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.     

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 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.     

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.     

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.     

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.