Abstract
Sufficient conditions are obtained for continuous dependence of solutions of boundary value problems of linear elasticity on internal constraints. Arbitrary hyperelastic materials with arbitrary (linear) internal constraints are included. In particular the results of Bramble and Payne, Kobelkov, Mikhlin for homogeneous, isotropic, incompressible materials are obtained as a special case. In the case of boundary value problem of place, a compatibility condition is obtained between the internal constraints and the boundary data which is necessary for the existence of solutions. With a further coercivity assumption on the compliance tensor, it is shown that the compatibility condition is also sufficient for existence. An orthogonal decomposition theorem for second order tensor fields modeled after Weyl's decomposition of solenoidal and gradient fields leads to the variational formulation of the problem and existence theorems.
Almost all the results here apply to materials both with or without internal constraints. For internally constrained materials however, the verification of certain hypothesis is surprisingly non-trivial as indicated by the computation in the appendix.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Bramble, J. H. and L. E.Payne, Effect of error in measurement of elastic constants on the solutions of problems in classical elasticity, J. Res. Nat. Bur. Standards 67B (1963) 157–167.
Dorn, W. S. and A.Schild, A converse to the virtual work theorems for deformable solids, Quart. J. App. Math. 14 (1956) 209–213.
Duvaut, G. and J. L.Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976.
Fichera, G., Existence Theorems in Elasticity, in S.Flügge, ed., Handbuch der Physik, Vol. VI a/2, Springer-Verlag, Berlin, 1972.
Friedrichs, K. O., On the boundary-value problems of the theory of elasticity and Korn's inequality, Ann. Math. 48 (1947) 441–471.
Gurtin, M. E., Variational problems in the linear theory of visco-elasticity, Arch. Ratl. Mech. Anal. (1963) 179–191.
Gurtin, M. E., The Linear Theory of Elasticity, in S.Flügge, ed., Handbuch der Physik, Vol. VI a/2, Springer-Verlag, Berlin, 1972.
Hlaváček, I. and J.Nečas. On inequalities of Korn's type, parts I and II, Arch. Ratl. Mech. Anal. 36 (1970) 305–311 and 312–334.
Hörmander, L., Linear Partial Differential Operators, Springer, 1969.
Hünlich, R. and J.Naumann, On general boundary value problems and duality in linear elasticity, I, Aplikace Matematiky 23 (1978) 208–230.
Kobel'kov, G. M., Concerning existence theorems for some problems of elasticity theory, Mat. Zametki 17 (1975) 599–609.
Lions, J. L. and E.Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol.I, Springer-Verlag, Berlin, 1972.
Mikhlin, S. G., The spectrum of a family of operators in the theory of elasticity, Russian Math. Surveys 28 (1973) 45–88.
Mikhlin, S. G., The Problem of the Minimum of a Quadratic Functional, Holden-Day, San Francisco, 1965.
Pipkin, A. C., Constraints in linearly elastic materials, J. Elasticity 6 (1976) 179–193.
Rostamian, R., Internal constraints in boundary value problems of continuum mechanics, Indiana Univ. J. Math. 27 (1978) 637–656.
Rostamian, R., Continuity properties of the stationary points of quadratic functions, to appear in Numerical Functional Analysis and Optimization.
Sokolnikoff, I. S., Mathematical Theory of Elasticity, 2nd Ed., McGraw-Hill, New York, 1956.
Temam, R., Ravier-Stokes Equations, North-Holland, 1978.
Ting, T. W., Problem of compatibility and orthogonal decomposition of second-order symmetric tensors in a compact Riemannian manifold with boundary, Arch. Ratl. Mech. Anal. 64 (1977) 221–243.
Truesdell, C. and W. Noll, The Non-Linear Field Theories of Mechanics, in S. Flügge, Ed., Handbuch der Physik, Vol. III/3, Springer-Verlag, 1965.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rostamian, R. Internal constraints in linear elasticity. J Elasticity 11, 11–31 (1981). https://doi.org/10.1007/BF00042479
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00042479