Wave propagation through a random array of pinned dislocations: Velocity change and attenuation in a generalized Granato and Lücke theory

A quantitative theory of the elastic wave damping and velocity change due to interaction with dislocations is presented. It provides a firm theoretical basis and a generalization of the Granato and Lücke model [ J. Appl. Phys. 27 583 (1956)]. This is done considering the interaction of transverse (T) and longitudinal (L) elastic waves with an ensemble of dislocation segments randomly placed and randomly oriented in an elastic solid. In order to characterize the coherent wave propagation using multiple scattering theory, a perturbation approach is used, which is based on a wave equation that takes into account the dislocation motion when forced by an external stress. In our calculations, the effective velocities of the coherent waves appear at first order in perturbation theory while the attenuations have a part at first order due to the internal viscosity and a part at second order due to the energy that is taken away from the incident direction. This leads to a frequency dependence law for longitudinal and transverse attenuations that is a combination of quadratic and quartic terms instead of the usual quadratic term alone. Comparison with resonant ultrasound spectroscopy (RUS) and electromagnetic acoustic resonance (EMAR) experiments is proposed. The present theory explains the difference experimentally observed between longitudinal and transverse attenuations [ Ledbetter J. Mater. Res. 10 1352 (1995)].