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.     

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 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.     

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.     

Quasi-Static Thermal Stresses Due to Heat Generation in a Thin Hollow Circular Disk

The present paper deals with the determination of displacement and thermal stresses in a thin hollow circular disk defined by a ≤ r ≤ b due to internal heat generation within it. Time dependent heat flux Q(t) is applied at the outer circular boundary (r = b), whereas inner circular boundary (r = a) is at zero heat flux. Also, initially the circular disk is at arbitrary temperature F(r). The governing heat conduction equation has been solved by the method of integral transform technique. The radial stress function σ_rr is zero at inner and outer circular boundaries (r = a and r = b). The results are obtained in a series form in terms of Bessel's functions. The results for displacement and stresses have been computed numerically and 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.     

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.     

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.     

Optimal Boundary Control of Thermal Stresses in a Plate Based on Time-Fractional Heat Conduction Equation

This article presents an optimal control problem for a fractional heat conduction equation that describes a temperature field. The main purpose of the research was to find the boundary temperature that takes the thermal stress under control. The fractional derivative is defined in terms of the Caputo operator. The Laplace and finite Fourier sine transforms were applied to obtain the exact solution. Linear approximation is used to get the numerical results. The dependence of the solution on the order of fractional derivative and on the nondimensional time is analyzed.     

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.     

An Investigation of the Thermal Stresses Induced in a Thin-Film Thermoelectric Cooler

This work aims at investigating the thermal stresses induced within a four-layered thin-film thermoelectric cooler. The one-dimensional (1D) temperature and thermal-stress distributions are firstly analyzed under the consideration of Joule heating, the conduction heat transfer as well as Thomson heating. Followed are two-dimensional (2D) calculations of the thermal stresses with the commercial software ANSYS. The validity of the 1D analytical model is then examined by a comparison of its predicted thermal stresses with the numerical ones obtained from the 2D model. In the 2D model, the thermoelectric element becomes curved due to the shrinkage and the fixed boundary conditions. The latter also causes huge values and rapid changes of thermal stresses near the ends. In the middle portion of the thermoelectric element where the thermal effect dominates, the thermal stresses predicted by the 1D model are not much different from those computed from the 2D model. Quantitative differences arise from the fact that the 1D model does not count the stresses induced by the non-zero Poisson's ratios. In addition, the normal-stress distributions are pretty uniform across the layer thickness (the variation is less than 1MPa within each layer in the worst case). These results verify the possibility of using the 1D model for a preliminary estimate of the thermal stresses induced within the layered thin-film thermoelectric element. The 1D model nonetheless fails to capture the behaviors near the ends of the thermoelectric element.     

Thermo-Elastic Analysis of a Cracked Half-Plane Under a Thermal Shock Impact Using the Hyperbolic Heat Conduction Theory

The transient thermal stresses around a crack in a thermo-elastic half-plane are obtained under a thermal shock using the hyperbolic heat conduction theory. Fourier, Laplace transforms and singular integral equations are applied to solve the temperature and thermal stress fields consecutively. The integral equations are solved numerically and the asymptotic fields around the crack tip are obtained. Numerical results show that the hyperbolic heat conduction have significant influence on the dynamic temperature and stress field. It is suggested that to design materials and structures against fracture under thermal loading, the hyperbolic model is more appropriate than the Fourier heat conduction model.     

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.     

Computational Methods for Inclined Cracks in Orthotropic Functionally Graded Materials Under Thermal Stresses

This article sets forth two different computational methods developed to evaluate fracture parameters for inclined cracks lying in orthotropic functionally graded materials, that are under the effect of thermal stresses. The first method is based on the J _k -integral, whereas the second entails the use of the J ₁-integral and the asymptotic displacement fields. The procedures introduced are implemented by means of the finite element method and integrated into a general purpose finite element analysis software. Numerical results are generated for an inclined edge crack in an orthotropic functionally graded layer subjected to steady-state thermal stresses. Comparisons of the mixed-mode stress intensity factors computed by the use of the proposed methods to those calculated by the displacement correlation technique point out that both approaches lead to numerical results of high accuracy. Further results are provided in order to illustrate the influences of inclination angle, material property gradation, and crack length upon the thermal fracture parameters.     

Thermal Stresses Around Two Parallel Interface Cracks Between a Nonhomogeneous Bonding Layer and Two Dissimilar Elastic Half-Planes Under Uniform Heat Flux

Composite materials consisting of two dissimilar elastic half-planes bonded by a nonhomogeneous elastic layer contain two interface cracks; one is situated at the lower interface between the layer and the lower half-plane, while the other is situated at the upper interface between the layer and the upper dissimilar half-plane. The stress intensity factors are solved under uniform heat flux normal to the cracks. The material properties of the bonding layer vary continuously from the lower half-plane to the upper half-plane. The boundary conditions are reduced to dual integral equations using the Fourier transform technique, and they are satisfied outside the cracks by expanding the differences in temperature and displacements at the crack surfaces using a series of functions that vanish outside the cracks. The unknown coefficients in each series are evaluated using the Schmidt method. The stress intensity factors were calculated numerically for selected crack configurations.     

Thermal Stresses Around Two Collinear Interface Cracks between a Nonhomogeneous Bonding Layer and One of the Two Dissimilar Elastic Half-Planes Under Uniform Heat Flux

In composite materials, in which two dissimilar elastic half-planes are bonded by a nonhomogeneous elastic layer, two collinear cracks are situated at the interface between the nonhomogeneous elastic layer and one of the two dissimilar half-planes. The stress intensity factors are solved under uniform heat flux normal to the cracks. The material properties of the bonding layer vary continuously from the lower half-plane to the upper half-plane. The boundary conditions are reduced to dual integral equations using the Fourier transform technique. In order to satisfy the boundary conditions outside the cracks, the differences in temperature and displacements at the crack surfaces are expanded in a series of functions that vanish outside the cracks. The unknown coefficients in each series are evaluated using the Schmidt method. The stress intensity factors were calculated numerically for selected crack configurations.