New trends in couple-stress hyperelasticity

Abstract

This communication reconsiders and reorganises parts of existing knowledge that refers to Cosserat-type hyperelasticity formulations in a manner that respects and pays tribute to the early development pattern of the couple-stress theory and, also, associates with it more recent findings and relevant discoveries related to a certain type of polar hyperelastic behaviour of fibrous composites. It further shows that these different branches of couple-stress theory can both emanate, as special cases, from a common, more general, advanced theoretical hyperelasticity framework. It thus reveals that proper completion of any relevant theoretical formulation requires in advance specification of a pair of kinematic parameters, each of which (1) is related, in a virtual or actual manner, to the observed global, macroscopic elastic deformation of the material, but (2) is principally relevant to the polar part of the observed material response. The first of these parameters represents a pseudovector field whose gradient is energetically reciprocal to the emerging couple-stress field, while the second represents either a vector or a pseudovector field that serves specific constitutional needs characterising the source of the anticipated polar material response. Different versions of couple-stress theory formulations, thus, are obtained by appropriately choosing or suitably tuning that pair of kinematic parameters. It is also seen that, regardless of the employed couple-stress theoretical model, full solution of a relevant well-posed boundary value problem is generally achievable with use of a two-step solution process. The first step includes determination of the deviatoric couple-stress and the actual spin vector of the global material deformation. In the second step, these initial findings enable formation of an additional differential equation whose solution leads to determination of the spherical part of the couple-stress.