Scattering of Plane Harmonic Waves in an Infinite Elastic Solid by an Arbitrary Configuration of Two-dimensional Obstacles

The problem of the multiple scattering of plane harmonic wavesin an infinite elastic solid by arbitrary configurations oftwo-dimensional obstacles is considered. Sets of integral equationsrelating the multiple scattering functions to the single scatteringfunctions are derived. The integral equations are replaced bya set of algebraic equations which are then solved by an iterativeprocedure. Some results are given for the scattering of a planeharmonic P or SV wave by an identical pair of parallel circularcylindrical cavities.     

On Surface Waves and Deformations in a Pre-stressed Incompressible Elastic Solid

The propagation of infinitesimal surface waves on a half-spaceof incompressible isotropic elastic material subject to a generalpure homogeneous pre-strain is considered. The secular equationfor propagation along a principal axis of the pre-strain isobtained for a general strain-energy function, and conditionswhich ensure stability of the underlying pre-strain are derived.The influence of the pre-stress on the existence of surfacewaves is examined and, in particular, it is found that, undera certain range of hydrostatic pre-stress, a unique wavespeedexists and is bounded above by a limiting speed which correspondsto the shear wave speed in an infinite body. The secular equationis analysed in detail for particular deformations and, for anumber of specific forms of strain-energy function, numericalresults are used to illustrate the dependence of the wave speedon the pre-strain. Particular attention is focused on pre-strainscorresponding to loss of stability, in which case the infinitesimalstrain is time-independent (the wave speed being zero). Thetheory described here encompasses previous work on surface wavesand instabilities in incompressible isotropic elastic materialsand provides a clear delimitation of the range of deformationsfor which surface waves exist.     

弹性和电-微拉伸广义热弹性固体中的平面波在不完全边界上的传播

讨论了平面波入射在弹性固体(介质(M))和电-微拉伸广义热弹性固体(介质M)的界面上时,波的反射和透射问题.介质M中存在5种反射波(纵向位移(LD)波、热(T)波、纵向微拉伸(LM)波和2种横向耦合位移和微转动波(CD(Ⅰ)和CD(Ⅱ)波));介质(M)中存在2种透射波(纵(P)波和横(SV)波).得到不完全边界上不同反射波和透射波的振幅比,并导出法向力刚度、切向力刚度和完全粘接时的振幅比.对LD波和CD(Ⅰ)波,图示出振幅比随不同入射角的变化.显示出反射波和透射波的振幅比受到介质的刚度、电场、拉伸和热特性的影响.推演出一些有价值的特例.     

一类弹性系统的精确L2-能控性

该文研究带有压电驱动器的Rayleigh梁系统的精确能控性.先用算子半群方法和提升结果[9]建立了Rayleigh梁方程解的正则性;再用Hilbert唯一性方法结合Diophantine逼近理论的某些结果得到了系统的精确L2-能控性.     

弹性固体和充满两种互不相溶粘性流体的多孔固体之间界面的反射和折射及其衰减波

在充满两种互不相溶粘性流体的多孔固体中,研究弹性波的传播.用3个数性的势函数描述3个纵波的传播,用1个矢性的势函数单独描述横波的传播.根据这些势函数,在不同的组合相中,定义出质点的位移.可以看出,可能存在3个纵波和1个横波.在一个弹性固体半空间与一个充满两种互不相溶粘性流体的多孔固体半空间之间,研究其界面上入射纵波和横波所引起的反射和折射现象.由于孔隙流体中有粘性,折射到多孔介质中的波,朝垂直界面方向偏离.将入射波引起的反射波和折射波的波幅比,作为非奇异的线性代数方程组计算.进一步通过这些波幅比,计算出各个被离散波在入射波能量中所占的份额.通过一个特殊的数值模型,计算出波幅比和能量比系数随入射角的变化.超过SV波的临界入射角,反射波P将不再出现.越过界面的能量守恒原理得到了验证.绘出了图形并对不同孔隙饱和度以及频率的变化,讨论它们对能量分配的影响.     

Rotational Effects on Surface Waves

In the classical theory of water waves only irrotational motions are considered. In the present paper a class of wave problems associated with inviscid rotational motions is examined and the consequences explored.     

Interaction of Elastic Waves

We derive and analyze asymptotic equations for the interaction of weakly nonlinear elastic waves. We show that there are resonant triads involving two transverse and one longitudinal wave provided the wave speeds satisfy a certain irrationality condition. We study initial value and signaling problems, and the interaction of sawtooth wave packets.     

Finite-amplitude Rotational Waves in Viscous Shear Flows

Growing finite-amplitude initially spanwise-independent two-dimensional rotational waves and their nonlinear interaction with unidirectional viscous shear flows of various strengths are considered. Both primary and secondary instabilities are studied, but only secondary instabilities are permitted to vary in the spanwise direction. A generalized Lagrangian-mean formulation is employed to describe wave-mean interactions, and a separate theory is constructed to account for the back effect of the developing mean flow on the wave field. Viscosity is seen to significantly complicate calculation of the back effect. The primary instability is seen to act as a platform for, and catalyst to, secondary instabilities. The analysis leads to an eigenvalue problem for the initial growth of the secondary instability, this being a generalization of the eigenvalue problem constructed by Craik for inviscid neutral waves. Two inviscid secondary instability mechanisms to longitudinal vortex form are observed: the first has as its basis the Craik–Leibovich type 2 mechanism. The second, which is as yet unproven, requires that both the wave and flow field distort in concert at all levels of shear. Both mechanisms excite exponential growth on a convective rather than diffusive scale in the presence of neutral waves, but growing waves alter that growth rate.     

On Rotational Waves of Limit Amplitude

Functional Analysis and Its Applications - In this note we discuss some recent results on extreme steady waves under gravity. They include the existence and regularity theorems for highest waves on...     

Waves from a Deep Source of Longitudinal Waves in an Elastic Half Space with a Free Boundary

The nonstationary propagation of waves on the surface of an elastic half space from a deep expansion source (model of an explosion in a half space) is examined. Exact solutions are obtained in the form of integrals with finite limits and the general solution is calculated. Algebraic expressions are obtained for the Rayleigh wave. The transition of Rayleigh waves at the surface of the half space is studied. Calculations of Rayleigh waves from discontinuous pulsed sources are presented.     

Stability of Surface Rayleigh Waves in an Elastic Half‐Space

This paper is concerned with nonlinear analysis for the propagation of Rayleigh surface waves on a homogeneous, elastic half‐space of general anisotropy. We show how to derive an asymptotic equation for the displacement by applying the second‐order elasticity theory. The evolution equation obtained is a nonlocal generalization of Burgers' equation, for which an explicit stability condition is exhibited. Finally, we investigate examples of interest, namely, isotropic materials, Ogden's materials, compressible Mooney–Rivlin materials, compressible neo‐Hookean materials, Simpson–Spector materials, St Venant–Kirchhoff materials, and Hadamard–Green materials.     

弹性固体和微极多孔晶体界面及其结合松散度对弹性波传播的影响

在松散结合的弹性固体和微极多孔立方晶体半空间之间的界面上,研究周期平面波的反射和透射,假定界面性质类似于断层,断层上的牵引力保持连续,但允许产生有限的滑移.用图形给出各种反射波和透射波的振幅比.文中还演绎出某些重要的特殊情况.     

On the Response of an Elastic Solid to Droplet Impact

Expressions are obtained in closed form for the transient stressin a homogeneous isotropic elastic half-space whose surfaceis subjected to a constant uniform pressure over a circle whoseradius increases in proportion to the square root of time. Thepositions of the wave fronts are derived, together with themagnitudes of the discontinuities of the stress components acrossthem. It is shown that a Rayleigh surface wave appears at themoment when the radius of the area under load is increasingwith the Rayleigh wave-speed. The behaviour of the stress andstrain is described in the region of the stress singularityat its leading edge.     

Torsional Oscillations of a Rigid Convex Inclusion Embedded in an Elastic Solid

The problem discussed in this paper concerns a rigid axi-symmetricbody of convex form embedded in an infinite isotropic elasticsolid. When the inclusion is set in motion by an impulsive torqueapplied about the axis of symmetry it executes a damped torsionaloscillation and generates a shear pulse in the surrounding elasticmaterial. A representation of the elastic wave field in theform of a progressing wave expansion is shown to lead to anintegral equation of Volterra type for the angular motion ofthe inclusion. The exact solution obtained by Chadwick &Trowbridge (1967) for the case of a spherical inclusion is thenre-derived and used in developing an approximate method of solutionof the more general problem described above. Detailed resultsare worked out for spheroidal inclusions of oblate and prolateforms and numerical results are presented and discussed.     

Propagation of Creeping Waves Along the Curvilinear Surface of an Anisotropic Elastic Body

An analytic expression for the wave field of a creeping wave propagating along the curvilinear surface of an anisotropic elastic body is constructed by the boundary layer method. Bibliography: 14 titles.Dedicated to L. A. Molotkov on the occasion of his seventieth birthday__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 297, 2003, pp. 9–29.     

微极各向同性弹性板中的Rayleigh-Lamb波

研究了无应力作用条件下，均匀、各向同性、圆柱形微极结构弹性板中波的传播．导出了对称和斜对称模式下波传播的特征方程．对短波这一极端情况，无应力圆板中对称和斜对称模态波的特征方程退化为Pmyle曲表面波频率方程．并得到薄板的计算结果．给出了位移和微转动分量，并绘制了相应图形．给出了若干特殊情况的研究结果及对称和斜对称模态特征方程的图示．     

An Approximation for Slowly Varying Waves in Elastic Rods

A systematic method of approximation, involving series expansionsin terms of a small parameter , is used to study the propagationof slowly varying (long wave) disturbances in elastic rods.It is shown that the bar equations are the result in the limitas 0. This result is independent of the form of the cross-sectionof the rod. Further terms may be calculated, but these dependon this shape. Near the driven end this approximation breaksdown and a boundary layer is introduced. The boundary layerequations are displayed and discussed.     

Long Time Behavior of Nonlinear Strain Waves in Elastic Waveguides

The initial-boundary value problem of the propagation of nonlinear longitudinal elastic waves in an initially strained rod is considered. The rod is assumed to interact with the surrouding elastic and viscous external medium. The long time behavior of solutions is derived ancl global attractors in E_1 space is obtained.