The plane strain concept in couple-stress hyperelasticity

It has recently become understood that, in dealing with plane strain deformations in polar linearly isotropic elasticity, the couple-stress theory predicts that neither the trace of the couple-stress tensor nor the encountered couple-stress and non-symmetric stress fields are, or can generally be considered independent of the out-of-plane co-ordinate. The present communication proceeds beyond the bounds of polar linear elasticity and initiates an investigation regarding the extent to which these observations hold when polar material response is attributed to large plane strain elastic deformations. Clear distinction is made and separate attention is given to hyperelastic polar material behaviour that is due to inherent, probably unknown material properties, from possible relevant cases in which the source of such behaviour is known and well understood; such as the case that fibres embedded in a solid material are capable to resist deformation. The differences and similarities observed between these two categories of potential applications is also demonstrated with a brief discussion of a pair of fundamental boundary value problems, namely, the pure bending problem of an incompressible fibre-reinforced slab and the problem of pure azimuthal shear of an incompressible isotropic tube.