Scattering of elastic waves by a small inclined surface-breaking crack

I we examine the scattering of Rayleigh waves by an inclined two-dimensional plane surface-breaking crack in an isotropic elastic half-plane. We follow the method already introduced by the authors (A and W , 1992a, J. Mech. Phys. Solids 40, 1683) to obtain an analytical solution when the parameter , which is the ratio of a typical length scale of the imperfection to the incident radiation's wavelength, is small. The procedure employed is the method of matched asymptotic expansions, which involves determining the scattered wave field both away from and near the crack. The outer solution is specialized from the general expansion in the first part of this work (A and W , 1992a, J. Mech. Phys. Solids 40, 1683), and the inner problem is exactly solved by the Wiener-Hopf technique. The displacement field and scattered Rayleigh waves are found uniformly to third order in , and concluding remarks are made about the general method as well as the results presented here.     

Asymptotic models for optimizing the contour of multiply-connected cross-section of an elastic bar in torsion

An asymptotic solution is obtained for the problem of maximizing the torsional rigidity of elastic, multiply-connected cylindrical bars for a given area of cross-section. The shapes of the inner contours of the multiply-connected cross-section are specified while the outer contour is determined as a result of the shape optimization. We apply the method of matched asymptotic expansions to construct a first-order asymptotic model. The conditions for unique solvability of the asymptotic model have been established under some restrictions imposed on the location of the inner contours and their polarization matrices. The economy achieved by optimization is estimated.     

Asymptotic study of viscous compressible gas flow past a blunt body

Supersonic viscous gas flow past a blunt body is examined. A method is proposed which permits constructing the asymptotic expansion of any order in the small parameter , which characterizes the viscosity and thermal conductivity coefficients. The asymptotic solution is constructed, including terras of zero, first, and second orders of . Acomparison is made with results of other authors who have studied various particular aspects of the subject problem using the method of inner and outer expansions [1–3].     

Higher-order effects in natural convection flow over uniform flux inclined flat plates in porous media

Higher-order boundary-layer effects for natural convection flow along inclined flat plates (of both positive and negative inclinations) with a uniform heat flux surface condition in a saturated porous medium are studied. Using the method of matched asymptotic expansions the three terms in inner and outer expansions have been obtained. It is shown that the first eigenfunction for this considered problem coincides with 0(? ²)-term in the inner expansion.     

Approximate Solutions of the Cahn-Hilliard Equation via Corrections to the Mullins-Sekerka Motion

We develop an alternative method to matched asymptotic expansions for the construction of approximate solutions of the Cahn-Hilliard equation suitable for the study of its sharp interface limit. The method is based on the Hilbert expansion used in kinetic theory. Besides its relative simplicity, it leads to calculable higher order corrections to the interface motion. An erratum to this article is available at .     

Higher approximations to the free convection flow from a heated vertical flat plate

This paper is concerned with the problem of obtaining higher approximations for the free convection from a heated vertical flat plate to that represented by the well known solution of Schmidt and Beckmann. For large Grashof number, the perturbation problem is a singular one and the method of matched asymptotic expansions is used to construct inner and outer expansions for the velocity and temperature distributions. The small perturbation parameterε is the inverse of the fourth root of the Grashof number and the expansions are shown to involve only integral powers ofε. The first three terms in the expansion are calculated and numerical results are presented for the velocity, temperature, skin friction and heat transfer. The agreement with experiment is found to be excellent, and the theory fully explains the discrepancies which exist between boundary layer theory and experiment.     

The laminar boundary layer near a body corner point

A method is developed for calculating the characteristics of a laminar boundary layer near a body contour corner point, in the vicinity of which the outer supersonic stream passes through a rarefaction flow. In the study we use the asymptotic solution of the Navier-Stokes equations in the region with large longitudinal gradients of the flow functions for large values of the Reynolds number, the general form of which was used in [1].The pressure, heat flux, and friction distributions along the body surface are obtained. For small pressure differentials near the corner the solution of the corresponding equations for small disturbances is obtained in analytic form.The conventional method for studying viscous gas flow near body surfaces for large values of the Reynolds number is the use of the Prandtl boundary layer theory. Far from the body the asymptotic solution of the Navier-Stokes equations in the first approximation reduces to the solution of the Euler equations, while near the body it reduces to the solution of the Prandtl boundary layer equations. The characteristic feature of the boundary layer region is the small variation of the flow functions in the longitudinal direction in comparison with their variation in the transverse direction. However, in many cases this condition is violated.The necessity arises for constructing additional asymptotic expansions for the region in which the longitudinal and transverse variations of the flow functions are quantities of the same order. The general method for constructing asymptotic solutions for such flows with the use of the known method of outer and inner expansions is presented in [1].In the following we consider the flow in a laminar boundary layer for the case of a viscous supersonic gas stream in the vicinity of a body corner point. Behind the corner the flow separates from the body surface and flows around a stagnant zone, in which the pressure differs by a specified amount from the pressure in the undisturbed flow ahead of the point of separation. A pressure (rarefaction) disturbance propagates in the subsonic portion of the boundary layer upstream for a distance which in order of magnitude is equal to several boundary layer thicknesses. In the disturbed region of the boundary layer the longitudinal and transverse pressure and velocity disturbances are quantities of the same order. In this study we construct additional asymptotic expansions in the first approximation and calculate the distributions of the pressure, friction stress, and thermal flux along the body surface.     

Perturbation solutions for asymmetric laminar flow in porous channel with expanding and contracting walls

The cases of large Reynolds number and small expansion ratio for the asym- metric laminar flow through a two-dimensional porous expanding channel are considered. The Navier-Stokes equations are reduced to a nonlinear fourth-order ordinary differential equation by introducing a time and space similar transformation. A singular perturbation method is used for the large suction Reynolds case to obtain an asymptotic solution by matching outer and inner solutions. For the case of small expansion ratios, we are able to obtain asymptotic solutions by double parameter expansion in either a small Reynolds number or a small asymmetric parameter. The asymptotic solutions indicate that the Reynolds number and expansion ratio play an important role in the flow behavior. Nu- merical methods are also designed to confirm the correctness of the present asymptotic solutions.     

Scattering by penetrable acoustic targets

An acoustic target of constant density ?_t and variable index of refraction is imbedded in a surrounding acoustic fluid of constant density ?_a. A time harmonic wave propagating in the surrounding fluid is incident on the target. We consider two limiting cases of the target where the parameter ε ≡ ?_a/?_t → 0 (the nearly rigid target) or ε → ∞ (the nearly soft target). Wh en the frequency of the incident wave is bounded away from the ‘in-vacuo’ resonant frequencies of the target, the resulting scattered field is essentially the field scattered by the rigid target for ε = 0 or the soft target if ε → ∞. However, when the frequency of the incident wave is near a resonant frequency,the target oscillates and its interaction with the surrounding fluid produces peaks in the scattered field amplitude. In this paper we obtain asymptotic expansions of the solutions of the scattering problems for the nearly rigid and the nearly soft targets as ε → 0 or ε → ∞, respectively, that are uniformly valid in the incident frequency. The method of matched asymptotic expansions is used in the analysis. The outer and inner expansions correspond to the incident frequencies being far or near to the resonant frequencies, respectively. We have applied the method only to simple resonant frequencies, but it can be extended to multiple resonant frequencies. The method is applied to the incidence of a plane wave on a nearly rigid sphere of constant index of refraction. The far field expressions for the scattered fields, including the total scattering cross-sections, that are obtained from the asymptotic method and from the partial wave expansion of the solution are in close agreement for sufficiently small values of ε.     

Wake layer in a thermal turbulent boundary layer with pressure gradient

The open equations of thermal turbulent boundary layer subjected to pressure gradient have been analysed by method of matched asymptotic expansions at large Reynolds number. The flow is divided into outer wake layer and inner wall layer. The asymptotic expansions are matched by Millikan-Kolmogorov hypothesis. The temperature profile in overlap region yields composite law which reduce to log. law for moderate pressure gradient and inverse half power law for strong adverse pressure gradient. In case of a shallow thermal wake, the matching result of outer wake layer reduces to composite temperature defect law, which is more general than the classical log. law. The comparison of data for thermal boundary layer with strong adverse pressure gradient is also considered. Received on 26 May 1998     

A bending quasi-front generated by an instantaneous impulse loading at the edge of a semi-infinite pre-stressed incompressible elastic plate

A refined membrane-like theory is used to describe bending of a semi-infinite pre-stressed incompressible elastic plate subjected to an instantaneous impulse loading at the edge. A far-field solution for the quasi-front is obtained by using the method of matched asymptotic expansions. A leading-order hyperbolic membrane equation is used for an outer problem, whereas a refined (singularly perturbed) membrane equation of an inner problem describes a boundary layer, which smoothes a discontinuity predicted by the outer problem at the wave front. The inner problem is then reduced to one-dimensional by an appropriate choice of inner coordinates, motivated by the wave front geometry. Using the inherent symmetry of the outer problem, a solution for the quasi-front is derived that is valid in a vicinity of the tip of the wave front. Pre-stress is shown to affect geometry and type of the generated quasi-front; in addition to a classical receding quasi-front the pre-stressed plate can support propagation of an advancing quasi-front. Possible responses may even feature both types of quasi-front at the same time, which is illustrated by numerical examples. The case of a so-called narrow quasi-front, associated with a possible degeneration of contribution of singular perturbation terms to the governing equation, is studied qualitatively.     

Bending of a radially prestressed annular plate by tilting a central rigid inclusion

A thin annular plate contains a rigid, circular, central inclusion. The plate is subjected to a large axisymmetric radial load at its outer edge, where it is also restrained against transverse displacement and rotation. A couple applied to the rigid inclusion causes it to rotate about its diameter out of the plane of the plate. We use the method of matched asymptotic expansions to find an approximate expression for the applied couple as a function of the angle of rotation of the rigid inclusion. If the outer radius of the annulus is very large compared to the inner radius, then the couple required to rotate a truly rigid inclusion is 25% higher than the couple required to rotate an inclusion whose membrane strain stiffness is the same as that of the plate (cf. ref. [3]) through the same small angle.     

On axisymmetric motion of an incompressible elastic medium under impact loading

The method of matched asymptotic expansions was employed to obtain approximate solutions to the one-dimensional boundary-value problems of nonlinear dynamic elasticity theory of impact loading on the surface of a cylindrical cavity of an incompressible medium that causes antiplane motion or torsion of the medium. The expansion of the solution in the near-front region is based on solutions of evolution equations different from the equations for quasi-simple waves. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 144–151, November–December, 2006.     

The solution of large deflection problem of thin circular plate by the method of composite expansion

In this paper, the method of composite expansion in perturbation theory is used for the solution of large deflection problem of thin circular plate. In this method. the outer field solution and the inner boundary layer solution are combined together to satisfy all the boundary conditions. In this paper, Hencky’s membrane solution is used for the first approximation in outer field solution, and then the second approximate solution is obtained. The inner boundary layer solution is found on the bases of boundary layer coondinate. In this paper, the reciprocal ratio of maximum deflection and thickness of the plate is used as the small parameter. The results of this paper improves quite a bit in comparison with the results obtained in 1948 by Chien Wei-zang.     

Nonlinear aspects of high Reynolds number channel flows

This paper considers the flow in a two-dimensional channel at high Reynolds number, with wall deformations which can lead to flow separation. An asymptotic model is proposed by using the successive complementary expansion method with generalized asymptotic expansions. In particular, the model emphasizes the asymmetry of the channel geometry by introducing a change of variables. It is shown that the model is more general than the models developed with the method of matched asymptotic expansions. Comparisons with Navier–Stokes solutions show that the model is well founded and enables us to treat original problems.     

Electrically-excited (electroosmotic) flows in microchannels for mixing applications

We propose an asymptotic model for quite general liquid microchannel flows in the presence of electrical double layers (EDLs). The model provides an “inner” solution for the wall layer, which reflects the dominant balance between electrical forces and viscous forces (tangentially), respectively between electrical forces and pressure and viscous forces (normally). The electrically-neutral core of the flow is governed by the standard Navier–Stokes equations, providing the “outer” solution. The asymptotic matching of both solutions provides a method for the simplified numerical treatment of such EDLs. The superposition of the solutions in both regions then allows to infer an approximate solution, valid within the entire domain.Based on this model, we apply external oscillatory electrical fields to excite secondary flows (i) in microchannels with an internal obstacle or (ii) in folded (meander) microchannels. These secondary flows are demonstrated to greatly enhance the mixing of two liquids flowing in a layered fashion through these microchannels. Thus, electrical excitation has considerable potential if micromixers for ionic liquids are designed within electrically-insulating (e.g. plastics, glass) substrates.     

Asymptotic theory of a wing moving near a solid wall

In this study we use the method of matched asymptotic expansions to obtain an approximate solution of the problem of the nonstationary motion of a lifting surface near a solid wall. The region of flow is provisionally subdivided into characteristic zones, in which, using the appropriate coordinates, we construct asymptotic expansions for the velocity potential, which thereafter coalesce in the regions of common validity. In the first approximation (extremely small heights of flight) the problem reduces to the solution of a Poisson equation in a plane region bounded by the contour of the wing in the horizontal plane with boundary conditions established from the coalescence.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 115–124, November–December, 1977.     

Scattering of elastic waves by an arbitrary small imperfection in the surface of a half-space

T , the first of two articles, is concerned with the scattering of elastic waves by arbitrary surface-breaking or near surface defects in an isotropic half-plane. We present an analytical solution, by the method of matched asymptotic expansions, when the parameter , which is the ratio of a typical length scale of the imperfection to the incident radiation's wavelength, is small. The problem is formulated for a general class of small defects, including cracks, surface bumps and inclusions, and for arbitrary incident waves. As a straightforward example of the asymptotic scheme we specialize the defect to a two-dimensional circular void or protrusion, which breaks the free surface, and assume Rayleigh wave excitation ; this inner problem is exactly solvable by conformal mapping methods. The displacement field is found uniformly to leading order in , and the Rayleigh waves which are scattered by the crack are explicitly determined. In the second article we use the method given here to tackle the important problem of an inclined edge-crack. In that work we show that the scattered field can be found to any asymptotic order in a straightforward manner, and in particular the Rayleigh wave coefficients are given to O(²).     

Multi-Scale Asymptotic Expansion for a Small Inclusion in Elastic Media

The aim of this paper is to present an asymptotic expansion of the influence of a small inclusion of different stiffness in an elastic media. The applicative interest of this study is to provide tools which take into account this influence and correct the deformation without inclusion by additive terms that can be precalculated and which depend only on the shape of the inclusion. We treat two problems: an anti-plane linearized elasticity problem and a plane strain problem. On every expansion order we provide corrective terms modeling the influence of the inclusion using techniques of scaling and multi-scale asymptotic expansions. The resulting expansion is validated by comparing it to a test case obtained by solving the Poisson transmission problem in the case of an inclusion of circular shape using the separation of variables method. Proofs of existence and uniqueness of our fields on unbounded domains are also adapted to the bidimensional Poisson problem and the linear elasticity problem.     

Development of a distributed dislocation dipole technique for the analysis of multiple straight,kinked and branched cracks in an elastic half-plane

A distributed dislocation dipole technique for the analysis of multiple straight, kinked and branched cracks in an elastic half plane has been developed. The dipole density distribution is represented with a weighted Jacobi polynomial expansion where the weight function captures the asymptotic behaviour at each end of the crack. To allow for opening and sliding at crack kinking and branching the dipole density representation contains conditional extra terms which fulfills the asymptotic behaviour at each endpoint. Several test cases involving straight, kinked and branched cracks have been analysed, and the results suggest that the accuracy of the method is within 1% provided that Jacobi polynomial expansions up to at least the sixth order are used. Adopting even higher order Jacobi polynomials yields improved accuracy. The method is compared to a simplified procedure suggested in the literature where stress singularities associated with corners at kinking or branching are neglected in the representation for the dipole density distribution. The comparison suggests that both procedures work, but that the current procedure is superior, in as much as the same accuracy is reached using substantially lower order polynomial expansions.