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Differential Complexes in Continuum Mechanics

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Abstract

We study some differential complexes in continuum mechanics that involve both symmetric and non-symmetric second-order tensors. In particular, we show that the tensorial analogue of the standard grad-curl-div complex can simultaneously describe the kinematics and the kinetics of motion of a continuum. The relation between this complex and the de Rham complex allows one to readily derive the necessary and sufficient conditions for the compatibility of displacement gradient and the existence of stress functions on non-contractible bodies.We also derive the local compatibility equations in terms of the Green deformation tensor for motions of 2D and 3D bodies, and shells in curved ambient spaces with constant curvatures.

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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

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

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Communicated by M. Ortiz

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Angoshtari, A., Yavari, A. Differential Complexes in Continuum Mechanics. Arch Rational Mech Anal 216, 193–220 (2015). https://doi.org/10.1007/s00205-014-0806-1

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  • DOI: https://doi.org/10.1007/s00205-014-0806-1

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