Propagation of low-frequency elastic waves in double-porositymedia containing viscoelastic component in the pores

A self-consistent scheme named the effective field method (EFM) is applied for the calculation of the velocities and quality factors of elastic waves propagating in double-porosity media. A double-porosity medium is considered to be a heterogeneous material composed of a matrix with primary pores and inclusions that are represent by flat (crack-like) secondary pores. The prediction of the effective viscoelastic moduli consists of two steps. First, we calculate the effective viscoelastic properties of the matrix with the primary small-scale pores (matrix homogenization). Then, the porous matrix is treated as a homogeneous isotropic host where the large-scale secondary pores are embedded. Spatial distribution of inclusions in the medium is taken into account via a special two-point correlation function. The results of the calculation of the viscoelastic properties of double-porosity media containing isotropic fields of crack-like inclusions and double-porosity media with some non-isotropic spatial distributions of crack-like inclusions are presented.     

Effective elastic properties of solids with irregularly shaped inclusions

A unit rectangular cell is usually cut out from a medium for investigating fracture mechanism and elastic properties of the medium containing an array of irregularly shaped inclusions. It is desirable to clarify the geometrical parameters controlling the elastic properties of heterogeneous materials because they are usually embedded with randomly distributed particulate. The stress and strain relationship of the rectangular cell is obtained by an ad hoc hybrid-stress finite element method. By matching the boundary condition requirements, the effective elastic properties of composite materials are then calculated, and the effect of shape and arrangement of inclusions on the effective elastic properties is subsequently considered by the application of the ad hoc hybrid-stress finite element method through examining three types of rectangular cell models assuming rectangular arrays of rectangular or diamond inclusions. It is found that the area fraction (the ratio of the inclusion area over the rectangular cell area) is one dominant parameter controlling the effective elastic properties.     

Effects of Stress and Pore Pressure on Deformation of a Thin Crack-Like Inhomogeneity in a Poroelastic Medium

The influence of applied stresses and internal fluid pressure on deformation of a flat ellipsoidal inhomogeneity in a homogeneous poroelastic medium is studied. The inhomogeneity is filled with a porous material having a skeleton much softer than the corresponding to the surrounding medium (crack-like inclusion). The result of the calculation of stress and strain fields inside such inclusion is presented. Explicit formulas for the inclusion aspect ratio changing are obtained. They are relevant for velocities of seismic waves in porous rocks (Toksöz et al, 1976) their permeability (Gibson &; Toksöz, 1990) as functions of confining and pore pressures. Results are compared with the ones available in literature.     

Scattering of elastic waves by spherical inclusions with applications to low frequency wave propagation in composites

Scattering of plane elastic waves by a spherical inclusion is considered. A unified method of solution is presented which treats compressional and shear incidence on a similar basis. Explicit results are given for Rayleigh scattering. We apply the results of the single scattering problem to the propagation of low frequency waves in a composite containing a dilute concentration of spherical inclusions. Explicit formulae are given for the effective wave speeds and attenuations when the inclusions are voids. Both the compressional and shear wave speeds decrease initially as a function of frequency.     

Numerical simulation of acoustic band gaps in homogenized elastic composites

The dispersion of acoustic or elastodynamic waves in elastic composites are studied using the homogenized model. We consider heterogeneous periodic structures consisting of soft but heavy inclusions embedded in a stiffer matrix. By virtue of the asymptotic homogenization technique in conjunction with an appropriate scaling of the elasticity coefficients in the inclusions, the limit model exhibits the band gaps in wave propagation due to the negative effective mass. This phenomenon can be revealed by studying guided waves in discrete mass-spring structures with scale-dependent parameters. The main purpose of the paper is to justify the applicability of the homogenized model of the heterogeneous elastic continuum for prediction of the band gaps in structures featured by a finite scale of heterogeneities. We show the band gaps numerical identification and discus aspects of anisotropy, microstructure geometry and material contrast between the constituents in the context of the long wave dispersion.     

Combining self-consistent and numerical methods for the calculation of elastic fields and effective properties of 3D-matrix composites with periodic and random microstructures

The work is devoted to the calculation of static elastic fields in 3D-composite materials consisting of a homogeneous host medium (matrix) and an array of isolated heterogeneous inclusions. A self-consistent effective field method allows reducing this problem to the problem for a typical cell of the composite that contains a finite number of the inclusions. The volume integral equations for strain and stress fields in a heterogeneous medium are used. Discretization of these equations is performed by the radial Gaussian functions centered at a system of approximating nodes. Such functions allow calculating the elements of the matrix of the discretized problem in explicit analytical form. For a regular grid of approximating nodes, the matrix of the discretized problem has the Toeplitz properties, and matrix-vector products with such matrices may be calculated by the fast fourier transform technique. The latter accelerates significantly the iterative procedure. First, the method is applied to the calculation of elastic fields in a homogeneous medium with a spherical heterogeneous inclusion and then, to composites with periodic and random sets of spherical inclusions. Simple cubic and FCC lattices of the inclusions which material is stiffer or softer than the material of the matrix are considered. The calculations are performed for cells that contain various numbers of the inclusions, and the predicted effective constants of the composites are compared with the numerical solutions of other authors. Finally, a composite material with a random set of spherical inclusions is considered. It is shown that the consideration of a composite cell that contains a dozen of randomly distributed inclusions allows predicting the composite effective elastic constants with sufficient accuracy.     

Application of the boundary element method to elastic wave scattering by irregular defects

A time-harmonic boundary element formulation for elastic wave scattering in 3D is adapted to ultrasonic NDE. Defect classes addressed are volumetric voids and inclusions, and crack-like elliptical voids. For axisymmetric flaws, comparisons are made with method of optimal truncation (MOOT) and transition-matrix calculations. Comparison to experiment is made for more general shapes. For crack-like voids, comparisons are made with the Kirchhoff, geometric theory of diffraction (GTD), and quasistatic asymptotic approximations. The efficiency and usefulness of the boundary element method (BEM) in finding the bounds of applicability of these approximate theories are demonstrated. An example of a flaw characterization technique based on intermediate frequency scattering data simulated by BEM is given. The ability of BEM to handle nonplanar incident fields, as described by a transducer beam model, is shown. Other computational and modeling efficiencies of the BEM are noted.     

Two-dimensional and axisymmetric unit cell models in the analysis of composite materials

Unit cell models have been widely used for investigating fracture mechanisms and mechanical properties of composite materials assuming periodically arrangement of inclusions in matrix. It is desirable to clarify the geometrical parameters controlling the mechanical properties of composites because they usually contain randomly distributed particulate. To begin with a tractable problem this paper focuses on the effective Young’s modulus E of heterogeneous materials. Then, the effect of shape and arrangement of inclusions on E is considered by the application of FEM through examining three types of unit cell models assuming 2D and 3D arrays of inclusions. It is found that the projected area fraction and volume fraction of inclusions are two major parameters controlling effective elastic modulus of inclusions.     

S波由饱和土入射于弹性土时在界面上的反射与透射

从地震工程实际出发，借助Biot多孔介质中的波动方程，根据各种界面条件导出了S波从饱和土入射于弹性土时在交界面上反射与透射的一般计算公式。作为算例，数值计算分析了S波从饱和土入射于饱和土与弹性土交界面时，饱和土中P1、P2和S波的反射系数以及弹性土中P波、SV射系数与界面排水条件、入射角以及频率之间的关系。结果表明：各种波的反射、透射系数与入射角、入射频率以及界面排水条件有关系。     

Reflection of Magneto-Thermoviscoelastic Waves Under Generalized Thermoviscoelasticity

Considering each of the thermal fields, viscoelastic and electromagnetic fields contribute to the total deformation of a body and interact with each other. Reflection of magneto-thermoelastic waves under generalized thermoviscoelastic theories is employed to study the reflection of plane harmonic waves from a semi-infinite elastic solid in a vacuum. The expressions for the reflection coefficients, which are the ratios of the amplitudes of the reflected waves to the amplitudes of the incident waves, are obtained, and the reflection coefficient ratios variation with the angle of incidence under different conditions are shown graphically when acrylic plastic materials are considered.     

A polarization‐based FFT iterative scheme for computing the effective properties of elastic composites with arbitrary contrast

It is recognized that the convergence of FFT‐based iterative schemes used for computing the effective properties of elastic composite materials drastically depends on the contrast between the phases. Particularly, the rate of convergence of the strain‐based iterative scheme strongly decreases when the composites contain very stiff inclusions and the method diverges in the case of rigid inclusions. Reversely, the stress‐based iterative scheme converges rapidly in the case of composites with very stiff or rigid inclusions but leads to low convergence rates when soft inclusions are considered and to divergence for composites containing voids. It follows that the computation of effective properties is costly when the heterogeneous medium contains simultaneously soft and stiff phases. Particularly, the problem of composites containing voids and rigid inclusions cannot be solved by the strain or the stress‐based approaches. In this paper, we propose a new polarization‐based iterative scheme for computing the macroscopic properties of elastic composites with an arbitrary contrast which is nearly as simple as the basic schemes (strain and stress‐based) but which has the ability to compute the overall properties of multiphase composites with arbitrary elastic moduli, as illustrated through several examples. Copyright © 2011 John Wiley & Sons, Ltd.     

Scattering of antiplane shear waves in a fiber-reinforced composite medium with interfacial layers

Summary This paper deals with the scattering of antiplane shear waves in a metal matrix composite reinforced by fibers with interfacial layers. We assume same-size cylindrical inclusions and same-thickness interface layers with nonhomogeneous elastic properties. The effective complex wave numbers follow from the coherent wave equation which depends only upon the scattering amplitude of the single scattering problem. Effective elastic constants can be obtained from phase velocities of coherent waves. Numerical calculations for an SiC-fiber-reinforced Al composite are carried out, and the effect of interface properties on scattering cross section, phase velocity, attenuation of coherent plane wave, and effective elastic constant is shown graphically.     

Influence of parallelogram cells in the axial behaviour of fibrous composite

Effective longitudinal shear moduli closed-form analytical expressions of two-phase fibrous periodic composites are obtained by means of the asymptotic homogenization method (AHM) for a parallelogram array of circular cylinders. This work is an extension of previous reported results, where elastic, piezoelectric and magneto-electro-elastic composites for square and hexagonal arrays with perfect contact were considered. The constituents exhibit transversely isotropic properties. A doubly period-parallelogram array of cylindrical inclusions under longitudinal shear is studied. The behaviour of the anisotropic shear elastic coefficients is studied for several cell geometry arrays. Numerical examples and comparisons with other theoretical results demonstrate that the present model is efficient for the analysis of composites in which the periodic cell is rectangular, rhombic or a parallelogram. The effect of the arrangement of the cells on the shear effective property is discussed. The present method can provide benchmark results for other numerical and approximate methods.     

Calculation of electro and thermo static fields in matrix composite materials of regular or random microstructures

In the work, a numerical method for calculation of electro and thermo static fields in matrix composite materials is considered. Such materials consist of a regular or random set of isolated inclusions embedded in a homogeneous background medium (matrix). The proposed method is based on fast calculation of fields in a homogeneous medium containing a finite number of isolated inclusions. By the solution of this problem, the volume integral equations for the fields in heterogeneous media are used. Discretization of these equations is carried out by Gaussian approximating functions that allow calculating the elements of the matrix of the discretized problem in explicit analytical forms. If the grid of approximating nodes is regular, the matrix of the discretized problem proves to have the Toeplitz structure, and the matrix-vector product with such matrices can be calculated by the Fast Fourier Transform technique. The latter strongly accelerates the process of iterative solution of the discretized problem. In the case of an infinite medium containing a homogeneous in space random set of inclusions, our approach combines a self-consistent effective field method with the numerical solution of the conductivity problem for a typical cell. The method allows constructing detailed static (electric or temperature) fields in the composites with inclusions of arbitrary shapes and calculating effective conductivity coefficients of the composites. Results are given for 2D and 3D-composites and compared with the existing exact and numerical solutions.     

Wave Propagation in a Green–Naghdi Thermoelastic Solid with Diffusion

The Green and Naghdi theory of thermoelasticity is applied to study plane-wave propagation in an elastic solid with thermo-diffusion. The governing equations of an elastic solid with generalized thermo-diffusion are solved to show the existence of three coupled longitudinal waves and a shear vertical (SV) wave in a two-dimensional model of the solid with thermo-diffusion. The reflection of plane waves from a thermally insulated stress-free surface of an elastic solid with thermo-diffusion is also studied. A non-homogeneous system of four equations in reflection coefficients is obtained. The speeds of the plane waves are computed numerically and plotted against frequency for a particular range. The complex absolute values of the reflection coefficients of all reflected waves are computed numerically and plotted against the angle of incidence of the striking wave at the free surface. The effects of diffusion parameters are shown graphically for speeds and reflection coefficients of plane waves.     

Existence of longitudinal and transverse waves in anisotropic thermoelastic media

Four waves propagate in an anisotropic thermoelastic medium. The fastest among them is a quasi-longitudinal wave. The slowest of them is a thermal wave. The remaining two are called quasi-transverse waves. The prefix ‘quasi’ refers to their polarizations being nearly, but not exactly, parallel or perpendicular to the direction of propagation. The polarizations of these four waves are not mutually orthogonal. Hence, unlike anisotropic elastic media, the existence of a longitudinal wave may not imply the existence of a transverse wave, by default. The existence of a purely longitudinal wave in an anisotropic thermoelastic medium is ensured by the stationary characters of three expressions. These expressions involve components of phase direction with elastic (stiffness and coupling) and thermal coefficients of the thermoelastic medium. The existence of a purely transverse wave is ensured by the two equations restricting the choice of thermoelastic (stiffness and coupling) coefficients. The existence of longitudinal and transverse waves along the coordinate axes and in the coordinate planes are discussed for general anisotropy. The discussion is extended to orthotropic materials, and the existence of pure phases is explored along few specific phase directions.     

A self-consistent approach to dynamic effective properties of a composite reinforced by distributed spherical particles

Summary A self-consistent approach to dynamic effective properties of a composite reinforced by randomly distributed spherical inclusions is studied. The coherent plane waves propagating through the particle-reinforced composite are of attenuation nature. It implies that there is an analogy between the particle-reinforced composite and the effective medium with complex-valued elastic constants from the viewpoints of wave propagation. A composite sphere consisting of the inclusion, the matrix and the interphase between them is assumed embedded in the effective medium. The effective wavenumbers of the coherent plane waves propagating through the particle-reinforced composite are obtained by the dynamic self-consistent conditions which require that the forward scattering amplitudes of such a composite sphere embedded in the effective medium are equal to zero. The dynamic effective properties (effective phase velocity, effective attenuation and effective elastic constants) obtained by the present dynamic self-consistent approach for SiC-Al composites are compared numerically with that obtained by the effective field approach at various volume concentrations. It is found that there is a good agreement between the two approaches at a relatively low frequency and low volume concentration but the numerical results deviate from each other at a relatively high frequency and high volume concentration.     

Wave beams in crystals with dislocations and quadratic nonlinearity

Nonlinear elastic waves in dislocated crystals are considered in terms of equations with constant coefficients, which involve a quadratic nonlinearity. A nonlinear modulation equation with a cubic nonlinearity is obtained for the first harmonic amplitude, a dispersion equation is derived, and the absorption coefficient is determined. The stability and focusing (self-focusing) of a wave beam are studied.     

Effects of sand aggregate on ultrasonic attenuation in cement-based materials

Ultrasonic wave attenuation measurements have been successfully used to characterize the microstructure and mechanical properties of inhomogeneous materials; these ultrasonic techniques have the potential to provide for the in-situ characterization of microstructure changes in cement-based materials due to damage. Recent research has applied acoustic scattering models to quantitatively predict ultrasonic attenuation for evaluating the air void content in hardened cement paste. The objective of the current research is to investigate the influence of sand aggregate on the ultrasonic attenuation as a first step towards a full simulation of more realistic microstructures in real concrete. Hardened cement paste samples containing sand aggregate of varying volume fractions are considered. The research employs an independent scattering model and a self-consistent effective medium theory to predict the scattering-induced attenuation of longitudinal ultrasonic waves by the sand inclusions distributed in the cement paste matrix. The predicted attenuation coefficients are compared with measured ones. It is observed that at low volume fractions, both models provide a good estimate of the total attenuation in the specimens. These results indicate that it is possible to use a physics-based model to quantify the effect of sand aggregate on ultrasonic attenuation.     

A stepping scheme for predicting effective properties of the multi-inclusion composites

A stepping scheme is developed in the present paper to predict the effective properties of composites with high inclusion volume fraction and/or several kinds of inclusions. In this method the inclusions are treated step by step and the effective stiffness coefficients are calculated for the resulting composite in each step. The numerical results of the effective properties of multi-inclusion composites indicate that materials with a high volume fraction of inclusions or multiple inclusions can be dealt with precisely by the stepping scheme. For the case of a binary composite consisting of a matrix and one kind of inclusion, the present results agree with those from a differential scheme.