IOP PUBLISHING INVERSE PROBLEMS Inverse Problems 25 (2009) 045009 (14pp) doi:10.1088/0266-5611/25/4/045009 Detecting general inclusions in elastic plates Antonino Morassi 1 , Edi Rosset 2 and Sergio Vessella 3 1 Dipartimento di Georisorse e Territorio, Universit` a degli Studi di Udine, via Cotonificio 114, 33100 Udine, Italy 2 Dipartimento di Matematica e Informatica, Universit` a degli Studi di Trieste, via Valerio 12/1, 34127 Trieste, Italy 3 DIMAD, Universit` a degli Studi di Firenze, Via Lombroso 6/17, 50134 Firenze, Italy E-mail: antonino.morassi@uniud.it, rossedi@univ.trieste.it and sergio.vessella@dmd.unifi.it Received 4 November 2008 Published 17 February 2009 Online at stacks.iop.org/IP/25/045009 Abstract We consider the problem of determining, within an elastic isotropic thin plate, the possible presence of an inclusion made of different elastic material. We prove constructive upper and lower estimates of the area of the inclusion 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. 1. Introduction This paper continues with the line of research initiated in [16] and aims to identify an inclusion in a thin plate from measurements of a couple field and of the induced transversal displacement and its normal derivative taken at the boundary of the plate. Let us briefly introduce the mathematical formulation of the problem and the results obtained in [16]. Let  be the middle plane of a thin plate having uniform thickness h and let us denote by D,D ⊂⊂  and D measurable, the subset corresponding to the unknown inclusion. The plate is assumed to be made of linear elastic material, and an assigned couple field ˆ M = ˆ M τ n + ˆ M n τ is applied to ∂. Here, n and τ denote the unit outer normal and the unit tangent at the boundary ∂. Working in the framework of the Kirchhoff–Love theory, the transversal displacement w of the middle plane of the plate satisfies the following fourth-order Neumann boundary value problem: div(div((χ \D P + χ D  P)∇ 2 w)) = 0, in , (1.1) (P∇ 2 w)n · n =− ˆ M n, on ∂, (1.2) div(P∇ 2 w) · n + ((P∇ 2 w)n · τ), s = ( ˆ M τ ), s , on ∂. (1.3) 0266-5611/09/045009+14$30.00 © 2009 IOP Publishing Ltd Printed in the UK 1