Fracture Characteristics of Solids Containing Doubly-Periodic Arrays of Cracks

Fracture Characteristics of Solids Containing Doubly-Periodic Arrays of CracksAuthor(s): B. L. KarihalooSource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 360, No. 1702 (Apr. 4, 1978), pp. 373-387Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/79588 .

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Proc. R. Soc. Lond. A. 360, 373-387 (1978)

Printed in Great Britain

Fracture characteristics of solids containing doubly-periodic arrays of cracks

By B. L. KARIHALOO

Department of Civil Engineering, The University of Newcastle, N.S.W, 2308, Australia

(Communicated by Sir Alan Cottrell, F.R.S. - Received 5 April 1977, revised 2 August 1977)

The stress relaxation process from the tips of doubly-periodic (rectangular and diamond-shaped) arrays of slit-like cracks contained in an infinite elastic solid is studied under both plane and anti-plane strain conditions. The displacement discontinuities due to slit-like cracks are represented by distributions of suitable dislocations. The latter are determined from singular integral equations resulting from the satisfaction of the traction-free conditions at the crack faces. In the absence of a closed form solution, these equations are solved numerically after expanding the non-singular part of the kernel in a series of Chebyshev polynomials. Results are presented for the extent of spread of plasticity from each of the cracks and for the crack-tip opening displacement as functions of the horizontal and vertical crack spacings and the externally applied stress and discussed from the point of fracture initiation from an array of stress concentrations. It is shown that an array of cracks can have a detrimental or beneficial effect on the fracture characteristics of the solid depending on the far-field state of stress. Moreover, the crack-tip opening displacement is practically independent of the horizontal separation of cracks for small values of the distance of vertical separation and depends only on the latter.

1. INTRODUCTION

The importance of studying the elastic behaviour and the fracture characteristics of elastic bodies containing multiple cracks hardly needs to be stressed. An im- mediate application is in the geomechanical field. The presence of randomly oriented and distributed cracks of various scales and shapes can significantly alter the elastic and fracture properties of the solid, the extent of alteration depending on the spacing of the cracks, their relative orientation and the far-field state of stress. However, to render this physical situation mathematically manageable the randomness of the orientation and of the distribution and scale of the cracks must be sacrificed. Therefore, problems where cracks are of the same length and arranged in a periodic fashion are investigated. It is hoped that such idealized models can

reveal some important trends as far as the elastic and fracture properties of the

solids are concerned. Many authors, among them Louat (i 96z), Bilby, Cottrell & Swinden (1 963), Bilby,

[ 373 ]



374 B. L. Karihaloo

Cottrell, Smith & Swinden (I964), Smith (I964), Koiter (I959), Benthem & Koiter (I973), Paris & Sih (I965), Karihaloo (1977a, b, c), considered several such arrays of cracks under different modes of loading with a view to studying the change in the elastic constants and/or the fracture characteristics of the solid. It is interesting to note that, for mathematical expediency, a majority of these authors invoked the equivalence of slit-like cracks (displacement discontinuities) and distributions of straight dislocations, a technique now well established (see, for example, Bilby & Eshelby I968). The arrangements of cracks were essentially one-dimensional: a row of collinear cracks or cracks with a constant distance of vertical separation (a 'stack' of cracks). A more difficult problem of the weakening of elastic solids due to doubly-periodic (rectangular and diamond-shaped) arrays of cracks was studied in detail under all the modes of loading by Delameter, Herrmann & Barnett (I975), and Delameter & Herrmann (1974). However, the stress relaxation process from doubly-periodic arrays of cracks, a natural extension to the above class of problems, has only received an approximate, and at best a cursory, treatment by Isida (I972). It is the aim of the present work to study this process in detail under both plane and anti-plane strain conditions.

Thus, the problem is formulated in ? 2 by the well established technique of representing cracks by suitable continuous distributions of infinitesimal dislocations. Section 3 briefly describes an approximate numerical procedure used to solve the resulting singular integral equations. Finally, in ? 4 the results are presented graphically and the influence of arrays of cracks on the fracture characteristics of solids is discussed. For a better appreciation the results are, where possible, compared with that for an isolated crack. In particular, it is shown that the presence of multiple inhomogeneities (cracks) can have a beneficial or adverse effect, in comparison with an isolated crack, depending on the far-field state of stress. Moreover, it is shown that the crack-tip opening displacement (a measure of the fracture characteristics of the solids) is predominantly dependent on the distance of vertical separation for small values of this distance - a result which justifies the assumptions made by Budiansky & O'Connell (1976), and earlier by Salganik (I 973), in calculating approximately the elastic moduli of bodies containing randomly distributed flat cracks by using the self-consistent method.

2. FORMULATION OF THE PROBLEM

An infinite, isotropically elastic solid containing a doubly-periodic (rectangular and diamond-shaped) array of cracks is shown in figure 1.

In the rectangular configuration the traction-free cracks occupy the positions md1-c < x < md1 + c; y = nh1 (n, m = 0, ? 1, ? 2, ...), while the coplanar plastic regions are located at md1-a < x < md1-c and md1 + c < x < md1 + a; y =+ nh1. In the diamond-shaped configuration the traction-free parts occupy the positions md1-c < x < nmd1+c; y-=?+ nh (n-O, ? 2, ?4, ...) or

md, J -id,-c < x < md, J 1 d. + c; y ?nhi. (n-=J 1, ? 3,...),



Fracture of solids with arrays of cracks 375

x~ 101

it

I 0?~~~~~~~~~~~~~

0 0 0 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -

J i ^,s~~~~~~~~~~~Z

T E X= g~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~6

. 0S

dC,a

X * > V~~~~~~~~~~~~~~~~~~~~~~~C , i H ot~~~~~~~~~~

_~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C +5 ,

!:3 s tt~~~~~~~~~~~~~C

I 1w 1+ W: MlE~~~~~~~~~~~~~~~~~~~~~~~~~.

l l | |1 $ R 4 D~~~~~~~~~~~~~e



376 B. L. Karihaloo

and the coplanar plastic regions the locations md1 - a < x < md1 - c and

md1?c(x md1+a;y-+?nh, (n= 0,+2,+4,...)

or the locations md1 + d1 d-a < x < Imd - c, and

md1+ +d,+c < x < md1+ Id,+a; y- ? nh (n = + , ? 3,...).

It is evident that the effect of an array of cracks on the fracture characteristics of a solid will be different from that of an isolated crack because of the interaction of the stress fields of the former. It is here that the equivalence of slit-like cracks (displacement discontinuities) and continuous distributions of infinitesimal dislocations comes in handy. The type of dislocations suitable to represent a slit-like crack is dictated by the state of far-field stress. Thus, anti-plane strain state (mode 3) is modelled by screw dislocations with a constant Burgers vector parallel to the axis of displacement discontinuity (z-axis), while the states of plane strain (modes 1 and 2) are induced by edge dislocations whose Burgers vectors are of constant magnitude and parallel to the respective axes of displacement discontinuity (y- and x-axes). Likewise, the plastic regions ahead of the crack tips, and coplanar with them, may be represented by suitable dislocation distributions. The displacement discontinuity across each crack is related to the dislocation distribution function defined by a singular integral equation resulting from the satisfaction of traction-free conditions at the crack faces which require the resultant stress on any dislocation in the distribution to vanish when the whole system is in equilibrium.

To write an expression for the resultant stress on a dislocation we need to dis- tinguish between the dislocations representing the freely-slipping part of the cracks and those representing the plastic regions. This is because in the latter the dislocations are hindered in their movement by a resistance stress, which, for simplicity, is put equal to the yield stress of the material, oh,. Thus, the dislocations representing the freely-slipping cracks are subjected to only the relevant component of the uniform far-field stress oTn = C (o`v in mode 1, coy in mode 2 or o-v in mode 3), while the dislocations representing the plastic regions are under a net stress

a-?= - o-,. Each dislocation in the freely-slipping and the plastic regions is further subjected to interaction stresses from all the other dislocations in the distributions. Expressions for these stresses are readily obtained by considering the influence of a vertical array of relevant dislocations situated along a plane x = x' at a point x along one of the planes y = ? nh1. The resulting expressions are considerably simplified as the sum over the number of rows, n, is easily evaluated in terms of hyperbolic functions. Having imposed the traction-free requirement of the crack faces and introduced non-dimensional variables, the above-mentioned equilibrium condition for each dislocation in the distribution is mathematically expressed by the following singular integral equation in the distribution function

ff(x0):

f (x') {1(x - XI) +K(x', x)}dx'+ P(x) = O,(1




where P(x) takes the values o-/o-Y and (o-/o-, - 1) in the traction-free parts of the cracks and the plastic regions ahead of the crack tips, respectively. The spatial variables x and x' have been normalized by a, and the dislocation distribution function by o-Y/A, where the constant A takes the values jtb/2r(1 - v) and jtb/2nt in the plane strain modes 1 and 2, and the anti-plane strain mode 3, respectively. Here, It is the shear modulus, b the Burgers vector of each of the dislocations, and v the Poisson ratio. The decomposition of the kernel into singular and non-singular parts is dictated by the technique used to integrate (1).

The non-singular part of the kernel K(x', x) is given by:

rectangular array of cracks:

=-l/(x-x')+ E A(m) mode 3 M -0I

K(x', xX) -1/(x-x') + , B(m) mode 2 (2) M = - 0

- -1/(x-x')+ E [2A(m)-B(m)] mode 1; m= -

diamond-shaped array of cracks: 00

= -l/(x-x')+ E [A(m)+C(m)] mode 3 m= --

K(x', x) =l /(x - x') + E [B(m) - D(m)] mode 2 (3) M = - 0

=-l/(x-x')+ , [2A(m)-B(m)+20(m)+D(m)] model,

where A(m) = coth [(x -x' + md)/clh],

B(m) = (1/x2h2) (x -x' + md) cosh2 [(x-x' + md)/ah],

C(m) = (1/och)tanh[(x-x' +md+ ld)/ah], (4)

D(m) = (I/ac2h2) (x-x' +md + ld) sech2 [(x-x' +md +d)/lah], and h hll/c, d = d1/a and a = c/a. It is worth emphasizing that, despite the appearance of individually singular terms in (2) and (3), each of these expressions is non-singular and depends on the crack spacings h and d, and on the crack half- length. In fact, it is easy to show that as x->-x', K(x', x) -O0. Likewise, as h->ox and m = 0, K(x', x) -+0, and we recover the singular integral equation for an isolated relaxed crack.

3. METHOD OF SOLUTION

In the absence of a possible closed-form solution to the singular integral equation (1), two methods have been proposed for its numerical integration. The first method - a general Gauss-Jacobi formula (Erdogan & Gupta I97z; Krenk I975) -

approximates the principal part of (1) by p point interpolation as if f(x') were a



378 B. L. Karihaloo

polynomial of degree 2p, while the secondary part is approximated as if K(x', x)f (x') were a polynomial of degree (2p - 1) in x'. This method, besides being more convenient to programme on a computer, makes an explicit extraction of the singularity unnecessary.

The second method, which will be used here, is based on an expansion of the non-singular part of the kernel K(x', x) in a series of orthogonal polynomials. This method, although not as convenient as the first one, is preferred for two reasons. First, by making an explicit extraction of the singularity necessary, it allows us to recover the solution for an isolated relaxed crack. Indeed, as will be clear later, the singular solution of (1) is similar to that for an isolated crack. The second, and possibly more important, reason from a mathematical point of view is that the function P(x) in (1) is piece-wise continuous with a sharp discontinuity at the tips of the cracks IxI = a. In fact, it is because of this discontinuity that f(x) is singular at Jxl = a. In order to use the Gauss-Jacobi quadrature we would have to considerably modify the method as it now stands by making one of the p interpolation points coincide with lxl = a. It is not unlikely that its advantage over the second method would then be totally nullified.

The numerical integration method to be used here was first proposed by Erdogan (I969). In its original form it is applicable only to unrelaxed cracks, where the singularity in f (x') occurs at the ends of the interval. This method, adapted to the solution of relaxed crack problems, is outlined in (Karihaloo 1977 a), and only the most essential formulae are reproduced here.

The crux of the method lies in expanding the non-singular part of the kernel K(x', x) in a series of Chebyshev polynomials in the variable x, the coefficients of the series being functions of x'. Chebyshev polynomials are chosen because their orthogonality properties considerably simplify the algebra. With this expansion, and for given values of the crack spacings, h and d, and the externally applied stress or, the singular integral equation (1) may be rewritten as

?1 oo f{f (x')/(x -x')} dx + n a, Ti(x) + P(x) O, (5)

1 n==O

where an- f(x')An(x')dx', (6)

T.(x) is the nth Chebyshev polynomial of the first kind, and the coefficients An(x') are defined by the orthogonality properties of T,J(x).

For smooth closure of the tips of the plastic zones (x = + 1), the dislocation distribution function f(x) must vanish at these points. This gives the so-called consistency condition (Muskhelishvili 1953) for the solution of (5). This condition simplifies to

arecos (a) - o (o/o- + ao). (7)

Relation (7) specifies the distance to which the yield spreads under a given applied stress, oC. Note its similarity to the corresponding expression from an




isolated relaxed crack (Bilby et al. I963). However, it should be stressed that ao, the coefficient defined by (6), is a function of the applied stress, o, and the crack spacings, h and d. For a better appreciation of the change brought about by the presence of a doubly-periodic array of cracks, relation (7) is compared with that for an isolated crack, identified by the subscript 'iso', whereupon

ljso - cos ('it) ao[ -tan (t1n) (o-/oy) tan (Q T) a0]. (8)

It is clear that the coefficient ao plays a very important role in determining the sign of perturbation due to a doubly-periodic array of cracks.

Having satisfied the consistency condition (7), the solution of the singular integral equation (5), which can be shown to be an odd function, may be directly written (Muskhelishvili 1953) in terms of the odd Chebyshev polynomials of the second kind, U2,1,

f(x) =(x)-[V(l-X2)/7g] E bnU2.-1(X) (9) n=1

where bn= f(x')A2n(X')dx', (10)

and y(x) is identical in form to the solution for an isolated relaxed crack (Bilby et al. I963), except that ac is given by (7).

However, we have yet to determine the coefficients bn. To do so, we substitute (9) into (10) and arrive at an infinite system of linear algebraic equations in the coefficients bn.

Having solved this system of equations for given values of o, h and d and, hence, found the functionf (x), the coefficient ao appearing in (7) may be evaluated from (6).

In order to study the fracture behaviour of a solid containing an array of cracks we need to know the crack tip opening displacement (c.t.o.d.) necessary to accom- modate the dislocations between the crack tip, lxl = a, and the end of the plastic zone, IxI = 1. The number of dislocations is found by integrating the corresponding dislocation distribution functionf (x) (expression (9)) between the limits a <, I < 1. Having performed the necessary integrations, we get the following expression for non-dimensional c.t.o.d., A *(Q),

I= ln (1 /a)-(/2a) E bn J (1jx2) U2n-i(x) dx, (11) n=l J

where oc- c/a is specified by (7). A *(a) has been normalized by 4co(y K/ICt/, where the constant K takes the values 1 and (1 - v) under anti-plane and plane strain conditions, respectively. The first term on the right hand side is identical in form to that for an isolated crack.

4. RESULTS AND DISCUSSION

The coefficients bn were determined by solving the above-mentioned system of linear simultaneous equations for given values of the non-dimensional parameters c, h and d. The infinite system of equations was truncated at i = j = 12. This assured



380 B. L. Karihaloo

sufficient accuracy for all the cases treated. (The accuracy of the present technique was judged against known closed-form solutions (e.g. a row of collinear cracks, see Bilby et al. I964). The absolute maximum deviation was found to be within 5 ? The value of o-/crY1 was found in an inverse manner, whereby a value of a was assumed and o-/lo- evaluated from (7) at the end of the iterations after having computed the distribution function, f(x), and, hence, the coefficient ao appearing in (7).

Typical results for the extent of spread of plasticity, a, and the ratio of J *(c) to that for an isolated crack zi *(cis.) are presented graphically in figures 2-9. It is instructive to discuss the results for each loading mode and array configuration separately. However, for the sake of brevity mode 3 results, being qualitatively very similar to mode 1, are not included.

The choice of the values of the dimensionless distance of horizontal separation, d, needs some amplification. The manner in which this distance has been non- dimensionalized precludes coalescence between the adjacent plastic regions in any row of cracks, if d > 2. This is true for the rectangular array configuration. How- ever, in the diamond-shaped array the plastic regions in the adjacent rows will overlap each other, if 2 < d < 4. The effect of this overlapping will become apparent later (??4.1.2, 4.2.2).

4. t1. Mode 1 4.1.1. Rectangular array of cracks

In figure 2 is shown the extent of spread of plasticity from each crack tip. It may be mentioned that the results for very large h (not shown in the figure) were in very good agreement with the available analytical solution (Bilby et al. I964), while those for very large d coincided with those obtained by Karihaloo (I 977 b).

It is evident from the figure that, for a given externally applied stress, o, the plastic regions spread to a lesser extent (at increases) as the rows of cracks come closer (h decreases). This trend is quite consistent with the results obtained by the boundary collocation procedure (figs. 16-19, Isida I972). To compare the results obtained by the present method with those due to Isida (I972), let us consider the following numerical values.

Let /O-/0 = 4, h 1, d = 5. From figure I b, cx = c/a = 0.93. Corresponding to -/0TY = 4, iso = COS (to-1/2o-y)=0.924. Moreover, d = dl/a gives the dimensioned

distance of horizontal separation d1 = dc/el = 5.38c, which corresponds to i/b = 0. 372 in Isida's notation. Also, h = hll/tc means that the dimensioned distance of vertical separation h1 = 3.14c. In other words, h1/d1 (or c/b in Isida's notation) = 0.584. The length of plastic zone at each crack tip, s, is easily calculated, whereupon s/sis, (or s/&8. in Isida's notation) works out to be 0.915. The corresponding value of s/sq. due to Isida, obtained by interpolating the values for h1l/d = 0.7 and 0.5 (figs 18 and 19, Isida I 972), is approximately 0.9 which is close to the value predicted by the present technique.

To bring out the influence of both h and d on the c.t.o.d., figure 3 is drawn. One




of the many striking features of the figure is that the curves for various values of d tend to merge into the same curve as h gets smaller. In other words, A *(c)/ZI *(cis0)

is independent of the horizontal distance of crack separation for small values of the distance of vertical separation, h, and depends only on the latter. It is, thus, reasonable to conclude that the result for a single stack of cracks (Karihaloo

I977b, c) holds for all values of d as h 0. This trend is quite consistent with the results of Delameter et al. (I974).

(a) (b)

c/a h\\/I h \1/\r

LL1 d=5 X1 5 ~~~~~~~~~~5 0.5-

dzz2.5 dzz5

0 0.5 I/c 10 0.5 G-/oY 1

FIGURE 2. The extent of spread of plasticity, a, from a rectangular array of cracks in an infinite solid as a function of the applied stress o,,, = o- and the non-dimensional distance h for two selected values of distance d. (a) d = 2.5; (b) d = 5.

1 / "1

*

/~~~G CSy= -4 /C/G-Y=2

O 2.5 h 5 O 2.5 h 5

FIGURE 3. The ratio J *(c)/J *(c.,O) as a function of non-dimensional distances h and d for two selected values of o,, = o acting on an infinite solid containing a rectangular array of cracks. (a) o/o, = ; (b) o/o = , d = oo corresponds to a stack of relaxed cracks.



382 B. L. Karihaloo

The other most interesting conclusion which follows from an inspection of figure 3 a and b is that there exists for each value of d a corresponding value of h where the ratio a *(c)/a *(ciSO) is equal to unity. This is not surprising because the interactive effects between the cracks in a single stack and the cracks in a single row counter each other. Indeed, the interaction between the cracks in a single stack (identified by the index 'st') is such that A *(c,t)/A *(ciso) < 1.0 (Karihaloo I977b), but that between the collinear cracks in a single row (identified by the index 'row') is such that A*(crOw )/A*(cio) > 1.0 (Bilby et al. I964). From the figure it is easy to establish a graphical relationship between' d and h for several constant values of a *(c)/a *(ciSo). In particular, it is seen that the ratio A *(c)/A *(cisJ) is equal to unity when the distance separating the adjacent plastic zone tips in the same row, (d - 2), is much smaller than the distance h separating adjacent plastic zones in the same stack. Thus the influence of the stack, which normally tends to diminish a *(c)/a *(ciso), dominates the influence of the row, which normally tends to magnify this ratio. Moreover, if it is assumed that fracture occurs when the c.t.o.d. achieves a value which is greater than or equal to an experimentally determined critical value, then an infinite body containing a rectangular array of cracks and subjected to a given external stress, o, for which the point (h, d) lies on or below the asymptote a *(c)/A *(cis.) = 1 (see figure 3), will be of the same strength or stronger (i.e. will fracture at a higher level of applied stress) than a similarly loaded body containing only a single crack of the same length. This conclusion is consistent with the results obtained by Isida (1972, fig. 20).

4.1.2. Diamond-shaped array of cracks -

As in the previous case, the extent of spread of plasticity from each crack tip is shown in figure 4. As mentioned above, d = 2.5 in the present array configuration means that plastic zones in the adjacent rows will overlap with each other, whereas d = 5 precludes any such overlapping.

As with a rectangular array of cracks, with an increase in o, the rate of spread of plasticity decreases as the distance of vertical separation between the cracks decreases. However, in contrast to the rectangular array of cracks, the influence of the distance of horizontal separation on the rate of spread of plasticity is far stronger, the rate of spread of plasticity increasing with an increase in d as can be judged by comparing figure 4a and b.

Figure 5 gives a better understanding of the combined influence of h and d on the c.t.o.d. In the absence of overlapping of the plastic zones in the adjacent rows (d > 4), the curves corresponding to a *(c)/a *(ciso) for fixed values of d merge with the curve for a single stack (d = so) as h -o 0, as was also true for the rectangular array. However, in contrast to the latter, the ratio a *(c)/a *(clso) decreases with a decrease in d, at least for values of h < 5. When there is an overlapping of the plastic zones in the adjacent rows (d < 4), separate stacks of closely spaced cracks are not formed when h -o 0, and the values of a *(c)/A *(ciso) are quite different for small h, as can be seen from figure 5 a and b. In general, overlapping of the plastic zones




tends to decrease the value of J *(c)/l *(ciso) and, hence, to strengthen the solid in mode 1 deformation. As was also true for the rectangular array, there exists for each value of d a corresponding value of h where the ratio A *(c)/l *(cis0) is equal to unity.

1 (a) (b)

c/a h=l h4/w

2.5 ~~~~~~~~~~~~1

0.5- 5 2.5

d=2.5 d=5 5

0 0.5 ar/o; 1 0 0.5 or/oY 1

FIGURE 4. The extent of spread of plasticity, a, from a diamond-shaped array of cracks in an infinite solid as a function of the applied stress o-,, = o- and the non-dimensional distance h for two selected values of d. (a) d 2.5 (overlapping plastic regions in adjacent rows); (b) d = 5 (no overlapping).

(a) / |(by

/ci=2.5 /~~~~~~~~~_-. 3.5~~~~~~~6

0 2.5 h 5 0 2.5 h 5

FIGURE 5. The ratio A*(c)/A*(c180) as a function of non-dimensional distances h and d for two selected values of o-,, = 0- acting on an infinite solid containing a diamond-shaped array of cracks. (a) ul/o,, = i; (b) ./.,y = .



384 B. L. Karihaloo

4.2. Mode 2

4.2.1. Rectangular array of cracks

Let us consider now the mode 2 deformation state. As before, figure 6 shows the variation in the extent of spread of plasticity as a function of the externally applied stress, o.

The curves corresponding to h oc refer to a collinear row of cracks. As can be

(a) (b)

c/a

0.5 -1/Tr 1/Tt

d-z2.5 d-5

0 0.5 or/Y 1 0 0.5 o/roy 1

FIGURE 6. The extent of spread of plasticity, a, from a rectangular array of cracks in an infinite, solid as a function of the applied stress o- = o and the non-dimensional distance h for two selected values of d. (a) d = 2.5; (b) d = 5. h = -o corresponds to a row of collinear relaxed cracks.

(a) (b)

3- \ S~~0/0Y- 4, /OY -343

d002.5 3.5 5

1 IT I _____J_~~~~ ~~ II

0 2.5 h 5 0 2.5 h 5

FIGURE 7. The ratio z *(c)/A *(clO) as a function of non-dimensional distances h and d for two selected values of o -,= o. (a) o/oi, =; (b) /o, = . The infinite solid contains a rectangular array of cracks. d _ so corresponds to a stack of relaxed cracks.




seen from the figure, the rate of spread of plasticity increases with a decrease in h and an increase in d. This is just the opposite of what is observed in mode 1 deformation with a similar configuration of cracks.

It is known that for both a single row (Bilby et al. I964) and a single stack (Karihaloo 1977a) of cracks in mode 2, the cracks interact such that a *(c)/Z *(c1,O) increases with decreasing crack spacing, and therefore it would be expected that J *(c)/A *(ciso) increase monotonically with increasing crack density. From figure 7 a and b, in which is shown the ratio a *(c)/J *(ci80) as a function of h and d, it may

(a) (b)

cla

h~~~~~~5~- 0.5 1//Tr

dzz 2.5d5

0 0.5 o-/ocr 1 0 0.5 or/ay 1

FIGURE 8. The extent of spread of plasticity, a, from a diamond-shaped array of cracks in an infinite solid as a function of the applied stress o-,, = o- and the non-dimensional distance h for two selected values of distance d. (a) d = 2.5 (overlapping plastic regions in adjacent rows); (b) d = 5 (no overlapping).

3 - (a) (b)

* ~~~~~dm2.5

d-z2.5

5 ~~~3.5 003.5

5

, ~ ~ ~~~ ~ ~ ~ ~~~~~~I l I I I 0 2.5 h 5 0 2.5 h 5

FIGURE 9. The ratio z*(c)/a*(cq80) as a function of non-dimensional distances h and d for two selected values of ovX, = oC acting on an infinite solid containing a diamond-shaped array of cracks. (a) cr/ocy- 1; (b) o-/o_ = .



386 B. L. Karihaloo

be seen that this is indeed true for the rectangular array; A *(c)/l *(ci,o) tends to increase as both h and d decrease, and as h get small, the curves merge and tend to infinity along the same curve as for a single stack (Karihaloo 1977 a).

It may also be mentioned that, for small values of d and o/-o-, the value of A *(c)/A *(ci,O) is practically independent of the distance of vertical separation greater than or equal to 1.25 and tends to approach the value for a single row of cracks.

From the above discussion it may be concluded that a body containing a rectangular array of cracks and deforming in mode 2 will be weaker (i.e. will fracture at a lower level of applied stress) than a similar body containing only an isolated crack of the same length. This is exactly the opposite of what was observed in mode 1.

4.2.2. Diamond-shaped array of cracks

The picture is far more complex for a body containing a diamond-shaped array of cracks and deforming in mode 2. This is especially so for smaller values of d when the plastic regions in the adjacent rows overlap, as is evident from figure 8a. Although the general trend seems to resemble that for the corresponding case of the rectangular array, it is clear that there are (small) values of h when the rate of spread of plasticity actually decreases with a decrease in h. The same trend is also visible, albeit not as pronounced, when the selected value of d precludes any overlapping (figure 8b).

It is not surprising, therefore, that there are regions where the value of zi *(c)/J *(c1,0) actually diminishes with decreasing crack spacing (figure 9), although the general trend is for A *(c)/l *(ci,.) to increase with increasing crack density. Indeed, this ratio can be less than unity, indicating that the body is less susceptible to fracture than a similar body containing a single crack of the same length. This is in direct contrast to a body containing a rectangular array, when the value of a *(c)/4 *(ciso) is greater than unity for any combination of h and d. However, it should be mentioned that for small values of o/o-y, z *(c)/zI *(ci,0) does tend to approach the value for a single stack of cracks (d = oo) as h--?0 and that for a single row of cracks when h is very large.

In conclusion it may be said that the present work brings out clearly the strong influence of crack geometry and the far-field state of stress on the fracture characteristics of solids. More importantly, the effect is not uniformly adverse or otherwise but rather strongly dependent on the far-field state of stress. Moreover, it appears that the influence of the vertical distance of crack separation is far more pronounced than that due to the horizontal separation in the more frequently met mode 1 conditions.



Fracture of golidq tvith arrays of cracks 387

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Fracture Characteristics of Solids Containing Doubly-Periodic Arrays of Cracks