A Method Based on the Radon Transform for Three-Dimensional Elastodynamic Problems of Moving Loads

An integral transform procedure is developed to obtain fundamental elastodynamic three-dimensional (3D) solutions for loads moving steadily over the surface of a half-space. These solutions are exact, and results are presented over the entire speed range (i.e., for subsonic, transonic and supersonic source speeds). Especially, the results obtained here for the tangential loads (one of these loads is along the direction of motion and the other is orthogonal to the direction of motion) are quite new in the literature. The solution technique is based on the use of the Radon transform and elements of distribution theory. It also fully exploits as auxiliary solutions the ones for the corresponding plane-strain and anti-plane shear problems (the latter problems are 2D and uncoupled from each other). In particular, it should be noticed that the plane-strain problem here is completely analogous to the original 3D problem, not only with respect to the field equations but also with respect to the boundary conditions. This makes the present technique more advantageous than other techniques, which require first the determination of a fictitious auxiliary plane-strain problem through the solution of an integral equation. Our approach becomes particularly simple when there is no angular dependence in the boundary conditions (i.e., when axially symmetric problems regarding their boundary conditions are considered). On the contrary, no such constraint needs to be fulfilled as regards the material response (and, therefore, the governing equations of the problem) and/or also possible loss of axisymmetry due to the generation of shock (Mach-type) waves in the medium. Therefore, the solution technique can easily deal with general 3D problems having a largely arbitrary radial dependence in the boundary conditions and involving: (i) material anisotropy in static and dynamic situations, and (ii) asymmetry caused by changes in the nature of governing PDEs due to the existence of different velocity regimes (super-Rayleigh, transonic, supersonic) in dynamic situations. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

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.     

Thermoelastodynamic disturbances in a half-space under the action of a buried thermal/mechanical line source

The transient dynamic coupled-thermoelasticity problem of a half-space under the action of a buried thermal/mechanical source is analyzed here. This situation aims primarily at modeling underground explosions and impulsively applied heat loadings near a boundary. Also, the present basic analysis may yield the necessary field quantities required to apply the Boundary Element Method in more complicated thermoelastodynamic problems involving half-plane domains. A material response for the half-space predicted by Biots thermoelasticity theory is assumed in an effort to give a formulation of the problem as general as possible (within the confines of a linear theory) . The loading consists of a concentrated thermal source and a concentrated force (mechanical source) having arbitrary direction with respect to the half-plane surface. Both thermal and mechanical line sources are situated at the same location in a fixed distance from the surface. Plane-strain conditions are assumed to prevail. Our problem can be viewed as a generalization of the classical Nakano–Lapwood–Garvin problem and its recent versions due to Payton (1968) and Tsai and Ma (1991) . The initial/boundary value problem is attacked with one- and two-sided Laplace transforms to suppress, respectively, the time variable and the horizontal space variable. A 9×9 system of linear equations arises in the double transformed domain and its exact solution is obtained by employing a program of symbolic manipulations. From this solution the two-sided Laplace transform inversion is then obtained exactly through contour integration. The one-sided Laplace transform inversion for the vertical displacement at the surface is obtained here asymptotically for long times and numerically for short times.     

Static equilibrium of a thermoelastic half-plane: Green’s functions and solutions in integrals

This article presents in a closed form new influence functions of a unit point heat source on the displacements for three boundary value problems of thermoelasticity for a half-plane. We also obtain the corresponding new integral formulas of Green’s and Poisson’s types that directly determine the thermoelastic displacements and stresses in the form of integrals of the products of specified internal heat sources or prescribed boundary temperature and constructed already thermoelastic influence functions (kernels). All these results are presented in terms of elementary functions in the form of three theorems. Based on these theorems and on derived early by author the general Green-type integral formula, we obtain in elementary functions new solutions to two particular boundary value problems of thermoelasticity for half-plane. The graphical presentation of the temperature and thermal stresses of one concrete boundary value problems of thermoelasticity for half-plane also is included. The proposed method of constructing thermoelastic Green’s functions and integral formulas is applicable not only for a half-plane, but also for many other two- and three-dimensional canonical domains of different orthogonal coordinate systems.     

On the effect of the heat generated during a stress-induced thermoelastic phase transformation

In this paper we examine the stress-elongation response of a bar undergoing a thermoelastic phase transition. Attention is focussed on how this response is affected by the heat generated during the transformation. The analysis is based on a continuum model consisting of a two-well Helmholtz free-energy function, a kinetic relation and a nucleation criterion. The governing mathematical problem is related to one that describes a moving heat source, except that here, the strength and speed of the source are not knowna priori and the energy field equation involves coupling between thermal and mechanical effects. A finite difference solution of this moving boundary-value problem is carried out. The heat generated by the transformation is found to have a significant effect on the mechanical response whenever the prescribed elongation-rate is moderately large.     

GN理论下受到移动内热源的半空间问题的研究

运用无能量耗散的热弹性GN理论研究了受到移动内热源的半空间问题.通过势函数法使问题 转化成一组偏微分方程,采用Laplace变换和Fourier变换法得到问题在变换域内表面位移 精确解. 运用级数展开法得到在小时间范围内表面位移的近似解.给出近似解的适用范围,同时给出热 源固定不动和非耦合理论下问题的解.并对铜介质进行了数值计算.     

Thermal stress intensity factors for a crack in an anisotropic half plane

On the basis of the two-dimensional theory of anisotropic thermoelasticity, a solution is given for the thermal stress intensity factors due to the obstruction of a uniform heat flux by an insulated line crack in a generally anisotropic half plane. The crack is replaced by continuous distributions of sources of temperature discontinuity and dislocations. First, the particular thermoelastic dislocation solutions for the half plane are obtained; then the corresponding isothermal solutions are superposed to satisfy the traction-free conditions on the crack surfaces. The dislocation solutions are applied to calculate the thermal stress intensity factors, which are validated by the exact solutions. The effects of the uniform heat flux, the ply angle and the crack length are investigated.     

Analogue solution of two-dimensional thermoelastic problems with internal heat generation and heat transfer

The analogue procedure which the authors have proposed² is expanded to plane thermoelastic fields, with heat generation within the region or heat transfer from both surfaces of the thermoelastic field. A plate similar to the thermoelastic field is prepared and the plate-bending test is carried out with distributed loads over the plate and with a concentrated load on the inner flange. The strains are measured carefully for each bending test and these are combined to transform into the thermal stress of the original thermoelastic field. The results are compared with theoretical or numerical ones and good agreement is confirmed among them.     

广义热弹模型的热力学基础与瞬态响应

微纳科技的快速发展与超短脉冲激光技术的广泛运用, 对描述微纳尺度超快热冲击的广义热传导及其热弹耦合理论提出迫切需求. 基于拓展热力学原理, 本文建立了考虑热传导双相滞后效应和高阶热流率的广义热弹耦合理论. 类比于力学领域黏弹性本构关系的串联、并联模型, 并受Green-Naghdi (GN)广义热传导模型启发, 本文提出了热学“弹性”单元和“黏性”单元模型, 并采用串联、并联方法实现了Cattaneo-Vernotte (CV)、GN、双相滞后(DPL)和Moore-Gibson-Thompson (MGT) 热传导模型的重构. 理论推导进一步表明, 本文新建模型对应于热学Burgers模型, 并得到了新模型中各相位滞后中松弛时间之间的比例关系. 运用拉普拉斯变换方法, 研究了一维结构受边界热冲击和移动热源作用下的瞬态响应, 计算结果表明: 新模型克服了热波速度无限大的悖论; 仅有边界热冲击载荷时, 新模型得到的响应结果均较大, 响应范围最小; 相比于无热源作用情形, 受移动热源作用时, 新模型会产生更大的峰值响应. 新模型与经典弹性理论耦合构建了广义热弹性理论, 运用该理论, 可以清晰观察到在热波和弹性波波前的应力突变. 理论方面, 本文推动了拓展热力学与连续介质力学的结合, 对于远离平衡态极端力学基础理论问题的研究具有启发意义; 应用方面, 本文研究结果可为激光等移动热源作用下材料的瞬态响应分析提供理论基础和数值方法.      

磁_微极广义热弹性介质中轴对称变形的弹性动力学

The present investigation is concerned with an axi-symmetric problem in the electromagnetic micropolar thermoelastic half-space whose surface is subjected to the mechanical or thermal source. Laplace and Hankel transform techniques are used to solve the problem. Various types of sources are taken to illustrate the utility of the approach. Integral transforms are inverted by using a numerical technique to obtain the components of stresses, temperature distribution, and induced electric and magnetic fields. The expressions of these quantities are illustrated graphically to depict the magnetic effect for two different generalized thermoelasticity theories, i.e., Lord and Shulman （L-S theory） and Green and Lindsay （G-L theory）. Some particular interesting cases are also deduced from the present investigation.     

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.     

Fundamental solution for transversely isotropic thermoelastic materials

We use the compact harmonic general solutions of transversely isotropic thermoelastic materials to construct the three-dimensional fundamental solutions for a steady point heat source in an infinite transversely isotropic thermoelastic material and a steady point heat source on the surface of a semi-infinite transversely isotropic thermoelastic material by three newly introduced harmonic functions, respectively. All components of coupled field are expressed in terms of elementary functions and are convenient to use. Numerical results for hexagonal zinc are given graphically by contours.     

Thermoelastostatic equilibrium of a spatial quadrant: Green’s function and solution in integrals

In this study, a new Green??s function and a new Green-type integral formula for a 3D boundary value problem (BVP) in thermoelastostatics for a quarter-space are derived in closed form. On the boundary half-planes, twice mixed homogeneous mechanical boundary conditions are given. One boundary half-plane is free of loadings and the normal displacements and the tangential stresses are zero on the other one. The thermoelastic displacements are subjected by a heat source applied in the inner points of the quarter-space and by mixed non-homogeneous boundary heat conditions. On one of the boundary half-plane, the temperature is prescribed and the heat flux is given on the other one. When the thermoelastic Green??s function is derived, the thermoelastic displacements are generated by an inner unit point heat source, described by ??-Dirac??s function. All results are obtained in elementary functions that are formulated in a special theorem. As a particular case, when one of the boundary half-plane of the quarter-space is placed at infinity, we obtain the respective results for half-space. Exact solutions in elementary functions for two particular BVPs for a thermoelastic quarter-space and their graphical presentations are included. They demonstrate how to apply the obtained Green-type integral formula as well as the derived influence functions of an inner unit point body force on volume dilatation to solve particular BVPs of thermoelasticity. In addition, advantages of the obtained results and possibilities of the proposed method to derive new Green??s functions and new Green-type integral formulae not for quarter-space only, but also for any canonical Cartesian domain are also discussed.     

考虑应变率的广义压电热弹理论及其应用

工程中大量材料的形变介于弹性与黏性之间, 既具有弹性固体特性, 又具有黏性流体特点, 即为黏弹性. 黏弹性使得材料出现很多力学松弛现象, 如应变松弛、滞后损耗等行为. 在研究受热载荷作用的多场耦合问题的瞬态响应时, 考虑此类问题中的热松弛和应变松弛现象, 对准确描述其瞬态响应尤为重要. 针对广义压电热弹问题的瞬态响应, 尽管已有学者建立了考虑热松弛的广义压电热弹模型, 但迄今, 尚未计入应变松弛. 本文中, 考虑到材料变形时的应变松弛, 通过引入应变率, 在Chandrasekharaiah广义压电热弹理论的基础之上, 经拓展, 建立了考虑应变率的广义压电热弹理论. 借助热力学定律, 给出了理论的建立过程并得到了相应的状态方程及控制方程. 在本构方程中, 引入了应变松弛时间与应变率的乘积项, 同时, 分别在本构方程和能量方程中引入了热松弛时间因子. 其后, 该理论被用于研究受移动热源作用的压电热弹一维问题的动态响应问题. 采用拉普拉斯变换及其数值反变换, 对问题进行了求解, 得到了不同应变松弛时间和热源移动速度下的瞬态响应, 即无量纲温度、位移、应力和电势的分布规律, 并重点考察了应变率对各物理量的影响效应, 将结果以图形形式进行了表示. 结果表明: 应变率对温度、位移、应力和电势的分布规律有显著影响.      

A GENERALIZED PIEZOELECTRIC-THERMOELASTIC THEORY WITH STRAIN RATE AND ITS APPLICATION 1)

工程中大量材料的形变介于弹性与黏性之间, 既具有弹性固体特性, 又具有黏性流体特点, 即为黏弹性. 黏弹性使得材料出现很多力学松弛现象, 如应变松弛、滞后损耗等行为. 在研究受热载荷作用的多场耦合问题的瞬态响应时, 考虑此类问题中的热松弛和应变松弛现象, 对准确描述其瞬态响应尤为重要. 针对广义压电热弹问题的瞬态响应, 尽管已有学者建立了考虑热松弛的广义压电热弹模型, 但迄今, 尚未计入应变松弛. 本文中, 考虑到材料变形时的应变松弛, 通过引入应变率, 在Chandrasekharaiah广义压电热弹理论的基础之上, 经拓展, 建立了考虑应变率的广义压电热弹理论. 借助热力学定律, 给出了理论的建立过程并得到了相应的状态方程及控制方程. 在本构方程中, 引入了应变松弛时间与应变率的乘积项, 同时, 分别在本构方程和能量方程中引入了热松弛时间因子. 其后, 该理论被用于研究受移动热源作用的压电热弹一维问题的动态响应问题. 采用拉普拉斯变换及其数值反变换, 对问题进行了求解, 得到了不同应变松弛时间和热源移动速度下的瞬态响应, 即无量纲温度、位移、应力和电势的分布规律, 并重点考察了应变率对各物理量的影响效应, 将结果以图形形式进行了表示. 结果表明: 应变率对温度、位移、应力和电势的分布规律有显著影响.     

Numerical and analytical study of the propagation of thermoelastic waves in a medium with heat-flux relaxation

The thermoelastic problem of laser exposure of metals and dielectrics is studied taking into account the finite speed of propagation of thermal waves and using a numerical finite-difference algorithm. The resulting numerical solution is compared with the analytical one. The problem is solved in coupled and uncoupled formulations. The solutions of the hyperbolic thermoelastic problem are compared with the solutions of the classical problem. Analytical expressions are obtained for the propagation speeds of the thermoelastic wave components. Times are determined at which the difference between the solutions of the hyperbolic and classical thermoelastic problems can be detected experimentally.     

Fundamental solutions of the thermoelastic equations for transversely isotropic plates

The problem of thermoelasticity for transversely isotropic plates acted upon by concentrated heat sources is solved. The {1, 2}-order equations of thermoelasticity that incorporate the transverse shear and normal stresses are used. A bending heat source with symmetric heat transfer is considered. The dependence of thermal stress components on the thermal and thermomechanical parameters of transversely isotropic plates is studied     

Relaxation Effects of a Saturated Porous Media Using the Two-Dimensional Generalized Thermoelastic Theory

A model of the equations of a generalized thermoelasticity (GT) with relaxation times for a saturated porous medium is given in this article. The formulation can be applied to the GT theories: Lord–Shulman theory, Green–Lindsay theory, and Coupled theory for the porous medium. A two-dimensional thermoelastic problem that is subjected to a time-dependent thermal/mechanical source is investigated with the model of the generalized porous thermoelasticity. By using the Laplace transform and the Fourier transform technique, solutions for the displacement, temperature, pore pressure, and stresses are obtained with a semi-analytical approach in the transform domain. Numerical results are also performed for portraying the nature of variations of the field variables. In addition, comparisons are presented with the corresponding four theories.     

Generalized solutions of transient thermal shock problem with bounded boundaries

In this work, the generalized thermoelastic solutions with bounded boundaries for the transient shock problem are proposed by an asymptotic method. The governing equations are taken in the context of the generalized thermoelasticity with one relaxation time (L–S theory). The general solutions for any set of boundary conditions are obtained in the physical domain by the Laplace transform techniques. The corresponding asymptotic solutions for a thin plate with finite thickness, subjected to different sudden temperature rises in its two boundaries, are obtained by means of the limit theorem of Laplace transform. In the context of these asymptotic solutions, two specific problems with different boundary conditions have been conducted. The distributions of displacement, temperature and stresses, as well as the propagations, intersections and reflections of two elastic waves, named as thermoelastic wave and thermal wave separately, are obtained and plotted. These results are agreed with the results obtained in the existing literatures.