Elastic waves interacting with a thin, prestressed, fiber-reinforced surface film

Elastic surface waves propagating at the interface between an isotropic substrate and a thin, transversely isotropic film are analyzed. The transverse isotropy is conferred by fibers lying parallel to the interface. A rigorous leading-order model of the thin-film/substrate interface is derived from the equations of three-dimensional elasticity for prestressed, transversely isotropic films having non- uniform properties. This is used to study Love waves.     

The rotation of a rigid spheroidal inclusion embedded in a transversely isotropic medium

The exact three-dimensional elasticity solutions are given for two problems related to a rigid spheroidal inclusion embedded in bonded contact with an infinite transversely isotropic elastic medium. The first is of axisymmetric nature in which the inclusion is given a constant rotation about its axis of revolution which coincides with the axis of symmetry of the material. The second problem is asymmetric where the spheroidal inclusion is given a constant rotation about a direction that is perpendicular to the axis of elastic symmetry of the material. The displacement potential representation for the equilibrium of three-dimensional transversely isotropic bodies is used to solve the problem. In both cases, the moment-rotation relationship for the spheroidal inclusion and its limiting configurations are obtained in closed form. Numerical results are presented to show the effect of the aspect ratio of the spheroid on the rotational stiffness.     

Cylindrical Bending and Vibration of Polar Material Laminates

This article considers a plane strain problem, which is known in conventional linear elasticity as cylindrical bending of simply supported plates and cross-ply laminates. By considering fibrous composites containing fibers resistant to bending, it formulates and solves corresponding polar elasticity equations governing the static and dynamic behavior of beam-like components made of a homogeneous or layered transversely isotropic material; each layer has embedded a single family of fibers. Fiber bending stiffness is accounted for through involvement of an extra elastic modulus, which, unlike its conventional elasticity counterparts that have dimensions of stress, has dimensions of force. Its involvement in the analysis implies existence of some intrinsic material area or length parameter, which may be associated, for instance, with fiber thickness of fiber spacing. A considerable amount of relevant numerical results are presented for thick beam components made of either homogeneous or two-layered transversely isotropic material. For the static bending problem, these include a detailed presentation of through-thickness distributions of displacements, stresses, as well as couple-stress. For the dynamic problem, attention is focused on the influence that fiber bending stiffness exerts on fundamental frequency parameters.     

Dynamic crack problems in three-dimensional transversely isotropic solids

In this paper, a general boundary element approach for three-dimensional dynamic crack problems in transversely isotropic bodies is presented for the first time. Quarter-point and singular quarter-point elements are implemented in a quadratic isoparametric element context. The procedure is based on the subdomain technique, the displacement integral representation for elastodynamic problems and the expressions of the time-harmonic point load fundamental solution for transversely isotropic media. Numerical results corresponding to cracks under the effects of impinging waves are presented. The accuracy of the present approach for the analysis of dynamic fracture mechanics problems in transversely isotropic solids is shown by comparison of the obtained results with existing solutions.     

Transverse elasticity of a unidirectionally reinforced composite with an irregular fibre arrangement: Experiments,theory and computations

This paper examines the effective elasticity properties of a unidirectionally reinforced carbon fibre–polyester composite. A computational simulation of an experimentally determined fibre arrangement is used to derive the effective elasticity properties of the transversely isotropic composite. The computational estimates for the elastic constants are compared with several theoretical estimates for the effective elasticity properties that are based on regular arrangement of the reinforcing fibres, their volume fraction and the elasticity properties of the constituents.     

Dispersion of elastic waves in periodically inhomogeneous media

Propagation of time-harmonic elastic waves through periodically inhomogeneous media is considered. The material inhomogeneity exists in a single direction along which the elastic waves propagate. Within the period of the linear elastic and isotropic medium, the density and elastic modulus vary either in a continuous or a discontinuous manner. The continuous variations are approximated by staircase functions so that the generic problem at hand is the propagation of elastic waves in a medium whose finite period consists of an arbitrary number of different homogeneous layers. A dynamic elasticity formulation is followed and the exact phase velocity is derived explicitly as a solution in closed form in terms of frequency and layer properties. Numerical examples are then presented for several inhomogeneous structures.     

Wave propagation in a cylindrical viscous layer between two elastic shells

Propagation of longitudinal waves in a liquid-filled layer between two thin coaxial shells is investigated. Both liquid viscosity and elasticity of the shells are accounted for. Dynamics of the shells is treated using the Kirchhoff–Love approximation. The elastic deformations of the shells in the sound wave are coupled with the liquid flow in the gap through appropriate dynamic and kinematic boundary conditions. Hydrodynamics of the liquid is described using the quasi-one-dimensional (hydraulic) approach. It is assumed that the external and internal shells are composed of different isotropic elastic materials and have different widths. The dispersion equation for harmonic waves in the system is obtained; it is valid in the low frequency range where the wave length is greater than the external shell radius. In the limiting case for an ideal liquid the dispersion equation yields water hammer speed in the system. The analysis of the dispersion equation has shown strong influence of viscous losses on dispersion and attenuation of pressure signals in the low frequency region. The wave speed and attenuation are highly dependent on the geometrical parameters of the system and elastic properties of the shells.     

Small torsional vibration superposed on finitely deformed state of a circular cylindrical rod of transversely isotropic elastic material

Starting with a class of small deformations superposed on a finitely deformed state of a transversely isotropic elastic solid, we study a problem of small torsional vibration superposed on homogeneous finitely deformed state of a circular cylindrical rod made of transversely isotropic elastic material. It has been found that free vibration is possible and, due to anisotropy, the speed of propagation of waves of torsion along the cylinder is increased or decreased according as the initial stressed state is under tension or compression.     

The effects of surface elasticity and surface tension on the transverse overall elastic behavior of unidirectional nano-composites

The effects of surface elasticity and surface tension on the transverse overall behavior of unidirectional nano-scale fiber-reinforced composites are studied. The interfaces between the nano-fibers and the matrix are regarded as material surfaces described by the Gurtin and Murdoch model. The analysis is based on the equivalent inhomogeneity technique. In this technique, the effective elastic properties of the material are deduced from the analysis of a small cluster of fibers embedded into an infinite plane. All interactions between the inhomogeneities in the cluster are precisely accounted for. The results related to the effects of surface elasticity are compared with those provided by the modified generalized self-consistent method, which only indirectly accounts for the interactions between the inhomogeneities. New results related to the effects of surface tension are presented. Although the approach employed is applicable to all transversely isotropic composites, in this paper we consider only a hexagonal arrangement of circular cylindrical fibers.     

饱和横观各向同性分数导数黏弹性土中半封闭衬砌振动响应

在频率域内研究了饱和横观各向同性分数导数黏弹性土体中深埋圆形隧道半封闭衬砌振动响应问题。根据土体在长期沉积过程中存在各向异性的特点,将土骨架视为具有分数导数本构关系的横观各向同性黏弹性体,采用饱和多孔介质理论和弹性理论,利用衬砌内边界应力协调以及土体和衬砌界面处应力和位移连续,得到了简谐荷载作用下饱和横观各向同性黏弹性土和弹性衬砌的位移、应力和孔隙水压力解析表达式。考察了饱和经典弹性土、饱和分数导数性黏弹性土和饱和经典黏弹性土三种条件下饱和黏弹性土和衬砌各参数的影响,表明：横观各向同性面的弹性模量和衬砌厚度对系统动力响应的影响与分数导数阶数和土骨架的黏性有关;渗透系数较小时,系统存在明显的共振现象。另外,在三种条件下半封闭衬砌振动响应存在较大差异。     

Annular crack surrounding an elastic fibre in transversely isotropic elasticity

The axisymmetric problem of an infinitely long transversely isotropic elastic fibre perfectly bonded to a dissimilar transversely isotropic elastic matrix containing an annular crack is considered. The annular crack, surrounding the fibre, is subjected to prescribed longitudinal tension. A potential function approach is used to find the solution of the basic equations. The mixed boundary value problem is reduced to the solution of a singular integral equation, which is further reduced, by using Chebyshev polynomials, to a system of algebraic equations.     

On wave propagation in anisotropic elastic cylinders at nanoscale: surface elasticity and its effect

Material heterogeneity induced by a surface or interface may be neglected at macroscale since the surface-to-volume ratio is usually small. However, its effect can become significant for structures at nanoscale with a large surface-to-volume ratio. In this paper, we incorporate such surface material heterogeneity into wave propagation analysis of a nanosized transversely isotropic cylinder. This is achieved by using the concept of surface elasticity. Instead of directly using the well-known Gurtin–Murdoch (GM) surface elasticity, we develop here another general framework based on a thin layer model. A novel approach based on state-space formalism is used to derive the approximate governing equations. Three different sources of surface effect can be identified in the first-order surface elasticity, i.e., the elasticity effect, the inertia effect and the thickness effect. It is found that the derived theory becomes identical to the GM surface elasticity if the thickness effect is dropped and when the material is isotropic. The axisymmetric wave propagation in a transversely isotropic cylinder with surface effect is then studied, and an exact solution is presented. Numerical results are finally given to show that the surface effect will play a very pronounced role in wave propagation in cylinders at nanoscale.     

Effects of inhomogeneity on surface waves in anisotropic media

This paper investigates the effects of anisotropy and inhomogeneity on surface waves in elastic media. Exponential variation in properties are assumed for the elastic parameters and material density. The classical equations of motion for propagation of waves in an inhomogeneous transversely isotropic elastic solid are deduced. The equations of motion for surface waves are derived and general surface waves are investigated. This general theory is then utilized to investigate Rayleigh, Love and Stoneley waves. Results obtained in the above cases reduce to the corresponding well-known classical results when inhomogeneity and anisotropy are not present. It is seen that inhomogeneity has significant effects on dispersion characteristics. Numerical calculations are included for Love waves and some conclusions have been drawn from the above calculations.     

Elastic properties of powders during compaction. Part 2: elastic anisotropy

Isotropy in the elastic properties of powders undergoing uniaxial compaction in a cylindrical die was evaluated from in situ measurements of elastic wave speed. Shear and bulk longitudinal wave speeds were measured in both the axial (pressing) and radial directions. For the five different metal powders studied, wave speeds were generally higher in the axial direction. As such, the powder body was best described as a transversely isotropic material; complete isotropy was approached only when the powder was close to the loose packed state, or completely solid. Transversely isotropic elastic moduli analogous to the common isotropic ‘engineering’ moduli (Young’s modulus, Poisson’s ratio, etc.) were calculated by combining elastic wave speed measurements with the Saint-Venant approximation. Pseudo-isotropic elastic moduli (calculated from axial wave speed measurements and assuming elastic isotropy) were found to be only qualitatively similar to transversely isotropic elastic moduli for the axial plane.     

Analytical solutions for an infinite transversely isotropic functionally graded sectorial plate subjected to a concentrated force or couple at the tip

This paper considers the elastic responses of an infinite sectorial plate made of transversely isotropic functionally graded material (FGM), which is subjected to a concentrated force or couple at the tip. There is no load acting on the upper and lower surfaces, and the elastic coefficients can vary arbitrarily through the plate thickness. No constraint is required on the symmetry of the plate in the thickness direction. Based on the displacement assumption for the bending of an FGM plate and by using the complex variable method, this paper presents the general solutions to the basic equations governing transversely isotropic FGM plates, which are expressed in terms of four analytical functions (or complex potentials). The boundary conditions are a combination of those from the plane elasticity and those from the classical plate theory. For a particular boundary value problem, such as the ones considered here for a sectorial plate, with the specific conditions for determining solutions, the four analytic functions can be assumed in appropriate forms, which contain only some unknown constants. Once these constants are determined from the specific conditions, the complete solutions are readily derived too. Among the solutions presented here, the solutions for the infinite FGM sectorial plate under a concentrated couple are absolutely new to the literature, and they are also applicable to isotropic FGM sectorial plates. The solutions degenerate into the ones for a homogeneous sectorial plate, which coincide with the available solutions from the plane elasticity theory. There are three-dimensional correction terms in the mid-plane displacements.     

BEM analysis of wave scattering in transversely isotropic solids

A boundary element approach for wave propagation problems in transversely isotropic solids is developed in this paper. The procedure is based on the well‐known formulation for time‐harmonic elasticity and a new version of a recently obtained fundamental solution for transversely isotropic media. The fundamental solution is transformed to obtain new expressions which can be efficiently evaluated at any point. This fact allows for a drastic reduction of the computation time and makes possible the implementation of a general purpose three‐dimensional quadratic element code. To show the simplicity and accuracy of the approach, the diffraction of waves by a spherical cavity and the interaction between two cavities in a boundless domain are studied. The computed results show a very good agreement with the analytical solution in the simple case where such solution exists. Other geometries can be studied without difficulty. Copyright © 1999 John Wiley & Sons, Ltd.     

Surface/interface effects on dispersion relations of 2D phononic crystals with parallel nanoholes or nanofibers

Owing to the increasing ratio of surface to bulk volume, surface effects emerge in the mechanical performance of devices and materials when the characteristic size reduces to nanoscale. In this paper, the dispersion relations and the band gap properties of 2D phononic crystals with periodically arranged nanoholes or nanofibers are studied. The scattering matrix of a single scatterer is first derived from a set of nontraditional boundary conditions based on the surface elasticity theory. Then, the addition theorem of cylindrical waves and Bloch’s theorem of periodical structure are used to obtain the dispersion relations of an elastic wave polarized in the plane perpendicular to the axis of the nanoholes or nanofibers. It is found that the surface effect has remarkable influences on the dispersion relations and the band gaps for the phononic crystal with nanoholes. For the 2D phononic crystal with periodical arranged nanofibers, the interface effect is more evident for the softer fibers than for the stiffer fibers.     

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.     

Two-dimensional static deformation of an anisotropic medium

The problem of two-dimensional static deformation of a monoclinic elastic medium has been studied using the eigenvalue method, following a Fourier transform. We have obtained expressions for displacements and stresses for the medium in the transformed domain. As an application of the above theory, the particular case of a normal line-load acting inside an orthotropic elastic half-space has been considered in detail and closed form expressions for the displacements and stresses are obtained. Further, the results for the displacements for a transversely isotropic as well as for an isotropic medium have also been derived in the closed form. The use of matrix notation is straightforward and avoids unwieldy mathematical expressions. To examine the effect of anisotropy, variations of dimensionless displacements for an orthotropic, transversely isotropic and isotropic elastic medium have been compared numerically and it is found that anisotropy affects the deformation significantly.     

Sensitivities of Two-Dimensional Fracture Problems to the Near-tip Mesh Parameters

Analytical results for a penny-shaped crack with a plastic zone at the crack front are given. The crack is embedded in an infinite transversely isotropic elastic medium and is assumed to be subjected to two identical axisymmetric loads on the upper and lower crack faces. The size of the plastic zone at the crack front is determined by applying Dugdale hypothesis to the elasticity results for a penny-shaped crack. The size of the plastic zone is derived in terms of hyper-geometric functions. Expression of the normal stress outside the plastic zone is also given.