Data driven problems in elasticity

TL;DR

This paper introduces Data-Driven problems in elasticity, focusing on minimizing the distance between material data sets and compatible strain-stress fields, and explores convergence, relaxation, and differences from classical approaches.

Contribution

It formulates a new Data-Driven approach to elasticity problems, analyzing convergence, relaxation, and fundamental differences from classical energy-based methods.

Findings

Classical solutions are recovered in linear elasticity.

Conditions for convergence of Data-Driven solutions are identified.

Relaxation in the Data-Driven framework differs from classical energy relaxation.

Abstract

We consider a new class of problems in elasticity, referred to as Data-Driven problems, defined on the space of strain-stress field pairs, or phase space. The problem consists of minimizing the distance between a given material data set and the subspace of compatible strain fields and stress fields in equilibrium. We find that the classical solutions are recovered in the case of linear elasticity. We identify conditions for convergence of Data-Driven solutions corresponding to sequences of ap- proximating material data sets. Specialization to constant material data set sequences in turn establishes an appropriate notion of relaxation. We find that relaxation within this Data-Driven framework is fundamentally different from the classical relaxation of energy functions. For instance, we show that in the Data-Driven framework the relaxation of a bistable material leads to material data sets that are not graphs.