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.      

Secular Equations for Rayleigh and Stoneley Waves in?Exponentially Graded Elastic Materials of General Anisotropy under the Influence of Gravity

The Stroh formalism is employed to study Rayleigh and Stoneley waves in exponentially graded elastic materials of general anisotropy under the influence of gravity. The 6×6 fundamental matrix N is no longer real. Nevertheless the coefficients of the sextic equation for the Stroh eigenvalue p are real. The orthogonality and closure relations are derived. Also derived are three Barnett-Lothe tensors. They are not necessarily real. Secular equations for Rayleigh and Stoneley wave speeds are presented. Explicit secular equations are obtained when the materials are orthotropic. In the literature, the secular equations for Stoneley waves in orthotropic materials are obtained without using the Stroh formalism. As a result, it requires computation of a 4×4 determinant. The secular equation presented here requires computation of a 2×2 determinant, and hence is fully explicit. A Rayleigh or Stoneley wave exists in the exponentially graded material under the influence of gravity if the wave can propagate in the homogeneous material without the influence of gravity. As the wave number k????, the Rayleigh or Stoneley wave speed approaches the speed for the homogeneous material.     

Perturbation Formula for Phase Velocity of Rayleigh Waves in Prestressed Anisotropic Media

Herein we consider Rayleigh waves propagating 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 comparative ‘unperturbed’, unstressed and isotropic state, as formally caused by the initial stress and by the anisotropic part of the incremental elasticity tensor, be small. No assumption, however, is made on the material anisotropy of the incremental elasticity tensor. With the help of the Stroh formalism, we derive a first-order perturbation formula for the shift of phase velocity of Rayleigh waves from its comparative isotropic value. Our perturbation formula does not agree totally with that which was derived some years ago by Delsanto and Clark, and we provide another argument as further support for our version of the formula. According to our first-order formula, the anisotropy-induced velocity shifts of Rayleigh waves, taken in totality of all propagation directions on the free surface, carry information only on 13 elastic constants of the anisotropic part of the incremental elasticity tensor. The remaining eight elastic constants are those which would become zero if were monoclinic with the two-fold symmetry axis normal to the free surface of the material half-space in question.     

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.     

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。     

The completeness and a new derivation of the Stroh formalism of anisotropic linear elasticity

In this paper we present a new, simpler and unified derivation of the Stroh formalism of anisotropic linear elasticity, for both nondegenerate and degenerate cases. It is based on the potential representation and Jordan canonical representation theorems. The completeness of the Stroh formalism is proved in the derivation process itself. This new approach is also extended to piezoelastic problems. Besides, we show that the eigenvalues of the fundamental elastic matrix in planar anisotropic elasticity are always distinct, except for the case of isotropy. The project supported by the National Natural Science Foundation of China (19525207 and 19891180).     

Stroh-like formalism for the coupled stretching–bending analysis of composite laminates

Due to the two-dimensional nature of thin plates, the lamination theory considering the composite laminates with in-plane and plate bending problems coupling each other is treated in this paper by using complex variable formulation. By following the steps of Stroh formalism for two-dimensional linear anisotropic elasticity, a displacement complex variable formalism developed by the other researchers was introduced and re-derived in a different but more Stroh-like way. In addition, a brand-new mixed formalism (mixed use of displacements and stresses as basic functions) is established to compensate the displacement formalism. In order to transfer all the related formulae and mathematical techniques of the Stroh formalism to these two formalisms, the general solutions for the basic equations of lamination theory and their associated eigenrelations have been purposely arranged in the form of Stroh formalism. Moreover, by using the presently developed mixed formalism, the explicit expressions for the fundamental matrix and eigenvectors are obtained first time for the most general composite laminates. Furthermore, letting the coupling stiffness vanish, the formalism has been reduced to the case of symmetric laminates and checked by a recently developed Stroh-like formalism for the plate bending problems. The comparison between Stroh formalism for two-dimensional problem, Stroh-like formalism for plate bending problem, displacement formalism and mixed formalism is then made at the end of this paper, and through their connection some useful relations are obtained.     

Some explicit expressions of extended Stroh formalism for two-dimensional piezoelectric anisotropic elasticity

Since the extended Stroh formalism for two-dimensional piezoelectric anisotropic elasticity preserves essential features of Stroh formalism for pure elastic materials, it becomes important to get the corresponding explicit expressions of some important matrices frequently appeared in Stroh formalism. In this paper, explicit expressions are obtained for the fundamental matrix N, material eigenvector matrices A and B, and Barnett–Lothe tensors L, S and H. Although the explicit expressions are presented under the generalized plane strain and short circuit condition, by suitable replacement of the material constants they are still valid for the other two-dimensional states. To provide a clear picture of these expressions, two typical examples are presented, which are piezoelectric ceramics with two different poling axes.     

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.     

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.     

各向异性双材料界面裂纹扩展裂尖场

应用界面断裂力学理论和Stroh方法,研究了广义平面变形下动态裂纹沿着各向异性双材料界面扩展时的裂尖奇异应力及动态应力强度因子.双材料界面的动态裂尖区域特性主要由两个实矩阵W和D确定,且裂尖奇异应力和动态应力强度因子可以由包含这两个矩阵的柯西奇异积分方程确定,同时给出了动态应力强度因子和能量释放率的显示表达式.算例得出当裂纹以小速度扩展时,裂尖振荡因子ε与静态时几乎相同,当界面裂纹扩展速度接近瑞利波速时,ε趋于无穷大;同时得出应力强度因子及能量释放率随裂纹扩展速度的变化关系.     

Cosserat (micropolar) elasticity in Stroh form

The theory of defects in Cosserat continua is sketched out in strict analogy to the theory of line defects in anisotropic elasticity (Stroh theory). This rewrite of the second order equilibrium equations of elasticity in a 3-dimensional space as first order equations in a 6-dimensional space is analogous to replacing the Laplace equation by the Riemann–Cauchy equations. For generalized plane strain of anisotropic micropolar (Cosserat) elasticity one obtains a 14-dimensional coupled linear system of differential equations of first order and for plane strain of anisotropic micropolar (Cosserat) elasticity we obtain a 6-dimensional coupled linear system of differential equations of first order.     

含刚性线夹杂及裂纹的各向异性压电材料耦合场分析

采用各向异性弹性力学中Ｓｔｒｏｈ方法对含刚性线夹杂及裂纹的无限大各向异性压电材料耦合的弹性场和电场进行了分析。并得到夹杂和基体界面间耦合场的实型显函表达式及夹杂尖端的１／２阶奇异性。     

Surface and interfacial waves in anisotropic elastic quasicrystals

We present the Stroh formalism for two-dimensional subsonic steady-state motion of anisotropic quasicrystals. Using this new formalism and a series of identities and properties which follow, we investigate subsonic surface and interfacial waves in anisotropic quasicrystals. Our results suggest that there exist at most three subsonic surface wave speeds. This interesting observation is quite different from the unique surface wave speed known for anisotropic crystals. The degenerate case of decagonal quasicrystalline materials is discussed in detail.     

各向异性弹性体光滑接触界面上的滑移脉冲波

利用Stroh公式,Fourier分析和奇异积分方程技术研究了两各向异性弹性半空间光滑接触可分离界面上滑移脉冲波的存在及其传播特性。结果表明,如果至少能在一种介质中存在Rayleigh波,且其波速小于两种介质中的最小极限速度,则滑移脉冲波就可以存在。这种脉冲波传播速度不确定,可在最小极限波速与较低的Rayleigh波速之间取值,而该取值范围又取决于无界面分离情况下的第一、第二滑移波的解。分离区大小取决于扰动的强度,界面法向力和质点速度在分离区两端有 1 /2奇异性。     

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.     

Generalized stroh formalism for anisotropic elasticity for general boundary conditions

The Stroh formalism is most elegant when the boundary conditions are simple, namely, they are prescribed in terms of traction or displacement. For mixed boundary conditions such as there for a slippery boundary, the concise matrix expressions of the Stroh formalism are destroyed. We present a generalized Stroh formalism which is applicable to a class of general boundary conditions. The general boundary conditions include the simple and slippery boundary conditions as special cases. For Green's functions for the half space, the general solution is applicable to the case when the surface of the half-space is a fixed, a free, a slippery, or other more general boundary. For the Griffith crack in the infinite space, the crack can be a slit-like crack with free surfaces, a rigid line inclusion (which is sometimes called an anticrack), or a rigid line with slippery surface or with other general surface conditions. It is worth mention that the modifications required on the Stroh formalism are minor. The generalized formalism and the final solutions look very similar to those of unmodified version. Yet the results are applicable to a rather wide range of boundary conditions.     

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.     

各向异性弹性力学一般边值问题的广义Stroh公式

当边值问题是简单的,即是应力边值问题时,Stroh公式是很有效的。对于混合边值问题,倒如滑动边界条件,Stroh公式中的简洁的矩阵表达式就失效了。我们提出了一个广义的Stroh公式,它可应用于一大类一般的边界条件。简单的边界条件和滑动边界条件是这一类一般边界条件的特殊情形。值得指出的是,这个关于Stroh公式所作的修正并不大。广义的公式和最后的解答看起来很类似于未修正的原公式和原来的解。然而这个修正却可应用于相当广的边界条件。