Data-Driven Finite Elasticity

TL;DR

This paper extends the Data-Driven approach to finite elasticity, establishing a well-posed framework for existence of solutions using material data sets and specific conditions, applicable to 2D and 3D cases.

Contribution

It develops a new framework for Data-Driven finite elasticity ensuring well-posedness and solution existence, with concrete examples fitting physical and frame indifference requirements.

Findings

Framework guarantees existence of solutions under certain conditions.

Material data sets can be constructed to satisfy frame indifference.

Examples demonstrate applicability in 2D and 3D elasticity.

Abstract

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 $F \in L^p(\Omega;\mathbb{R}^{n\times n})$ and first Piola-Kirchhoff stresses $P \in L^q(\Omega;\mathbb{R}^{n\times n})$, 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.