Degenerate scale problem for plane elasticity in a multiply connected region with outer elliptic boundary

This paper investigates the degenerate scale problem for plane elasticity in a multiply connected region with an outer elliptic boundary. Inside the elliptic boundary, there are many voids with arbitrary configurations. The problem is studied on the relevant homogenous boundary integral equation. The suggested solution is derived from a solution of a relevant problem. It is found that the degenerate scale and the non-trivial solution along the elliptic boundary in the problem are same as in the case of a single elliptic contour without voids. The present study mainly depends on integrations of several integrals, which can be integrated in a closed form.     

Solution of the Reynolds Equation for Viscous-Fluid Flow in the Clearance between a Rotating Ellipsoid and a Surface

Slow viscous-fluid flows in the narrow clearance (i) between a moving ellipsoid and a straight tube of elliptic cross section and (ii) between a rotating ellipsoid and a toroidal tube, including the case of an ellipsoid near a plane, are considered. A solution of the boundary-value problem for the Reynolds equation describing the flow in the clearance is found. The similarity of the pressure profiles in the “ellipsoid-plane” and “ cylinder-plane” systems is indicated.     

Some Inverse Problems of Deformation and Fracture of Physically Nonlinear Inhomogeneous Media

The following two types of physically nonlinear inhomogeneous media are considered: linear-elastic plane with nonlinear-elastic elliptic inclusions and linear-viscous plane with elliptic inclusions from a material that possesses nonlinear-creep properties. The problem is to determine infinitely distant loads that produce a required value of the principal shear stress (in the first case) or principal shear-strain rate (in the second case) for two arbitrary inclusions. Conditions for the existence of solutions of these problems for incompressible media under plane strains are obtained.     

Spatial Estimates for the Constrained Anisotropic Elastic Cylinder

This paper establishes spatial estimates in a prismatic (semi-infinite) cylinder occupied by an anisotropic homogeneous linear elastic material, whose elasticity tensor is strongly elliptic. The cylinder is maintained in equilibrium under zero body force, zero displacement on the lateral boundary and pointwise specified displacement over the base. The other plane end is subject to zero displacement (when the cylinder is finite, say). The limiting case of a semi-infinite cylinder is also considered and zero displacement on the remote end (at large distance) is not assumed in this case. A first approach is developed by considering two mean-square cross-sectional measures of the displacement vector whose spatial evolution with respect to the axial variable is studied by means of a technique based on a second-order differential inequality. Conditions on the elastic constants are derived that show the cross-sectional measures exhibit alternative behaviour and in particular for the semi-infinite cylinder that there is either at least exponential growth or at most exponential decay. A second approach considers cross-sectional integrals involving the displacement and its gradient and furnishes information upon the spatial evolution, without restricting the range of strongly elliptic elastic constants. Such models are principally based upon a first-order differential inequality as well as on one of second order. The general results are explicitly presented for transversely isotropic materials and graphically illustrated for a cortical bone.     

Internal-strain measurements of longitudinal pulses in conical bars

Longitudinal stress waves in a truncated 20-deg solid cone were investigated using embedded semiconductor strain gages. The cone, composed of an aluminum-filled epoxy, was struck normally at its small end with a 1/2-in.-diam steel ball traveling at a velocity of 170 jps. The results show the magnitude of the resulting stress wave to be nonuniform over a plane cross section perpendicular to the cone axis, the strain being greater at the center of the cone than near the surface, and the nonuniformity to increase with distance of travel from the impact end. The surface-strain measurements were compared with the one-dimensional theory of longitudinal waves in cones developed by Landon and Quinney as solved by Kenner and Goldsmith for a onehalf cycle sine-squared input pulse, and found to be in qualitative agreement with this theory, but to vary significantly in strain magnitude due to the strain nonuniformity over plane cross sections. The nonuniformity was compared with the Pochhammer-Chree theory for stress waves in cylindrical bars when that theory was evaluated for a cross section equivalent to the cone cross section. The trends of the deviations were similar, but the variations measured in the cone were consistently greater than that predicted by the theory.     

Fundamental solution for bonded materials with a free surface parallel to the interface. Part II: Solutions of concentrated force acting at the interior of the substrate and the case when the force acting at the interface

In this work, the exact analyses are presented for the plane problem of a coating material subjected to a concentrated force acting at the interior of the substrate and the case when the force at the interface. The stress functions are constructed as an infinite series form by utilizing the method of image. According to the orders of the image points from lower to higher, the terms in the stress functions series have the recursive relationships. For the case when the force acting at the substrate, the first two terms are the original stress functions for a homogenous infinite plane subjected to a concentrated force, which are known and simple. For the case when the force acting at the interface, the fundamental solution is obtained for two bonded dissimilar semi-infinite plane. The stress functions in this solution can be used as the first two terms for the problem considered in this paper. Therefore, all other terms can be derived by the recurrence equations explicitly. Also, through comparisons between the theoretical results and the numerical results by FEM, it is verified that the convergence rate of the solutions is very rapid. In most practical cases only the first several image points can ensure the solutions with satisfactory accuracy.     

Bifurcation and stability analysis of laminar flow in curved ducts

The development of viscous flow in a curved duct under variation of the axial pressure gradient q is studied. We confine ourselves to two‐dimensional solutions of the Dean problem. Bifurcation diagrams are calculated for rectangular and elliptic cross sections of the duct. We detect a new branch of asymmetric solutions for the case of a rectangular cross section. Furthermore we compute paths of quadratic turning points and symmetry breaking bifurcation points under variation of the aspect ratio γ (γ=0.8…1.5). The computed diagrams extend the results presented by other authors. We succeed in finding two origins of the Hopf bifurcation. Making use of the Cayley transformation, we determine the stability of stationary laminar solutions in the case of a quadratic cross section. All the calculations were performed on a parallel computer with 32×32 processors. Copyright © 2009 John Wiley & Sons, Ltd.     

An image treatment of elastostatic transmission from an interface layer

A basic theorem for representing the Airy stress function for two perfectly bonded semi-infinite planes in terms of the corresponding Airy function for the unbounded homogeneous plane is applied in a systematic stepwise fashion to generate the corresponding Airy stress function for a three-phase composite comprising two semi-infinite planes separated by a thick layer. The loading of the three-phase composite is arbitrary, and may be in or near the interface layer. The basic theorem is first illustrated by applying it to an elastic medium which is bounded by two unloaded straight edges which intersect at an angle π/n, where n is a positive integer. This example illustrates a case of a finite system of images, while the plane-layered medium problem leads to an infinite series of images.     

Bifurcation phenomena in the plane compression test

A basic, compression, bifurcation problem is studied by methods similar to those used by R. Hill and J. W. Hutchinson (1975) for the corresponding tension problem. Bifurcations from a state of homogeneous in-plane compression loading are investigated for a rectangular block of incompressible material constrained to undergo plane deformations. The sides of the block are tractionfree, and it is loaded compressively by a uniform, shear-free, relative displacement of its ends. For a wide class of incrementally-linear time-independent materials only two instantaneous moduli enter into the analysis. Diffuse modes of both symmetric and antisymmetric bifurcation are examined in the elliptic regime of the governing equations, and the possibility of localized modes is considered both inside and outside this regime. Lowest bifurcation stresses are computed for essentially the entire range of possible combinations of material properties and geometry, and these are compared with results obtained by Hill and Hutchinson for the tension problem. The limiting case of large thickness (the semi-infinite block) is considered, confirming the results of M. A. Biot (1965).     

Saint-Venant end effects for plane deformations of linear piezoelectric solids

The recent developments in smart structures technology have stimulated renewed interest in the fundamental theory and applications of linear piezoelectricity. In this paper, we investigate the decay of Saint-Venant end effects for plane deformations of a piezoelectric semi-infinite strip. First of all, we develop the theory of plane deformations for a general anisotropic linear piezoelectric solid. Just as in the mechanical case, not all linear homogeneous anisotropic piezoelectric cylindrical solids will sustain a non-trivial state of plane deformation. The governing system of four second-order partial differential equations for the two in-plane displacements and electric potential are overdetermined in general. Sufficient conditions on the elastic and piezoelectric constants are established that do allow for a state of plane deformation. The resulting traction boundary-value problem with prescribed surface charge is an oblique derivative boundary-value problem for a coupled elliptic system of three second-order partial differential equations. The special case of a piezoelectric material transversely isotropic about the poling axis is then considered. Thus the results are valid for the hexagonal crystal class 6mm. The geometry is then specialized to be a two-dimensional semi-infinite strip and the poling axis is the axis transverse to the longitudinal direction. We consider such a strip with sides traction-free, subject to zero surface charge and self-equilibrated conditions at the end and with tractions and surface charge assumed to decay to zero as the axial variable tends to infinity. A formulation of the problem in terms of an Airy-type stress function and an induction function is adopted. The governing partial differential equations are a coupled system of a fourth and third-order equation for these two functions. On seeking solutions that exponentially decay in the axial direction one obtains an eigenvalue problem for a coupled system of fourth and second-order ordinary differential equations. This problem is the piezoelectric analog of the well-known eigenvalue problem arising in the case of an anisotropic elastic strip. It is shown that the problem can be uncoupled to an eigenvalue problem for a single sixth-order ordinary differential equation with complex eigenvalues characterized as roots of transcendental equations governing symmetric and anti-symmetric deformations and electric fields. Assuming completeness of the eigenfunctions, the rate of decay of end effects is then given by the real part of the eigenvalue with smallest positive real part. Numerical results are given for PZT-5H, PZT-5, PZT-4 and Ceramic-B. It is shown that end effects for plane deformations of these piezoceramics penetrate further into the strip than their counterparts for purely elastic isotropic materials.     

Theory of electrostatic focusing of intense beams of charged particles

The mathematical formulation of the problem of determining the electrodes for the formation of intense beams of charged particles reduces to the solution of the Cauchy problem for the Laplace equation. One can proceed either by separating the variables [1] or on the basis of the theory of analytic continuation [2–5], This approach can be used for plane or axisymmetric flows. An algorithm for the construction of the analytic solution, which can also be used in the threedimensional case, is given below. It is assumed that the beam boundary coincides with the coordinate surface x¹=0 of an orthogonal system xⁱ (i=1,2,3). The solution is put in the form of a series in x¹ with coefficients dependent on x² and x³, determined from recurrence relations. The case of emission limited by space charge and temperature generally gives rise to difficulties due to the divergence of the series which makes it impossible to calculate the zero equipotential by the indicated method.As an example, the formation of beams with an elliptic cross section is considered in the following cases: (1) periodic variation of the z- component of the velocity; (2) nonmonotonic variation of the potential in one-dimensional flow between planes z=const; (3) a beam accelerated in accordance with a 3/2 law.In the construction of the expansions the conditions on the boundary are satisfied exactly by the first two terms of the series.     

Acoustic scattering of a plane wave by a circular penetrable cone

The paper is devoted to the diffraction of a plane wave by an acoustically transparent semi-infinite cone. The problem of diffraction is reduced to a singular integral equation in the framework of the incomplete separation of variables. The Fredholm property and regularization of the integral equation are discussed. Some important integral representations of the wave field are considered. The detailed study of the far-field asymptotics is given. Expressions for the diffraction coefficient of the spherical wave scattered from the vertex of the cone are considered. The reflected from the conical surface waves and those transmitted across the surface are also discussed.     

Three-dimensional instability of an elliptic Kirchhoff vortex

The instability of a Kirchhoff vortex [1–3] with respect to three-dimensional perturbations is considered in the linear approximation. The method of successive approximations is applied in the form described in [4–6]. The eccentricity of the core is used as a small parameter. The analysis is restricted to the calculation of the first two approximations. It is shown that exponentially increasing perturbations of the same type as previously predicted and observed in rotating flows in vessels of elliptic cross section [4–9] appear even in the first approximation. As distinct from the case of plane perturbations [1-3], where there is a critical value of the core eccentricity separating the stable and unstable flow regimes, instability is predicted for arbitrarily small eccentricity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 40–45, May–June, 1988.     

Scattering of elastic waves by an inclusion with an interface crack

A method of perturbation is used to derive an integral representation of the displacement field for the scattering of a plane wave from an inclusion with an interface crack. In the long-wave approximation it is shown that the solution of only an associated static problem is required and formal expressions are derived for the scattered far field amplitudes and scattering cross section. In the case of a cylindrical inclusion the solution of the associated static problem is then used to find in a closed form the corresponding expressions for plane incident P- and S-waves.     

Shock tensors and shock polars

A refraction law for the velocity at an oblique shock in a compressible fluid is derived in dyadic form similar to that for refraction of light rays at an interface. The shock tensor embodies only the assumptions of conservation of mass and equality of tangential velocity components. Given the shock inclination and density ratio, a quadratic equation in the ratio of the flow speeds can be found with flow turning angle as a parameter. Analysis of the two solutions shows that they lie on a circle in the polar plane, a result independent of the equation of state or other conservation laws. If the density ratio is allowed to vary, a pencil of circles is generated in the hodograph plane ; or, equivalently a right, elliptic cone with two nappes appears in the three-space formed when the density ratio coordinate is added at right angles to the hodograph plane. The further requirements that momentum and energy be conserved taken together with weak restrictions on the functional form of the equation of state are sufficient to permit the development of a general theory of shock polars. The allowed shock states are seen to lie on the space curve formed by intersection of a surface called the Hugoniot cylinder with the elliptic cone. The projection of this space curve on the hodograph plane is the shock polar. The theory is applied to the special case of a polytropic gas by way of illustration.     

The seismic pulse in a semi-infinite medium

Summary In a foregoing paper the present author developed methods for studying the transient field from a vertical electric antenna placed in the vicinity of the plane boundary of two semi-infinite dielectric media.As the theory involved is applicable to the comparable elastodynamic pulse problem the present paper deals with the field from a buried transient longitudinal source in an elastic half space.The method appears to be relatively simple and is also applicable to the more general problem in which two elastic semi-infinite solids are separated by a plane boundary.     

On the Convergence to Equilibrium for Degenerate Transport Problems

We give a counterexample which shows that the asymptotic rate of convergence to the equilibrium state for the transport equation, with a degenerate cross section and in the periodic setting, cannot be better than t ^?1/2 in the general case. We suggest, moreover, that the geometrical properties of the cross section are the key feature of the problem and impose, through the distribution of the forward exit time, the speed of convergence to the stationary state.     

Three-dimensional laminar boundary layer on a blunt body

We consider the Prandtl laminar boundary layer which occurs with stationary flow about a blunted cone at an angle of attack. The solution of the Prandtl equations is sought using a finite difference method. It is found that a smooth solution of the problem exists only in the vicinity of the rounded nose of the body, while far from the nose the solutions acquire a singularity; in the problem symmetry plane (on the downwind side) there is a discontinuity of the first derivatives of the velocity components and the density. In the study of the Prandtl boundary layer in the problem of stationary flow about a pointed cone at an angle of attack, it has been shown [1] that the self-similar solution (dependent on two independent variables) of the Prandtl equations has a discontinuity of the first derivatives in the problem symmetry plane (on the downwind side of the cone). The suggestion has been made that in the three-dimensional problem of flow about a blunt cone at an angle of attack the solutions of the Prandtl equations may also be discontinuous. The present study was carried out to clarify the nature of the behavior of the solutions of the three-dimensional Prandtl equations. To this end we considered stationary supersonic flow of an ideal gas past a blunted cone. The results of this study (as well as those of [1]) were obtained using a numerical, finite-difference method. However, an analysis of the numerical results (investigation of the scheme stability, reduction of step size, etc.) shows that the properties of the solutions of the finite-difference equations are not in this case a result of numerical effects, but reflect the behavior of the solutions of the differential equations. The mathematical problem on the boundary layer which is considered in this study will be formulated in §2; this formulation is due to K. N. Babenko.     

应力脉冲在变截面SHPB锥杆中的传播特性

为研究混凝土材料的动态力学性能 ,将原有的SHPB装置改装成直锥变截面式SHPB。系统分析了应力脉冲在直锥变截面杆中的传播特性 ,讨论了大小端杆径、过渡段长度以及锥角等对波传播的影响 ,为合理设计直锥变截面式SHPB装置提供了理论依据。     

基于局部偏转吻切方法的多级压缩乘波体设计

乘波体因优异的气动特性,被认为是突破现有"升阻比屏障"的有效途径之一,已成为高超声速飞行器气动设计的研究热点.针对常规单级压缩乘波前体压缩量不足的问题,基于局部偏转吻切方法提出一种多级压缩乘波体设计方法,实现了多道非轴对称激波的逆向乘波设计.通过引入多道非轴对称激波,可充分发挥乘波前体的预压缩效果,并为复杂外形条件下的...