Temperature distributions in a periodically stratified layer with a vertical dug well

The stationary problems of heat conduction in a periodically stratified layer with a vertical cylindrical hole is considered. The lateral surface of the hole is kept at a given piece-wise constant temperature. The boundary planes at zero temperature are taken into account, and the ideal thermal contact between the composite constituents is assumed. The problem within the homogenized model with microlocal parameters is solved.     

On axisymmetrical problem of layer with temperature dependent properties

The paper deals with the axisymmetrical problem of thermoelastic layer with mechanical properties dependent on temperature. The boundary planes of the body are kept at constant but different temperatures. Moreover, the layer is assumed to be ideal fixed to a rigid foundation. The upper boundary plane is loaded by normal forces dependent on the radius. The considered stationary problem is solved according with the following scheme: (1⁰) firstly the distribution of temperature is found, (2⁰) secondly, assuming that the Young modulus is a power function of temperature and Poisson ratio is constant, the displacements and stress are calculated from adequate boundary value problem. The obtained results in the form of Hankel integrals are analysed numerically for the case of linear dependence of Young modulus on the temperature.     

Action of an elliptic punch on a transversally isotropic half-space

The 3D contact problem on the action of a punch elliptic in horizontal projection on a transversally isotropic elastic half-space is considered for the case in which the isotropy planes are perpendicular to the boundary of the half-space. The elliptic contact region is assumed to be given (the punch has sharp edges). The integral equation of the contact problem is obtained. The elastic rigidity of the half-space boundary characterized by the normal displacement under the action of a given lumped force significantly depends on the chosen direction on this boundary. In this connection, the following two cases of location of the ellipse of contact are considered: it can be elongated along the first or the second axis of Cartesian coordinate system on the body boundary. Exact solutions are obtained for a punch with base shaped as an elliptic paraboloid, and these solutions are used to carry out the computations for various versions of the five elastic constants. The structure of the exact solution is found for a punch with polynomial base, and a method for determining the solution is proposed.     

边界膜剪切弹性模量对于微接触性能的影响

建立接触模型,理论分析了微接触中边界膜剪切弹性模量对于接触性能的影响. 接触区由两平行平面形成,属一维接触. 上接触表面为粗糙表面,具有矩形微凸体. 下接触表面为光滑平面. 两接触表面均处理成刚性表面. 微接触区中充满流体. 它分成两个子区,在微接触的出口区由于极小的接触间隙充满边界膜,在微接触的入口区由于接触间隙较大充满流体膜. 边界膜和流体膜行为决定整个微接触性能. 当膜厚较大时,这里边界膜可看成纳米级薄膜. 由于上接触表面处有限的剪应力承受能力,边界膜可于上接触表面滑移. 设下接触表面处剪应力承受能力很大而边界膜在下接触表面不滑移. 由于边界膜-接触表面间相互作用,边界膜黏度、密度和剪切弹性模量均沿膜厚变化,在理论分析中使用它们的等效值,这些值与边界膜厚度有关. 流体膜在两个接触表面均不发生滑移,分析中不考虑流体膜剪切弹性模量. 流体膜采用传统分析法. 给出了理论分析和若干变工况参数下的计算结果.      

Mode III stress intensity factors for edge-cracked circular shafts,bonded wedges,bonded half planes and DCB’s

Antiplane shear deformation of several edge-cracked geometries is considered. Analytical expressions are derived for the mode III stress intensity factor (SIF) of circular shafts with edge cracks, bonded half planes containing an interfacial edge crack, bonded wedges with an interfacial edge crack and also DCB’s. The results are extracted for simple isotropic materials as well as anisotropic materials and also bonded dissimilar materials and it is shown that the same expressions are obtained for the SIF under the same geometries but with different above-mentioned material properties. Different boundary conditions are assumed and the SIF relations are derived in each case. As the special cases, the SIF’s of the two bonded quarter planes containing an edge crack at the interface and infinite strip with a semi-infinite edge crack are extracted which coincide with the results cited in the literature.     

Interfacial fracture analysis of bonded dissimilar strips with a functionally graded interlayer under antiplane deformation

This paper provides the solution to the problem of dissimilar, homogeneous semi-infinite strips bonded through a functionally graded interlayer and weakened by an embedded or edge interfacial crack. The bonded system is assumed to be under antiplane deformation, subjected to either traction-free or clamped boundary conditions along its bounding planes. Based on the Fourier integral transform, the problem is formulated in terms of a singular integral equation which has a simple Cauchy kernel for the embedded crack and a generalized Cauchy kernel for the edge crack. In the numerical results, the effects of geometric and material parameters of the bonded system on the crack-tip stress intensity factors are presented in order to quantify the interfacial fracture behavior in the presence of the graded interlayer.     

Effect of a permeable partition on convective instability of a horizontal liquid layer

We consider the effect of a thin permeable partition on the static stability of a horizontal liquid layer heated from underneath. The permeable partition is assumed to be plane and situated parallel to the boundary planes in the center of the layer. The resistance of the partition to the flow of liquid from one part of the layer to another leads to an increase in the static stability. We investigate the dependence of the minimum critical Rayleigh number-on the resistance of the partition and the form of the critical motions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 157–159, January–February, 1977.     

Doubly curved shallow shells with rectangular base elastically supported by edge arched beams and tie-rods (I)

The equations of equilibrium of shallow shells with rectangular base elastically supported with edge arched beams are obtained through the variational principle together with corresponding boundary conditions and corner conditions. It is assumed that edge arched beams are of narrow plate form, so that only the rigidities in their own planes are taken into consideration, torsional rigidities and bending rigidities out of their own planes are neglected. In this paper, two kinds of corner conditions are discussed. First of these is pinned corner conditions. Second of these is simply supported corner conditions, such that the corner can be moved freely in horizontal directions. The former corresponds to the conditions of those with heavy tension beams, in which the tension rigidities of the rods can be assumed infinite. The latter corresponds to the conditions of elastically supported edge arched beams without tension rods. In the former case, the edge tangential displacement of shallow shells is assumed to be zero everywhere, so that the vertical displacement of the edge arched beams gives the only elastic supported forces. This kind of supporting conditions is a good approximation for practical roof design.In this paper, the solutions of the problem of shallow spherical shell of square base supported elastically by edge arched beams and tie-rods under the conditions, such that the corners are restricted, are solved by the method of double trigonometric series. The edge conditions are integrated along their respective edge, and the conditions at corner are satisfied by proper choices of integral constants. The integrated edge conditions are then used to determine the unknown constants in the double trigonometric series. The result of this paper gives the tension in the tie-rods directly, which is an important quantity in the shell roof design practice.The method of integrated form of boundary conditions used in this paper in general is useful for the treatment of problem of plates and shells elastically supported by edge frames and tie-rods or by other means.This paper also gives the results of numerical calculation based upon the method of double trigonometric series on the problem of shallow spherical shell with square bases elastically supported by arched beams. The corner are pinned supported or simply supported. The calculated results for =11.5936 show that the trigonometric series converges rapidly. The effect of elastic deformation in the arched beams to the components of membrane tensions, moments, and deflections of the shell are given.     

Numerical simulation of two-dimensional convection: Role of boundary conditions

The problem of two-dimensional, periodic in the horizontal coordinate, convection of an incompressible fluid heated from below between two horizontal planes is considered. The problem is solved in two formulations: with (stress-)free and hard (no-slip) boundary conditions on the horizontal planes. It is shown that at small supercriticalities the two-dimensional convection calculation leads to more correct results with hard than with free boundary conditions. It is established that the difference between the free and hard conditions is most strongly manifested in the pulsations of the vertical velocity component, whereas the dependence of the Nusselt number and the pulsations of the horizontal velocity component on the boundary conditions is more weakly expressed.     

Laminar boundary layer on an oscillating wedge

A study is made of the nonstationary laminar boundary layer on a sharp wedge over which a compressible perfect gas flows; the wedge executes slow harmonic oscillations about its front point. It is assumed that the perturbations due to the oscillations are small, and the problem is solved in the linear approximation. It is also assumed that the thickness of the boundary layer is small compared with the thickness of the complete perturbed region. Then in a first approximation the influence of the boundary layer on the exterior inviscid flow can be ignored, and the parameters on the outer boundary of the boundary layer can be taken equal to their values on the body for the case of inviscid flow over the wedge. They are determined from the solution to the inviscid problem that is exact in the framework of the linear formulation. The wall is assumed to be isothermal. The dependence of the viscosity on the temperature is linear. Under these assumptions, the problem of calculating the nonstationary perturbations in the boundary layer on the wedge is a self-similar problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 146–151, July–August, 1980.     

Analytical solutions to the 2D elasticity and thermoelasticity problems for inhomogeneous planes and half-planes

This paper presents a method for analytical solving the plane elasticity and thermoelasticity problems for planes and half-planes which exhibit inhomogeneous material properties in one of the planar directions (in the case of half-plane, the inhomogeneity direction is assumed to be perpendicular to the boundary.) With the aid of direct integration of equilibrium equations, the original problems are reduced to the set of governing, harmonic equations with corresponding conditions. In the transform domain of the integral Fourier-transform, the governing equations are reduced to an integral equation which is solved by simple iteration technique. The rapid convergence of the iterative procedure is established to perform the numerical calculation. The authors gratefully acknowledge the financial support of this research by the National Science Council (Republic of China) under Grant NSC 95-2221-E-002-048-MY2. The co-author, Yu Tokovyy, gratefully acknowledges the particular financial support of this research by the grant of the President of Ukraine for young scientists (No GP/F13/0094).     

Boundary linear integral method for compressible potential flows

A boundary linear integral method based on Green function theory has been developed to solve the full potential equation for subsonic and transonic flows. In this integral method, potential values in the flow region are determined by potential values represented by boundary integrals and a volume integral. The boundary potential values are obtained by implementing the boundary integrals along boundary segments where a linear potential relation is assumed. The volume integral is evaluated in a grid generated by finite element discretization. The volume integral is evaluated only outside the body. Therefore there is no extra boundary treatment required for evaluation of the volume integral. The source term is assumed to be constant in an element integral volume. The volume integral needs to be evaluated only once and can be stored in computer memory for further usage.     

The unsteady boundary layer flow in a convergent channel

An incompressible fluid is at rest in a converging channel between intersecting planes when, at some moment of time, the flow towards a sink at the point of intersection is set up impulsively. The motion is taken to be of the boundary layer type, and it is found that the transient part of the flow decays algebraically as the steady state motion becomes dominant. The final decay takes place at the edge of the boundary layer, with the viscous forces affecting the unsteady flow in a region that eventually lies outside the steady state boundary layer.     

Three-dimensional features of a Mach 2.1 shock/boundary layer interaction

2D particle image velocimetry was used to study the three-dimensionality of the shock-boundary layer interaction generated by a small 20° compression ramp in a low aspect ratio continuously operated wind tunnel. High-resolution data were taken in four streamwise-wallnormal planes: three planes located in the sidewall boundary layer and one near the tunnel centerline. The incoming boundary layer was found to show three-dimensionality, with significant overshoot in the velocity profiles observed near the sidewall. The size of the wedge influenced the interaction, which was weaker than that observed in the case of a large compression wedge. The flow turning angle was ≈8° near the tunnel centerline and changed significantly across the span. Measurements behind the compression wedge in the centerline plane showed that both velocity and turbulence properties were nearly fully recovered ≈14δ behind the compression corner. The shock angle varied with spanwise position, and a multi-shock structure was observed in the sidewall planes. The size of the interaction decreased in the sidewall boundary layer. Non-monotonic variations in both velocity and turbulence profiles across the sidewall planes suggest the presence of significant spanwise flows, possibly corner vortices.     

Fundamental solutions to contact problems of a magneto-electro-elastic half-space indented by a semi-infinite punch

This paper presents the fundamental contact solutions of a magneto-electro-elastic half-space indented by a smooth and rigid half-infinite punch. The material is assumed to be transversely isotropic with the symmetric axis perpendicular to the surface of the half-space. Based on the general solutions, the generalized method of potential theory is adopted to solve the boundary value problems. The involved potentials are properly assumed and the corresponding boundary integral equations are solved by using the results in literature. Complete and exact fundamental solutions are derived case by case, in terms of elementary functions for the first time. The obtained solutions are of significance to boundary element analysis, and an important role in determining the physical properties of materials by indentation technique can be expected to play.     

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.     

Investigation on the choice of boundary conditions and shape functions for flexible multi-body system

The objective of this investigation is to examine the correctness and efficiency of the choice of boundary conditions when using assumed mode approach to simulate flexible multi-body systems. The displacement field due to deformation is approximated by the Rayleigh-Ritz assumed modes in floating frame of reference (FFR) formulation. The deformations obtained by the absolute nodal coordinate (ANC) formulation which are transformed by two sets of reference coordinates are introduced as a criterion to verify the accuracy of the simulation results by using the FFR formulation. The relationship between the deformations obtained from different boundary conditions is revealed. Numerical simulation examples demonstrate that the assumed modes with cantilevered-free, simply-supported and freefree boundary conditions without inclusion of rigid body modes are suitable for simulation of flexible multi-body system with large over all motion, and the same physical deformation can be obtained using those mode functions, differ only by a coordinate transformation. It is also shown that when using mode shapes with statically indeterminate boundary conditions, significant error may occur. Furthermore, the slider crank mechanism with rigid crank is accurate enough for investigating boundary condition problem of flexible multi-body system, which cost significant less simulating time.     

Analysis of shear deformable plates with combined geometric and material nonlinearities by boundary element method

In this paper a boundary element formulation for analysis of shear deformable plates with combined geometric and material nonlinearities by boundary element method is presented. The dual reciprocity method is used in dealing with the geometric nonlinearity and domain discretization is implemented in dealing with material nonlinearity. The material is assumed to undergo large deflection with small strains. The von Mises criteria is used to evaluate the plastic zone and an elastic perfectly plastic material behaviour is assumed. An initial stress formulation is used to formulate the boundary integral equations. A total incremental method is applied to solve the nonlinear boundary integral equations. Numerical examples are presented to demonstrate the validity and the accuracy of the proposed method.     

Pressure field around a well in a stratified inhomogeneous bed

The steady axisymmetric flow of an incompressible fluid into a vertical well hydrodynamically perfectly drilled into a stratified inhomogeneous half-space consisting of three layers with different permeabilities is considered. The boundaries of the layers are assumed to be horizontal planes and the roof of the upper layer is assumed to be impermeable. The flow obeys a linear Darcy’s law. The pressure distribution on the well is assumed to be given, which is the main obstacle to finding an exact solution of the problem. Beginning with the classical studies of Muskat and Charnyi [1, 2], approximate solutions of such problems have been constructed as a superposition of flows generated by point sources with given intensities, distributed along the well axis in accordance with a fairly simple law. In the present study, this approach is used to obtain an integral equation for the source density distribution, which is then solved numerically. Comparison with the known exact solution for a thin elongated ellipsoid (“needle”) shows that this approach makes it possible to ensure an accuracy which at any rate is sufficient for applications.     

Control functions and grid qualities measurements in the elliptic grid generation around arbitrary surfaces

This paper describes the three‐dimensional elliptic grid generation. The two‐dimensional approach for the control functions obtained by modifying the Thomas–Middlecoff method is applied on the planes perpendicular to the main flow direction co‐ordinate, which is assumed to be the function of only one corresponding co‐ordinate in the computational domain. The grid orthogonality is improved by about 40 per cent compared with that of the algebraic initial grid. Copyright © 2000 John Wiley & Sons, Ltd.