Abstract
Let Ω be a bounded open connected subset of ℝ3 with a sufficiently smooth boundary. The additional condition ∫ det ▽ψ dx ≦ vol ψ(Ω) is imposed on the admissible deformations ψ: ¯Ω → ℝ of a hyperelastic body whose reference configuration is ¯Ω. We show that the associated minimization problem provides a mathematical model for matter to come into frictionless contact with itself but not interpenetrate. We also extend J. Ball's theorems on existence to this case by establishing the existence of a minimizer of the energy in the space W 1,p(Ω;ℝ3), p > 3, that is injective almost everywhere.
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Ciarlet, P.G., Nečas, J. Injectivity and self-contact in nonlinear elasticity. Arch. Rational Mech. Anal. 97, 171–188 (1987). https://doi.org/10.1007/BF00250807
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DOI: https://doi.org/10.1007/BF00250807