Internal constraints in linear elasticity

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.