Rapid Sliding Indentation with Friction of a Pre-Stressed Thermoelastic Material

A rigid indentor travels with a constant speed over the surface of an isotropic thermoelastic half-space. Friction exists between the indentor and half-space, and the latter is initially in equilibrium at a uniform temperature under a uniform normal pre-stress. This pre-stress, below but near yield, is assumed to produce deformations that dominate the additional deformations due to indentation. Thus, the process is treated as one of small deformations superposed upon (relatively) large. The governing equations for the superposed deformation are those of nonisotropic coupled thermoelasticity. A steady-state two-dimensional study uses robust asymptotic analytical solutions to reduce the associated mixed boundary value problem to a classical singular integral equation which can be solved analytically. The solutions show that the pre-stress-induced de facto nonisotropy alters the values of the rotational and dilatational wave and Rayleigh speeds in the half-space and, in the case of a compressive pre-stress, generates a second, lower, critical speed. In addition, pre-stress generates noncritical sliding speeds at which the friction-dependent integral equation eigenvalue changes sign. For purposes of illustration, expressions for the half-space surface temperature change and its average over the contact zone, the equations necessary to determine contact zone size and location, the resultant moment on the indentor, and the maximum compressive stress on the contact zone are presented for a parabolic indentor. This revised version was published online in August 2006 with corrections to the Cover Date.     

Rapid sliding motion of a rigid frictionless indentor with a flat base over a thermoelastic half-space

The three-dimensional, rapid sliding indentation of a deformable half-space by a rigid indentor of a flat elliptical base is treated in this paper. The response of the material that fills the half-space is assumed to be governed by coupled thermoelasticity. The indentor translates without friction on the half-space surface at a constant sub-Rayleigh speed and the problem is treated as a steady-state one. An exact solution is obtained that is based on a Green’s function approach, integral equations, and Galin’s theorem. A closed-form expression for the distributed contact pressure under the elliptical base of the indentor is derived. Representative numerical results are given illustrating the effects of the indentor velocity, indentor geometry, and parameters of the thermoelastic solid on the contact displacement. Since there is an analogy between the steady-state theories of thermoelasticity and poroelasticity, the present results carry over to the latter case directly.     

Rolling without Slip on a Transversely Isotropic Thermoelastic Half-space

Rolling without slip by a rigid cylinder on a transversely isotropic, coupled thermoelastic half-space at constant subcritical speed is studied. The cylinder is of infinite length, surface heat convection is neglected, and a dynamic steady state of plane strain is treated. The unmixed problem of traction applied to a translating surface strip is addressed first. A robust asymptotic form of the exact transform solution, valid when Fourier heat conduction dominates any thermal relaxation effect, is extracted, and inverted analytically. Use of material characterization and identification of parameters that vanish in the isotropic limit or are invariant under an isothermal–thermoelastic transformation result in compact full-field solutions. These expressions are used to construct analytical solutions that satisfy the mixed boundary value problem and auxiliary conditions of rolling contact. For the hexagonal material zinc, calculations are made for contact zone width and temperature increases near onset of zone yield. Mathematics Subject Classifications (2000) 73B30, 73C25, 73C30, 73C35.     

半无限大体广义电磁热弹耦合的二维问题

基于Lord和Shulman广义热弹性理论,研究了热、电可导的半无限大体电磁热弹耦合的二维问题。半无限大体受热和外加恒定磁场的作用,文中建立了电磁热弹性耦合的控制方程,利用正则模态法求解得到了所考虑物理量的解析解,并用图形反映了各物理量的分布规律,从分布图上可以看出,介质中出现了电磁热弹耦合效应,各物理量的非零值仅在一个有限的区域内。     

Finite amplitude one-dimensional wave propagation in a thermoelastic half-space

The problem of finite wave propagation in a nonlinearly thermoelastic half-space is considered. The surface of the half-space is subjected to a time-dependent thermal and normal mechanical loading. The solution is obtained by a numerical procedure, which is shown to furnish accurate results, and linear dynamic thermoelastic problems are obtained as special cases. The accuracy of the results is checked by comparison with some known analytical solutions which can be obtained in some special cases of both the linear and the nonlinear problems. In those cases where the solution contains shocks, it is shown that the numerical results satisfy the necessary jumps conditions which need to hold across such discontinuities.     

Dynamic thermoelastic effects for half-planes and half-spaces with nearly-planar surfaces

The effects of non-planarity on the dynamic surface temperature changes induced for plane-strain and 3D problems on the nearly-planar surfaces of, respectively, coupled thermoelastic half-planes and half-spaces by surface heat fluxes are treated. The nearly-planar nature of the surfaces allows the problem solutions to be written, following a standard perturbation scheme, as series expansions in a dimensionless surface contour amplitude parameter. The first, or zero-order, terms represent the ideal (planar) surface solutions, while the second, or first-order, terms represent corrections for non-planarity.Because the characteristic thermoelastic time is of O(10^–7)s, large-time asymptotic forms of the exact integral transform solutions can be used. These can be inverted exactly and used in Green's function operations to yield analytic, or integrals of analytic, expressions. Two types of thermal loading for the half-plane and yet a third type of thermal loading for the half-space are considered. Comparison of the zero- and first-order surface temperature changes for each case indicate that non-planarity gives rise for large times to changes in surface regions beyond those predicted by an ideal surface analysis. Moreover, the magnitudes of these changes can be more significant than the ideal surface results.     

Wave propagation in dual-phase-lag anisotropic thermoelasticity

The present paper is concerned with the propagation of plane waves in a transversely isotropic dual-phase-lag generalized thermoelastic solid half-space. The governing equations are solved in x–z plane to show the existence of three plane waves. Reflection of these plane waves from thermally insulated as well as isothermal stress-free surfaces is studied to obtain a system of three non-homogeneous equations in reflection coefficients of reflected waves. For numerical computations of speeds and reflection coefficients, a particular material is modeled as transversely isotropic dual-phase-lag generalized thermoelastic solid half-space. The speeds of plane waves are computed numerically for a certain range of the angle of propagation and are shown graphically against the angle of propagation for the cases of dual-phase-lag (DPL) thermoelasticity, coupled thermoelasticity and Lord–Shulman generalized thermoelasticity. Reflection coefficients of various reflected plane waves are computed numerically for thermally insulated as well as isothermal cases and are shown graphically against the angle of incidence for the cases of DPL thermoelasticity, coupled thermoelasticity and Lord–Shulman generalized thermoelasticity.     

Rapid sliding indentation with friction on a transversely isotropic thermoelastic half-space

A rigid insulated die slides at a constant sub-critical speed on a transversely isotropic half-space in the presence of friction. In a two-dimensional analysis of the dynamic steady-state, the coupled equations of thermoelasticity are invoked. All elements of the Coulomb friction model are strictly enforced, thus giving rise to auxiliary conditions, including two unilateral constraints.Robust asymptotic forms of an exact solution to a related problem with unmixed boundary conditions lead to analytical solutions for the sliding indentation problem. The solution expressions, abetted by calculations for zinc, show the role of frictional heating on the half-space surface. The effects of friction and sliding speed on contact zone size and location and average contact zone temperature are also studied.The analysis is aided by factoring procedures that simplify the complicated forms that arise in anisotropic elasticity. A scheme that renders expressions for roots of certain irrational functions analytic to within a single quadrature also plays a role.     

The three-dimensional steady-state thermo-elastodynamic problem of moving sources over a half space

A procedure based on the Radon transform and elements of distribution theory is developed to obtain fundamental thermoelastic three-dimensional (3D) solutions for thermal and/or mechanical point sources moving steadily over the surface of a half space. A concentrated heat flux is taken as the thermal source, whereas the mechanical source consists of normal and tangential concentrated loads. It is assumed that the sources move with a constant velocity along a fixed direction. The solutions obtained are exact within the bounds of Biot’s coupled thermo-elastodynamic theory, and results for surface displacements are obtained over the entire speed range (i.e. for sub-Rayleigh, super-Rayleigh/subsonic, transonic and supersonic source speeds). This problem has relevance to situations in Contact Mechanics, Tribology and Dynamic Fracture, and is especially related to the well-known heat checking problem (thermo-mechanical cracking in an unflawed half-space material from high-speed asperity excitations). Our solution technique fully exploits as auxiliary solutions the ones for the corresponding plane-strain and anti-plane shear problems by reducing the original 3D problem to two separate 2D problems. These problems are uncoupled from each other, with the first problem being thermoelastic and the second one pure elastic. In particular, the auxiliary plane-strain problem is completely analogous to the original problem, not only with regard to the field equations but also with regard to the boundary conditions. This makes the technique employed here more advantageous than other techniques, which require the prior determination of a fictitious auxiliary plane-strain problem through solving an integral equation.     

Two-temperature generalized thermoelasticity under ramp-type heating by finite element method

In this work, a general finite element model is proposed to analyze transient phenomena in thermoelastic half-space filled with an elastic material, which has constant elastic parameters. The governing equations are taken in the context of the two-temperature generalized thermoelasticity theory (Youssef in IMA J. Appl. Math. 71(3):383–390, 2006). A linear temperature ramping function is used to more realistically model thermal loading of the half-space surface. The medium is assumed initially quiescent. A finite element scheme is presented for the high accuracy numerical purpose. The numerical solutions of the non-dimensional governing partial differential equations of the problem have been shown graphically and some comparisons have been shown in figures to estimate the effect of the ramping parameter of heating and the parameter of two-temperature.     

Wave propagation in thermally conducting linear fibre-reinforced composite materials

The propagation of plane waves in a fibre-reinforced, anisotropic, generalized thermoelastic media is discussed. The governing equations in x–y plane are solved to obtain a cubic equation in phase velocity. Three coupled waves, namely quasi-P, quasi-SV and quasi-thermal waves are shown to exist. The propagation of Rayleigh waves in stress free thermally insulated and transversely isotropic fibre-reinforced thermoelastic solid half-space is also investigated. The frequency equation is obtained for these waves. The velocities of the plane waves are shown graphically with the angle of propagation. The numerical results are also compared to those without thermal disturbances and anisotropy parameters.     

Multiple-zone sliding contact with friction on an anisotropic thermoelastic half-space

The paper studies a class of multiple-zone sliding contact problems. This class is general enough to include frictional and thermal effects, and anisotropic response of the indented material. In particular, a rigid die (indenter) slides with Coulomb friction and at constant speed over the surface of a deformable and conducting body in the form of a 2D half-space. The body is assumed to behave as a thermoelastic transversely isotropic material. Thermoelasticity of the Green–Lindsay type is assumed to govern. The solution method is based on integral transforms and singular integral equations. First, an exact transform solution for the auxiliary problem of multiple-zone (integer n > 1) surface tractions is obtained. Then, an asymptotic form for this auxiliary problem is extracted. This form can be inverted analytically, and the result applied to sliding contacts with multiple zones. For illustration, detailed calculations are provided for the case of two (n = 2) contact zones. The solution yields the contact zone width and location in terms of sliding speed, friction, die profile, and also the force exerted. Calculations for the hexagonal material zinc illustrate effects of speed, friction and line of action of the die force on relative contact zone size, location of maximal values for the temperature and the compressive stress, and the maximum temperature for a given maximum stress. Finally, from our general results, a single contact zone solution follows as a simple limit.     

Response of welded thermoelastic solids to the rapid motion of thermomechanical sources parallel to the interface

The analysis of rapidly-moving thermomechanical surface sources is extended to the study of buried thermomechanical sources that move parallel to the interface of two welded dissimilar thermoelastic half-spaces at a constant subcritical speed. The sources are manifest as body force line loads in the coupled equations of thermoelasticity, and a 2-D steady-state situation is treated. Exact integral transform solutions are obtained, and expressions for the displacements and temperature changes are generated by analytical inversion of robust asymptotic versions of the transforms.These expressions show that thermoelastic coupling effects increase with source speed, and that the thermal source is always manifest in combination with a component of the mechanical source, i.e. an effective thermal source term exists. The expressions also exhibit component functions that are in effect hybrids of functions that are seen in purely thermal and isothermal elastic solutions.The critical source speed is defined as the minimum of the two asymptotic thermoelastic Rayleigh speeds in the half-spaces and, when it exists, the asymptotic thermoelastic Stoneley speed. Exact expressions for these speeds are given, and used to present some typical values.     

Elliptical crack and punch problems solved by integral equation method for piezoelectric media

The problem of an elliptical crack embedded in an unbounded transversely isotropic piezoelectric media with the crack-plane parallel to the plane of isotropy of the media and subjected to remote normal mechanical as well as electric loading is considered first. The problem has been successfully reduced to a pair of coupled integral equations that are suitable for the application of an integral equation method developed earlier for three-dimensional problems of LEFM. Solution to the mechanical displacement and electric potentials are obtained for prescribed uniform loadings and expressions for corresponding intensity factors and crack opening displacement are deduced. The above method has further been applied to solve the problem of a rigid flat-ended elliptical punch indenting a transversely isotropic piezoelectric half-space surface with the plane of isotropy parallel to the surface. Solutions to mechanical stress and electric displacement are obtained for prescribed constant normal displacement and constant electric potential interior to the elliptical region and expression for the total force required to maintain a prescribed indentation is deduced.     

Elasticity solutions for constant and linearly varying loads applied to a rectangular surface patch on the elastic half-space

The solution of the point load problem in the half-space is well known in the theory of elasticity. Using direct integration, the point solution can theoretically be used to develop the solution for loading various contact areas with a variety of loading profiles. Unfortunately, anything more complicated than constant pressure loading has previously required numerical integration, and hence, no closed form solution was obtainable. Partial solutions, i.e. solutions valid only on the surface of the half-space have also been available. This paper presents the methodology to generate complete solutions to the integrals for constant and linearly varying loads applied in both the normal and tangential directions everywhere in the half-space.     

A nonlinearly thermoelastic half-space under time-dependent normal and shear loading

A nonlinearly thermoelastic half-space is subjected to combined time-dependent normal and shear loading. The solution is obtained by a numerical method which is shown to yield accurate results by comparison with some known analytical solutions which can be obtained in some special cases. When shocks are involved, it is shown that the numerical results satisfy all the Rankine-Hugoniot jump conditions as well as the entropy condition across the shock.     

Reflection and transmission of plane waves at the interface of an elastic half-space and a fractional order thermoelastic half-space

The problem of reflection and transmission due to longitudinal and transverse waves incident obliquely at a plane interface between uniform elastic solid half-space and fractional order thermoelastic solid half-space has been studied. It is found that the amplitude ratios of various reflected and refracted waves are functions of angle of incidence and frequency of incident wave and are influenced by the fractional order thermoelastic properties of media. The expressions of amplitude ratios and energy ratios have been computed numerically for a particular model. The variation of amplitude and energy ratios with angle of incidence is shown graphically. The conservation of energy at the interface is verified.     

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.     

磁-电-弹性半空间在轴对称热载荷作用下的三维问题研究

考虑力-电-磁-热等多场耦合作用,基于线性理论给出了磁-电-弹性半空间在表面轴对称温度载荷作用下的热-磁-电-弹性分析,并得到了问题的解析解.利用Hankel积分变换法求解了磁-电-弹性材料中的热传导及控制方程,讨论了在磁-电-弹性半空间在边界表面上作用局部热载荷时的混合边值问题,利用积分变换和积分方程技术,通过在边界表面上施加应力自由及磁-电开路条件,推导得到了磁-电-弹性半空间中位移、电势及磁势的积分形式的表达式.获得了磁-电-弹性半空间中温度场的解析表达式并且给出了应力,电位移和磁通量的解析解.数值计算结果表明温度载荷对磁-电-弹性场的分布有显著影响.当温度载荷作用的圆域半径增大时,最大正应力发生的位置会远离半无限大体的边界;反之当温度载荷作用的圆域半径减小时,最大应力发生的位置会靠近半无限大体的边界.电场和磁场在温度载荷作用的圆域内在边界表面附近有明显的强化,而磁-电-弹性场强化区域的强化程度跟温度载荷的大小和作用区域大小相关.本研究的相关结果对智能材料和结构在热载荷作用下的设计和制造具有指导意义.     

Three-dimensional analytical elasticity solution for loaded functionally graded coated half-space

In the present paper, a three-dimensional problem of elasticity of normal and tangential loading of surface of the functionally graded coated half-space is considered. In case when Poisson's ratio is constant and the Young's modulus is a power or exponential function of the distance from the surface of the half-space, analytical solution using Fourier transform is obtained. Stress field due to Hertz contact pressure in an elliptical region are studied as a function of the parameter b/a (where a and b are axes of the contact ellipse) and coating thickness.