Surface waves in an elastic half-space with voids

Summary Surface waves of general type propagating in a plane-faced, homogeneous and isotropic linear elastic halfspace containing a distribution of vacuous pores (voids) are studied. Assuming that the face of the halfspace is free of external loads, it is found that the motion is not necessarily two-dimensional and that all physical variables associated with the waves are derivable from a single scalar function. The phase speed equation reveals that, unlike in the classical problem, the waves are dispersive, in general. If the frequency is very small, the motion is qualitatively similar to that in the classical problem, but the wave speed is modified due to the presence of voids.     

Effect of surface stresses on surface waves in elastic solids

This paper deals with the propagation of surface waves in homogeneous, elastic solid media whose free surfaces or interfaces of separation are capable of supporting their own stress fields. The general theory for the propagation of surface waves in a medium which supports surface stresses is first deduced, and then this theory is employed to investigate the particular cases of surface waves, viz. (a) Rayleigh waves, (b) Love waves and (c) Stoneley waves. It is seen that the Rayleigh waves become dispersive in nature; and, in case of low frequency with residual surface tension, a critical wavelength exists, below which the propagation of Rayleigh waves is not possible. This critical wave length is directly proportional to the surface tension. Some numerical calculations have been made in the case of Love waves and conclusions have been drawn.     

Free surface and surface tension effects on submerged bodies

Summary The Oseen problem for the steady motion of an object beneath a free surface with surface tension under the action of gravity is formulated. The Green's tensor for the problem is used to convert the boundary value problem to a coupled pair of integral equations for the stresses which the fluid exerts on the object. For the special case of a flat plate, these integral equations are analyzed asymptotically for small velocities and deep immersion. This yields a Fredholm equation of the second kind with Cauchy kernel, which hasa well-known solution. The results indicate the effect of surface tension on the stress singularities at the edges of the plate, and modification of the lift and drag due to the free surface.     

Plane waves in a rotating elastic solid with voids

Plane waves in a linear, homogeneous and isotropic unbounded clastic solid containing a distribution of vacuous pores (voids) and rotating with a uniform angular velocity are studied. The effects of rotation of the body on the phase speed, energy loss and decay coefficient are analysed in some detail for small and large frequency dilatational waves influenced by the presence of voids. Results obtained earlier are recovered as particular cases of the more general results obtained here.     

Surface waves in fiber-reinforced anisotropic general viscoelastic media of higher orders with voids,rotation, and electromagnetic field

In this article, we investigated an effect of external magnetic field on the propagation of surface waves in a perfect electrically conducting fiber-reinforced anisotropic general viscoelastic media of rational and higher orders with voids in a rotating medium. The general surface wave speed is derived to investigate effect of electromagnetic field and rotation on surface waves in the presence of voids and viscosity. Boundary conditions are applied to obtain the secular equation for generalized types of waves. Particular cases of Stoneley, Love and Rayleigh waves are derived. The results obtained are more general in the sense that some earlier published results are obtained from our result as special cases. In the absence of voids, the results for viscoelasticity of order zero are in good agreement with the fiber-reinforced materials. Also by neglecting the reinforced elastic parameters, the results reduce to a well-known isotropic medium. It is observed that surface waves cannot propagate in a strong initially applied electromagnetic field and rotation. Numerical results for particular cases have been obtained and displayed graphically. The results indicate that the effects of voids, anisotropic, fiber-reinforcement, rotation and electromagnetic field are very pronounced and applicable for the phenomena.     

Quasi-brittle dynamic fracture with damping due to crack edge failure zones

A model for dynamic crack propagation involving quasi-brittle fracture is studied in which adjacent material points in crack edge failure zones do not break completely apart instantaneously, but first undergo relative motion resisted by relative velocity-dependent stresses. An exact analysis for such a crack extending in an unbounded elastic plane under uniform shear and tension stresses indicates that the stress singularity orders are damped below the brittle fracture square-root value, and vanish at the Rayleigh wave speed. The power generated in the failure zones is also damped below the brittle fracture values, but the effect is not order-of-magnitude. Indeed, for small zone/crack size ratios, the approximate brittle fracture value can be spread out over the zone. In this case, however, the stresses are only weakly singular.     

Cavitation erosion as a kind of dynamic damage

The purpose of this work is to show that the spherical shock waves arising in a liquid during cavitation bubble collapse can lead to formation of deep needle-like pits on the solid surface. The nature of dynamic damage during cavitation erosion is the spallation caused by interference of rarefaction waves. Rarefaction at spherical wave impact arises when the velocity of contact surface boundary becomes less than the speed of sound in a target. If the tension caused by the focused rarefaction wave exceeds the spall strength of material, channel spall cracks can arise. At low pulsed loading, spall cracks are formed in a dynamic fatigue mode. Needle-like damage arises upon focusing rarefaction waves. In terms of our model, a system of cylindrical spall cracks is consecutively formed around a deeper axial spall needle-like crack. Upon subsequent loading, each crack acts as a source of new rarefaction wave. Newly formed cylindrical spall cracks suppress the growth of the cracks of previous generation and give birth to the cracks of next generation. A distinctive feature is that the cracks are first formed at the periphery of damageability zone, subsequent cracks having a lower depth.     

Dynamic dip-slip fault rupture in a layered geological medium: Broken symmetry of seismic motion

As recorded recently on 22 November 2014 during the Nagano-ken Hokubu (Kamishiro Fault), Japan, earthquake, one important observation in dynamic rupture (fracture) related to a shallow dip-slip earthquake (usually assumed as mode-II shear crack propagation from depth towards a free surface) is the broken symmetry of seismic motion in the vicinity of the fracturing fault plane. Generally, the strong motion (particle motion) on the hanging wall is much larger than that on the footwall, but the physical mechanisms causing this asymmetry have not been fully clarified yet. Here, utilising the techniques of finite difference calculations and dynamic photoelasticity in conjunction with high speed cinematography, we investigate the fracture dynamics of a dip-slip fault plane situated near a free surface and try to explain the mechanics behind the asymmetry, numerically as well as experimentally. In our two-dimensional crack-like rupture models, we prepare a flat fault plane, which dips either vertically or at an angle, in a monolithic (scenario (1)) or layered (scenario (2)) linear elastic medium (representing rocks). In the basic scenario (1), when the primary fault rupture initiated at some depth approaches the free surface, four Rayleigh-type waves may be induced: two of them propagate along the free surface as Rayleigh surface waves into the opposite directions to the far field, and the other two travel back downwards along the fractured fault plane as interface waves into depth. If the fault plane is inclined, in the hanging wall, the interface and Rayleigh waves may interact with each other and a shear wave carrying concentrated energy (corner wave) can be produced to cause stronger disturbances. The corner waves, generated by primary fault rupture, may exist only when the fault plane is inclined, i.e. only when the geometry considered is asymmetric. In the scenario (2), however, we indicate that (anti-)symmetry of mode-II seismic motion can be easily broken even in geometrically symmetric models if the secondary fracture is allowed at an interface between layers. If primary vertical dip-slip fault rupture in an (anti-)symmetric model moves from depth and interacts with a horizontal interface that follows, for example, a tensile fracture criterion, basically only the interface segments where the primary rupture induces dynamic tension (in the relatively subsiding footwall) may be fractured and the segments in compression (in the rising hanging wall) may remain unbroken. In this case, in the hanging wall, the dynamic stresses in the upper layer above the interface become relatively large because the compressive parts of the primary rupture-induced wave (rupture front wave) can propagate from the lower layer across the still bonded interface into the upper layer, with less reflection back to the lower one. On the contrary, in the footwall, much of the energy carried by the rupture front wave in the lower layer is reflected at the broken interface and dynamic disturbances in the upper layer tend to become smaller. Thus, the strong motion on the hanging wall is expected to become larger than that on the footwall not only in geometrically asymmetric cases but also in symmetric ones.     

Higher order surface couplings in elastic ferromagnets

The “classical” theory of elastic ferromagnetic materials is extended to the case of media able to support hyperstresses (double stresses, couple stresses), so as to take account of surface effects (curvature, role played by the magnetic anisotropy on the surface, boundary layer effect and to offer a larger choice of boundary conditions as far as magnetic spins are concerned (pinning, nonzero exchange torque,…). The field equations and the associated boundary conditions are obtained by using the generalized formulation of the principle of virtual power based on the requirement of objectivity for internal forces. Constitutive equations for nonlinear thermoelastic, magnetically saturated insulators and equations linearized with respect to small deformations are established. The set of equations obtained will permit a finer study of coupled magnetoelastic surface waves and of the resulting resonance phenomena in thin ferromagnetic deformable layers.     

Thermocapillary Convection Problem for Two Compressible Immiscible Fluids

A study is made of the motion of two compressible barotropic fluids separated by a closed interface where the surface tension depending on the temperature is taken into account. We have the so-called Marangoni effect which plays the leading role in thermodynamics under low gravity. Local (in time) unique solvability of the problem is obtained in Hölder spaces of functions with power-like decay at infinity. After the passage to Lagrangian coordinates, we arrive at a nonlinear, noncoercive initial boundary—value problem which is equivalent to the original one for small time. We establish the existence theorem for this problem on the basis of the solvability of the problem for two compressible fluids with constant surface tension coefficient on the interface. All results are obtained under some restrictions on fluid densities and viscosities which mean that the fluids are not so different from each other.     

Surface tension effects on the nonlinear behavior of long waves in a two layer flow

 The propagation of long waves of finite amplitude at the interface of two viscous fluids in the presence of interfacial tension is examined. The effect of capillarity on the shape of the waves at the interface of two superposed fluids is investigated for a wide range of density differences, viscosity ratios and imposed pressure gradients. It is found that in planar geometry surface tension stabilizes the interfacial disturbances. Attention is given to the case in which the upper fluid is more dense and comprises a thin film above the lower fluid. With the heavier fluid on the top the flow pattern is always unstable when surface tension effects are neglected. In this case the interfacial waves do not grow forever and reach a finite amplitude only when the interfacial tension is greater than a critical value.     

The free surface fluid flow over a step of an arbitrary shape in a channel

A boundary integral technique has been developed for the two-dimensional, free surface fluid flow in an arbitrary shaped channel. For steady, ideal, irrotational fluid flow a system of boundary integral equations are derived which are based on the Riemann-Hilbert technique for mixed boundary value problems of an analytical function. The boundary integral equations are solved for the fluid speed on both the solid boundary and the free surface, the shape of the free surface and for the critical Froude number for which waves first occur.     

Morphological evolution of edge-hillocks on single-crystal films having anisotropic drift-diffusion under the capillary and electromigration forces

The morphological evolution of hillocks at the unpassivated sidewalls of single-crystal metallic thin film interconnects is investigated via computer simulations using the free-moving boundary value problem. The effect of drift-diffusion anisotropy on the development of surface topographical scenarios is fully explored under the action of electromigration and capillary forces, utilizing numerous combinations of the surface texture, the drift-diffusion anisotropy and the direction of the applied electric field. The simulation studies yield analytical relationships for the velocity of the surface solitary waves and the drift velocity of electromigration-induced internal voids as a function of the applied current densities, which contain intrinsic and structural properties of the single-crystal thin films. The threshold value of the applied current density, above which electromigration-induced internal voids can be formed and may cause the catastrophic failure of interconnects by breaching, also appears explicitly in this relationship.     

轴向运动超薄梁的非局部动力学分析

基于非局部弹性理论,建立了两端受初始张力的轴向运动超薄梁横向振动的控制方程。与现有的一些仅仅在控制方程中考虑非局部效应的研究不同,该文同时将非局部效应引入到两种典型的边界条件中,考察了非局部参数对超薄梁横向振动行为尤其是固有频率和临界速度的影响。结果表明:超薄性使得轴向运动梁的自由振动固有频率及临界速度降低,经典弹性理论高估了纳米尺度结构的弯曲刚度,轴向运动超薄梁的动力学行为存在明显的非局部尺寸效应。     

Effect of platinum on the critical current density of fluorine-doped YBa2Cu3Ox superconductors

The critical current density, Jc, in fluorine-doped YBa2Cu3Ox bulk superconductors has been measured at several temperatures. The measured Jc value is 108 A m-2 at 77 K and 0.5 T. A sample with a larger critical current density can be produced by the addition of a small amount of platinum during the sintering process. The added platinum has the effect of evenly distributing Y2BaCuO5 particles and voids in the matrix in comparison with the undoped sample. These distributions of Y2BaCuO5 particles and voids are reflected in an increase in the critical current density. However, these distributions are not directly related to the observed peak effect in the critical current density. If the fluorine-doped YBa2Cu3Ox superconductor consists of the matrix and a phase with a Ginzburg–Landau parameter, that is slightly different from that of the matrix, then the occurrence of the peak effect can be explained.     

Viscosity and Surface-Tension Effects on Wave Generation by a Translating Body

Various theoretical and experimental studies have been carried out to examine the generation of waves ahead of a translating body. Not all issues pertaining to this wave-motion problem are, however, fully resolved. In particular, mechanisms pertaining to generation of white-water instability and inception of vortices in the bow region are not fully understood. In this paper, the two-dimensional, unsteady, nonlinear, viscous-flow problem associated with a translating surface-piercing body is solved by means of finite-difference algorithm based on boundary-fitted coordinates. Effects of surface tension and surfactants are examined. Results of this work resolve certain classic issues pertaining to bow flows. A continuous generation of short and steepening bow waves is observed at low (draft) Froude number, a nonlinear phenomenon uncovered recently in the case of inviscid fluid also. This indicates that, steady-state nonlinear bow-flow solution may not exist, even at low speed. It is postulated that these short bow waves are responsible for the white-water instability commonly observed ahead of a full-scale ship. The amplitudes of these short bow waves are suppressed by surface tension, which is, possibly, the reason why white-water instability is not distinctly observed in laboratory-scale experiments. The presence of surfactants on the free surface is found to intensify the generation of free-surface vorticity, thus resulting in the formation of bow vortices. The accumulation of surface-active contaminants at the bow is hence responsible for the generation of bow vortices observed in laboratory experiments at low Froude number. At high Froude number, an impulsive starting motion of the body results in the generation of a jet-like splash at the bow and a gentle start an overturning bow wave, as previously observed in the case of inviscid bow flow.     

Flexural- and capillary-gravity waves due to fundamental singularities in an inviscid fluid of finite depth

Wave motion due to line, point and ring sources submerged in an inviscid fluid are analytically investigated. The initially quiescent fluid of finite depth, covered by a thin elastic plate or by an inertial surface with the capillary effect, is assumed to be incompressible and homogenous. The strengths of the sources are time-dependent. The linearized initial-boundary-value problem is formulated within the framework of potential flow. The perturbed flow is decomposed into the regular and the singular components. An image system is introduced for the singular part to meet the boundary condition at the flat bottom. The solutions in integral form for the velocity potentials and the surface deflexions due to various singularities are obtained by means of a joint Laplace-Fourier transform. To analyze the dynamic characteristics of the flexural- and capillary-gravity waves due to unsteady disturbances, the asymptotic representations of the wave motion are explicitly derived for large time with a fixed distance-to-time ratio by virtue of the Stokes and Scorer methods of stationary phase. It is found that the generated waves consist of three wave systems, namely the steady-state gravity waves, the transient gravity waves and the transient flexural/capillary waves. The transient wave system observed depends on the moving speed of the observer in relation to the minimal and maximal group velocities. There exists a minimal depth of fluid for the possibility of the propagation of capillary-gravity waves on an inertial surface. Furthermore, the results for the pure gravity and capillary-gravity waves in a clean surface can also be recovered as the flexural and inertial parameters tend to zero.     

On periodic and solitary pure gravity waves in water of infinite depth

Non-linear two-dimensional waves propagating at a constant velocity at the surface of a fluid of infinite depth are considered. The fluid is assumed to be inviscid and incompressible and the flow irrotational. Gravity is taken into account, but surface tension is neglected. Accurate solutions are computed by boundary integral equation methods. Previous results on irregular waves are confirmed and extended. The computations strongly suggest the existence of non-periodic waves. These waves resemble generalised solitary waves in the sense that they are characterised by a train of oscillations in the far field.     

部分充有两种不相溶流体转子系统振动和稳定性的实验研究

实验研究了部分充有两种互不相溶混合流体转子系统的振动和稳定性，分析了充液量对转子系统振动和稳定性的影响。结果表明，部分充有两种互不相溶混合流体转子系统的失稳过程比充有单一流体转子系统的失稳过程更为复杂，其动力特性不仅与自由表面有关，而且还与不同流体分界面上旋转波的特性密切相关。充有两种互不相溶混合流体的转子系统与充有单一流体的转子系统一样，一般在转速超过了其一阶临界转速后就会出现不稳定。转子在不稳定区内既不以一个固定值或转子系统的某阶固有频率进行涡动，也不以转速的某恒比率频率进行涡动。随着转速的增大，转子在不稳定区的涡动频率随之增大。     

A note on the porous-wavemaker problem

Summary A complete analytical solution is obtained, by using an integral transform method, for the porous-wavemaker problem, when the effect of surface tension is taken into account on the free surface of water of finite-depth in which surface waves are produced by small horizontal oscillations of a porous vertical plate. The final results are expressed in the form of convergent integrals as well as series and known results are reproduced when surface tension is neglected.