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Optimal Control of Inclusion and Crack Shapes in Elastic Bodies

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Abstract

The paper is concerned with the control of the shape of rigid and elastic inclusions and crack paths in elastic bodies. We provide the corresponding problem formulations and analyze the shape sensitivity of such inclusions and cracks with respect to different perturbations. Inequality type boundary conditions are imposed at the crack faces to provide a mutual nonpenetration between crack faces. Inclusion and crack shapes are considered as control functions and control objectives, respectively. The cost functional, which is based on the Griffith rupture criterion, characterizes the energy release rate and provides the shape sensitivity with respect to a change of the geometry. We prove an existence of optimal solutions.

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Notes

  1. Taking the same sequence of parameters λ n as in (20), one can easily see that the weak limits in (20) and (31) coincide.

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Acknowledgements

The work was partially supported by RFBR (10-01-00054) and FTP Kadry (Π597). It was further supported by the DFG in the frame of the SFB/TR TRR 30 “Prozessintegrierte Herstellung funktional gradierter Strukturen auf der Grundlage thermomechanisch gekoppelter Phänomene.” The first two authors also acknowledge support by the DFG-EXC 315: “Engineering of Advanced Materials.”

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Authors and Affiliations

  1. Lavrentyev Institute of Hydrodynamics, Russian Academy of Sciences, Novosibirsk, 630090, Russia

    A. Khludnev

  2. Department Mathematik, Lehrstuhl II Universität Erlangen, Nürnberg Cauerstr. 11, 91058, Erlangen, Germany

    G. Leugering

  3. University of Kassel, Heinrich Plett Str. 40, 34132, Kassel, Germany

    M. Specovius-Neugebauer

Authors
  1. A. Khludnev
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  2. G. Leugering
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  3. M. Specovius-Neugebauer
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Correspondence to G. Leugering.

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Communicated by Hans-Josef Pesch.

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Khludnev, A., Leugering, G. & Specovius-Neugebauer, M. Optimal Control of Inclusion and Crack Shapes in Elastic Bodies. J Optim Theory Appl 155, 54–78 (2012). https://doi.org/10.1007/s10957-012-0053-2

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  • DOI: https://doi.org/10.1007/s10957-012-0053-2

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