The Lagrange multipliers and hyperstress constraint reactions in incompressible multipolar elasticity theory
by
R.L. Fosdick and G. Royer
in
J. Mech. & Phys. Solids, 50, 1627-1647, 2002.
Category: Journal Article
Keywords: Lagrange multipliers; Constraint reactions; Multipolar elasticity theory; Hyperstress; Intrinsic
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Abstract:
Multipolar elasticity is a pseudo spatial-dependent theory which allows for higher deformation gradients to a5ect 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 7elds to lie in a Sobolevspace Wn;p(B;R3), 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 7rst variation condition for a su9ciently smooth minimizer y∗ ∈Wn;p(B;R3) in a su9ciently 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 W1;p(B;R), which we consider in Lemma 4.1. The strange nature of the space dual to W1;p(B;R) requires certain special technical considerations and these lead us to propose a natural scheme for constructing the unique Lagrange multiplier 7elds. 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 7elds. We show how these 7elds depend upon the length scale
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