DYNAMIC WEIGHT FUNCTIONS FOR A MOVING CRACK II. SHEAR LOADING. University of Bath. Bath BA2 7AY, U.K. University of Cambridge

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1 DYNAMIC WEIGHT FUNCTIONS FOR A MOVING CRACK II. SHEAR LOADING A.B. Movchan and J.R. Willis 2 School of Mathematical Sciences University of Bath Bath BA2 7AY, U.K. 2 University of Cambridge Department of Applied Mathematics and Theoretical Physics Silver Street, Cambridge CB3 9EW, U.K. ABSTRACT Dynamic weight functions are constructed for general time{dependent shear loading of a plane semi{innite crack propagating with constant speed in an innite isotropic elastic body. The use of Fourier transforms reduces the problem to the analysis of a matrix Wiener{Hopf equation. The solution of the Wiener{Hopf problem is presented. An expression is derived for the rst{order perturbation to the stress intensity factors induced by a small time{dependent deviation from straightness of the crack front. The asymptotic procedure requires consideration of two terms of the asymptotic expansions of the displacement and stress tensor components in a neighbourhood of the crack front.. INTRODUCTION This is a continuation of a study of dynamic weight functions for three-dimensional loading of a plane, semi-innite crack, moving with speed V. Part I Willis and Movchan, 994 presented the basic framework and developed the solution in detail, for the case of Mode I loading. This required the solution of a scalar equation of Weiner-Hopf type. In contrast, even though the crack edge is straight, threedimensional shear loading inevitably produces coupling between Modes II and III. The Fourier transform formulation correspondingly generates, for shear loading, a

2 2 2 matrix Wiener-Hopf problem. This problem has particular structure that permits its solution in closed form; this is obtained. Next, the Fourier transform of the weight function matrix is employed, in a manner similar to that presented in Part I, to obtain a Fourier transformed relation between the rst-order perturbations to the Mode II and Mode III stress intensity factors, induced by a small deviation from straightness of the crack front. Inversion of the transforms is performed employing a technique, alternative to that of Cagniard 939, developed by Willis 973, to obtain the relation in integral form. A by-product of the analysis is a set of explicit relations between second-order terms in the expansions of relative displacement just behind, and traction just ahead of, the crack edge, in the three-dimensional time-dependent case. It is conrmed in an Appendix that, in the limiting case of statics, the formulae derived reproduce results that are already known Gao and Rice, 986. Since the basic formulation has already been presented in Part I, formulae are usually not repeated: reference to I, 3.3, for example, means equation 3.3 of Part I. However, just to set the scene, the reader is reminded that the basic problem is to solve the equations of isotropic elastodynamics, for an innite space which contains a crack occupying the region X := x? V t < 0;? < x 2 < ; x 3 = 0: : The crack faces are loaded, with tractions prescribed as i3 t; X; x 2 ; 0 =?P i t; X; x 2 for X < 0. Attention is focussed, initially, on the relative displacement of the crack faces, [u i ]t; X; x 2 for X < 0 and the traction components on the plane x 3 = 0, i3 t; X; x 2 ; 0 for X > 0. In particular, the weight functions generate the associated stress intensity factors. Later, the crack is taken to occupy the region X < "t; x 2,? < x 2 <, x 3 = 0, and the perturbations to the stress intensity factors are found, to rst order in ". 2. THE MATRIX WIENER{HOPF PROBLEM Part I see Willis and Movchan 994 introduced the notation [U i ] and i3 for the relative displacement and traction components on the plane x 3 = 0, associated with a weight function, and derived, in Appendix A, a relation between their Fourier transforms [ ~ U i ] and ~ i3 with respect to t; x ; and x 2. Mode I uncouples from Modes II and III and was discussed in Part I. For the coupled Modes II and III, the relation 2

3 takes the form given by I, A.6, which is equivalent to ~ 3 ~ 0 ; ; 2 = ? [U R t DR ~ ] 23 [U~ 2 ] where and D denotes the diagonal matrix 0 ; ; 2 ; 2. R = 2 ; 2.2? 2 id D = diag 0 ; ; 2 i 2 0 =b 2 0 ; =b 2? jj 2 =2 0 =b2? jj 2 =2 2 : 2.3 Here, notation is as dened in Part I: in particular, jj = =2 and D is given by I, 4.2. The matrix Wiener{Hopf problem is obtained, as described in Part I, by introducing transforms i3 and [U i ], which are the Fourier transforms of i3 and [U i ] with respect to variables t; X = x? V t and x 2. It is easily veried that, for a function ft; x ; x 2, f; ; 2 = ~ f? V ; ; 2 : 2.4 Thus, 2. relates i3 and [U i ]; i = ; 2; if 0 =? V. Furthermore, since [U i ] = 0 when X < 0 and i3 = 0 when X > 0, it follows that i3 is an analytic function of for each xed with Im > 0 and each real 2 in a half{plane Im < k +, while [U i ] is analytic in a half{plane Im > k?, for some k? ; k + satisfying?k? < 0 < k +. In preparation for the solution of the Wiener{Hopf equation, factorize D? V ; ; 2 as follows: D? V ; ; 2 = D + D? ; 2.5 where D are the diagonal matrices D + = diag [ + V=c 2 + iq c ]T + [ + V=b 2 + iq b ] ; [ =2 + V=b 2 + iq b ] =2 ; D? =? 2 diag + V=c 2? iq c b 2 RV T? ; [ V 2 [ + V=b 2? iq b ] =2 + V=b 2? iq b ] =2 :2.6 Notation is as dened in Part I; D + is analytic in Im > k? and D? is analytic in Im < k +. 3

4 Thus, in the strip k? < Im < k +,? = ? R t D + D? RU + ; 2.7 where? := 3 ; 23 t and U + := [U ]; [U 2 ] t. The objective is to deduce from 2.7 the vector functions?, analytic in Im < k +, and U +, analytic in Im > k?, such that U + is a homogeneous vector function of degree?=2 and, correspondingly,? is homogeneous of degree =2 in ; ; 2 : 3. SOLUTION OF THE WIENER{HOPF EQUATION It is noted that, as 2 0 or, equivalently, as =j 2 j, R I; where I is the identity matrix, and the Wiener{Hopf equation 2.7 uncouples, asymptotically, to give? D + D? U + ; 3. for which a candidate solution with the desired homogeneity is? D? C; U + D? + C; 3.2 for some vector vector function C of and 2, which is homogeneous of degree zero. This provides at least partial motivation for seeking a solution of 2.7 in the form? = D? C + R t? ; U + = D? + C 2 + R t u + ; 3.3 where C and C 2 are vector functions of and 2, homogeneous of degree zero. Substituting 3.3 into 2.7 and multiplying by R from the left give RD? C ? = D + D? RD +? C D + D? u + ; 3.4 or, upon rearrangement, ? C? C F; ; 2 + D +?? = D + u + ; 3.5 4

5 where with F; ; 2 = D?? ED? C? D + ED +? C 2 ; 3.6 : 3.7 E = 0? 0 To construct a general solution, two cases are considered: i C 2 = 0; C 6= 0 and ii C = 0; C 2 6= 0. First, suppose that C 2 = 0: Equation 3.5 takes the form " 2? I + 2 D?? ED? #C + D??? = D + u + : 3.8 The following additive split is obvious: " 2 + ij 2 j +? ij 2 j? sign 2 2i? sign 2 2i # + sign 2 D?? ED? 2i? ij 2 j " D?? ED?? fd?? ED? g;?ij 2 j; 2 fd?? ED? g;?ij 2 j; 2 + ij 2 j The usual Wiener{Hopf argument then gives u + = 2 + ij 2 j D +? + ij 2 j C = D + u +? D??? : 3.9 I + i sign 2 fd?? ED? g;?ij 2 j; 2 # C ; 3.0? =? 2 D?? ij 2 j? i sign 2D?? ED?? ij 2 j and, consequently, U + = Rt D +? 2 + ij 2 j +i sign 2 D?? ED?? fd?? ED? g;?ij 2 j; 2 + ij 2 j C ; 3. I + i sign 2 fd?? ED? g;?ij 2 j; 2 C ; 3.2 5

6 ? = 2 + ij 2 j Rt D? I + i sign 2 fd?? ED? g;?ij 2 j; 2 C : 3.3 It can be checked by direct calculation that the expression 3.3 does represent a \?" function: it does not have a singularity at =?ij 2 j, on account of the particular form of R. It should be mentioned that the rank of the matrix D := I + i sign 2 fd?? ED? g;?ij 2 j; is the same as that of I + i sign 2 E and so is equal to one. Thus, this solution contains, in eect, only one free parameter. For the case when C = 0, equation 3.5 can be rewritten as follows ? " I + 2 D + ED +? #C 2 = D?? which is equivalent to " # 2 + ij 2 j +? sign 2? ij 2 j 2i + sign 2 2i + sign 2 2i " D+?? D + u + ; 3.5 D + ED +? + ij 2 j ED +?? fd + ED +? g; ij 2 j; 2? ij 2 j fd + ED +? g;?ij 2 j; 2? ij 2 j Similarly to 3.2, 3.3 it follows in this case that and? = U + = Rt D +? 2? ij 2 j C 2 = D???? D + u + : 3.6 I? i sign 2 fd + ED +? g; ij 2 j; 2 2? i j 2 j Rt D? I? i sign 2 D + ED +? # ; ij 2 j; 2 C 2 ; 3.7 C 2 : 3.8 The particular form of R ensures that U + has no singularity at = ij 2 j: As in 3.4, the rank of the matrix D 2 := I? i sign 2 D + ED +? ; ij 2 j;

7 is equal to one. The solution of the Wiener{Hopf equation 3.5 is now U + = 2 Rt D +?" + ij 2 j D C +? = 2 Rt D? " + ij 2 j D C + # #? ij 2 j D2 C 2? ij 2 j D2 C 2 ; 3.20 : 3.2 Although C and C 2 are arbitrary vectors independent of, the fact that D and D 2 have rank one implies that this solution is expressible in terms of two arbitrary scalar parameters. Since D can be written in the form D = D?? its eigenvalues are readily evaluated as " I + i sign 2 E = 0; 2 = 2; ; X 2 = # D? ; and the corresponding eigenvectors are D?? X ; D?? X 2 ; where X = i sign2 i sign 2 Similarly, D 2 has eigenvalues = 0; 2 = 2 and corresponding eigenvectors D + X 2 ; D + X : Therefore, in the solution given by 3.20, 3.2 there is no loss in choosing C = c 2 D?? ;?ij 2 j; 2 X 2 ; C 2 = c 2 2 D +; ij 2 j; 2 X ; 3.22 where c ; c 2 are arbitrary scalar functions of ; 2 ; homogeneous of degree zero. Then, 3.20 takes the form U + = R t D +? where a ; b are dened so that 0 B@ i sign 2 + ij 2 ja? a +? ij 2 j + ij 2 jb? i sign 2b +? ij 2 j : CA c c 2 ; 3.23 diagfa? ; b? g := D? ;?ij 2 j; 2 ; diagfa + ; b + g := D + ; ij 2 j; 2 :

8 It is convenient to introduce, starting from 3.23, a \fundamental matrix" solution, with prescribed behaviour as : This is done by expressing c ; c 2 t in the form c a? b? i sign2 b =? +?a + l ; 3.25 c 2 a + a? + b + b??=b? i sign 2 =a? l 2 for some constant l ; l 2. Then U + = R t D +? a + a? + b + b? and, as, 0 B@ b+b? + ij 2 j + a +a?? ij 2 j? 2a?b ?=2 l U + + 0i l 2 2a + b? a + a? + ij 2 j + b +b?? ij 2 j CA l l 2 ; 3.26 : 3.27 The expression for? is similar: it is necessary only to replace D +? by D? on the right side of The matrix multiplying l ; l 2 t in 3.26 is the fundamental matrix that was sought on account of its simple form as : This section is concluded with the observation that, in correspondence with 3.27, the asymptotic form of U + as X 0 is U + t; X; x 2?i l =2 i HXtx 2 : 3.28 l 2 X 4. STRESS{INTENSITY FACTORS It was shown in Part I that [U i ] i3? k3 [u k ] t; X 0 ; x 0 2 = 0; 4. where i3 ; k3 denote stress components for u and U respectively, evaluated at x 3 = 0; as before, [:] denotes the jump across x 3 = 0 of the quantity indicated, and signies convolution with respect to t; X and x 2. Implications of the identity 8

9 4. are now explored in the case that [U ]; [U 2 ] t = U + ; 3 ; 23 t =? and [U 3 ] = 33 = 0. The convolution 4. will rst be evaluated when X 0 > 0. Since [u k ] and k3 are? functions, their convolution has the value zero and 4. reduces to U t + t; X 0 ; x 0 2 = 0; 4.2 where represents 3 ; 23 t, evaluated when x 3 = 0. Now, i3 t; X; x 0 2 ; 0 =?P it; X; x 2 H?X + + i3 t; X; x 2 ; 4.3 where P i t; X; x 2 i = ; 2 are prescribed. Furthermore, as X 0; t; X; x 2 2X?=2 KII K III HX: 4.4 The convolution in 4.2 is now readily evaluated, in the limit as X 0 +0; employing the asymptotic forms 3.28 and 4.4: i =2 i l ; l 2 2 KII t; x 2 K III t; x 2 + fu t + P? gt; 0; x 2 = 0; 4.5 where P? = P ; P 2 t H?X: This result motivates dening a weight function matrix, still called U +, so that it has the Fourier transform 2i U =2 + = R t D +? a + a? + b + b? 0 B@ b+b? + ij 2 j + a +a?? ij 2 j? 2a?b a + b? a + a? + ij 2 j + b +b?? ij 2 j 4.6 This is nothing but the fundamental matrix introduced in the preceding section, multiplied by 2i =2. The corresponding matrix? has Fourier transform? = 0 2i =2 R t D? a + a? + b + b? b+b? + ij 2 j + a +a?? ij 2 j? 2a?b a + b? a + a? + ij 2 j + b +b?? ij 2 j CA : CA : 4.7 9

10 The result 4.5 then gives KII t; x 2 = U t + P? t; 0; x K III t; x 2 2 : 4.8 It is also possible to develop the relation 4.8 by evaluating the convolutions U t + + and t? [u], as X 0; from the corresponding asymptotic forms of the transforms as ; as was done in Part I for the case of Mode-I loading. This will emerge from the more general reasoning presented next. 5. ASYMPTOTIC EXPANSIONS TO HIGHER ORDER As noted in Part I, the identity 4. induces a corresponding identity for its Fourier transform, [U i ] i3? k3 [u k ] = 0; 5. or U t + +? t?[u] =?U t + P? : 5.2 The identity 5.2 will now be expanded as. It is noted rst that, as X 0; + K2X?=2? P + AX =2 HX; 5.3 and [u] V K?2X= =2? B?X 3=2 H?X; 5.4 for some vectors P ; A and B, and some matrix V. Here, K denotes K II ; K III t : Correspondingly, as ; + i 2 =2 K ip? =2 + 0i + 0i? =2 A ; 5.5 2i =2 3=2 + 0i and Next, K [u]? i=2 3i? 3=2 2? 0i 4 i =2 B : 5.6 5=2? 0i U t + P? Ft; x 2? 22= =2 Lt; x 2 X =2 HX + =2 =2 Mt; x 2 X; 5.7 0

11 where Ft; x 2 = U t + P? t; 0; x 2 ; 5.8 Lt; x 2 = Pt; 0; x 2 ; 5.9 and Mt; x 2 = U t + P0? t; 0; x 2 ; 5.0 with P 0? Fourier transforming 5.7 gives, as ; # U t + P? i F" + 0i? L? 2i=2? 0i + 0i # 3=2 +i=2 M" =2 + 0i? : 5. 2? 0i 2 All of these results are direct counterparts of corresponding results in Part I for the case of Mode I loading. Now corresponding expansions of U + and? are required. First, for the diagonal matrices D, as ; A D?? 2? 0i =2? +? 0i A2? ; 5.2 D? + + 0i I?= i B2 + ; 5.3 where A? = diagfb; g; 5.4 A 2? = diag B V c? iq 2 c? 2 V b? iq b + t? ; 2 2 V b? iq b ; with and B 2 + =?diag V c 2 + iq c? 2 V b 2 + iq b + t + ; 2 V b 2 + iq b t =? 2i ; 5.6 B = b 2 RV =V 2 ; 5.7 Z C ln T ; 0 ; 2d 0 : 5.8

12 Notation not explicitly dened is as in Part I. It follows now, from 3.26 and 5.3, that where with and where U + 2i =2 + 0i?=2 q = a +a?? b + b? a + a? + b + b? 0 0? i I + ; + 0i Q 5.9 iq = B 2 + ij 2 jq + 2 2f? E; 5.20 ; 5.2 f = a + a? + b + b?? 0 a + b??a? b + 0?? I? 2 E AV? : 5.22 Inspection of the Wiener{Hopf equation 2. reveals that, correct to order j j?=2 ; " #? I + 2 AV = diag 2V V=b 2 2 ; RV V U + ; 5.23 : 5.24 E Thus, as,??2i =2? 0i =2 [AV ] I? +? i0 A0 V [AV ]? + 2 E? AV E[AV ]? + iq : 5.25 These expressions suce to expand the identity 5.2 to three terms as : ik + 0i? i2i=2 P + 0i 3=2? =2=2 A + Q t K + 0i 2? i=2=2 [AV ]? K? 0i i=2 =2 [AV ]? A 0 V? 2 E? [AV ]? EAV + iq t[av ]? K?? 0i 2 2

13 " # + 3=2=2=2 [AV ]? B = if? 0i 2 + 0i? L? i2i=2? 0i + 0i # 3=2? =2 M" =2 + 0i? : ? 0i 2 It follows, therefore, that K = F; = 2 =2 AV ; A = M? 2 Q t =2 K; B = 2 3 AV A + i 2 =2? 2 E? AV E[AV ]? " t# [AV ]? A 0 V K : 5.27 The rst of relations 5.27 conrms the result 4.8. The second is quoted, for example, by Freund 990. The expressions for A and B were not known previously to the authors. 6. PERTURBATION OF THE CRACK FRONT As in the Mode I analysis of Part I, the identity 5. can also be applied when the stress and displacement elds solve the problem of a crack occupying the region X = x? V t < "t; x 2 ;? < x 2 < ; x 3 = 0; 6. where " is a small positive constant and the function is smooth and bounded. The displacement and stress elds for this problem are written as u + u; + ; where u; solve the problem for the unperturbed crack " = 0: Since 5. is satised for both the perturbed and unperturbed elds, it follows that U t +? t?[u] = 0: 6.2 This identity applies for all and all "; however, information on the rst{order perturbation of the stress{intensity factors will be sought by considering the lowest{ order non{trivial asymptotic expansion of 6.2 as and " 0: 3

14 As in Part I, since corresponds to X 0, it suces to note that, as X 0 and " 0; and [u + u] + K + K [2X? "] =2 + X? "=2 A + A? P + P + " 0 E K 2X =2? P + X =2 A HX? " "? X =2 + K + K? "? X 3=2 B + B +" 0 E?X =2 K??X 3=2 B HX; 6.3 H"? X H?X: 6.4 In comparison with the Mode I problem the expansions 6.3, 6.4 include additional terms which arise from the local rotation of the crack edge. Fourier transforming 6.3, 6.4 and performing formal expansions with respect to " and = gives and i 2=2 + 0i?=2 K + "E 0 K + i" K? =2 i + 0i P + "E 0 P? 2i =2 + 0i 3=2A + i" A + "E 0 A; 6.5 [u]?i=2 K + "E 2? 0i 0 K + K + i" 3=2 K? 3i 4 i =2? 0i 5=2 B + "E 0 B + i" B These formulae are vector analogues of I, 8.6 and I, 8.7. The asymptotic relations 5.9 and 6.5 yield i U t +?"K + + 0i?"Q t K? 2 " K + "E 0 K =2 "A # : 6.6 ; 6.7 4

15 and 5.25 and 6.4 give " i t?[u]?"k + K + "[AV ]? EAV? 0i 0 K + =2 =2 [AV ]? K +" i[av ]? A 0 V? i 2 E? [AV ]? EAV? Q tk? " 3 2 =2 [AV ]? B 2 # : 6.8 The terms that are independent of in 6.7 and 6.8 are the same; hence, they cancel, upon substitution into 6.2. It also follows, from the identities 5.27, that the coecient of + 0i? in 6.7 is the same as the coecient of? 0i? in 6.8. Hence, the identity 6.2 implies to rst order in ", that K = " Q t K? E 0 K + =2 =2 A ; 6.9 where A is given by the third of relations Inverting the transforms yields K = " Q t K? E 0 K + =2 =2 M? Q t K : 6.0 Here, the convolution is over the variables t; x 2 and, since Q; 2 is homogeneous of degree one, it follows from I, 8.2 that " Qt; x 2 2 Q?x2 # + 0i; t + Qx 2 + 0i;?t Ht : 6. 2?x 2 + 0i 2 x 2 + 0i 2 The result 6.0 is the mixed Mode II/III analogue of the Mode I formula I, 8.. The Appendix considers the static limit of the solution. This is obtained by setting V = 0 and allowing to tend to zero. This static limit was addressed directly by Gao and Rice 986; the agreement that is found provides a partial check on the present analysis. 5

16 7. CONCLUDING REMARKS Parts I and II together provide a complete weight function matrix for any loading of the semi{innite crack in uniform motion. The matrix partitions into a scalar component corresponding to Mode I, and a 22 matrix relating to the coupled Modes II and III. Fourier transforms were employed to derive the weight function matrix. Treatment of the Mode I component required the solution of a scalar Wiener{Hopf problem whose solution is routine; the coupled Modes II and III generated a 2 2 matrix Wiener{Hopf problem for which a closed{form solution was derived. Use of a reciprocal theorem permitted the deduction of the perturbation to the stress intensity factors induced by a small time{ and space{dependent deviation from straightness of the crack front. The perturbation is expressed by I, 8. for Mode I, and by 6.0 of the present paper for Modes II and III. Implications of these formulae will be followed up elsewhere. 6

17 REFERENCES Cagniard, L., 939, Reexion et Refraction des Ondes Seismiques Progressives, Gauthier{ Villars, Paris. Freund, L.B., 990, Dynamic Fracture Mechanics, Cambridge Univ. Press. Gao, H. and Rice, J.R., 986, Shear Stress Intensity Factors For a Planar Crack With Slightly Curved Front, ASME Journal of Applied Mechanics, 53, 774{778. Uyand, Y.S., 965, Survey of Articles on the Application of Integral Transforms in the Theory of Elasticity, North Carolina State University, Department of Applied Mathematics, File No. PSR-24/6, Raleigh, N.C. Willis, J.R., 973, Self{Similar Problems in Elastodynamics, Phil. Lond. A274, 435{49. Trans. R. Soc. Willis, J.R. and Movchan, A.B., 994, Dynamic Weight Functions For a Moving Crack. I. Mode I Loading, J. Mech. Phys. Solids. 7

18 APPENDIX The static limit can be obtained from the full dynamic problem by setting V = 0 and allowing to tend to zero. The Wiener{Hopf relation 2. then reduces to where the diagonal matrix is dened as? =? RU ; A. =2Rt = diagf?? ; g; A.2 with denoting Poisson's ratio. Comparison with the full dynamic equation 2.7 shows that D =? =2 ; and, from 2.6, D + + ij 2 j =2 I; D?? 2? ij 2 j =2 : A.3 The static limit of the weight function 4.6 is easily calculated, upon noting that Substituting these values into 4.6 gives U + 22i=2 2? + ij 2 j?=2 Inversion of the transforms then yields U + = p 2x Hx 3=2 x 2 + x 2 2 a + 2ij 2 j =2 ; b + 2ij 2 j =2 ; a??i 2ij 2j =2 2? ; b??i 2 2ij 2j =2 : A.4 0 B@ + 0 B@ 2? + ij 2 j? 2 + ij 2 j 2 2? x 2 =x 2? 4 2?? 2 + ij 2 j 2? + + ij 2 j x 2 =x 4 + x 2 =x 2 2? + x 2 =x 2 x 2 =x? 2? x 2 =x 2 + x 2 =x 2 2? + x 2 =x 2 8 CA : CA : A.5 A.6

19 Allowing for the replacement of x by?x because of the present use of notation that permits the use of convolutions, this agrees with the weight function given by equation 5 of Gao and Rice 986, and attributed by them originally to Uyand 965. Similarly, in the relation 6.9 between K and, the matrix Q reduces to Q =? j 2j 2? i 2 0 : A.7 22? ? 0 Hence, inverting the transform with respect to 2 ; Q = 22? 2? x 2 2 2? the generalized function =x 2 2 part integrals. Substitution of A.8 into 6.0 then gives Kx 2 = " 22? + 2 2? 0 0 x 0 2 ; A.8 being interpreted so as to generate Hadamard nite 2? ?? 0 Z? 0 x 2 Kx Kx 0 2[x 0 2? x 2 ] x 2? x dx 0 2 =2 x 2 Mx 2 ; A.9 This agrees precisely with equation 5 of Gao and Rice 986, except that the term involving M was absent from their work, through their choice of local coordinates. 9