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.
R.A. Adams and J.J.F. Fournier. Sobolev Spaces. Academic Press, Second edition, 2003.
J.M. Ball. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal, 63:337–403, 1977.
J.M. Ball. Strict convexity, strong ellipticity, and regularity in the calculus of variations. Proc. Camb. Phil. Soc, 87:501–513, 1980.
J.M. Ball. Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Phil Trans. Royal Soc. London A, 306:557–611, 1982.
J.M. Ball. Singularities and computation of minimizers for variational problems. In R. DeVore, A. Iserles, and E. Suli, editors, Foundations of Computational Mathematics. Cambridge University Press, 2001.
J.M. Ball. Some open problems in elasticity. In Geometry. Mechanics, and Dynamics, pages 3-59. Springer, New York, 2002.
J.M. Ball, B. Kirchheim, and J. Kristensen. Regularity of quasiconvex envelopes. Calculus of Variations and Partial Differential Equations, 11:333–359, 2000.
J.M. Ball and J.E. Marsden. Quasiconvexity at the boundary, positivity of the second variation, and elastic stability. Arch. Rational Mech. Anal, 86:251–277, 1984.
J.M. Ball and V.J. Mizel. One-dimensional variational problems whose minimizers do not satisfy the Euler-Lag range equations. Arch. Rational Mech. Anal, 90:325–388, 1985.
J.M. Ball and F. Murat. W 1, p-quasiconvexity and variational problems for multiple integrals. J. Functional Analysis, 58:225–253, 1984.
J.C. Bellido and C. Mora-Corral. Approximation of Hölder continuous homeomorphisms by piecewise affine homeomorphisms. arXiv:0806.3366vl [math.CA]
B. Dacorogna. Quasiconvexity and relaxation of non convex variational problems. J. Fund. Anal, 46:102–118, 1982.
CM. Dafermos. Hyperbolic Conservation Laws in Continuum Physics. Second Edition, volume 325 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 2005.
L.C. Evans. Quasiconvexity and partial regularity in the calculus of variations. Arch. Rational Mech. Anal, 95:227–268, 1986.
G.A. Francfort and J.-J. Marigo. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids, 46:1319–1342, 1998.
Y. Grabovsky and T. Mengesha. Sufficient conditions for strong local minima: the case of C1 extremals. Trans. Amer. Math. Soc, 361:1495–1541, 2009.
D. Henao and C. Mora-Corral. Invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity, in preparation.
D. Henao and C. Mora-Corral. Fracture surfaces and the regularity of inverses for BV deformations, in preparation.
B. Kirchheim and J. Kristensen. Differentiability of convex envelopes. C. R. Acad. Sei. Paris Sér. I Math., 333:725–728, 2001.
J.K. Knowles and E. Sternberg. On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics. J. Elasticity, 8:329–379, 1978.
J. Kristensen. On the non-locality of quasiconvexity. Ann. Inst. H. Poincaré, Anal. Non Linéaire, 16:1–13, 1999.
J. Kristensen and G. Mingione. The singular set of Lipschitzian minima of multiple integrals. Arch. Rational Mech. Anal., 184:341–369, 2007.
J. Kristensen and A. Taheri. Partial regularity of strong local minimizers in the multi-dimensional calculus of variations. Arch. Rational Mech. Anal, 170:63–89, 2003.
C. Mora-Corral. Approximation by piecewise homeomorphisms of Sobolev homeomorphisms that are smooth outside a point. Houston J. Math., in press.
C.B. Morrey. Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J. Math., 2:25–53, 1952.
C.B. Morrey. Multiple Integrals in the Calculus of Variations, Springer, 1966.
S. Müller, T. Qi, and B.S. Yan. On a new class of elastic deformations not allowing for cavitation. Ann. Inst. Henri Poincaré, Analyse Nonlinéaire, 11:217–243, 1994.
J. Nečas. Example of an irregular solution to a nonlinear elliptic system with analytic coefficients and conditions for regularity. In Theory of Nonlinear Operators, pages 197–206, Berlin, 1977. Akademie-Verlag.
T. Qin. Symmetrizing the nonlinear elastodynamic system. J. Elasticity, 50:245–252, 1998.
E.N. Spadaro. Non-uniqueness of minimizers for strictly polyconvex functional. Arch. Rational Mech. Anal, 2009, in press.
V. Šverák. Quasiconvex functions with subquadratic growth. Proc. Roy. Soc. Lond. A, 433:723–732, 1991.
V. Sverak and X. Yan. A singular minimizer of a smooth strongly convex functional in three dimensions. Calc. Var. Partial Differential Equations, 10:213–221, 2000.
C. Truesdell and W. Noll, The non-linear field theories of mechanics. In Handbuch der Physik, Vol. III/3m ed. S. Flügge, Springer, Berlin, 1965.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 CISM, Udine
About this chapter
Cite this chapter
Ball, J.M. (2010). Progress and puzzles 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_1
Download citation
DOI: https://doi.org/10.1007/978-3-7091-0174-2_1
Publisher Name: Springer, Vienna
Print ISBN: 978-3-7091-0173-5
Online ISBN: 978-3-7091-0174-2
eBook Packages: EngineeringEngineering (R0)