On the identifiability of the stored energy function of hyperelastic materials from sensor data at the boundary

Abstract

This article addresses the inverse problem of reconstructing the stored energy function of a certain class of hyperelastic materials from partial Cauchy data. It is motivated by so-called structural health monitoring systems, whose idea is to disclose defects in elastic, anisotropic structures from data which are recorded at sensors that are applied at the boundary of the object. Damage affects the spatially varying stored energy function and thus its reconstruction reveals defects of the structure. The dynamic behavior of hyperelastic materials is described by Cauchyʼs equation of motion, a second order, non-linear system of partial differential equations, and thus we get an identification problem for this PDE system. We prove conditions under which this identification problem is uniquely solvable and that the solution continuously depends on the measured data as well as on the initial conditions of the structure. An important assumption is that the stored energy function is a conic combination of finitely many given functions. Moreover, we show that in the linear case such a conic decomposition of the elasticity tensor exists if its entries are in a finite dimensional subspace of the space of continuous functions and some spectral conditions are satisfied. In the case of a homogeneous, isotropic medium this decomposition is explicitly known. As a consequence, we get the result that two sensors are sufficient to identify the two independent material parameters, the Lamé coefficients, of such a medium.