Anisotropic Elastic Half-space Research Articles

Waves in stratified anisotropic poroelastic media: a transfer matrix approach

The problem of propagation of waves in multilayered anisotropic poroelastic medium is studied using the technique of transfer matrix. The model consists of a stack of anisotropic poroelastic solid layers overlying an anisotropic elastic solid layered half-space. The whole layered medium is lying under fluid half-space. The transfer matrices for anisotropic poroelastic layer and anisotropic elastic solid layer are derived. Using these matrices, reflection and transmission coefficients for the waves propagating in considered multilayered model are evaluated. The interface between the two stacks is considered to be imperfect and modified boundary conditions are applied thereat. Numerical computations are done for a particular model. The effects of porosity, anisotropy of the layers, thickness of the layers and imperfect bonding are studied on the wave propagation through the considered model. The results indicate that anisotropy of half-space is a very dominant property influencing the reflection–refraction phenomenon. Also, it is noticed that the imperfection of interface is accompanied by the dissipation of energy and the interface acts as an absorbing boundary.

Reflection and transmission of SH-waves at a corrugated interface between two laterally and vertically heterogeneous anisotropic elastic solid half-spaces

This paper is concerned with the reflection and transmission coefficients of SH-waves at a corrugated interface between two anisotropic heterogeneous elastic solid half spaces. Both the half spaces are taken transversely isotropic and laterally and vertically heterogeneous. The Rayleigh’s method of approximation is adopted and expressions for reflection and transmission coefficients are obtained in closed form for the first-order approximation of the corrugation. In Rayleigh’s method, expressions in boundary conditions containing the function defining the corrugated boundary are expanded in Fourier series and unknown coefficients in the solutions are determined to any given order of approximation in terms of a small parameter characteristic of the boundary. The analytical expressions of these coefficients show that they depend upon corrugation of the interface and are strongly influenced by the anisotropy and heterogeneity of the half-spaces. Numerical computations are performed for the case of a particular corrugated interface: ζ =c cos k* x showing that the effect of heterogeneity on the reflection and transmission coefficients is minimum near the normal incidence and dominance of this effect increases with the angle of incidence. For incident wave striking at 45°, the effect of the corrugation is found significant on the reflection and transmission coefficients. The maximum effect of transverse isotropy on the reflection and transmission coefficients is observed at normal incidence when the values of the anisotropy parameters are 0.5 and 0.8 for the upper and lower half-spaces, respectively. The effect of frequency of the incident wave is observed on all reflected and refracted waves. The analytical expressions derived by Tomar and Saini (1997), Gupta (1987) and Asano (1960) are obtained as particular cases with our formulation.

Subsonic slip waves along the interface between two anisotropic elastic half-spaces in sliding contact with separation

The Stroh sextic formalism, together with the Fourier analysis and singular integral equation technique, is used to study propagation of possible slip waves in presence of local separation at the interface between two contact anisotropic solids. The existence of such waves is discussed in details. It is found that such waves may exist if at least one medium admits a Rayleigh wave below the minimum limiting speed of the two media. The wave speed is not fixed. It can be of any value in some regions between the lower Rayleigh wave speed and minimum limiting speed, depending on the existence of the first and second slip-wave solutions without interfacial separation studied by Barnett et al. [Proc. Roy. Soc. Lond. A 415 (1988) 389]. The waves have no free amplitude directly, but involve arbitrary size of the separation zone which depends on the intensity of the motion. The interface normal traction and the particle velocities involve square-root singularities at both ends of the separation zones.

Nonuniformly Moving Loads on an Anisotropic Elastic Half-Space

ABSTRACTThe transient motion in an anisotropic elastic half-space due to a moving surface line load is considered. The load is applied suddenly on the surface and moves off in a fixed direction with nonuniform speed. Integral expressions for the displacements are derived using the reciprocal theorem. The waves generated by the moving load are discussed. Special attention is paid to the singularities in surface displacements generated as the load moves through the Rayleigh wave speed. Explicit expression is obtained for the particle velocity due to a constant load moving with constant speed.

Transient motion of an interfacial line force or dislocation in an anisotropic elastic bimaterial

A full-field dynamic solution of an interfacial line force or dislocation in bonded anisotropic elastic half-spaces is presented. The form of the solution resembles that of the corresponding static solution. The evaluation of the solution requires only the calculation of certain eigenvalue problems. Particular attention is given to the singular feature in the response associated with the interfacial Stoneley wave. Numerical examples are given to illustrate the characteristics of a pair of bonded misoriented half-spaces of GaAs.

Surface waves of non-Rayleigh type

Existence of surface waves of non-Rayleigh type propagating on some anisotropic elastic half-spaces is proved. Conditions for originating the non-Rayleigh type waves are analyzed. An example of a transversely isotropic material admitting a surface wave of the non-Rayleigh type is constructed.

Explicit secular equations for surface waves in monoclinic materials with the symmetry plane x 1 = 0, x 2 = 0 or x 3 = 0

The traditional way of deriving the secular equation for surface waves propagating in the direction of the x1axis in an anisotropic elastic halfspace x20 is to find a general steadystate solution f...

The surface motion due to a line force or dislocation within an anisotropic elastic half-space.

An explicit solution of the surface displacements due to a line force or a line dislocation within an anisotropic half-space is presented. The surface displacements are derived from the solution corresponding to a suddenly applied surface line force using the reciprocal theorem. The solution is in a closed form for isotropic media. For anisotropic solids, only an eigenvalue problem needs to be solved numerically for a given time and position to calculate the surface response. Numerical results are given for silicon.

On Nix's Theorem for Two Skew Dislocations in Anisotropic Elastic Half-Spaces and Bimaterials

In the earlier papers, the net interaction force F between two skew dislocations with Burgers vectors b, b separated by a distance h in an infinite anisotropic elastic space, in a half space, or in a dissimilar bimaterial has been presented. In this paper, the authors employ Green's functions to obtain new explicit expressions of F. The new solutions provide new physical interpretations. The authors show that, for the infinite space and the half-space with a traction-free surface, F is independent of b2, b2 and that F = 0 if b or b is along the x2-axis. F is independent of the skew angle Oif the projections of b, b on the plane X2 = 0 lie along a pair of conjugate radii of an ellipse. For other half-spaces and bimaterials, F(on b) = 0 when b and b x v are along the x2-axis where v is the axial vector of a skew-symmetric matrix. Hence b is arbitrary if v = 0, which is the case for the half-space with a slippery surface and for certain bimaterials. F(on b) is independent of 0 when b is along the x2-axis and b is coplanar with v and the xi-axis. More physical interpretations are presented in the paper.

A new modified Lekhnitskii formalism àla Stroh for steady-state waves in anisotropic elastic materials

For the analysis of a two-dimensional steady-state motion such as the surface wave in an anisotropic elastic half-space, the Stroh formalism has always been employed. The solutions are in terms of the elastic stiffnesses C αβ . The Lekhnitskii formalism for elastostatics that provides the solutions in terms of the reduced elastic compliances s αβ ′ is not applicable for two-dimensional steady-state motion. We present a new modified Lekhnitskii formalism in the style of Stroh that can be employed for analyzing two-dimensional steady-state motion. In contrast to the Stroh formalism for which one computes the eigenvector b in terms of the eigenvector a, the new modified Lekhnitskii formalism can compute the eigenvector b without computing the vector a. This feature is attractive in the study of surface waves because the vector b is related to the surface traction. The vanishing of the surface traction at the boundary of the half-space is the key in the surface wave theory. Application to one-component surface waves shows that the conditions for such waves are easily deduced. Motivated by the new modified Lekhnitskii formalism we show that an eigenrelation for the vector b can also be derived for the Stroh formalism.

The net interaction force between two skew dislocations in an anisotropic linear elastic bimetallic medium

Barnett, in 1998, considered the problem of two skew dislocations in an anisotropic elastic half-space, each dislocation being parallel to the half-space boundary (which was taken as traction-free). He showed that the net interaction force of one dislocation on the other is independent of the separation of the dislocations. Rather incredibly, if the dislocation closest to the half-space boundary is denoted as (1) and the other skew dislocation is denoted as (2), Barnett deduced that the net interaction force of (1) on (2) vanishes, while the total interaction force of (2) on (1) is exactly twice the net force which (2) exerts on (1) in an infinite medium! The present work examines two skew dislocations in two perfectly bonded anisotropic half-spaces (one half-space may be doubly dislocated or each half-space may contain one dislocation). We show that there is an additional contribution (beyond that given by the half-space result) to the net interaction force exerted by one dislocation on the other, which is also separation-independent; essentially each dislocation experiences an additional contribution to the net interaction force which is the same as if the other dislocation were located at the interface between the two half-spaces. This additional contribution to the net force is ‘almost’ explicit, requiring only a computation of the admittance matrices of the Stroh formalism for each half-space.

Stress intensification due to an edge crack in an anisotropic elastic solid

Integral transform techniques are used to determine the stress intensity factors of a crack at the edge of an anisotropic elastic half space under generalized plane strain conditions. Numerical results are given for a carbon fibre reinforced epoxy in uniaxial tension.

Independence of the Presence of Cracks or Inclusions of the Net Interaction Force Acting on a Dislocation by a Skewed Line Singularity

Orlov and Indenbom [1] have shown that the net (integrated) interaction force F between two skew dislocations with Burgers vectors \(\hat b,b\) separated by a distance h in an infinite anisotropic elastic medium is independent of h. Nix [2] computed numerically the net interaction force F between two skew dislocations that are parallel to the traction-free surface X2=0 of an isotropic elastic half-space. His numerical results showed that F was independent of h; a partial result of what Barnett [3] called Nix"s theorem. The separation-independence portion of Nix"s theorem has been proved to hold for a general anisotropic elastic half-space with a traction-free, rigid, or slippery surface, and for bimaterials [3-5]. In this paper, we show that the net interaction force \(F\left( {on \hat b} \right)\) is independent of the presence of inclusions. We will consider the case in which the line dislocation b is a more general line singularity which can include a coincident line force with strength f per unit length of the line singularity. An inclusion is an infinitely long dissimilar anisotropic elastic cylinder of an arbitrary cross-section whose axis is parallel to the line singularity (f, b). The (skew) line dislocation \(\hat b\) does not intersect the inclusion. The special cases of an inclusion are a void, crack, or rigid inclusion. There can be more than one inclusion of different cross sections and different materials. The line singularity (f, b) can be outside the inclusions or inside one of the inclusions. The inclusions and the matrix need not have a perfect bonding. One can have a debonding with or without friction.

Three-Dimensional Interface Cracks in Anisotropic Bimaterials: The Non-Oscillatory Case

Two-dimensional interface cracks in anisotropic bimaterials have been studied extensively in the literature. However, solutions to three-dimensional interface cracks in anisotropic bimaterials are not available. In this paper, a penny-shaped crack on the interface between two anisotropic elastic half-spaces is considered. A formal solution is obtained by using the Stroh method in two-dimensional elasticity in conjunction with the Fourier transform method. Fracture mechanics parameters such as the stress intensity factor, crack-opening displacement, and energy release rate are obtained in terms of the interfacial matrix M. To illustrate the solution procedure, a circular delaminations in a unidirectional and a cross-ply composite are considered. Numerical results for the stress intensity factors and energy release rate along the crack front are presented.

The net interaction force between two skew dislocations in an elastically anisotropic half-space

In the present work the author shall show that the independence of the total interaction force on separation remains valid for two skew straight dislocations in an anisotropic linear elastic half-space (each dislocation being parallel to the traction-free half-space boundary). If he refers to the dislocation closest to the half-space boundary as (1) and to the other dislocation as (2), he obtains the remarkable result that the total interaction force of (1) on (2) vanishes, while the total interaction force of (2) on (1) is precisely twice the net interaction force which (2) exerts on (1) in an infinite medium. The author calls this set of results Nix`s Theorem after a friend and colleague W.D. Nix, whose numerical computations for isotropic half-spaces motivated the interest in seeking its proof for half-spaces of arbitrary anisotropy. The fact that the two net interaction forces in the half-space problem are not equal and opposite (unless they both vanish) is due to the lack of translational invariance normal to the half-space boundary.

Surface Response of a Fluid Loaded Anisotropic Solid to an Impulsive Point Force: Application to Scanning Acoustic Microscopy

An algorithm is presented for calculating the dynamic Green's function of two joined anisotropic elastic half-spaces. It is used to determine the surface response of a fluid loaded anisotropic solid to an impulsive point force. Calculated images, representing the dependence of the normal displacement on time and direction, are in good agreement with published images of a number of crystals obtained with a configuration of two point focus acoustic lenses. The images show prominent features due to leaky Rayleigh and pseudosurface waves, as well as sharper lateral wave structures associated with bulk wave arrivals.

Hypersingular integral equations for arbitrarily located planar cracks in an anisotropic elastic bimaterial

Hypersingular integral equations are derived for the problem of arbitrarily located planar cracks lying in the interior of two dissimilar anisotropic elastic half-spaces which adhere perfectly to each other. The unknown functions in the integral equations are the crack-opening displacements. The integral equations are solved numerically for specific examples involving particular transversely isotropic materials in order to compute physical quantities of interest such as the crack tip stress intensity factors or the crack energy.

Self–orthogonal sextic formalism for simple reflection and mode conversion at the free surface of an anisotropic elastic half–space

Simple reflection and mode conversion of acoustic waves at a free surface of a semi–infinite anisotropic elastic medium are investigated using the Stroh sextic formalism. By introducing new sets of self–orthogonal wavefunctions in the self–orthogonal sextic formalism, we establish a series of linear dependence relations which enable us to analyse various kinds of simple reflection and mode conversion under a unified theoretical framework. It is shown that the properties of simple reflection and mode conversion can be described in terms of three independent constants.

Reflection coefficients for weak anisotropic media

SUMMARY The interaction of plane elastic waves with a plane boundary between two anisotropic elastic half-spaces is investigated. The anisotropy dealt with in this study is of a general type. Explicit expressions for energy-related reflection and transmission coefficients are derived. They represent an approximation which is valid for a small deviation of the elastic parameters from isotropy. Classical perturbation theory is applied on a 6x 6 non-symmetric real eigenvalue problem to calculate first-order corrections for the polarization and stress of the plane waves. The explicit solution of the isotropic problem is used as a reference case. Degenerate perturbation theory is used to consider the splitting of the isotropic S-wave into two anisotropic qS-waves. The boundary conditions for two half-spaces in welded contact lead to a 6x6 system of linear equations. A correction to the isotropic solution is calculated by linearization. The resultant coefficients are functions of horizontal slowness, Lam6 parameters and densities of the reference media, and of the perturbation of the elasticity tensors from isotropy.

Multiple interacting planar cracks in an anisotropic multilayered medium under an antiplane shear stress: a hypersingular integral approach

The problem of determining the stress field around an arbitrary number of arbitrarily-located planar cracks in an anisotropic elastic half-space which adheres perfectly to an infinitely-long elastic strip is considered. The strip is made up of several layers of anisotropic materials which are perfectly bonded to one another. The multilayered medium is assumed to undergo an antiplane deformation. Suitable integral expressions are used to represent the displacement and the stress, leading to a system of hypersingular integral equations to be solved. For a specific example of the problem, which involves particular transversely-isotropic materials, the hypersingular integral equations are solved numerically, in order to calculate the relevant crack tip stress intensity factors.