Explosive collapse of vapor film under conditions of intensive heat fluxes

The stability is investigated (linear and nonlinear analysis) of the interface between a thin vapor film and a layer of liquid in the presence of a steady heat flux from a metal surface heated to a high temperature to the vapor film and then from vapor to subcooled liquid. In view of thermal disequilibrium which takes into account the temperature dependence of saturation pressure, boundary conditions on the vapor-liquid interface are derived, which generalize the known correlations on the free surface of liquid in the gravity field. A number of new effects arise in the problem under consideration, as distinct from the classical problem. The thermal processes, which occur on the phase boundary and are possible in the absence of the force of gravity as well, lead to the generation of weakly decaying periodic waves of low amplitude, whose velocity may exceed significantly that of gravity waves. The heat flux through the interface may cause on this surface periodic waves of small length (ripple) which are not capillary. The processes of phase transition on the interface are capable of providing for the stability of vapor film under the layer of liquid in the gravity field. Along with periodic waves and solitons, the mode of explosive instability may arise in the nonlinear stage because of a weak variation of the film thickness, where the amplitude of an initially low-amplitude plane wave rises to infinity during a finite period of time.     

An exact method for cracked elastic strips under concentrated loads — time-harmonic response

An exact method is presented for the determination of the near-tip stress field arising from the scattering of SH waves by a long crack in a strip-like elastic body. The waves are generated by a concentrated anti-plane shear force acting on each face of the crack. Time-harmonic variation of the external loading is assumed. The problem has two characteristic lengths, i.e. the strip width, and the distance between the point of application of the concentrated forces and the crack tip. It is well-known that the second characteristic length introduces a serious difficulty in the mathematical analysis of the problem: a non-standard Wiener-Hopf (W-H) equation arises, one that contains a forcing term with unbounded behavior at infinity in the transform plane. In addition, the presence of the strip's finite width results in a complicated W-H kernel. Nevertheless, a procedure is described here which circumvents the aforementioned difficulties and holds hope for solving more complicated problems (e.g. the plane-stress/strain version of the present problem) having similar features. The method is based on integral transform analysis, an exact kernel factorization and usage of certain theorems of analytic function theory. Numerical results for the stress-intensity-factor dependence upon the ratio of characteristic lengths and the external load frequency are presented.     

An analytical/numerical approach for cracked elastic strips under concentrated loads — transient response

An analytical/numerical approach is presented for the determination of the near-tip stress field arising from the scattering of SH waves by a long crack in a strip-like elastic body. The waves are generated by a concentrated anti-plane shear force acting suddenly on each face of the crack. The problem has two characteristic lengths, i.e. the strip width, and the distance between the point of application of the concentrated forces and the crack tip. It is well-known that the second characteristic length introduces a serious difficulty in the mathematical analysis of the problem. In particular, a non-standard Wiener-Hopf (W-H) equation arises, that contains a forcing term with unbounded behaviour at infinity in the transform plane. In addition, the presence of the strip's finite width results in a complicated W-H kernel introducing, therefore, further difficulties. Nevertheless, a procedure is described here which circumvents the aforementioned difficulties and holds hope for solving more complicated problems (e.g. the plane-stress/strain version of the present problem) having similar features. Our method is based on integral transform analysis, an exact kernel factorization, usage of certain theorems of analytic function theory, and numerical Laplace-transform inversion. Numerical results for the stress-intensity-factor dependence upon the ratio of characteristic lengths are presented.     

Transient evolution of weakly nonlinear sloshing waves: an analytical and numerical comparison

The problem of water waves generated in a horizontally oscillating basin is considered, with specific emphasis on the transient evolution of the wave amplitude. A third-order amplitude evolution equation is solved analytically in terms of Jacobian elliptic functions. The solution explicitly determines the maximum amplitude and nonlinear beating period of the resonated wave. An observed bifurcation in the amplitude response is shown to correspond to the elliptic modulus approaching unity and the beating period of the interaction approaching infinity. The theoretical predictions compare favorably to fully nonlinear simulations of the sloshing process. Due to the omission of damping, the consideration of only a single mode, and the weakly nonlinear framework, the analytical solution applies only to finite-depth, non-breaking waves. The inviscid numerical simulations are similarly limited to finite depth.     

Bleustein-Gulyaev SAW and its associated quasi-particle

Following a previous work that dealt with the case of Rayleigh surface waves in pure elasticity, here it is shown that a quasi-particle with Newtonian point-like mechanics (equation of inertial motion, expression of the kinetic energy) can be associated with the celebrated Bleustein-Gulyaev surface waves of linear piezoelectricity. This association is based on the integration over a vertical band of the sagittal plane of the canonical balance laws that accompany, via Noether’s theorem, the basic field equations. It accounts for the boundary conditions at the limiting surface, the periodicity of the solution in propagation space, and the vanishing of all fields at infinity in the substrate or in the outside vacuum. The proof benefits from the fact that the average (over one wavelength of propagation) of the Lagrangian density at the limiting surface is proportional to the satisfied “dispersion relation”, and hence is zero. The expression found for the “mass” of the said quasi-particle is informative in that it contains information about the frequency, the amplitude of the signal, and the electromechanical coupling.     

On the Nonuniqueness of Solutions to the Nonlinear Equations of Elasticity Theory

One-dimensional selfsimilar problems for waves in an elastic half-space generated by a sudden change of the boundary stress (the “piston” problem) and problems of disintegration of an arbitrary discontinuity are considered. For the case when small-amplitude waves are generated in a medium with small anisotropy a qualitative analysis shows that these problems have nonunique solutions when it is assumed that the solutions involve Riemann waves and evolutionary discontinuities. The above-mentioned problems are considered as limits of properly formulated problems for visco-elastic media when the viscosity tends to zero or (what is the same) that time tends to infinity. It is numerically found that all above-mentioned inviscid solutions can represent the asymptotics of visco-elastic solutions. The type of asymptotics depends on those details of the visco-elastic problem formulation which are absent when formulating inviscid selfsimilar problems. Similar considerations are made for elastic media with dispersion along with dissipation which are manifested in small-scale processes. In such media the number of available asymptotics (as t→∞) for the above-mentioned solutions depends on a relation between dispersion and dissipation and can be large. Thus, two possible causes for the nonuniqueness of solutions to the equations of elasticity theory are investigated.     

Splitting up an optical beam in a polarized component added to an unpolarized component

The decomposition of an optical beam in a polarized part added to an unpolarized part was studied by G.G. Stokes among numerous other works. Today, the problem is no longer a trigonometric manipulation proper to monochromatic waves, but a problem handling stationary processes with band spectra. In literature, the question seems to be: given some spectral properties and some propagation medium, can we obtain a decomposition? Furthermore, in the case of a positive answer, we have to provide devices for exhibiting solutions. In a linear framework, the problem always has a solution (and even an infinity) whatever the chosen polarization direction. In this paper, we study the links which appear most often between the members of the decomposition.     

A note on penny-shaped cracks in transversely isotropic materials

This note contains some further discussion of the problem of a penny-shaped crack in a transversely isotropic solid. For uniform applied stress at infinity, the problem is solved using Eshelby's method. Particular attention is given to the interaction energy and the crack opening displacements. The results are given in a form which is convenient in the study of cracked solids.     

One-dimensional plastic materials with work-hardening

Summary Acceleration waves in one-dimensional plastic materials are investigated by the theory of singular points. The unloading wave propagates with a constant velocity, while the propagation velocity of the loading wave is less than that of the unloading wave and the velocity depends upon the stress and the work-hardening. The growth and decay of the amplitude of the waves are also analyzed. The unloading wave propagates with a constant amplitude. The amplitude of the loading wave may grow or decay and the choice between the two depends upon the stress, the work-hardening and whether the wave is compressive or expansive. In the case of growth the amplitude tends to infinity in finite time, that is, the blow time, and the acceleration wave coalesces into a shock wave. In the case of decay the amplitude tends to zero as the time tends to infinity. The propagation velocity, the blow time and the blow distance are calculated and plotted against the strain.     

Transient scattering of elastic waves by three dimensional non-axisymmetric dipping layers

Scattering of elastic plane waves by three dimensional non-axisymmetric multiple dipping layers embedded in an elastic half-space is investigated by using a boundary method. The dipping layer is subjected to incident Rayleigh waves and oblique incident SH, SV and P waves. For the steady state problem, spherical wave functions are used to express the unkown scattered field. These functions satisfy the equation of motion and radiation conditions at infinity but they do not satisfy the stress free boundary conditions on the surface of the half-space. The boundary and continuity conditions are imposed locally in the least-square sense at points on the layer interfaces and on the surface of the half-space. The transient response is constructed from the steady state solution by using Fourier synthesis. Numerical results are presented for both steady state and transient problems. Steady state problems include solutions for two non-axisymmetric dipping layers in the form of a prolate. Transient responses are presented for one and two dipping layer models subjected to incident wave signals in the shape of a Ricker wavelet. It is shown that change in azimuthal orientation of the incident wave may significantly change the surface response of the dipping layer. For the transient problem, response comparison of one and two dipping layers indicates that the addition of an extra layer may also completely change the response characteristics of the alluvium. In particular, the delay in arrival of much larger amplitude surface waves by two dipping layers in comparison with other geometrically compatible models demonstrates the importance of the detailed three dimensional modelling of layered irregularities.     

Quarter-plane problem of a floating elastic plate

An investigation is made into the hydro-elastic behavior of a floating elastic plate, which occupies a quarter plane to infinity and is excited by water waves. A boundary-integral equation based on the Green function for this problem is shown for the case of finite water depth, as well as for the case of shallow water. The solution of the quarter-plane problem is composed of the corner effect and the solution of the half-plane problem. The corner effect is divided into two parts. The first part is the end effect of the forcing term of the integral equation, which is analytically estimated and its asymptotic form is derived. The second part is the local contribution whose asymptotic form is also obtained. The asymptotic form of the corner effect is confirmed by a numerical evaluation.     

Phase constraint for the waves diffracted by lossless symmetrical gratings at Littrow mount

The energy conservation of grating diffraction is analyzed in a particular condition of incidence in which two incident waves reach a symmetrical grating from the two sides of the grating normal at the first-order Littrow mounting. In such a situation the incident waves generate an interference pattern with the same period as the grating. Thus in each direction of diffraction, interference occurs between two consecutive diffractive orders of the symmetrical incident waves. By applying only energy conservation and the geometrical symmetry of the grating profile to this problem it is possible to establish a general constraint for the phases and amplitudes of the diffracted orders of the same incident wave. Experimental and theoretical results are presented confirming the obtained relations.     

INFINITE DOMAIN CORRECTION FOR IN-PLANE BODY WAVES IN A TWO-DIMENSIONAL BOUNDARY ELEMENT ANALYSIS

A method is described in this article to correct for the error that arises with the discretization of domains that include boundaries that extend to infinity. Typically when open domains are discretized, part of the boundary is excluded from the calculation resulting in a truncated region. Of particular interest in this article are earthquake wave amplification problems through zoned media. In these type of problems, the boundary element discretization scheme typically results in truncated regions. Correction for truncation in anti-plane wave problems has already been addressed in a previous article by Heymsfield. In this article, truncation correction for in-plane body waves in a damped material will be discussed. To prove the validity of the proposed technique, the method is checked by calculating the soil amplification of a unit in-plane SV wave through a soil layer resting on a rock half-space. Since an analytic solution exists for this problem, the problem serves as a good basis to compare results with and without the corrections for truncation. Results for this particular problem compare the analytic solution with the numerical solution considering (1) no truncation correction, (2) only layer correction, and (3) both layer and half-space corrections. © 1997 by John Wiley & Sons, Ltd.     

周向导波在不同连续性条件多层圆筒中的传播

通过对多层厚壁圆筒中周向导波频散曲线研究,发现周向导波在多层圆筒中传播时,会发生明显的频散现象和模态干涉现象,第1阶模态在高频时接近于无频散的Rayleigh面波。通过对不同连续性条件的多层筒与单层筒的频散曲线比较,以及筒的层数对频散曲线的影响,都充分说明了检测层间界面缺陷的周向导波应该集中在低阶模态上。最后对不同连续性条件多层筒的位移研究,最终推断出第2阶模态具有在层间界面附近集中能量的特征,可应用于层间界面缺陷的周向导波无损检测技术中。     

Propagation of a Mode-II crack in a viscoelastic medium with different bulk and shear relaxation

A solution for a crack propagating under shear-loading in an isotropic viscoelastic medium with different relaxation under volume and shear deformations is presented. The medium is infinite and the semi-infinite crack propagates along the x ₁-axis at constant speed V, which may take any value up to the speed of dilatational waves. The requisite Riemann–Hilbert problem for the steady-state case has been solved and the asymptotics of the stress component σ₁₂ directly ahead of the crack and at infinity have been obtained.     

Focusing of electromagnetic radiation by hyperboloidal and ellipsoidal lenses

A solution to the problem of plane electromagnetic waves focused by an ellipsoidal or a hyperboloidal lens is derived from the Stratton-Chu integral by solving a boundary-value problem. The current method is more rigorous than those hitherto published in the literature. Results show that for linearly polarized incident illumination and in the vicinity of the focus, the distribution of the time-averaged electric energy density is almost fully transverse electric.     

位移型耗能减震结构优化设计

提出了一种应用于耗能减震体系参数设计的新型优化数学模型。模型目标函数体现了明确的设计意义,在保证几种地震波作用下结构最大层间位移角接近设计值的前提下,尽可能使得应用的耗能体系刚度最小。新模型同时解决了不同地震波作用下优化结果各不相同,优化参数难以选择的问题,是一种有效的优化设计方法。     

Receptivity of boundary layers to distributed wall vibrations

Linear three-dimensional receptivity of boundary layers to distributed wall vibrations in the large Reynolds number limit (Re-->infinity) is studied in this paper. The fluid motion is analysed by means of the multiscale asymptotic technique combined with the method of matched asymptotic expansions. The body surface is assumed to be perturbed by small-amplitude oscillations being tuned in resonance with the neutral Tollmien-Schlichting wave at a certain point on the wall. The characteristic length of the resonance region is found to be O(Re-3/16), which follows from the condition that the boundary-layer non-parallelism and the wave amplitude growth have the same order of magnitude. The amplitude equation is derived as a solvability condition for the inhomogeneous boundary-value problem. Investigating detuning effects, we consider perturbations in the form of a wave packet with a narrow O(Re-3/16) discrete or continuous spectrum concentrated near the resonant wavenumber and frequency. The boundary-layer laminarization based on neutralizing the oncoming Tollmien-Schlichting waves (or wave packets) is also discussed.     

Interaction effect of two arbitrarily oriented cracks — Part I

The elastosatic problem solved in this paper is of an isotropic homogeneous infinite plate, with two arbitrarily oriented cracks of different lengths, subjected to uniform uniaxial tension at infinity. The problem is formulated in the complex plane using the Kolossoff-Muskhelishvili stress functions and further the Schwarz's alternating method is used to solve the problem of the doubly connected region. The mode I and mode II stress intensity factors at all the four crack tips for various crack length ratios, crack angles and crack spacings are found, and are in good agreement with those obtained by other research workers. The fracture angles at the four crack tips are evaluated using the strain energy density theory and maximum tangential stress theory. The minimum strain energy density factor is also found at all the tips.     

Crack problem in two-dimensional elasticity theory

Summary In this paper the stresses and displacement round a crack under various conditions have been investigated. The appropriate complex potentials are derived for four types of loading. These are: constant normal pressure on both sides of the crack with zero shear on the crack with zero principal stress at infinity, shear round the crack with zero finite stress at infinity, simple tension at infinity at a given angle to x-axis, shear at infinity. The case when the crack is subject to shear which is the case for nucleation of slip in fundamental metallurgical problem has been studied in detail. The problem has been solved after converting it into a Hilbert's Problem. Curves showing ischromatic lines (lines of constant stress) are sketched in.