The relative dislocation density of aluminum and copper samples is quantitatively measured using linear Resonant Ultrasound Spectroscopy (RUS). For each metallic group, four samples were prepared with different thermomechanical treatments in order to induce changes in their dislocation densities. The RUS results are compared with Nonlinear Resonant Ultrasound Spectroscopy (NRUS) as well as Second Harmonic Generation (SHG) measurements. NRUS has a higher sensitivity by a factor of two to six and SHG by 14–62%. The latter technique is, however, faster and simpler. As a main result, we obtain a quantitative relation between the changes in the nonlinear parameters and the dislocation density variations, which in a first approximation is a linear relation between these differences. We also present a simple theoretical expression that explains the better sensitivity to dislocation content of the nonlinear parameters with respect to the linear ones. X-Ray diffraction measurements, although intrusive and less accurate, support the acoustics results.

A canonical quantization procedure is applied to elastic waves interacting with pinned dislocation segments (“strings”) of length $L$ via the Peach-Koehler force. The interaction Hamiltonian, derived from an action principle that classically generates the Peach-Koehler force, is a power series of creation and annihilation operators. The leading term is quadratic, and keeping only this term the observable quantities of scattering processes are computed to all orders in perturbation theory. The resulting theory is characterized by the magnitude of $kL$, with $k$ the wave number of an incident phonon. The theory is solved for arbitrary $kL$, and different limits are explored. A significant result at this level is the scattering cross section for phonons by dislocation segments. As a function of frequency, this cross section has a much richer structure than the linear-in-frequency behavior that is inferred from scattering by an infinite, static, dislocation. The rate of spontaneous phonon emission by an excited dislocation is computed as well. When many dislocations are present, an effective mass operator is computed in the weak and independent scattering approximation. The contribution of the cubic terms is computed to leading order in perturbation theory. They allow for a comparison of the scattering of a phonon by a string and the three-phonon scattering, as well as studying the dependence of scattering amplitudes on the temperature of the solid. It is concluded that the effect of dislocations of Burgers vector 0.5 nm and length 50 nm will dominate for relatively modest dislocation densities: ${10}^8{\mathrm{cm}}^{-2}$. Finally, the full power series of the interaction Hamiltonian is considered. The effects of quantum corrections, i.e., contributions proportional to Planck's constant, are estimated, and found to be controlled by another wave-number-dependent parameter $k{d}_q$, where $d_q$ is a length proportional to $\sqrt{\hslash }$. The possibility of using the results of this paper in the study of the phononic thermal properties of two- and three-dimensional materials is noted and discussed.

In this work we present an implementation of the analytical string theory recently applied to the description of glasses. These are modeled as continuum media with embedded elastic string heterogeneities, randomly located and randomly oriented, which oscillate around a straight equilibrium position with a fundamental frequency depending on their length. The existence of a length distribution is reflected then in a distribution of oscillation frequencies which is responsible for the boson peak in the glass density of states. Previously, it has been shown that such a description can account for the elastic anomalies reported at frequencies comparable with the boson peak: the strong phonon scattering and the negative dispersion in the sound velocity, as a result of the interference of the string oscillations with propagating sound plane waves. Here we start from the generalized hydrodynamics to determine the dynamic correlation function $S(k,\omega )$ associated with the coherent, dispersive, and attenuated sound waves resulting from such interference. We show that once the vibrational density of states has been measured, we can use it for unambiguously fixing the string length distribution inherent to a given glass. The density-density correlation function obtained using such distribution is strongly constrained, and able to account for the experimental data collected on two prototypical glasses with very different microscopic structure and fragility: glycerol and silica. The obtained string length distribution is compatible with the typical size of elastic heterogeneities previously reported for silica and supercooled liquids, and the atomic motion associated with the string dynamics is consistent with the soft modes recently identified in large-scale numerical simulations as nonphonon modes responsible for the boson peak. The theory is thus in agreement with the most recent advances in the understanding of the glass-specific dynamics and offers an appealing, simple understanding of the microscopic origin of the latter, while raising new questions on the universality or material specificity of the string distribution properties.

The normal modes of a continuum solid endowed with a random distribution of line defects that behave like elastic strings are described. These strings interact with elastic waves in the bulk, generating wave dispersion and attenuation. Explicit formulas are provided that relate these properties to the density of string states. For a density of states that mimics the boson peak (BP) in amorphous materials, the attenuation as a function of frequency $\omega$ behaves as ${\omega }^4$ for low frequencies, and, as frequency increases, crosses over to ${\omega }^2$ near the BP, and then to linear in $\omega$. An Ioffe-Regel criterion is satisfied at the BP. Dispersion is negative in the frequency range where attenuation is quartic and quadratic in frequency, with effective velocity reaching a minimum near the BP. Continuum mechanics can thus be applied to both crystalline materials and their amorphous counterparts at similar length scales. The possibility of linking this model with the microstructure of amorphous materials is discussed.

The attenuation of ultrasound in polycrystalline materials is modeled with grain boundaries considered as arrays of dislocation segments, a model valid for low angle mismatches. The polycrystal is thus studied as a continuous medium containing many dislocation "walls" of finite size randomly placed and oriented. Wave attenuation is blamed on the scattering by such objects, an effect that is studied using a multiple scattering formalism. This scattering also renormalizes the speed of sound, an effect that is also calculated. At low frequencies, meaning wavelengths that are long compared to grain boundary size, then attenuation is found to scale with frequency following a law that is a linear combination of quadratic and quartic terms, in agreement with the results of recent experiments performed in copper [Zhang et al., J. Acoust. Soc. Am. 116(1), 109-116 (2004)]. The prefactor of the quartic term can be obtained with reasonable values for the material under study, without adjustable parameters. The prefactor of the quadratic term can be fit assuming that the drag on the dynamics of the dislocations making up the wall is one to two orders of magnitude smaller than the value usually accepted for isolated dislocations. The quartic contribution is compared with the effect of the changes in the elastic constants from grain to grain that is usually considered as the source of attenuation in polycrystals. A complete model should include this scattering as well.