Effects of grain boundary sliding on creep-constrained boundary cavitation and creep deformation

Mechanics of Materials 11 (1991) 43-62 Elsevier

43

Effects of grain boundary sliding on creep-constrained boundary cavitation and creep deformation K.J. Hsia, D.M. Parks and A.S. Argon Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 3 May 1990; revised version received 3 August 1990

Creep of polycrystaUine materials was modeled using a periodic array of plane strain hexagonal grains. Effects of grain boundary sliding on such an array containing a distribution of boundary facet cracks were investigated using the finite element method. The facet crack opening rate in a creeping material with grain boundary sliding and the energy release rate due to the development of a facet crack were studied parametrically in terms of the creep exponent n, equivalent stress triaxiality S / o e and crack density p. The results were generalized to the 3-D case of penny-shaped cracks. As an application, the grain boundary cavity growth rate under a creep-constrained condition and the creep constitutive equation, all with grain boundary sliding, were obtained and compared with the non-sliding boundary case. The results show that both cavity growth rate and creep strain rate are substantially increased, and become more sensitive to stress triaxiality if grain boundary sliding is considered.

1. Introduction

Creep deformation and fracture of polycrystalline materials at elevated temperatures and low stresses are often associated with intergranular cavitation and intergranular microcracking. Nucleation of grain boundary (G-B) cavities is usually caused by high stress concentration at non-deformable second phase perticles on G-Bs, as these perturb the smooth sliding of such G-Bs (Argon et al., 1980; Lau et al., 1983). It has been found that for materials with high creep ductility, the fracture process goes through several stages, starting with the forming of individual G-B facet cracks due to coalescence of individual cavities, and the final linkage of these facet cracks (Argon et al., 1985). Thus, in order to model intergranular creep fracture under these circumstances, the following two questions have to be answered: (a) what will be the creep behavior of a polycrystal with G-B sliding and a distribution of boundary facet cracks; (b) how will the G-B sliding and the presence of boundary facet cracks affect cavitation on other boundaries? 0167-6636/91/$03.50 © 1991 - Elsevier Science Publishers B.V.

The effect of G-B sliding on the overall creep rate of a non-cavitating polycrystal has been studied by Hart (1967) by spring dashpot models, by Crossman and Ashby (1975) and Ghahremani (1980) for a two-dimensional array of hexagonal grains using the finite element method (FEM); by Chen and Argon (1979) using a self-consistent method; by Anderson and Rice (1985) for a three-dimensional Wigner-Seitz cell, by applying a minimum principle to an approximate stress distribution; and by Rodin and Dib (1989) for a Wigner-Seitz cell using the FEM. Values of stress enhancement factors for various cases have been calculated in these papers. But the additional effect due to the presence of a distribution of G-B microcracks has not been considered. On the other hand, He and Hutchinson (1981) have obtained the solution of a single microcrack normal to the direction of principal extension in an infinite body of a power law creeping material, which can be used as an approximation for a dilute concentration of facet microcracks. In their analysis, the material was considered to be homogeneous and isotropic, with no grain boundary sliding being

44

K.J. Hsia et al. / Grain boundary sliding

taken into account. Anderson and Rice (1985) and Tvergaard (1985) have studied the load-shedding effect due to creep-constrained G-B cavitation with G-B sliding, but only for the limiting case, when all boundary facets, approximately normal to the applied tensile stress are uniformly cavitated. In this case, however, the G-B cavitation is not likely to be creep-constrained, as the experiments of Capano et al. (1989) on Astroloy have demonstrated. Two major effects of G-B sliding have been widely recognized. First, the overall creep strain rate will be enhanced by G-B sliding, a phenomenon which can be characterized by the stress enhancement factor defined later in this paper. Second, as shown in Fig. 1, in a cracked homogeneous body (with no G-B sliding) under a compressive stress parallel to the crack plane, the crack will produce no perturbation, while in the presence of G-B sliding, the crack will be wedged open and the stress and strain fields will be disturbed by the crack. Thus, intergranular creep fracture, with G-B sliding, will be more sensitive to the stress triaxiality. In the present communication, we report the effects of G-B sliding and the presence of a distribution of boundary facet cracks on creep deformation and boundary cavitation using finite element techniques. The interactions between boundary facet cracks have been studied by implementing different crack densities in the model. The effects of different degrees of stress triaxiality and different values of creep exponent have also been studied. These effects have already been incorporated into a creep damage model that has been used to investigate the evolution of such

T

non-sliding boundaries

sliding boundaries

Fig. I. Different creep behavior for non-sliding and sliding grain boundaries under transverse compression.

damage around sharp cracks and blunt notches (Hsia et al., 1991).

2. Problem formulation The polycrystalline material was modeled as an array of two-dimensional hexagonal grains as shown in Fig. 2a. The finite element method has been used in a plane strain configuration to study the effects of G-B sliding. As pointed out by Anderson and Rice (1985), the three-dimensional geometry is more constrained than the two-dimensional one. Therefore, the influence of a grain boundary facet crack should be less in the threedimensional configuration, than in the corresponding two-dimensional configuration, where the influence should be more far reaching. Because of this, the present plane strain computation overestimates the G-B sliding effects. This difference between 2-D and 3-D configurations will be considered in more detail later in Section 4 below. The finite element mesh for a specific facet crack density is shown in Fig. 2b where one-half of the crack is given by OR in the lower left comer of the figure. A fan section of triangular elements is used at comers of each grain to pick up the angular variation of stress and strain fields at the triple point junctions. A finer grid is used in the grains around the facet crack region (lower-left comer), because the disturbance introduced by a facet crack is of primary interest in this study. Since there is a stress singularity at the triple points as developed by Lau et al. (1983), use of this mesh will only give a first approximation to the stress and strain rate concentration there. In the present model, grain boundaries are considered to be free to slide which is the case when the temperature is sufficiently high, and the stress level, as well as the creep strain rate, is low. To simulate the free boundary sliding, the corresponding nodes on both sides of a grain boundary were constrained to have the same displacements in the direction normal to the boundary but were otherwise free to move in the tangential direction. The nodes of three adjacent grains at triple points were pinned together to meet the compatibility

45

K.J. Hsia et al. / Grain boundary sliding

8

s

(

T

4_

.

9

( O

R

A

(u) (a) Fig. 2. Two-dimensional array of hexagonal grains: (a) general view outlining dashed rectangle to be taken for study; (b) finite element mesh superimposed on dashed rectangle of (a), showing position of one facet half crack of length, OR.

conditions there. The boundary conditions were such that the nodes on boundary BC in Fig. 2b were constrained to have the same y-displacement while those on AC to have the same x-displacement. Furthermore, by symmetry, boundaries OB and RA were not allowed to have x and y displacements, respectively. The applied loads were an average uniform tensile stress S on boundary BC and a transverse stress T on AC, where S is the maximum principal stress. A R a m b e r g - O s g o o d constitutive equation was employed for the grain interiors, which in a onedimensional case takes the form n--I o

for non-linear, rate independent response, where E is the Young's modulus, n is the hardening exponent, % is a reference stress and a is the pre-exponential scale factor. These constants were chosen so that the elastic strain c = o/E is negligible compared to the non-linear part under the far field average stress state. Under a multi-axial stress

state, this equation can be readily generalized to the form,

E• = (1 + p)~r' + (1 - 2V)OmI + ~ot k °o ]

o',

(2) where oe is the Mises equivalent stress, Om = 1(Oll + 022 + 033) is the negative pressure, and o ' = o % I is the deviatoric stress tensor. By direct analogy, the solutions obtained through the use of this model for a nonlinear plastic material inside the grains can be easily converted to obtain solutions for the problem with power law creeping, rigid material inside the grains, by neglecting the first two terms on the R H S of equation (2) which give linear elastic response, and by replacing the resulting displacements and strains with displacement rates and strain rates in the remaining terms. A G-B facet crack can be implanted by letting the facet O R be entirely free of tractions. Different crack densities can be obtained by having different numbers of grains in the model. In the

K.J. Hsia et al. / Grain boundary sliding

46

present study, two crack densities were considered. The dimensionless parameter p = N R 2 was used to measure the facet crack density in which N is the number of facet cracks per unit area and R is the half-length of the plane strain facet crack (radius for penny-shaped facet crack in subsequent 3-D generalizations). It is to be noted that the problem solved was for a periodic cell with one crack in (rn 1 × m 2 ) g r a i n s ( m 1 = m 2 = 8 for the model in Fig. 2b). Therefore, for the grains of area AG, the crack density can be readily obtained as O = R 2 / m l m 2 A ~ in the present 2-D model. To evaluate the energy release rate, it is necessary to consider both the configurations with a facet crack and the ones without facet cracks. Symmetry allows the consideration of only a quarter of the whole configuration, where OA and OB are symmetry axes.

3. Numerical results 3.1. Overview

In what follows we first present the results of altered elastic response due to G-B sliding with and without G-B facet cracks. The equivalent elas-

(a)

tic constants and the stress enhancement factor will be calculated and compared with the results of some previous work. For creep response, two quantities will be evaluated to determine the G-B cavity growth rate under the creep-constrained growth condition due to G-B sliding, and to derive the macroscopic creep constitutive equation for material containing a distribution of facet microcracks with G-B sliding. These two quantities are: (1) the area growth rate (volume growth rate in the 3-D case) of a boundary facet crack; and (2) the energy release rate due to the development of a boundary facet crack. The functional forms of the G-B facet opening area and the associated energy release rate in terms of the creep exponent n and the stress triaxiality will be determined and compared with the results of the non-sliding case. The typical plane strain creep displacement fields under a far field uniform tensile stress %y without and with G-B facet cracks are shown in Figs. 3a and 3b, respectively. The centers of grains can be readily identified by the absence of the typical corner net. The displacement discontinuity in the tangential direction across the grain boundaries can be clearly seen in both figures. The facet crack in Fig. 3b has opened up as a result of the deformation inside grains and the sliding across

(b)

Fig. 3. Typical displacement fields under a remote uniform tensile stress in the vertical direction: (a) in the presence of G-B sliding without facet cracks; (b) in the presence of G-B sliding with a facet crack over length OR.

K.J. Hsia et at / Grain boundary sfiding

grain boundaries. In the case without any facet crack, as shown in Fig. 3a, because of the symmetry boundary conditions that have been employed, no G-B sliding occurs across horizontal boundaries. Whereas in the case with a facet crack shown in Fig. 3b, some horizontal boundaries in the vicinity of the facet crack slide against each other as a result of the disturbance, and broken symmetry condition.

3.2. Elastic deformation Elastic deformation of grains (for a = 0 in Eq. (2) and p = 0.3) under a plane strain tensile stress

47

S and in the presence of G-B sliding was studied first. The distributions of different stress components without any G-B facet crack are shown in Figs. 4a-4d. Due to G-B sliding, the deformation is no longer homogeneous throughout the grain. The contours of Mises equivalent stress over the whole configuration in Fig. 4a show that the stress distribution is purely periodic, which is also the case for other stress components. The slight difference between the contours at the triple points around the lower left c o m e r of the plate and those around the rest of the plate is due to different mesh fineness. The contours around the lower left c o m e r represent the stress distributions more ac-

3

2

f (a)

(b)

3

3

2

5

4

(o)

(d)

Fig. 4. Stress contours for elastic (r = 0.3) deformation with G-B sliding but no G-B facet crack: (a) overview of contours of Mises equivalent stress; (b) magnified local distribution of M i s e s s t r e s s o e (contour No./(oe/o~y): 1/0.146; 2/0.468; 3/0.790; 4/1.112; 5/1.434; 6/1.755); (c) contours of normal stress Oxx (contour No./(oxx/o~y): 1 / - 0 . 2 9 3 ; 2 / - 0 . 1 7 6 ; 3 / - 0 . 0 5 9 ; 4/0.059; 5/0.176; 6/0.293); (d) contours of normal stress Oyy (contour No./(Oyy/O~y): 1 / - 1 . 4 6 3 ; 2 / - 0 . 5 8 5 ; 3/0.293; 4/1.170; 5/2.048; 6/2.926).

K.J. Hsia et aL / Grain boundary sliding

48

curately because of a finer mesh. In Figs. 4b-4d, it is seen that all stress components are concentrated around the triple points, in the sense that they reach the largest extreme values at the triple points. The Oyy contours in Fig. 4d show that, at the triple points, while the material in the

120 o sector of a grain is showing significant positive stress concentration across the grain boundary normal to the y direction, the ayy stress is negative in the protected 60 o half sector of the adjoining grain. At the present level of mesh refinement the m a x i m u m Oyy value is as high as about 3 times the

k._/

--

<

2

0

4

R

(~)

(b)

(c) (d) Fig. 5. Stress contours for elastic (v = 0.3) deformation with G-B sliding and the presence of G-B facet crack at OR: (a) overview of contours of Mises equivalent stress; (b) magnified local distribution of Mises equivalent stress in the neighborhood of the facet crack OR (contour No./(oe/a~y): 1/0.146; 2/0.463; 3/0.790; 4/1.112; 5/1.434; 6/1.755; (c) contours of normal stress a~,~, (contour No./(Oxx/%,~): 1 / - 0 . 2 9 3 ; 2 / - 0 . 1 7 6 ; 3 / - 0 . 0 5 9 ; 4/0.059; 5/0.176; 6/0.293); (d) contours of normal stress avv (contour No./(avy/O~y): 1 / - 1 . 4 6 3 ; 2 / - 0 . 5 8 5 ; 3/0.293; 4/1.170; 5/2.048; 6/2.926).

49

K.J. Hsia et aL / Grain boundary sliding

average far field stress. All stresses at the triple point are, of course, singular as determined by the asymptotic analysis of Lau et al. (1983). The elastic stress distributions in the assembly of grains with sliding boundaries with a boundary facet crack are shown in Figs. 5a-5d. The oe contours over the whole configuration in Fig. 5a, and at a somewhat higher magnification in Fig. 5b, show that the disturbances induced by the facet crack decay rapidly away from the crack tip over a distance of about 2 grain diameters. This fast decaying behavior is partly a result of pinning all nodes together at the triple point junctions which impedes the spreading of the disturbance. In reality, this restriction is lessened by opening up of wedge cracks or the development of local plastic flow (often by the formation of deformation kinks). All these, however, are inelastic phenomena that must be considered separately. In a general 3-D case, geometrical constraints also play a very important role in impeding the spreading of disturbances induced by a facet crack, as we discuss in Section 6 below. The detailed contours of all stress components in the vicinity of the facet crack show the complicated stress distribution induced by the facet crack. While the stresses in the

1.0

l

'

,

,

l

l

,

l

l

,

(

l

l

i

i

l

(

l

l

i

,

,

,

I

cracked grain are somewhat released, the stresses in the adjacent grains are increased to bear the extra "load". As a check of the response of the solution, the equivalent elastic constants for polycrystalline material with G-B sliding were computed for different values of v. Under the plane strain condition ((~ = 0), the effective Young's modulus Eeff and the effective Poisson's ratio veff were given by Ghahremani (1980) as 1

Eef f = (yy/ayy + v 2 / E ,

Peff =

(3a)

X"eff I ~yy + E

(4a)

,

where E and v are the Young's modulus and the Poisson's ratio for the material inside grains, ixx, iyy and ~yy are the volume average strain and stress components throughout the body. The values of the effective Young's modulus and effective Poisson's ratio as a function of the actual v, obtained by the present finite element computations are shown in Figs. 6a and 6b, respectively. These values compare very well with Ghahremani's

1.0

'

'

'

'

i

'

'

,

,

,

,

0.9

,

,

,

i

,

,

,

,

,

,

,

,

,

Ghahremani's expression

__.___x__--~~ x

x

Present result

0.8 0.5 L~ 0.7 Ghahremani's expression 0.6

Present result

(a) 0.5

,

0.0

,

,

,

i

0.1

. . . .

I

0.2

. . . .

r

0.3

,

,

,

,

t

0.4

(b)

. . . .

0.0

0.5

. . . .

0.0

i

0.1

,

,

,

,

I

0.2

,

~

,

,

I

0.3

. . . .

i

0.4

,

~

,

i

0.5

Fig. 6. Calculated effective elastic constants of an assembly of grains with sliding G-Bs, as a function of the Poisson's ratio of grain matrix, compared with results of Ghahremani: (a) Young's modulus; (b) Poisson's ratio.

K.J. Hsia et al. / Grain boundary sliding

50

(1980) results characterized by his expressions obtained by curve fitting, Eel f

T

--

0.83v - 0.86 ,,- 1

uefr -- 0.83v + 0.14,

1.2

(3b) (4b)

in spite of the fact that the number of elements per grain used in our model is much less than that in his computation.

l.l

X

)(

3.3. Stress enhancement factor in power law creep

One of the major effects of G-B sliding is to enhance the overall creep rate of the material. Fundamental considerations demand that the stress strain-rate relation on log-log coordinates should show an S-shaped curve. The creep strainrate becomes a little higher in the low stress, low strain-rate regime at high temperatures because of G-B sliding (Chen and Argon, 1979). If the creep constitutive equation for the grain interiors can be characterized by the power law equation Oe

the creep constitutive equation of the bulk material with G-B sliding will take the form ~e~ = i0

f ve

o0

'

(6)

where o0 is a reference stress, i0 is a temperaturedependent reference strain rate, oe and ie are the local Mises equivalent stress and strain rate inside grains, o~ and i ~ are the global Mises equivalent stress and strain rate averaged over the bulk material, and f ( f > 1) is usually called the stress enhancement factor, which represents the volume average stress elevation due to G-B sliding. Stress enhancement factors have been determined by several investigators (Crossman and Ashby (1975), Chen and Argon (1979), Ghahremani (1980), Beere (1982), and Rodin and Dib (1989)), among whom Crossman and Ashby (1975) and Ghahremani (1980) have used a finite d e m e n t technique similar to ours to study the G-B sliding effects. As in all continuum analyses of homoge-

A • 0 ×

1.0

,

0

L

,

,

)(

Present result Ghahremani (fine mesh) Ghahremani (coarse mesh) Crossman and Ashby

~

5

,

,

,

,

10

I1 Fig. 7. Calculated stress enhancement factor f as a function of creep stress exponent n, compared with results of Ghahremani, and Crossman and Ashby.

neous properties, no "length" scale enters in the evaluation of f. Thus, f does not depend on grain size, but could depend on geometrical parameters related to grain shape and on the creep exponent n.

The value of the stress enhancement factor as a function of creep exponent n obtained by the present computation is plotted in Fig. 7 and is compared with the results of Crossman and Ashby (1975) and Ghahremani (1980). It is no surprise that the present results agree broadly with Ghahremani's, as both show that the stress enhancement factor increases slightly with increase in n. These results differ, however, from Crossman and Ashby's (1975) result, who stated that the stress enhancement factor f is independent of n, and has a value of 1.1+0.01 for 1 < n < 8 . 8 . The discrepancy between our results and Ghahremani's on the one hand, and those of Crossman and Ashby (1975) on the other, is probably due to the simple shear mode of deformation chosen by the latter, lacking symmetry, and employing somewhat unrealistic boundary conditions. It can be expected that the results for general 3-D configurations would differ from the above 2-D investigations. Proper selection of a geometri-

K.J. Hsia et al.

/ Grain boundary sliding

cal parameter which can correlate the stress enhancement factors of these two cases will be discussed later in Section 6 below together with a parameter proposed by Rodin and Dib (1989).

51

15

s/~.'=2/%[3 He and Hutchinson's result x A

i0

3.4. Grain boundary facet opening area

/

FE result without G-B slidingA / FE result with G-B sliding/

? p=O.O071

/V

/ /

A p=0.0126./ ST/ / ,/ / /

8

The problem of the opening area of a twodimensional plane strain crack in an infinite body of homogeneous power law creeping material has been solved by He and Hutchinson (1981). Their solutions apply to the non-sliding grain boundary case. Using a perturbation method, they obtained the functional form of the facet opening area in terms of the creep exponent n and the equivalent stress triaxiality S / o ~ as AA ( S ) ~ R 2 = h 1 n, 0 7

3~rv~S 2 02 '

(7)

where R is the half length of the facet crack, S is the far field maximum principal stress and o~ is the far field Mises equivalent stress given by o~ = ½¢~-(S- T) in the plane strain case. It should be noted that this solution will not be valid for high equivalent stress triaxiality because of the perturbation method they employed. As was mentioned above, due to G-B sliding, the facet opening area is increased and becomes more sensitive to stress triaxiality. Figure 8 shows the present computational results for the normalized facet crack opening area under remote plane strain tension ( S / o ~ = 2/v/-3-) as a function of creep exponent n, in which the solid curve is He and Hutchinson's (1981) perturbation result, and the two dotted curves are the results from the present model for two different facet crack densities. To compare our computational model with that of He and Hutchinson, the relative displacement between grains across G-Bs was constrained to prevent G-B sliding. The resulting behavior without G-B sliding is shown by the crosses, which compare very well with He and Hutchinson's perturbation solution. As was expected, as a result of G-B sliding, the facet opening area was found to be substantially larger than the opening without G-B sliding. The plot also shows that the higher

.J

//"

< <3

i

i

i

i

I

5

L

i

~

i

10

n Fig. 8. Normalized facet crack opening area as a function of creep stress exponent n for a given stress triaxiality ratio, S / o ~ = 2/v/3, compared with results of He and Hutchinson for a single G-B crack without sliding G-Bs, and showing results for sliding G-Bs, with two facet crack densities p.

the facet crack density, the larger is the increase due to G-B sliding. Figure 9 shows the normalized facet crack opening area as a function of the equivalent stress triaxiality S / o ~ for n = 5. Again, our solution without G-B sliding (crosses) compares very well with He and Hutchinson's perturbation solution (solid line) for low stress triaxialities. The openings with G-B sliding for two different facet crack densities are again much larger than that for the non-sliding case. The relationship between the opening area and the equivalent stress triaxiality obtained by our FE computation is no longer linear, although it can still be approximated well by a straight line for low values of stress triaxiality. In addition to the increase in the facet crack opening area, a qualitative difference between the sliding and non-sliding cases is observed from these computational results. When the equivalent stress triaxiality equals zero, which is the case when the maximum principal stress S is zero and the plate is under transverse compression, t h e result shows that the facet crack will not open at

K.J. Hsia et al. / Grain boundary sliding

52 20

....

Oe~)1"09-1"19 and G] ~x n °'55-°'6. The same procedure is used below to obtain functional forms for the energy release rate.

j n=5 He a n d H u t c h i n s o n ' s r e s u l t

15

x

×

/

FE r e s u l t w i t h o u t G B sliding FE r e s u l t with G-B sliding

v V

p=O.O071

/ ~

A p=0.0126

10

/5 /

/ /~/'f

.C

3.5. Energy release rate

./

The energy release rate due to the development of a boundary facet crack under plane strain condition is defined as (He and Hutchinson, 1981)

J

/ /z/V"

zGv ~ .< <3

/%7" /~H j

1 d(PE) 2 dR =

J

1 A(PE) 2 AR '

(9)

where PE(R) is the potential energy per unit thickness of the cracked body of area A R subject to the traction T ~ on its boundary SR, O

I

2

3

4

Fig. 9. Normalized facet crack opening area as a function of triaxiality ratio S / o ~ for a given creep stress exponent n = 5, compared with results of He and Hutchinson for a single G-B crack without sliding G-Bs, and showing results for sliding G-Bs with two facet crack densities p.

all without considering grain boundary sliding; while with G-B sliding, the facet crack does open up even if the average normal stress acting on the horizontal cross section is zero. The opening area at S/o~ = 0 varies with the creep exponent n and is expected to depend more strongly on grain shape as well. The dependence on n and S/o~ and the crack density p of the computational results of facet crack opening area were fitted to a functional form, using the least-squares fitting technique. The functional form that emerged is S

f~R2 = h i

n, _-7-~, oe P

ee(R)=fA W(,)dA-fs T~GdS, R

)

= 60(.) + G,(., ° ) 0 7 ' (8a)

where W(c) is the strain energy density which is evaluated at each material point as a function of strain state. The differential of the total potential energy A(PE) can be obtained by the difference in the values of PE between the configuration with a facet crack and the one without the crack subjected to the same load. It should be noted that the line integral definition of J ( R i c e , 1968) is not readily applicable to the present model of a facet crack with G-B sliding. Nevertheless, we will continue to use the symbol " J " for the energy release rate in the present problem. The functional form of the energy release rate (per unit time) for a plane strain crack in an infinite homogeneous body of power law creeping material was obtained by He and Hutchinson (1981) using the perturbation method, in which this energy release rate (per unit time) was normalized by the product of the far field Mises equivalent stress o~ and the far field creep strain rate (e'°° and the half crack length R as, J

where

o~i~R

G O = 4 . 0 n 0"65,

(8b)

G1 = 4.0(2.05 + 15.5p)Vcff.

(8c)

In the above functional fit some approximations were made to obtain simple expressions. More accurate fits to the numerical results are possible by a choice of h i - Go(n)(x ( S /

(10)

R

( S )

h2

n,~

3~-

4

(S)

2"

(11)

The numerical values of the energy release rate with G-B sliding under remote plane strain tension obtained using our finite element model are shown in Fig. 10 as a function of the creep exponent n. Again, He and Hutchinson's (1981) perturbation solution is shown as the solid curve in

K.J. Hsia et al. / Grain boundary sliding

the figure, which is in very good agreement with our finite dement result of the non-sliding boundary case represented by the crosses. The results incorporating the grain boundary sliding effects show a much higher energy release rate. The values of energy release rate as a function of the equivalent stress triaxiality due to G-B sliding for a creep exponent n = 5 are shown in Fig. 11. The essential difference between the sliding and non-sliding grain boundary cases for S/o~ = 0 can also be observed in this computation, i.e. the energy release rate due to G-B sliding does not vanish when S--> 0, whereas it is zero for the non-sliding grain boundary. Similarly, using the least-squares fitting technique, an approximate functional form for the normalized energy release rate can be obtained as J

o~i~g

=

(s)

h 2 n,

= Ho(n,

~'

p

p)+

Hl(n,P)(o-~)

200

,,,i

53 I ....

j , , , , q , , , , ] , , , ,

n=5.0 He a n d Hutchinson's ×

FE result without G - B

~7

A

FE r e s u l t w i t h G - B s l i d i n g

8

V

.J 8

result

×

p=0.0071

A

sliding

p=0.0126

i00

.

o -

, , ,~r ~ 0

~

/

/

-

~ I

~

'

~

.......

3

2

4

) Fig. 11. Normalized energy release rate (per unit time) as a function of triaxiality ratio S/o~ for a given creep stress exponent n = 5, compared with results of He and Hutchinson for a single G-B crack without sliding G-Bs, and showing results for sliding G-Bs with two facet crack densities p.

+ H2(n, O)(o-~ ) 2,

(12a) where

Ho(n, p) = (1.399

i00

+ 2 . 9 0 9 p ) n 0"5838-3"636p,

(12b)

Hi(n, p) = (2.699 + 32.80p) S/or,'= 2 / ~ -

+ (1.732 + 2.527p)n He a n d H u t c h i n s o n ' s x

x

--V----dr-

and

FE r e s u l t w i t h o u t G - B s l i d i n g FE r e s u l t w i t h G - B s l i d i n g

~7 p=O.0071

A

H2(n, p) = (3.006 - 61.16p)

p=0.0126

8

.J 8

(12c)

result

+ (1.054 + 38.94p)n.

50 J L.../

~ _ _ _ _ . - - - - - - -

5

10

n Fig. 10. Normalized energy release rate (per unit time) as a function of creep stress exponent n for a given triaxiality ratio S / o 2 = 2/vc3, compared with results of He and Hutchinson for a single G-B crack without sliding G-Bs, and showing results for sliding G-Bs with two facet crack densities p.

(12d)

The numerical results above show that the effect of G-B sliding is to increase both the facet crack opening area and the energy release rate. These increases become more substantial for larger creep exponents n and higher triaxiality S/o~. Both the opening area and the energy release rate increase with facet crack density, although for the two values of densities studied, the difference is not very dramatic. When the equivalent stress triaxiality S/o~ = 0 (S = 0, T < 0), an essential difference between the sliding and non-sliding boundary cases exists, where the trivial solutions of opening area and energy release rate for the

K.J. Hsh~et al. / Grainboundarysflding

54

non-sliding case change into the non-zero solutions for sliding grain boundaries. This difference is a result of the inhomogeneity of the deformation field, and therefore, should be a function of grain shape.

4. Extension from 2-D to 3-D

The results of the facet opening area and the energy release rate for a 3-D configuration should qualitatively be the same as those for a 2-D geometry. As was mentioned before, however, due to the fact that there are more kinematic constraints in 3-D configurations, the effect of G-B sliding is less substantial in 3-D. One way to obtain an estimate of the 3-D behavior from the 2-D computational results given in the previous section is to assume that the relationship between the 3-D and 2-D solutions for the sliding boundary case is the same as the relationship for the non-sliding boundary case. He and Hutchinson (1981) have obtained the solution of a single penny-shaped crack in an infinite body of homogeneous power law creeping material under maximum principal stress S normal to the crack face and radial stress T parallel to crack plane. This solution can be considered as a first approximation to the general 3-D case for a dilute concentration of pennyshaped facet cracks. Based on our assumption, the approximate solutions of facet crack opening rate and energy release rate in the 3-D case for sliding G-Bs should then satisfy, h 1,3-D FE

h HH 1,3-D

h FE

h HH - f l ( n ) = 3~r ~ / - ~ ~ '

1,2-D

16

1

(13) FE

h2,2-D

HH h2,3_ D --

..

h2,E-O

8

1

=fz(n) = ~2 v ~ ~

AV -- hFE (

~R 3 -

S

1,3-D n , V '

p

)

---f,(n) ao(n) + c1( , 0 ) 0 7 ,

J

1,2-D

FE h2,3_ D

plane strain crack (2-D) and a penny-shaped crack FE D (3-D). Furthermore, functions hFE1,2-D and h2,2_ are those given in equations (8a) and (12a) for a plane strain facet crack with G-B sliding, and FE D denote the (unknown) calibration hFEa,3-Dand h2,3_ functions for a facet crack in a 3-D configuration with G-B sliding. Because H H only considered the problem of a single crack in an infinite body, the effect of crack density does not enter the relationship between 2-D and 3-D cases. For low and moderate crack density, this could be considered as a reasonable approximation. It is also noted that both ratios in equations (13) and (14) are functions of n only, which implies that these two quantities have the same stress triaxiality dependence for the plane strain crack and the penny-shaped crack. It can be expected that if G-B sliding effects are considered, the results should be more sensitive to the stress triaxiality, but it is not clear if the stress triaxiality dependence will be different for 2-D configurations and for 3-D configurations. However, based on the limited knowledge about the stress triaxiality dependence for a 3-D configuration with G-B sliding, we simply assume that equations (13) and (14) will hold for both sliding and non-sliding boundary cases. Thus, general 3-D solutions of the facet crack opening volume and the energy release rate with G-B sliding can be given as

FE(S)

O e ~ R --h2,3. D

n, U'

p

' (14)

where

h HH HH h HH 1,2-D, h 1,3-D, 2,2-D, and h Hn 2,3-D a r e f u n c t i o n s of creep exponent n and stress triaxiality

S/o~ in He and Hutchinson's (HH) perturbation solutions of the facet crack opening area or opening volume h 1 and energy release r a t e h 2 for a

where the functions of Go, G 1, H o, H 1 and H 2 have been defined in equations (8b), (8c), (12b), (12c) and (12d) respectively, and R is the equivalent radius of the G-B facet crack.

K.J. Hsia et al. / Grain boundary sliding

5. Applications of the grain boundary sliding results

5.1. Cavity growth rate under creep-constrained growth conditions A direct application of these solutions with the effects of grain boundary sliding is to estimate the cavity growth rate on grain boundaries under creep-constrained growth conditions first discussed by Dyson (1976). Following the steps of Rice's (1981) development, we can evaluate the G-B cavity growth rate under constrained growth conditions with G-B sliding, and compare the results with the non-sliding case. The argument for creep-constrained growth is, that if only a small fraction of grain boundaries are cavitating at one time, the thickening rate of the boundary facet due to diffusive cavity growth must be compatible with the overall creep rate of the background material. Considering the cavitating G-B facet crack subject to the far field tensile stress S and equivalent stress o~ as well as a crack surface traction o F, the volumetric growth rate of this facet crack in a power law creeping material with G-B sliding is given by the finite element results,

ar;'= TS3fx( n )( Go(n ) "+ Gl(n,

P)

S--OFI 0¢°° 1"

The effective thickening rate of the G-B facet due to power law creep of the grains is then

i~= Af," ,rrR2

= l iTRf~(n)(Go(n)

. S--OF~ -{" GI ( n , P ) - - ~ ). (18)

On the other hand, the thickening rate of the G-B facet due to the diffusional growth of G-B cavities is (Rice, 1981) 4D OF- (1 - A ) o s

[J= b 2

Q(A)

parameter, DB8 B is the boundary diffusivity, 12 is the atomic volume, kT has its usual meaning, b is the average half spacing between cavities, and os is the sintering stress, which we assume to be negligible in this analysis. Furthermore, for cavities of average radius a, the area fraction of cavity coverage on the cavitating boundary is given by A = a2/b 2, and finally Q(A) is a function of A only, given by

Q(A) = I n ( l / A ) - ½(3 -.A)(1 - A ) .

(19a)

'

where D = DaSBI2/kT is the G-B diffusion

(19b)

From the compatibility condition, these two thickening rates derived from two different considerations must be equal. This yields an expression for the facet stress,

G,(n, P) + - ~

OF o~

=

4,nD

Q(A) (20)

o~

b2R iVf,(n)Gl(n, P) + Q(A) The cavity growth rate for the quasi-equilibrium diffusional growth mode is then given as, •

D

OF/O~

as"d'ng= aZh(+) Q(A) o~ =

(17)

55

D Go/G 1 + S / o ~ o °° a2h( + ) 4,a) oy e, bER iTflG---~l+ Q(A)

(21)

where h ( + ) = [1/(1 + cos +) - ½cos +]/sin + is a function of the dihedral half angle + at the cavity tip. We note that in these two equations, as long as the Mises equivalent stress is non-zero, the facet stress o F and the cavity growth rate will not vanish when G-B sliding is taken into account, even if the far field maximum principal stress S equals zero. The cavity growth rate without G-B sliding has been given by Rice (1981) under a uniaxial tensile stress as D a non-sliding

a 2h ( + )

S

4DS C°asb2(ZR)

+Q(A) (22)

K.J. Hsia et aL / Grain boundary sliding

56

where a s is a numerical constant of the order of unity. Under fully constrained creep growth and for a cavity coverage A which is not very small, it is known that the term Q(A) in the denominators of equations (21) and (22) is negligibly small compared to the first term in the denominators and can be ignored. We note also that under uniaxial tension, the far field equivalent stress o : equals the applied tensile stress S, and the average creep strain rate with and without G-B sliding has the relationship £.o~ sliding = ~0( fS/oo )" = f "g n~on-sliding (Section 3.3). On rearranging the expressions and comparing the results of sliding and non-sliding grain boundary cases, the ratio of the cavity growth rates in these two cases under a uniaxial tensile stress is obtained as

15

.

.

.

.

.

.

p=O.05

.

non-sliding

t0(n) GI(n, p)

+

11 f l ( f l ) G l ( n ' ]

20ts~ff

p)

~sliding C non-sliding

to(n) + l)fl(n)Gl(n,2as~r P) f,. Gi(n, P)

/

p=O00 / .//

=o e.

•/

/

/

/

/

I

/ /

/

/

/

/.// /

/ / (~././//

0

i

i

i

0 sliding

//

p=O.lO .

i

~ 5

i

i

i

i 10

I2 Fig. 12. Effect of G-B sliding on intergranular constrained cavity growth rate, as a function of creep exponent n and three facet crack densities P (P = 0 means a single isolated facet crack).

(23)

Taking some typical values of a s = 0.9 and f = 1.1, the ratio of the cavity growth rates for these two cases is plotted in Fig. 12 as a function of n. It is seen that because of G-B sliding, the constrained growth rate of G-B cavities becomes much higher even for zero facet crack density (i.e. a single facet crack in an infinite body). The result also shows that the higher the non-linearity exponent n, the more substantial is the effect of G-B sliding.

crack, characterized by an energy release rate due to the development of the crack, Hutchinson (1983) obtained a stress potential function ~no,-sliding for a power law creeping material with a dilute distribution of microcracks. Following an argument of Rodin and Parks (1986), Hutchinson's function can be modified to meet the convexity requirements as (see also Hsia et al. (1991)), ~non-sliding

_ 0.0 (o:)

5.2. Stress potential and creep strain rate

n+lk.0

In a previous section, we showed that the effect of G-B sliding on a material without any facet microcracks is to enhance the overall creep strain rate. In this section, the effects of G-B sliding on the stress potential function and the creep constitutive equation for a material with a distribution of grain boundary facet microcracks will be presented. Considering each of the G-B facet cracks to be penny-shaped, and superposing the effect of each

X

n+l

1 -t- Unon_sliding

NrR3, n, ~

,

(24a) where

= ~'rrUfR3h~.I~,-(D n, -oS. ) ,

(24b)

57

K.J. Hsia et al. / Grain boundary sliding

Nf is the number of cracks per unit volume, R is the facet crack radius, G0 and o0 are a reference strain rate and reference stress, respectively, and h HH 2,3-D~.In S/off) is the normalized energy release rate given by He and Hutchinson (1981). Replacing He and Hutchinson's (1981) approximate expression of the energy release rate h HH 2,3-Dktn , S / o f ) with our modified expression in equation (16) by taking G-B sliding into account, and defining the crack density parameter for the 3-D case as p = NfR 3, we obtain

where [~]

- 1 + Usliding p, /'/, ~

(27b)

,

,

Usliding(P, n, ~S )

+142(,,, 01 o 7

• (25)

Considering further the convexity condition, a modified, convex stress potential function can be constructed as Go°o ~sliding = ( n T ] ) f

and ~ = 3S/Oa = 8,,iSmsei ® ej (no summation on m is to be taken; where m is the maximum principal stress direction) is a second-rank tensor whose dot product with the stress tensor gives the maximum principal stress, o ' is the deviatoric stress tensor, and e~'s are eigenvectors of the current coordinate system. An examination of equations (27a) and (27b) reveals that the deformation is no longer incompressible even when S / o f vanishes. The volumetric expansion rate of the material Gm ~- Gll + G22 + G33 is directly contributed by the volumetric growth rate of G-B facet cracks. For S / o f = 0 , the material is compressible only if G-B sliding is accounted for. Under uniaxial tensile loading with a far field average stress of o, equation (27a) becomes

GsCliding= G0 (;oo) f

f-e o0

x [1 +

n + 141 +

142)]

(26)

(28)

that since Ho(n, 0), increasing functions of the crack density 0 for n > 2, the modified stress potential function (/)sliding becomes more sensitive to the crack density 0. The interaction between the microcracks is then partially taken into account by this higher sensitivity with respect to crack density 0. The modified creep constitutive equation with the effect of G-B sliding can now be derived by differentiating this potential function with respect to the stress tensor o as,

Equation (28) indicates that when there are no micro-defects (such as G-B facet microcracks) in the material, the creep constitutive equation becomes of a pure power law type. But note that G-B sliding still exists, which, we know from our results in Section 3.3, gives an equivalent stress elevation by a factor f , equal to the stress enhancement factor. To see how creep strain rate is increased by G-B sliding for a material with a distribution of facet microcracks, the ratio of creep strain rate with G-B sliding to one without G-B sliding under uniaxial tension is obtained as

× l+Us,ding It

should be

O, n , ~

noted

tIl(n, O) and H2(n, O) are

•

•c = Esliding

%

e

f

-~0

(n -

1)/2

[.;¢g-]

Gsliding Gnon-sliding

f " [1 + ]~pfz(H0 + H 1 + H2)] '"+a)/2 8p 1 + (1 +

+~of2[½Hl+I-I2(o-@)]~ },

(27a)

)(~+

1)/2

3/n) 1/2 (29)

K.J. Hsia et aL / Grain boundary sliding

58 100

n=7

......

n=5 ~

/l //

n=3

~ .we

/ /

50

/

~ ~ .~

/ / /

0

i

0.00

I

i

i

I .05

J

/

/

./

/,

i

i

i 0.10

P Fig. 13. Effect of G-B sliding on uniaxial creep strain rate as a function of facet crack density #, for three creep stress exponents.

The values of this ratio for some typical values of n as a function of the crack density P are plotted in Fig. 13. Although the creep strain rate due to G-B sliding is only about 2-3 times higher than that without G-B sliding when there are no boundary facet microcracks (O = 0), the effects become much more dramatic when the crack density increases. For a two-dimensional array of hexagonal grains, at p = 0.1 every grain has one cracked G-B facet. Thus, it is expected that severe interactions between facet cracks will be in effect when p approaches 0.1. Figure 13 shows that when p ~ 0.1, the creep strain rate with G-B sliding is increased by almost two orders of magnitude. Of course, when the cracking is random, for p = 0.1 many cracked facets will become near neighbors and the load carrying capacity of the assembly is likely to be nearly completely lost. It is also clear from the figure that the enhancement of creep strain rate is a very strong function of the creep exponent n.

6. Discussion

Our present study of G-B sliding effects employing 2-D plane strain hexagonal grains is clearly

only an approximation to the behavior of a real polycrystalline material. Therefore, it is important to know how much the behavior of the real 3-D configuration differs from the 2-D simulation. Since the finite element investigation of a general 3-D grain structure with a distribution of G-B facet cracks would be an extremely time consuming task, we have considered an axisymmetrical model similar to that used by Tvergaard (1985) to study the G-B sliding effects, in the hope that the axisymmetrical model would possess some of the characteristics of the 3-D geometrical constraints. As shown in Fig. 14 we choose as a representative geometry a conical grain embedded into an annular surrounding with a central cross section that resembles the situation of a plane strain configuration. The results of the stress enhancement factor for this axisymmetrical geometry are given in Table 1, and are compared first with the results for the plane strain case. Although the values of the stress enhancement factor for this axisymmetrical geometry are slightly lower than those for the plane strain geometry, the differences are not at all substantial. It is also interesting to note that

I.

R2

,,,.J

h

l R1

~1

Fig. 14. Axisymmetrical conical grain to study grain b o u n d a r y sliding effects in a 3-D a p p r o x i m a t i o n ( R J R = 3; R 2 / R = 2; h / R = 1.732).

K.J. Hsm et al. / Grain boundary sfiding Table 1 Comparison of stress enhancement factor of different geometries Plain strain hexagonal Pea = 0.577

Axisymmetrical hexagonal PGB = 0.577

f

f/PGB

f

f/PGa

1.155 1.166 1.174 1.181

2.002 2.020 2.034 2.047

1.116 1.099 1.096 1.095 1.094 1.088

1.933 1.906 1.900 1.898 1.895 1.886

Wigner-Seitz cell (Rodin and Dib, 1989) PGB = 0 . 4 6 4

1 3 5 7 10 20

f

f/pcB

1.31 1.26 1.24 1.23

2.823 2.716 2.672 2.640

the stress enhancement factor for the axisymmetrical geometry decreases with an increase in the creep exponent n while for the plane strain geometry it increases with n. Rodin and Dib (1989) have performed a similar finite element study for the stress enhancement factor for a 3-D WignerSeitz unit cell, which also showed the decreasing behavior as n increases. While such a trend may be a likely result of a fundamentally different behavior between the elastic response and the plastic limit load response of these different geometries, this could not be readily verified. Although the stress enhancement factor f has been evaluated by several authors (Crossman and Ashby, 1975; Ghahremani, 1980; Chen and Argon, 1979; Anderson and Rice, 1985; Rodin and Dib, 1989; Beere, 1982) for different grain geometries, the characterization of it by a proper parameter has not been very successful. Rodin and Dib (1989) have tried to characterize f using a dimensionless geometrical parameter proposed by Budiansky and O'Connell (1976), defined as =

2.,

i=1

/L-7'

(30)

where N is the number of grain boundaries in a representative volume element V, A i is the area of the ith grain boundary, and L i is the perimeter of the i th grain boundary. They found that the stress enhancement factor can be well correlated with the parameter POB and gave a result where f/P~a --- 2.8 for both the periodic arrays of cubic grains and Wigner-Seitz grains for n = 1.

59

We have tested our results for a similar correlation. Evaluations of the dimensionless parameter show that PrB = 0.577 for both plane strain hexagonal grains and axisymmetrical hexagonal conical grains in our consideration. But from Table 1 it is clear that f/PGB is roughly 2.0 for both configurations we have considered for all n values. This seems to indicate that the dimensionless parameter PGa may not be able to characterize the stress enhancement factor uniquely, or, at least, f will not be proportional to PGB" Since G-B sliding is associated with a very complicated deformation mode within the grains, it is not intuitively clear that a single geometrical parameter can completely characterize this behavior. The two facet crack densities considered in the present study are rather dilute. Although the effects of interactions between facet cracks should be well represented by our numerical solutions, these solutions are valid only for the values of crack density not too different from the values we used. The interactions become very strong when the crack density approaches its maximum value, corresponding to the case that every boundary normal to the maximum principal stress direction becomes a boundary facet crack. In their approximate solution of 3-D Wigner-Seitz grains, Anderson and Rice (1985) have shown that, in the case when every boundary facet normal to the applied stress becomes a facet crack, the creep strain rate will increase by about a factor of 100 above that of the uncavitated sliding G-Bs. For materials with large creep ductility such as 304 stainless steel in which the boundary cavitation is non-uniform, it is very likely that a macroscopic crack is formed well before such a high microcrack density is reached, thus the continuum approach employed here would no longer be applicable to the cases of high microcrack density. On the other hand, for some materials with little creep ductility such as Astroloy, in which cavitation is nearly uniform on all grain boundaries as in the case considered by Anderson and Rice (1985) and Tvergaard (1985), Capano et al. (1989) have shown that cavity growth will not be constrained. In another communication (Hsia et al., 1991), we pointed out that in developing the creep constitutive equation for the material with cavita-

K.J. Hsia et al. / Grain boundary sliding

60

tional damage, we considered the contributions of damage to the macroscopic creep rate differently from Tvergaard (1985). While Tvergaard considered only the contributions of the cavitating grain boundaries to the creep rate, and modeled the cavitating boundary facets as traction-transmitting microcracks with facet stress equal to o F, by contrast, we considered only the contributions of the traction free facet microcracks to the creep rate. Nevertheless, these two models should yield the same instantaneous creep rate for the same crack density p if o F = 0 in Tvergaard's model. The comparison of the two models with the effects of G-B sliding accounted for is made in the Appendix, which gives the ratio of the creep strain rates of the present study over that of Tvergaard's model under remote uniaxial tensile stress as (see Appendix),

(~1)Tvergaard

[1 + ~'rr#f2( H 0 + H1 + 112)] ('+1)/2

1 + G(1

[.. o

v

-LP v

4

2 /

0

i

ooo

~

L

i

I

o5

i

k

i

i

OlO

P Fig. 15. Comparison of prediction of present model for uniaxial tensile strain rate with results of Tvergaard (n = 5, "/= 0.5, C 1 = 1, and C2 = 4) as a function of crack density o, for vanishing facet stress (o F = 0 in Tvergaard's model).

(~l)p ..... t

=

6

+

G) p/(1

(31)

+

Figure 15 shows the values of this ratio for different crack densities. The two creep rates predicted by different models are the same when p = 0, which is the case when there is no facet crack and only the effects of free sliding boundaries are taken into account. However, for # > 0, the creep rates by our model are higher than that by Tvergaard's model. The difference is substantial, particularly for high crack density cases. Since in these cases, the facet cracks interact with each other strongly and the influence of the reduction of the net section area is large, Tvergaard's model cannot account for this dramatic increase in creep rate. On the other hand, our model is derived from the computational results for two dilute concentrations of facet cracks. Although the extrapolation of the results near these values of the crack density should be fairly accurate, the accuracy of the results for high crack densities is not guaranteed. It is known that the effect of stress triaxiality on a crack is to enhance its volumetric growth rate as well as its energy release rate. He and Hutchin-

son (1981) have shown that the stress triaxiality dependence of these quantities for a homogeneous material is proportional to S / o ~ and ( S / o ~ ) 2, respectively. On the other hand, our computational results with G-B sliding indicate that, in addition to this effect, a state of stress triaxiality with its value higher than that of uniaxial tensile stress state (i.e. T > 0) tends to suppress the volumetric growth rate and energy release rate of a crack, while a stress triaxiality lower than uniaxial tension ( T < 0) tends to further enhance them. The final volumetric growth rate and energy release rate of a crack are determined by the sum of these two effects which is not a simple superposition due to the non-linear material behavior. It is noted in Fig. 3a that for a perfectly periodic array of grains, no G-B sliding across the boundaries normal to the m a x i m u m principal stress direction (horizontal) boundaries) occurs. But experiments show that cavitation is most severe on these grain boundaries, while it is well known that G-B sliding is a necessary condition for cavity nucleation. Several reasons can be cited to explain this seemingly conflicting behavior. First, no real material possesses a perfectly regular

K.J. Hsia et al. / Grain boundary sliding

a n d p e r i o d i c g r a i n structure. I n a real m a t e r i a l a n y t h i n g t h a t b r e a k s the s y m m e t r y of the structure such as a b o u n d a r y facet c r a c k (as shown in Fig. 3b) results in G - B sliding across the b o u n d a r i e s n o r m a l to the m a x i m u m p r i n c i p a l stress direction; F u r t h e r m o r e , it is k n o w n that c a v i t a t i o n involves b o t h n u c l e a t i o n a n d growth, a n d while n u c l e a t i o n is the direct c o n s e q u e n c e of G - B sliding, diffusional g r o w t h of cavities, which c o n t r i b u t e s m o r e to c a v i t a t i o n a l d a m a g e t h a n nucleation, is driven b y the g r a d i e n t of c h e m i c a l p o t e n t i a l which is o v e r w h e l m i n g l y favored b y the n o r m a l stress on the grain b o u n d a r y .

Acknowledgement This research has been s u p p o r t e d b y the N S F / M R L t h r o u g h the M.I.T. C e n t e r for M a t e r i als Science a n d Engineering u n d e r G r a n t No. DMR-87-19217. T h e F E M c o m p u t a t i o n s have benefited m a r k e d l y f r o m the a v a i l a b i l i t y of a n A1liant F X - 8 c o m p u t e r m a d e p o s s i b l e t h r o u g h a DARPA/ONR U R I p r o g r a m o n s i m u l a t i o n of p o l y m e r p r o p e r t i e s u n d e r C o n t r a c t N o . N0001486-K-0768. T h e A B A Q U S finite e l e m e n t p r o g r a m was m a d e available u n d e r a c a d e m i c license f r o m H i b b i t t , K a r l s s o n , a n d Sorensen, Inc., Providence, RI. T h e a u t h o r s also a c k n o w l e d g e some useful discussions with Professor G. R o d i n o f the U n i versity of T e x a s at Austin.

References Anderson, P.M. and J.R. Rice (1985), Constrained creep cavitation of grain-boundary facets, Acta Metall. 33, 409. Argon, A.S., I.-W. Chert and C.W. Lau (1980), Intergranular cavitation in creep: theory and experiments, in: Creep-Fatigue-Enoironment Interactions, R.M. Pelloux and N.S. Stoloff, eds., AIME, New York, p. 46. Argon, A.S., C.W. Lau, B. Ozmat and D.M. Parks (1985), Creep crack growth in ductile alloys, in: Fundamentals of Deformation and Fracture, K.J. Miller, ed. Cambridge Univ. Press, Cambridge, p. 189. Beere, W. (1982), Stress redistribution due to grain-boundary sliding during creep, Metal Sci. 16~ 223.

61

Budiansky, B. and R.J. O'ConneU (1976), Elastic moduli of a cracked solid, Int. J. Solids Struct. 12, 81. Capano, M.A., A.S. Argon, and I.-W. Chen (1989), Intergranulax cavitation during creeping in Astroloy, Acta Metall. 37, 3195. Chen, I.-W. and A.S. Argon (1979), Grain boundary and interphase boundary sliding in power-law Creep, Acta Metall. 27, 749. Crossman, F.W. and M.F. Ashby (1975), The non-uniform flow of polycrystals by grain-boundary sliding accommodated by power-law creep, Acta Metall. 23, 425. Dyson, B.F. (1976), Constraints on diffusional cavity growth rates, Metal. Sci. 10, 349. Ghahremani, F. (1980), Effect of grain-boundary sliding on anelasticity of polycrystals, Int. J. Solids Struct. 16, 825. Ghahremani, F. (1980), Effect of grain boundary sliding on steady creep of polycrystals, Int. J. Solids Struct. 16, 847. Hart, E.W. (1967), A theory for flow of polycrystals, Acta Metall. 15, 1545. He, M.Y. and J.W. Hutchinson (1981), The penny-shaped crack and the plane strain crack in an infinite body of power-law material, J. Appl. Mech. 48, 830. Hsia, K.J., A.S. Argon and D.M. Parks, (1991), Modeling of creep damage evolution around blunt notches and sharp cracks, Mech. Mater. 11, 19. Hutchinson, J.W. (1983), Constitutive behavior and crack tip fields for materials undergoing creep-constrained grainboundary cavitation, Acta Metall. 31, 1079. Lau, C.W., A.S. Argon and F.A. McClintock (1983), Stress concentrations due to sliding grain-boundaries in creeping alloys, in: Elastic-Plastic Fracture Mechanics, ASTM STP 803, C.F. Shih and J.P. Gudas, Eds., American Society for Testing and Materials, p. 1-551. Rice, J.R. (1968), Mathematical analysis in the mechanics of fracture, in: Fracture: An Advanced Treatise, H. Liebowitz, ed., Vol. 2, Academic Press, New York, p. 247. Rice, J.R. (1981), Constraints on the diffusive cavitation of isolated grain boundary facets in creeping polycrystals, Acta Metall. 29, 675. Rodin, G.J. and D.M. Parks (1986), Constitutive models of a power-law matrix containing aligned penny-shaped cracks, Mech. Mater. 3, 221. Rodin, G.J. and M.W. Dib (1989), Effective properties of creeping solids undergoing grain-boundary sliding, in: Advances in Fracture Research, K. Salama et al., eds., Vol. 2, Pergamon Press, Oxford, p. 1835. Tvergaard, V. (1985), Influence of grain-boundary sliding on material failure in the tertiary creep range, Int. J. Solids Struct. 21, 279. Tvergaard, V. (1985), Effect of grain-boundary sliding on creep constrained diffusive cavitation, J. Mech. Phys. Solids 33, 447.

62

K.J. Hsia et al. / Grain boundary sliding

Appendix By considering the effects of G-B sliding, and modeling the cavitating grain boundaries as traction-transmitting penny-shaped cracks with crack surface traction equal to o v, Tvergaard obtained the modified creep constitutive equations as (see Tvergaard (1989), equation (3.9)),

+v)

( ~c] )Tvergaard = .[3

+o

n - - 1 sij

771

2 n+l

S* - o ¢ %

and S is the applied remote tensile stress normal to the facet crack, T is the radial stress in the crack plane, rn,s=aS/aoij, s~j the deviatoric stress tensor, G,j the metric tensor in the current configuration, and 1' = 0.5, C 1 = 1, and C 2 = 4 are constants used by Tvergaard. Under remote uniaxial tension S, and when o v = 0, the creep rate becomes,

(~l)Tvergaard =lEO Oee [(1-~-'Y) -}- C2(1-[- CI)2p]"

2-oe (A1)

where

(A2) From equation (28), the creep rate under uniaxial tension predicted by the present model is

(~l)p ..... t =i°

f~00

[1 + %rpf2(Ho+H 1 +H2)] 2("+''/3

S* = S + C I ( S - T ) ,

(Ala)

(13)

T),

(Alb)

We should note that f " in equation (A3) and 1 + 3, in equation (A2) are equal for n = 5, then the ratio of these two creep rates is easily obtainable and given by equation (31).

T * = T-~- C l ( S -

rn,'~ = (1 + 3 C , ) m i j - ½C,Gi.,,

(Alc)

O* = C2P

(Ald)