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Some remarks on the theory of elasticity for compressible Neohookean materials. (English) Zbl 1114.74004
In compressible Neohookean elasticity one minimizes functionals which are composed by the sum of the \(L^2\) norm of the deformation gradient and a nonlinear function of the determinant of the gradient, namely, the energy of the form \(\int_{\Omega}|\nabla u|^2+\varphi(\det\nabla u)\), where \(\varphi\) is a convex function on \((0,\infty)\) with superlinear growth and approaching infinity at zero. The minimizers are sought among deformations \(u\) which map \(\Omega\subset\mathbb R^3\) into \(\mathbb R^3\) and satisfy some notions of invertibility and a Dirichlet boundary condition. S. C. Müller and S. J. Spector [Arch. Ration. Mech. Anal. 131, No. 1, 1–66 (1995; Zbl 0836.73025)] gave a condition of invertibility (INV) which strongly relies on topological arguments. In the present paper, the authors consider (a version of) the condition INV to the case \(p=2\) when \(n=3\) and characterize this condition in terms of Cartesian currents. In addition, the authors exhibit a sequence of bilipschitz maps with the “bad” behaviour and make some remarks on the consequences of such a behaviour.

74B20 Nonlinear elasticity
35A15 Variational methods applied to PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
49J45 Methods involving semicontinuity and convergence; relaxation
74G65 Energy minimization in equilibrium problems in solid mechanics
Zbl 0836.73025
Full Text: EuDML
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