Data-Driven Problems in Elasticity
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 approximating 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.
© 2018 Springer-Verlag GmbH Germany, part of Springer Nature. Received: 16 August 2017; Accepted: 19 December 2017; First Online: 20 January 2018. This work was partially supported by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 1060 "The mathematics of emergent effects".
Submitted - 1708.02880.pdf