On elastic compliances of irregularly shaped cracks

Transcription

1 International Journal of Fracture 114: , Kluwer Academic Publishers. Printed in the Netherlands. On elastic compliances of irregularly shaped cracks I. SEVOSTIANOV 1 and M. KACHANOV 2 1 Department of Mechanical Engineering, New Mexico State University, Las Cruces, NM 88003, U.S.A. 2 Department of Mechanical Engineering, Tufts University Medford, MA 02155, U.S.A. Received 22 June 2001; accepted in revised form 7 January 2001 Abstract. Several methods are suggested to estimate compliances of irregularly shaped cracks quantities that determine the increase in the overall compliance of a solid due to introduction of such a crack. Besides, the compliance of an annular crack is given a result that may be of interest on its own. Key words: Crack, compliance, effective properties, irregular. 1. Introduction The problem of effective elastic properties of cracked solids has been addressed in literature under the assumption that the cracks have simple shapes circular or elliptical (see the pioneering work of Bristow, 1960, for penny-shaped cracks; Budiansky and O Connell, 1976, for the elliptical cracks and Kachanov, 1993, for a review). However, in many applications (rocks, sprayed coatings, etc), cracks, while being approximately planar, may have irregular in-plane shapes. To estimate the effective elastic properties in such cases, one needs to estimate the contribution of one crack into the overall compliance (crack compliance). To our knowledge, no systematic methodology that addresses this problem has been developed. A step in this direction is made in the present work. We start with general observations on compliances of planar cracks (notably, on the existence of the principal compliance axes for a crack of an arbitrary shape). Then we suggest several methods to approximately estimate the compliances of irregularly shaped cracks. They are based on utilizing results for several comparison shapes circular, elliptical and annular (the compliance of the annular crack is derived in the course of this work a result that may be of interest on its own). The discussion is illustrated by examples. 2. General relations for crack compliances. Principal axes of the crack shape For a material with pores, the key quantity for finding the effective properties is the compliance contribution tensor H of a single pore, such that the strain (per certain reference volume V ) due to the pore is H ij kl σ kl where σ kl is the applied stress (H tensors for pores of various shapes were calculated by Kachanov et al., 1994). For a planar crack with unit normal n, this tensor has the following structure: H 1 ( V nbn in components, H ij kl = 1 ) V n ib jk n l, (2.1) where the appropriate symmetrization of H ij kl (with respect to i j, k l, and, as implied by the existence of the elastic potential, with respect to ij kl) is imposed. Second rank

2 246 I. Sevostianov and M. Kachanov tensor B can he called COD (crack opening displacement) tensor of a crack (Kachanov, 1992): it relates the average crack opening displacement (displacement discontinuity) vector to vector t of uniform traction applied at the crack faces: u + u =B t/s (in components, u + i u i =B ij t j /S), (2.2) where S is the crack area. Thus, component B nn is the normal crack compliance and B ss (where s is some unit vector in the crack plane) is the shear crack compliance in the direction s. Tensor B is symmetric (as follows from applying Betti s reciprocity theorem to loadings of a crack by uniform tractions t of different directions). Therefore, it has the principal representation in terms of three mutually orthogonal unit vectors n, s and t: B = B nn nn + B ss ss + B tt tt. (2.3) In the case of the isotropic matrix, n is a normal to the crack and s and t lie in the crack plane; B nn, B ss and B tt are crack compliances in the corresponding directions. Representation (2.3) implies that, in the case of the isotropic matrix, an arbitrary crack shape possesses two principal axes in the crack plane, s and t, such that a traction applied in either of these two directions generates the average displacement discontinuity vector u + u that is collinear to it (B st = 0). These axes can be called the principal axes of the crack compliance. For an irregularly shaped crack, the existence of such axes may not be intuitively obvious. Remark 1. In the case of an anisotropic matrix, principal representation (2.3) is still valid, but the principal vectors n, s and t are not necessarily normal/parallel to the crack. Remark 2. It may seem that symmetry of B implies symmetry of H ij kl = n i B jk n l with respect to j k, so that, with the account of the usual symmetrization of H ij kl (with respect to i j, k l, ij kl), tensor H is fully symmetric (with respect to all rearrangements of indices i, j, k, 1). However, this is incorrect: imposition of the mentioned symmetrization makes H ij kl non-symmetric with respect to j k. 3. Circular (penny-shaped) and elliptical cracks 3.1. CIRCULAR CRACK For a circular crack of radius a in the isotropic matrix (with Young s modulus E and Poisson s ratio ν) [ nn + B = 16 ( 1 nu 2) a 3 3E ] 1 (I nn), (3.1) 1 ν/2 where coefficients at nn and I-nnrepresent crack s normal, B nn and shear B tt, compliances. They are relatively close (note that in the 2-D case of a rectilinear crack they are equal) ELLIPTICAL CRACK Unit vectors s and t along semiaxes a and b are, obviously, the principal axes of the crack compliance. Utilizing calculations of Budiansky and O Connell (1976) based on Eshelby s results for an ellipsoidal inclusion, B-tensor is found as follows:

3 On elastic compliances of irregularly shaped cracks 247 [ B = ξ nn + η + ζ (I nn) + η ζ ] (ss tt) (3.2) 2ξ 2ξ with ξ = B nn = 8 ( 1 ν 2) πab 2 3E E(k) = 32 1 ν 2 S 2 3π E P, (3.3) where P = 4aE(k) and S = πab are ellipse s perimeter and area. Coefficients η, ζ are obtained from ξ by changing E(k) to Q and R, respectively, given by the following formulas: Q(k, ν) = k [( 2 k 2 + ν νk 2) E(k) ν(1 k 2 )K(k) ], R(k,ν) = k [( 2 k 2 ν ) E(k) + ν(1 k 2 )K(k) ] (3.4), where E(k) and K(k) are the complete elliptic integrals of the first and the second kinds of argument k =[1 (b/a)] 1/2. The first two terms in (3.2) are similar to B for the circular crack. The third term vanishes for the circular crack (and, more generally, for the isotropic shapes, like all the equilateral polygons, excluding the square). Remark. As follows from (3.3), all elliptical cracks (and, in particular, the circular one) with the same ratio S 2 /P have the same normal compliance B nn (although the shear compliances B ss and B tt will, obviously, differ) REPLACEMENT OF ELLIPTICAL CRACKS BY EQUIVALENT CIRCULAR CRACKS In the case of many elliptical cracks, having random orientational distribution of eccentricities, their effect on the overall compliances can be obtained, with good accuracy, by replacing them by equivalent circular cracks, as follows. We define the equivalent circular crack as having the same ratio S 2 /P as the elliptical crack, i.e. as having radius r = ( 2S 2 /πp ) 1/3. Thus defined equivalent circular crack has the same normal compliance B nn as the elliptical one. A simple calculation shows that its shear compliance is quite close to the average, over the in-plane directions, shear compliance of the elliptical crack. (The difference between the mentioned average and the shear compliance of the circular crack increases very slowly with ellipse s eccentricity, from 0.7% for a/b = 2to2.5%fora/b = 6 and to 3.2% for a/b = 58.) Therefore, for the purpose of finding the effective elastic properties of a solid with any orientational distribution of elliptical cracks random (overall isotropy) or non-random (overall anisotropy) these cracks can be replaced by the equivalent circular ones (as long as the ellipses eccentricities are randomly oriented). Remark. The expression for ellipse s perimeter in terms of the elliptical integral can be well approximated (for all eccentricities) by an elementary function: P π(a + b)(64 3κ 4 )/(64 16κ 2 ) where κ = (a b)/(a + b). 4. Annular crack We consider an annular crack bounded by two concentric circles, of radii a and a c (Figure 1). This configuration (discussed in greater detail by Sevostianov and Kachanov, 2001) is relevant for the cases when crack faces are in contact along certain areas ( islands ). We estimate normal compliance B nn of the annular crack by:

4 248 I. Sevostianov and M. Kachanov Figure 1. Annular crack and SIF at the outer edge.

5 On elastic compliances of irregularly shaped cracks 249 using the general relation between the elastic compliances of cracked solids and stress intensity factors (SIFs) along crack edges; utilizing the available result for the SIF at the outer edge of the annular crack. The SIF at the outer edge of the annular crack (Figure 1), was given in the numerical form (with claimed accuracy of 1%) by Smetanin (1968) (See also Rooke and Cartwright, 1976). It can be approximated by the following expression (where λ = c/a): K I = σ 33 πa 2 ( 2 2 π λ λ + λ(1 + λ) ). (4.1) We now utilize the general relation between SIFs and the effect of cracks on the overall elastic compliances derived by Rice (1975). For a reference volume V containing one or several cracks that may propagate, the increment ds ij kl of the overall compliance due to incremental advance dl on all propagating crack fronts, collectively denoted by L, can be represented in the form: ) dl, (4.2) ds ij kl = 1 ( 1 K q K r c qr dl V 4 L σ ij σ kl where coefficients c qr relate the near tip displacement discontinuity to SIFs: [u i ]=c ij K j r/2π. (4.3) In the case of the isotropic matrix, there is no coupling between mode I and modes II and III, so that, for the normal displacement discontinuity [u 3 ], the only relevant coefficient is c 31 ; for the (locally) plane strain conditions, c 31 = 8 ( 1 ν 2) /E. SinceK I depends only on component σ 33 of the applied stress, we have ( ds 3333 = d 1 ) = 1 2 ( 1 ν 2) [ ( ) 2 K1 dl] dl. (4.4) E 3 V E L σ 33 and, since n = e 3,wehavedS 3333 = (1/V)B nn. In our case of the annular crack, we treat this crack as having grown from an infinitesimally thin circular line outwards (with a circular island remaining in the final configuration). Then L is the crack s outer edge and the integrand is a purely geometrical factor obtained from (4.1). Hence, calculation of the compliance change due to a certain amount of crack growth and of the normal crack compliance B nn reduces to integration of the mentioned factor over the area of the advance: B nn = 1 V 2π(1 ν 2 ) E a 3 ( λ π 1 ) 2 ( ln(1 λ) + λ2 (2 λ) + 2λ3 2 2 ) 2 π 1. (4.5) Figure 2a shows B nn in terms of the relative area of the island, Figure 2b gives radius R eff of the equivalent circular crack that has the same normal compliance B nn. 5. Rigorous bounds for normal compliance B nn of an irregularly shaped crack We remark that B nn can be interpreted as the volume of a crack loaded by uniform normal traction of unit intensity. Therefore, bounding B nn may he visualized as bounding this volume.

6 250 I. Sevostianov and M. Kachanov Figure 2. (a) compliance of the annular crack as a function of the relative area of the island; (b) radius of the equivalent circular crack.

7 On elastic compliances of irregularly shaped cracks 251 Figure 3. Bounding of the compliance of a crack with slightly jagged contour by the ones of the inscribed/circumscribed elliptical cracks. Any crack that is inscribed provides an obvious lower bound for B nn ; similarly, any crack that is circumscribed provides an obvious upper bound. Albeit trivial, these bounds have a useful implication: if the crack contour is slightly jagged (Figure 3), this factor can be ignored and the contour can be smoothed, without a significant effect on the crack compliance B nn (we emphasize that the contour should be jagged only slightly, for the smoothing not to generate substantial errors, since B nn of the inscribed and circumscribed cracks is proportional to the third power of their linear dimensions). In order to actually construct bounds from the inscribed and circumscribed crack shapes, one has to know B nn for these shapes. However, since the database of shapes for which B nn is known is presently limited to the elliptical and annular ones, these bounds may be quite wide if the considered crack has a strongly non-elliptical shape. Somewhat tighter rigorous bounds can be constructed as follows. We first identify the best inscribed and circumscribed cracks of the elliptical shape. Then we fill the gap between the inscribed crack and the actual one by smaller elliptical cracks (for the best results, choosing a fewer number of larger cracks wherever possible). The sum of B nn of all the inscribed cracks constitutes the obvious lower bound. Similarly, we fill the gap between the circumscribed crack and the actual one by smaller elliptical cracks. The difference between B nn of the main circumscribed crack and the sum of B nn of the smaller cracks filling the gap constitutes the upper bound. In both bounds, we take volumes of smaller cracks filling the gaps by treating them as isolated ones, so that these volumes could be easily found. Since the coplanar interactions are of the amplifying nature, their neglection does not violate the upper/lower character of the bounds. Note that a similar procedure was used by Huet et al. (1991) for estimation of the effective properties of a material with inclusions. Although these bounds are somewhat tighter than the above mentioned trivial bounds, the improvement due to small cracks filling the gaps may be only slight-to-moderate. This is best visualized in terms of the volume of the crack under the uniform pressure. Figure 4 provides a schematic illustration showing that the total volume of a number of inscribed cracks is substantially smaller than the volume of the actual crack. This procedures can be illustrated by the example of Figure 5. Compliance B nn of the irregularly shaped crack (shaded area) can be bounded as follows: B III nn + BIV nn + BV nn <B nn <B ext nn ( B I nn + BII nn ) (5.1) where Bnn I,..., BV nn are compliances of the elliptical (or circular) cracks shown. Compliance Bnn ext of the exterior circumscribed elliptical crack is the trivial upper bound. Subtraction in the right-hand part of (5.1) is the improvement due to subtraction of two smaller elliptical cracks

8 252 I. Sevostianov and M. Kachanov Figure 4. A substantial underestimation of the crack volume by a sum of inscribed crack volumes. Figure 5. An irregular crack (shaded area), the circumscribed ellipse, smaller elliptical cracks filling the gap (I and II) and the inscribed cracks (III, IV and V ). (I and II) that fill the gaps. For the particular shape shown in Figure 5, inequality (5.1) takes the form: 3E 3.10 < 16(1 ν 2 ) B nn < (5.2) The bounds are relatively wide. It is seen that the subtraction in the right-hand part provides only a 12% improvement. 6. Approximate estimates of B nn We suggest several estimates (called estimates to distinguish them from rigorous bounds) of crack compliance B nn that appear to be reasonable on the physical grounds, but cannot be rigorously proved by the present authors GENERAL ESTIMATES (A) We hypothesize that the compliance of a circular crack of the same area as the considered irregular one provides an upper estimate. In other words, for all cracks of a given area, the crack volume under uniform normal traction is maximal for the circular shape.

9 On elastic compliances of irregularly shaped cracks 253 Figure 6. Perturbed circular crack. (B) We hypothesize that a lower estimate of B nn is provided by B nn of a circular crack having the same ratio S 2 /P as the considered irregular crack. The underlying idea can be clarified on the following example. As mentioned above, all the elliptical cracks with the same S 2 /P have the same compliance B nn. Consider now an irregular crack obtained by a perturbation of the elliptical one that leaves the area unchanged. The irregularity of the shape generally leads to an increase in P and thus to a reduction in S 2 /P.Theelliptical crack having the reduced S 2 /P will be smaller than the original one and will have a lower B nn. This reduction is hypothesized to be sufficient to constitute a lower estimate for the irregular crack. In fact, if the perturbed ellipse has a sufficiently wavy contour, the increase in P is even sufficient for the reduced ellipse to become the inscribed shape - and thus to provide a rigorous lower bound. For a simple illustration, we consider a circular crack of radius a perturbed as shown in Figure 6 (with the area conserved). The perimeter increase is given by h 2 + l 2 /l ψ. For h/a = 0.25, if the ratio l/a exceeds 0.08, then a circle with the same S 2 /P becomes an inscribed figure and thus provides a rigorous lower bound. Estimates (A) and (B) can be further illustrated by the following two examples. Example 1. Square shape (with side a). Estimates (A) and (B) yield a relatively narrow interval: 0.16a 3 a3 2π < 3E 16(1 ν 2 ) B nn < a3 π 3/2 0.18a3 (6.1) Example 2. Sectorial shape (Figure 7). Estimates (A) and (B) provide a narrow interval for the plotted range 0 ϕ π/2: ϕ 2 2π(ϕ + 2) R3 < It is seen that in both examples B (A) nn 3E ( ϕ ) 3/2 16(1 ν 2 ) B nn < R 3. (6.2) 2π >B(B) nn and that these two estimates are relatively close.

10 254 I. Sevostianov and M. Kachanov Figure 7. Bounds (A) and (B) for the sectorial crack CRACK SHAPES THAT CONTAIN AN APPROXIMATELY ELLIPTICAL DOMINANT SHAPE We now consider a special class of crack shapes the ones that consist of a dominant subshape, that approximately resembles an ellipse, plus some attachments (in the case of several approximately elliptical sub-shapes of roughly the same size, either of them can be treated as the dominant one). Figures 8 and 9 provide examples. For the shapes of this kind, we suggest the following lower and upper estimates, that may be tighter than the general estimates (A) and (B). (C) For the lower estimate, we construct several, possibly intersecting, elliptical shapes, one of them approximating the dominant sub-shape (as in Figures 8 and 9), whose envelope constitutes an inscribed shape. The envelope s compliance is difficult to find, but its lower estimate the lower bound of its volume under uniform pressure can be given as follows. We take the volume of the largest elliptical crack and add the volumes of the remaining intersecting cracks, with the parts of the latter volumes corresponding to the intersection area subtracted. Clearly, such a sum provides a lower estimate of the actual volume of the original crack it somewhat underestimates the volume corresponding to the intersection areas. It is also clear, geometrically, that such an underestimation will, typically, be small. The drawback of this construction is that it involves some calculations related to subtraction of the volumes located above the intersection area (shaded area in Figure 9). (D) For the upper estimate of B nn we enlarge the dominant elliptical shape, as to reflect the presence of other attachments. We suggest the enlargement that retains the aspect ratio of the dominant elliptical shape and increases its area to the area of the entire irregular crack. For an example, we consider the crack shape shown in Figure 8 (ellipse with two attached parts). Estimates (C) and (D) yield

11 On elastic compliances of irregularly shaped cracks 255 Figure 8. Example of a shape possessing a dominant elliptical sub-shape Figure 9. Irregularly shaped crack formed by two ellipses intersecting at angle ϕ. 3E 16(1 ν 2 ) BC nn = 3E 23.6a2 ; 16(1 ν 2 ) BD nn = 25.9a2. (6.3) They are narrower than the general estimates (A) and (B): 3E 16(1 ν 2 ) BA nn = 31.5a2 ; 3E 16(1 ν 2 ) BB nn = 22.5a2. (6.4) 6.3. COMPARISON OF THE ESTIMATES WITH NUMERICAL RESULTS. We evaluated all the suggested estimates (A) (D) on the configuration of Figure 9, that covers a range of irregular geometries obtained by varying angle ϕ. This configuration was studied numerically by FEM and the numerical results are compared with the estimates (A) (D) in Figure 10. It is seen that estimate (C) yields results that are very close to the numerically obtained ones, and that estimates (B) and (D) are also satisfactory.

12 256 I. Sevostianov and M. Kachanov Figure 10. Comparison of estimates (A) (D) with numerical results (line 1). 7. On the shear crack compliances B tt and the effective elastic properties of solids with multiple irregularly shaped cracks For the elliptical crack, the average, over the in-plane directions shear compliance is very close to the one of the equivalent circular crack, with radius chosen in such a way as to match the normal compliance of the elliptical crack (Kachanov, 1993). Therefore, the effective compliances of a solid with multiple elliptical cracks of any orientational distribution (random, parallel, parallel with scatter, etc.) are very close to the ones of solid with equivalent circular cracks, provided the ellipses eccentricities are random (do not have any preferential orientations). One can reasonably hypothesize that, in the case of multiple cracks of arbitrary irregular shapes, the overall compliances will also be close to the ones of a solid with circular cracks, with radii chosen as to match the normal compliances B nn of the irregular cracks (provided the shape irregularities of different cracks are randomly oriented). This fact was confirmed, to some extent, by the examination of geophysical data related to wavespeeds in rocks done by Sayers and Kachanov (1995): the observed wavespeed patterns seemed to correspond to the assumption that, on average, the shear compliances of cracks were indeed close to the normal ones. 8. Conclusions Several relatively simple estimates for the crack compliances quantities that determine the contribution of a crack into the overall compliance of a solid are constructed for irregularly shaped cracks.

13 On elastic compliances of irregularly shaped cracks 257 We first discuss the trivial bounds for the crack, based on the inscribed and circumscribed comparison shapes (for which the compliances are known). Since, however, such comparison shapes are limited to the elliptical ones, the bounds obtained this way may be wide if the considered crack has a substantially non-elliptical shape. Some tightening of these bounds may be achieved by filling the gaps between the considered crack and the inscribed/circumscribed shapes with smaller elliptical cracks. However, the improvement is, typically, only moderate. Several lower and upper estimates of the crack compliance that are of non-rigorous nature and that are substantially tighter are suggested. Two of them apply to the crack shapes that contain a dominant sub-shape of roughly elliptical shape. The suggested estimates are compared with numerical results obtained by FEM for a range of irregular crack geometries. Based on this comparison, the estimates appear to give sufficiently accurate values of the crack compliance. Other results given in the present work, that may be of interest of their own, are the expression for the compliance of the annular crack and establishing the fact that any irregular crack shape in the isotropic matrix possesses two principal axes. References Bristow, J.R. (1960). Microcracks, and the static and dynamic elastic constants of annealed heavily cold-worked metals. British Journal of Applied Physics 11, Budiansky, B. and O Connell R. (1976). Elastic moduli of a cracked solid. International Journal of Solids and Structures 12, Huet, C., Navi, P. and Roelfstra, P.E. (1991). A homogenization technique based on Hill s modification theorem. In: Continuum Models and Discrete Systems (edited by Maugin, G.), vol. 2, Kachanov, M. (1992). Effective elastic properties of cracked solids: critical review of some basic concepts. Applied Mechanics Review 45, Kachanov, M. (1993). Elastic solids with many cracks and related problems, In: Advances in Applied Mechanics (Edited by Hutchinson, J. and Wu, T.), Academic Press, New York, Kachanov, M., Tsukrov, I. and Shafiro, B. (1994). Effective moduli of solids with cavities of various shapes. Applied Mechanic Reviews 47, S151 S174. Rice, J. (1975). Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanisms. In: Constitutive Equations in Plasticity (edited by Argon, A.) MIT Press, New York, Rooke, D.P. and Cartwright, D.J. (1976). Compendium of Stress Intensity Factors. Her Majesty Stationery Office, London. Sayers, C. and Kachanov, M. (1995). Microcrack-induced elastic wave anisotropy of brittle rock. Journal of Geophysical Research 100, Sevostianov, I. and Kachanov, M. (2001). Elastic compliance of an annular crack. International Journal of Fracture 110, L51 L54. Smetanin, B.I. (1968). The problem of extension of an elastic space containing a plane annular slit. PMM (Translation of Applied Mathematics and Mechanics) 32,