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Data-Driven Problems in Elasticity

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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 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.

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Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    S. Conti, S. Müller & M. Ortiz

  2. Hausdorff Center for Mathematics, Endenicher Allee 60, 53115, Bonn, Germany

    S. Conti, S. Müller & M. Ortiz

  3. Division of Engineering and Applied Science, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA, 91125, USA

    M. Ortiz

Authors
  1. S. Conti
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  2. S. Müller
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  3. M. Ortiz
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Correspondence to S. Müller.

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Communicated by R. James

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Conti, S., Müller, S. & Ortiz, M. Data-Driven Problems in Elasticity. Arch Rational Mech Anal 229, 79–123 (2018). https://doi.org/10.1007/s00205-017-1214-0

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