On the mystery of Calderón’s formula for the geometry of an inclusion in elastic materials

Abstract

I write this paper to pay homage to Marie-Louise Steele and in honor of Charles R. Steele. I have had the pleasure and the honor to serve their journals IJSS and JoMMS, with George Herrmann. They have made Solids & Structures and now Material Sciences a subject of nobility to all of us.We consider the nonlinear inverse problem of determining an inclusion in an elastic body, in antiplane shear loading. The perturbation of the shear modulus due to the inclusion was determined by Calderón (1980) in the case of a small amplitude of perturbation. For the general nonlinear case, the problem is decomposed into two linear problems: a source inverse problem, which determines the geometry of the inclusion, and a Volterra integral equation of the first kind for determining the amplitude. In this paper, we deal only with the determination of the inclusion geometry in the two-dimensional problem. We derive a simple formula for determining the inclusion geometry. This formula enables us to investigate the mystery of Calderón's solution for the linearized perturbation h 0 , raised by Isaacson and Isaacson (1986), in the case of axisymmetry. By using a series method for numerical analysis, they found that the supports of the perturbation, in the linearized theory and the nonlinear theory in the axisymmetric case, are practically the same. We elucidate the mystery by discovering that both theories give exactly the same support of the perturbation, supp(h 0 ) ≡ supp(h), for the general case of geometry and loadings. Then, we discuss an application of the geometry method to locate an inclusion and solve the source inverse problem, which gives an indication of the amplitude of the perturbation.