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On the compatibility equations of nonlinear and linear elasticity in the presence of boundary conditions

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Abstract

We use Hodge-type orthogonal decompositions for studying the compatibility equations of the displacement gradient and the linear strain with prescribed boundary displacements. We show that the displacement gradient is compatible if and only if for any equilibrated virtual first Piola–Kirchhoff stress tensor field, the virtual work done by the displacement gradient is equal to the virtual work done by the prescribed boundary displacements. This condition is very similar to the classical compatibility equations for the linear strain. Since these compatibility equations for linear and nonlinear strains involve infinite-dimensional spaces and consequently are not easy to use in practice, we derive alternative compatibility equations, which are written in terms of some finite-dimensional spaces and are more useful in practice. Using these new compatibility equations, we present some non-trivial examples that show that compatible strains may become incompatible in the presence of prescribed boundary displacements.

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Authors and Affiliations

  1. School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Arzhang Angoshtari

  2. School of Civil and Environmental Engineering, The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Arash Yavari

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  1. Arzhang Angoshtari
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  2. Arash Yavari
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Correspondence to Arash Yavari.

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Angoshtari, A., Yavari, A. On the compatibility equations of nonlinear and linear elasticity in the presence of boundary conditions. Z. Angew. Math. Phys. 66, 3627–3644 (2015). https://doi.org/10.1007/s00033-015-0575-1

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  • DOI: https://doi.org/10.1007/s00033-015-0575-1

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