Arbitrary Elastic Anisotropy Research Articles

Cauchy Six-Dimensional Formalism for Lamb Waves in Multilayered Plates

Propagation of Lamb waves in multilayered elastic anisotropic plates is studied in the framework of combination of the six-dimensional Cauchy formalism and the transfer matrix method. The closed form secular equations for dispersion curves of Lamb waves propagating in multilayered plates with arbitrary elastic anisotropy are obtained.

Elastic solutions with arbitrary elastic inhomogeneity and anisotropy

An efficient numerical algorithm is proposed to accurately compute the elastic fields in two-dimensional (2D) or three-dimensional (3D) microstructures with arbitrary elastic inhomogeneity and anisotropy. It combines the equivalent inclusion method of Eshelby, the microelasticity theory of Khachaturyan, and the spectral iterative perturbation method of Hu and Chen. Its efficiency is compared with those of existing approaches in the literature. The method can be conveniently implemented in phase-field modeling of stress-dependent microstructure evolution and/or of mass/electrical transport.

A Spectral Iterative Method for the Computation of Effective Properties Of Elastically Inhomogeneous Polycrystals

AbstractWe report an efficient phase field formalism to compute the stress distribution in polycrystalline materials with arbitrary elastic inhomogeneity and anisotropy The dependence of elastic stiffness tensor on grain orientation is taken into account, and the elastic equilibrium equation is solved using a spectral iterative perturbation method. We discuss its applications to computing residual stress distribution in systems containing arbitrarily shaped cavities and cracks (with zero elastic modulus) and to determining the effective elastic properties of polycrystals and multilayered composites.

Modelling of Lamb wave propagation in composite plate excited by surface bonded piezoelectrical actuators

AbstractTransient propagation of Lamb waves in a multilayered infinite composite plate with the arbitrary elastic anisotropy of each layer is considered in this paper. Integral representations of wave fields, an algorithm of computing the Green's matrix of the multilayered medium in Fourier domain and some methods of the Fourier transform inversion are used as basic analytical research instruments. The piezoelectrical transducers frequently used for Lamb wave excitation are modelled in this paper as surface sources with stresses concentrated on the boundaries of actuators. The contribution of each wave mode to a whole amplitude, focusing of Lamb waves and influence of the geometry of the surface source are discussed in this paper by numerical examples. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Intermediate and Bragg acousto-optic interaction in elastically anisotropic medium

The Raman–Nath, intermediate and Bragg regimes of acousto-optic interaction in an acoustically anisotropic medium are theoretically examined in the paper. The general model of the interaction is extended over the case of an elastically anisotropic medium. Basic results of computer modeling of diffraction processes taking place in crystals are presented in the paper. The interaction of light and ultrasound is investigated in media with arbitrary elastic anisotropy in a wide range of the Klein–Cook and Raman–Nath parameters. The influence of the acoustic anisotropy on the process of light diffraction, especially on the light intensity in diffraction maxima, is also discussed.

A phase-field model of stress effect on grain boundary migration

We developed a phase-field model to study the stress-driven grain boundary migration in elastically inhomogeneous polycrystalline materials with arbitrary elastic inhomogeneity and anisotropy. The dependence of elastic stiffness tensor on grain orientation is taken into account, and the elastic equilibrium equation is solved using the Fourier spectral iterative-perturbation method. We studied the migration of planar and curved grain boundaries under an applied stress. The relation between grain boundary migration velocity and driving force is found to be linear in the steady-state regime. Our study shows that the stress distribution depends on the relative misorientation between the grains and the nature of the applied load. As a consequence, the mechanism of grain boundary migration is different when the load is applied parallel or perpendicular to a grain boundary. The bulk mechanical driving force for grain boundary migration is provided by the difference in the level of stress in the adjoining grains which arise due to difference in elastic moduli. We further show that under certain conditions an applied stress may act as a precursor to abnormal grain growth.

Anisotropic bodies with inhomogeneities: The case of a set of cracks

We use the Betti theorem to obtain the integral equations of the dynamic theory of elasticity for a multilayer convex body with an arbitrary elastic anisotropy of layers containing plane infinitely thin cracks. The systems of integral equations relate the displacement jumps to the stresses on the crack lips and are stated numerically in terms of Fourier transforms. For the case of plane-parallel layers with a set of plane cracks on the interfaces between the layers, we propose a simple numerical-analytic method for constructing the Fourier symbol, i.e., the matrix of the kernel of the system of integral equations. The method is stable for an arbitrary combination of continuous and discontinuous conditions on the layer boundaries. Numerical examples are given for a packet of four heterogeneous anisotropic layers.

Fundamental and singular solutions of Lamé equations for media with arbitrary elastic anisotropy

Fundamental and singular solutions of Lamé equations for media with arbitrary elastic anisotropy are constructed on the basis of multipolar expansions (expansions in spherical harmonics) of symbols and the corresponding operators. Theorems of convergence are formulated. A posteriori error estimates are presented.

Two exact micromechanics-based nonlocal constitutive equations for random linear elastic composite materials

A Hashin–Shtrikman–Willis variational principle is employed to derive two exact micromechanics-based nonlocal constitutive equations relating ensemble averages of stress and strain for two-phase, and also many types of multi-phase, random linear elastic composite materials. By exact is meant that the constitutive equations employ the complete spatially-varying ensemble-average strain field, not gradient approximations to it as were employed in the previous, related work of Drugan and Willis (J. Mech. Phys. Solids 44 (1996) 497) and Drugan (J. Mech. Phys. Solids 48 (2000) 1359) (and in other, more phenomenological works). Thus, the nonlocal constitutive equations obtained here are valid for arbitrary ensemble-average strain fields, not restricted to slowly-varying ones as is the case for gradient-approximate nonlocal constitutive equations. One approach presented shows how to solve the integral equations arising from the variational principle directly and exactly, for a special, physically reasonable choice of the homogeneous comparison material. The resulting nonlocal constitutive equation is applicable to composites of arbitrary anisotropy, and arbitrary phase contrast and volume fraction. One exact nonlocal constitutive equation derived using this approach is valid for two-phase composites having any statistically uniform distribution of phases, accounting for up through two-point statistics and arbitrary phase shape. It is also shown that the same approach can be used to derive exact nonlocal constitutive equations for a large class of composites comprised of more than two phases, still permitting arbitrary elastic anisotropy. The second approach presented employs three-dimensional Fourier transforms, resulting in a nonlocal constitutive equation valid for arbitrary choices of the comparison modulus for isotropic composites. This approach is based on use of the general representation of an isotropic fourth-rank tensor function of a vector variable, and its inverse. The exact nonlocal constitutive equations derived from these two approaches are applied to some example cases, directly rationalizing some recently-obtained numerical simulation results and assessing the accuracy of previous results based on gradient-approximate nonlocal constitutive equations.

Linearized inverse scattering of teleseismic waves for anisotropic crust and mantle structure: 1. Theory

In this study we examine the recovery of vertically varying anisotropic crust and upper mantle structure using converted teleseismic waves recorded on three‐component seismograms from individual broadband stations. Our analysis is cast in terms of inverse scattering theory for a one‐dimensional medium and begins with a derivation of the plane wave Green's function for a homogeneous medium exhibiting arbitrary elastic anisotropy. This Green's function is employed within the single‐scattering approximation to derive formulae that relate the amplitude of the scattered wave in time to perturbations in material properties at corresponding depths. Inclusion of seismograms from multiple events representing a range of azimuths and incident angles leads to the construction of a linear system of equations that is readily solved using singular value decomposition. A useful by‐product of this “amplitude‐versus‐slowness” approach is the identification of simple and compact expressions for linearized reflection and conversion coefficients in anisotropic media that are accurate for small contrasts at near‐vertical angles of incidence. We demonstrate the accuracy of the linearized P‐to‐S transmission coefficient for an idealized upper mantle model. In a companion paper we will examine the application of this approach to field data recorded at stations of the Canadian National Seismograph Network.

Rayleigh waves in anisotropic porous media

Rayleigh waves represent waves propagating in a half-space and attenuating exponentially with depth. During the last 30 years considerable progress has been achieved in developing both analytical and numerical methods for analysis of Rayleigh waves propagating in homogeneous media with arbitrary elastic anisotropy. The propagation of Rayleigh waves in heterogeneous media has not been studied nearly as often. Very few works are devoted to the analysis of speed and attenuation of Rayleigh waves in media (mainly isotropic) with uniformly distributed pores and/or microcracks. The advantage of applying Rayleigh waves to the nondestructive determination of concentration and the preferred orientation of pores lies in its high sensitivity to variation of the material elastic parameters due to preexisting pores. The developed approach for analysis of Rayleigh waves in porous or cracky anisotropic media is based on a combination of six- and three-dimensional complex formalism and the two-scale asymptotic analysis. In its turn, the latter utilizes a newly developed spatially periodic boundary integral equation method. This is used for determination of the effective characteristics of heterogeneous media containing isolated uniformly distributed pores. Numerical data are discussed.

Some singular solutions of the theory of elasticity

Singular solutions corresponding to “double forces without moment”, an “extension-compression centre”, and “double forces with moment”, obtained previously for an isotropic medium [2, 3] by differentiation of the Kelvin fundamental solution, are constructed for media with an arbitrary elastic anisotropy using the method of multiple expansions [1].

Direct boundary integral equation method in the theory of elasticity

The Direct Boundary Integral Equation Method (BIEM), which leads to second-kind singular integral equations for all types of commonly used boundary-value problems of the theory of elasticity, is formulated. The exposition is applied to anisotropic bodies with arbitrary elastic anisotropy.

Influence of non-classical elastic-plastic constitutive features on shock wave existence and spectral solutions

T he influence of non-classical elastic-plastic constitutive features on dynamically moving discontinuities in stress, strain and material velocity is investigated. Non-classical behavior here includes non-normality of the plastic strain increment to the yield surface, plastic compressibility, pressure sensitivity of yield and dependence of the elastic moduli on plastic strain. D rugan and S hen's (1987) analysis of dynamically moving discontinuities with strain as well as stress jumps in classical materials is shown to be valid for a broad class of non-associative material models until deviation from normality exceeds a critical (non-infinitesimal) level. For these non-classical materials, an inequality that bounds the magnitude of the stress jump is derived, which is information not obtainable from a standard spectral analysis of a shock. For the special case of stress discontinuities with continuous strain or for quasi-static deformations, this inequality is shown to rule out jumps in specific projections of the stress tensor unless the non-normality is sufficiently large. These results invalidate a recent claim in the literature that an infinitesimal amount of non-normality permits moving surfaces of discontinuity in stress (with no strain jump) near the tip of a dynamically advancing crack tip. Using a very general plastic constitutive law that subsumes most non-classical (and classical) descriptions currently in use, a complete closed form solution is obtained for the plastic wave speeds and eigenvectors. A novel feature of the analysis is the clarity and completeness of the solutions. If the elastic part of the response is isotropic, one plastic wave speed equals the elastic shear wave speed, while the other two possible wave speeds depend in general on the stress and plastic strain within the shock transition layer. Concise necessary and sufficient conditions for real eigenvalues and for vanishing eigenvalues are derived. The real eigenvalues are classified by numerical sign and ordering relative to the elastic eigenvalues. The geometric multiplicity of plastic eigenvectors associated with elastic eigenvalues is shown to depend on the stress state within the shock transition layer. These solutions, several of which hold for arbitrary elastic anisotropy, are also applicable to acceleration waves and localization problems and to materials with dependence of the elastic moduli on plastic strain. Such elastic-plastic coupling is shown to imply a non-self-adjoint fourth order tangent stiffness tensor even if the plastic constitutive law is associative.

A direct version of the method of boundary integral equations in the theory of elasticity

A version of the direct method of boundary integral equations is considered which leads to systems of singular integral equations of the second kind for solving all the main types of boundary value problems of the theory of elasticity. Media with an arbitrary elastic anisotropy are discussed.

The Eshelby factor for cubic crystals with arbitrary elastic anisotropy

The Eshelby factor for cubic crystals has been calculated for arbitrary anisotropy. One of the two integrations needed could be performed analytically. An expansion up to 4th order in the anisotropy and exact results for special cases are also given.

Strain and stress field calculations for precipitates and dislocation loops using affine transformations

Starting from the covariant formulation of the linear theory of elasticity the elastic strain and stress fields inside and outside precipitates are calculated using appropriate affine coordinates. Explicit integral formulae well suited for numerical evaluation are derived for ellipsoidal, cylindrical, polyhedral, and plate-like particles as well as rhomboidal and elliptical dislocation loops, allowing for an arbitrary homogeneous stress-free strain and elastic anisotropy. The application of the results in precipitate structure analysis by diffraction contrast image matching will be considered in a subsequent paper.

Thermal diffuse scattering in cubic powder patterns

The one-phonon thermal diffuse scattering in powder patterns of monatomic cubic materials has been investigated using a model that gives the correct frequencies and polarization vectors of the long-wavelength phonons for materials of arbitrary elastic anisotropy. Computer calculations have been made of the intensity distribution of this scattering and of the correction for its inclusion in measured integrated intensities of powder pattern reflections. Elastic anisotropy is found to produce marked differences, despite the powder pattern averaging; and, contrary to the Chipman–Paskin approximation, the integrated intensity correction is found generally not to vary smoothly with (h2 + k2 + l2) or linearly with scan length even for isotropic materials, as Suortti also noted. A much simpler method for calculating the integrated intensity correction has also been developed, based on a modified Warren model, that gives reasonably accurate values under most conditions even for very anisotropic materials and is several orders of magnitude faster than the primary method.