Explicit expressions of S(v), H(v) and L(v) for anisotropic elastic materials

The three matrices L(v), S(v) and H(v), appearing frequently in the investigations of the two-dimensional steady state motions of elastic solids, are expressed explicitly in terms of the elastic stiffness for general anisotropic materials. The special cases of monoclinic materials with a plane of symmetry at x₃ = 0, x₁ = 0, and x₂ = 0 are all deduced. Results for orthotropic materials appearing in the literature may be recovered from the present explicit expressions.     

Surface effects in the deformation of an anisotropic elastic material with nano-sized elliptical hole

We examine the effect of surface energy on an anisotropic elastic material weakened by an elliptical hole. A closed-form, full-field solution is derived using the standard Stroh formalism. In particular, explicit expressions for the hoop stress, normal, in-plane tangential and out-of-plane displacement components along the edge of the hole are obtained. These expressions clearly demonstrate the effect of elastic anisotropy of the bulk material on the corresponding field variables. When the material becomes isotropic, the hoop stress agrees with the well-known result in the literature while both the in-plane tangential and out-of-plane displacements vanish and the normal displacement is constant along the entire boundary of the elliptical hole.     

Explicit conditions for the existence of exceptional body waves and subsonic surface waves in anisotropic elastic solids

It is known that a subsonic surface (Rayleigh) wave exists in an anisotropic elastic half-space x₂  0 if the first transonic state is not of Type 1. If the first transonic state is of Type 1 but the limiting wave is not exceptional, a subsonic surface wave exists. If the first transonic state is of Type 1 and the limiting wave is exceptional, a subsonic surface wave exists when . It is shown that an exceptional body wave is necessarily an exceptional transonic wave, and could be an exceptional limiting wave. Only two wave speeds are possible for an exceptional body wave. We present explicit conditions in terms of the reduced elastic compliances for the existence of an exceptional body wave. If an exceptional body wave exists, conditions are given for identifying whether the transonic state is of Type 1. Hence, through the existence of an exceptional body wave we provide explicit conditions for the existence of a subsonic surface wave with the exception when needs to be computed.     

Surface instability of a semi-infinite anisotropic elastic body under surface van der Waals forces

We investigate the surface instability of an anisotropic elastic half-plane subjected to surface van der Waals forces due to the influence of another rigid contactor by means of the Stroh formalism. It is observed that the surface of a generally anisotropic elastic half-plane subjected to van der Waals forces from another rigid flat is always unstable. The wave number of the surface wrinkling is only reliant on the positive M₂₂ component of the 3 × 3 surface admittance tensor M, the van der Waals interaction coefficient β and the surface energy γ of the elastic half-plane. The decay rate of surface perturbation along the direction normal to the surface of the anisotropic half-plane is different from the wave number, a phenomenon different from that observed for an isotropic half-plane.     

Explicit expression of polarization vector for surface waves,slip waves,Stoneley waves and interfacial slip waves in anisotropic elastic materials

We present explicit expression of the polarization vector for surface waves and slip waves in an anisotropic elastic half-space, and Stoneley waves and interfacial slip waves in two dissimilar anisotropic elastic half-spaces. An unexpected result is that, in the case of interfacial slip waves, the polarization vector for the material in the half-space _x2≥0$x_2\ge 0$ does not depend explicitly on the material property in the half-space _x2≤0$x_2\le 0$. It depends on the material property in the half-space _x2≤0$x_2\le 0$ implicitly through the interfacial slip wave speed υ$\upsilon$. The same is true for the polarization vector for the material in the half-space _x2≤0$x_2\le 0$.     

Surface waves in an exponentially graded, general anisotropic elastic material under the influence of gravity

In a recent paper Destrade [1] studied surface waves in an exponentially graded orthotropic elastic material. He showed that the quartic equation for the Stroh eigenvalue p is, after properly modified, a quadratic equation in p² with real coefficients. He also showed that the displacement and the stress decay at different rates with the depth x₂ of the half-space. Vinh and Seriani [2] considered the same problem and added the influence of gravity on surface waves. In this paper we generalize the problem to exponentially graded general anisotropic elastic materials. We prove that the coefficients of the sextic equation for p remain real and that the different decay rates for the displacement and the stress hold also for general anisotropic materials. A surface wave exists in the graded material under the influence of gravity if a surface wave can propagate in the homogeneous material without the influence of gravity in which the material parameters are taken at the surface of the graded half-space. As the wave number k → ∞, the surface wave speed approaches the surface wave speed for the homogeneous material. A new matrix differential equation for surface waves in an arbitrarily graded anisotropic elastic material under the influence of gravity is presented. Finally we discuss the existence of one-component surface waves in the exponentially graded anisotropic elastic material with or without the influence of gravity.     

Slip pulse along an interface between two anisotropic elastic half-spaces in sliding contact with separation

The Stroh sextic formalism, together with Fourier analysis and the singular integral equation technique, is used to study the propagation of a possible slip pulse in the presence of local separation at the interface between two contact anisotropic solids. The existence of such a pulse is discussed in detail. It is found that the pulse may exist if at least one medium admits Rayleigh wave below the minimum limiting speed of the two media. The pulse-propagating speed is not fixed; it can be of any value in some regions between the lower Rayleigh wave speed and minimum limiting speed. These speed regions depend on the existence of the first and second slip-wave solutions without interfacial separation studied by Barnett, Gavazza and Lothe (Proc. R. Soc. Lond. 1988, A415, 389–419). The pulse has no free amplitude directly but involves the arbitrary size of the separation zone that depends on the intensity of the motion. The interface normal traction and the particle velocities involve a square-root singularity at both ends of the separation zones that act as energy source and sink.     

Edge waves in asymmetrically laminated plates

We study edge waves propagating along the edge of an asymmetrically laminated elastic plate for which the out-of-plane component of displacement is coupled with the in-plane components. A Stroh-like formulation is used to show that such a plate can support at most two edge waves. An efficient method for computing the edge-wave speeds is proposed and explained through examples.     

Subinterface crack in an anisotropic piezoelectric bimaterial

In this paper, the problem of a subinterface crack in an anisotropic piezoelectric bimaterial is analyzed. A system of singular integral equations is formulated for general anisotropic piezoelectric bimaterial with kernel functions expressed in complex form. For commonly used transversely isotropic piezoelectric materials, the kernel functions are given in real forms. By considering special properties of one of the bimaterial, various real kernel functions for half-plane problems with mechanical traction-free or displacement-fixed boundary conditions combined with different electric boundary conditions are obtained. Investigations of half-plane piezoelectric solids show that, particularly for the mechanical traction-free problem, the evaluations of the mechanical stress intensity factors (electric displacement intensity factor) under mechanical loadings (electric displacement loading) for coupled mechanical and electric problems may be evaluated directly by considering the corresponding decoupled elastic (electric) problem irrespective of what electric boundary condition is applied on the boundary. However, for the piezoelectric bimaterial problem, purely elastic bimaterial analysis or purely electric bimaterial analysis is inadequate for the determination of the generalized stress intensity factors. Instead, both elastic and electric properties of the bimaterial’s constants should be simultaneously taken into account for better accuracy of the generalized stress intensity factors.     

Kinetic modeling of multiple scattering of elastic waves in heterogeneous anisotropic media

In this paper we develop a multiple scattering model for elastic waves in random anisotropic media. It relies on a kinetic approach of wave propagation phenomena pertaining to the situation whereby the wavelength is comparable to the correlation length of the weak random inhomogeneities—the so-called weak coupling limit. The waves are described in terms of their associated energy densities in the phase space position  ×$\times$  wave vector. They satisfy radiative transfer equations in this scaling, characterized by collision operators depending on the correlation structure of the heterogeneities. The derivation is based on a multi-scale asymptotic analysis using spatio-temporal Wigner transforms and their interpretation in terms of semiclassical operators, along the same lines as Bal (2005). The model accounts for all possible polarizations of waves in anisotropic elastic media and their interactions, as well as for the degeneracy directions of propagation when two phase speeds possibly coincide. Thus it embodies isotropic elasticity which was considered in several previous publications. Some particular anisotropic cases of engineering interest are derived in detail.     

An integral representation of the surface-impedance tensor for incompressible elastic materials

A fundamental result in anisotropic elasticity and surface-wave theory is the integral representation for the surface-impedance tensor first derived by Barnett and Lothe in 1973. However, this representation is only valid for compressible materials but not valid for incompressible materials. In this paper the corresponding integral representation for the surface-impedance tensor valid for incompressible materials is derived and is used to establish the uniqueness of surface-wave speed and to obtain an expression for the tensor Green's function for the infinite space. Mathematics subject classifications (2000) 74B05, 74B15, 74B20, 74J15     

Transient Waves on the Free Surface of a Viscoelastic Anisotropic Half-Space

The Stroh formalism is employed to discuss the existence of transient surface waves on a viscoelastic anisotropic hall-space. The compatibility conditions, obtained using the integral formulation of Lothe and Barnett [13, 14], are examined on the basis of an asymptotic expansion of the viscoelastic kernel and a separation of space variables. Some previous results on elastic media are extended to viscoelasticity, exploiting the consequences of the second law of thermodynamics. It is found that all the allowed transient surface modes take the form of inhomogeneous plane waves whose amplitude exponentially decays along the propagation direction on the surface. Special solutions are derived explicitly for one-component surface waves where transient modes are admitted also in those cases in which stationary waves cannot occur. Mathematics Subject Classifications (2000) 74D05, 74J15.     

Surface waves and surface stability for a pre-stretched, unconstrained, non-linearly elastic half-space

An unconstrained, non-linearly elastic, semi-infinite solid is maintained in a state of large static plane strain. A power-law relation between the pre-stretches is assumed and it is shown that this assumption is well motivated physically and is likely to describe the state of pre-stretch for a wide class of materials. A general class of strain-energy functions consistent with this assumption is derived. For this class of materials, the secular equation for incremental surface waves and the bifurcation condition for surface instability are shown to reduce to an equation involving only ordinary derivatives of the strain-energy equation. A compressible neo-Hookean material is considered as an example and it is found that finite compressibility has little quantitative effect on the speed of a surface wave and on the critical ratio of compression for surface instability.     

Stroh Formalism and Rayleigh Waves

The Stroh formalism is a powerful and elegant mathematical method developed for the analysis of the equations of anisotropic elasticity. The purpose of this exposition is to introduce the essence of this formalism and demonstrate its effectiveness in both static and dynamic elasticity. The equations of elasticity are complicated, because they constitute a system and, particularly for the anisotropic cases, inherit many parameters from the elasticity tensor. The Stroh formalism reveals simple structures hidden in the equations of anisotropic elasticity and provides a systematic approach to these equations. This exposition is divided into three chapters. Chapter 1 gives a succinct introduction to the Stroh formalism so that the reader could grasp the essentials as quickly as possible. In Chapter 2 several important topics in static elasticity, which include fundamental solutions, piezoelectricity, and inverse boundary value problems, are studied on the basis of the Stroh formalism. Chapter 3 is devoted to Rayleigh waves, for long a topic of utmost importance in nondestructive evaluation, seismology, and materials science. There we discuss existence, uniqueness, phase velocity, polarization, and perturbation of Rayleigh waves through the Stroh formalism.

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The Table of Contents and Index are also provided as Electronic Supplementary Material for online readers at doi:      

Analysis of 2D infinite anisotropic elastic strips and semi-strips

Summary  Using Stroh's formalism and the theory of analytic functions, simple and explicit solutions for a line dislocation in an infinite anisotropic elastic strip are obtained. The two boundaries of the strip are free of traction. The problem of a dislocation in an anisotropic elastic semi-infinite strip with traction-free boundaries is also studied. A set of singular integral equations governing the unknown functions is derived. When the medium is orthogonal anisotropic and the coordinate axes x ₁ x ₂ x ₃ are coincident with the material principal axes, all the eigenvalues of the material coefficient matrix are pure imaginary. Explicit expressions of the unknown functions are given for this case. The results obtained are valid not only for plane and anti-plane problems but also for coupled problems between in-plane and out-of-plane deformations. Received 30 October 2000; accepted for publication 28 March 2001     

Analysis of dynamic instability of interfacial slip waves based on the surface impedance tensor

A new method relying on the Stroh formulism and the theory of the surface impedance tensor was developed to investigate the dynamic instability of interfacial slip waves. The concept of the surface impedance tensor was extended to the case where the wave speed is of a complex value, and the boundary conditions at the frictionally contacting interface were expressed by the surface impedance tensor. Then the boundary value problem was transformed to searching for zeroes of a complex polynomial in the unit circle. As an example, the steady frictional sliding of an elastic half-space in contact with a rigid flat surface was considered in details. A quartic complex characteristic equation was derived and its solution behavior in the unit circle was discussed. An explicit expression for the instability condition of the interfacial slip waves was presented.     

Periodic elliptic holes in anisotropic elasticity

The problem of collinear periodic elliptic holes in an anisotropic medium is examined in this paper. By means of Stroh formalism and the conformal mapping method, explicit full domain solutions for the periodic hole problems are presented. The solutions are valid not only for plane problems but also for antiplane problems and the problems whose implane and antiplane deformations are coupled. The stress concentration around the holes is analysed.     

Implicit and explicit secular equations for Rayleigh waves in two-dimensional anisotropic media

This paper is concerned with the derivation of implicit and explicit secular equations for Rayleigh waves polarized in a plane of symmetry of an anisotropic linear elastic medium. It has been confirmed, in accord with Ting’s paper [2], that the Rayleigh waves propagate with no geometric dispersion. Numerical evaluations of both the implicit and explicit equations give the same values of Rayleigh wave velocities. In the case of orthotropic material (thin composites) it has been found that Rayleigh wave velocity depends significantly (as with bulk waves) on the directions of principal material axes. For the same material model the analytical solutions, based on implicit and explicit secular equations, were compared against the finite element and experimental data that had been published by Cerv et al. [4] in 2010. It emerged that the theory was in accordance with the experiment.     

Causal Stroh formalism for uniformly-moving dislocations in anisotropic media: Somigliana dislocations and Mach cones

In this work, Stroh’s formalism is endowed with causal properties on the basis of an analysis of the radiation condition in the Green tensor of the elastodynamic wave equation. The modified formalism is applied to dislocations moving uniformly in an anisotropic medium. In practice, accounting for causality amounts to a simple analytic continuation procedure whereby to the dislocation velocity is added an infinitesimal positive imaginary part. This device allows for a straightforward computation of velocity-dependent field expressions that are valid whatever the dislocation velocity–including supersonic regimes–without needing to consider subsonic and supersonic cases separately. As an illustration, the distortion field of a Somigliana dislocation of the Peierls–Nabarro–Eshelby-type with finite-width core is computed analytically, starting from the Green’s tensor of elastodynamics. To obtain the result in the form of a single compact expression, use of the modified Stroh formalism requires splitting the Green’s function into its reactive and radiative parts. In supersonic regimes, the solution obtained displays Mach cones, which are supported by Dirac measures in the Volterra limit. From these results, an explanation of Payton’s ‘backward’ Mach cones (Payton, 1995) is given in terms of slowness surfaces, and a simple criterion for their existence is derived. The findings are illustrated by full-field calculations from analytical formulas for a dislocation of finite width in iron, and by Huygens-type geometric constructions of Mach cones from ray surfaces.     

General formalism for plane guided waves in transversely inhomogeneous anisotropic plates

The sextic approach to plane waves in infinite (visco)elastic plates of arbitrary anisotropy and transverse inhomogeneity is outlined. A particular thrust is set on continuous inhomogeneity when the propagator is defined by the Peano expansion. Despite underlying explicit intricacy, the basic framework of the pursued formalism is little affected by a through-plate variation of material. To make it evident, the principal algebraic symmetry of the propagator for unattenuated waves and the ensuing arrangement of the impedance as a Hermitian matrix with specific traits are inferred directly from energy considerations. Staying the same as for homogeneous plates, those features yield useful developments in the broader context of inhomogeneity. The formalism may be expressed in either pair picked among velocity, frequency and wavenumber, but different choices of a dispersion variable are shown to entail analytical dissimilarities. In addition, the impact of the profile symmetry and of the horizontal plane of crystallographic symmetry is examined. The surface-impedance method and some other aspects of the numerical treatment are discussed.