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

Abstract

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 ͑1956͔͒. This is done considering the interaction of transverse ͑T͒ and longitudinal ͑L͒ 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 ͑RUS͒ and electromagnetic acoustic resonance ͑EMAR͒ experiments is proposed. The present theory explains the difference experimentally observed between longitudinal and transverse attenuations ͓Ledbetter, J. Mater. Res. 10, 1352 ͑1995͔͒.

Propagation of guided waves through weak penetrable scatterers

“…42,43 The dependence of T rand NN on the length L of the slab containing the inclusions is linear, a consequence of the linearity of our equation in the single scattering approximation, which causes the prediction on the transmission unreliable for large L. The limit of T rand NN for a large slab is less problematic: Including a small attenuation in the host medium that produces a small imaginary part in k N , and the limit L ! 1 becomes possible [namely e 2ik N L vanishes in Eq.…”

Attenuation of ultrasound in severely plastically deformed nickel

“…This estimation showed that the contribution of dislocation damping to attenuation is the most important effect for the cold working metal and that the phenomenological Granato–Lücke model can be employed [20]. A further development of the theory includes averaging of the dislocation orientation and Burgers vectors but gives similar results [21]. Practically the most important result is that the decrease in the dislocation loop length shifts the maximum of the ultrasonic attenuation to higher frequencies.…”

Shear modulus of an elastic solid under external pressure as a function of temperature: The case of helium

“…In the case where many of these dislocations are present, with probability p(L)dL of having a length between L and L + dL, Maurel et al 2 used multiple-scattering theory to compute an effective, complex index of refraction whose real part gives a renormalized velocity of propagation for both longitudinal and transverse waves, and whose imaginary part provides an attenuation, also for both longitudinal and transverse waves. This is done in the following way: An homogeneous, isotropic, elastic medium is described by u i ( x,t), with the displacement of a particle at time t from its equilibrium position x, whose time derivative v i ≡ ∂u i /∂t obeys, in the presence of many dislocation segments, the inhomogeneous wave equation…”