Nonstationary heat transfer in anisotropic half-space under the conditions of heat exchange with the environment having a specified temperature

An analytical solution to the problem of heating anisotropic half-space by the environment with a spatially and temporally variable temperature (boundary conditions of the third kind at an anisotropic body) has been obtained for the first time based on the construction of the boundary influence function (the Green’s function), which is determined using the Fourier and Laplace integral transforms. Nonstationary temperature fields in anisotropic blunt bodies have been found under the conditions of aero-gas-dynamic heating of hypersonic aircrafts with different heat-transfer coefficients and incoming-flow temperatures. The solution obtained is recommended for determining the state of thermal protection fabricated from composites which are generally anisotropic.     

Symmetrical frictionless indentation over a uniformly expanding contact region—I. Basic analysis

The study of two-dimensional wave propagation in a half-space due to indentation by a rigid smooth indentor of a general shape at a non-uniform velocity is approached by assuming that the indentor shape and displacement history can be represented by polynomial curves in, respectively, a spatial variable and the time. As a first step symmetric indentation over a contact region expanding at a constant sub-Rayleigh wave speed is considered. Since superposition will yield more general forms attention is confined to polynomials homogeneous of degree $\text{n ? 1}$ with n + 1 terms and arbitrary coefficients. By homogeneous function techniques all the field variables in the half-space for any $\text{n}$ are obtained as single integrals. Conditions for the existence of singularities in the stresses and particle velocities are examined and some general results with bearing on the indentation problem are discussed.     

Model of the interaction of electromagnetic and thermal fields in delay media

We have solved the problem on the propagation of a high-frequency electromagnetic field in a half-space filled with a medium with delayed electric and magnetic polarizations and conduction currents under the action on the half-space of an arbitrary combination of plane fields, as well as of surface currents and charges distributed over the half-space surface. A model of the heating of the half-space medium as a result of the electromagnetic-to-thermal and vice versa energy conversion has been considered. A numerical study of the heating depending on the delay time has been made. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 82, No. 1, pp. 177–183, January–February, 2009.     

Analytical solution for the electric potential due to a point source in an arbitrarily anisotropic half-space

A very large class of important theory and applications in geophysics requires analytical solutions for the determination of the electric potential due to a point source in an arbitrarily anisotropic half-space. In this paper, a very clear and simple solution to the half-space problem has been developed from consideration of an arbitrarily anisotropic whole space. For the first time, the method of images is used to generate the solution for an arbitrarily anisotropic three dimensional half-space. Based on traditional extreme value theory the image source point has been determined and calculated for the half-space case.     

Two-dimensional problem for a half-space with axi-symmetric distribution in the theory of generalized thermoelastic diffusion

In this work, we study a two-dimensional problem of axi-symmetric distribution of temperatures in a half-space with a permeating substance in contact with the bounding plane in the context of the theory of generalized thermoelastic diffusion with one relaxation time. The surface of the half-space is taken as traction free and subjected to axi-symmetric time-dependent thermal shock. The chemical potential is also assumed to be a known function of time on the bounding plane. The Laplace and Hankel transform techniques are used. The analytical solution in the transform domain is obtained by using a direct approach. The inverse of the double transform is obtained by using a numerical method based on Fourier expansion techniques. Numerical results for the temperature, displacement, stress, concentration, and chemical potential are carried out and represented graphically.     

Dynamical (time-harmonic) axisymmetric interface stress field in the finite pre-strained half-space covered with the finite pre-stretched layer

Within the framework of the piecewise homogeneous body model with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies (TLTEWISB), the dynamical (time-harmonic) stress field in the initially finite strained half-space covered with an initially and finitely stretched layer is investigated. It is assumed that on the upper free face of the covering layer the point located force which acts is harmonic with respect to time. The corresponding boundary contact problem is solved by employing the Hankel integral transformation. Moreover, it is assumed that the material of the layer and half-space are incompressible and elastic relations for those are given through the Treloar’s potential. In the case where the initial strains are absent in the layer and half-space the considered problem formulation and solution to that coincide with the corresponding ones of the classical linear theory of elasticity for an incompressible body. The algorithm for obtaining numerical results is proposed. The numerical results regarding the stresses acting on the interface plane are presented. These results are obtained for the case where the stiffness (distortion wave velocity) of the covering layer material is less (greater) than that for the half-space material. In this case, the main attention is focused on the dependencies between the values of the stresses and frequency of the external force and also the influence of the initial strains on these dependencies. In particular, it is established that the “resonance” values of the frequency of the external force increase, but the absolute maximum values of the stresses decrease significantly with the amount of the initial tension of the covering layer.     

On the complex potentials of the linear thermoelasticity with microlocal parameters

Summary The linear theory of thermoelasticity with microlocal parameters was proposed lately in [1], [2], [3] as a certain alternative approach to the modelling of microperiodic composites. In this paper we adopt the complex variable method for two-dimensional problems of the periodically layered thermoelastic composites treated within the framework of the linear thermoelasticity with microlocal parameters. The presented complex potentials reduce two-dimensional static problems of the microperiodically layered thermoelastic composites to the boundary values problems for analytical functions. As an example illustrating this method the plane-strain problem of stress and temperature distribution in the microperiodic two-layered half-space subjected to general loading and temperature on the boundary is considered.With 3 Figures     

Computational modeling of the indentation of a cracked poroelastic half-space

The paper examines the computational modelling of the surface identation of a poroelastic half-space region which is weakened either by a cylindrical crack or a penny-shaped crack. The axisymmetric problems associated with those situations are examined using a finite element procedure where special singularity elements are incorporated at the crack tip and appropriate interaction conditions are incorporated on the faces of the crack. The results presented in the paper illustrate the influence of the extent of fracture and the pore pressure boundary conditions on the various surfaces, on the time dependent evolution of the stress intensity factors and the time dependent consolidation settlement of the axisymmetric indentor. The analysis is extended to the consideration of crack extension in poroelastic materials where displacement, traction and pore water pressure boundary conditions are alerted to take into account the evolving crack. The path of crack extension is established by mixed mode crack extension criteria applicable to porous fabric. The computational procedure associated with this approach is used to examine the problem of the surface indentation of a half-space by a rigid circular indentor. This revised version was published online in July 2006 with corrections to the Cover Date.     

Torsional impact of a layer or a cylinder bonded to an elastic half-space

In this paper two torsional impact problems are considered. The first problem deals with the solution of a layer bonded to an elastic half-space when the layer is driven by the torsional impact over a bonded rigid circular disc. In the second problem sudden torsion by a rigid disc attached over the plane face of a circular cylinder is considered and the rest of the plane surface of the cylinder is stress free. The cylinder is bonded to the half-space, making use of Laplace and Hankel transforms the solution of each problem is reduced into Fredholm integral equations of the second kind. A numerical Laplace inversion technique is then used to recover the time depencence of the solution. The numerical values for the applied torque at the surface of rigid disc are calculated for each problem and then are displayed graphically.     

On the dispersion equation of Love waves in a porous layer

Summary The paper is concerned with the propagation of Love waves in a porous layer over an inhomogeneous half-space due to a point source. The dispersion equation for Love waves in a porous layer has been derived. The effects of porosity and inhomogeneity in the half-space have been shown graphically on the dispersion curves. The theory pointed out by Biot for the porous medium has been applied to solve the problem with the help of Green's function technique.With 4 Figures     

Heating of a half-space by a heat source in the shape of a rectangular frame

A solution is presented for the inverse nonstationary spatial coefficient problem of heat conduction for a half-space heated through a domain in the shape of a rectangular frame on its surface by an arbitrary heat source with respect to time.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 57, No. 2, pp. 322–326, August, 1989.     

A two-dimensional generalized thermoelastic diffusion problem for a half-space subjected to harmonically varying heating

A two-dimensional problem for a thermoelastic half-space is considered within the context of the theory of generalized thermoelastic diffusion with one relaxation time. The upper surface of the half-space is taken to be traction free and subjected to harmonically varying heating with constant angular frequency of thermal vibration. Laplace and Fourier transform techniques are used. The solution in the transformed domain is obtained by a direct approach. Numerical inversion techniques are used to obtain the inverse double transforms. Numerical results are discussed and represented graphically.     

Slow vertical motions of an elliptic punch on an elastic half-space

The dynamic contact problem for a homogeneous, isotropic elastic half-space and a punch of an arbitrary base shape is studied. During the motion of the punch it is assumed that the contact area is fixed and there is no friction under the punch. An approximate solution to the problem is obtained under the assumption that the contact pressure under the punch is slightly varied during the time of travel of the Rayleigh wave along the distance equal to the diameter of the contact area. The solution of the problem of slow motions of a punch on an elastic half-space is reduced to the solution of the recurrent system of integral equations of the static contact problem. Asymptotic models of vertical motions of a flat-ended punch with an arbitrary base are constructed. The case of an elliptic punch is considered in detail.     

Generalized electromagneto-thermoviscoelastic in case of 2-D thermal shock problem in a finite conducting medium with one relaxation time

Summary. This paper studies the problem of two-dimensional electromagneto-thermovisco-elasticity based on Lord-Shulman theory for a thermally and electrically conducting half-space solid whose surface is subjected to a thermal shock. There acts an initial magnetic field parallel to the plane boundary of the half-space. The medium deforms because of thermal shock and due to the application of the magnetic field, it results in induced magnetic and electric fields in the medium. The normal mode analysis is used to obtain the exact expressions for the considered variables. The distributions of the temperature, displacement, stress, induced magnetic and electric fields are represented graphically. Comparisons are made with the results predicted by both the coupled theory and with the theory of generalized thermo-viscoelasticity with one relaxation time.     

The indentation of a half-space of hexagonal elastic material by a circular punch of arbitrary end-profile

Summary The problem is considered of the indentation by a smooth rigid punch of a half-space composed of linear elastic material of hexagonal symmetry whose plane boundary is parallel to the basal planes. The case is considered in which the area of contact between the punch and the half-space is circular, the end of the punch with is in contact with the half-space having an arbitrary profile. An integral equation is formulated and solved for the boundary value of the normal displacement in the half-space, and an expression is derived for the distribution of pressure under the punch.     

Dynamic fracture analysis of an inclined subsurface crack subjected to dynamic moving loadings

The transient response of a half-space containing a subsurface inclined semi-infinite crack excited by a dynamic moving antiplane loading on the surface of the half-space is investigated in this study. The solutions of dynamic stress intensity factors are derived for all load speeds (subsonic and supersonic) and are determined by superposition of a proposed fundamental solution in the Laplace transform domain. The fundamental problem is the problem of applying an exponentially distributed traction in the Laplace transform domain on crack faces. The method of analysis is based on integral transform techniques and the Wiener-Hopf technique. The exact closed form transient solutions of dynamic stress intensity factors are expressed in very compact formulations in this study. These solutions are valid for an infinite length of time and have accounted for all the contributions of infinitely many waves. Numerical results of the transient stress intensity factor are obtained and the results of the limit case of zero load speed is compared with the corresponding static values.     

A Thick Plate Problem in the Theory of Generalized Thermoelastic Diffusion

In this work, the problem of a thermoelastic thick plate with a permeating substance in contact with one of the bounding planes is considered in the context of the theory of generalized thermoelastic diffusion with one relaxation time. The bounding surface of the half-space is taken to be traction free and is subjected to a time-dependent thermal shock. The chemical potential is also assumed to be a known function of time on the bounding plane. Laplace transform techniques are used. The solution is obtained in the Laplace transform domain by using a direct approach. The solution of the problem in the physical domain is obtained numerically using a numerical method for the inversion of the Laplace transform based on Fourier expansion techniques. The temperature, displacement, stress, and concentration as well as the chemical potential are obtained. Numerical computations are carried out and represented graphically.     

1D applications on fractional generalized thermoelasticity associated with two relaxation times

Here, this work is concerned with some different one-dimensional (1D) problems of distribution of the thermal stresses and temperature in fractional generalized thermoelastic material, as follows: (i) 1D problem for a half-space of elastic material in the presence of heat sources has been solved by using Laplace transform and state space techniques; and (ii) 1D problem of distribution of thermal stresses and temperature in infinite medium with a spherical cavity subjected to a sudden change in the temperature of its internal boundary, which is assumed to be traction free, has been solved by Laplace transform and direct approach techniques. In the preceding problems, the numerical results for dimensionless variable fields are given and illustrated graphically. According to the numerical results, some comparisons have been shown in figures to estimate the effect of the fractional derivative parameter α about the new theory on all variable fields and discussion has been established for a copper-like material.     

Propagation of transient SH-waves in a dipping structure by finite- and boundary element methods

Summary The boundary and the finite element formulations for the equations of elasticity are presented and applied to the problem of propagation of transient SH-waves in dipping layers overlying a half-space. When the finite element formulation is used, appropriate boundary conditions are imposed on the additional boundary dividing the half-space into a finite and an infinite region. These conditions ensure the transmission of waves across this boundary. When the boundary element method is applied, it is necessary to satisfy the radiation conditions. Theoretical seismograms for the displacement on the surface of the half-space are presented. They show that, for a specific case, the agreement between the two methods is satisfactory. The results can be compared with those found by the exact method of generalized rays in order to check the validity of the finite and the boundary element methods for the specific problem studied in this paper.     

Multiple cracks in a half-space under contact loading

This paper develops a semi-analytic solution for multiple cracks in an isotropic half-space under contact loading. The solution takes into account interactions among all the cracks as well as the interactions between the cracks and the loading body. In formulating the governing equations for the subsurface crack problem, each crack of mixed modes I and II is modeled as a continuous distribution of climb and glide dislocations with unknown densities. Such a treatment converts the original contact problem concerning an inhomogeneous half-space into a homogeneous half-space contact problem, for which governing equations with unknown surface contact areas and normal pressure and tangential tractions within the areas can be conveniently formulated. All the unknowns in the governing equations are determined by means of iteration. The iterative process is performed until the convergence of the half-space surface displacements, which are the sum of the displacements due to the surface contact load and subsurface cracks. The solution is validated by the finite element method. Numerical examples are calculated to demonstrate the generality of the solution.