Abstract
We give several examples of modeling in nonlinear elasticity where a quasiconvexification procedure is needed. We first recall that the three-dimensional Saint Venant-Kirchhoff energy fails to be quasiconvex and that its quasiconvex envelope can be obtained by means of careful computations. Second, we turn to the mathematical derivation of slender structure models: an asymptotic procedure using T-convergence tools leads to models whose energy is quasiconvex by construction. Third, we construct an homogenized quasiconvex energy for square lattices.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
E. Acerbi and N. Fusco. Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal, 86:125–145, 1984.
E. Acerbi, G. Buttazzo, and D. Percivale. A variational definition for the strain energy of an elastic string. J. Elasticity, 25:137–148, 1991.
R. Alicandro and M. Cicalese. A general integral representation result for continuum limits of discrete energies with superlinear growth. SIAM J. Math. Anal., 36:1–37, 2004.
O. Anza Hafsa and J.-Ph. Mandallena. The nonlinear membrane model: Variational derivation under the constraint det ∇u ≠ 0. J. Maths Pures Appl, 86:100–115, 2006.
E. Attouch. Variational Convergence for Functions and Operators. Pitman, 1984.
J.M. Ball. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal, 63:337–403, 1977.
M. Bousselsal and H. Le Dret. Relaxation of functionals involving homogeneous functions and invariance of envelopes. Chin. Ann. of Math., 23: 37–53, 2002.
A. Braides and M.S. Gelli. Continuum limits of discrete sytems without convexity hypotheses. Math. Mech. Solids, 7:41–66, 2002.
D. Caillerie, A. Mourad, and A. Raoult. Cell-to-muscle homogenization. Application to a constitutive law for the myocardium. Math. Model. Num. Anal, 37:681–698, 2003.
D. Caillerie, A. Mourad, and A. Raoult. Discrete homogenization in graphene sheet modeling. J. Elas., 84:33–68, 2006.
P.G. Ciarlet. Mathematical Elasticity. Vol I: Three-dimensional Elasticity. North-Holland, 1987.
P.G. Ciarlet and Ph. Destuynder. A justification of the two-dimensional plate structure. J. Mécanique, 18:315–344, 1979.
B. Dacorogna. Direct Methods in the Calculus of Variations, 2nd edition. Springer, 2007.
G. Dal Maso. An Introduction to Y-convergence. Birkäuser, 1993.
E. De Giorgi and T. Franzoni. Su un tipo di convergenza variazionale. Atti. Accad. Naz. Lincei, 58:842–850, 1975.
I. Ekeland and R. Temam. Analyse Convexe et Problèmes Variationnels. Dunod, 1974.
I. Fonseca and S. Müller. A-quasiconvexity, lower semicontinuity and Young measures. SIAM J. Math. Anal, 30:1355–1390, 1999.
D. Fox, A. Raoult, and J.C. Simo. A justification of nonlinear properly invariant plate theories. Arch. Rational Mech. Anal., 124:157–199,1993.
E. Friesecke, R.D. James, and S. Müller. A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity. Comm. Pure Appl. Math., 55:1461–1506, 2002.
E. Friesecke, R.D. James, and S. Müller. A hierarchy of plate models derived from nonlinear elasticity by γ-convergence. Arch. Rational Mech. Anal, 180:183–236, 2006.
H. Le Dret. Sur les fonctions de matrices convexes et isotropes. C.R. Acad. Sei. Paris, I, 310:617–620, 1990.
H. Le Dret and A. Raoult. Variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results. Arch. Rational Mech. Anal, 154:101–134, 2000.
H. Le Dret and A. Raoult. Remarks on the quasiconvex envelope of stored energy functions in nonlinear elasticity. Comm. Applied Nonlinear An., 1:85–96, 1994.
H. Le Dret and A. Raoult. The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl, 75: 551–580, 1995a.
H. Le Dret and A. Raoult. The quasiconvex envelope of the Saint Venant-Kirchhoff stored energy function. Proc. Roy. Soc. Edinburgh, 125A:1179–1192, 1995b.
H. Le Dret and A. Raoult. Quasiconvex envelopes of stored energy densities that are convex with respect to the strain tensor. In C. Bandle, J. Bemelmans, M. Chipot, J. Saint Jean Paulin, and I. Shafrir, editors, Progress in Partial Differential Equations. Pont-à-Mousson. 1994-Pitman, 1995c.
C.B. Morrey. Quasiconvexity and the semicontinuity of multiple integrals. Pacific J. Math., 2:25–53, 1995.
A. Mourad. Description Topologique de l’Architecture Fibreuse et Modélisation Mécanique du Myocarde. Doctoral dissertation, Institut National Polytechnique de Grenoble, 2003.
O. Pantz. Quelques Problèmes de Modélisation en Elasticité non Linéaire. Doctoral dissertation, Université Pierre et Marie Curie, 2001.
A.C. Pipkin. Convexity conditions for strain-dependent energy functions for membranes. Arch. Rational Mech. Anal, 121:361–376, 1993.
A.C. Pipkin. Relaxed energies for large deformations of membranes. IMA J. Appl. Math., 52:297–308, 1994.
A. Raoult. Non-polyconvexity of the stored energy function of a Saint Venant-Kirchhoff material. Aplikace Matematiky, 31:417–419, 1986.
V. Sverak. Rank-one convexity does not imply quasiconvexity. Proc. Roy. Soc. Edinburgh, 120A: 185–189, 1992.
R.C. Thompson and L.J. Freede. Eigenvalues of sums of hermitian matrices. J. Research Nat. Bur. Standards B, 75B:115–120, 1971.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 CISM, Udine
About this chapter
Cite this chapter
Raoult, A. (2010). Quasiconvex envelopes in nonlinear elasticity. In: Schröder, J., Neff, P. (eds) Poly-, Quasi- and Rank-One Convexity in Applied Mechanics. CISM International Centre for Mechanical Sciences, vol 516. Springer, Vienna. https://doi.org/10.1007/978-3-7091-0174-2_2
Download citation
DOI: https://doi.org/10.1007/978-3-7091-0174-2_2
Publisher Name: Springer, Vienna
Print ISBN: 978-3-7091-0173-5
Online ISBN: 978-3-7091-0174-2
eBook Packages: EngineeringEngineering (R0)