We outline the mathematical model of the time-harmonic ultrasonic response of wet cortical bone. Using two-scale asymptotics, we derive an effective model of acoustic wave propagation in a two-phase medium modeling a fine mixture of linear piezo-elastic solid and a viscous, Newtonian, ionic bearing fluid. Following the works of Moyne and Murad, and Lemaire et. al. for the quasi-static case, we develop a two-scale homogenization method for the dynamical system, albeit the time harmonic case. The idea is to connect the bulk pressure to the small displacement by an assumption used in acoustics, i.e. the pressure p ≈ - ρfa2f divu; where ρf is the fluid density, af the speed of sound in the fluid, and u is the displacement. We investigate several time scales; one is associated with high frequency domination which leads to different hierarchies. The ratio ε of a typical size of the microstructural inhomogeneity and the macroscopic length scale is a small parameter of the problem. Another possibly small parameter is the Peclet number which influences the type of effective equations which are obtained. A brief asymptotic analysis is presented.

We study the problem of derivation of an effective model of acoustic wave propagation in a two-phase, non-periodic medium modeling a fine mixture of linear elastic solid and a viscous Newtonian fluid. Bone tissue is an important example of a composite material that can be modeled in this fashion. We extend known homogenization results for periodic geometries to the case of a stationary random, scale-separated microstructure. The ratio ε between a typical size of microstructural inhomogeneity and the macroscopic length scale is a small parameter of the problem. We employ stochastic two-scale convergence in the mean to pass to the limit ε → 0 in the governing equations. The effective model describes a biphasic, viscoelastic material with long-time history dependence. Homogenized system describes macroscopically anisotropic media and is more general than the Biot system.

The scattering of acoustic waves by a sphere in a shallow ocean waveguide is investigated. Expressions for the scattered waves are given. Numerical values for a quantity called the far-field form function for various depths are presented in graphical forms. Also we investigate the unknown body problem in a waveguide. The Rayleigh conjecture states that every point on an illuminated body radiates sound from that point as if the point lies on its tangent sphere. This conjecture is the cornerstone of the intersecting canonical body approximation (ICBA) for solving the unknown body inverse problem. Therefore, the use of the ICBA requires that an analytical solution be known exterior to the sphere in the waveguide, which leads us to analytically compute the exterior solution for a sphere between two parallel plates. A least-squares matching of theoretical acoustic fields against the measured, scattered field permits a reconstruction of the unknown object.

We investigate a three-dimensional mixed initial-boundary value problem arising in the dynamical solid–fluid interaction theory. A 3D domain occupied by an incompressible and viscous Stokes fluid may be bounded or unbounded, while a domain occupied by an elastic body immersed in the fluid is assumed to be bounded. On the basis of the results obtained for an elastic inclusion of an arbitrary geometrical shape, we derive a special model and analyze in detail the case when an elastic inclusion is a thin prismatic shell, in particular a plate of variable thickness. Here, we apply I. Vekua’s dimension reduction method in the elastic part which reduces 3D solid–3D fluid interaction problems to the 2D solid–3D fluid interaction problems and which is important from the practical point of view since it takes into account intrinsic differences of the dimensions of solid and fluids part. The main goal of the paper was to study the strain–stress state of the elastic part under the action of the Stokes flow. The corresponding mechanical model is described mathematically as a transmission problem for the linear Stokes system and the dynamical Lamé equations in the corresponding domains with appropriate initial conditions along with the boundary and interface conditions. For 3D solid–3D fluid dynamical interaction problems, we prove the uniqueness and existence theorem. Further, considering the case when the elastic inclusion is a thin prismatic shell of variable thickness, we apply the N = 0 approximation of Vekua’s hierarchical model for the elastic field in the solid part. In contrast to the usual classical streamline conditions, in the case under consideration, on the cut surface, there appear non-local boundary conditions. We prove unique solvability of the non-classical boundary value problem that leads to the existence results for the solid–fluid interaction problem with a thin elastic inclusion.

This paper deals with the application of the multiscale finite element method for simulating the cancellous bone. For this purpose, two types of biphasic representative volume elements are proposed. In the first one, the solid frame consists of thin walls simulated by shell elements. On the other hand, the solid phase of the second model is made up of columns consisting of eight-node brick elements. This choice of representative volume elements is motivated by experimental investigations reporting on the existence of plate-like and rode-like types of cancellous bone and possible conversions between them. The proposed representative volume elements are used to calculate effective material tensors and parameters and to investigate their change in terms of increasing porosity, which is typical for osteoporosis. As a first example, changes in the geometry of the representative volume elements are used to explore material anisotropy. In the end, the final example considers wave propagation through the bone treated as a homogenized medium.

We study the problem of derivation of an effective model of acoustic wave propagation in a two-phase medium composed of a linear Kelvin–Voight viscoelastic solid and a shear-thinning non-Newtonian fluid. Bone tissue is an important example of such composite materials. The microstructure is modeled as a periodic arrangement of fluid-saturated pores inside the solid matrix. The ratio ε of the macroscopic length scale and the size of the microstructural periodicity cell is a small parameter of the problem. We employ two-scale convergence and some other weak convergence techniques to pass to the limit ε→0 in the nonlinear governing equations. The effective model is a two-velocity system for the effective velocity v¯ and a corrector velocity w. The latter describes the influence of the high-frequency oscillations on the effective wave propagation. The effective constitutive equation provides an explicit dependence of the effective stress on e(v¯)+ey(w).

A dynamical problem in the (0, 0) approximation of elastic cusped prismatic beams is investigated when stresses are applied at the face surfaces and the ends of the beam. Two types of cusped ends are considered when the beam cross-section turns into either a point or a straight line segment. Correspondingly, at the cusped end either a force concentrated at the point or forces concentrated along the straight line segment is applied. We prove the exists and uniqueness theorems in appropriate weighted Sobolev spaces.

The acoustic response of a rigid-frame porous plate with a periodic set of inclusions is investigated by a multipole method. The acoustic properties, in particular, the absorption, of such a structure are then derived and studied. Numerical results together with a modal analysis show that the addition of a periodic set of high-contrast inclusions leads to the excitation of the modes of the plate and to a large increase in the acoustic absorption.

We study the well posedness of boundary value problems for elastic cusped prismatic shells in the Nth approximation of I. Vekua's hierarchical models under (all reasonable) boundary conditions at the cusped edge and given displacements at the non‐cusped edge and stresses at the upper and lower faces of the shell. Copyright © 2008 John Wiley & Sons, Ltd.

The Biot model is widely used to model poroelastic media. Several authors have tested its applicability to cancellous bone, but to do so requires a priori estimation of the parameters of the Biot model, which is an uncertain and expensive endeavor. A method of computing acoustic pressure in the low 100 kHz range is developed.

A domain integral method employing a specific Green's function (i.e., incorporating some features of the global problem of wave propagation in an inhomogeneous medium) is developed for solving direct and inverse scattering problems relative to slab-like macroscopically inhomogeneous porous obstacles. It is shown how to numerically solve such problems, involving both spatially-varying density and compressibility, by means of an iterative scheme initialized with a Born approximation. A numerical solution is obtained for a canonical problem involving a two-layer slab.

We study nonexistence of the solutions to quasilinear elliptic differential equations arising from nonisothermal, non-Newtonian Hele-Shaw flows. The proof is based on the trial function method developed by Pohozaev without recourse to comparison theorems and to the maximum principle.

The paper is devoted to study of acoustic wave propagation in a partially consolidated composite material containing loose particles. Friction of particles against the consolidated part of the material causes mechanical energy dissipation. This situation is modelled by assuming that the medium has a periodic microstructure changing rapidly on the small scale ε. Each of the periodic microscopic cells is composed of a viscoelastic matrix containing a rigid particle in frictional contact with the matrix. We use the methods of two‐scale convergence to obtain effective acoustic equations for the homogenized material. The effective equations are history‐dependent and contain the body force term, reminiscent of the well‐known Stokes drag force. Copyright © 2004 John Wiley & Sons, Ltd.

In this paper we formulate a non-isothermal, non-Newtonian Hele-Shaw flow with nonlinear thermal conductivity from the injection molding. Then we study the existence of the resulting nonlinear system. Copyright (c) 2005 John Wiley & Sons, Ltd.

In this paper we investigate the unknown body problem in a wave guide where one boundary has a pressure release condition and the other an impedance condition. The method used in the paper for solving the unknown body inverse problem is the intersection canonical body approximation (ICBA). The ICBA is based on the Rayleigh conjecture, which states that every point on an illuminated body radiates sound from that point as if the point lies on its tangent sphere. The ICBA method requires that an analytical solution be known exterior to a canonical body in the wave guide. We use the sphere of arbitrary centre and radius in the wave guide as our canonical body. We are lead then to analytically computing the exterior solution for a sphere between two parallel plates. We use the ICBA to construct solutions at points ranging over the suspected surface of the unknown object to reconstruct the unknown object using a least-squares matching of computed, acoustic field against the measured, scattered field. Copyright (c) 2005 John Wiley & Sons, Ltd.

In this paper we present some results from our research on the interrogation of cancellous bone using ultrasound. Cancellous bone is known to be poroelastic in structure and, hence, Biot's equations may be applicable. In order to do the interrogation we consider using focused, ultrasonic, waves, symmetric about the axis of propagation. This suggests that we formulate the Biot equations using cylindrical coordinates. Our results provide relations involving the velocities of the transmitted waves and the Biot coefficients and these have clinical applications to the determination of osteoporosis of the cancellous bone.

We discuss an idealized model for compression molding, assuming a compressible flow. Existence theorems are established for this system.

This paper develops algorithms for solving an undetermined coefficient problem for a wave equation. The algorithms are based on an integral representation for the solution to the wave equation obtained by using transmutation. The convergence of the algorithm is studied and numerical experiments are performed.

We study acoustic wave propagation in a two-phase medium in which the solid phase is a linear elastic material, and the fluid phase is assumed to be a compressible Newtonian barotropic fluid. Assuming that properties of the medium change rapidly on the small scale @e, we analyze the microscopic nonlinear Navier-Stokes equations and show that they can be linearized when @e tends to zero. Using a variant of Tartar's method of oscillating test functions, we derive effective acoustic equations which turn out to be viscoelastic. In order to treat disordered materials occurring in nature, we develop a new approach to describing geometry of a nonperiodic medium with length scale separation. Our approach is not based on probabilistic considerations. Instead, we postulate that certain inequalities hold uniformly on the microscale.