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
,
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,
, 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.
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.
Keywords
nonlinear inverse problem, inclusion geometry, antiplane
problem
