Griffith energies as small strain limit of nonlinear models for nonsimple brittle materials

Manuel Friedrich; Manuel Friedrich

doi:10.3934/mine.2020005

Mathematics in Engineering

2020, Volume 2, Issue 1: 75-100. doi: 10.3934/mine.2020005

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Griffith energies as small strain limit of nonlinear models for nonsimple brittle materials

Manuel Friedrich ^,

Applied Mathematics Münster, University of Münster, Einsteinstrasse 62, 48149 Münster, Germany

Received: 24 June 2019 Accepted: 15 September 2019 Published: 26 November 2019

We consider a nonlinear, frame indifferent Griffith model for nonsimple brittle materials where the elastic energy also depends on the second gradient of the deformations. In the framework of free discontinuity and gradient discontinuity problems, we prove existence of minimizers for boundary value problems. We then pass to a small strain limit in terms of suitably rescaled displacement fields and show that the nonlinear energies can be identified with a linear Griffith model in the sense of Γ-convergence. This complements the study in ^[39] by providing a linearization result in arbitrary space dimensions.
- brittle materials,
- variational fracture,
- nonsimple materials,
- free discontinuity problems,
- Griffith energies,
- Γ-convergence,
- functions of bounded variation and deformation
Citation: Manuel Friedrich. Griffith energies as small strain limit of nonlinear models for nonsimple brittle materials[J]. Mathematics in Engineering, 2020, 2(1): 75-100. doi: 10.3934/mine.2020005

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Abstract

We consider a nonlinear, frame indifferent Griffith model for nonsimple brittle materials where the elastic energy also depends on the second gradient of the deformations. In the framework of free discontinuity and gradient discontinuity problems, we prove existence of minimizers for boundary value problems. We then pass to a small strain limit in terms of suitably rescaled displacement fields and show that the nonlinear energies can be identified with a linear Griffith model in the sense of Γ-convergence. This complements the study in ^[39] by providing a linearization result in arbitrary space dimensions.

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