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Uniqueness and Complementary Energy in Nonlinear Elastostatics

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Abstract

Global uniqueness of the smooth stress and deformation to within the usual rigid-body translation and rotation is established in the null traction boundary value problem of nonlinear homogeneous elasticity on a n-dimensional star-shaped region. A complementary energy is postulated to be a function of the Biot stress and to be para-convex and rank-(n-1) convex, conditions analogous to quasi-convexity and rank-(n-2) of the stored energy function. Uniqueness follows immediately from an identity involving the complementary energy and the Piola-Kirchhoff stress. The interrelationship is discussed between the two conditions imposed on the complementary energy, and between these conditions and those known for uniqueness in the linear elastic traction boundary value problem.

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Authors and Affiliations

  1. Department of Mathematics, Heriot-Watt University, Edinburgh, EH14 4AS, U.K

    R.J. Knops

  2. Dipartimento di Matematica Applicata “U.Dini”, Università di Pisa, V. Bonanno 25/B, 56125, Pisa, Italy

    C. Trimarco

  3. Jaguar Engineering Centre, Coventry, CV3 4LF, U.K

    H.T. Williams

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  1. R.J. Knops
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  2. C. Trimarco
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  3. H.T. Williams
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Knops, R., Trimarco, C. & Williams, H. Uniqueness and Complementary Energy in Nonlinear Elastostatics. Meccanica 38, 519–534 (2003). https://doi.org/10.1023/A:1024775130090

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