Torsional stiffness of a layer bonded to an elastic half space

The static solution to the problem of a layer bonded to an elastic half-space, where the layer is driven by the torsional rotation of a bonded rigid circular disk, is considered here. An iterative solution, perturbing on that given for the elastic half-space, is obtained as a convergent power series, provided the ratio of the stratum depth to the radius of the disk is large. An equation for the applied static torque at the surface of the rigid disk is also calculated and compared, under limiting cases, with known results.     

Torsional oscillations of a layer bonded to an elastic half-space

The problem of a layer bonded to an elastic half-space, where the layer is driven by torsional oscillations of a bonded rigid circular disk, is solved by means of integral transform techniques. Using a standard technique, the problem is reduced to a Fredholm integral equation of the second kind, the kernel of which involves the calculation of principal value integrals. Dynamic stiffnesses are developed for a range of layer thicknesses, material properties, and frequencies.     

Stress analysis for a layer of finite length bonded to a half-space of identical material

The problem in the plane theory of elasticity of an elastic layer bonded to an elastic half-space of the same material is considered. The formulation is achieved by means of integral transforms and the problem is reduced to the solution of a system of singular integral equations. A numerical solution is accomplished for the half-space and layer by use of the collocation scheme developed by Erdogan and Gupta.     

Plane waves in a fractional order micropolar magneto-thermoelastic half-space

This paper deals with the problem of magneto-thermoelastic interactions in an unbounded, perfectly conducting half-space whose surface suffers a time harmonic thermal source in the context of micropolar generalized thermoelasticity with fractional heat transfer allowing the second sound effects. The medium is assumed to be unstrained and unstressed initially and has uniform temperature. The Laplace–Fourier double transform technique has been used to solve the resulting non-dimensional coupled field equations. Expressions for displacements, stresses and temperature in the physical domain are obtained using a numerical inversion technique. The effects of fractional parameter, magnetic field and micropolarity on the physical fields are noticed and depicted graphically. For a particular model, these fields are found to be significantly affected by the above mentioned parameters. Some particular cases of interest have been deduced from the present problem. Numerical results predict finite speed of propagation for thermoelastic waves.     

Semi-infinite medium with a cylindrical inclusion twisted by an annular die

The problem considered is that of torsion of an elastic half-space consisting of two materials which are separated by a cylindrical surface. It is assumed that there is perfect bonding at the common cylindrical surface. The torque is applied to the half-space through an annular rigid disc which is bonded to both materials. By use of integral transforms and the theory of dual integral equations the problem is reduced to that of solving a pair of simultaneous Fredholm integral equations of the second kind. These simultaneous integral equations are solved numerically. The numerical results obtained are exhibited graphically.Research for this paper was partially supported by the Natural Sciences and Engineering Research Council of Canada through NSERC Grant No. A4177.     

A Simple Hysteretic Constitutive Model for Unsaturated Flow

The present study covers the problem of rotation of a porous disk under a viscous incompressible fluid that fills the half-space above the disk, which is the generalization of the von Karman’s problem. It is found that, instead of solving the exact problem, which is rather complicated by coupling the motions of the free fluid and that contained inside the permeable disk, it is sufficient to solve a much simpler problem of the motion of the free fluid placed onto a permeable plane. Assuming the flow in the permeable disk is described by the Brinkman equations, we obtain a self-similar formulation of the problem. Employing this formulation, we also show that the boundary condition associated with continuity of the tangential strains and tangential velocity components is satisfied at the fluid–porous body interface. The coefficient for the vertical velocity component is furthermore obtained. Various extreme cases are identified.     

Two-dimensional generalized thermoelasticity problem for a half-space subjected to ramp-type heating

In this paper, we will consider a half-space filled with an elastic material, which has constant elastic parameters. The governing equations are taken in a unified system from which the field equations for coupled thermoelasticity as well as for generalized thermoelasticity can be easily obtained as particular cases. A linear temperature ramping function is used to more realistically model thermal loading of the half-space surface. The medium is assumed initially quiescent. Laplace and Fourier transform techniques are used to obtain the general solution for any set of boundary conditions. The general solution obtained is applied to a specific problem of a half-space subjected to ramp-type heating. The inverse Fourier transforms are obtained analytically while the inverse Laplace transforms are computed numerically using a method based on Fourier expansion techniques. Some comparisons have been shown in figures to estimate the effect of the ramping parameter of heating with different theories of thermoelasticity.     

The longitudinal harmonic excitation of a circular bar embedded in an elastic half-space

This investigation is concerned with the dynamic response of a circular elastic bar of finite length partially embedded in a half-space of distinct elastic properties. The bar is perpendicular to the free surface of the embedding medium and supports a mass which is harmonically excited in the direction of the bar's longitudinal axis. Two bonding conditions are considered: fully bonded wherein the bar completely adheres to the embedding medium throughout the surface of contact, and loosely bonded wherein the bar is secured through its terminal cross section alone. Of primary importance is the energy dissipation due to the spatial characteristics of the embedding medium and accordingly the system is interpreted as a frequency-dependent spring-dashpot.The determination of the effective spring constant and damping coefficient is achieved by modeling the bar with a one-dimensional theory and using three-dimensional theory for a region which approximates the embedding medium, namely the full half-space. Lamé potentials and Hankel transforms enable a basic half-space problem to be solved which in turn allows integral representations for the spring constant and damping coefficient to be established. For the fully-bonded problem these integral representations involve a bar-force term which must be determined from an integral equation. In both cases the solutions are evaluated numerically over a range of forcing frequencies and for various bar/half-space configurations.     

Green’s functions of a surface-stiffened transversely isotropic half-space

Green’s functions of a transversely isotropic half-space overlaid by a thin coating layer are analytically obtained. The surface coating is modeled by a Kirchhoff thin plate perfectly bonded to the half-space. With the aid of superposition technique and employing appropriate displacement potential functions, the Green’s functions are expressed in two parts; (i) a closed-form part corresponding to the transversely isotropic half-space with surface kinematic constraints, and (ii) a numerically evaluated part reflecting the interaction between the half-space and the plate in the form of semi-infinite integrals. Some limiting cases of the problem such as surface-stiffened isotropic half-space, Boussinesq and Cerruti loadings, and extremely flexible and rigid plates are also studied. For the classical Cerruti problem in transversely isotropic materials, the effects of incompressibility are highlighted. Numerical results are provided to show the effects of material anisotropy, relative stiffness factor, and load buried depth. The obtained Green’s functions play a key role in treating further mixed-boundary-value problems in surface stiffened transversely isotropic half-spaces.     

The effect of fractional parameter on a perfect conducting elastic half-space in generalized magneto-thermoelasticity

Recently, Sherief et al. (Int. J. Solids Struct. 47:269–275, 2010) proposed a model in generalized thermoelasticity based on the fractional order time derivatives. The propagation of electro-magneto-thermoelastic disturbances in a perfectly conducting elastic half-space is investigated in the context of the above fractional order theory of generalized thermoelasticity. There acts an initial magnetic field parallel to the plane boundary of the half-space. Normal mode analysis together with the eigenvalue approach technique is used to solve the resulting non-dimensional coupled governing equations of the problem. The obtained solution is then applied to two specific problems for the half-space, whose boundary is subjected to (i) thermally isolated surfaces subjected to time-dependent compression and (ii) a time-dependent thermal shock and zero stress. The effects of fractional parameter and magnetic field on the variations of different field quantities inside the half-space are analyzed graphically.     

The Action of an Arbitrary Load on the Surface of an Orthotropic Half-Space

The exact solution to the first boundary-value problem for a half-space is constructed on the basis of the general solution of the equilibrium equations for an orthotropic medium (nine elastic constants). The stress–strain state of an orthotropic half-space whose surface is under an arbitrarily applied concentrated force is described as an example. The well-known solution for the isotropic case is obtained by the same scheme, which confirms the reliability of the result.     

Uniform indentation of the elastic half-space by a rigid rectangular punch

The problem considered is that of a rigid flat-ended punch with rectangular contact area pressed into a linear elastic half-space to a uniform depth. Both the lubricated and adhesive cases are treated. The problem reduces to solving an integral equation (or equations) for the contact stresses. These stresses have a singular nature which is dealt with explicitly by a singularity-incorporating finite-element method. Values for the stiffness of the lubricated punch and the adhesive punch are determined: the effect of adhesion on the stiffness is found to be small, producing an increase of the order of 3%.     

Love wave dispersion in pre-stressed homogeneous medium over a porous half-space with irregular boundary surfaces

The paper investigates the existence of Love wave propagation in an initially stressed homogeneous layer over a porous half-space with irregular boundary surfaces. The method of separation of variables has been adopted to get an analytical solution for the dispersion equation and thus dispersion equations have been obtained in several particular cases. Propagation of Love wave is influenced by initial stress parameters, corrugation parameter and porosity of half-space. Velocity of Love waves have been plotted in several figures to study the effect of various parameters and found that the velocity of wave decreases with increases of non-dimensional wave number. It has been observed that the phase velocity decreases with increase of initial stress parameters and porosity of half-space.     

On the relationship between the solutions of static contact problems of elasticity and electroelasticity for a half-space

The paper establishes the relationship between the static contact problems of elasticity and electroelasticity (in the absence of friction) for a transversely isotropic half-space whose surface is the isotropy plane. This makes it possible to avoid solving the electroelastic problem by finding all the characteristics of electroelastic contact from known cases of purely elastic interaction. Moreover, the electroelastic state of the half-space can be fully described using a known harmonic function, which is a solution of the purely elastic problem. The approach is exemplified by solving contact problems of electroelasticity for flat, elliptic, two circular, conical, and paraboloidal (circular and elliptic in plan) punches __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 11, pp. 69–84, November 2006.     

Bonded contact and general crack problems in generalized materials and their relationships

The term of generalized material is introduced here as the material, whose state is described by the second order homogeneous differential equations with constant coefficients. The generalized point sources are described by homogeneous expressions, containing the first order derivatives with constant coefficients. The Green’s function for a half-space, made of generalized material, subjected to the action of generalized point sources, is derived in the form of a single integral over a circle. Some of the components of the surface Green’s function are presented in finite form, no computation of any integral is needed. The bonded contact problem is described as mixed–mixed boundary value problem for a half-space, with normal and tangential displacements prescribed inside domain S and vanishing tractions outside this domain on the plane \(x_{3}=0\). A set of governing integral equations for bonded contact problem was derived, with some of the kernels defined in finite form. The general crack problem is defined as that of a flat crack of arbitrary shape S in the plane \(x_{3}=0\), with normal tractions \(\sigma _{33}\) applied symmetrically to the crack faces and tangential tractions \(\sigma _{31}\) and \(\sigma _{23}\) applied anti-symmetrically to the crack faces. A set of 3 governing integral equations is derived, with some of the kernels presented in the finite form. The relationship between the integrands in Fourier transform of the kernels of the sets of the integral equations of both problems is established: they are related in such a way, as if they were the coefficients in the schematic sets of algebraic equations. This relationship can also be presented as a general relationship between matrices of arbitrary rank n, with their components being determinants of certain matrices of rank q, with \(q\ge n\).     

周期界面裂纹的弹性波散射问题研究

本文研究了分布于两个关元限空间的周期界面对垂直入射Ｐ波及ＳＨ波的散射问题，文中利用有限Ｆｏｕｒｉｅｒ变换将一个周期带内散射场的边值问题转化为求解一个带周期核的奇异积分方程，并对ＳＨ波入射的情形进行了详细的分析，求解了相应的异积分方程，最后给出裂纹尖端的应力强度因子的计算公式及远离裂纹时散射位移场的渐进形式，并对散场的动态特性进行了数值分析。     

Analytical solution of the spherical indentation problem for a half-space with gradients with the depth elastic properties

The contact problem for the impression of spherical indenter into a non-homogeneous (both layered and functionally graded) elastic half-space is considered. Analytical methods for solving this problem have been developed. It is assumed that the Lame coefficients vary arbitrarily with the half-space depth. The problem is reduced to dual integral equations. The developed methods make it possible to find the analytical asymptotically exact problem solution, suitable for a PC. The influence of the Lame coefficients variation upon the contact stresses and size of the contact zone with different radius of indenter as well as values of the impressing forces are studied. The effect of the non-homogeneity is examined. The developed method allows to construct analytical solutions with presupposed accuracy and gives the opportunity to do multiparametric and qualitative investigations of the problem with minimal computation time expenditure.     

Effects due to body-forces and body-couples in the interior of a micropolar elastic half-space—I

A general solution of the equilibrium equations is obtained for a half-space with a fixed boundary and arbitrary but axisymmetric distribution of body forces and body couples in the interior of the half-space. Few particular cases have been investigated in detail. The stresses have been obtained at the boundary and the displacements have been obtained in the interior of the half-space. Numerical results have been displayed graphically.

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Zusammenfassung Eine allgemeine lösung für Gleichgewicht eines Halbraums mit festem Rand und einer beliebigen, aber antisymmetrischen Verteilung von Körperkräften und Körpermomenten im Inneren wird gefunden. Einige Spezialfälle wurden ansführlich erforscht. Die Spannungen am Rand und die Verschiebungen im Inneren des Halbraums wurden erhalten. Numerische Resultate wurden graphisch dargestellt.

     

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.     

Study into the Interaction of Cracks in an Elastic Half-Space under a Shock Load by Means of Boundary Integral Equations

The effect of a shock load on the interaction of circular cracks in an elastic half-space is studied. In the space of Fourier time transforms, the problem is reduced to a system of two-dimensional boundary integral equations in the form of the Helmholtz potential with unknown densities characterizing the discontinuities in the displacements of the opposite crack faces. Discrete analogs of those equations are constructed. As an example, two cracks are considered whose faces are under the action of shock tensile loads varying in time as the Heaviside function. The time dependences of the dynamic stress intensity factors are obtained. Their dependence on the relative position of the cracks in the half-space is analyzed.