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Cauchy’s stress theorem for stresses represented by measures

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Abstract

A version of Cauchy’s stress theorem is given in which the stress describing the system of forces in a continuous body is represented by a tensor valued measure with weak divergence a vector valued measure. The system of forces is formalized in the notion of an unbounded Cauchy flux generalizing the bounded Cauchy flux by Gurtin and Martins (Arch Ration Mech Anal 60:305–324, 1976). The main result of the paper says that unbounded Cauchy fluxes are in one-to-one correspondence with tensor valued measures with weak divergence a vector valued measure. Unavoidably, the force transmitted by a surface generally cannot be defined for all surfaces but only for almost every translation of the surface. Also conditions are given guaranteeing that the transmitted force is represented by a measure. These results are proved by using a new homotopy formula for tensor valued measure with weak divergence a vector valued measure.

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Authors and Affiliations

  1. Institute of Mathematics of the AV ČR, Žitná 25, 115 67, Prague 1, Czech Republic

    M. Šilhavý

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  1. M. Šilhavý
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Correspondence to M. Šilhavý.

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Communicated by L. Evans

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Šilhavý, M. Cauchy’s stress theorem for stresses represented by measures. Continuum Mech. Thermodyn. 20, 75–96 (2008). https://doi.org/10.1007/s00161-008-0073-1

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  • DOI: https://doi.org/10.1007/s00161-008-0073-1

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