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Abstract. We consider Cauchy's equation of motion for hyperelastic materials. The solution of this nonlinear initial-boundary value problem is the vector field which discribes the displacement which a particle of this material perceives when exposed to stress and external forces. This equation is of greatest relevance when investigating the behaviour of elastic, anisotropic composites and for the detection of defects in such materials from boundary measurements. Thus results on unique solvability and continuous dependence from the initial values are of large interest in materials research and structural health monitoring. In this article we present such a result, provided that reasonable smoothness assumptions for the displacement field and the boundary of the domain are satisfied, for a certain class of hyperelastic materials where the first Piola-Kirchhoff tensor is written as a conic combination of finitely many given tensors.

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.

Dispersion curves of elastic guided waves in plates can be efficiently computed by the Strip-Element Method. This method is based on a finite-element discretization in the thickness direction of the plate and leads to an eigenvalue problem relating frequencies to wavenumbers of the wave modes. In this paper we present a rigorous mathematical background of the Strip-Element Method for anisotropic media including a thorough analysis of the corresponding infinite-dimensional eigenvalue problem as well as a proof of the existence of eigenvalues.