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Numerical Study of Microstructures in Multiwell Problems in Linear Elasticity

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Variational Views in Mechanics

Part of the book series: Advances in Mechanics and Mathematics ((ACM,volume 46))

Abstract

The free energy density of the prototypical model for solid-to-solid phase transformations with two distinct phases in the low-temperature range is given by the minimum of two quadratic potentials whose minima are attained in distinct stress-free states. Such an energy typically fails to be quasiconvex in the sense of Morrey, so that minimizers may fail to exist and minimizing sequences converge to minimizers of the corresponding relaxed problem. In this contribution, strategies for the derivation of relaxed energies are reviewed and illustrated at well-chosen model problems. More importantly, a new algorithmic approach for the computation of relaxed energies in linearized elasticity under minimal assumptions is proposed and validated based on the model examples discussed before. Additionally, phase diagrams for the relaxed energies in situations which are not included in the analytical formulas are presented and illustrate the predictive power of the algorithm. This contribution continues the discussion of a numerical scheme for the computation of relaxations in nonlinear elasticity presented in Conti and Dolzmann (J Optim Theory Appl 184:43–60, 2020).

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Acknowledgements

This work was partially supported by the Deutsche Forschungsgemeinschaft (DFG German Research Foundation) through SFB 1060, “The mathematics of emergent effects,” project 211504053, and SPP 2256 “Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials,” projects 441211072 and project 441468770.

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Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Bonn, Bonn, Germany

    Sergio Conti

  2. Fakultät für Mathematik, Universität Regensburg, Regensburg, Germany

    Georg Dolzmann

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  1. Sergio Conti
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  2. Georg Dolzmann
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Correspondence to Sergio Conti .

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Editors and Affiliations

  1. DICEA, University of Florence, Florence, Italy

    Paolo Maria Mariano

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Conti, S., Dolzmann, G. (2021). Numerical Study of Microstructures in Multiwell Problems in Linear Elasticity. In: Mariano, P.M. (eds) Variational Views in Mechanics. Advances in Mechanics and Mathematics(), vol 46. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-90051-9_1

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