ELASTIC COMPLEX ANALYSIS AND ITS APPLICATIONS IN FRACTURE MECHANICS. N. D. Aparicio. Oxford Brookes University, Gipsy Lane, Oxford OX3 OBP, UK.

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1 EASTIC COMPEX ANAYSIS AND ITS APPICATIONS IN FRACTURE MECHANICS N. D. Aparicio School of Computing & Mathematical Sciences, Oxford Brookes University, Gipsy ane, Oxford OX3 OBP, UK. April 9, 998 Abstract The key point in this paper is the introduction of elastic analytic functions. An elastic analytic function is a function of the form u : C! C which is dierentiable and satises equations which are analogous to the Cauchy-Riemann equations of traditional complex analysis such that the following conditions hold: rst the real and imaginary part of the rst complex component of u satisfy the Navier equation of plane elasticity, and second, the derivative along a line of the real and imaginary part of the second complex component of u is proportional to the applied tractions along that line. Algebraical operations have been dened on elastic analytic functions such that they constitute a commutative algebra over the real eld and a module over the set of analytic functions. Next a derivative and an integral of elastic analytic functions are introduced such that they behave in a similar way to complex dierentation and integration of analytic functions, in particular we have properties such as: the integral of an elastic analytic function around a contour is zero, a Cauchy-like integral formula and Plemelj-like formulae. These properties can be very useful in tackling problems of plane elasticity involving cracks through the boundary element method. It is also proved that path independent integrals in plane elasticity that are derived from Noether's theorem, whose integrand only depends on the position and gradient of displacements, can be written as the integral of an elastic analytic function. Introduction The usefulness of complex analysis in the resolution of problems governed by the aplace equation, for example antiplane elastostatic problems, is well known. The key point is that the real and imaginary part of an analytic function in complex analysis satisfy the aplace equation. If we are given a potential that satises the aplace equation, we can always add its harmonic conjugate to build up an analytic function and then solve a particular problem by using all the tools of complex analysis, for example: Cauchy integral formula, analytic continuation, expansion in series, Hilbert

2 problem, path independent integrals, etc. Furthermore, the normal derivative of the potential, which is normally associated with \load type" boundary conditions, is equal to the derivative of the harmonic conjugate along a line (multiplied by? depending on the conventions). For an example of how powerful this technique can be, the reader is referred to two recent works by Atkinson & Aparicio (994) and Aparicio & Pidcock (996). The techniques of complex analysis can also be applied to solve plane elastostatic problems governed by the Navier equations. This is normally done by writing the displacements and stresses as functions of complex analytic potentials. The best known ways of doing this are probably those given by MacGregor (935), Westergaard (939) and Muskhelishvili (963). However, the analogy between the use of these methods in plane elastostatic problems and the use of complex analysis in antiplane elastostatic problems is apparently not obvious. For example, the displacements of a plane problem are not written as part of an \analytic function" but a combination of analytic functions and their conjugates. This brings a gap between the mathematical armoury associated with the complex potentials and the physical interpretation of the problem. As an example, let us consider the Muskhelishvili potential (z). Since it is an analytic function within the region where a plane elastic state exists, we can apply to it the Cauchy integral formula or other tools of complex analysis, but what all that mean in relation to the plane elastic problem itself is rather unclear. On the other hand, if we take the complex potential whose real part is the solution of an antiplane problem in a particular region and apply to it the Cauchy integral formula, then we have a weak Green's fundamental formula that will allow us, for example, to solve the problem using the boundary element method. The intention of this work is to construct a new complex analysis that will interact with plane elastostatic problems in a manner that is similar to the interaction between normal complex analysis and antiplane problems. For this reasons it is called elastic complex analysis to distinguish it from normal complex analysis. Furthermore, the adjective \elastic" will be used throughout this paper to distinguish the mathematical operations dened within elastic complex analysis from those of normal complex

3 analysis (i.e. elastic analytic function, elastic derivative, etc.). In the same way that the key feature in the use of complex analysis in antiplane elastostatic problems is the concept of an analytic function, the key feature in the use of elastic complex analysis in plane elastostatic problems is the concept of an elastic analytic function. An elastic analytic function is a function u : C! C. The real and imaginary part of the rst complex component of an elastic analytic function satisfy the Navier equations of plane elasticity, in other words, they work as the displacements of an elastic state. The derivative of the real and imaginary part of the second complex component along a line is proportional to the applied tractions. The basic arithmetic operations will be dened such that the set of elastic analytic functions becomes a commutative algebra over the real eld under these operations. This includes the fact that the sum and product of elastic analytic functions are also elastic analytic functions. However, the set of elastic analytic functions with the sum and product so dened will not be an integral domain unlike the set of analytic functions. A division operation will also be introduced that will allow certain elastic analytic functions to be inverted. The product of elastic analytic functions has been introduced only as a formalism and has not been proved so far to be very useful in practical applications. A more interesting tool is the product of an elastic analytic function and an analytic function. Under this product the set of elastic analytic functions is, in general, a module over the set of analytic functions (it is, however, an algebra over the integral domain of analytic functions under a more restricted denition). Other useful tools are the elastic derivative and its inverse the elastic integral. As in normal complex analysis, the elastic integral around a domain of a function which is elastic analytic in is zero. This property allows us to construct path independent integrals. It will be proved that path independent integrals of the J type can be written as the elastic integral of an elastic analytic function. A formula similar to the Cauchy integral formula will also be introduced. As the Cauchy integral formula can be seen as a weak form of Green's fundamental formula, this new integral can be seen as a weak form of Somigliana's integral formula which is widely used in the resolution of elasticity problems through 3

4 boundary elements. The equivalent of Plemelj formulae for this new integral formula as well as a closed form inversion will also be studied as this can be particularly useful in the resolution through boundary elements of plane elastostatic problems involving cracks. Finally, it will be seen that elastic analytic functions can also be expanded in Taylor series and analytically continued. Indeed, they have most of the properties of normal analytic functions. Elastic analytic functions The stresses and displacements of an elastic state can be written in terms of the Muskhelishvili potentials (z) and (z) (Muskhelishvili 963) as xx + yy = 0 (z) + 0 (z) yy? xx + i xy = z 00 (z) + 0 (z) u = u x + iu y = G [k(z)? z0 (z)? (z)]; () where k = ( + 3G)=( + G) is a real elastic constant greater than and G > 0 and > 0 are the ame constants. We introduce the traction potential p dened by p = G [(z) + z0 (z) + (z)]: () It is called traction potential because its derivative along a line is proportional to the applied tractions dp ds = i G [T x(n) + it y (n)]; where n is a unit vector orthogonal to that has been obtained by rotating 90 degrees clockwise a unit vector tangent to that points to the direction where the arc length s increases. 4

5 Denition. A function u : C! C of the form u = (u; p) is an elastic analytic function in an open subset of C if it is dierentiable in and satises the @z : (4) Equations (3) and (4) can be regarded as the equivalent of Cauchy Riemann equations of complex analysis. They simply state that the Muskhelishvili potentials (z) and (z) must be analytic functions, i.e. their partial derivative with respect to z must vanish. From these equations we can obtain the conditions that u and p must satisfy in order to be the components of an elastic analytic function. Dierentiating (4) with respect to z and using (3), we = p : (5) Dierentiating (4) with respect to z and using (3) we u = p : (6) Subtracting the conjugate of (5) from (6) we nd, after some minor p = 0: (7) By proceeding in a similar way but eliminating p rather than u, we + = 0: (8) Equation (8) is actually the Navier equation of plane elasticity as can easily be seen if it is @z + = 0 and we use the following relationships 5

6 @u = r:(u x; u = 4 r : In order to simplify the notation in the forthcoming sections, the following matrices have been introduced A = + k " k k # " A =? k + k?k # " A 3 = + k?? # : These matrices have the following multiplication table and also satisfy A = A A A = 0 A A 3 = 0 A A = 0 A =?A A A 3 =?A 3 A 3 A = A 3 A 3 A = 0 A = 0 3 A? A = I; where I is the identity matrix. If u is an elastic analytic function, then the following relationships = 0; = 0 A = 0: If u = (u ; p ) and u = (u ; p ) are elastic analytic functions and a and b are real numbers, then it is clear that au + bu = (au + bu ; ap + bp ) is also an elastic analytic function (i.e. it satises equations (3) and (4)). The set of elastic analytic functions therefore constitutes a vector space over the real eld with the usual operations of addition and multiplication by scalars. 6

7 3 The product between an elastic analytic function and an analytic function Denition. et be an open subset of C, u an elastic analytic function in, f an analytic function in and z o a complex number. The translated product between u and f with respect to z o is dened as (u z f)(z; z) = (A f(z)? A f(z))u(z; z)? ( )A 3 f 0 (z)u(z; z): () When the translated product is written as, we mean that it is applied to the same variable as that used by the analytic function that follows it. In the particular case where z o = 0 or f is a constant, we will denote the product of an elastic analytic function and an analytic function as u f. The following properties can be veried using the multiplication table for A i matrices. u f is elastic analytic in.. If u and u are elastic analytic functions in then (u + u ) f = u f + u f 3. If f and f are analytic functions in then u (f + f ) = u f + u f 4. u = u. 5. If f and g are analytic functions in, then (u f) g = (u g) f = u (fg): 6. From the above property we have, in particular, that if f is an analytic function in that does not vanish anywhere in and u = u f then u = u f 7

8 7. If a and b are real numbers then (au) (bf) = (ab)(u f) Properties to 5 establish that the set of elastic analytic functions is a module over the set of analytic functions. If u is an elastic analytic function, there is an elastic state associated with it with Muskhelishvili potentials (z) and (z). Since u f is also elastic analytic, we can associate to it Muskhelishvili potentials (z) and (z). Equation () must therefore relate and to, and the analytic function f. After some manipulation, it can be proved that these relationships are (z) = f(z)(z); (z) = f(z) (z)? z o f 0 (z)(z): Properties 5 and 6 above can be proved more easily using these relationships between Muskhelishvili potentials. Notice that when z o = 0, the mathematical meaning of this operation is simply the multiplication by f of the Muskhelishvili potentials associated with u. 4 The elastic derivative and elastic integral of an elastic analytic function Denition. et be an open subset of C and u an elastic analytic function in. We dene the elastic derivative of u with respect to z as eu ez = + The elastic derivative has the following properties:. eu=ez is elastic analytic : (). If a and b are real numbers and u and u are elastic analytic functions, then e ez [au + bu ] = a eu ez + b eu ez : Notice that if a is a complex number and u is elastic analytic, au is not necessarily elastic analytic. 8

9 3. If z o is a complex number and f is an analytic function in, then e ez (u f) = eu ez f + u (if 0 ): 4. If the elastic derivative of an elastic analytic function u is zero in an open set, then u is a constant in. etting equation () equal to zero, we see = = 0: Combining this result with equations (9) and (0) we nd that both u + p and kp? u are constants and therefore u is a constant in. This means that the elastic derivative is a kind of total derivative rather than a partial derivative when dealing with elastic analytic functions. This is similar to complex dierentiation of analytic functions. The eect of the elastic derivative on the Muskhelishvili potentials associated with an elastic analytic function is very simple. It simply dierentiates the Muskhelishvili potentials associated with u and multiplies them by i. Denition. If is a rectiable curve in C (that is is of bounded variation) and u is a function u : C! C, then the elastic integral of u along is dened by u dz =?i (A udz + A udz + A 3 udz): (3) The elastic integral has the following properties:. If a and b are real numbers, then (au + bu ) dz = a u dz + b u dz. If is a closed curve that surrounds a region where u is elastic analytic, then I u dz = 0: 9

10 To prove this, it is sucient to show that the integrand of the above expression is an exact dierential, (A u + A 3 (A u): This is straightforward to prove using the rst equation in (9) and the conjugate of (0). 3. If is an open subset of C where u is elastic analytic and is an open rectiable curve in, then eu dz = u; ez where u denotes the dierence between the value of u at the nal point of and the value of u at the initial point of according to the direction of integration. To prove this assumption, we simply substitute () into (3). By using the multiplication table of the A i matrices, we arrive at eu ez dz dz? dz : Adding = 0 and noting the conjugate of (0) we get eu ez dz = = (A? dz + (A? @z dz = u: 5 A Cauchy integral formula for elastic complex analysis Theorem. If is an open subset of C with boundary, u is an elastic analytic function in and z o is a complex number in, then I u(z; z) z dz = u(z o ; z o ); (4) where the integral is carried out anticlockwise along. 0

11 Proof. et us consider a small circle of radius centered at z o. Then, using the property of path independence, we have I I u(z; z) z dz = u(z; z) z dz I u(z o ; z o ) z dz as! 0 + : Rewriting this integral by substituting the denition of the translated product () into the denition of the elastic integral (3) and using the multiplication table for the A i matrices, we obtain I u(z o ; z o ) z dz = I A dz =?i + A dz z? u(z o ; z o ) + A 3 u(z o ; z o )d = [A ()? A ()]u(z o ; z o ) = u(z o ; z o ): 6 Plemelj like formulae for elastic complex analysis Theorem. et be a smooth line not intersecting itself and u :! C a function of the form u = (u; p) such that: both u and p satisfy the Holder condition on, u + p vanishes at the end points of, is dierentiable on and its derivative satises the Holder condition on. Then the following function is elastic analytic everywhere in C except on. p(z o ; z o ) = u(z; z) z dz (5) Proof. The function (z) = G(u + p)=( + k) vanishes at the end points of, is dierentiable on and its derivative satises the Holder condition on. This means that the following functions are analytic everywhere in C except on (z o ) = (z)dz ; 0 (z o ) = 0 (z)dz = i i i (z)dz ( )

12 On the other hand, the function (z) = Gp?? z 0 also satises the Holder condition on and therefore is also analytic in C?. (z o ) = (z)dz We introduce the following notation R (u; p) = u; R (u; p) = p: Substituting the expression of u = (u; p) in terms of its Muskhelishvili potentials (see () and ()) into the right hand side of (5), we obtain " G(i) R u(z; z) z dz = k dz + z0 + z? # dz + d = " G(i) k dz + z o 0 + # dz + d Similarly = G (k(z o)? z o 0 (z o )? (z o )): R u(z; z) z dz = G ((z o) + z o 0 (z o ) + (z o )): Thus, p(z o ; z o ) is the elastic analytic function associated with the Muskhelishvili potentials (z o ) and (z o ). Theorem. et be a simple smooth oriented curve, let z o be a point on, let u j = (u j ; p j ) be a function u j :! C satisfying the Holder condition on such that u + p vanishes at the end points of, is dierentiable on and its derivative satises the Holder condition on. Then if U + (z o ; z o ) and U? (z o ; z o ) are respectively the limits of

13 U(z ; z ) = u j (z; z) z dz z z? z as z approaches z o on the left and right of respectively, we have u a (z o ; z o ) = U +(z o ; z o ) + U? (z o ; z o ) = PV u j (z; z) z dz; (6) u j (z o ; z o ) = U + (z o ; z o )? U? (z o ; z o ); where PV means that the integral must be interpreted in the sense of its Cauchy principal value. Proof. The proof can be carried out in a similar way to the previous theorem by taking into account Plemelj formulae (Muskhelishvili 958) + (z o ) +? (z o ) = i PV (z)dz ; + (z o )?? (z o ) = (z o ): Equation (6) can be inverted in closed form in a similar way as for complex functions in Muskhelishvili (958). We simply state here the result (see appendix for a complete proof). u j (z o ; z o ) = PV (u a (z; z) z g + (z)) z dz g + (z o ) + + A g + (z o ) ; (7) where A is a constant in C and g + (z) the limit of the function g() g() = (? z ) s? z? z as approaches a point z on on the left. In the above formula the branch cut is chosen along and z, z are the initial and nal points of respectively. 3

14 Equation (6) can be seen as the superposition of two cases by making u j = u j + u j, where R u j = 0 and R u j = 0 on. The rst case corresponds to a distribution of dislocations along which produces stresses that are continuous across. The second case corresponds to a distributed load and therefore displacements are continuous across. These cases were already studied by Aparicio & Atkinson (997) in the determination of weight functions for curved cracks. Equation (6) can be very useful in the resolution of plane elastostatic problems involving cracks. Only two of the four components of (6) need to be considered when constructing the system of equations. The choice of which complex component is more appropriate will depend on which complex component of u a can be written in terms of either the variables of the problem or known quantities. For external boundaries, either u or p will be appropriate as nodal variables depending on the boundary conditions; while along cracks, where stresses are known, u j = u +? u? will be a better choice for nodal variables, and the second complex component of (6) should be considered. After the system of equations has been solved, then (6) can be used again to determine either u + or u? along cracks. Notice that (6) only needs to be evaluated once at each node along the crack. This approach to solving elastostatic problems has an advantage and a disadvantage. The disadvantage is that the traction potential is the integral of the tractions and therefore has an unknown constant involved that must be included in the list of nodal variables. There is a constant for each disjointed boundary in the problem (including cracks as part of the boundaries). It is a worthy task to gure out the best ways this constant can be included into the system of equations. Since the number of these constants is probably much smaller than the number of nodal variables, its main drawback is not that it is adding more equations into the system, but rather that we must nd the most appropriate equations to include them. Work dealing with these constants have already been published (see Chen 993 for example). The advantage of using (6) to solve elastostatic problems numerically is that (6) is a singular integral equation with a Cauchy type kernel whose singularity is straight forward to remove, namely 4

15 PV u j (z; z) z dz = PV (u j (z; z)? u j (z o ; z o )) z dz+ + PV u j (z o ; z o ) z dz; (8) where is for example a boundary element. The rst integral on the right hand side of (8) is non-singular, the second integral can be evaluated in closed form. If q (z o ; z o ) = e ez [u j(z o ; z o ) z (?i ln ( ))] dz = (9) = fu j (z o ; z o ) z [?i ln ( )]g; where ln means the logarithm has the branch cut on the side of, so the branch cut does not intersect in (9) as z o is also on the side of. Then the last integral in (8) will be equal to (q + + q? )=. Since ln + ( )? ln? ( ) = i; this result can be written in terms of one kind of logarithm (see Aparicio & Atkinson (997) for more details in a numerical implementation of this procedure). Another alternative that has appeared in the literature is integrating (6) by parts. This is possible because e ez [u j(z; z) z (?i ln euj (z; z) ( ))] = ez +u j (z; z) z : z (?i ln ( )) + Then the unknown is associated with eu=ez (which means that a condition of closure of the crack is needed) and the singularity is of a logarithmic type (see Chen 994 and Chan & Mear 995 for recent works on this topic). 5

16 7 The product of two elastic analytic functions As it was mentioned earlier, the product of two elastic analytic functions does not appear to have any useful application and we merely introduce it here as a formalism. The aim is to dene a product between two elastic analytic functions such that this product and the addition of elastic analytic functions introduced previously constitute a commutative ring (this includes the fact that the product of two elastic analytic functions is also an elastic analytic function); and the elastic derivative of the product of two elastic analytic functions operates in a similar way to the derivative of the product of two analytic functions. et and be two analytic functions. In order to simplify the notation, we dene [(z); (z)] = G (k(z)? z0 (z)? (z); (z) + z 0 (z) + (z)): In other words, [; ] is the elastic analytic function associated with the Muskhelishvili potentials and. We also have as a consequence of this u = G R (A 3 u); R A u? za 3 Denition. If u = [ ; ] and u = [ ; ] are two elastic analytic functions dened on the same open subset of the complex plane, then the product of these functions is dened as u u = [ ; ]: The product of two elastic analytic functions satisfy the following properties:. The product and addition dened for elastic analytic functions constitute a commutative ring. The neutral element for the addition is 0 = (0; 0) and the neutral element for the product is = (k? ; )=(G) = [; ].. e(u u )=ez = (eu =ez)u + u (eu =ez). This follows from the fact that e ez [ ; ] = [i( ) 0 ; i( ) 0 ] 6

17 = [(i 0 ) + (i 0 ); (i 0 ) + (i 0 )] = [i 0 ; i 0 ][ ; ] + [ ; ][i 0 ; i 0 ]: 3. (u u ) f = u (u f) = (u f)u. This last property proves that when the translated product is applied with z o = 0, the set of elastic analytic functions forms an algebra over the integral domain of analytic functions. A product between two elastic analytic functions can be dened such that it will be an algebra over the integral domain of analytic functions with the more general translated product [ ; ] [ ; ] = [ ; ( + z o 0 )( + z o 0 )? z o ( ) 0 ]: (0) The set of elastic analytic functions is also a commutative ring under the above product with 0 = (0; 0) and = [; ] and the property above is also satised. Denition. The transpose of an elastic analytic function u = [; ] is dened as u T = [ ; ]. Denition. The determinant of an elastic analytic function u = [; ] is dened as det(u) =. The following properties can easily be veried. The transpose of an elastic analytic function is also an elastic analytic function. The determinant of an elastic analytic function is an analytic function.. uu T = det(u). From this property we can conclude that if an elastic analytic function in has a determinant that does not vanish anywhere in, then it has an inverse given by u? = u T det(u) which is also an elastic analytic function in. 7

18 3. det(u T ) = det(u). 4. det(u? ) = =det(u). 5. (u T ) T = u. 6. (u f) T = u T f. 7. (u f)? = u? (=f). 8. (u u ) T = u T u T. 9. det(u u ) = det(u )det(u ). 0. (u u )? = u? u?.. If u = u T, then u can be written as u = f where f is an analytic function. In this case it is said that u is symmetrical. A typical example of a symmetrical elastic analytic function is uu T.. eu T =ez = (eu=ez) T. 3. eu? =ez = (eu=ez)(?u? ). In the case of the product dened in (0), the transpose, determinant and inverse of an elastic analytic function are given by [; ] T = [ + z o 0 ;? z o ( + z o 0 ) 0 ] det[; ] = ( + z o 0 ) [; ]? = ; + z o 0 + z o 0 : All properties mentioned above are equally satised by considering the product (0), the denitions above and the translated product between an elastic analytic function and an analytic function. 8

19 The set of elastic analytic functions in a certain open subset of the complex plane can also be written as the direct sum of two principal ideals I and I generated by = [; 0]; = [0; ]; respectively. We conclude this section by pointing out that the product of elastic analytic functions is not an integral domain. If u u = 0, it does not necessarily mean that either u = 0 or u = 0. However, if u has a non-vanishing determinant then it can be concluded from u u = 0 that u = 0. 8 Taylor expansions of elastic analytic functions The property that elastic analytic functions can be expanded in Taylor series is a direct consequence of the fact that Muskhelishvili potentials associated with it can be expanded in Taylor series. et us assume that an elastic analytic function u has an expansion u(z; z) = X n=0 a n z () n : () Taking the elastic derivative n times and evaluating the resulting expression for z = z o, we obtain e n u ez n (z o; z o ) = a n (n!i n ): Substituting the above expression into () we obtain the Taylor expansion of an elastic analytic function u(z; z) = X n=0 en u n! ez (z o; z n o ) z 9 Path independent integrals z? i n : The subject of path independent integrals in the theory of elasticity has interested the scientic community working in the area since the introduction of the J (or F l ) integral 9

20 by Eshelby (956) and Rice (968). The main implication of this path independent integral lies in its interpretation as the \force on a defect" and its link with the energy release rate and the stress intensity factor in cracks. Gunther (96) and Knowles & Sternberg (97) derived two more path independent integrals by applying a restricted version of Noether's theorem (Noether 98) to the equations of elasticity (see also Eshelby (975)). Edelen (98) correctly suggested that more path independent integrals could be derived from variational principles in elastostatics. Olver (984) proved that in three dimensions there is a nite number of non-trivial conservation laws depending on the position, displacement and its gradient, while in two dimensions there is an innite number of families of independent symmetries and conservation laws. Some of these path independent integrals were already derived by Tsamasphyros & Theocaris (98) using the property that analytic compositions of Muskhelishvili potentials are analytic functions. For other works on path independent integrals, the reader is referred to Ioakimidis & Anastasselou (993), Dong (994) and Aparicio & Atkinson (997). In the case of a two-dimensional elastostatic medium which is also linear homogeneous and istotropic, Olver (984) proved that all path independent integrals that depend only on the derivatives of the displacements are of the form A = G( + ( + G)iB + C; () where A is the complex density of the path independent integral, i.e. the integral is of the form I [Re(A )n x + Im(A )n y ]ds = ia dz = 0; is given =? G (z00 (z) + 0 (z)); is given by = i( + G)0 (z) ; + G 0

21 and B and C are analytic functions of. In order to write () in the same notation that has been used throughout this paper, we introduce the functions F ( 0 ) =?( + G)B() F ( 0 ) =?ic() A = ia : By substituting the above expression into () we obtain A = (z 00 (z) + 0 (z))f 0 (0 ) + F ( 0 ) + F ( 0 ): (3) It can easily be veried is a real function and therefore, from Green's theorem in complex form Re Adz d we obtain I Re Adz = 0: In the more general case where the symmetry conservation law is also allowed to depend on the position, equation (3) becomes A = (z 00 (z) + 0 (z; F (z; 0 ) + F (z; 0 ) + (z; 0 ) : Here, both F (z; 0 ) and F (z; 0 ) are analytic functions in each of their arguments. It will be proved in the remaining part of this section that path independent integrals of the form (4) with a complex density given by (5) can be written as the real or imaginary part of R u dz = 0 where u is an elastic analytic function in. Since F (z; 0 ) and F (z; 0 ) are arbitrary analytic functions, without loss of generality, (5) can be written as

22 A = (z 00 (z) + 0 (z; F (z; 0 ) + F (z; 0 ) + F (z; 0 ) + (z; 0 ) : We introduce the conjugate expression of (6) A y =?(z 00 (z)+ 0 (z; 0 0 +F (z; 0 )+F (z; 0 )?F (z; 0 (z; 0 ) : It can easily be veried that ia y is also the complex density of a path independent integral of the form (4). In other words, we can construct a complex path independent integral I I Re Adz + ire ia y dz Through elementary manipulations, the above integral can also be written as " A + A y! dz + A? Ay! dz # = 0: Substituting (6) and (7) into the above expression we obtain ( (F + F )dz + " (z F et u be the elastic analytic function in dened by # dz ) = 0: (8) u = G if ; 0 + if : (9) Substituting the above expression into (3) and comparing the result with (8), we conclude that I I I Re Adz + ire ia y dz = R u dz = @ An immediate implication from this result is that conservation laws with a complex density of the form (6) can be seen as the equilibrium condition of a \transformed" elastic state. If u is an elastic analytic function in and z o a complex number in, then we can dene another elastic analytic function p in as

23 z p(z; z) = u(; ) d where the integral is carried along any path entirely contained in which goes from z o to z. If u is associated with a particular conservation law as described above, then equation (30) can be seen as the equilibrium of forces in a closed subset of associated with the elastic analytic function p. 0 The asymptotic behaviour of an elastic analytic function near a crack tip et us consider a crack with tip at z o and whose tangent at its tip makes an angle with respect to the horintal ( = 0 means the crack goes to the right of z o ). The asymptotic behaviour of an elastic analytic function near the crack tip is given by u(z; z) u(z o ; z o ) + p k z p z? (3) as z! z o. Here k is a constant function given by k = 3?(K II + ik I )e i= ; K II? i K I e?3i= where K I and K II are the mode I and II stress intensity factors respectively. It can be veried that (3) leads to the well known asymptotic behaviour of a in-plane elastic state near a crack tip (Freund 990). The asymptotic behaviour of the Muskhelishvili potentials of an in-plane elastic state near the crack tip of an inclined crack can be seen explicitly written in Aparicio & Atkinson (997). On the other hand, the J and J path independent integrals can be written in the form (6) and (7) respectively by making + F (z; 0 k ) =? i( 0 ) ; F (z; 0 ) = 0: 4G Substituting the above expressions into (9), we obtain u = + k [(0 ) ; 0 0 ] = ( + k)g eu ez ( + ) R A 3 eu ez ; (3) 3

24 where u = [; ], and therefore J + ij = R u dz = ( + k)gr From (3) we obtain that eu ez ( + ) R A 3 eu ez dz: and R A 3 eu ez = i(k II + ik I )e i= 4G p p ; (33) eu ez ( + ) = p "? i(k II + ik I )e i= p z? ; i(3k II? ik I )e?3i= p z? #? iz o(k II + ik I )e p i= : (34) (z? ) 3 Substituting (33) and (34) into (3) we obtain u =? + k 6 h(k I? K II? ik I K II )e i ; (K I + 3K II + ik I K II )e?i i z Integrating along any that surrounds z o : and applying the Cauchy Integral formula (4) we nd the values of the J i integral around a crack tip I J + ij = R u dz =? + k 8G (K I + K II? ik I K II )e i This formulation is consistent with the equation J i = G ij m j m = (? cos ;? sin ) where is a unit vector tangent to the crack at its tip and pointing to the direction of crack advance and G ij is the energy release rate tensor introduced by Atkinson & Aparicio (998) which is dened by G = G = + k 8G (K I + K II ) G =?G = + k 8G (K IK II ): In the particular case where the crack is horintal starting at its tip towards the left ( = and m = (; 0)) and under mode I loading, we obtain the well-known formula for the value of J around the crack tip. J = ( + k)k I 8G 4

25 Conclusions We have dened elastic analytic functions, that is, functions of the kind u : C! C that obey equations (3) and (4). We have established that the real and imaginary part of the rst complex component satisfy the Navier equation of plane elasticity (homogeneous, linear and isotropic) and the real and imaginary part of the second complex component behave like a traction potential (its derivative along a line is proportional to the applied tractions on it) and satisfy the Navier equation for plane strain with a Poisson coecient equal to unity. Algebraical operations have been dened such that the set of elastic analytic functions constitutes a commutative algebra over the real eld and a module over the set of analytic functions. Additionally, a derivative and an integral have been dened such that they will interact with elastic analytic functions like complex dierentiation and integration do with normal analytic functions. We have in particular that the elastic integral around a contour, enclosing a region where the integrand is elastic analytic, vanishes. We also have derived a formula similar to the Cauchy integral formula and Plemelj formulae for elastic analytic functions. Finally we have established that path independent integrals of the J-type (that is, path independent integrals that can be derived from Noether's theorem whose integrand only depends on the position and derivatives of the displacements) can be written as the elastic integral of an elastic analytic function. Elastic complex analysis can provide additional tools in the resolution of two dimensional elastostatic problems in linearly homogeneous and isotropic media. It can be particularly useful in two dimensional problems involving cracks since the Cauchy integral formula for elastic analytic functions is a weak version of Somigliana integral formula with a Cauchy type singularity and has the equivalent of Plemelj formulae, therefore, it can be a powerful tool in the resolution of crack problems through the boundary element method. Elastic complex analysis also provides a better understanding of the theory behind path independent integrals in two dimensional elasticity. 5

26 Acknowledgment. The author would like to thank Prof. M.K. Pidcock for revising this work and Dr. W.R.B. ionheart for his useful comments on the algebra of elastic analytic functions. 6

27 Appendix A Derivation of the inverse formula (7). et be an open smooth oriented curve not intersecting itself that goes from z to z, let g be a complex function dened as g(z) = (z? z ) s z? z z? z ; where the branch cut of the square root has been chosen to be along. et u j = (u j ; p j ) be a function u j :! C that satises the Holder condition on such that u + p vanishes at the end points of, is dierentiable on and its derivative satises the Holder condition on. It is the aim of this section to nd a closed form inversion formula for the integral equation u a (z o ; z o ) = PV u j (z; z) z et us dene the function q(z o ; z o ) = u(z o ; z o ) g(z o ) = u j (z; z) z dz: dz g(z o ) q is a function which is elastic analytic in C?. We can therefore apply to it formula (4) in a region surrounded by a contour that encloses in the clockwise direction and a circular contour centered at some point near and a large radius. The result is q + z dz? q? z dz+ I + q z dz = q(z o ; z o ): (35) The integral on the right hand side of (5) has a Cauchy type singularity, and hence, u(z; z) is of order O(=z) as z!. Since g(z) is of order O(z) as z!, we conclude that q(z; z) is of order O() as z!. This means that the last integral in the above equation tends to a constant A as!. By noting that q +? q? = u + g +? u? g? = (u + + u? ) g + = u a g + 7

28 since g? = g +, we obtain from (35) (u a g + ) z Using (6), we establish that u + g + + u? g? and since u + g + + u? g? = u j g + we obtain (u j g + )(z o ; z o ) = PV dz + A = (u g)(z o ; z o ): = PV (u a g + ) z dz + A; (u a g + ) z dz + A: Equation (7) follows from the above equation by multiplying it by =g +. 8

29 References [] Aparicio, N.D. & Pidcock, M.K. 996 The boundary inverse problem for the aplace equation in two dimensions. Inverse Problems., [] Aparicio, N.D. & Atkinson, C. 997 Ancillary solutions for curved crack stress analysis. Proc. R. Soc. ond. A 453, [3] Atkinson, C. & Aparicio, N.D. 994 An inverse problem method for antiplane crack detection in elastic materials. Proc. R. Soc. ond. A 445, [4] Atkinson, C. & Aparicio, N.D. 998 Fracture detection problems: applications and limitations of the energy momentum tensor and related invariants. Int. J. Solids Struct. To appear. [5] Chang, C. & Mear, M.E. A boundary element method for two dimensional linear elastic fracture analysis. Int. J. Fract. 74, 9-5. [6] Chen, Y New Fredholm integral-equation for multiple crack problem in plane elasticity and antiplane elasticity. Int. J. Fract. 64, [7] Chen, Y.. & Cheung, Y.K. 994 A new boundary integral-equation for notch problem of plane elasticity. Int. J. Fract. 66, [8] Dong, X.Y. 994 Representation of stress intensity factors by path-integrals of the rst stress invariant or antiplane displacement for general -dimensional static problems. Int. J. Fract. 68, [9] Edelen, D.G.B. 98 Aspects of variational arguments in the theory of elasticity: fact and folklore. Int. J. Solids Struct. 7, [0] Eshelby, J.D. 956 The continuum theory of lattice deects. In Solid State Physics. (ed. F. Seitz & D. Turnbull), vol. 3, pp. 79 { 44. New York: Academic Press. [] Eshelby, J.D. 975 The elastic energy-momentum tensor. Journal of Elasticity. 5,

30 [] Freund,.B. 990 Dynamic Fracture Mechanics. Cambridge University Press. Cambridge, UK. [3] Gunther, W. 96 Abh. braunschw. wiss. Ges. 4, 54. [4] Ioakimidis, N.I. & Anastasselou, E.G. 993 Application of the Green and the Rayleigh-Green reciprocal identities to path-independent integrals in two- and three-dimensional elasticity. Acta Mechanica. 98, [5] Knowles, J.K. & Sternberg, E. 97 On a class of conservation laws in linearized and nite elastostatics. Arch. Rational Mech. Anal. 44, [6] MacGregor, C.W. 935 The potential function method for the solution of twodimensional stress problems. Transactions of the American Mathematical Society. 38, [7] Muskhelishvili, N.I. 958 Singular Integral Equations, Wolters-Noordho Publishing, Groningen, The Netherlands. [8] Muskhelishvili, N.I. 963 Some Basic Problems of the Mathematical Theory of Elasticity, 4th edn. Wolters-Noordho Publishing, Groningen, The Netherlands. [9] Noether, E. 98 Invariante Variationsprobleme. Math.-Physik. Kl., [0] Olver, P.J. 984 Conservation laws in elasticity. I. General results. Arch. Rational Mech. Anal. 85, - 9. [] Olver, P.J. 984 Conservation laws in elasticity. II. inear homogeneus isotropic elastostatics. Arch. Rational Mech. Anal. 85, [] Rice, J.R. 968 A path independent integral and the approximate analysis of strain concentrations by notches and cracks. J. appl. Mech. 35, [3] Tsamasphyros, G.J & Theocaris, P.S. 98 A new concept of path-independent integrals for plane elasticity. J. Elast., [4] Westergaard, H.M. 939 Bearing pressures and Cracks. ASME Journal of Applied Mechanics. 6, A49 - A53. 30