Fracture dynamics of fusion of two anti-plane cracks

Transcription

1 Proc. Nati. Acad. Sci. USA Vol. 82, pp , April 1985 Applied Mathematical Sciences Fracture dynamics of fusion of two anti-plane cracks (stress redistribution/stress-intensity factor/faulting) A. K. CHATTERJEE AND L. KNOPOFF Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA Contributed by L. Knopoff, October 22, 1984 ABSTRACT The problem of the fracture dynamics of two two-dimensional coplanar anti-plane shear cracks is considered. Both cracks extend under a critical stress-intensity fracture criterion. The two cracks are initiated at different locations and times. A solution developed by Kostrov [Kostrov, B. V. (1966) J. Appl. Math. Mech. 30, for isolated cracks is used to determine the tearing loci of the cracks up to the time at which the individual cracks begin to interact. In the interaction interval prior to the fusion of the two cracks, both the stresses and the fracture tip loci are determined sequentially by applying the solution to Abel's equation twice iteratively. This method can be used to solve problems of the fusion of any number of coplanar cracks. The problems of dynamic crack propagation in an elastic solid have been extensively studied with a view toward various engineering and geophysical applications. In the latter arena, the modeling of the earthquake source requires that the problems of dynamical crack propagation take into account a complex process of crack fusion. It has been suggested that the development of an extended crack can be modeled in terms of the occurrence of smaller independent cracks; in this model, the generation of a new crack in the medium occurs under conditions that do not depend on the dynamics of other contemporaneous dynamical cracks (1). This model may be valid, but it requires verification in view of the fact that the development of a crack changes the stress distribution in the medium so as to influence the future initiation of cracks and their rupture histories. For a single isolated plane crack imbedded in an elastic medium, the stress distribution near the crack tip varies as r-1/2 where r is the distance from the tip of the crack, if the crack terminates in a line and a transition zone at the edge is absent. We are aware that stresses that vary as r-1/2 are impossible to support in real materials, and for this reason transition zones around crack edges were proposed as a means of distributing the singular stresses (2, 3). However, we shall use the singular form for mathematical convenience. If the points of initiation of new cracks are remote from existing cracks, we can assume that the histories of these new cracks are independent of the older ones. But, in general, if cracks initiate under the condition that the external shear stress is equal to the breaking strength of the medium at the point of initiation, then the initiation and subsequent growth of new or additional cracks usually depend on the history of formation of and the stresses due to earlier fractures. In general, subsequent fractures cannot be assumed to be independent events. The analysis of the problem of general three-dimensional interactive ruptures is complex. In general, an earthquake fault consists of a distribution of cracks located in the neighborhood of some plane that we take to be the fault plane. The physical properties in the vicinity of the fault plane have a The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C solely to indicate this fact. nonuniform distribution. The medium in which the cracks are located is nonuniform and the cracks are themselves geometrically arranged three dimensionally. To simplify the mathematics we assume that the incipient or existing cracks are situated in a common plane and that the properties of the medium are those of elasticity, homogeneity, and isotropy except for the cracks themselves. The stresses are two-dimensional and anti-plane in nature-i.e., fracture takes place in the mode III configuration. Thus we avoid consideration of the geometrical features of the two- and three-dimensional problems of crack bifurcations, echeloning, and such. However, we do assume that the breaking strength and dynamical friction on the fault plane will have a nonuniform distribution. Except for certain special cases such as the problems of self-similar cracks, the problem of the extension of a spontaneous isolated crack under the conditions cited above is difficult to solve because of the interactive nature of the two crack tips. One iterative solution to this problem has been given by Chatterjee and Knopoff (4). If cracks are triggered because some critical stress has been exceeded, then the assumption that only a single crack develops requires that nowhere outside the rupture zone of the crack can the critical stress be exceeded. This condition imposes some restrictions on the distribution of the nonhomogeneous physical properties on the fault plane. However in general, for a given distribution of the breaking strength along the fault plane, the following cases arise. (i) The crack encounters a region with such high rupture strength that it will stop. (ii) A propagating crack may trigger a second crack outside its own rupture zone; the two cracks do not merge. (iii) A propagating crack may trigger a second crack outside its own rupture zone and the two cracks merge dynamically. In the last case, fusion of the two cracks takes place to form a bigger crack; we can also consider examples in which the extension of a dynamic crack engulfs a preexisting stationary crack that lies in its path. In all these cases we can imagine that breakthrough of previously unbroken material will generate wave motions with high-frequency components. Simple fault models often do not provide an adequate treatment of the high-frequency problem (5, 6). A combination of all of the possibilities above may be necessary for the construction of a model that will generate a satisfactory description of the high-frequency spectrum in the theoretical seismogram (7, 8). In this paper, we present a method to solve the problem of the fusion of two dynamic cracks; the technique can be easily extended to the fusion of multiple cracks. This represents the completion of a missing link among the analytical tools that are needed to generate complex two-dimensional fault models and theoretical seismograms from such models. Formulation We consider the problem of the fracture dynamics of two two-dimensional coplanar anti-plane shear cracks that initi- 1869

2 1870 Applied Mathematical Sciences: Chatterjee and Knopoff Proc. NatL Acad Sci. USA 82 (1985) ate at different locations and different times. The cracks lie in the plane x = 0. The first crack initiates at 0 (y = 0, t = 0) while the second crack initiates at A (y = yo, t = to) (Fig. 1); both extend in the ±y directions. The displacement vector u = (0, 0, w), w = w(x, y, t) satisfies the wave equation Wxx + Wyy = W,tt, where the shear wave velocity has been set equal to 1 and the medium surrounding the plane x = 0 is presumed to be homogeneous. In the anti-plane problem, TXOZ = r is the only nonzero component of the prestress in the medium; w(x, y, t) is the displacement in the medium relative to the state at t = 0. The nonzero components of stress in the medium are rx, To = +.Lwx, Tyz = uwy, where a is the shear modulus. On the plane x = 0, Tyz = 0. The displacement w(x, y, t) can be derived if w(0, y, t) is known (9, 10). We restrict our study to the determination of w(0, y, t). The solution of the differential equation (1) on the crack plane x = 0 is (10) where (c', w(0, y,t) = w(v', t') 1f,d[2] W'X~~f, n)d~drn'q)112' 27Jo Jo( _ - 2(, - q)1/2 q') are the characteristic coordinates I = t + y, Iq = t - y. Let a;(j') and e2(n') be the loci of the upper (OC'F') and lower (OBC) edges of the first crack (Fig. 1). Similarly, let 6(71) and t4(iq) represent the loci of the upper (ADC) and lower (AEFG) edges of the second crack, where f = a' - to - yo and -q = 7' - to + yo are characteristic coordinates referred to A. The point B is the intersection of the trace of the lower edge of the first crack and the line = 0. The point C is the point of fusion of the two cracks. Let R be the entire (y, t) plane; U' be the region to the right Ill of jof0'-i.e., it is the causal cone for the point 0; U be the region to the right of qa6-i.e., it is the causal cone for the point A; and S be the region to the right of F'OCAG-i.e., it is the region of slip. The boundary conditions on the crack plane x = 0 are (i) w = 0 in R-S, (ii) w,x = -T/pg in S, [31 where T is the stress drop in S; we assume T has a known distribution. Solution On OD' and OB the crack tip histories for the first crack do not depend on the existence of the second crack; these histories can be determined using known techniques for an isolated crack. The procedure for the determination of the fracture loci in these regions is briefly sketched below following Chatterjee and Knopoff (4). Eqs. 2 and 3 give, for (ff, Q) E U' - S, 6o 0 W, ~II r')df'dn' { {o (np- o)1/2(o - fo > f2'(ykx) or 77k > n7l(f0), where i7t(f') is the inverse of 4i(ri'). This integral is separable into the expressions (9) and f w,x(6,)df 0 1(fo~f)l 0 X(fo ')d71' = 0, b qot - T)1/2 GI > e2l(w > nl(ft [4al I '170 i 0) [4b]. These are Abel equations for the determination of the unknown stresses in the unfractured region and have the solution FIG. 1. Crack tip loci in the y-t plane. Fusion of the two cracks occurs at C.

3 Applied Mathematical Sciences: Chatteree and Knopoff 7fp- 0tV I~- et to0 > f2l(0k (et, 6) = -1 t 4wx(f, Qd I wx(4:6,,x(09) i)= =ir(a 4 f"1 71' )/2 ( 6-7' o)w2 1/2~ I d711i - 7)1)2, 716> 71i(60). [5] If we take the limit in Eq. 5 as fo6 and 4i approach f2: and vi, respectively, we obtain the stress intensities at the crack tips. We use a critical stress-intensity fracture criterion. To start the iteration we assume that the stress in the interval between the crack edge and the characteristic through the origin is linearly distributed when scaled with respect to the stress at the edge and that estimates of OD' and OB can be obtained. Solutions for the curves OD' and OB, and estimates of the stresses at the opposite edges, can be iterated simultaneously until the solution converges. Stress Distribution in ABD Let the displacement w(f, 71) in U be w = w1 + W2, [6] where w1 represents the displacement on the crack plane that would have arisen had the second crack been absent. The procedure for obtaining w, is outlined above; we assume in what follows that w, and wl,x have known distributions in U'. From Eqs. 6 and 3 it follows that w2(f, 7) = 0 and w2,x(4, 71) = 0 for (4, 71) E U' - U. [7] For a point (64, no) in ABD, we have, from Eq. 3, 0 = -2sirw(f, 7) = J., - 1 [8] From Eq. 6, it follows that Eq. 8 is valid when w, is replaced by wlx. Hence, in view of Eq. 7, Eq. 8 can be rewritten as f Jo[(0 ow2,x(, 77)d:d1 =, - 4:)1/2(no - 71)1/ 77 where (4O, no) are the values of (CO:, i6) in the coordinate system Af47. Since Eq. 9 is valid for all a in 0 4-' oc(no), f W2,(&) 7),d7 d 0l =, The solution to Eq. 10 is In Eq. 11, w2,7 Proc. NatL Acad ScL USA 82 (1985) ) 1/2-0 '-& "-f(1) W2,(64), n) = -1 (6))} A )2x(Oj {m(6)_ }2d the boundary conditions 3-i.e., [10] [ill] for 0-' 71-' qn3(f) can be evaluated by using T w2,x = -- -wl,x, 74(40)-C 7 C m(0: [12] provided we can determine the crack tip paths AD and AE and also the back edge stresses in the almost triangular regions ABD and AEL. We can repeat the above calculations for (4:, no) in AEL. Thus the solutions for the stresses in ABD and AEL, and the crack tip loci AD and AE, can be derived using the techniques outlined in refs. 3 and 4 for the problem of a single crack, except that we calculate the stress drop in the fractured region DAE using Eq. 12. Tip Loci and Stresses in the Interactive Region In BCD, the stress distribution is complex because of the interaction between the elastic waves radiated from the two regions of slip. We modify the techniques of solution for single cracks to determine the unknown stress in BCD and the crack tip loci BC and DC. Let BN be the tip locus for the first crack in the absence of the second crack. Then BN lies above BC as shown because the presence of the second crack increases the stress near the tip of the first crack over the value it would have had, had the second crack been absent. Thus the first crack extends faster in the presence of the second crack than it would in its absence. The condition for zero displacement for a point P(4:o, 71o) in BCD is (Fig. 2) 7, 4:2(X7) -N FIG. 2. HH2, etc. Crack tip loci in the complex interaction region BCD. Solutions for the crack tip loci BC and DC are computed in the steps BH, DH1,

4 1872 Applied Mathematical Sciences: Chatterjee and Knopoff Proc. NatL Acad Sd USA 82 (1985) Xo o (4o - 1/2(o -1)V2 = 0, (go, 7o) in BCD. [13] Then JO f(lo, 1)d Jo(,qo -i~where f(4: 77)o=(e) - )1/2 [14] We write Eq. 14 as I10 f(co, 71)di `13(e) f(4o, q)dq 3(fo) (710-1)1/ "-. (71o -q 711/2 [16] Since Eq. 16 is in the form of Abel's equation, its solution is f'3 fn o, 77)( d To- )1/2 Jo 71 - [17] ir(,qo- 3) R - I f(4:01 71o) 4 We let the right-hand side of Eq. 17 be the function Gl(4:, 10). Eqs. 15 and 17 give C W2,x(e ) =G(4 71) [18] (4: - 4)1/2 We write Eq. 18 as G = G i:,~j(f, - m (: 4)1/2 Up to a factor of the modulus of elasticity, the quantity G is an "effective" stress drop and allows us to calculate the stress intensity factor of the first crack in the presence of the second at points on the edge locus segment BC. Then, W2,a1- {G(4, 71o)}d4 W2,X(fo, n) = X{2 (4 [19] Eq. 19 determines the stress at P(40, 71o) (Fig. 2), provided G can be determined. But the determination of G depends on our knowledge of the crack tip loci BC and AC. Eq. 19 determines the state of stress in BDH (Fig. 2), provided we know the crack tip paths BH and AD. But AD is already known. Hence we only need to find BH. We calculate BH from the critical stress intensity factor criterion. To find the dynamic stress intensity factor along BH, we use Eq. 17 where G1 is known for (4:, Xo) in BDH, since AD and AE are known along with the stresses in the back edge regions ABD and AEH. In the limit as 4o 4:C2 along 71 = 71o, Eq. 17 becomes og2 W2,(f, 710)d[: where the, =rs intensity factor (at -o)1/2 [20] where K2, the stress intensity factor at (4:2,, 1o), is defined by = K2(7_o) w,x(4:, 710) = W2,x(4:2, 71) = f- 42)1/2' [21] To evaluate the second integral on the right-hand side of Eq. 20, we observe that W2,x = -TIMu - wax in 0 6 ' 42, 71= qo [22] from the boundary conditions 3. Let f = flv(rq) be the curve BN (Fig. 1). Then, since wl,, = c -TIA in 0 SI 4 fn, 7 = no, we have, from Eq. 22, so that Eq. 21 yields W2,x = 0 in 0-'4: _ 4:141, [23] [15] irk2(4:2, no) = G1(f2, no) + f {T [151 +wi( 7o)}df.[24j N (6~~~~~~~~~~~- :) 112 Since the right-hand side of Eq. 24 is a known function of (42, no) along BH, it follows that the crack tip locus BH is determined from the known critical stress intensity distribution in the (y, t) plane. Once BH is known, we can compute the crack tip locus DH1 (Fig. 2) by using a similar approach. DH1 in turn gives HH2, etc. Thus the solution for the stress in BDC as well as the crack tip loci BC and DC can be derived as an iteration. To obtain the solution to the crack tip paths D'E' and EF (Fig. 1) we need to know the stress distribution in BDC as well as the crack tip paths BC and DC and hence it can be derived completely by using the above procedure. Beyond F and E', methods of solution for a single crack can again be applied. Thus we have derived the complete solution for the problem of two propagating coalescing coplanar anti-plane shear cracks. To derive the nature of the stresses in the complex interaction region BCD, we write Eq. 24 in an alternative manner. Since wl,, has a singularity at 4: = {N. we can write wl,x(4, 7no) Kl(4N, i7o) = x(e, B0), (4: WM~ W1,(:1) [25] where wl,x is a regular function of (4, 71). Eqs. 24 and 25 give K2(42, 710) = K1(4N, 710) + - G1(42, no) it + 1 f {T/M + WV,X(f:, 710)} d4 (4:2-0:1/2. [26] The dynamic stress-intensity factor at the tip of the first crack K2 is the sum of (i) the dynamic stress-intensity factor in the absence of the second crack for the same value of a, (ii) the contribution from the slip in the torn region of the second crack, and (iii) the contribution of the additional slip of the first crack due to interaction with the second crack. Comments In the absence of the second crack we can set f(4:, 71)-=0 in Eq. 14 in the unfractured region (9); hence, G1 0 and 42 4N and K2 -* K1. Thus the solution reduces to the solution for a single crack. The method of solution for the problem of the dynamic interaction of two propagating mode III cracks can be extended to that of three or more propagating cracks in the same plane. To do this we redefine w appropriately. For example, in the presence of three coplanar cracks undergoing fusion, we define W = W2 + W3, where w2 is the solution for w in the absence of the third crack. Hence W3 can be found in an identical manner to that given above for the problem of two interacting cracks.

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