Remarks on the theory of elasticity

Conti, Sergio; de Lellis, Camillo

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 3, pp. 521-549.

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. Non-interpenetrability of matter is then represented by additional invertibility conditions. An existence theory which includes a precise notion of invertibility and allows for cavitation was formulated by Müller and Spector in 1995. It applies, however, only if some $L^p$-norm of the gradient with $p>2$ is controlled (in three dimensions). We first characterize their class of functions in terms of properties of the associated rectifiable current. Then we address the physically relevant $p=2$ case, and show how their notion of invertibility can be extended to $p=2$. The class of functions so obtained is, however, not closed. We prove this by giving an explicit construction.

[1] E. Acerbi - G. Dal Maso, New lower semicontinuity results for polyconvex integrals, Calc. Var. Partial Differential Equations 2 (1994), 329-371. | MR | Zbl

[2] L. Ambrosio - N. Fusco - D. Pallara, “Functions of bounded variation and free discontinuity problems", Oxford Mathematical Monographs, Clarendon Press, Oxford, 2000. | MR | Zbl

[3] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1977), 337-403. | MR | Zbl

[4] J. M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. Roy. Soc. London 306 A (1982), 557-611. | MR | Zbl

[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. | MR | Zbl

[6] H. Brezis - L. Nirenberg, Degree theory and BMO: Part 1, compact manifolds without boudaries, Selecta Math. (N.S.) 1 (1995), 197-263. | MR | Zbl

[7] P. G. Ciarlet - J. Nečas, Injectivity and self-contact in nonlinear elasticity, Arch. Rational Mech. Anal. 97 (1987), 171-188. | MR | Zbl

[8] B. Dacorogna - J. Moser, On a partial differential equation involving the jacobian determinat, Ann. IHP Anal. Non Lin. 7 (1990), 1-26. | Numdam | MR | Zbl

[9] C. De Lellis, Some fine properties of currents and applications to distributional Jacobians, Proc. Roy. Soc. Edinburgh 132 A (2002), 815-842. | MR | Zbl

[10] H. Federer, “Geometric measure theory", Classics in Mathematics, Springer Verlag, Berlin, 1969. | MR | Zbl

[11] I. Fonseca - W. Gangbo, “Degree theory in analysis and applications", Oxford Lecture Series in Mathematics and its Applications, 2, Clarendon Press, Oxford, 1995. | MR | Zbl

[12] M. Giaquinta - G. Modica - J. Souček, “Cartesian currents in the calculus of variations”, Vol. 1, 2, Springer Verlag, Berlin, 1998. | MR | Zbl

[13] J. Malý, Weak lower semicontinuity of polyconvex integrals, Proc. Roy. Soc. Edinburgh 123 A (1993), 681-691. | MR | Zbl

[14] J. Malý, Lower semicontinuity of quasiconvex integrals, Manuscripta Math. 85 (1994), 419-428. | MR | Zbl

[15] S. Müller - S. Spector, An existence theory for nonlinear elasticity that allows for cavitation, Arch. Rat. Mech. Anal. 131 (1995), 1-66. | MR | Zbl

[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. | MR | Zbl

[17] V. Šverák, Regularity properties of deformations with finite energy, Arch. Rat. Mech. Anal. 100 (1988), 105-127. | MR | Zbl