Prediction of negative dispersion by a nonlocal poroelastic theory

Dynamics in Explicit Gradient Elasticity: Material Frame-Indifference, Boundary Conditions and Consistent Euler-Bernoulli Beam Theory

The paper is concerned with the boundary conditions of explicit gradient elasticity of Mindlin's type in dynamics. It has been argued in an earlier paper that acceleration terms should not be present in the boundary tractions because of objectivity arguments. This is discussed in the present paper in more detail, and it is supplemented by assuming the validity of the principle of material frame indifference. Furthermore, new examples are discussed in order to illustrate that significant differences exist in the responses predicted by boundary tractions with and without acceleration terms.

Application of an effective medium theory for modeling ultrasound wave propagation in healing long bones

Quantitative ultrasound has recently drawn significant interest in the monitoring of the bone healing process. Several research groups have studied ultrasound propagation in healing bones numerically, assuming callus to be a homogeneous and isotropic medium, thus neglecting the multiple scattering phenomena that occur due to the porous nature of callus. In this study, we model ultrasound wave propagation in healing long bones using an iterative effective medium approximation (IEMA), which has been shown to be significantly accurate for highly concentrated elastic mixtures. First, the effectiveness of IEMA in bone characterization is examined: (a) by comparing the theoretical phase velocities with experimental measurements in cancellous bone mimicking phantoms, and (b) by simulating wave propagation in complex healing bone geometries by using IEMA. The original material properties of cortical bone and callus were derived using serial scanning acoustic microscopy (SAM) images from previous animal studies. Guided wave analysis is performed for different healing stages and the results clearly indicate that IEMA predictions could provide supplementary information for bone assessment during the healing process. This methodology could potentially be applied in numerical studies dealing with wave propagation in composite media such as healing or osteoporotic bones in order to reduce the simulation time and simplify the study of complicated geometries with a significant porous nature.

Phase velocity and attenuation predictions of waves in cancellous bone using an iterative effective medium approximation

The quantitative determination of wave dispersion and attenuation in bone is an open research area as the factors responsible for ultrasound absorption and scattering in composite biological tissues have not been completely explained. In this study, we use the iterative effective medium approximation (IEMA) proposed in [1] so as to calculate phase velocity and attenuation in media with properties similar to those of cancellous bones. Calculations are performed for a frequency range of 0.4-0.8 MHz and for different inclusions' volume concentrations and sizes. Our numerical results are compared with previous experimental findings so as to assess the effectiveness of IEMA. It was made clear that attenuation and phase velocity estimations could provide supplementary information for cancellous bone characterization.

A numerical study on the propagation of Rayleigh and guided waves in cortical bone according to Mindlin's Form II gradient elastic theory

Cortical bone is a multiscale heterogeneous natural material characterized by microstructural effects. Thus guided waves propagating in cortical bone undergo dispersion due to both material microstructure and bone geometry. However, above 0.8 MHz, ultrasound propagates rather as a dispersive surface Rayleigh wave than a dispersive guided wave because at those frequencies, the corresponding wavelengths are smaller than the thickness of cortical bone. Classical elasticity, although it has been largely used for wave propagation modeling in bones, is not able to support dispersion in bulk and Rayleigh waves. This is possible with the use of Mindlin's Form-II gradient elastic theory, which introduces in its equation of motion intrinsic parameters that correlate microstructure with the macrostructure. In this work, the boundary element method in conjunction with the reassigned smoothed pseudo Wigner-Ville transform are employed for the numerical determination of time-frequency diagrams corresponding to the dispersion curves of Rayleigh and guided waves propagating in a cortical bone. A composite material model for the determination of the internal length scale parameters imposed by Mindlin's elastic theory is exploited. The obtained results demonstrate the dispersive nature of Rayleigh wave propagating along the complex structure of bone as well as how microstructure affects guided waves.

Velocity dispersion and attenuation in granular marine sediments: comparison of measurements with predictions using acoustic models

The large velocity dispersion recently reported could be explained by a gap stiffness model incorporated into the Biot model (the BIMGS model) proposed by the author. However, at high frequencies, some measured results have been reported for negative velocity dispersion and attenuation proportional to the first to fourth power of frequency. In this study, first, it is shown that the results of velocity dispersion and attenuation calculated using the BIMGS model are consistent with the results measured in two kinds of water-saturated sands with different grain sizes, except in the high-frequency range. Then, the velocity dispersion and attenuation in six kinds of water-saturated glass beads and four kinds of water-saturated silica sands with different grain sizes are measured in the frequency ranges of 80-140 and 300-700 kHz. The measured results are compared with those calculated using the BIMGS model plus some acoustic models. It is shown that the velocity dispersion and attenuation are well predicted by using the BIMGS model in the range of kd ≤ 0.5 (k: wavenumber in water, d: grain diameter) and by using the BIMGS model plus multiple scattering effects in the range of kd ≥ 0.5 in which negative velocity dispersion appears.

The effect of surface stress on the propagation of Lamb waves

This work investigates the possibility of the propagation of Lamb waves in thin solid layers with external traction free surfaces, in the presence of surface elasticity, inertia and residual stress. It is demonstrated that such waves do exist and that their characteristics can be quite different from their classical counterparts. The governing equations with non-classical boundary conditions involving the bulk and surface stress are solved exactly in the frequency-wavenumber domain. This solution is utilized to compute the Lamb wave modes for different layer thicknesses. An efficient strategy to capture all the modes of Lamb waves within a given frequency window is outlined. It is shown that the effect of surface elasticity and inertia becomes significant with increasing frequency and decreasing layer thickness, where the number of modes participating within a given frequency window is more than that permitted by the classical theory. Further, it is observed that the nature of the Lamb wave modes (in terms of negative dispersion) in the presence of surface stress is similar to what predicted by the nonlocal theory and microstructure based continuum theory.

Velocity dispersion of guided waves propagating in a free gradient elastic plate: application to cortical bone

The classical linear theory of elasticity has been largely used for the ultrasonic characterization of bone. However, linear elasticity cannot adequately describe the mechanical behavior of materials with microstructure in which the stress state has to be defined in a non-local manner. In this study, the simplest form of gradient theory (Mindlin Form-II) is used to theoretically determine the velocity dispersion curves of guided modes propagating in isotropic bone-mimicking plates. Two additional terms are included in the constitutive equations representing the characteristic length in bone: (a) the gradient coefficient g, introduced in the strain energy, and (b) the micro-inertia term h, in the kinetic energy. The plate was assumed free of stresses and of double stresses. Two cases were studied for the characteristic length: h=10(-4) m and h=10(-5) m. For each case, three subcases for g were assumed, namely, g>h, g<h, and g=h. The values of g and h were of the order of the osteons size. The velocity dispersion curves of guided waves were numerically obtained and compared with the Lamb modes. The results indicate that when g was not equal to h (i.e., g not equal h), microstructure affects mode dispersion by inducing both material and geometrical dispersion. In conclusion, gradient elasticity can provide supplementary information to better understand guided waves in bones.

High-frequency dispersion from viscous drag at the grain-grain contact in water-saturated sand

Shear viscous drag within the thin fluid film at the contact between grains in water-saturated sand is an important loss mechanism for high-frequency sound in the Biot-Stoll plus contact squirt flow and shear viscous drag (BICSQS) model [J. Acoust. Soc. Am. 116, 2011-2022 (2004)]. Couette flow was assumed for the shear drag but it breaks down when inertial effects within the film become significant. Using Biot's method, a correction is derived for the shear drag and inserted into the BICSQS model. The result is a prediction of negative sound speed dispersion, consistent with dynamic theories of fluid-filled poroelastic bodies.

Interference between wave modes may contribute to the apparent negative dispersion observed in cancellous bone

Previous work has shown that ultrasonic waves propagating through cancellous bone often exhibit a linear-with-frequency attenuation coefficient, but a decrease in phase velocity with frequency (negative dispersion) that is inconsistent with the causality-imposed Kramers-Kronig relations. In the current study, interfering wave modes similar to those observed in bone are shown to potentially contribute to the observed negative dispersion. Biot theory, the modified Biot-Attenborogh model, and experimental results are used to aid in simulating multiple-mode wave propagation through cancellous bone. Simulations entail constructing individual wave modes exhibiting a positive dispersion using plausible velocities and amplitudes, and then summing the individual modes to create mixed-mode output wave forms. Results of the simulations indicate that mixed-mode wave forms can exhibit negative dispersion when analyzed conventionally under the assumption that only one wave is present, even when the individual interfering waves exhibit positive dispersions in accordance with the Kramers-Kronig relations. Furthermore, negative dispersion is observed when little or no visual evidence of interference exists in the time-domain data. Understanding the mechanisms responsible for the observed negative dispersion could aid in determining the true material properties of cancellous bone, as opposed to the apparent properties measured using conventional data analysis techniques.