The Lagrange multipliers and hyperstress constraint reactions in incompressible multipolar elasticity theory

Abstract

Multipolar elasticity is a pseudo spatial-dependent theory which allows for higher deformation gradients to affect the value of the stored energy function and, in particular, it introduces the concept of multipolar traction as a source of contact interaction between adjacent surfaces in material bodies and at their boundaries. As a result of the presence of higher deformation gradients in the constitutive structure, the mathematical set-up for such a theory of material behavior generally requires deformation fields to lie in a Sobolev space W n,p( B, R 3) , where n is the order of the highest deformation gradient that is present. The standard such norm is nonuniformly scale dependent because of the presence of higher gradients and so we introduce an equivalent weighted norm which, for n=2, requires the introduction of a single intrinsic length scale l. For this case, we then study the necessary first variation condition for a sufficiently smooth minimizer y ∗∈W n,p( B, R 3) in a sufficiently smooth domain and prove a related Lagrange multiplier theorem in Theorem 4.1. This theorem depends upon the validity of a “Riesz-like” representation theorem for a continuous linear functional on W 1,p( B, R) , which we consider in Lemma 4.1. The strange nature of the space dual to W 1,p( B, R) requires certain special technical considerations and these lead us to propose a natural scheme for constructing the unique Lagrange multiplier fields. Finally, in the last section of this work, we solve an elementary example problem and apply our earlier conclusions on existence and uniqueness to uniquely determine the constraint reaction stress and hyperstress fields. We show how these fields depend upon the length scale l.