Second-gradient plane deformations of ideal fibre-reinforced materials: implications of hyper-elasticity theory

The influence that some new, second-gradient, effects introduced in a recent publication (Spencer and Soldatos, Int J Non-linear Mech 42:355-368, 2007) have on finite plane deformations of ideal fibre-reinforced hyper-elastic solids is investigated. The second-gradient effects are due to the ability of the fibres to resist bending but, in the present case, the constraints of material incompressibility and fibre inextensibility associated with this ideal class of materials offer considerable theoretical simplification. In agreement with its conventional counterpart, where inextensible fibres are perfectly flexible, the present new theoretical development is still associated with kinematics and reaction stresses that are largely independent of the specific type of material behaviour considered. Static equilibrium considerations reveal therefore a manner in which relevant, non-symmetric stress distributions can be determined by solving two simultaneous, first-order linear differential equations. However, the principal interest of this investigation remains within the class of hyper-elastic materials for which two sets of relatively simple constitutive equations are obtained. At this stage of early theoretical development, immediate interest is directed towards the simplest of those sets, namely the set associated with problems where only gradients relevant to the change of the deformed fibre direction are of principal importance. These developments are applied (i) to the classical problem of plane-strain bending of a rectangular block reinforced by a family of straight fibres running parallel to one of its sides; and (ii) to the problem of "area-preserving" azimuthal shear strain of a circular cylindrical tube having its cross-section reinforced by a family of strong fibres. In the particular case in which the fibres are initially straight and aligned with the radii of the tube cross-section, the solution of the latter problem, which is new in the literature, reveals that fibres resist local bending completely. Instead, they remain straight during deformation and force the tube cross-section to undergo area-preserving azimuthal shear by changing their direction.