On the characterisation of polar fibrous composites when fibres resist bending

This study aims to initiate research for the invention of methods appropriate for characterisation of fibre-reinforced materials that exhibit polar material behaviour due to fibre bending resistance. It thus focuses interest in the small strain regime, where there are examples of particular deformations for which non-polar linear elasticity fails to distinguish clearly the nature of a fibrous composite or even to account for the presence of fibres. Particular attention is accordingly given to the solution of the polar material version of the pure bending problem of transverse isotropic or special orthotropic plates with embedded fibres resistant in bending. It is seen that pure bending deformation enables polar fibre-reinforced materials to generate constant couple stress-field which, in turn, endorses uniqueness of the solution of the corresponding boundary value problem. In this context, by appropriately extending the validity of Clapeyron's theorem within the regime of polar linear elasticity for fibre-reinforced materials, it is shown that the solution of well-posed linear elasticity boundary value problems that generate a constant couple-stress field is unique. The well-known uniqueness of solution of conventional, non-polar linear elasticity boundary value problems is, in fact, a particular case in which the generated constant value of the couple-stress field is zero.