Data-Driven Finite Elasticity

We extend to finite elasticity the Data-Driven formulation of geometrically linear elasticity presented in Conti, M"uller, 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 $F\in L^p(\mathrm{\Omega };{\mathbb{R}}^{n\times n})$ and first Piola-Kirchhoff stresses $P\in L^q(\mathrm{\Omega };{\mathbb{R}}^{n\times n})$, with $(p,q)\in (1,\mathrm{\infty })$ and $1\mathrm{/}p+1\mathrm{/}q=1$. 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.