Remarks on the theory of elasticity

Conti, Sergio; de Lellis, Camillo

Conti, Sergio; de Lellis, Camillo

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 3, p. 521-549

Abstract

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.

MR 2020859 | Zbl 1114.74004 | 2 citations in Numdam

Classification: 74B20, 35D05, 46E35, 49J45

@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},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 2},
     number = {3},
     year = {2003},
     pages = {521-549},
     zbl = {1114.74004},
     mrnumber = {2020859},
     language = {en},
     url = {http://www.numdam.org/item/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/

References

[1] E. Acerbi - G. Dal Maso, New lower semicontinuity results for polyconvex integrals, Calc. Var. Partial Differential Equations 2 (1994), 329-371. | MR 1385074 | 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. | MR 1857292 | Zbl 0957.49001

[3] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1977), 337-403. | MR 475169 | 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. | MR 703623 | 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. | MR 1135972 | 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. | MR 1354598 | Zbl 0852.58010

[7] P. G. Ciarlet - J. Nečas, Injectivity and self-contact in nonlinear elasticity, Arch. Rational Mech. Anal. 97 (1987), 171-188. | MR 862546 | 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. | Numdam | MR 1046081 | 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. | MR 1926918 | Zbl 1025.49029

[10] H. Federer, “Geometric measure theory", Classics in Mathematics, Springer Verlag, Berlin, 1969. | MR 257325 | 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. | MR 1373430 | Zbl 0852.47030

[12] M. Giaquinta - G. Modica - J. Souček, “Cartesian currents in the calculus of variations”, Vol. 1, 2, Springer Verlag, Berlin, 1998. | MR 1645086 | Zbl 0914.49001

[13] J. Malý, Weak lower semicontinuity of polyconvex integrals, Proc. Roy. Soc. Edinburgh 123 A (1993), 681-691. | MR 1237608 | Zbl 0813.49017

[14] J. Malý, Lower semicontinuity of quasiconvex integrals, Manuscripta Math. 85 (1994), 419-428. | MR 1305752 | 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. | MR 1346364 | 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. | MR 1816651 | Zbl 0977.74005

[17] V. Šverák, Regularity properties of deformations with finite energy, Arch. Rat. Mech. Anal. 100 (1988), 105-127. | MR 913960 | Zbl 0659.73038