Mixed Boundary Value Problem in a Non-homogeneous Medium Under Steady Distribution of Temperature

This paper is concerned with a mixed boundary value problem of a non-homogeneous medium under steady distribution of temperature whose elastic constants are exponential functions of y. By using Fourier cosine transforms the mixed boundary value problem of heat conduction is reduced to a Fredholm integral equation of the second kind. Then the elastic problem of the non-homogeneous semi-infinite half-plane under distribution of load over a plane face is solved. The influence of the non-homogeneity of the material on the thermal stress distribution is illustrated graphically.     

Scattering of sound by two parallel semi-infinite screens

A plane sound wave is incident upon two semi-infinite rigid plates, lying along y = 0, x > 0 and y = -h, x < 0, respectively, where (x, y) are two-dimensional Cartesian coordinates. The problem is formulated into a matrix Wiener-Hopf equation which is uncoupled by the introduction of an infinite sum of poles. The exact solution is then easily obtained in terms of the coefficients of the poles, where these coefficients are shown to satisfy a linear system of algebraic equations. The far-field solution is obtained and an asymptotic approximation to the total potential is determined in the limit as h, the plate spacing, becomes small compared to the wavelength of the incident wave. The algebraic system is solved numerically in this limit and the results are shown to agree with those obtained by the method of matched asymptotic expansions.     

Reissner-Sagoci problem for a homogeneous coating on a functionally graded half-space

This paper is concerned with the axisymmetric elastostatic problem related to the rotation of a rigid punch which is bonded to the surface of a nonhomogeneous half-space. The half-space is composed of an isotropic homogeneous coating in the form of layer, which is attached to the functionally graded half-space. The shear modulus of the FGM is assumed to vary in the direction of axis Oz normal to the boundary as μ₁(z) = μ₀(1 + αz)^β, where μ₀, α, β are positive constants. The punch undergoes rotation due to the action of the internal loads. By using Hankel's integral transforms, the mixed boundary value problem is reduced to dual integral equations, and next, to a Fredholm's integral equation of the second kind, which is solved numerically for the case of β = 2. The final results show the effect of non-homogeneity on the shear stresses and an unknown moment of punch rotation.     

The reflection of a symmetric Rayleigh-Lamb wave at the fixed or free edge of a plate

A semi-infinite plate of homogeneous isotropic, linearly elastic material occupies the region x≥0, |y|≤1, -∞<z<∞; the faces y=±1 are free of tractions, the end x=0 may be either fixed or traction free, and there are no body forces. A plane strain, time-harmonic, symmetric Rayleigh-Lamb wave propagates in the plate and is normally incident upon the end x=0. The problem of determining the resulting reflected wave field is solved by the “method of projection”, a method developed by the authors for solving corresponding problems in elastostatics. The solutions obtained for the dynamic problem fully satisfy the equations and boundary conditions of the linear theory, and (in the fixed-end case) proper account is taken of the singularities of the stress field at the corners x=0, y=±1. In each case the division of energy between the various reflected modes is found, and the dynamical stress intensity factors at the corners are determined in the fixed-end case. The existence of an “edge-mode” for the free-end case at a single isolated value of the frequency is confirmed, but a careful search revealed no similar phenomenon for the fixed-end case.     

The penny-shaped crack at a bonded plane with localized elastic non-homogeneity

This paper examines the axisymmetric problem pertaining to a penny-shaped crack which is located at the bonded plane of two similar elastic halfspace regions which exhibit localized axial variations in the linear elastic shear modulus, which has the form G(z)=G₁+G₂e^±ζz. The equations of elasticity governing this type of non-homogeneity are solved by employing a Hankel transform technique. The resulting mixed boundary value problem associated with the penny-shaped crack is reduced to a Fredholm integral equation of the second kind which is solved in a numerical fashion to generate the crack opening mode stress intensity factor at the tip.     

Image decomposition method for the analysis of a mixed dislocation in a general multilayer

The elastic solutions for a mixed dislocation in a general multilayer with N dissimilar anisotropic layers are obtained via a generalized image decomposition method. The original problem is decomposed into N homogeneous subproblems with strategically placed continuously distributed image (virtual) dislocations which satisfy the consistency conditions for degenerate N − M (M < N) layer problems. The image dislocations are used to satisfy the interface or free surface conditions, and represent the unknowns of the problem. The resulting singular Cauchy integral equations are transformed into non-singular Fredholm integral equations of the second kind using certain H- and I-integral transforms. The Fredholm integral equations are then solved via the classical Nyström method. The general decomposition and the elimination of all singular integrals yield an exact formulation of the problem; the approximation arises only in the Nyström method. The dislocation mixity and the number of layers dissimilar in thickness and elastic anisotropy can be handled without difficulty, constrained only by the number of linear algebraic equations in the Nyström method for large N. For the numerical study, image forces on a dislocation in two- and three-layer systems are calculated. The accuracy of the results is verified by checking the boundary conditions and by comparison with previous results. The dependence of the image force on the dislocation position and mixity, and on the layer thicknesses and elastic anisotropies, is also illustrated via numerical investigations.     

Computations of hydrodynamic coefficients, displacement-amplitude ratios and forces for a floating cylinder due to wave diffraction and radiation

Computations of the hydrodynamic coefficients, displacement-amplitude ratios and loadings on floating vertical circular cylinder due to diffraction and radiation are presented here. The boundary value problem (BVP) is solved in terms of diffraction potential and three potentials due to radiation, two translational motions about x-axis (surge) and about z-axis (heave), one rotational motion about y-axis (pitch). The analytical expressions for the hydrodynamic coefficients, displacement-amplitude ratios and loadings for this case were obtained previously by Bhatta and Rahman [1]. In this paper, we present the computational aspects of those analytical results for different depth to radius and draft to radius ratios. JMSL (Java Mathematical and Statistical Library) is used to compute special functions and solve complex matrix equations.     

Rolling of a Rigid Cylinder over a Half-Plane with Initial Stresses

The Mehler–Fock transformation method is used to study the rolling of a rigid cylinder over a half-plane with initial stresses. The problem is reduced to a system of two dual integral equations, which are then reduced to a system of two integral Fredholm equations of the second kind. The system of integral equations is solved by the method of degenerate kernels. The dependences of the normal and tangential stresses on the elongation are plotted.     

Heat Convection of Compressible Viscous Fluids: I

The stationary problem for the heat convection of compressible fluid is considered around the equilibrium solution with the external forces in the horizontal strip domain z ₀ < z < z ₀ + 1 and it is proved that the solution exists uniformly with respect to z ₀ ≥ Z ₀. The limit system as z ₀ → + ∞ is the Oberbeck–Boussinesq equations.     

Interaction between a screw dislocation and a viscoelastic piezoelectric bimaterial interface

The complex variable method is employed to derive analytical solutions for the interaction between a piezoelectric screw dislocation and a Kelvin-type viscoelastic piezoelectric bimaterial interface. Through analytical continuation, the original boundary value problem can be reduced to an inhomogeneous first-order partial differential equation for a single function of location z = x + iy and time t defined in the lower half-plane, which is free of the screw dislocation. Once the initial, steady-state and far-field conditions are known, the solution to the first order differential equation can be obtained. From the solved function, explicit expressions are then derived for the stresses, strains, electric fields and electric displacements induced by the piezoelectric screw dislocation. Also presented is the image force acting on the screw dislocation due to its interaction with the Kelvin-type viscoelastic interface. The derived solutions are verified by comparing with existing solutions for the simplified cases, and various interesting features are observed, particularly for those associated with the image force.     

A Partially Built-in Plate under Uniform Load

A thin plate has the form of the infinite strip ?∞<x<∞, 0≤y≤aand has the edge y=abuilt-in. The edge y=0 has its right half 0<x<∞ built-in while the left half ?∞<x<0 is free. The whole plate is now subjected to a uniform load p ₀applied to its upper surface. What is the resulting deflection of the plate and what are the induced moment and shear resultants? We present a solution to this classical problem based on eigenfunction expansions. In the right and left halves of the strip, the deflection can be expanded as separate eigenfunction expansion series, but these are difficult to match across the line x=0 because of the singularity at (0,0) induced by the boundary conditions. We adopt the novel technique of expanding the field near the centre of the strip in its correct form as a series of Williams polar eigenfunctions, and then linking this expansion to the right and left eigenfunction expansions by using a special form of elastic reciprocity. These right and left reciprocity conditions give two infinite systems of linear equations satisfied by the polar expansion coefficients, and we prove that these equations are sufficient to determine these coefficients. Further applications of reciprocity give closed form expressions for the right and left eigenfunction expansion coefficients so that the whole solution is then determined. The method yields accurate results using small systems of linear equations. We present numerical results for the deflection of the plate and the induced moment and shear resultants.     

Calculation of the boundary layer on the electrically conducting wall of a plane channel

There are presently available quite a large number of works devoted to the study of the motion of an electrically conducting fluid in boundary layers formed on electrodes or on the nonconducting walls of various MHD devices. However, the methods of solving the boundary layer equations in these studies are based on various simplifying assumptions which allow the problem to be reduced to the solution of a system of ordinary differential equations. Thus, in [1] there is imposed on the flow the special magnetic fieldH1/x, which enables the problem to be reduced to the self-similar form, while in the studies of other authors [2, 3] either the solution is sought in the form of expansions in x, or it is assumed that the problem is locally self-similar [4]. In the present paper we construct the solution of the MHD boundary layer equations which is obtained by one of the numerical methods which has long been used for solving the boundary layer equations for a nonconducting fluid.     

Static analysis of anisotropic functionally graded magneto-electro-elastic beams subjected to arbitrary loading

This paper considers the analytical and semi-analytical solutions for anisotropic functionally graded magneto-electro-elastic beams subjected to an arbitrary load, which can be expanded in terms of sinusoidal series. For the generalized plane stress problem, the stress function, electric displacement function and magnetic induction function are assumed to consist of two parts, respectively. One is a product of a trigonometric function of the longitudinal coordinate (x) and an undetermined function of the thickness coordinate (z), and the other a linear polynomial of x with unknown coefficients depending on z. The governing equations satisfied by these z-dependent functions are derived. The analytical expressions of stresses, electric displacements, magnetic induction, axial force, bending moment, shear force, average electric displacement, average magnetic induction, displacements, electric potential and magnetic potential are then deduced, with integral constants determinable from the boundary conditions. The analytical solution is derived for beam with material coefficients varying exponentially along the thickness, while the semi-analytical solution is sought by making use of the sub-layer approximation for beam with an arbitrary variation of material parameters along the thickness. The present analysis is applicable to beams with various boundary conditions at the two ends. Two numerical examples are presented for validation of the theory and illustration of the effects of certain parameters.     

The stress state of a plate subjected to a piecewise homogeneous load

We consider the stress-strain state of a plate having a doubly connected domain S bounded from the outside by a circle of radius R and from the inside by an ellipse with two rectilinear cuts. The cuts lie symmetrically on the x-axis. The plate is subjected to various forces: the hole contour (the ellipse) is under the action of uniformly distributed forces of intensity q, and the cut shores are free of loads; at the points ±ib of the imaginary axis, the plate is under the action of a lumped force P.The solution of the problem is reduced to determining two analytic functions φ(z) and ψ(z) satisfying certain boundary conditions (depending on the type of the acting loads).We use the Kolosov-Muskhelishvili method to reduce the problem to a system of linear algebraic equations for the coefficients in the expansions of the functions φ(z) and ψ(z). The solution thus obtained is illustrated by numerical examples.     

The derivation of a thick and thin plate formulation without ad hoc assumptions

For the plate formulation considered in this paper, appropriate three-dimensional elasticity solution representations for isotropic materials are constructed. No a priori assumptions for stress or displacement distributions over the thickness of the plate are made. The strategy used in the derivation is to separate functions of the thickness variable z from functions of the coordinates x and y lying in the midplane of the plate. Real and complex 3-dimensional elasticity solution representations are used to obtain three types of functions of the coordinates x, y and the corresponding differential equations. The separation of the functions of the thickness coordinate can be done by separately considering homogeneous and nonhomogeneous boundary conditions on the upper and lower faces of the plate. One type of the plate solutions derived involves polynomials of the thickness coordinate z. The other two solution forms contain trigonometric and hyperbolic functions of z, respectively. Both bending and stretching (or in-plane) solutions are included in the derivation.     

Wiener-Hopf analysis of an acoustic plane wave in a trifurcated waveguide

In this paper, we have made Wiener-Hopf analysis of an acoustic plane wave by a semi-infinite hard duct that is placed symmetrically inside an infinite soft/hard duct. The method of solution is integral transform and Wiener-Hopf technique. The imposition of boundary conditions result in a 2 × 2 matrix Wiener-Hopf equation associated with a new canonical scattering problem which is solved by using the pole removal technique. In the solution, two infinite sets of unknown coefficients are involved that satisfy two infinite systems of linear algebraic equations. These systems of linear algebraic equations are solved numerically. The graphs are plotted for sundry parameters of interest. Kernel functions are also factorized.     

A fractional differential constitutive model for dynamic stress intensity factors of an anti-plane crack in viscoelastic materials

Fractional differential constitutive relationships are introduced to depict the history of dynamic stress inten- sity factors （DSIFs） for a semi-infinite crack in infinite viscoelastic material subjected to anti-plane shear impact load. The basic equations which govern the anti-plane deformation behavior are converted to a fractional wave-like equation. By utilizing Laplace and Fourier integral transforms, the fractional wave-like equation is cast into an ordinary differential equation （ODE）. The unknown function in the solution of ODE is obtained by applying Fourier transform directly to the boundary conditions of fractional wave-like equation in Laplace domain instead of solving dual integral equations. Analytical solutions of DSIFs in Laplace domain are derived by Wiener-Hopf technique and the numerical solutions of DSIFs in time domain are obtained by Talbot algorithm. The effects of four parameters α, β, b1, b2 of the fractional dif- ferential constitutive model on DSIFs are discussed. The numerical results show that the present fractional differential constitutive model can well describe the behavior of DSIFs of anti-plane fracture in viscoelastic materials, and the model is also compatible with solutions of DSIFs of anti-plane fracture in elastic materials.     

Asymmetric problem of a row of revolutional ellipsoidal cavities using singular integral equations

This paper deals with numerical solution of singular integral equations of the body force method in an interaction problem of revolutional ellipsoidal cavities under asymmetric uniaxial tension. The problem is solved on the superposition of two auxiliary loads; (i) biaxial tension and (ii) plane state of pure shear. These problems are formulated as a system of singular integral equations with Cauchy-type singularities, where the unknowns are densities of body forces distributed in the r, θ, z directions. In order to satisfy the boundary conditions along the ellipsoidal boundaries, eight kinds of fundamental density functions proposed in our previous papers are applied. In the analysis, the number, shape, and spacing of cavities are varied systematically; then the magnitude and position of the maximum stress are examined. For any fixed shape and size of cavities, the maximum stress is shown to be linear with the reciprocal of squared number of cavities. The present method is found to yield rapidly converging numerical results for various geometrical conditions of cavities.     

Proving conditional attractivity of a closed set with a family of liapunov functions

Considering a closed set M of some x-space and a solution x(t), y(t) of a differential system x = X(x, y, t), y = Y(x, y, t), we give sufficient conditions in order that x(t) approaches M. We use several auxiliary functions and employ Salvadori's method of a one parameter family of Liapunov functions. An application is given to the two-body problem in the presence of some friction forces and when the reference frame is non-inertial.     

Eccentric crack in a piezoelectric strip bonded to half planes

We consider the problem of determining the singular stresses and electric fields in a piezoelectric ceramic strip containing an eccentric Griffith crack off the centre line bonded to two elastic half planes under anti-plane shear loading using the continuous crack-face condition. Fourier transforms are used to reduce the problem to the solution of two pairs of dual integral equations, which are then expressed to a Fredholm integral equation of the second kind. Numerical values on the stress intensity factor and energy release rate are obtained.