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.     

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.     

Solution in Elementary Functions to a BVP of Thermoelasticity: Green's Functions and Green's-Type Integral Formula for Thermal Stresses within a Half-Strip

This article presents new elementary Green's functions for displacements and stresses created by a unit heat source applied in an arbitrary interior point of a half-strip. We also obtain the corresponding new integration formulas of Green's and Poisson's types which directly determine the thermal stresses in the form of integrals of the products of internal distributed heat source, temperature, or heat flux prescribed on boundary and derived thermoelastic influence functions (kernels). All these results are presented in terms of elementary functions in the form of a theorem. Based on this theorem and on derived early by author general Green's type integral formula, we obtain a new solution to one particular boundary value problem of thermoelasticity for half-strip. The graphical presentation of thermal stresses created by a unit point heat source and of thermal stresses for one particular boundary value problem of thermoelasticity for half-strip is also included. The proposed method of constructing thermoelastic Green's functions and integration formulas are applicable not only for a half-strip but also for many other two- and three-dimensional canonical domains of Cartesian system of coordinates.     

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.     

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.     

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.     

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.     

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.     

GREEN'S FUNCTION FOR A THERMOMECHANICAL MIXED BOUNDARY VALUE PROBLEM OF AN INFINITE PLANE WITH AN ELLIPTIC HOLE

This article derives the Green's function for a thermomechanical mixed boundary value problem of an infinite plane with an elliptic hole under a pair of heat source and sink. To derive the Green's function in closed form, the Cauchy integral method and a basic Green's function for an external force boundary value problem with a pair of heat source and sink are employed. Illustrative numerical results for temperature, heat flux, and stress along the hole edge and stress intensity factors when the hole collapses into a crack are presented graphically.     

Normal Mode Method for Two-Temperature Generalized Thermoelasticity Under Thermal Shock Problem

The theory of two-temperature generalized thermoelasticity, based on Youssef's theory, was used to solve boundary value problems of one-dimensional generalized thermoelasticity half-space by heating its boundary with different types of heating. The governing equations are solved using new mathematical methods within the purview of the Lord-?hulman (L-S) theory and the classical dynamical coupled theory (CD). The general solution obtained is applied to a specific problem of a half-space subjected to one type of heating—thermal shock type. The separation of variables method is used to get the exact expressions for distributions of displacement, the stresses, and temperature distribution. Variations of the considered functions through the horizontal distance are illustrated graphically. Comparisons are made with results between the two theories. Numerical work is also performed for a suitable material and results are discussed, specifically the conductive temperature, the dynamical temperature, and the stress and strain distributions are shown graphically when discussed.     

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.     

Green's Functions for a Point Heat Source in Circularly Cylindrical Layered Media

Within the framework of the linear theory of thermoelasticity, the problem of circularly cylindrical layered media subjected to an arbitrary point heat source is considered and solved in this paper. Based on the method of analytical continuation in conjunction with the alternating technique, the solutions to heat conduction and thermoelasticity problems for a three-phase multilayered cylinder are first derived. A rapid convergent series solution for both the temperature and stress functions, which is expressed in terms of an explicit general term of the complex potential of the corresponding homogeneous problem, is obtained in an elegant form. Numerical results are provided for some particular examples to investigate the effect of material combinations on the interfacial stresses.     

Determination of skin temperature distribution and heat flux during simulated fires using Green's functions over finite-length scales

One of the current practices for measuring heat flux during flash fire testing, forest fires, and other industrial cases focuses on the use of semi-infinite models to predict the heat flux during exposure through surface temperature measurements on simulated skin sensors. For short time frames, these models can be shown to have acceptable accuracy. However, when considering longer time exposures at reduced heat fluxes, such as with firefighters in a forest fire, the accuracy of these models could be brought into question. A one-dimensional, finite length scale, transient heat conduction model was developed using a Green's function approach on a rectangular sensor. The model was developed using transient temperature boundary conditions to avoid the use of complicated radiation and convection conditions at each boundary. For comparison, a semi-infinite model utilizing the same boundary condition on the exposed face was solved using both the Laplace transform method and Green's function method. Experimental data was obtained during exposure to a cone calorimeter. All measurements were taken for a minimum duration of 2 min. This temperature data was used to develop appropriate curves for the boundary conditions and validate the analytical models. It was found that the temperature obtained from the one-dimensional transient heat conduction model based on Green's functions agreed well with the experimental results over longer exposure times, and with reduced error when compared with the semi-infinite model. This suggests that modeling the problem on a finite-length scale will produce more accurate or more conservative temperature and heat flux results over extended periods of exposure in high heat load applications.     

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.     

Numerical solution for the hyperbolic heat conduction problems in the radial–spherical coordinate system using a hybrid Green's function method

The hyperbolic heat conduction problems in the radial–spherical coordinate system are investigated by the hybrid Green's function method. The present method combines the Laplace transform for the time domain, Green's function for the space domain and ?-algorithm acceleration method for fast convergence of the series solution. Three different examples problems have been analyzed by the present method. It is found that the present method does not exhibit numerical oscillations at the wave front and the numerical solutions are stable.     

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.     

Heterogeneous Problems in Plane Thermoelasticity

Within the framework of the linear theory of thermoelasticity, the heterogeneous problem associated with multiple inclusions, circularly cylindrical layered media and plane layered media is considered and solved in this paper. The number of inclusions and layers is arbitrary and the system is subjected to arbitrary loading (singularities). The solutions to heat conduction (or antiplane deformation) and thermoelasticity problems are derived by the heterogenization technique that allows us to write down the solution explicitly in terms of the solution of a corresponding homogeneous problem subjected to the same loading. A rapid convergent series solution for both the temperature (or antiplane displacement) and stress functions, which is expressed in terms of an explicit general term of the complex potential of the corresponding homogeneous problem, is obtained in an elegant form. Numerical results are provided for some particular examples to investigate the effect of material combinations and geometrical configurations on the interfacial stresses.     

Method to derive thermoelastic Green’s functions for cylindrical domains

This article presents a new method to derive Green’s functions for boundary value problems (BVPs) of steady-state thermoelasticity for domains described in cylindrical system of coordinate. The proposed method is based on new integral representations for main thermoelastic Green’s functions (MTGFs) in terms of Green’s functions for incompressible Lamé equations written in a cylindrical system of coordinates. The method is demonstrated on a BVP for cylindrical half-wedge for which MTGFs and Green-type integral formula are derived. The obtained MTGFs for half-wedge are validated by MTGFs for respective BVP for thermoelastic wedge that are obtained earlier using ΘG convolution method (ΘGCM). New MTGFs for octant, quarter-space, and half-space as particular cases of the cylindrical half-wedge also can be easily written. The advantages of the proposed method, called method of incompressible cylindrical integral representations (MICIR), in comparison with ΘGCM, are: (a) it is not necessary to construct influence functions for elastic volume dilatation Θ^(q), caused by unit point body force; and (b) it is not necessary to compute complicated convolution (volume integral of product between Θ^(q) and Green’s function G_T) in heat conduction equation.     

Thermal Stresses Induced by a Variable Heat Source in a Rectangle and Variable Pressure at Its Boundary by Finite Fourier Transform

This paper deals with the two-dimensional, non-homogeneous boundary value problem for static, isotropic and thermoelastic material occupying an infinitely long cylinder with a rectangular cross-section. The cylinder is surrounded by a given temperature and subjected to variable pressures at its boundaries. We deal with static, uncoupled, linear thermoelasticity. The equations of heat conduction and mechanical problem are considered separately. The technique of the finite Fourier transform is used for the solution. The thermoelastic behavior, due to an internal heat generation within the domain, is discussed. The results for displacement and stresses have been computed from the Airy stress function and are illustrated graphically.     

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.