Further studies on edge waves in anisotropic elastic plates

A modified Stroh-type formalism for edge waves in unsymmetrical anisotropic plates is derived. Explicit expressions of the fundamental matrices for the formalism are presented. The existence conditions for one or two subsonic edge waves in the unsymmetrical anisotropic plates are discussed based on the formalism, and a procedure for finding an explicit secular equation for the edge-wave speed is proposed.     

Comments on the relation between surface wave theory and the theory of reflection

The sextic Stroh formalism, previously extensively used in the analysis of subsonic phenomena, has been used for the analysis of reflection phenomena and leaky surface waves in the first transsonic range of velocities. In particular the behaviour of the reflection problem at the limiting velocity is studied. It is shown that when the condition of free surface can be satisfied without the inhomogeneous partial wave, a situation which would appear to be the natural limiting case of a surface wave of infinite penetration, the body wave alone satisfies the condition of free surface. This result illuminates the Barnett-Lothe existence theorem for subsonic surface waves. The close connection between the reflection problem and the leaky surface wave problem becomes very apparent in the formalism used. It is shown that for a point on a branch of leaky waves where the solution is undamped, the conditions for simple reflection, i.e. reflection only involving the two body waves, are also present. In the vicinity of such a point reflection is accompanied by resonance excitation of leaky waves. The paper concludes with some explicit calculations for transversely isotropic solids.     

二维各向异性弹性力学的Stroh公式及其推广

Stroh公式是解决各向异性弹性体二维变形的强有力的工具。该公式形式上简洁优美、应用上方便有力。它的推广形式也日益在混合边值问题、压电材料、非对称弯曲板、三维各向异性弹性体、稳态运动、多自由度系统振动等方面得到广泛应用。本文简要介绍Stroh公式及其推广形式的研究状况与进展。     

Existence of Surface Waves and Band Gaps in Periodic Heterogeneous Half-spaces

We find a sufficient condition for the existence of surface (Rayleigh) waves based on the Rayleigh-Ritz variational method. When specialized to a homogeneous half-space, the sufficient condition recovers the known criterion for the existence of subsonic surface waves. A simple existence criterion in terms of material properties is obtained for periodic half-spaces of general anisotropic materials. Further, we numerically compute the dispersion relation of the surface waves for a half-space of periodic laminates of two materials and demonstrate the existence of surface wave band gaps.     

The matrix sign function for solving surface wave problems in homogeneous and laterally periodic elastic half-spaces

The matrix sign function is shown to provide a simple and direct method to derive some fundamental results in the theory of surface waves in anisotropic materials. It is used to establish a shortcut to the basic formulas of the Barnett–Lothe integral formalism and to obtain an explicit solution of the algebraic matrix Riccati equation for the surface impedance. The matrix sign function allows the Barnett–Lothe formalism to be readily generalized for the problem of finding the surface wave speed in a periodically inhomogeneous half-space with material properties that are independent of depth. No partial wave solutions need to be found; the surface wave dispersion equation is formulated instead in terms of blocks of the matrix sign function of i$i$ times the Stroh matrix.     

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.     

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.     

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.     

十次对称二维准晶材料的位错动力学问题

论文采用Bak准晶弹性动力学模型研究了十次对称二维准晶材料的位错动力学问题。将Stroh公式推广到了准晶材料动力学问题的研究中,给出了位移和应力一般解的表达式。讨论了位错移动速度、声子场和相位子场耦合弹性常数对声子场、相位子场位移和应力的影响。     

Jet and vortex flow induced by anisotropic blast wave: experimental and computational study

This paper discusses gas-dynamic aspects of intense explosions in uniform environments. In experiments, the energy of a laser is almost instantaneously released in a volume of air shaped as either a flattened or stretched cylinder generating a blast wave. Its shape evolves in time and ultimately becomes spherical. But momentum transferred to the air when the blast wave is strongly nonspherical is anisotropic. As a result, a subsonic jet and a vortex are induced and propagate along the symmetry axis or along the perpendicular plane, depending on the initial configuration of the blast wave. Simulations based on a free-Lagrangian method for a nonviscous gas are in good agreement with the experiments. Velocities, circulation, and positions of fluid particles found in computations give an insight into the causes and details of the flow. Two simultaneous and contrary processes take place – vorticity production by the anisotropic shock wave and baroclinical generation of vorticity at the boundary of the heated gas – which give rise to net circulation. Received 21 April 1997 / Accepted 27 June 1997     

Solutions for hydrodynamics of 5- and 10-fold symmetry quasicrystals

This study carries out a complete analysis of time-space solution of hydrodynamics of pentagonal/decagonal quasicrystals. The behaviors of wave propagation for phonons and diffusion for phasons and coupling between phonon-phason fields are explored explicitly. Comprehensive discussion on physical time-space variations of all hydrodynamic field variables of the alloy quasicrystals is given. The computational specimen is simple, convenient in testing computational results, and provides a possibility that is easy to test experimentally. The quantitative results of mass density, viscosity velocities, phonon displacements, phason displacements, phonon stresses, phason stresses, viscosity stresses, and their time-space variations help us understand the motion of solid quasicrystals in a hydrodynamic condition (long-wavelength and low-frequency). The analysis presented in this paper can be used for octagonal and dodecagonal quasicrystals and is easily extended to other two-dimensional quasicrystals and three-dimensional icosahedral quasicrystals. Some problems explored by the computational results are also discussed.     

Stroh formulation for a generally constrained and pre-stressed elastic material

The Stroh formalism is essentially a spatial Hamiltonian formulation and has been recognized to be a powerful tool for solving elasticity problems involving generally anisotropic elastic materials for which conventional methods developed for isotropic materials become intractable. In this paper we develop the Stroh/Hamiltonian formulation for a generally constrained and prestressed elastic material. We derive the corresponding integral representation for the surface-impedance tensor and explain how it can be used, together with a matrix Riccati equation, to calculate the surface-wave speed. The proposed algorithm can deal with any form of constraint, pre-stress, and direction of wave propagation. As an illustration, previously known results are reproduced for surface waves in a pre-stressed incompressible elastic material and an unstressed inextensible fibre-reinforced composite, and an additional example is included analyzing the effects of pre-stress upon surface waves in an inextensible material.     

The Stroh formalism and the reciprocity properties of reflection-transmission problems in crystal piezo-acoustics

The paper is concerned with the Helmholtz-Rayleigh reciprocity, which implies invariance of mode-into-mode transformation with respect to interchange of incident mode and reflected or transmitted mode. This concept is considered for a wide range of acoustic reflection-transmission problems in anisotropic piezoelectric media. Resorting to the ideas of the Stroh formalism and casting the wave solutions of a boundary problem into a self-orthogonal and complete set, we develop the common approach which allows us to prove the reciprocity properties in a similar fashion for reflection-transmission for various boundary conditions.     

Surface waves in a rotating anisotropic elastic half-space

The Stroh formalism for surface waves in an anisotropic elastic half-space is extended to the case when the half-space rotates about an axis with a constant rotation rate. The sextic equation for the Stroh eigenvalues, the eigenvectors, the orthogonality and closure relations are obtained. The Barnett–Lothe tensors are no longer real, but two of them are Hermitian. Taziev’s equation is generalized and used to derive the polarization vector and the secular equation without computing the Stroh eigenvalues and eigenvectors. An alternative derivation using the method of first integrals by Mozhaev and Destrade yields new invariants that relate the displacement and stress and are independent of the depth from the free surface. Explicit expression of the polarization vector and the secular equation for monoclinic materials with the symmetry plane at x₃ = 0 with the rotation about the x₃-axis obtained by Destrade is re-examined, and new results are presented. Also presented is the one-component surface wave in the rotating half-space.     

Crack solutions and weight functions for plane problems in three-dimensional quasicrystals

The plane problems of an elliptic hole and a crack in three-dimensional quasicrystals subject to far-field loadings are studied. The generalized Stroh formalism is adopted here, and the explicit solutions for the coupled fields are obtained in the closed form. When the elliptic hole reduces to a crack, the analytical expressions for both the entire fields and the asymptotic fields near the crack tip are determined. The crack theory of quasicrystals, including the determination of the field intensity factors, crack opening displacements, crack tip energy release rates and so on, is a prerequisite. Applying Betti’s theorem of reciprocity, the weight functions for a quasicrystal body with a crack are derived. The weight functions provide a means of calculating the intensity factors for the crack when both phonon and phason point forces are imposed at arbitrary locations.     

Perturbation Formulas for Polarization Ratio and Phase Shift of Rayleigh Waves in Prestressed Anisotropic Media

This paper completes an earlier study (Tanuma and Man, Journal of Elasticity, 85, 21–37, 2006) where we derive a first-order perturbation formula for the phase velocity of Rayleigh waves that propagate along the free surface of a macroscopically homogeneous, anisotropic, prestressed half-space. We adopt the formulation of linear elasticity with initial stress and assume that the deviation of the prestressed anisotropic medium from a suitably-chosen, comparative, unstressed and isotropic state be small. No assumption, however, is made on the material anisotropy of the incremental elasticity tensor. With the help of the Stroh formalism, here we derive first-order perturbation formulas for the changes in polarization ratio and phase shift of Rayleigh waves from their respective comparative isotropic value. Examples are given, which show that the perturbation formulas for phase velocity and polarization ratio can serve as a starting point for investigations on the possible advantages of using Rayleigh-wave polarization, as compared with using wave speed, for acoustoelastic measurement of stress.      

具有表面效应的压电半空间中的表面波

纳米科技的快速发展使压电纳米结构在纳米机电系统中得到广泛应用,形成了诸如纳米压电电子学等新的研究方向.与传统的宏观压电材料相比,在纳米尺度下压电材料往往呈现出不同的力学特性,而造成这种差异的原因之一便是表面效应.本文基于Stroh公式、Barnett-Lothe积分矩阵及表面阻抗矩阵,研究计入表面效应的任意各向异性压电半空间中的表面波传播问题,导出了频散方程.针对横观各向同性压电材料,假设矢状平面平行于材料各向同性面,发现Rayleigh表面波和B-G波解耦,并得到各自的显式频散方程.结果表明,Rayleigh表面波的波速小于偏振方向垂直于表面的体波,而B-G波的波速小于偏振方向垂直于矢状平面的体波.以PZT-5H材料为例,用数值方法考察表面残余应力和电学边界条件对表面波频散特性的影响发现:表面残余应力只对第一阶Rayleigh波有明显的影响;电学开路情形的B-G波比电学闭路情形的B-G波传播快.本文工作可为纳米表面声波器件的设计或压电纳米结构的无损检测提供理论依据.     

A state space formalism for anisotropic elasticity. Part I: Rectilinear anisotropy

A state space formalism for anisotropic elasticity including the thermal effect is developed. A salient feature of the formalism is that it bridges the compliance-based and stiffness-based formalisms in a natural way. The displacement and stress components and the thermoelastic constants of a general anisotropic elastic material appear explicitly in the formulation, yet it is simple and clear. This is achieved by using the matrix notation to express the basic equations and grouping the stress in such a way that it enables us to cast neatly the three-dimensional equations of anisotropic elasticity into a compact state equation and an output equation. The homogeneous solution to the state equation for the generalized plane problem leads naturally to the eigen relation and the sextic equation of Stroh. Extension, twisting, bending, temperature change and body forces are accounted for through the particular solution. Based on the formalism the general solution for generalized plane strain and generalized torsion of an anisotropic elastic body are determined in an elegant manner.     

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.     

MOVING SCREW DISLOCATION IN CUBIC QUASICRYSTAL

The elasticity theory of the dislocation of cubic quasicrystals is developed. The governing equations of anti-plane elasticity dynamics problem of the quasicrystals were reduced to a solution of wave equations by introducing displacement functions, and the analytical expressions of displacements, stresses and energies induced by a moving screw dislocation in the cubic quasicrystalline and the velocity limit of the dislocation were obtained. These provide important information for studying the plastic deformation of the new solid material.