Fundamental solutions for dilute distributions of inclusions embedded in microcracked solids

Mechanics of Materials 16 (1993) 281-294 Elsevier

281

Fundamental solutions for dilute distributions of inclusions embedded in microcracked solids K.X. Hu, A. C h a n d r a a n d Y. H u a n g Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZ 85721, USA

Received 23 October 1992; revised version received 31 March 1993

Dilute distributions of inclusions such as second-phase particles, foreign grains, or other inhomogeneities exist in a variety of materials. In the neighborhood of an inclusion, there is always a possibility of crack initiation due to stress concentration. The problem of a solid containing an inclusion surrounded by multiple cracks is investigated in this paper. The present analysis provides fundamental solutions for crack tip behavior and damage evaluation for a solid containing cracks and dilute distributions of inclusions. A distribution of dislocations approach is pursued to model crack-crack and crack-inclusion interactions accurately and efficiently. Both transformation loading from the inclusion and remote loading on the matrix are considered, and the effects of crack-crack and crack-inclusion interactions are investigated in detail. The numerical results for radial crack systems reveal that crack-inclusion interactions produce stress retardation for the harder inclusions only if the matrix is subject to remote loading. Unlike the remote loading on the matrix, transformation loading on the inclusion produces stress amplification for hard inclusions. The opposite holds for softer or weaker inclusions. It is also observed that crack-crack interactions can produce either retardation or amplification of stress intensity factors, but retardation prevails as the number of radial cracks increases. The competition between crack-crack and crack-inclusion interactions for amplification or retardation is also examined.

1. Introduction C r a c k s a n d i n h o m o g e n e i t i e s exist in v a r i o u s types o f m a t e r i a l s , such as rocks, c e r a m i c c o m pounds, and intermetallics. For example, various ceramic components and intermetallics contain s e c o n d - p h a s e p a r t i c l e s i n t r o d u c e d by alloying p r o c e s s e s . I n rocks, g e o l o g i c a l e v o l u t i o n introd u c e s a v a r i e t y o f f o r e i g n grains. F o r m a t e r i a l s c o n t a i n i n g f o r e i g n inclusions o r i n h o m o g e n e i t i e s , m i c r o c r a c k s f o r m w h e n r e s i d u a l stress o f sufficient m a g n i t u d e d e v e l o p s , even in t h e a b s e n c e o f e x t e r n a l loads. T h e r e s i d u a l stress c a n b e d u e to several sources, such as s t r e s s - i n d u c e d t r a n s f o r mation of thermal mismatch between the matrix a n d t h e inclusion. A c c o r d i n g l y , a f r a c t u r e m e Correspondence to: Prof. A. Chandra, College of Engineering and Mines, Department of Aerospace and Mechanical Engineering, The University of Arizona, Aero Building 16, Tucson, AZ 85721, USA.

chanics a p p r o a c h j s p u r s u e d in t h e p r e s e n t w o r k to investigate i n t e r a c t i o n s a m o n g m u l t i p l e cracks a n d a n inclusion. A s a first a t t e m p t , i n c l u s i o n - i n clusion i n t e r a c t i o n s a r e n e g l e c t e d a n d t h e seco n d - p h a s e p a r t i c l e s a r e m o d e l e d as d i l u t e inclusions. T h e c r a c k - c r a c k a n d c r a c k - i n c l u s i o n interactions, however, can b e very strong, a n d t h e s e a r e i n v e s t i g a t e d in d e t a i l in t h e p r e s e n t work. A d i s t r i b u t i o n o f dislocations a p p r o a c h (Bilby a n d Eshelby, 1968; A t k i n s o n , 1972; a n d E r d o g a n et al., 1974) is a d a p t e d to f o r m u l a t e t h e p r o b l e m as a system o f s i n g u l a r i n t e g r a l e q u a t i o n s , with kernels b e i n g p r o p e r l y a u g m e n t e d t h r o u g h D u n d u r s a n d M u r a ' s (1964) s o l u t i o n o f a p o i n t d i s l o c a t i o n acting in a solid c o n t a i n i n g a n e m b e d d e d inclusion. B o t h t r a n s f o r m a t i o n l o a d i n g f r o m t h e inclusion a n d r e m o t e l o a d i n g o n t h e m a t r i x a r e conside r e d . Effects o f c r a c k - c r a c k a n d c r a c k - i n c l u s i o n i n t e r a c t i o n s on stress a m p l i f i c a t i o n s a n d r e t a r d a tions a r e i n v e s t i g a t e d in detail. T h e n u m e r i c a l results reveal t h a t t h e n a t u r e o f stress r e t a r d a t i o n

0167-6636/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

282

K~X. Hu et al. / Inclusions embedded in microcracked solids

and amplification depends on the type of loading and the ratio of inclusion to matrix moduli. When a matrix is subject to remote loading, crack-inclusion interactions produce local stress retardations for harder inclusions and stress amplification for weaker inclusions. It is also observed for a radial crack system that crack-crack interactions can produce either retardation or amplification of stress intensity factors, depending strongly on the number and distribution of cracks. Stress amplification or retardation effects due to various crack-crack and crack-inclusion interactions are also investigated in this paper. In the past, Erdogan et al. (1974) investigated the interactions between a circular inclusion and an arbitrarily oriented crack using an integral equation approach, Horii and Nemat-Nasser (1985) developed a pseudo-traction approach for investigating crack interactions, and Kachanov (1987) proposed a useful simplification of the pseudotraction approach for the approximate analysis of crack interactions. The present paper attempts to develop an integral equation approach capable of handling very close packing of microcracks in the presence of inclusions. In various ceramic materials and composites, damage is initiated as microcracks around an inclusion, and the results from the present work provide the foundation for evaluating the state of damage for such cases (e.g., Huang et al., 1992) as well as to predict growth of such microcracks and their eventual coalescence to cause macro-scale damage in a body.

2. Multiple cracks near an inclusion

2.1. Problem description In this section, a fracture mechanics approach is pursued to investigate the interactions among an inclusion and cracks. In addition to standard local stress analysis, this approach provides the fundamental solution for determination of the macroscopic strain increase and modulus reduction due to microcracks in a solid. To be specific, it is assumed that an inclusion of a circular shape with radius R, shear modulus G 2 , and Poisson's

u

1

Fig. 1. Schematic diagram of an inclusion surrounded by cracks.

ratio 1-'2 is perfectly bonded to an infinitely extended, microcracked matrix material with shear modulus G~ and Poisson's ratio u t. Without loss of generality, the center of the inclusion is assumed to be at the origin of a rectangular Cartesian coordinate system (see Fig. 1). In the matrix, there are N cracks surrounding the inclusion. The associated polar coordinate system is denoted by r and 0, and a local normal-tangential coordinate system associated with the ith crack is represented by s ~i) and t ~°. The occupancy of the ith crack is taken as - a ~° < t( ° < a ~°. The geometry of the ith crack is specified by the center coordinates (x~ °, y~i)), orientation angle O~i), and the half length of the crack a ~i). The external loading applied to the system can be an arbitrary remote loading, a thermal mismatching, or a transformation strain from the inclusion. The solution of such a system must meet the following requirements: (1) the equilibrium equation must be satisfied; (2) the displacements and tractions on the inclusion-matrix interface must be continuous; (3) the crack surface must be traction free (neglecting any effects of crack clo-

K.X. Hu et al. / Inclusions embedded in microcracked solids

sure). The boundary conditions for cracks require that = 0,

0,

i = 1, 2 , . . . , N for - a ~i)< t (i)
(1)

The continuity conditions on the interface lead to

trrr,( r = R, O) = O'rr2(r

=

R, 0),

trrO,(r + R , O) =trroz(r + R, 0), 0 ~<0 ~<2~r,

(2a)

u ~ , ( r = R , O) = u , 2 ( r = R , 0), u o l ( r = R , O) =Uo2(r=R, O), 0 ~<0 ~<2rr,

(2b)

where Crrr1, tr~r2, trr01, and try02 stand for normal and tangential stresses on the interface of the matrix and inclusion, respectively; and U~l, u~2, uol and Uo2 stand for the radial and loop displacements of the matrix and inclusions. We shall refer to the problem described above as the original problem.

////

2.2. Superposition technique The difficulty in solving the original problem lies in the fact that we must account for the effects of crack-crack and crack-inclusion interactions. Various techniques have been developed to model this type of interactions (see, for example, Chen (1984), Horii and Nemat-Nasser (1983, 1985), and Horii and Nemat-Nasser (1987) for multiple holes or cracks, Kachanov (1987) for multiple cracks, Hu and Chandra (1993) for multiple cracks and rigid lines, and Hu et al. (1993) for multiple voids and cracks). The superposition technique is used in all these various approaches. As in many crack problems, the original problem can be superimposed as a homogeneous problem and perturbed problem. The homogeneous problem deals with the infinitely extended matrix embedded with the inclusion, subject to the same external loading as the original problem, but in the absence of any cracks. The perturbed problem is concerned with N cracks in the neighborhood of the inclusion but subject to no external

////

@ /12222" +Z crack

i

@

Fig. 2. Schematic diagram of superposition.

283

284

ICX. Hu et al. / Inclusions embedded in microcracked solids

loading. As a result, the perturbed problem represents only the localized stress contribution in the matrix. Following Bilby and Eshelby (1968), Atkinson (1972), and Erdogan et al. (1974), a crack can be modeled as a continuous distribution of dislocations. We further subdivide the perturbed problem into N subproblems, each having only a single crack in the matrix and near the inclusions but with an unknown distribution of dislocations (see Fig. 2). The unknown distribution of dislocations in each crack will need to be determined in such a way that all conditions for the original problem are satisfied. Based on the superposition approach discussed above, an elasticity solution for the problem of an infinitely extended solid embedded with an inclusion and subject to a point dislocation will be required in the current analysis. This solution is given by Dundurs and Mura (1964) and Erdogan et al. (1974) for a dislocation of components b x and by acting at a point (c, 0) (c > R). The stresses outside the inclusion can be written as

orL

-rr(K, + 1) Gl

[Ixxx(X, y; c)b +Ixxy(X , y; c)by],

o'by -- 'ff(K 1 + 1) G1

+ 1) G1

Let us now consider the effects of all cracks on the mth crack. For consistency, the stress fields associated with different cracks are transformed to the local tangential-normal coordinate system (t (m), S (m)) for the mth crack. For example, the stress field associated with the ith crack may be transformed as

ors~m) = or[i) sin20(mi) -- 2or//) sin 0 (mi) cos 0 (mi)

q- or(/) COS20(mi), ort~m, = (or~/)- or[])) sin 0 (mi)

(4a) COS

0 (mi)

+ Or(i)(cos20 (mi) -- sin20(mi)),

(4b)

where 0 (mi) is the angle between axes t (i) and t(m).

The N cracks are now represented by their corresponding distribution of dislocations. The contribution of the ith crack to the stresses at the presumed location of the mth crack can be obtained by integrating the effects of the distributed dislocation over the ith crack. Summing the effects of all cracks on the mth crack and imposing traction-free conditions on the mth crack, we get

N ~]

[ Ixyx( X, y; c)bx +Ixyy(x, r; c)by],

or 5_

(3a)

2.3. Integral equations

fa(" [Kll (t(m) ' t(i))b~i)(t(i) )

i = 1 a _a(i)~

(3b)

+ K 1 2 ( t (m), t(i))b(si)(t(i))] d/(i) + O'(sm0)(t(m)) = 0, m = 1, 2 . . . . .

[Iyyx(X, y; c ) b x

N,

(5a)

N +Iyyy(X, y; c)by],

(3c)

where/(1 = 3 - 4v 1 for plane strain and K1 = (3 vl)/(1 + v 1) for plane stress; functions Ixx x through Iyyy are given in the Appendix. It should be noted that the elasticity solution shown in Eq. (3) satisfies the equilibrium equation and the continuity condition on the interface (the stresses and the derivable displacements inside the inclusion are not presented here because they are not relevant to the formulation). The fundamental elastic solution bears a singularity of the order of r as the distance r between the field point (x, y) and the source point (c, 0) approaches zero.

fa(O [ g21( t(m), t(i))b}i)( t(i) ) i = 1 ~ -a(i)~

WK22 ( t (m), t (i)) b(si)( t(i))] dt (i) 4- or(sin°)(t (m)) = O, m = 1, 2 , . . . , N ,

(5b)

where or~0) and or[~0) represent the stress components at the presumed location of the mth crack, for the infinitely extended matrix embedded with the inclusion but in the absence of all cracks, i.e., stresses from the homogeneous problem. Here, b~i) and b~i) are the components of the ith crack dislocation density vector. The tangential and normal surface movements of the ith

285

K.X. Hu et al. / Inclusions embedded in microcracked solids

crack are denoted by Atti)(t
b~i)( t (i)) = ~

where angles qJ and ¢ depend on the geometries of the ith and mth cracks, integral point coordinate t (i), and collocation point t (m) in the following fashion: = fl - O~ i),

(8a)

= 0 (m) -- /3,

(8b)

and b~i)( t (i))

= ~

0

[ A(s/)(t(i))] .

(6)

Here, the kernels Kxl through K22 are obtained by casting the fundamental solution, Eq. (3), into the proper stress coordinate transformation, Eq. (4). The results can be expressed in terms of crack geometry and fundamental solution functions: g l l ( t (m), t (i))

sin ~b) cos2~b

- 2 ( l x y x cos O - l y y r sin #) sin ~b cos ~b,

(7a) gl2( t(m) , t (i))

= (Ix~ sin ~ + Ixxy COS ~b) sin2~b + (Iyr x sin ~ + Iyry cos ~b) cos2~b - 2 ( I x y x sin ff + I~yy cos ~b) sin ¢ cos ¢,

(7b) K 2 1 ( t (m), t (i))

= [(i.x

sin o)

cos

sin/3 = x J r p , cos/3 = yp/rp

(9)

and

COS o(i)c ,

( lOa)

yp = yc~i) + t
(lOb)

rp = x~p2 + y 2 .

(10c)

Xp = X (i) + t (i)

Finally, Ixxx, etc., in Eq. (6) are obtained. For example,

= (Ix~ x cos qJ - I ~ y sin ~b) sinZ¢ + ( I y y x coS ~ b - Iyyy

with/3 being defined as

Ix~y =Ixxy(X, y; c),

(11)

with x, y and c substituted as x = r ~m) cos ( y - / 3 ) + t ('m coS(0c°'°-/3),

(12a)

y = r
(12b)

c = rp,

(12c)

where r (m) is the distance between the origin and the center of the ruth crack and 3' is the angle between the x-axis and the line connecting the origin and the center of the ruth crack. The single valuedness of the displacements requires that the following additional conditions be satisfied:

- ( I ~ x cos ~ - Ix~ r sin qt)] sin $ cos

+ (I~yx cos ~ - I~yy sin ~O)(cosZ¢- sinZ¢), (7c) K 2 2 ( t (m), t (i))

= [(I.x

-(I~

sin

+ I.,

aO) b(i)(t(i)~ dt(i ) = 0 a(i) t k

]

and f_a(i) a(i)b(i)(t(i)~ s x i dt (i) = O,

cos

sin ~p + I ~ y sin ~)] sin ¢ cos ~b

+ (I~y~ sin gJ + I~y r cos ~p)(sin24~ - cosZ¢),

(7d)

i = 1, 2, • • • , N.

(13)

Equations (5a) and (5b), along with the additional conditions in Eq. (13), now provide an integral equation representation of the original problem. It is noted that the field equations and the interface continuity conditions are exactly sat-

286

K.XI Hu et al. / Inclusions embedded in microcracked solids

isfied by virtue of the fundamental solution equation as given in Eqs. (3). The traction-free conditions on the crack surfaces, as presented in Eqs. (5a) and (5b), will have to be satisfied through numerical solution of the integral equations. A detailed discussion of the treatment of singular integral equations is given by Erdogan et al. (1973). The Gauss-Chebychev quadrature proposed by Erdogan et ai. is adapted for the current analysis. Once dislocation distributions for all cracks are obtained from the numerical scheme, stresses and displacements can be obtained through definite integrations of appropriate kernels. Since the purpose of our analysis is to investigate local behaviors such as stress intensity factors, strain increase, and modulus reduction

t

Fig. 3. Schematic diagram of an array of radial cracks.

0.9 ===

%

o

,6

o.8

O.7

e-

~

0.6

E 2/E 1=4.0 E=/E 1=1.0 EJE 1 = 0 . 2 5

ffl

.~

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

0.5

ooo ..o.o°°* .o.O"

C C

"0 .~

0.4 ...........

0

......

o

Z

0.3

I

0.1

I

0.3

I

I

0.5

I

I

0.7

I

0.9

Nondimensional geometry parameter,~l=a/(d-R) Fig. 4. Variation of the normalized inner tip stress intensity factor with parameter 61: dilatational transformation loading and A, = 1.5.

K.X. Hu et al. / Inclusions embedded in microcracked solids

287

and ~t(~°)(t (')) in the integral equations (Eqs. (5a) and (5b)), can be written as

due to cracks, it is sufficient to know just the dislocation distribution. Other local quantities such as SIFs (Krenk, 1975), as well as global parameters such as modulus reduction (Shum and Huang, 1990), can be easily obtained from the dislocation distribution.

R2

~r~=-Cro~-T,

(r > R ) ,

R2

°'o = or0 .2 '

(r>R),

1

(r>R),

cry0 = 0, 3. Transformation loading of inclusions

where

We consider the case where the inclusion initiates a dilatational strain ~T, i.e., e~r= coo = E T and ~rO = O. The stresses induced in the matrix, which determine the forcing terms cr~°)(t ° ~ )

eT or0 =

1 + v1

l--v2

1

el

-

0.38

E=/E 1=4.0 E=/EI=I.0 E=/E 1=0.25

0.a4_

.

0.3 o

/ .

.

-

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

S~/S//J

j j S ~/

•°"

o.2e-

C C "

SSS J J S

U) -

._o.

0.22

o

-

•°

0.18-

C t'-

•o

"O ID

0.14

g o Z

0.1

.°

-

o••o.°ooO••*°•

._~

°°oo•o°°O°°•°°° -

. . . . .

•

..................... I

0.1

l

0.3

I

I

0.5

!

I

0.7

0.9

Nondimensional geometry parameter,61 =a/(d-R) Fig. 5. Variation o f the n o r m a l i z e d inner tip stress intensity factor with p a r a m e t e r St: dilatational transformation loading and A1 = 3.0.

K.X. Hu et al. / Inclusions embedded in microcracked solids

288

An array of radial cracks is considered in Fig. 3. The cracks are assumed to be of equal length, a, and to be spaced apart from each other in an equal angle. For fixed material properties, the transformation-induced stress intensity factor (SIF) of the inner crack tip, normalized with respect to E 1 e T ~ / ~ -, can be expressed as K I EIET1/~

respectively. The hardest inclusion, E 2 / E 1 = 4.0, enhances the SIFs in all three cases. As A l becomes bigger (or the crack moves away from the inclusion), the SIFs decrease. This was expected since transformation-induced stresses are localized in nature. It is noted that E 2 / E l = 1.0 in each figure represents the case of the inclusion material being identical to the matrix, therefore the deviation of the SIF corresponding to E 2 / E l = 4.0 or E 2 / E 1 = 0.25 from that for E 2 / E 1 = 1.0 is indicative of the interaction between the inclusion and cracks. It is seen that the weakest inclusion ( E 2 / E 1 = 0.25) produces a stronger inclusion-crack interaction than the hardest inclusion ( E z / E 1 = 4.0). It is also observed that the deviation of the SIF for E 2 / E ~ = 0.25 from that for E 2 / E 1 = 1.0 changes slightly with 61 for all three values of Ar This probably implies decoupling of material and geometric properties, as observed by Shum and H u a n g (1990) for an annular crack

F(A 1, t51, N ) ,

where A1 = d / a , 61 = a / ( d - R ) , and N is the number of cracks. In the following calculations, the Poisson's ratios of the matrix and inclusions are taken as /')1 = 1"2 = 0.3. Three cases, E 2 / E 1 = 0.25, 1.0, and 4.0, are considered, and the number of cracks is N = 8. Shown in Figs. 4, 5 and 6 are the variations of the normalized SIF at the inner tip with parameters 6~ for various A1 = 1.5, 3.0, and 6.0, 0.2

E2/E ~=4.0 E2/E~=I.0 E2/E 1=0.25 . . . . . . . . . . . . . . . . . . .

u.T "~'-x,," 0.16

-

o ~

0.14 -

~

0.12 -

t-

•

~

_

c"

to

/ / I

//t//

.s /

0.1 -

°°"

I j

.°"

s j

.-0-

0.o8

er.-

"-

......,....=.,."=""

0.06

°'"

J S d i JlJl/,..°.,.o,.°,," .° ,,,''°''°* J ,,"s °.o ..." °"

S .~."

0.04

......-''"

o Z

.....

0.02

I

0.1

I

0.2

I

I

0.3

I

- ..... l

0.4

.I

I

0.5

I

I

0.6

I

I

0.7

I

I

0.a

I

0.9

Nondimensional geometry parameter, $1=a/(d-R) Fig. 6. V a r i a t i o n of the n o r m a l i z e d i n n e r tip stress intensity factor w i t h p a r a m e t e r t~l: d i l a t a t i o n a l t r a n s f o r m a t i o n l o a d i n g and A, = 6.0.

K.X.Hu et aL / Inclusionsembeddedin microcrackedsolids

289

gral equations (Eqs. (5a) and (5b)), can be written as (Muskhelishvili, 1953)

surrounding a spherical inclusion, still holds here for the weakest inclusion. However, this is not true for the hardest inclusion. Finally, contrary to what will be seen for remote loading on the matrix, the transformation loading from the particle produces stress amplification for the hardest inclusion and stress retardation for the weakest inclusion.

~ [ O'rr =

c2R2

1

r2

+ 1

r2

r4

cos 20 ,

(r>R), _~[ 4. R e m o t e loading of matrix

(reo =

c2R2

1

r2

1

3c3R4 r2 ) cos 20],

(r > R ) ,

Remote loading is considered in this section. For a solid embedded with an inclusion subject to a uniaxial tension, tr0, in the x-direction, the stresses in the matrix, which determine the forcing terms (~(~°)(t(m)) and o'/m°)(t("O), in the inte-

tr° ( Grro=-- T

clR2 3c3R4 / l + - - 7 - + r------T--] sin 20,

(r>R),

1.7 [~

1.6

N=2

Q

t,,.

o •'~

1.4

ca C

1.3

\ \

N=8

.....

._~ ca

1.2

.g-

1.1

C

"-~ "O

1

O

~° ,

°°°-°°O.o

-*o .......................

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

° ..................

%

E o Z

0.9

°°

0.8

I

I

2

I

I

3 Nondimensional

I

I

4 geometry

5 parameter,

I

6

k1=d/a

Fig. 7. Variation of the normalized inner tip stress intensity factor with parameter ) t l : biaxial remote loading, A2 = 1.0, I~1 = /"2 = 0 . 3 ,

and

E 2 / E 1=

0.25.

K.X. Hu et al. / Inclusions embedded in microcracked solids

290

where cl, c2, and c 3 are the constants related to material properties:

tor at the inner crack tip (normalized with respect to o-0f~a-) can be expressed as KI a0v/-~- = F ( A 1, A2, N ) ,

2(G2- Gi)

cl

G1 + KIG 2 ' G,(K 2 -

c2 =

1) - G2(K 1

- -

2G2 + Gn(K 2 - 1)

where A1 = d / a , ~2 : R / a , and N is the number of cracks. Figure 7 shows the variation of the inner tip SIF with the nondimensional geometrical parameter/~1 (~--" d / a ) for a case where /~2 = 1.0, u I = / 2 2 = 0.3, and E2/E 1 = 0.25. The SIF is enhanced as A 1 decreases (the cracks get closer to the inclusion). However, the extent of the enhancement strongly depends on the number of cracks. The c r a c k - c r a c k interaction becomes stronger as the number of cracks increases. The c r a c k - c r a c k interaction for a larger number of radial cracks tends to shield the stress intensity factors. The

1) '

G 2 - G l

C3

G1 + KIG 2 "

It is noted that the solution for other remote loading situations, i.e. biaxial and shear, can be easily obtained by rotating the coordinates. An array of radial cracks with the same geometry as shown in Fig. 3 is considered first. Assuming that the system is subject to biaxial remote loading (o-xx = cryy = o'0) , the stress intensity fac-

1.05

0

~Z 0 . e s ..°....°o--'°'"

t,=.

o

-~

._~

.°.oo..-° .°°.o..°°'"

0.9°.

e,-

~

°°

.o°

0.85 -

°°.°

o°"

°°-°'"

~-

oO ~

°°

°

o

°

°"

-"

,°

°"

¢h

o ~

e,l

°

°.~°°~

~

~

~'"

0.8N=2

.I cc "0 ®

°~ o ° ~ "

0.75 -

N

~

°,

N--4

~° . f

-

N=6 . . . . . . . . . . . . . . . . . . . .

0.7

N=8

o~.,o~" o Z 0.65

I

2

3

I

I

I

4

I

5

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c o m b i n e d effect of the c r a c k - i n c l u s i o n interaction (which leads to amplification in SIF for w e a k e r inclusions) and the c r a c k - c r a c k interaction (which leads to retardation in SIF for a large n u m b e r of cracks) can be clearly seen in Fig. 7. F o r N = 6 and 8, one can find a small value of A1 (or a position where the cracks are close to the inclusion) that makes the normalized SIF value equal to 1.0. T h e competition between the amplification and the retardation effects is responsible for this result. O f course, the values of SIF tend to be 1.0 as A] b e c o m e s larger (neither c r a c k crack nor c r a c k - i n c l u s i o n interactions exist when the cracks are spaced far apart f r o m the inclusion). Figure 8 shows the variation of the inner tip SIF with the nondimensional geometrical p a r a m eter A1 (d/a) for a case where A 2 --- 1.0, v 1 = v 2 = 0.3, and E2/E 1 = 1.0. T h e inclusion has the same

material properties as the matrix in this case. As one may expect, the inner tip S I F b e c o m e s shielded for N = 4, 6, and 8, but amplification exists only for collinear cracks ( N = 2). T h e shielding b e c o m e s stronger as the cracks move closer. Figure 9 shows the variation of the inner tip SIF with the nondimensional geometrical p a r a m eter h 1 ( = d/a) for a case where h 2 = 1.0, v x = v 2 = 0.3, and E2/E 1 = 4.0. R e t a r d a t i o n occurs in this case. It is observed that retardation b e c o m e s stronger as the cracks get closer to the inclusion. O n e can conclude that, for h a r d e r inclusions, the crack-inclusion interaction produces stress retardation (particle shielding). It is also observed that retardation increases if the n u m b e r of cracks increases• As before, increasing c r a c k - c r a c k interactions are responsible for this result• T h e c o m b i n e d effects of both c r a c k - c r a c k and c r a c k -

292

K.X. Hu et al. / Inclusions embedded in microcracked solids

inclusion interactions produce significant retardation in the stress intensity factors. Comparisons between Figs. 7 and 8 and between Figs. 8 and 9 reveal that crack-inclusion interactions greatly affect the nature of local stress distributions.

5. Conclusions The effects of interactions among multiple cracks and an inclusion on crack tip stress amplification and retardation are studied in this paper. The numerical results for radial crack systems reveal that crack-inclusion interactions produce stress retardation for harder inclusions only if the matrix is subject to remote loading. Unlike remote loading on the matrix, transformation loading on the inclusion produces stress amplification for hard inclusions. The competition between crack-crack and crack-inclusion interactions for amplification or retardation is also investigated in this paper. Many real-life materials contain inhomogeneities that, in turn, provide possibilities for matrix cracking. The current analysis provides fundamental solutions needed to continuum modeling of constitutive behavior of materials that contain dilute distributions of inclusions and undergo microcracking. Work on implementations of the current approach to determine overall material response of microcracked composites is currently in progress at The University of Arizona.

Acknowledgments The authors gratefully acknowledge the financial support provided under Grant No. DMC 8657345 from the U.S. National Science Foundation.

References Atkinson, C. (1972) The interaction between a crack and an inclusion, Int. J. Eng. Sci. 10, 127-136. Bilby, B.A. and J.D. Eshelby (1968), Dislocation and the theory of fracture, in: H. Liebowitz, ed., Fracture; An Advanced Treatise, Vol. I, Academic Press, New York, pp. 99-182. Chert, Y.Z. (1984), General case of multiple crack problems in an infinite plate, Eng. Fract. Mech. 20, 591-597. Dundurs, J. and T. Mura (1964), Interaction between an edge dislocation and a circular inclusion, J. Mech. Phys. Solids 12, 177-189. Erdogan, F., G.D. Gupta and T.S. Cook (1973), Numerical solution of singular integral equations, in G.C. Sih, ed., Methods of Analysis and Solutions of Crack Problems Noordhoff, Leiden, The Netherlands, pp. 386-425. Erdogan, F. G.D. Gupta and M. Ratwani (1974), Interaction between a circular inclusion and an arbitrarily oriented crack, J. Appl. Mech. 41, 1007-1013. Hori, M. and S. Nemat-Nasser (1987), Interacting micro-cracks near the tip in the process zone of a macro-crack, J. Mech. Phys. Solids 35, 601-629. Horii, H. and S. Nemat-Nasser (1983), Estimate of stress intensity factors for interacting cracks, Advances in Aerospace Structures, Materials and Dynamics, ASME, New York, pp. 111-117. Horii, H. and S. NematoNasser (1985), Elastic fields of interacting inhomogeneities, Int. Z Solids Struct. 21,731-745. Hu, K.X. and A. Chandra (1993), Interactions among general systems of cracks and anticracks: An integral equation approach, J. Appl. Mech., (in press). Hu, K.X., A. Chandra and Y. Huang (1993), Multiple voidcrack interaction, Int. J. Solids Struct. 30, 1473-1489. Huang, Y., K.X. Hu and A. Chandra (1992) Damage evaluation for solids containing dilute distributions of inclusions and microcracks, J. Appl. Mech., (submitted). Kachanov, M. (1987), Elastic solids with many cracks; A simple method of analysis, Int. J. Solids Struct. 23, 23-43. Krenk, S. (1975), On the use of the interpolation polynomial for solutions of singular integral equations, Q. Appl. Math. 32, 479-484. Muskhelishvili, N.I. (1953), Some Basic Problems of the Mathematical Theory of Elasticity, Noorhoff, Ltd., Groningen, The Netherlands. Shum, D.K.M. and Y. Huang (1990), Fundamental solutions for microcracking induced by residual stress, Eng. Fract. Mech. 37, 107-117.

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