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.

@article{ASNSP_2003_5_2_3_521_0, author = {Conti, Sergio and de Lellis, Camillo}, title = {Remarks on the theory of elasticity}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {521--549}, publisher = {Scuola normale superiore}, volume = {Ser. 5, 2}, number = {3}, year = {2003}, zbl = {1114.74004}, mrnumber = {2020859}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2003_5_2_3_521_0/} }

TY - JOUR AU - Conti, Sergio AU - de Lellis, Camillo TI - Remarks on the theory of elasticity JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2003 SP - 521 EP - 549 VL - 2 IS - 3 PB - Scuola normale superiore UR - http://www.numdam.org/item/ASNSP_2003_5_2_3_521_0/ LA - en ID - ASNSP_2003_5_2_3_521_0 ER -

%0 Journal Article %A Conti, Sergio %A de Lellis, Camillo %T Remarks on the theory of elasticity %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2003 %P 521-549 %V 2 %N 3 %I Scuola normale superiore %U http://www.numdam.org/item/ASNSP_2003_5_2_3_521_0/ %G en %F ASNSP_2003_5_2_3_521_0

Conti, Sergio; de Lellis, Camillo. Remarks on the theory of elasticity. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 3, pp. 521-549. http://www.numdam.org/item/ASNSP_2003_5_2_3_521_0/

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

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

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

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

,[5] Maximal smoothness of solutions to certain Euler-Lagrange equations from nonlinear elasticity, Proc. Roy. Soc. Edinburgh 119 A (1991), 241-263. | MR | Zbl

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

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

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

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

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

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

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

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

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

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

- ,[16] On the optimal location of singularities arising in variational problems of nonlinear elasticity, J. of Elast. 58 (2000), 191-224. | MR | Zbl

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

,