THE LINEARIZATION METHOD IN GEOMETRICAL INVERSE PROBLEMS OF THE THEORY OF ELASTICITY

Both rigorous and linearized formulations of the boundary-geometrical inverse problem (BIP) of the theory of elasticity are presented for an elastic half-space with a defect modelled by an elastic inclusion or cavity. When formulating the linearized BIP it is assumed that the bou ndary of the defect can be specified in a local system of coordinates connected with the boundary of the defect The aim of the solution of t he linearized problem is to find a scalar function, namely, the shape variation, i.e. the normal distance between the known boundary and the points of the boundary to be found. The plane and the anti-plane prob lems of the theory of elasticity are considered. Using the boundary in tegral equations the rigorous BIP can be reduced to a non-linear syste m of integro-differential equations, and the linearized BIP can be red uced to a linear system of integral equations. It is found that the li nearized boundary integral equations preserve the order of singulariti es characteristic of the ordinary boundary integral equations. To solv e the rigorous inverse problem an iteration procedure of successive ap proximations of the shape of the defect is proposed. In this case it s uffices to solve a linearized BIP at each iteration step. (C) 1997 Els evier Science Ltd. All rights reserved.