Purely transverse waves in elastic anisotropic media

Formulas are obtained for decompositions of the third- and fourth-rank tensors symmetric in the last two and three indices, respectively, into irreducible parts invariant relative to the orthogonal group of coordinate transformation. The corresponding parts of the decompositions are orthogonal to each other. These decompositions are used to obtain a general representation of the displacement vectors of plane transverse waves in elastic isotropic and anisotropic solids. It is shown that the displacement vectors of transverse waves are second-, third-, and fourth-degree homogeneous polynomials of the wave normal. Special orthotropic materials are found that transmit purely transverse waves for any direction of the wave normal. The eigenmoduli, eigenstates, and engineering constants (bulk moduli, Youngs moduli, Poissons ratios, shear moduli, and Lame constants of the closest isotropic materials) are determined for these materials.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 160–172, January–February, 2005     

On the Existence of Longitudinal Plane Waves in General Elastic Anisotropic Media

We give a new proof of Kolodner's result that longitudinal waves can propagate in at least three directions in a hyperelastic anisotropic medium. We give examples of an orthotropic hyperelastic tensor with exactly three such directions, of a monoclinic elastic (but not hyperelastic) tensor with only one, and of a monoclinic elastic (elliptic, but not uniformly elliptic) tensor with no direction for longitudinal waves. This revised version was published online in August 2006 with corrections to the Cover Date.     

Solving Stress Problems for Elastic Systems of Anisotropic Shells of Revolution with Regard for Transverse Shear and Reduction

An approach is proposed for refined solution of stress problems for elastic systems consisting of coaxial shells of revolution. Transverse shear and reduction are taken into account. Multivariant calculations made for orthotropic cylindrical shells with elliptical end-plates allow us to analyze the influence of the semiaxis ratio and intermediate supports on the stress–strain state of the shell systems under consideration     

Anisotropic Elastic Materials that Uncouple All Three Displacement Components, and Existence of One-Displacement Green's Function

It was shown in an earlier paper that, under a two-dimensional deformation, there are anisotropic elastic materials for which the antiplane displacement u ₃ and the inplane displacements u ₁, u ₂ are uncoupled but the antiplane stresses σ₃₁, σ₃₂ and the inplane stresses σ₁₁, σ₁₂, σ₂₂ remain coupled. The conditions for this to be possible were derived, but they have a complicated expression. In this paper new and simpler conditions are obtained, and a general anisotropic elastic material that satisfies the conditions is presented. For this material, and for certain monoclinic materials with the symmetry plane at x ₃ = 0, we show that the unnormalized Stroh eigenvectors a _k for k = 1, 2, 3 are all real. The matrix A =[a ₁, a ₂, a ₃] is a unit matrix when the material has a symmetry plane at x ₂ = 0. Thus any one of the u ₁, u ₂, u ₃ can be the only nonzero displacement, and the solution is a one-displacement field. Application to the Green's function due to a line of concentrated force f and a line dislocation with Burgers vector v in the infinite space, the half-space with a rigid boundary, and the infinite space with an elliptic rigid inclusion shows that one can indeed have a one-displacement field u ₁, u ₂ or u ₃. One can also have a two-displacement field polarized on a plane other than the (x ₁, x ₂)-plane. The material that uncouples u ₁, u ₂, u ₃ is not as restrictive as one might have thought. It can be triclinic, monoclinic, orthotropic, tetragonal, transversely isotropic, or cubic. However, it cannot be isotropic. This revised version was published online in August 2006 with corrections to the Cover Date.     

Evolution Equations for Plane Cubically Nonlinear Elastic Waves

Consideration is given to the nonlinear theory of elastic waves with cubic nonlinearity. This nonlinearity is separated out, and the interaction of four harmonic waves is studied. The method of slowly varying amplitudes is used. The shortened and evolution equations, the first integrals of these equations (Manley–Rowe relations), and energy balance law for a set of four interacting waves (quadruplet) are derived. The interaction of waves is described using the wavefront reversal scheme     

Wave Motion Localization Effects in Elastic Waveguides

Dynamic effects characteristic of elastic bodies and associated with local disturbances in finite elastic bodies and inhomogeneous waveguides are analyzed and systematized. The physical causes of such disturbances are analyzed. It is shown that these disturbances are due to energy transfer from longitudinal to transverse waves and back, when they are reflected from the free surface of an elastic body __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 9, pp. 38–45, September 2005.     

Stroh Formalism and Rayleigh Waves

The Stroh formalism is a powerful and elegant mathematical method developed for the analysis of the equations of anisotropic elasticity. The purpose of this exposition is to introduce the essence of this formalism and demonstrate its effectiveness in both static and dynamic elasticity. The equations of elasticity are complicated, because they constitute a system and, particularly for the anisotropic cases, inherit many parameters from the elasticity tensor. The Stroh formalism reveals simple structures hidden in the equations of anisotropic elasticity and provides a systematic approach to these equations. This exposition is divided into three chapters. Chapter 1 gives a succinct introduction to the Stroh formalism so that the reader could grasp the essentials as quickly as possible. In Chapter 2 several important topics in static elasticity, which include fundamental solutions, piezoelectricity, and inverse boundary value problems, are studied on the basis of the Stroh formalism. Chapter 3 is devoted to Rayleigh waves, for long a topic of utmost importance in nondestructive evaluation, seismology, and materials science. There we discuss existence, uniqueness, phase velocity, polarization, and perturbation of Rayleigh waves through the Stroh formalism.

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The Table of Contents and Index are also provided as Electronic Supplementary Material for online readers at doi:      

弹性SHPB试件内部纵向应力波传播及所产生的横向约束应力

采用考虑横向惯性效应的Rayleigh-Love杆理论分析了一个弹性试件在分离式霍普金森压杆(SHPB)加载过程中的内部弹性波传播过程,运用Laplace变换和反变换方法,得到了试件内部各点的变形、速度、应变和应力解析解.通过数值计算,得到梯形入射波加载情况下,纵向应力在试件内部的连续变化过程,以及波传播所伴随的横向附加应力.计算表明：在试件/入射杆界面附近,初次加载所产生的横向附加应力最大,可达入射波平台的12%;在大部分试件区域,纵向应力波传播将造成入射波平台4%～6%的横向附加应力;材料的泊松比越大,或者杆/试件声阻抗比越小,所伴随的横向附加应力越大;梯形波的上升时间和试件长径比对横向附加应力影响不大.     

Comparing the Evolution Characteristics of Waves in Nonlinearly Elastic Micro- and Nanocomposites with Carbon Fillers

A method is proposed for studying the evolution of plane waves in micro- and nanocomposite materials. This method permits comparing the evolutions of harmonic waves and produces results that are in agreement with data obtained earlier and with the metaphysical reasoning on the nanomechanics of composite materials     

Influence of prestress on the velocities of plane waves propagating normally to the layers of nanocomposites

The paper is concerned with longitudinal and transverse waves propagating at a right angle to the layers of a nanocomposite material with initial (process-induced residual) stresses. The composite consists of alternating layers of two dissimilar materials. The materials are assumed nonlinearly elastic and described by the Murnaghan potential. For simulation of wave propagation, a problem is formulated within the framework of the three-dimensional linearized theory of elasticity for finite prestrains. It is established that the relative velocities of waves depend linearly on small prestresses. In some composite materials, however, the thicknesses of the layers may be in a ratio such that the wave velocities are independent of the prestress level __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 7, pp. 3–22, July 2006.     

微机化的纵横波螺栓轴向应力检测仪研制

介绍了一种既可测安装过程中的螺栓应力又可测已紧固螺栓轴向应力的一种仪器,该仪器是以单片机和计算机为核心,用超声纵横波声时进行在线检测;利用超声波沿轴向传播的波速同轴向应力以及三阶弹性常数的关系导出了应力同纵横波声时比值及温度的关系,考虑了受应力作用时温度对声速的影响,简化了测量及计算过程.仪器采用了声时、声幅衰减及数字处理技术联合排除偶合错误及噪声干扰等方法,增加了仪器的稳定性;仪器采用了高精度测声时方法,提高了应力测量精度;计算机提供了数字及图形显示并提供了数据查询及图形还原等功能,极大地方便了用户操作及管理.实验结果表明应力低于250 MPa,夹紧距离大于30 mm时,应力超声测量的绝对误差小于±8 MPa.     

Reflection and Transmission of Elastic Waves at the Interface Between Water and a Double Porosity Solid

The reflection and transmission of an incident plane wave at an interface between water and a fluid-saturated double porosity solid are investigated. The properties of the three compressional waves and one shear wave in double porosity media are discussed in detail. The effect of the viscosity, permeability, and porosity on the phase velocity and attenuation of four bulk waves is presented. Comparison between the double porosity theory and the Biot theory reveals that there exists obvious difference in the phase velocity, attenuation and the reflection and transmission coefficients. Two cases of boundary conditions, i.e., the sealed-pore and the open-pore boundaries, are introduced in the numerical calculation. At last, the effect of the fracture permeability and porosity on the reflection and transmission coefficients considered. It is revealed that the amplitudes of the reflected and transmitted waves strongly depend the boundary condition, fracture permeability, and porosity.     

小内力平板弹性波散射问题的摄动方法

基于平板小挠度弯曲波动方程,采用摄动方法具有纵向内力作用下的平板开孔弹性波的散射问题进行了研究,得么了传播稳态波时此种平板弯曲波动问题的分析解,分析了均匀纵向内力对弹性波散射结果的影响,作为算例,本文给出了平板圆形开孔的动应力集中系数的数值结果,并对计算结果进行了讨论。     

Elastoplastic stress-strain state of flexible layered shells made of isotropic and transversely isotropic materials with different moduli and subjected to axisymmetric loading

The paper proposes a numerical technique for analysis of the elastoplastic stress-strain state of flexible layered shells of revolution under axisymmetric loading. It is assumed that the shells are made of isotropic and transversely isotropic materials with different moduli in tension and compression. The technique is based on a geometrically nonlinear theory of shells that takes into account the squared angles of rotation and the Kirchhoff-Love hypotheses for a layer stack. The deformation of isotropic materials is described using the theory of deformation along paths of small curvature. The deformation of transversely isotropic materials is described using the theory of elasticity with different moduli in tension and compression. The problem is solved by the method of successive approximations. A numerical example is given __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 11, pp. 31–42, November 2007.     

拉压模量不同材料的圆孔扩张问题

本文建立了拉、压模量不同材料的圆孔扩展理论，分析了有关参数对圆孔扩张时应力、位移以及塑性区的发展规律的影响。     

New Transforms for the Resolving Equations in Elastic Theory and New Integral Transforms, with Applications to Boundary-Value Problems of Mechanics

This paper reviews results from an analysis of exact boundary-value solutions of static and dynamic elasticity obtained by the method of integral transforms. Consideration is given to the solutions of problems for a half-space and a blunted hollow cone, the heat conduction problem for a cone, and the diffraction problem for a cone with a hole along the generatrix. Solutions of mechanics problems are analyzed and the method of integral transforms is generalized     

Elastoplastic Axisymmetric Stress-Strain State of Laminated Shells Made of Isotropic and Transversely Isotropic Materials with Different Moduli

A method is developed for analysis of the elastoplastic stress-strain state of laminated shells of revolution under axisymmetric loading. The shells are made of isotropic and transversally isotropic materials with different moduli. The method is based on the Kirchhoff-Love hypotheses for the whole laminate, the theory of deformation along paths of small curvature (for isotropic materials), and the theory of elasticity with different tensile and compressive moduli (transversely isotropic materials). The problem is solved by the method of successive approximations. Numerical examples are given __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 8, pp. 88–96, August 2005.     

HAMILTONIAN SYSTEM AND SIMPLETIC GEOMETRY IN MECHANICS OF COMPOSITE MATERIALS (Ⅱ)——PLANE STRESS PROBLEM

Fundamental theory presented in Part (I)^[8] is used to analyze anisotropic plane stress problems. First we construct the generalized variational principle to enter Hamiltonian system and get Hamiltonian differential operator matrix; then we solve eigen problem; finally, we present the process of obtaining analytical solutions and semi-analytical solutions for anisotropic plane stress problems on rectangular area.     

Reflection and Transmission of Elastic Waves from the Interface of a Fluid-saturated Porous Solid and a Double Porosity Solid

The reflection and transmission characteristics of an incident plane P1 wave from the interface of a fluid-saturated single porous solid and a fluid-saturated double porosity solid are investigated. The fluid-saturated porous solid is modeled with the classic Biot’s theory and the double porosity medium is described by an extended Biot’s theory. In a double-porosity model with dual-permeability there exist three compressional waves and a shear wave. The effects of the incident angle and frequency on amplitude ratios of the reflected and transmitted waves to the incident wave are discussed. Two boundary conditions are discussed in detail: (a) Open-pore boundary and (b) Sealed-pore boundary. Numerical results reveal that the characteristics of the reflection and transmission coefficients to the incident angle and the frequency are quite different for the two cases of boundary conditions. Properties of the bulk waves existing in the fluid-saturated porous solid and the double porosity medium are also studied.