Magic angles for fibrous incompressible elastic materials. (21st March 2018) - Horgan, C. O.; Murphy, J. G.
Abstract : In the analysis of the mechanical behaviour of fibre-reinforced incompressible elastic bodies, there is a special angle of orientation of the fibres which leads to a particular mechanical response. This angle has been called a 'magic angle' due to its appearance as if by magic in many different aspects of the mechanics of fibrous solids including several examples in biology. It occurs most commonly not only in structural elements composed of circular cylindrical tubes or cylinders reinforced by helically wound fibres but also in flat thin sheets reinforced by fibres in the plane. The occurrence of such a special angle was classically demonstrated using a simple purely geometric analysis in the context of a lattice composed of a single family of helically wound inextensible fibres. Recently, the magic angle concept has been discussed in the framework of nonlinear hyperelasticity for anisotropic materials with detailed constitutive modelling. Our purpose here is to describe some other contexts in which the magic angle occurs starting from earlier work in a special theory of linear elasticity for inextensible fibres and proceeding to relatively accessible models of hyperelasticity. We discuss the role of the magic angle in the quasi-isotropic mechanical response of fibre-reinforced composites as well as the implications for material instability.