Conservation laws in nonhomogeneous plane elastostatics

Conservation laws in nonhomogeneous elastostatics are studied by means of a special version of Noether's theorem on invariant variational problems. The investigation is restricted to isotropic, linearly elastic bodies with smoothly varying elastic moduli and subjected to plane strain deformation fields. By applying Lie's infinitesimal criterion on the invariance of the action integral under a continuous transformation group, it is found that the materials admitting conservation laws are those for which the elastic parameters satisfy a first-order linear partial differential equation. While the general solution to this equation is readily available, a special emphasis is placed on extending Rice's J-integral to certain classes of materials with varying Young's modulus in the direction of a crack line. The extended integrals are related to stress intensity factors at the crack tip, and their use is illustrated by applying them to two simple examples. Also, the circumstances under which these integrals can be generalized to the three-dimensional case are briefly discussed. (C) 1997 Elsevier Science Ltd.