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We extend to finite elasticity the Data-Driven formulation of geometrically linear elasticity presented in Conti, Muller, Ortiz, Arch. Ration. Mech. Anal. 229, 79-123, 2018. The main focus of this paper concerns the formulation of a suitable framework in which the Data-Driven problem of finite elasticity is well-posed in the sense of existence of solutions. We confine attention to deformation gradients $F in L^p(Omega;mathbb{R}^{ntimes n})$ and first Piola-Kirchhoff stresses $P in L^q(Omega;mathbb{R}^{ntimes n})$, with $(p,q)in(1,infty)$ and $1/p+1/q=1$. We assume that the material behavior is described by means of a material data set containing all the states $(F,P)$ that can be attained by the material, and develop germane notions of coercivity and closedness of the material data set. Within this framework, we put forth conditions ensuring the existence of solutions. We exhibit specific examples of two- and three-dimensional material data sets that fit the present setting and are compatible with material frame indifference.
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This paper is devoted to studying impedance eigenvalues (that is, eigenvalues of a particular Dirichlet-to-Neumann map) for the time harmonic linear elastic wave problem, and their potential use as target-signatures for fluid-solid interaction problems. We first consider several possible families of eigenvalues of the elasticity problem, focusing on certain impedance eigenvalues that are an analogue of Steklov eigenvalues. We show that one of these families arises naturally in inverse scattering. We also analyse their approximation from far field measurements of the scattered pressure field in the fluid, and illustrate several alternative methods of approximation in the case of an isotropic elastic disk.
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