Lower-dimensional long wave dynamic models for idealised fibre-reinforced elastic structures

Abstract

The dispersion of harmonic waves in an idealised fibre-reinforced elastic layer is investigated. Guided by a numerical and asymptotic long-wave investigation of the dispersion relation, appropriate scales are introduced to help elucidate features of long wave high- and low-frequency motion. In the former case, the stress–strain-state is determined in terms of the long-wave amplitude, appropriate leading-order and refined second-order governing equations being obtained from the second- and third-order problems, respectively. At each order the dispersion relation associated with the governing equation agrees with the appropriate expansion of the “exact” dispersion relation. With respect to low-frequency motion, the long wave limit of anti-symmetric motion is non-zero. This contrasts with the classical case and also indicates that inextensible fibres preclude classical bending. The asymptotic long-wave low frequency stress–strain-state is determined in terms of the governing extensions and mid-surface deflection in the symmetric and anti-symmetric cases, respectively. Appropriate leading and second-order governing equations are also found for these functions. The second-order equations act both to refine the stress–strain-state and also provide the leading-order governing equation in the vicinity of the appropriate quasi wave front. This phenomenon is illustrated by considering a problem concerning shock edge loading of a semi-infinite layer.