Archive of Applied Mechanics 71 (2001) 110 ± 122 Ó Springer-Verlag 2001

A micromechanical model for polycrystalline creep with grain boundary cavitation* P. A. Fotiu, F. Ziegler

110

Summary A micromechanical model is developed to describe effects such as combined powerlaw creep and diffusion, grain boundary sliding and cavitation in polycrystals. Several aspects of creep-constrained cavitation are taken into account such as diffusion in a cage of creeping matrix material and cavitating facets in a cage of creeping grains. Grain boundary sliding is modelled by distributed micro-shearcracks. It is shown that the different physical mechanisms and their interactions are functions of a well-de®ned material parameter k, which can be related to the material length scale L introduced by Rice. Key words Micromechanics, creep, grain boundary cavitation, sliding, diffusion

1 Introduction Creep deformation of polycrystalline metals at elevated temperatures is governed by a variety of mechanisms on the microscale. At high stresses, the main contribution to creep comes from thermally activated dislocation climb within the grains, commonly called power-law creep. At low stresses, deformations are caused mainly by diffusion along grain boundaries, accompanied by the formation of cavities. After nucleation, these cavities grow by vacancy diffusion and by extensive creep deformation in the vicinity of the voids. After coalescence, these voids form microcracks which, at lower stress levels, favourably grow along grain boundaries, leading ®nally to intergranular creep fracture. At moderate stress levels, both mechanisms of diffusion and power-law creep contribute to the overall deformation of the polycrystal and, as will be shown later, this interaction leads to a signi®cant ampli®cation of the void growth process. The basic theory of cavity growth by grain boundary diffusion was formulated by Hull and Rimmer [1], who assumed the grains to be rigid. This model has been extended and improved in [2] and [3]. Coupling of diffusion and dislocation creep has been studied on the basis of the Hull±Rimmer model in [4] and [5]. Needleman and Rice [6] presented a detailed numerical treatment of coupling effects on grain boundary cavitation. Subsequent extensions of their work to various levels of stress triaxiality have been given in [7±9]. At elevated temperatures, the effect of grain boundary sliding becomes important, leading to a further increase of the creep strain rate. This mechanism has been studied numerically by several authors [10±13]. In this paper, we propose a micromechanical model for the description of combined diffusion ± power-law creep and cavity growth in plane strain. Coupling of power-law creep Received 18 January 2000; accepted for publication 17 May 2000 P. A. Fotiu (&) Fachhochschule Wiener Neustadt, Johannes Gutenbergstrasse 3, A-2700 Wiener Neustadt, Austria e-mail: [email protected] F. Ziegler Technische UniversitaÈt Wien, Wiedner Hauptstrasse 8-10, A-1040 Wien, Austria e-mail: [email protected] * Dedicated to Professor Kolumban Hutter on the occasion of his 60th birthday.

and diffusion in the cavitation process is taken into account in two different ways, which have been addressed in [5]. First, there will be a short-range effect due to an interaction between neighbouring cavities, each of them being located in a diffusion zone, which is caged in a creeping matrix. On the other hand, diffusion on a cavitating facet is constrained by surrounding creeping grains. This will give rise to a long-range effect (typically several grain sizes), compared to the previously mentioned short-range effect with a length of an average cavity spacing. For the ®rst short-range effect, an analytical expression for a critical stress is derived where the extension of the diffusion zone starts to shrink. Interactions of cavitating facets are taken into account via a dilute distribution of microcracks in a creeping matrix, which has been suggested ®rst by Rice, [14]. Within such a formulation, grain boundary sliding can be introduced by a shear crack model, [15, 16].

2 Short range creep constrained cavitation at a single grain boundary facet Consider a periodic array of cylindrical, cap shaped voids along a grain boundary, described in a local coordinate system x01 x02 , Fig. 1. The void geometry is determined by the radius a and the angle w, and the cavities are separated by a distance 2c. Throughout the following calculations, we assume that the void shape remains constant, that is, we restrict ourselves to equilibrium growth models. Nonequilibrium void growth leads to shape changes of the cavity such as pronounced elongation along the boundary and the width remaining small. Such a crack-like growth behavior of voids is treated in [3]. The extension of the diffusion zone is given by the radius b which is yet to be determined. By b , we denote the average stress acting on the boundary facet. As pointed out in [17], r b is not r 0 equal to the applied stress which we denote by r . The following steps will be similar to the analysis in [4]. The governing equations for the boundary ¯ux Jb and for the thickening rate d_n of the boundary are given by XJb ˆ D

orb20 20 ; ox01

d_n ˆ

X

oJb ˆ ox01

D

o2 rb20 20 ; ox02 1

…1†

where X is the atomic volume and D ˆ Db db X=kT is the boundary diffusivity. From the Hull± Rimmer assumption, d_n …x01 † = const. and Eq. (1) we obtain o2 rb20 20 =ox02 1 = const. Hence, rb20 20 …x01 † is quadratic in x01 between a and b. Mass continuity requires the additional constraint between d_n and the void volume growth rate V_ ˆ 4h…w†aa_

d_n ˆ

2h…w† V_ _ a; ˆ 2…b a† b=a 1

h…w† ˆ

w

sin w cos w : sin2 w

…2†

In order to obtain analytic expressions for b and the average normal stress rb20 20 on the boundary, we have to employ the following boundary conditions, see Fig. 1:

rb20 20 …x01 ˆ a† ˆ rsint ˆ

sin w l; a s

orb20 20 =ox01 …x01 ˆ b† ˆ 0 ;

…3†

Fig. 1. Coupling of the diffusion and the power-law creep between voids along the grain boundary. Stress distribution according to the Hull-Rimmer model, [1]

111

where rsint is the (usually small) sintering stress, and ls denotes the surface free energy. In addition to Eq. (3), we have to account for a constraint relation between the normal extension rate d_n within the diffusion zone and the corresponding deformation in the creeping material, such as

d_n ˆ 2ae_b20 20 :

112

…4†

The reference length 2a in Eq. (4) is an assumption based on asymptotic compatibility. In _ Hence, this should be the case also for Eq. (4) in order to be Eq. (2), d_n is proportional to a. compatible with Eq. (2) for small a. The normal creep strain rate e_b20 20 at the boundary, which we assume to follow a power law according to

e_bij

 b n 1 b sij 3 r ˆ e_0 e : r0 r0 2

…5†

b In Eq. (5), r0 and e_0 denote a reference stress and strain rate,  respectively, s is the stress q b b b deviator and the effective stress is given by re ˆ …3=2†sij sij . Assuming a nearly uniaxial stress state rb20 20 in the boundary material between the voids, we write e_b20 20 in the form

e_b20 20

 b n r20 20 …x01 ˆ b† ˆ e_0 ; r0

…6†

and obtain, after some lengthy algebra (see Appendix A), for the size of the diffusion zone

b

a c

 ˆ

1 n‡1

kn f n tn

1

 1

f :

…7†

In Eq. (7), f ˆ a=c is the void fraction along the grain boundary and the nondimensional parameters

kˆ

r0 D ; e_0 c3

tˆ

h…w†a_ ; e_0 c

…8†

describe creep-diffusion interaction and void growth, respectively. The material parameter k may be compared to the material length scale L ˆ …Dr0e =e_0e †1=3 introduced by Rice, [14],

 0 n 1  3 re L kˆ : r0 a

…9†

It might be argued that the power-law creep is governed by a single parameter B ˆ e_0 =rn0 rather than by e_0 and r0 separately. Then, we may write instead of Eq. (8)

kˆb kr10 n ;

t ˆ btr0 n ;

…10†

with new parameters depending only on B

D b k ˆ 3; Bc

bt ˆ

h…w†a_ : Bc

…11†

Note that b k and bt are no longer nondimensional. In the following derivations, we use the nondimensional parameters k and t according to Eq. (8). However, formulations depending on B can be easily found by introducing relations (10) and subsequently cancelling the stress parameter r0 . With parameters k; t the average normal stress rb20 20 over the grain boundary is found as

rb20 20

ˆ …1

ft f †rsint ‡ k



kn f n tn

" 1 n‡1

1

1

f

1 3



kn f n tn

# 1 n‡1

1

r0 :

…12†

If the diffusion zone extends over the entire half spacing, then b ˆ c and the equality holds in Eq. (7)



1 n‡1

kn f n tn

1

ˆ1

f ;

…13†

and introducing this into Eq. (12), we obtain for t

b ˆ c:

tˆ

k~ rn ; f …1 f †2

…14† 113

with the nondimensional boundary normal stress

r~n ˆ

3 rb20 20 2

…1 f †rsint : r0

…15†

~n , Equation (14) equals the solution obtained in [3], but it holds only up to a critical value r which will be de®ned subsequently. At ®rst, it should be noted that Eq. (13) represents an upper bound for kn =…f n tn 1 †, that is, for any value of t rendering the left-hand side of (13) larger than 1 f it has to be set to 1 f . We de®ne the growth rate, where Eq. (13) is exactly satis®ed by t , as

kn f n …1 f †n‡1

t ˆ

!n 1 1 ;

…16a†

and the corresponding stress

r~n

 ˆ

k …1 f

n 3

f†

n 1 1

;

…16b†

~n , is the critical stress. For normal stresses found by introducing t into (14) and solving for r ~n , the diffusion zone will no longer extend along the entire grain boundary, i.e. exceeding r b < c. Note that in case of linear viscoelasticity, n ˆ 1, there is no de®nite critical stress. ~n > r ~n and t > t , a close approximation In case of a reduced diffusion zone b < c, where r of the void growth rate t can be found from Eq. (12) (see Appendix B for details)

b < c:

tˆ

!n‡1 2

2 3

t1 ;

…t =t1 †p…n†

…17†

with

 t1 ˆ

1

~n r

s n‡1 2 k ; f f

…18†

4…n 1† : p…n† ˆ …n ‡ 1†…n ‡ 3†

~n =~ Introducing a new nondimensional stress measure s ˆ r rn , we can derive the following  equations for the relative growth rate t=t :

t  t ; s  1: 

t  t ; s  1:

t ˆ s; t  t ˆ t 3

2s s2…1

n†=…3‡n†

n

:

…19†

Fig. 2. Dependence of the void growth on the boundary normal stress

114

~n =~ Figure 2 shows the dependence of t=t on s ˆ r rn , and clearly indicates the substantial impact of diffusion zone reduction on the cavity growth rate for supercritical stresses s  1, i.e. ~n . While t grows linearly as long as r ~n  r ~n , the growth rate increases exponentially in ~n > r r the supercritical range with a strong dependence on the creep exponent n.

3 Long range creep-diffusion coupling As we already mentioned, there is also a creep constraint between boundary facets and the  b which then are surrounding creeping grains. This affects the average boundary stresses r 0 different from the applied stress r . This problem has been addressed by Rice [14], who considered each facet as a crack in a power-law creeping matrix, opened up by the far®eld stress r0 and with rb20 20 acting at the crack faces. The average opening rate of such a crack can be found from the analysis given in [18]. By comparing this opening rate with the result for d_n according to Eq. (2), a constraint relation between the normal components rb20 20 and r020 20 can be found. From the results for the plane strain crack, [18], we ®nd for the average normal opening rate p  n 1 0 p b20 20 pm n n 1 0 n 1 0 d_n 3pm n r0e r20 20 r an …~ ˆ ˆ rn † …~ rn ce_0 r0 r0 4 2

r~n † ;

…20†

where

r~0n ˆ

3 r020 20 2

…1 f †rsint ; r0

…21†

m is the number of cavities per facet and

an ˆ

…3=2†…r0n

r0e …1

f †rsint †



r0e ; …3=2†r0n

r0n > 0 ;

…22†

denotes the ratio of the equivalent applied stress to its component normal to the grain boundary, if rsint is neglected. For uniaxial tension along x2 we have an ˆ 2=…3 cos u† with u being the angle between the x1 axis and the grain boundary. Usually, signi®cant void growth will appear only on boundaries with small angles, u < 10 . From (2) and (7), we ®nd the following expressions for the average opening rate:

  d_n kn ˆ 2f t n n 1 ce_0 f t

1 n‡1

;

and, with Eqs. (14) and (17)

…23†

d_n 2f t 2k~ rn 2f ˆ ˆ rn †n 1 r~n ; ˆ n …~ 3 ce_0 1 f …1 f † …1 f † n   d_n f 2 n‡1 2f 2~ rn t ˆ 2f ˆ n ce_0 k …1 f † 3 …~ rn =~ rn †2…1

b ˆ c: b < c:

…24†

!n n†=…3‡n†

:


…n‡1†=2 In Eq. (24/2), we used Eq. (17) for t together with the equivalence t1 =t ˆ r~n =~ rn . Comparing (24) with (20) and using the nondimensional stresses s ˆ r~n =~ rn , s0 ˆ r~0n =~ rn yields the following expressions for the creep constrained average boundary stresses s:

b ˆ c:

sˆ 

b < c:

nn …s0 †n 1 ‡ nn …s0 †n 2s

3

s2…1

115

1;

n

n†=…3‡n†

…25† 0 n 1

‡ nn …s †

0 n

s ˆ nn …s † ;

where

nn ˆ

p pm n…1 4f

f †n

ann

1

:

…26†

Relation (25/1) is equivalent to an expression introduced by Rice [14], while the second constraint (25/2) embodies the short-range effect of a reduced diffusion zone. Hence, the shortrange and the long-range creep constraints are not independent of each other.

4 Coupling of creep and grain boundary sliding In a similar way, we can model the effect of grain boundary sliding. Here, the shear mode sliding rate must be compared to the relative velocity of the crack faces due to rb20 10 calculated from the viscous sliding law of the grain boundaries. If free sliding is assumed, then we have simply rb20 10 ˆ 0. Otherwise, we derive the average boundary shear stress from a simple assumption of a piecewise constant shear distribution between the voids, Fig. 3. Then, the average shear stress is given by rb20 10 ˆ

c

a c

rb20 10 ˆ …1

f †rb20 10 :

…27†

We assume a nonlinear viscous sliding relation within the boundary

e_b20 10

 b k r00 ˆ c_ 0 2 1 ; s0

…28†

Fig. 3. Shear stress assumption in modelling grain boundary sliding

where s0 ; c_ 0 denote a reference stress and strain rate, respectively, and k is the creep exponent of grain boundary sliding. According to (28), the relative displacement across the boundary will be approximately (comp. Eq. (4))

d_s ˆ 2ae_b20 10 ;

…29†

and with (28) this amounts to

116

d_s 2f ˆ …~ rs †k ; cc_ 0 …1 f †k

r~s ˆ

rb20 10 : s0

…30†

This result can be compared with the average relative displacement of the faces of a shear crack in a power-law material, which has been deduced in [15]

  p  n 1 0 p d_s 3pm n r0e r20 10 rb20 10 3pm n n 1 s0 n 0 n 1 0 as ˆ ˆ …~ rs † …~ rs ce_0 r0 r0 r0 4 4

r~s † ;

…31†

with

as ˆ

r0e ; r0s

r0s 6ˆ 0;

r~0s ˆ

r020 10 : s0

…32†

Equating (31) to (30) yields the following dependence between the applied stress and the average boundary shear:

…~ rs †k ‡ ns …~ r0s †n 1 r~s ˆ ns …~ r0s †n ;

…33†

where

ns ˆ

p 3pm n…1 8f

f †k

gasn

1

;

…34†

and

  e_0 s0 n gˆ ; c_ 0 r0

…35†

is a nondimensional material parameter relating power-law creep and grain boundary viscosity.

5 Void growth In coupled diffusion power-law creep, the growth of cavities is governed by both mechanisms. Contributions from diffusion are given by 4t V_ d ; ˆ 2 e_0 a f

…36†

where t has to be taken either from Eqs. (14) or (17). Void growth in a power-law deforming matrix has been the subject of a large number of studies, see [7, 8, 19±22]. All these models are of the form

 0 n re V_ c ˆ F…R; n; f † ; 2 r0 e_0 a

…37†

where R, de®ned by

Rˆ

r0m ; r0e

r0m ˆ

r0ii ; 3

…38†

denotes the triaxiality factor of the applied far®eld stress. In domains where both mechanisms of diffusion and creep are signi®cant, Sham and Needleman [7] suggested the total growth rate to be given by the sum of each contribution, i.e. V_ ˆ V_ c ‡ V_ d , and a possible reduction of the diffusion zone is incorporated in their model by a special choice of the parameter f . The model in [7] describes the case of elevated triaxiality, while a complementary formula for low triaxialites has been presented in [8]. A detailed numerical study in [9] showed good agreement between numerical results and a combination of the model of Budiansky et al. [19], and a modi®ed version of this model especially designed to account for the void arrangement along well-separated grain boundary facets. We present their equations for void growth due to the power-law creep, although the model is designed for an axisymmetric ¯ow mode rather than plane strain

_ ˆ max‰jV_ cL j; jV_ cH jŠ ; jVj

…39†

with

V_ cL ˆ e_0 V

(

n 3 2 d …an jRj ‡ bn …d†† ; n 3 2 …an ‡ bn …d†† R;

jRj  1 jRj < 1 , 8 h
in 3 1 d > < d a jRj ‡ ; jRj  1 n 3=n 2 n V_ cH 1 …0:87f † ˆ h i
n >3 e_0 V : 1 d R; jRj < 1 , 3=n an ‡ n 2

…40†

1 …0:87f †

and

3 …n 1†‰n ‡ g…d†Š ; bn ; d ˆ sign R; 2n n2 g…1† ˆ 0:4319; g… 1† ˆ 0:4031 : an ˆ

…41†

6 Macroscopic creep strain In order to ®nd an estimate for the overall creep strain rate, we consider cavitating and/or sliding grain boundaries as microcracks. In a composite material, the total strain rate is assumed as the weighted average e_ ˆ …1

F†e_ M ‡ Fe_ I ;

…42†

where an overbar denotes an averaged quantity and superscripts (M) and (I) indicate that the average is to be taken over the matrix volume V M or the inclusion volume V I , respectively. By F, we understand the volume fraction of inclusions, F ˆ V I =V; V ˆ V M ‡ V I . If the inclusion becomes a crack, F tends to zero, whereas e_ I grows to in®nity, such that the product Fe_ I stays M ˆ r0 , and ®nite. In that case, the average matrix stress is equal to the uniform far®eld stress, r the matrix strain rate is therefore given by the power law

 0 n 1 0 s _eM ˆ 3 e_0 re : r0 r0 2

…43†

The strains within the crack can be written as, [23],

1 e_ I ˆ I V

Z VI

e_ I dV ˆ

1 2FV

Z _ n ‡ n ‰uŠ†dS _ …‰uŠ :

…44†

SI

_ denoting the crack opening The surface integral in Eq. (44) extends over all crack faces with [u] displacement rate and n being the unit vector normal to the crack face. If there are K classes of cracks with equal size and orientation, each class r; r ˆ 1; . . . ; K, counting a number of N …r† cracks per unit volume, we may write instead of (44)

117

Fig. 4. Grain boundary modelled as a crack in a local coordinate system x01 x02

118

Fe_ I ˆ

K X

…r† _ …r†

F e

ˆ

rˆ1

Z K X N …r† rˆ1

2

_ n ‡ n ‰uŠ†dS _ …‰uŠ :

…45†

S…r†

…r† Based on the analysis in [18], we ®nd the normal strain rate e_ 20 20 within the r-th class of cracks with length 2l…r† in a local coordinate system x01 x02 , Fig. 4,

…r† F …r† e_ 20 20

ˆN

Zl…r†

…r†

l…r†

p  0 n 1 0 2 r 0 0 3p n r rb20 20 ‰u_ 20 Šdx10 ˆ N …r† l…r† 2 2 : e_0 e r0 r0 2

…46†

A similar relation can be obtained for the shear strains, using the results from [15] for a shear crack in a power-law material, …r† F …r† e_ 20 10

ˆN

Zl…r†

…r†

l…r†

p  0 n 1 0 2 r 0 0 3p n r rb20 10 ‰u_ 10 Šdx10 ˆ N …r† l…r† 2 1 : e_0 e r0 r0 4

…47†

It should be emphasized that Eqs. (45)±(47) are based on the assumption of a dilute distribution of cracks neglecting any interactions. However, void growth due to normal stresses occurs predominantly on boundaries approximately normal to the largest principal stress. Usually, boundaries with the same orientation are typically one grain size apart which justi®es the simple assumption of noninteracting cracks. On the other hand, signi®cant grain boundary sliding may be activated at boundaries with common junctions. Several micromechanical models have been investigated in [24] in order to describe the overall stiffness of an elastic grain assemblage with freely sliding boundaries. They found that, in spite of the elevated crack density, the dilute distribution model in plane strain still gave suf®ciently accurate results, at least in an averaged sense. With the results (46) and (47), we may now give an estimate of the overall creep strain rate of a material undergoing combined diffusion ± power-law creep deformation including grain boundary sliding

#  0 n 1 " 0 K 0 b sij p X 2  r r 3 re …r† kl e_ ij ˆ e_0 ; ‡p n N …r† l…r† Hijkl kl r0 r0 r 2 0 rˆ1

…48†

with …r†

…r†

…r†

…r†

…r†

…r†

Hijkl ˆ Qim0 Qjn0 Qkp0 Qlq0 Hm0 n0 p0 q0 ; …r†

H20 20 20 20 ˆ 1; …r†

…r†

…r†

…r†

…r†

…r†

H10 20 10 20 ˆ H10 20 20 10 ˆ H20 10 20 10 ˆ H20 10 10 20 ˆ 1=4;

Q110 ˆ Q220 ˆ cos u…r† ;

…r†

…49†

…r†

Q210 ˆ Q120 ˆ sin u…r† :

Consider, for example, an arrangement of regular hexagonal grains, each side of the hexagon having a length 2l, Fig. 5. Then, there are three classes of cracks, K ˆ 3, with, [24],

Fig. 5. Unit cell in a hexagonal grain assemblage with three classes of `cracks'

l…r† ˆ l; u…1† ˆ 0 ;

2 2 1 l ˆ p ; r ˆ 1; . . . ; 3 ; l1 l2 6 3 ˆ 60 ; u…3† ˆ 60 :

119

2

N …r† l…r† ˆ u…2†

…50†

By a proper choice of parameters l…r† , N …r† and u…r† , arbitrary grain geometries can be modelled. If cavitation is assumed on a fraction of boundaries only, this fraction may be considered as a separate class of cracks. Hence, the behavior of materials with sparsely cavitating boundaries can be easily included into this model.

7 Conclusion A model for the combined action of power-law creep, grain boundary diffusion and boundary sliding is presented, where creep constraints on diffusion and sliding processes are taken into account. Interactions between creep and diffusion appear along individual facets (short-range effect) allowing for a reduced diffusion zone with a signi®cant increase in the void growth rate. In addition, creep effects along the boundary interact with the creeping bulk material necessitating an averaged formulation due to the different behavior of grain cells and boundaries. The long-range effect can be modelled most simply by an averaging procedure, where the grain boundaries are taken as microcracks with averaged boundary stresses rbij acting on their faces. These stresses are found from the constraint relations (25) and (33). References

1. Hull, D.; Rimmer, D.E.: The growth of grain boundary voids under stress. Phil Mag 4 (1959) 673±687 2. Raj, R.; Ashby, M.F.: Intergranular fracture at elevated temperature. Acta Met 23 (1975) 653±666 3. Chuang, T.J.; Kagawa, K.I.; Rice, J.R.; Sills, L.B.: Non-equilibrium models for diffusive cavitation of grain interfaces. Acta Met 27 (1979) 265±284 4. Beere, W.; Speight, M.V.: Creep cavitation by vacancy diffusion in plastically deforming solids. Met Sci 12 (1978) 172±176 5. Edward, G.H.; Ashby, M.F.: Intergranular fracture during power-law creep. Acta Met 27 (1979) 1505±1518 6. Needleman, A.; Rice, J.R.: Plastic creep ¯ow effects in the diffusive cavitation of grain boundaries. Acta Met 28 (1980) 1315±1332 7. Sham, T.L.; Needleman, A.: Effects of triaxial stressing on creep cavitation of grain boundaries. Acta Met 31 (1983) 919±926 8. Tvergaard, V.: Constitutive relations for creep in polycrystals with grain boundary cavitation. Acta Met 32 (1984) 1977±1990 9. Van der Giessen, E.; Van der Burg, M.W.D.; Needleman, A.; Tvergaard, V.: Void growth due to creep and grain boundary diffusion at high triaxialities. J Mech Phys Solids 43 (1995) 123±165 10. Crossman F.W.; Ashby, M.F.: The non-uniform ¯ow of polycrystals by grain boundary sliding accommodated by power-law creep. Acta Met 23 (1975) 425±440 11. Ghahremani, F.: Effect of grain boundary sliding on steady creep of polycrystals. Int J Solids Struct 16 (1980) 847±862 12. Tvergaard, V.: Effect of grain boundary sliding on creep constrained diffusive cavitation. J Mech Phys Solids 33 (1985) 447±469 13. Van der Giessen, E.; Tvergaard, V.: A creep rupture model accounting for cavitation at sliding boundaries. Int J Fract 48 (1991) 153±178 14. Rice, J.R.: Constraints on the diffusive cavitation of isolated grain boundary facets in creeping polycrystals. Acta Met 29 (1981) 675±681 15. Riedel, H.: Cavity nucleation at particles on sliding grain boundaries. A shear crack model for grain boundary sliding in creeping polycrystals. Acta Met 32 (1984) 313±321

120

16. Onck, P.; van der Giessen, E.: In¯uence of microstructural variations on steady state creep and facet stresses in 2-D freely sliding polycrystals. Int J Solids Struct 34 (1997) 703±726 17. Dyson, B.F.: Constraints on diffusional cavity growth rates. Met Sci 10 (1976) 349±353 18. He, M.Y.; Hutchinson, J.W.: The penny-shaped crack and the plane strain crack in an in®nite body of power-law material. J Appl Mech 48 (1981) 830±840 19. Budiansky, B.; Hutchinson, J.W.; Slutsky, S.: Void growth and collapse in viscous solids. In: Hopkins, H.G.; Sewell, M.J. (eds.) Mechanics of Solids: The R. Hill 60th Anniversary Volume, pp. 13±45. Oxford, Pergamon Press, 1982 20. Cocks, A.C.F.: Inelastic deformation of porous materials. J Mech Phys Solids 37 (1989) 693±715 21. Duva, J.M.; Hutchinson, J.W.: Constitutive potentials for dilutely voided nonlinear materials. Mech Mat 3 (1984) 41±54 22. Sofronis, P.; McMeeking, R.M.: Creep of power-law material containing spherical voids. J Appl Mech 59 (1992) S88±S95 23. Nemat-Nasser, S.; Hori, M.: Micromechanics: Overall Properties of Heterogeneous Materials. Amsterdam, North-Holland, 1993 24. Fotiu, P.A.; Heuer, R.: Overall stiffness of an elastic polycrystal with relaxed grain boundaries. Z Angew Math Mech 77 (1997) S465±S468

Appendix A Derivation of Eq. (7). As a quadratic function satisfying the conditions (3), rb20 20 …x01 † has the form

a†2

rb20 20 …x01 † ˆ A…x01

a†…x01

2A…b

a† ‡ rsint ;

…A:1†

and

rb20 20 …x01 ˆ b† ˆ

o2 rb20 20 ˆ ox02 1

a†2 ‡ rsint ;

A…b

2DA :

…A:2†

By comparing the two expressions (1)2 and (2)1 for d_n , we obtain the constant A as

h…w†a a_ : …b a†D

Aˆ

…A:3†

Next, we compare the thickening rate due to diffusion (2)1 to the extension rate due to power law creep (4), with e_b20 20 given by (6), as

 h…w†a 2ae_0 …b Dr0

a†a_

n

ˆ

2h…w†a a_ ; b a

…A:4†

where we have tacitly neglected the sintering stress in (A.2)1 . Equation (A.4) can be solved for b a, giving

b



a

ˆ

c

r0 D e_0 c3 …a=c†

n  n‡1

e_0 c h…w†a_

nn‡11

a ; c

1

…A:5†

which equals Eq. (7) with the nondimensional parameters k; t de®ned by (8). Appendix B Derivation of Eq. (17) With Eqs. (15) and (16a), we can rewrite (12) in the following form:

 1 3 f t2 n‡1 r~n ˆ …1 2 k

" f† 1

 n 1 # 1 t n‡1 ; 3 t

…B:1†

giving

 tˆ

1

r~n

s" n‡1 2 k f f 3

#n‡1 2

2 …t =t†…n

1†=…n‡1†

" ˆ t1

3

#n‡1 2

2 …t =t†…n

1†=…n‡1†

:

…B:2†

This is an implicit equation for t. Since Eq. (B.2) is valid only for t  t , the term in brackets remains in the interval [2/3, 1]. We approximate Eq. (B.2) by an explicit equation, replacing on the right-hand side t by t1 and using a different exponent p…n† according to

tˆ

2 3

p…n† ˆ

!n‡1 2

…t =t1 †p…n† 4…n 1† : …n ‡ 1†…n ‡ 3†

t1 ;

…B:3† …B:4†

Since there is always t1 > t, Eq. (B.3) is asymptotically convergent to (B.2) for large t, and at t ˆ t both equations correctly yield t ˆ t1 . The gradient dt=dt1 of Eq. (B.2) can be evaluated at t ˆ t to give

Fig. 6. Comparison of Eqs. (B.2) and (B.3) for the computation of coupled diffusive void growth

121

dt 4 : …t ˆ t † ˆ dt1 3‡n

…B:5†

Now, we determine the exponent p…n† in (B.3) such that the gradient of (B.3) at t ˆ t equals the result (B.5). Expression (B.4) follows from this constraint. Figure 6 shows a comparison of function (B.2) and its approximation (B.3) for several values of the creep exponent n.

122