Antonino Morassi

Stable Determination of an Inclusion in an Elastic Body by Boundary Measurements

We consider the inverse problem of identifying an unknown inclusion contained in an elastic body by the Dirichlet-to-Neumann map. The body is made by linearly elastic, homogeneous, and isotropic material. The Lame moduli of the inclusion are constant and different from those of the surrounding material. Under mild a priori regularity assumptions on the unknown defect, we establish a logarithmic stability estimate. Main tools are propagation of smallness arguments based on three-spheres inequality for solutions to the Lame system and a refined asymptotic analysis of the fundamental solution of the Lame system in presence of an inclusion which shows surprising features.

Lipschitz continuous dependence of piecewise constant Lamé coefficients from boundary data: the case of non-flat interfaces

We consider the inverse problem of determining the Lame moduli for a piecewise constant elasticity tensor , where is a known finite partition of the body Ω, from the Dirichlet-to-Neumann map. We prove that Lipschitz stability estimates can be derived under regularity assumptions on the interfaces.

Identification of an open crack in a beam with variable profile by two resonant frequencies

We consider the identification of a single open crack in a simply supported beam having nonuniform smooth profile and undergoing infinitesimal in-plane flexural vibration. The profile is assumed to be symmetric with respect to the mid-point of the beam axis. The crack is modeled by inserting a rotational linearly elastic spring at the damaged cross-section. We establish sufficient conditions for the unique identification of the crack by a suitable pair of natural frequency data, and we present a constructive algorithm for determining the damage parameters. The result is proved under a technical a priori assumption on the zeros of a suitable function determined in terms of the eigenfunctions of the problem. Extensions to beams under different sets of end conditions are also discussed. Theoretical results are confirmed by an extensive numerical investigation, both on simulated and experimental data.

Doubling inequalities for anisotropic plate equations and applications to size estimates of inclusions

We prove the upper and lower estimates of the area of an unknown elastic inclusion in a thin plate by one boundary measurement. The plate is made of non-homogeneous linearly elastic material belonging to a general class of anisotropy and the domain of the inclusion is a measurable subset of the plate. The size estimates are expressed in terms of the work exerted by a couple field applied at the boundary and of the induced transversal displacement and its normal derivative taken at the boundary of the plate. The main new mathematical tool is a doubling inequality for solutions to fourth-order elliptic equations whose principal part P(x, D) is the product of two second-order elliptic operators P1(x, D), P2(x, D) such that P1(0, D) = P2(0, D). The proof of the doubling inequality is based on the Carleman method, a sharp three-spheres inequality and a bootstrapping argument.

Global stability for an inverse problem in soil–structure interaction

We consider the inverse problem of determining the Winkler subgrade reaction coefficient of a slab foundation modelled as a thin elastic plate clamped at the boundary. The plate is loaded by a concentrated force and its transversal deflection is measured at the interior points. We prove a global Hölder stability estimate under (mild) regularity assumptions on the unknown coefficient.

Size estimates for fat inclusions in an isotropic Reissner-Mindlin plate

In this paper we consider the inverse problem of determining, within an elastic isotropic thick plate modelled by the Reissner-Mindlin theory, the possible presence of an inclusion made of a different elastic material. Under some a priori assumptions on the inclusion, we deduce constructive upper and lower estimates of the area of the inclusion in terms of a scalar quantity related to the work developed in deforming the plate by applying simultaneously a couple field and a transverse force field at the boundary of the plate. The approach allows to consider plates with boundary of Lipschitz class.