Abstract
The simplest theories for quasilinear elliptic equations for an unknown scalar-valued function u are based upon coercivity conditions that require some energy-like term to exceed const |u|P where p > 1. There are many physical applications of such theories. Theories for quasilinear elliptic systems for a vector-valued u can likewise be based on analogous coercivity conditions. If, however, the system is to consist of the equilibrium equations of nonlinear elasticity with u(z) denoting the position of a material point z, then this kind of coercivity condition is singularly inadequate on both physical and mathematical grounds. This paper first describes the mathematical structure of the quasilinear equations of elastostatics, focusing; on the far more intricate kinds of coercivity conditions that are needed to meet reasonable physical restrictions and to promote mathematical analysis. The role of these new coercivity conditions is then illustrated by an analysis of the existence, regularity, and qualitative behavior of solutions of specific boundary value problems.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alexander, J. C., and Antman, S. S.: 1982, The ambiguous twist of Love. Quart. Appl. Math. 40, pp. 83–92.
Antman, S. S.: 1983a, Regular and singular problems for large elastic deformations of tubes, wedges, and cylinders. Arch. Rational Mech. Anal. To appear.
Antman, S. S.: 1983b, Constitutive restrictions permitting cavitation in nonlinearly elastic cylinders. In preparation.
Ball, J. M.: 1977a, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63, pp. 337–403.
Ball, J. M.: 1977b, Constitutive inequalities and existence theorems in nonlinear elastostatics, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. I, Pitman Research Notes in Math. 17, London, pp. 187–241.
Ball, J. M.: 1980, Strict convexity, strong ellipticity, and regularity in the calculus of variations. Math. Proc. Camb. Phil. Soc. 87, pp. 501–513.
Ball, J. M.: 1982, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Proc. Roy. Soc. London. To appear.
Brezis, H.: 1968, Equations et inequations non lineaires dans les espaces vectoriels en duality. Ann. Inst. Fourier, 18, pp. 115–175.
Hughes, T. J. R., Kato, T., and Marsden, J. E.: 1977, Well-posed quasilinear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Arch. Rational Mech. Anal. 63, pp. 273–294.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1983 D. Reidel Publishing Company
About this chapter
Cite this chapter
Antman, S.S. (1983). Coercivity Conditions in Nonlinear Elasticity. In: Ball, J.M. (eds) Systems of Nonlinear Partial Differential Equations. NATO Science Series C: (closed), vol 111. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-7189-9_14
Download citation
DOI: https://doi.org/10.1007/978-94-009-7189-9_14
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-009-7191-2
Online ISBN: 978-94-009-7189-9
eBook Packages: Springer Book Archive