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A Derivation of Linearized Griffith Energies from Nonlinear Models

Abstract

We derive Griffith functionals in the framework of linearized elasticity from nonlinear and frame indifferent energies in a brittle fracture via \({\Gamma}\)-convergence. The convergence is given in terms of rescaled displacement fields measuring the distance of deformations from piecewise rigid motions. The configurations of the limiting model consist of partitions of the material, corresponding piecewise rigid deformations and displacement fields which are defined separately on each component of the cracked body. Apart from the linearized Griffith energy the limiting functional also comprises the segmentation energy, which is necessary to disconnect the parts of the specimen.

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Acknowledgements

Open access funding provided by University ofVienna. I am very grateful to Bernd Schmidt formany stimulating discussions and valuable comments from which the results of this paper and their exposition have benefited a lot. Moreover, I am gratefully indebted to the referee for her/his careful reading of the manuscript and many helpful suggestions.

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Correspondence to Manuel Friedrich.

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Communicated by G. M. Ortiz

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Friedrich, M. A Derivation of Linearized Griffith Energies from Nonlinear Models. Arch Rational Mech Anal 225, 425–467 (2017). https://doi.org/10.1007/s00205-017-1108-1

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  • DOI: https://doi.org/10.1007/s00205-017-1108-1