A novel boundary element approach to time-harmonic dynamics of incremental nonlinear elasticity: The role of pre-stress on structural vibrations and dynamic shear banding

A new boundary element technique is developed to analyze two-dimensional, time-harmonic, small-amplitude vibrations, superimposed upon a homogeneously pre-stressed, orthotropic and incompressible elastic solid. New expressions for the Green’s functions for incremental applied tractions are obtained, in which ‘static’ and ‘dynamic’ contributions are uncoupled. The dynamic contributions are regular, whereas the static terms are strongly singular and are solved in closed-form expressions, particularly useful for numerical calculations. As a consequence of the static/dynamic de-coupling, these expressions turn out to be useful also for quasi-static deformation. The formulation is tested for different boundary value problems. These include a problem with certain boundary conditions investigated by Ryzhak in the static case, for which an analytic solution is proposed here in the time-harmonic regime. The effect of pre-stress is shown to strongly influence the vibrational response of elastic structures. It is shown that natural frequencies strongly decrease when pre-stress approaches quasi-static bifurcation loads and wave propagation speeds tend to vanish when the boundary of ellipticity is approached. Near this boundary (but still within the elliptic region) we observe a focussing of vibrations along plane waves parallel to the shear bands.