Optimal identification of cavities in the Generalized Plane Stress problem in linear elasticity

Antonino Morassi,Edi Rosset,Sergio Vessella

doi:10.4171/jems/1190

Abstract

For the Generalized Plane Stress (GPS) problem in linear elasticity, we obtain an optimal stability estimate of logarithmic type for the inverse problem of determining smooth cavities inside a thin isotropic cylinder from a single boundary measurement of traction and displacement. The result is obtained by reformulating the GPS problem as a Kirchhoff–Love plate-like problem in terms of the Airy function, and by using the strong unique continuation at the boundary for a Kirchhoff–Love plate operator under homogeneous Dirichlet conditions, which has recently been obtained in \[G. Alessandrini et al., Arch. Ration. Mech. Anal. 231 (2019)].

Highlights

IntroductionIn this paper we consider the inverse problem of detecting cavities inside a thin isotropic elastic bounded domain in R2 plate Ω ×
In this paper we consider the inverse problem of detecting cavities inside a thin isotropic elastic bounded domain in R2 plate Ω × − h 2, h 2, where the middle planeΩ is a and h is the constant thickness, subject to a single experiment consisting in applying in-plane boundary loads and measuring the induced displacement at the boundary
Where the middle plane is a and h is the constant thickness, subject to a single experiment consisting in applying in-plane boundary loads and measuring the induced displacement at the boundary

Summary

Introduction

In this paper we consider the inverse problem of detecting cavities inside a thin isotropic elastic bounded domain in R2 plate Ω ×. Aiming at obtaining a strong unique continuation property at the boundary (SUCB) for solutions to the GPS elliptic system, in this paper we have exploited the two dimensional character of the problem (1.1)–(1.6) by using the classical Airy’s transformation, which (locally) reduces the GPS system with homogeneous Neumann boundary conditions to a scalar fourth order Kirchhoff-Love plate’s equation under homogeneous Dirichlet boundary conditions This reformulation allows us to use the finite vanishing rate at the boundary for homogeneous Dirichlet boundary conditions recently obtained in [A-R-V] in the form of a three spheres inequality at the boundary with optimal exponent, and in [M-R-V3] in the form of a doubling inequality at the boundary.

Notation

Airy’s transformation

Proof of the main result

Conclusion

Generalized Plane Stress problem