On the acoustic-radiation-induced strain and stress in elastic solids with quadratic nonlinearity (L)

Abstract

This letter demonstrates that an eigenstrain is induced when a wave propagates through an elastic solid with quadratic nonlinearity. It is shown that this eigenstrain is intrinsic to the material, but the mean stress and the total mean strain are not. Instead, the mean stress and total means strain also depend on the boundary conditions, so care must be taken when using the static deformation to measure the acoustic nonlinearity parameter of a solid.

Scattering of time-harmonic elastic waves by an elastic inclusion with quadratic nonlinearity

“…Both the near and far fields of the scattered second harmonic waves are obtained in terms of the Green's function of the matrix material. Major findings of this investigation include: (1) the scattered second harmonic fields depend on the nonlinear properties of the inclusion through two independent acoustic nonlinearity parameters, b L and b S , both of which are functions of the second and third order elastic constants; (2) under longitudinal wave incidence, the scattered wave fields contain both longitudinal and shear second harmonic waves, and the second harmonic shear wave disappears when b S ¼ 0; (3) in the near field, a "static" displacement corresponding to the radiation induced eigenstrain (Qu et al, 2011) is present. This static displacement vanishes in the far field.…”

“…This static displacement gives rise to the radiation-induced eigenstrain (Qu et al, 2011). Since the integrands decay with r À2 in the far field, one can conclude that the static displacement vanishes far away from the inclusion.…”

Finite-size effects on the quasistatic displacement pulse in a solid specimen with quadratic nonlinearity

“…3, this is difficult to reconcile, since it suggests that information can be carried by an elastic disturbance faster than the speed of sound; an observer stationed at x ¼ L cannot instantaneously determine that the tone burst was turned off at time t ¼ s at a transmitter located at x ¼ 0, since this information is not available at the point of observation (x ¼ L) before t ¼ L/c þ s. Yost and Cantrell conducted carefully designed and executed measurements to validate their analytical predictions in the [110] crystallographic direction of single crystal silicon and isotropic vitreous silica. 2 Later, Cantrell et al successfully used the slope of the quasistatic displacement pulse a) Author to whom correspondence should be addressed.…”

“…9 would mean that the energy density must increase uniformly with propagation distance, thus violating the law of energy conservation. Recently, Qu et al 3,11 studied this controversial issue of quasistatic pulse shape by obtaining an analytical solution for the propagation of tone burst in elastic solids with quadratic nonlinearity. They showed that as the eigenstrain pulse produced by the nonlinearity moves through the medium at the speed of sound, it continuously generates a quasistatic elastic wave, like an airplane flying at the speed of sound, which results in a cumulative effect and produces a quasistatic displacement pulse that is proportional to the propagation length, but independent of the duration of the tone burst.…”

“…12 prompted us to carry out a careful re-examination of our early work. 3,11 Our objectives are twofold. First, we want to check the correctness of our analytical solutions, specifically checking if they violate the law of energy conservation.…”