Polar plasticity of thin-walled composites made of ideal fibre-reinforced material

The present study investigates the influence that polar material response has on the plastic behaviour of thin-walled structures made of ideal fibre-reinforced materials (Spencer, 1972); or, equivalently, on the response of thin-walled fibrous composites within the first branch of the matrix dominated form (MDM) of the bimodal theory of plasticity (Soldatos, 2011, Dvorak and Bahei-El-Din, 1987). The plasticity studies mentioned above assume that fibres are infinitely thin and, therefore, perfectly flexible. They possess no bending stiffness and, hence, their negligible bending resistance cannot influence the developed stress state, which is accordingly described by a symmetric stress tensor. In contrast, the present study considers that if fibres resistant in bending are embedded in a material at high volume concentrations, their flexure produces couple-stress and, as a result of this kind of polar material behaviour, the stress tensor becomes non-symmetric. Under plane stress conditions that dominate behaviour of thin-walled structures, the stress-space and, therefore, conditions of plastic yield and relevant yield surfaces are thus four-dimensional. However, shapes and properties of initial yield surfaces relevant to the f₁-branch of MDM are studied comprehensively by considering their projection on particular planes of such a four-dimensional stress-space. It then becomes easier understood that, in the regime of polar material response, a thin-walled structure made of ideal fibre-reinforced material deforms plastically when suitable combinations of shear stress values are reached simultaneously, rather than when only one of two unequal shear stress components reaches some maximum absolute value. Thus, polar material plasticity dismisses the conventional concept of material yield stress in shear and replaces it with a pair of two independent yield moduli. Existence of the latter is perceived as a theoretical justification of the expectation that, due to the presence of fibres, two rather than one shear yield parameters of the composite should be present and accountable for. The non-zero values of those parameters are shown to exert paramount influence on the form of the yield surface of the ideal fibre-reinforced material of interest.