. Part Itheoretical. Proc. Roy. Soc.(London) A145, 362387.

Taylor, G.I., 1938. Plastic strain in metals. J. Inst. Metals 62, 307324.

Tvergaard, V., 1990. Material failure by void growth to coalescence. Adv. Appl. Mech. 27, 83147.

Tvergaard, V., Huang, Y., Hutchinson, J.W., 1992. Cavitation instabilities in a power hardening elasticplastic

solid. Euro. J. Mech. A/Solids 11, 215231.

Wiedersich, H., 1964. Hardening mechanisms and the theory of deformation. J. Metals 16, 425430.

Xue, Z., Huang, Y., Li, M., 2002a. Particle size eect in metallic materials: a study by the theory of

mechanism-based strain gradient plasticity. Acta Mater. 50, 149160.

Xue, Z., Saif, T.M.A., Huang, Y., 2002b. The strain gradient eect in micro-electro-mechanical systems

(MEMS). J. Microelectromechanical Systems 11, 2735.

Zhang, S., Hsia, K.J., 2001. Modeling the fracture of a sandwich structure due to cavitation in a ductile

adhesive layer. ASME Trans. J. Appl. Mech. 68, 93100.Zhang, K.S., Bai, J.B., Francois, D., 1999. Ductile fracture of materials with high void volume fraction. Int.

J. Solids Struct. 36, 34073425.

Documents

The Size Effect on Void Growth in Ductile Metals