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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.

MSC:
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
Citations:
Zbl 0836.73025
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References:
[1] E. Acerbi - G. Dal Maso, New lower semicontinuity results for polyconvex integrals, Calc. Var. Partial Differential Equations 2 (1994), 329-371. Zbl0810.49014 MR1385074 · Zbl 0810.49014
[2] L. Ambrosio - N. Fusco - D. Pallara, “Functions of bounded variation and free discontinuity problems”, Oxford Mathematical Monographs, Clarendon Press, Oxford, 2000. Zbl0957.49001 MR1857292 · Zbl 0957.49001
[3] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1977), 337-403. Zbl0368.73040 MR475169 · Zbl 0368.73040
[4] J. M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. Roy. Soc. London 306 A (1982), 557-611. Zbl0513.73020 MR703623 · Zbl 0513.73020
[5] P. Bauman - D. Phillips - N. C. Owen, Maximal smoothness of solutions to certain Euler-Lagrange equations from nonlinear elasticity, Proc. Roy. Soc. Edinburgh 119 A (1991), 241-263. Zbl0744.49008 MR1135972 · Zbl 0744.49008
[6] H. Brezis - L. Nirenberg, Degree theory and BMO: Part 1, compact manifolds without boudaries, Selecta Math. (N.S.) 1 (1995), 197-263. Zbl0852.58010 MR1354598 · Zbl 0852.58010
[7] P. G. Ciarlet - J. Nečas, Injectivity and self-contact in nonlinear elasticity, Arch. Rational Mech. Anal. 97 (1987), 171-188. Zbl0628.73043 MR862546 · Zbl 0628.73043
[8] B. Dacorogna - J. Moser, On a partial differential equation involving the jacobian determinat, Ann. IHP Anal. Non Lin. 7 (1990), 1-26. Zbl0707.35041 MR1046081 · Zbl 0707.35041
[9] C. De Lellis, Some fine properties of currents and applications to distributional Jacobians, Proc. Roy. Soc. Edinburgh 132 A (2002), 815-842. Zbl1025.49029 MR1926918 · Zbl 1025.49029
[10] H. Federer, “Geometric measure theory”, Classics in Mathematics, Springer Verlag, Berlin, 1969. Zbl0874.49001 MR257325 · Zbl 0874.49001
[11] I. Fonseca - W. Gangbo, “Degree theory in analysis and applications”, Oxford Lecture Series in Mathematics and its Applications, 2, Clarendon Press, Oxford, 1995. Zbl0852.47030 MR1373430 · Zbl 0852.47030
[12] M. Giaquinta - G. Modica - J. Souček, “Cartesian currents in the calculus of variations”, Vol. 1, 2, Springer Verlag, Berlin, 1998. Zbl0914.49001 MR1645086 · Zbl 0914.49001
[13] J. Malý, Weak lower semicontinuity of polyconvex integrals, Proc. Roy. Soc. Edinburgh 123 A (1993), 681-691. Zbl0813.49017 MR1237608 · Zbl 0813.49017
[14] J. Malý, Lower semicontinuity of quasiconvex integrals, Manuscripta Math. 85 (1994), 419-428. Zbl0862.49017 MR1305752 · Zbl 0862.49017
[15] S. Müller - S. Spector, An existence theory for nonlinear elasticity that allows for cavitation, Arch. Rat. Mech. Anal. 131 (1995), 1-66. Zbl0836.73025 MR1346364 · Zbl 0836.73025
[16] J. Sivaloganathan - S. Spector, On the optimal location of singularities arising in variational problems of nonlinear elasticity, J. of Elast. 58 (2000), 191-224. Zbl0977.74005 MR1816651 · Zbl 0977.74005
[17] V. Šverák, Regularity properties of deformations with finite energy, Arch. Rat. Mech. Anal. 100 (1988), 105-127. Zbl0659.73038 MR913960 · Zbl 0659.73038
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