Size-dependent effective dynamic properties of unidirectional nanocomposites with interface energy effects

This article studies the size effect on wave propagation characteristics of plane longitudinal and transverse elastic waves in a two-phase nanocomposite consisting of transversely isotropic and unidirectionally oriented identical cylindrical nanofibers embedded in a transversely isotropic homogeneous matrix. The surface elasticity theory is employed to incorporate the interfacial stress effects. The effect of random distribution of nanofibers in the composite medium is taken into account via a generalized self-consistent multiple scattering model. The phase velocities and attenuations of longitudinal and shear waves along with the associated dynamic effective elastic constants are calculated for a wide range of frequencies and fiber concentrations. The numerical results reveal that interface elasticity at nanometer length scales can significantly alter the overall dynamic mechanical properties of nanofiber-reinforced composites. Limiting cases are considered and excellent agreements with solutions available in the literature have been obtained.     

Axisymmetric crack terminating at the interface of transversely isotropic dissimilar media

In this paper, the axisymmetric elasticity problem of an infinitely long transversely isotropic solid cylinder imbedded in a transversely isotropic medium is considered. The cylinder contains an annular or a penny shaped crack subjected to uniform pressure on its surfaces. It is assumed that the cylinder is perfectly bonded to the medium. A singular integral equation of the first kind (whose unknown is the derivative of crack surface displacement) is derived by using Fourier and Hankel transforms. By performing an asymptotic analysis of the Fredholm kernel, the generalized Cauchy kernel associated with the case of `crack terminating at the interface' is derived. The stress singularity associated with this case is obtained. The singular integral equation is solved numerically for sample cases. Stress intensity factors are given for various crack geometries (internal annular and penny-shaped cracks, annular cracks and penny-shaped cracks terminating at the interface) for sample material pairs.     

Rayleigh-type wave propagation on a transversely isotropic viscoelastic layer with yielding and rigid foundations

The present study investigates the propagation of Rayleigh-type wave in a transversely isotropic viscoelastic layer in effect of yielding foundation and rigid foundation in two different cases for two considered models. Numerical computation and graphical demonstration have been carried out for the case when layer is comprised of transversely isotropic viscoelastic Mesaverde clay shale material (Model I) and simply isotropic viscoelastic material (Model II). Closed-form expression of phase velocity and damped velocity for both the cases are deduced analytically. Obtained result is found in well agreement to the established standard results existing in the literature. Significant effect of dilatational viscoelasticity, volume viscoelasticity and yielding parameter on phase and damped velocities for both the considered models has been traced out. The comparative study has been performed to unravel the effect of viscoelasticity over elasticity and anisotropy over isotropy in the context of the present problem. Moreover, comparison of phase and damped velocities for the case of layer with stress-free foundation, layer with rigid foundation and layer with yielding foundation serve as a major highlight of the present work.     

Transducer-modeling in general transversely isotropic media via point-source-synthesis: Theory

Based on a theory of elastic wave propagation in arbitrarily oriented transversely isotropic media, which has been presented recently, the radiation characteristics of ultrasonic transducers in these media are determined. Using the directivity patterns for normal and transverse point sources on the free surface of such (semi-infinite) materials—the derivation is based on the reciprocity theorem—the radiated wave fields are obtained by the method of point-source-synthesis, i.e., by superposing the wave fields of numerous point sources located within the transducer aperture. Since ultrasonic inspection of anisotropic materials, especially weld material in nuclear power plants, suffers from the well-known effects of beam splitting, beam distortion, and beam skewing, valuable information in view of an optimized inspection is provided. Focusing on transversely isotropic weld material specimens, numerical evaluation is performed for several grain orientations with respect to the transducer-normal. The approach presented is particularly useful in view of an appropriate extension to inhomogeneous welds and the consideration of time-dependent RF-impulse functions.     

Wave propagation in an asymmetric fibre-reinforced laminated plate

Summary This paper examines the propagation of waves in a 2-ply laminate composed of a transversely isotropic elastic material with the fibre direction in the two plies being at right angles to each other. Attention is confined to waves propagating in the fibre direction in one of the plies. The dispersion equation, relating the wave speed to the wavelength, is derived and is solved numerically for a number of different values of the ratio of the thickness of the two layers. The long and short wavelength behaviour is examined analytically. Comparisons are made between this asymmetric composite and a symmetric 3-ply laminate composed of the same material. Results are plotted for the fundamental mode and first harmonic of the dispersion relation of the asymmetric laminate. The high frequency limit of the dispersion curve for the fundamental mode is the velocity of a Rayleigh surface wave in the cross-fibre direction, whilst that of the first harmonic is the velocity of a shear wave in the cross-fibre direction. Corresponding results are obtained for the ideal material which is inextensible in the direction of transverse isotropy, and these are compared with those for the extensible material.With 3 Figures     

On surface wave propagation in incompressible, transversely isotropic, pre-stressed elastic half-spaces

The problem of surface wave propagation for a transversely isotropic incompressible pre-stressed half-space is investigated in respect of a general in-plane fibre direction and a general angle of propagation. It is shown that the surface wave speed crucially depends on the normal static stress in the finite primary deformation. Some special cases leading to considerable simplification are pointed out. An important particular case of the fibres assumed almost inextensible is tackled both numerically and asymptotically to reveal an idealised fibre-reinforced material as a leading order approximation.     

An elasticity solution for transversely isotropic,functionally graded circular plates

This article presents a new elasticity solution for transversely isotropic, functionally graded circular plates subject to axisymmetric loads. It is assumed that the material properties vary along the thickness of a circular plate according to an exponential form. By extending the displacement function presented by Plevako to the case of transversely isotropic material, we derived the governing equation of the problem studied. The displacement function was assumed as the sum of the Bessel function and polynomial function to obtain the analytical solution of a transversely isotropic, functionally graded circular plate under different boundary conditions. As a numerical example, the influence of the graded variations of the material properties on the displacements and stresses was studied. The results demonstrate that the graded variations have a significant effect on the mechanical behavior of a circular plate.     

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.     

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.     

On free vibration of piezoelectric nanospheres with surface effect

Surface effect responsible for some size-dependent characteristics can become distinctly important for piezoelectric nanomaterials with inherent large surface-to-volume ratio. In this paper, we investigate the surface effect on the free vibration behavior of a spherically isotropic piezoelectric nanosphere. Instead of directly using the well-known Huang-Yu surface piezoelectricity theory (HY theory), another general framework based on a thin shell layer model is proposed. A novel approach is developed to establish the surface piezoelectricity theory or the effective boundary conditions for piezoelectric nanospheres employing the state-space formalism. Three different sources of surface effect can be identified in the first-order surface piezoelectricity, i.e. the electroelastic effect, the inertia effect, and the thickness effect. It is found that the proposed theory becomes identical to the HY theory for a spherical material boundary if the transverse stress components are discarded and the electromechanical properties are properly defined. The nonaxisymmetric free vibration of a piezoelectric nanosphere with surface effect is then studied and an exact solution is obtained. In order to investigate the surface effect on the natural frequencies of piezoelectric nanospheres, numerical calculations are finally performed. Our numerical findings demonstrate that the surface effect, especially the thickness effect, may have a particularly significant influence on the free vibration of piezoelectric nanospheres. This work provides a more accurate prediction of the dynamic characteristics of piezoelectric nanospherical devices in nano-electro-mechanical systems.     

On dispersion relations of Rayleigh waves in a functionally graded piezoelectric material (FGPM) half-space

For propagation of Rayleigh surface waves in a transversely isotropic graded piezoelectric half-space with material properties varying continuously along depth direction, the Wentzel–Kramers–Brillouin (WKB) technique is employed for the asymptotic analytical derivations. The phase velocity equations for both the electrically open and shorted cases at the free surface are obtained. Influences of piezoelectric material parameters graded variations on Rayleigh wave dispersion relations, particles’ displacements magnitude and corresponding decay properties are discussed. Results obtained indicate that coupled Rayleigh waves can propagate at the surface of the graded piezoelectric half-space, and their dispersion relations and the particles displacements ellipticity at the free surface are dependent upon the graded variation tendency of the material parameters. By the Rayleigh surface waves phase velocities relative changing values combined with the relationship between the wave number and the material graded coefficient, a theoretical foundation can be provided for the graded material characterization by experimental measurement.     

Limiting State of Transversely Isotropic Plates with Surface and Internal Cracks

We propose a method for the analysis of the limiting state of transversely isotropic plates with surface and internal cracks and obtain a new simple approximate formula for the evaluation of the stress intensity factors. The numerical calculations are based on the improved version of the analysis of stress-strain states in defect-free transversely isotropic plates. The failure is initiated as soon the damage ratio near the crack front in the direction of crack propagation attains a certain critical value.     

Free vibration analysis of double-bonded isotropic piezoelectric Timoshenko microbeam based on strain gradient and surface stress elasticity theories under initial stress using differential quadrature method

The size-dependent effect on free vibration of double-bonded isotropic piezoelectric Timoshenko microbeams using strain gradient and surface stress elasticity theories under initial stress is presented. This article is developed for isotropic piezoelectric material. Due to the high surface-to-volume ratio, surface stress has an important role with micro- and nanoscale materials. Thus, the Gurtin–Murdoch continuum mechanic approach is used. Governing equations of motion are derived by Hamilton's principle and solved by the differential quadrature method. The effects of pre-stress load, surface residual stress, surface mass density, surface piezoelectrics, Young's modulus of surface layers, three material length scale parameters, thickness to material length scale parameter ratios, various boundary conditions, and two elastic foundation coefficients are investigated. It is concluded that the effect of pre-stress load in greater modes is negligible for higher aspect ratios and this effect is similar to lower aspect ratios. Also, the size-dependent effect on the dimensionless natural frequency for strain gradient theory is higher than that for modified couple stress theory and classical theory, which is due to increasing stiffness of the Timoshenko microbeam model. Moreover, the results show that dimensionless natural frequency affects more by considering the material length scale parameters with respect to surface effect. The results are compared with the obtained results from the literature and show good agreement between them. It is concluded that the amplitude of the transverse displacements (w₀) for a microbeam (MB) is more than the transverse displacements (w₁) for a piezoelectric microbeam (PMB). On the other hand, using a piezoelectric layer for PMB, the amplitude of the transverse displacements (w₁) reduces considerably with respect to MB, in which this effect leads to increase the stiffness of the microbeam and stability of microstructures. With considering the piezoelectric layer, the obtained results can be used to control the amplitude and vibration of microstructures, prevent the resonance phenomenon, design smart structures, and can be employed for micro-electro-mechanical systems and nano-electro-mechanical systems.     

Effect of anisotropy on steady state creep in functionally graded cylinder

The steady state creep in transversely isotropic functionally graded cylinder, operating under internal and external pressures, has been investigated. The cylinder is composed of functionally graded material (FGM) containing silicon carbide whiskers in a matrix of 6061Al. The creep behavior of the FGM has been described by a threshold stress based creep law. The effect of anisotropy on creep stresses and creep rates in the FGM cylinder has been analysed and compared with an isotropic FGM cylinder. The anisotropy is represented by a parameter α, defined as the ratio of radial (or axial) and tangential yield strength. The study reveals that in an anisotropic FGM cylinder i.e. when α deviates from unity, radial and tangential stresses are marginally affected whereas axial and effective stresses are significantly affected as compared to those in an isotropic FGM cylinder. The strain rates as well as inhomogeneity in strain rates in the FGM cylinder decrease significantly when α reduces from 1.3 to 0.7. The magnitude of stresses, strain rates and inhomogenity in strain rates in the FGM cylinder, subjected to internal pressure alone, could be significantly reduced by subjecting it to both internal and external pressures though the stress inhomogenity in the FGM cylinder increases.     

Wave propagation at the imperfect boundary between transversely isotropic thermodiffusive elastic layer and half-space

The aim of the present investigation is to study surface wave propagation at the imperfect boundary between two transversely isotropic thermodiffusive elastic objects: a layer of finite thickness and a half-space, in the context of the Green–Lindsay theory. The secular equation for surface waves in a compact form is derived after the mathematical model has been developed. The phase velocity and attenuation coefficient are obtained for stiffness (normal and tangential), thermal conductance, and for the case of a welded contact. The dispersion curves for these quantities are presented to depict the effect of stiffness and thermal relaxation times. The amplitudes of displacements, temperature, and concentration at the free plane boundary, as well as the specific loss of energy, are obtained and presented graphically. The special cases are considered and the results are compared with the known ones.     

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.     

Effect of An Initial Stress on SH-Type GuidedWaves Propagating in a Piezoelectric Layer Bonded on A Piezomagnetic Substrate

Propagation of SH-type guided waves in a layered structure with an invariant initial stress is studied, where a piezoelectric thin layer is perfectly bonded on a piezomagnetic substrate. Both the layer and the substrate possess transversely isotropic property. The dispersion relations of SH waves are obtained for four kinds of different electro-magnetic boundary conditions. The effects of initial stress, thickness ratio and electro-magnetic boundary conditions on the propagation behaviors are analyzed in detail. The numerical results show that: 1) The positive initial stresses make the phase velocity increasing, while the negative initial stresses decrease the phase velocity; 2) The smaller the thickness ratio of a piezoelectric layer to a piezomagnetic substrate, the larger the phase velocity of SH-type guided wave propagating in the corresponding layered structure; 3) The electrical boundary conditions play a dominant role in the propagating characteristics. Moreover the phase velocities for the electrically shorted surface are smaller than that for the open case. The obtained results are useful for understanding and design of the electromagnetic acoustic wave and microwave devices.     

Transient waves in inhomogeneous elastic shells of variable thickness

Summary This paper considers the problem of transient wave propagation in Cosserat shells of variable thickness, the inhomogeneous material of which is linearly elastic and isotropic. We do not say that the shells are isotropic because varying thickness causes behaviour characteristic of anisotropy despite the material's being isotropic. The methods of rays and of singular wave curves are combined to find and integrate the transport equations governing growth-decay behaviour of the six possible wave modes. Conditions on material parameters, thickness variation, and wave geometry are obtained for various different uncouplings of the wave modes. Some special cases of propagation conditions and of decay equations are worked out in detail.     

求解弹性波有限差分法中自由边界处理方法的对比

自由边界条件在计算方法中的数值表征是地震波模拟中的一个重要内容,表征的有效性直接关系到所得波场能否代表地表介质特性的真实响应。该文评估了交错网格有限差分法中5 种常用自由边界处理方法：直接法、应力镜像法、改进应力镜像法、横向各向同性介质替换法和声学边界替换法,并与有限元法模拟结果进行了对比,波形曲线直观比较及波幅比与相关系数定量比较显示横向各向同性介质替换法与有限法模拟结果一致性最好。进一步的层状介质模型弹性波数值模拟结果表明：横向各向同性介质替换法的精度和可靠性最高,能真实表征地表介质中的地震波传播。