Valery M. Levin

Effective medium method in the problem of axial elastic shear wave propagation through fiber composites

The effective medium method (EMM) is applied to the solution of the problem of monochromatic elastic shear wave propagation through matrix composite materials reinforced with cylindrical unidirected fibers. The dispersion equations for the wave numbers of the mean wave field in such composites are derived using two different versions of the EMM. Asymptotic solutions of these equations in the long and short wave regions are found in closed analytical forms. Numerical solutions of the dispersion equations are constructed in a wide region of frequencies of the incident field that covers long, middle and short wave regions of the mean wave field. Velocities and attenuation factors of the mean wave fields in the composites obtained by different versions of the EMM are compared for various volume concentrations and properties of the inclusions. The main discrepancies in the predictions of different versions of the EMM are indicated, analyzed and discussed.

Propagation of electroacoustic waves in the transversely isotropic piezoelectric medium reinforced by randomly distributed cylindrical inhomogeneities

Abstract The propagation of electroacoustic waves in a piezoelectric medium containing a statistical ensemble of cylindrical fibers is considered. Both the matrix and the fibers consist of piezoelectric transversely isotropic material with symmetry axis parallel to the fiber axes. Special emphasis is given on the propagation of an electroacoustic axial shear wave polarized parallel to the axis of symmetry propagating in the direction normal to the fiber axis. The scattering problem of one isolated continuous fiber (“one-particle scattering problem”) is considered. By means of a Green’s function approach a system of coupled integral equations for the electroelastic field in the medium containing a single inhomogeneity (fiber) is solved in closed form in the long-wave approximation. The total scattering cross-section of this problem is obtained in closed form and is in accordance with the electroacoustic analogue of the optical theorem. The solution of the one-particle scattering problem is used to solve the homogenization problem for a random set of fibers by means of the self-consistent scheme of effective field method. Closed form expressions for the dynamic characteristics such as total cross-section, effective wave velocity and attenuation factor are obtained in the long-wave approximation.

Dynamic potentials and Green's functions of a quasi-plane piezoelectric medium with inclusion

The dynamic potentials of a two–dimensional (2D) quasi–plane piezoelectric infinite medium of transversely isotropic symmetry containing an inclusion of arbitrary shape is derived in terms of scalar solutions of the 2D Laplace and Helmholtz equations. Closed–form expressions for the space–frequency representation of this dynamic potential are obtained for the case when the spatial source distribution is characterized by a region occupied by a circular inclusion embedded in a quasi–plane transversely isotropic matrix. The results are used to solve the dynamic Eshelby problem of a circular inclusion (plane region with the same material characteristics as the matrix) undergoing uniform eigenstrain and eigenelectric field. In contrast to the static case, the dynamic electroelastic fields inside the circular inclusion are non–uniform in the space–frequency representation. The derived dynamic piezoelectric potentials are basic quantities for the description of the dynamic properties of micro–inhomogeneous quasi–plane piezoelectric material systems (e.g. fibre–reinforced piezocomposites).

Eshelby's formula for an ellipsoidal elastic inclusion in anisotropic poroelasticity and thermoelasticity

Eshelbys formula that relates the strain inside of an ellipsoidal inclusion in an unbounded elastic medium to the uniform strain imposed at infinity is generalized to the cases of poroelastic and thermoelastic materials. This result holds for an arbitrary anisotropy of the inclusion and of the host material.

Evolution of the Mechanical Properties of Ceramics during Drying

The evaporative drying is the main step of many methods of the ceramic material manufacturing. One of the main problems here is the cracking and possible fracture due to the shrinkage during drying. The change of mechanical properties of the body during the process presents a real challenge to study the fracture of the drying body. The methods of the mechanics of composites are used here to estimate the effective properties of ceramics during manufacturing. Two models of the drying body are used: viscoelastic liquid matrix containing a set of elastic solid inclusions, that corresponds to the first stage of the process, and liquid inclusions distributed in the elastic matrix, corresponding to the final stage, when the solid network is formed. The obtained results are in good agreement with experimental results available in the literature.

On the Effective Constants of Inhomogeneous Poroelastic Medium

The arbitrary anisotropic micro-inhomogeneous (composite) poroelastic medium is considered, containing a random set of ellipsoidal inclusions with other poroelastic characteristics. The properties of these constituents are described by the linear poroelastic theory by Biot [1]. One of the self-consistent schemes named effective field method (EFM) is used to develop the explicit expressions for the effective poroelastic characteristics (tensor of the frame elastic module C ijkl * , Biot’s effective stress coefficient tensor α ij * and Biot’s constant M *) of the static poroelastic theory. For the two constituents composite porous materials these expressions satisfy the generalized Gassmann relations obtained in [2,3].

Scattering of acoustoelectric waves on a cylindrical inhomogeneity in the transversely isotropic piezoelectric medium

The scattering of acoustoelectric waves on a continuous cylindrical fiber embedded in a piezoelectric medium of hexagonal (transversely isotropic) symmetry is considered. It is assumed that both the matrix and the fiber has hexagonal symmetry with different material characteristics but with the same axis of symmetry which coincides with the cylinder axis of the fiber. Expressions for scattering amplitudes of the acoustoelectric waves follow from a system of integral equations for the electroelastic fields in the medium containing an inhomogeneity. This system is obtained in terms of Greens function of the coupled dynamic electroelastic problem. Explicit expressions are obtained for the components of the Greens function and scattering amplitudes for the quasiplane dynamic problem in the transversely isotropic piezoelectric medium. General formulae are derived for the total scattering cross-section of acoustoelectric waves propagating in the direction normal to the fiber axis. Finally, explicit expressions are obtained for the scattering amplitudes and scattering cross-sections of three acoustoelectric waves in the long wave limit. The results may be useful for future applications to various acoustoelectric inhomogeneity problems in piezoelectric media with hexagonal symmetry.

Green's Function for the Infinite Two-Dimensional Orthotropic Medium

The elastostatic Greens function (fundamental solution) for displacements of the two-dimensional infinite medium with orthotropic symmetry is derived in closed form in terms of elementary functions.

Scattering of monochromatic longitudinal waves on a planar crack of arbitrary shape in a fluid-saturated poroelastic medium

Scattering of monochromatic longitudinal waves on a planar crack of arbitrary shape in a saturated poroelastic medium is considered. The medium is described by Biot’s constitutive equations, the crack sides are fluid permeable. The problem is reduced to a two-dimensional integral equation for the crack opening vector. Gaussian approximating functions are used for discretization of this equation. For such functions, the elements of the matrix of discretized problem are combinations of four standard one-dimensional integrals that can be tabulated. As a result, numerical integration is not needed. For regular grids of approximating nodes, this matrix has Toeplitz’s structure, and matrix-vector products can be calculated by the fast Fourier transform technique. The latter accelerates substantially the process of iterative solution of the discretized problem. Calculation of crack opening vectors, differential, and total cross-sections of circular and elliptic cracks are performed for longitudinal incident waves orthogonal to the crack surfaces. Dependencies of these characteristics on the medium permeability and wavefrequency are studied. Comparison of a crack in the poroelastic medium and in a dry elastic medium with the same porosity and skeleton elastic properties is presented.

Propagation of axial shear waves in elastic composites reinforced with random or regular sets of unidirected fibers

This work is devoted to the problem of axial shear elastic wave propagation in composites reinforced with a set of unidirected cylindrical fibers. The effective field method and quasicrystalline approximation are used for the development of the dispersion equation for the wavenumber of the mean (coherent) wave field propagating in the composite. This dispersion equation serves for all frequencies of the incident field, properties and volume concentrations of the fibers. In the method, peculiarities in spatial distributions of inclusions are taken into account via a specific correlation function of the fiber set. Such a function may be constructed for random as well as for regular sets of fibers. Thus, wave propagation in random composites as well as in composites with regular fiber arrangements may be considered. In the case of a random set of fibers, different branches of the wave propagation in the composites with an epoxy matrix and glass fibers are obtained and analyzed. Such branches are constructed also for composites with regular lattices of fibers, and the position of the pass bands, where waves can propagate, and stop bands, where waves exponentially attenuate, in the frequency region are indicated. The comparison of the obtained approximate solutions with some exact solutions of the wave propagation problem for regular composites is presented.