Application of Betti’s reciprocal work theorem to the location of cracks in three-dimensional elasticity
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Date
1990-04
Authors
Ioakimidis, Nikolaos
Journal Title
Journal ISSN
Volume Title
Publisher
Kluwer Academic Publishers
Abstract
In this note, the solution of the problem of location of a crack C of arbitrary shape inside a finite three-dimensional isotropic elastic medium D is proposed through a combination of experimental and numerical methods and on the basis of the classical Betti’s reciprocal work theorem. More explicitly, the position, shape and orientation of the crack C can be characterized by a set of n unknown quantities a_j. Next, for the aforementioned medium D under consideration both the tractions t_i0 (generally known in advance) and the displacements u_i0 on the whole boundary S of D can become available by using one of the methods of experimental stress analysis. Next, a number m of loadings t_ik is assumed applied on the surface S of D and the corresponding displacements u_ik on the same surface S are computed by using an appropriate numerical method such as here the boundary element method. Under these circumstances and by appropriately using Betti’s reciprocal work theorem we can derive a system of m nonlinear algebraic equations with respect to the n unknown quantities a_j (evidently with m greater than or equal to n). The approximate solution of this system permits the computation of the unknown quantities a_j and, therefore, the determination of the position, shape and orientation of the crack C inside D. Generalizations of the present experimental–computational approach to more complicated crack problems, problems of arrays of cracks, holes, inclusions, etc. inside D are also easily possible.