Foundation of Polar Linear Elasticity for Fibre-Reinforced Materials II: Advanced Anisotropy

This paper presents a complete formulation of the linear theory of elasticity for polar fibre reinforced materials with two families of embedded fibres by building upon previous theoretical background available for transversely isotropic fibre-reinforced materials (Soldatos in J. Elast. 114:155–178, 2014). Polar material behaviour stems from the postulate that fibres of either family are not perfectly flexible. During deformation of the fibrous composite, they instead act as slender Euler-Bernoulli beams and, therefore, their individual deformation generates couple-stress and non-symmetric stress. For simplicity, the outlined formulation is based on the plausible consideration that the fibre bending mode is the most dominant and, therefore, the most significant one to be accounted for when compared with its fibre stretching and fibre twist counterparts. The simplification achieved with this consideration becomes evident in the particular case of transversely isotropic materials, when comparisons are made against the relevant full developed theory (Soldatos in J. Elast. 114:155–178, 2014). Association of this simplifying consideration with the presence of two families of fibres resistant in bending furnishes the present model with the ability to embrace groups of advanced material anisotropy, such as the case of locally monoclinic materials which represent the most advanced group. The particular cases of local and plane orthotropy, where the two families of fibres are perpendicular to each other, are also deduced and considered in detail. Because of their importance in modelling thin- and/or moderately thick-walled structural components and laminates, particular attention is also given to the so-called cases of general and special orthotropy, where the fibres of both families are straight. The obtained governing equations are non-elliptic and this feature of the model is connected with the manner that solutions involving second-gradient weak discontinuity surfaces within the material may complement corresponding solutions associated with continuous displacements having their derivatives of all orders also continuous. The manner that second-gradient weak discontinuity surfaces are sought and found is finally demonstrated in detail for the case of specially orthotropic materials.