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New Trends in Shape Optimization pp 141–197Cite as

Birkhäuser

Shape- and Topology Optimization for Passive Control of Crack Propagation

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Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 166))

Abstract

In this review article the theoretical foundations for shape-topological sensitivity analysis of elastic energy functional in bodies with nonlinear cracks and inclusions are presented. The results obtained can be used to determine the location and the shape of inclusions which influence in a desirable way the energy release at the crack tip. In contrast to the linear theory, where in principle, crack lips may mutually penetrate, here we employ nonlinear elliptic boundary value problems in non-smooth domains with cracks with non-penetration contact conditions across the crack lips or faces. A shape-topological sensitivity analysis of the associated variational inequalities is performed for the elastic energy functional. Topological derivatives of integral shape functionals for variational inequalities with unilateral boundary conditions are derived. The closed form results are obtained for the Laplacian and linear elasticity in two and three spatial dimensions. Singular geometrical perturbations in the form of cavities or inclusions are considered. In the variational context the singular perturbations are replaced by regular perturbations of bilinear forms. The obtained expressions for topological derivatives are useful in numerical method of shape optimization for contact problems as well as in passive control of crack propagation.

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Acknowledgments

This work has been supported by the DFG EC315 ’Engineering of Advanced Materials’ and by the ANR-12- BS01-0007 Optiform.

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Authors and Affiliations

  1. Institute of Applied Mathematics 2, Friedrich-Alexander University Erlangen-Nürnberg, Cauerstrasse 11, 91058, Erlangen, Germany

    Günter Leugering

  2. Institut Élie Cartan, UMR 7502 Nancy-Université-CNRS-INRIA, Laboratoire de Mathématiques, Université Henri Poincaré Nancy 1, BP 239, 54506, Vandoeuvre Lès Nancy Cedex, France

    Jan Sokołowski

  3. Systems Research Institute of the Polish Academy of Sciences, ul. Newelska 6, 01-447, Warszawa, Poland

    Antoni Zochowski

Authors
  1. Günter Leugering
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  2. Jan Sokołowski
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Correspondence to Günter Leugering .

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Editors and Affiliations

  1. Department Mathematik, Universität Erlangen‐Nürnberg, Erlangen, Germany

    Aldo Pratelli

  2. Department Mathematik, Universität Erlangen-Nürnberg, Erlangen, Germany

    Günter Leugering

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Leugering, G., Sokołowski, J., Zochowski, A. (2015). Shape- and Topology Optimization for Passive Control of Crack Propagation. In: Pratelli, A., Leugering, G. (eds) New Trends in Shape Optimization. International Series of Numerical Mathematics, vol 166. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-17563-8_7

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