A Penalty Interpretation for the Lagrange Multiplier Fields in Incompressible Multipolar Elasticity Theory

Abstract

In a recent paper we considered the characterization of the incompressibility constraint for multipolar hyperelastic materials. An existence theorem was given for the associated two Lagrange multiplier fields, representative of the stress and “hyper-stress” constraint reactions, but a congenital non-uniqueness in the splitting of these constraint reactions was encountered. Based upon purely mathematical arguments (representation theorems for functionals defined on Sobolev spaces), a strategy was advanced in order to overcome such indeterminacy. This consisted in selecting, among all possible choices, the unique pair of fields that minimized the appropriately defined norm. Here, we provide a physical justification for the so-called “minimum norm criterion” by interpreting incompressibility through the limit of a sequence of auxiliary minimum problems where changes in volume are progressively penalized by a “penalty term” which is added to the potential energy. The idea is known as the penalty method in the area of numerical analysis. Through the notion of a “modified Lagrangian”, a one-to-one correspondence is established between the solution of each penalty problem in the sequence and an approximation to the two Lagrange multiplier fields for the original constrained problem. In fact, a major effect of the penalty term is that it brings into focus the underlying presence of the constraint manifold of incompressibility as an “attractor” and the “shape” of this manifold identifies a “direction” in which the sequence of approximations approaches its limit. Ultimately, the approximation converges to one and only one choice. We find that the unique limiting Lagrange multiplier fields must necessarily minimize the same norm that was introduced adscititiously in our aforementioned recent paper, but there only as a purely mathematical scheme. This provides justification for the “minimum norm criterion” as a suitable procedure for sorting out the constraint reaction contributions to stress and hyperstress in incompressible multipolar theory.