Abstract
We consider the equations of linear homogeneous anisotropic elasticity admitting the possibility that the material is internally constrained, and formulate a simple necessary and sufficient condition for the fundamental boundary value problems to be well-posed. For materials fulfilling the condition, we establish continuous dependence of the displacement and stress on the elastic moduli and ellipticity of the elasticity system. As an application we determine the orthotropic materials for which the fundamental problems are well-posed in terms of their Young's moduli, shear moduli, and Poisson ratios. Finally, we derive a reformulation of the elasticity system that is valid for both constrained and unconstrained materials and involves only one scalar unknown in addition to the displacements. For a two-dimensional constrained material a further reduction to a single scalar equation is outlined.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Agmon, A. Douglis, & L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Comm. Pure Appl. Math. 17 (1964), 35–92.
S. Antman, General solutions for plane extensible elasticae having nonlinear stressstrain laws, Quart. Appl. Math. 29 (1968), 35–47.
D. Arnold & R. Falk, Continuous dependence on the elastic coefficients for a class of anisotropic materials, Preprint 165, Institute for Mathematics and its Applications (1985).
J. Bramble & L. Payne, Effect of error in measurement of elastic constants on the solution of problems in classical elasticity, J. Res. Nat. Bur. Standards 67 B (1963), 157–167.
F. Brezzi, On the existence, uniqueness, and the approximation of saddle point problems arising from Lagrangian multipliers, RAIRO Anal. Numér. 8 (1974), 129–151.
J. Debongnie, Sur la formulation de Herrmann pour l'étude des solides incompressibles, J. Méc. 17 (1978), 531–558.
A. Douglis & L. Nirenberg, Interior estimates for elliptic systems of partial differential equations, Comm. Pure Appl. Math. 8 (1955), 503–538.
R. E. Gibson & G. C. Sills, Settlement of a strip load on a nonhomogeneous orthotropic incompressible elastic half-space, Quart. J. Mech. Appl. Math. 28 (1975), 233–243.
R. F. S. Hearmon, The elastic constants of anisotropic materials, Rev. Modern Phys. 18 (1946), 409–440.
L. R. Herrmann, Elasticity equations for incompressible and nearly incompressible materials by a variational theorem, AIAA J. 3 (1965), 1896–1900.
M. H. Holmes, A mathematical model of the dynamics of the inner ear, J. Fluid Mech. 116 (1982), 59–75.
S. Key, A variational principle for incompressible and nearly incompressible anisotropic elasticity, Internat. J. Solids and Structures 5 (1969), 951–964.
G. M. Kobel'kov, Concerning existence theorems for some problems of elasticity theory, Math. Notes 17 (1975), 356–362.
M. I. Lazarev, Solution of fundamental problems of the theory of elasticity for incompressible media, J. Appl. Math. Mech. 44 (1980), 611–616.
S. G. Lekhnitskii, Theory of Elasticity of an Anisotropic Elastic Body, HoldenDay, 1963.
J. L. Lions & E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, 1972.
S. G. Mikhlin, The spectrum of a family of operators in the theory of elasticity, Russian Math. Surveys 28 (1973), 45–88.
A. Pipkin, Constraints in linearly elastic materials, J. Elasticity 6 (1976), 179–193.
R. Rostamian, Internal constraints in linear elasticity, J. Elasticity 11 (1981), 11–31.
B. W. Shaffer, Generalized plane strain of pressurized orthotropic tubes, Trans. ASME, J. Engrg. Ind. 87 (1965), 337–343.
A. J. M. Spencer, Finite deformation of an almost incompressible solid, in SecondOrder Effects in Elasticity, Plasticity, and Fluid Dynamics, M. Reiner & D. Abir, eds., Pergamon Press, 1962, 200–216.
R. Taylor, K. Pister, & L. Herrmann, On a variational theorem for incompressible and nearly-incompressible orthotropic elasticity, Internat. J. Solids and Structures 4 (1968), 875–883.
Author information
Authors and Affiliations
Additional information
Communicated by H. Weinberger
This paper is dedicated to Professor Joachim Nitsche on the occasion of his sixtieth birthday
Rights and permissions
About this article
Cite this article
Arnold, D.N., Falk, R.S. Well-posedness of the fundamental boundary value problems for constrained anisotropic elastic materials. Arch. Rational Mech. Anal. 98, 143–165 (1987). https://doi.org/10.1007/BF00251231
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00251231