Two-Dimensional Green's Function for Thermal Stresses in a Semi-Layer Under a Point Heat Source

We present specific new expressions for thermal stresses as Green's functions for a plane boundary value problem of steady-state thermoelasticity for a semi-layer. We also obtain new integration formulas of Green's type, which determine the thermal stresses in the form of integrals of the products of the given distributed internal heat source, boundary temperature, and heat flux and derived kernels. Elementary functions results obtained are formulated in a theorem, which is proved using the harmonic integral representations method to derive thermal stresses Green's functions, which are written in terms of Green's functions for Poisson's equation. A new solution to particular two-dimensional boundary value problem for a semi-layer under a boundary constant temperature gradient is obtained in explicit form. Graphical presentations for thermal stresses Green's functions created by a unit heat source (line load in out-of-plane direction) and by a temperature gradient are also included.     

Steady-state Green's functions for thermal stresses within rectangular region under point heat source

This article presents new steady-state Green's functions for displacements and thermal stresses for plane problem within a rectangular region. These results were derived on the basis of structural formulas for thermoelastic Green's functions which are expressed in terms of Green's functions for Poisson's equation. Structural formulas are formulated in a special theorem, which is proved using the author's developed harmonic integral representation method. Green's functions for thermal stresses within rectangle are obtained in the form of a sum of elementary functions and ordinary series. In the particular cases for a half-strip and strip, ordinary series vanish and Green's functions are presented by elementary functions. These concrete results for Green's functions and respective integration formulas for thermoelastic rectangle, half-strip and strip are presented in another theorem, which is proved on the basis of derived structural formulas. New analytical expressions for thermal stresses to a particular plane problem for a thermoelastic rectangle under a boundary constant temperature gradient also are derived. Analytical solutions were presented in the form of graphics. The fast convergence of the infinite series is demonstrated on a particular thermoelastic boundary value problem (BVP). The proposed technique of constructing thermal stresses Green's functions for a rectangle could be extended to many 3D BVPs for unbounded, semibounded, and bounded parallelepipeds.     

Exact Elementary Green's Functions and Poisson-Type Integral Formulas for a Thermoelastic Half-Wedge with Applications

In this paper new exact Green's functions and new exact Poisson-type integral formula for a boundary value problem (BVP) in thermoelastostatics for a half-wedge with mixed homogeneous mechanical boundary conditions (the boundary angle is free of loadings and normal displacements and tangential stresses are prescribed on the boundary quarter-planes) are derived. The thermoelastic displacements are produced by a heat source applied in the inner points of the half-wedge and by mixed non-homogeneous boundary heat conditions (the temperature is prescribed on the boundary angle and the heat fluxes are given on the boundary quarter-planes). When thermoelastic Green's function is derived the thermoelastic displacements are generated by an inner unit point heat source, described by δ-Dirac's function. All results are obtained in terms of elementary functions and they are formulated in a special theorem. Analogous results for an octant and for a quarter-space as particular cases of the angle of the thermoelastic half-wedge also are obtained. The main difficulties to obtain these results are in deriving the functions of influence of a unit concentrated force onto elastic volume dilatation Θ^(q) and, also, in calculating a volume integral of the product of function Θ^(q) and Green's function in heat conduction. Exact solutions in elementary functions for two particular BVPs of thermoelasticity for a quarter-space and a half-wedge, using the derived Poisson-type integral formula and the influence functions Θ^(q) also are included. The proposed approach may be extended not only for many different BVPs for half-wedge, but also for many canonical cylindrical and other orthogonal domains.     

Recent Integral Representations for Thermoelastic Green's Functions and Many Examples of Their Exact Analytical Expressions

This article is devoted to derivation of new integral representations for the main thermoelastic Green's functions (MTGFs), based on the presentation of solutions of respective Lamé elliptic differential equations via Green's functions for the Poisson equation (GFPEs). The newly derived integral representations in Cartesian coordinates permitted the proof of a theorem about constructive formulas for MTGFs expressed in terms of respective GFPEs. The thermoelastic displacements are generated by a unitary heat source, applied in an arbitrary inner point of a generalized boundary values problem (BVP) of thermoelasticity for an octant at different homogeneous mechanical and thermal boundary conditions, prescribed on its marginal quadrants. According to the constructive formulas obtained, the derivation of MTGFs for about 20 BVPs for a plane, a half-plane, a quadrant, a space, a quarter-space, and an octant may be obtained by changing the respective well-known GFPEs. All results obtained are in terms of elementary functions with many examples of their validation. Two new MTGFs for quarter-space and octant, together with some of their graphical computer evaluations, are also included. The main advantages of the proposed approach in comparison with the GΘ convolution method for MTGFs constructing are: First, it is not necessary to derive the functions of influence of a unit concentrated force onto elastic volume dilatation - Θ⁽ⁱ⁾. Second, it is not necessary to calculate an integral of the product of the volume dilatation and Green's function in heat conduction. By using the proposed approach it is possible to extend obtained results for Cartesian domains onto areas of any orthogonal system of coordinates.     

Thermoelastic Equilibrium of Some Semi-Infinite Domains Subjected to the Action of a Heat Source

By using the integral representations for main thermoelastic Green's functions (MTGFs) we prove a theorem about new structural formulas for MTGFs for a whole class of boundary value problems (BVPs) of thermoelasticity for some semi-infinite Cartesian domains. According to these new structural formulas many MTGFs for a plane, a half-plane, a quadrant, a space, a quarter-space and an octant may be obtained by changing the respective well-known GFPE and their regular parts. The crucial moment of our investigation consists of elaboration of a new technique for calculating some generalized integrals containing products of two different GFPEs. Also, the types of boundary conditions for volume dilatation considered and GFPE for temperature differ on a single boundary only. As example of application of the obtained new structural formulas, the new MTGFs for a concrete BVP of thermoelaesticity for an octant are derived in elementary functions. The MTGFs obtained are validated on a known example for a BVP for half-space. Graphical computer evaluation of the derived in elementary functions new MTGFs is included.     

New Explicit Green's Function and Poisson's Integral Formula for a Thermoelastic Quarter-Space

In this paper a new Green's function and a new Poisson-type integral formula for a boundary value problem (BVP) in thermoelasticity for a quarter-space with mixed homogeneous mechanical boundary conditions are derived. The thermoelastic displacements are generated by a heat source, applied in the inner points of the quarter-space and by heat flux, prescribed on its boundary half-planes. All results are obtained in closed forms that are formulated in a special theorem. A closed form solution for a particular BVP of thermoelasticity for a quarter-space also is included. The main difficulties to obtain these results are in deriving the functions of influence of a unit concentrated force onto elastic volume dilatation Θ^(k) and, also, in calculating a volume integral of the product of function Θ^(k) and Green's function in heat conduction. Using the proposed approach it is possible to extend the obtained results not only for any canonical Cartesian domain, but also for any orthogonal one.     

Thermoelastic influence functions and solution to a locally mixed BVP for a strip

This article presents in closed form new influence functions for displacements and stresses to a boundary value problems (BVP) of thermoelasticity within a strip, caused by a unit point heat source. We also obtain the respective new integration formula of Green’s type, which directly determines the thermal stresses in the form of integrals of the products between specified internal heat sources, temperature, or heat flux prescribed on boundary and derived thermoelastic influence functions (kernels). The general Green’s type integral formula permits to derive new solution to one particular BVP of thermoelasticity for a strip in the form of elementary functions. Graphical representation of thermal stresses, created by an internal point heat source and by a boundary temperature, is included.     

NEW FORMULAE FOR DYNAMICAL THERMAL STRESSES

A generalization of the function of influence of a unit heat source to the displacements is suggested for the boundary value problems in the dynamical uncoupled thermoelasticity. This generalization is a convolution over time and bulk of two influence functions. One of them is a Green's function for the heat conduction problem. The other is a function of influence of unit concentrated forces onto bulk dilatation. Broad possibilities are shown in constructing these influence functions. In particular, the theorem on dilatation constructing is proved. To calculate the convolutions successfully the following properties of the introduced function are found to be useful. (1) In coordinates of the point of observation, the function satisfies the equations used to find the Green's functions in the problem of heat conduction, with the unit heat source being replaced by the influence function of concentrated force onto dilatation; and (2) in coordinates of the point of heat source application, it satisfies the boundary value problem used to find Green's matrix, with the unit concentrated forces being replaced by derivatives of Green's function in the problem of heat conduction. Based on the introduced influence function, some new integral formulae for displacements and stresses are obtained, which are a generalization of Mysel's formula in the theory of dynamical thermal stresses. The proposed formulae have certain advantages allowing us to unite the two-staged process of finding the solutions for boundary value problems in thermoelasticity in a single stage. It is established that, based on the obtained results it becomes possible to compile a whole handbook on the influence functions and integral solutions for boundary value problems in dynamical thermoelasticity. As examples, the solutions for two boundary value problems in the theory of dynamical thermal stresses for the half-space and quarter-space are presented.     

Three-dimensional thermal stresses Green’s functions for an unbounded parallelepiped under an inner point heat source

The aim of this study is to derive new constructive formulas and analytical expressions for Green’s functions (GFs) to 3D generalized boundary value problem (BVP) for an unbounded parallelepiped under a point heat source. These results were obtained using the developed harmonic integral representation method. On the base of derived constructive formulas it is possible to obtain analytical expressions for thermal stresses GFs to 16 BVPs for unbounded parallelepiped. An example of such kind is presented for a spatial BVP, GFs of which are presented in the form of the sum of elementary functions and double infinite series, containing products between exponential and trigonometric functions. An integration formula for thermal stresses, caused by the thermal data, distributed on the boundary strips at homogeneous locally mixed mechanical boundary conditions was also derived. The main di?culty to obtain these results was calculating an integral of the product between two GFs for Poisson’s equation. This integral taken on the base of the earlier established statement that main thermoelastic displacement Green’s functions (MTDGFs) satisfy the boundary conditions: (a) homogeneous mechanical conditions with respect to points of findings MTDGFs and (b) homogeneous thermal conditions with respect to points of the application of the heat source.     

Green's Function for Transversely Isotropic Thermoelastic Diffusion Bimaterials

The aim of the present article is to study the Green's function in transversely isotropic thermoelastic diffusion bimaterial. With this objective, first the three-dimensional general solution in transversely isotropic thermoelastic diffusion bimaterial is derived. On the basis of general solution, Green's function, with a concentrated heat source in steady state, is completely solved using harmonic functions. The components of displacement, stress, temperature distribution, and mass concentration are expressed in terms of elementary functions. The resulting quantities are computed numerically and illustrated graphically. A particular case of three-dimensional Green function in transversely isotropic thermoelastic bimaterial has been deduced from the present investigation.     

Thermal Green's Functions for a Heat Source in a Piezoelectric Solid with a Parabolic Boundary

Using the Stroh formalism combined with the analytical continuation principle of Muskhelishvili, the Green's functions for a line heat source in a piezoelectric solid with a parabolic boundary are obtained in closed form. The obtained Green's functions not only satisfy all the given boundary conditions, but also ensure the displacement and electric potential to be single-valued. As special cases, the solutions for a piezoelectric half-plane are also presented, and they are shown to be consistent with previous works.     

Green's function for infinite thin plate with elliptic hole under bending heat source and interaction between elliptic hole and crack under uniform bending heat flux

Green's function is derived for the bending problem of an infinite thin plate with an elliptic hole under a bending heat source. Then the interaction problem between an elliptic hole and a crack in a thin plate under uniform bending heat flux is analyzed. First, the complex variable method is developed for the thermoelastic problem of bending. Then an exact solution in explicit form is derived for the Green's function by using the complex variable method. Distributions of temperature moment, heat flux moments, bending moments along the hole edge are shown in figures. For solving the interaction problem, a solution for an infinite thin plate with an adiabatic elliptic hole under uniform bending heat flux, and two Green's functions of the plate under a bending heat source couple and a bending dislocation are given. The interaction problem then reduces into singular integral equations using the Green's functions and the principle of superposition. After the equations are solved numerically, the moment intensity factors at crack tips are presented in the figures.     

New Integral Representations in the Dynamic Uncoupled Thermoelasticity

New integral representations of homogeneous 3D uncoupled dynamic thermoelasticity for semi-infinite cylindrical domains with curvilinear surfaces placed at infinity and subject to mixed boundary conditions on the plane boundaries are obtained. The representations are given in the form of integral convolutions involving a Green's function for the parabolic heat conduction equation, as well as Green's function for the isothermal elastodynamics. A multi-integral representation of solution to a particular initial-boundary value problem for an infinite wedge is included.     

D'Alembert's Solution in Thermoelasticity – Impact of a Rod Against a Heated Barrier: Part II. A Case of Coupled Strain and Temperature Fields

The problem of impact of a thermoelastic rod against a heated rigid barrier is considered, in so doing lateral surfaces and free end of the rod are heat insulated, and free heat exchange between the rod and the rigid obstacle or ideal thermal contact occurs within contacting end. The rod's thermoelastic behavior is described by the Green–Naghdi theory of thermoelasticity. D'Alembert's method, which is based on the analytical solution of equations of the hyperbolic type describing the dynamic behavior of the thermoelastic rod, is used as the method of solution. This solution involves four arbitrary functions which are determined from the initial and boundary conditions and are piecewise constant functions. The procedure developed enables one to analyze the influence of thermoelastic parameters on the values to be found and to investigate numerically the longitudinal coordinate dependence of the desired functions at each fixed instant of the time beginning from the moment of the rod's collision with the barrier up to the moment of its rebound both without account for the stress and temperature fields coupling (in the companion paper, Part I) and in the case of coupling thermoelasticity (in this paper). As a numerical example, the impact of a thermoelastic rod against a heated barrier is considered with a small parameter of coupling between the strain and temperature fields. It has been shown that the presence of small coupling results in the generation of a new shock wave of small amplitude, namely: the reflected thermal wave from the incident elastic wave at the free rod's end. The rod's rebound may occur either at the moment of simultaneous arrival at the contact place of two reflected waves: elastic wave from the incident thermal wave and thermal wave from the incident elastic wave—or at the time when the reflected elastic wave from the incident elastic wave reaches the contact point.     

GENERALIZED THERMOELASTICITY: CLOSED-FORM SOLUTIONS

A study of the one-dimensional thermoelastic waves produced by an instantaneous plane source of heat in homogeneous isotropic infinite and semi-infinite bodies of the Green-Lindsay (G-L) type is presented. Closed-form Green's functions corresponding to the plane heat source are obtained using the decomposition theorem for a potential-temperature wave of the G-L theory. Qualitative analysis of the results is included.     

Axisymmetric and One-Dimensional Thermoelastic Fields of Radially Graded Bodies

In this paper, both Young's modulus and Poisson's ratio along with thermal expansion coefficient are allowed to vary across the radius in a solid ring and a curved beam. Effects of non-constant Poisson's ratio on the thermoelastic field in these graded axisymmetric and one-dimensional problems are studied. A governing differential equation in terms of stress function is obtained for general axisymmetric and one-dimensional problems. Two linearly independent solutions in terms of hypergeometric functions are then attained to calculate the stresses and the strains. Using Green's function method, a form of a solution for the stress functions in terms of integral equations for a curved beam and a solid ring are obtained. Specifically, closed form solutions for the stress functions, when Young's modulus and Poisson's ratio are expressed as power law functions across the radius, are calculated. The results show that the effect of varying Poisson's ratio upon the thermal stresses is considerable for the solid ring. In addition, a non-constant Poisson's ratio has significant influences on the thermal strain field in solid rings. The effect of varying Poisson's ratio upon the thermal stresses is negligible for the curved beam. However, non-constant Poisson's ratios have substantial effects on the thermal strain field in curved beams. Finally, the effects of varying Poisson's ratio on the thermal stresses in thick solid rings and curved beams are also investigated.     

D'Alembert's Solution in Thermoelasticity—Impact of a Rod Against a Heated Barrier: Part I. A Case of Uncoupled Strain and Temperature Fields

The problem of impact of a thermoelastic rod against a heated rigid barrier is considered, in so doing lateral surfaces and free end of the rod are heat insulated, while there is either free heat exchange between the rod and the rigid obstacle within contacting end or ideal thermal contact, as a particular case. The rod's thermoelastic behavior is described by the Green–Naghdi theory of thermoelasticity. D'Alembert's method, which is based on the analytical solution of equations of the hyperbolic type describing the dynamic behavior of the thermoelastic rod, is used as the method of solution. This solution involves four arbitrary functions which are determined from the initial and boundary conditions and are piecewise constant functions. The procedure developed enables one to analyze the influence of thermoelastic parameters on the values to be found, as well as to investigate numerically the longitudinal coordinate dependence of the desired functions at each fixed instant of the time beginning from the moment of the rod's collision with the barrier up to the moment of its rebound. The case of uncoupled stress and temperature fields is examined in the first part of the paper, while the case of coupling thermoelasticity is considered in detail in the companion paper. It has been shown that the possibility for generating the reflected thermal wave from the incident elastic wave at the free rod's end is unavailable in the case of the uncoupled strain and temperature fields, and that the rod's rebound may occur either at the moment of arrival at the contact place of the reflected elastic wave from the incident thermal wave or at the time when the reflected elastic wave from the incident elastic wave reaches the contact point.     

GREEN'S FUNCTION APPROACH TO THREE-DIMENSIONAL HEAT CONDUCTION EQUATION OF FUNCTIONALLY GRADED MATERIALS

A Green's function approach based on the laminate theory is adopted to solve the three-dimensional heat conduction equation of functionally graded materials (FGMs) with one-directionally dependent properties. An approximate solution for each layer is substituted into the governing equation to yield an eigenvalue problem. The eigenvalues and the corresponding eigenfunctions obtained by solving an eigenvalue problem for each layer constitute the Green's function solution for analyzing the three-dimensional transient temperature. The eigenvalues and the corresponding eigenfunctions are determined from the homogeneous boundary conditions at outer sides and from the continuous conditions of temperature and heat flux at the interfaces. A three-dimensional transient temperature solution with a source is formulated by the Green's function. Numerical calculations are carried out for an FGM plate, and the numerical results are shown in tables and figures.     

GREEN'S FUNCTIONS OF POINT HEAT SOURCE IN VARIOUS THERMOELASTIC BOUNDARY VALUE PROBLEMS

With an emphasis on derivation, this paper reviews Green's functions for a point heat source in various thermoelastic boundary value problems for an infinite plane with an inhomogeneity. The inhomogeneity boundary conditions considered are external force, displacement, and mixed boundaries. The derivation is accomplished with three kinds of function for mapping, temperature, and stress. Stress functions for different boundary value problems are presented, and stress distributions along the inhomogeneity boundary are plotted in figures.     

KIRCHHOFF'S FORMULA FOR THERMOELASTIC SOLID

A generalization of Kirchhoff's theorem for the classical wave equation to the case of a central system of field equations of coupled thermoelasticity is established in this paper. The theorem asserts that a pair (Φ, θ), where Φ and θ denote the thermoelastic displacement potential and temperature, respectively, can be expressed by surface integrals over the boundary of a thermoelastic solid whose kernels have the form of an infinite series satisfying the wave-like and heat-like equations occurring in the decomposition theorem for the central system of equations ([3]).