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.     

Effect of uniform heat flux on stress distribution around a triangular hole in anisotropic infinite plates

The presence of a hole in an anisotropic plate under uniform heat flux causes thermal stress around the hole. In this study, on the basis of two-dimensional thermoelastic theory and using Lekhnitskii’s complex variable technique, the stress analysis of an anisotropic infinite plate with a circular hole under a uniform heat flux is developed to the plate containing a triangular hole. For this purpose, an infinite plate containing a triangular hole is mapped to the outside of a unit circle using a conformal mapping function. Stress and displacement distributions around the triangular holes in an anisotropic infinite plate are investigated in thermal steady-state condition. The plate is under uniform heat flux at infinity and Neumann boundary conditions and thermal-insulated condition on the hole boundary are considered. The rotation angle of the hole, fiber angle, the angle of heat flux, bluntness, and the aspect ratio of hole size are investigated in the present study. The accuracy of the analytical results is also confirmed by finite element analysis.     

SOLUTION OF DISPLACEMENT BOUNDARY VALUE PROBLEM UNDER UNIFORM HEAT FLUX

The general solution of the displacement boundary value problem is obtained for an infinite plate with an arbitrary shaped hole under uniform heat flux in any direction. The complex stress functions, the dislocation method, and a rational mapping function are used and the closed solution is obtained. An infinite plate with a circular hole and a slit is analyzed under the condition of the constrained displacements. The singularity at the tip of the slit of the constrained displacement is investigated     

Study of the effective parameters on stress distribution around triangular hole in metallic plates subjected to uniform heat flux

On the basis of the steady-state two-dimensional theory of thermoelasticity, stress field around a triangular hole in an infinite isotropic plate is discussed. A metallic plate subjected to uniform heat flux and thermal-insulated condition along the hole boundary is assumed. The method used for this study is the expansion of Goodier and Florence's method. They used the complex variable method for stress analysis of infinite isotropic plates with an elliptical or circular hole. The rotation angle of the hole, bluntness, aspect ratio of hole size, and angle of heat flux are important parameters considered in this paper.     

APPLICATION OF AN ALTERNATING METHOD TO THE THERMOELASTIC PROBLEMS WITH MULTIPLE CIRCULAR HOLES IN AN INFINITE DOMAIN

This paper presents a new efficient procedure to analyze the thermoelastic problems with multiple circular holes in two-dimensional infinite domain using an alternating method. To achieve this purpose, the analytical solutions including the temperature and associated thermal stress for the arbitrary heat flux across the single circular hole boundary in an infinite domain are first derived. Both the temperature and thermal stress fields in the thermal problems are simultaneously solved by these analytical solutions with the successive iterative superposition process. Compared with the solution of the conventional finite-element technique, the present method has been more accurate and has more advantages. Effects of the distributions and sizes of the holes on the stress concentration in the thermal problems also are evaluated in detail herein.     

Thermal stress analysis of infinite anisotropic plate with elliptical hole under uniform heat flux

Lekhnitskii’s complex variable method was developed to investigate the effect of uniform heat flux on perforated anisotropic plate with elliptical hole. The Cauchy’s integral formula was simplified by conformal mapping, and infinite area external to the hole was represented by the area outside the unit circle. In this article, Neumann boundary conditions and thermal-insulated condition along with the hole boundary were considered. Important parameters affecting stress distribution and displacement were those of rotation angle of hole, aspect ratio of hole size, and fiber angle. Results determined in this article were verified by finite element analysis.     

GENERALIZED THERMOELASTICITY IN AN INFINITE SOLID WITH A HOLE

This paper deals with one-dimensional generalized thermoelasticity based on the theories of Lord and Shulman and of Green and Lindsay. A formulation of generalized thermoelasticity that combines both generalized theories is derived. The generalized thermoelastic problems for an infinite solid with a cylindrical hole and an infinite solid with a spherical hole are analyzed by means of the Laplace transform technique. Numerical calculations for temperature, displacement, and stresses under the generalized formulation are carried out and compared with those of classical dynamic coupled theory.     

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.     

SENSITIVITY ANALYSIS AND OPTIMAL DESIGN FOR STEADY CONDUCTION PROBLEM WITH RADIATIVE HEAT TRANSFER

ABSTRACT

A steady heat conduction problem is considered, that is described by the heat conduction equation and the thermal boundary conditions (i.e., Dirichlet, Neumann, Henkel, and radiation conditions on the external boundary, and radiation condition on the hole boundary). An arbitrary behavioral functional is defined and its first-order sensitivity is derived using both the direct and the adjoint approaches. The shape optimization problem is next formulated and two optimization functionals are discussed. The simple numerical example is presented.     

Transient heat conduction in an infinite medium subjected to multiple cylindrical heat sources: An application to shallow geothermal systems

In this paper, we introduce analytical solutions for transient heat conduction in an infinite solid mass subjected to a varying single or multiple cylindrical heat sources. The solutions are formulated for two types of boundary conditions: a time-dependent Neumann boundary condition, and a time-dependent Dirichlet boundary condition. We solve the initial and boundary value problem for a single heat source using the modified Bessel function, for the spatial domain, and the fast Fourier transform, for the temporal domain. For multiple heat sources, we apply directly the superposition principle for the Neumann boundary condition, but for the Dirichlet boundary condition, we conduct an analytical coupling, which allows for the exact thermal interaction between all involved heat sources. The heat sources can exhibit different time-dependent signals, and can have any distribution in space. The solutions are verified against the analytical solution given by Carslaw and Jaeger for a constant Neumann boundary condition, and the finite element solution for both types of boundary conditions. Compared to these two solutions, the proposed solutions are exact at all radial distances, highly elegant, robust and easy to implement.     

THERMAL SHOCK FRACTURE IN AN EDGE-CRACKED PLATE

In this paper the transient thermal stress problem for an elastic strip with an edge crack is investigated. The elastic medium is assumed to be insulated on one face and cooled by surface convection on the face contaning the edge crack. Using the principle of superposition, the formulation results in a mixed boundary value problem, with the thermal stresses calculated from the thermoelasticity solution for an uncracked strip utilized as the necessary crack surface tractions. The resulting singular integral equation is of a well-known type and is solved numerically. In this paper, inertia effects are assumed negligible and possible temperature dependence of thermoelastic constants is not considered. The numerical results presented, include the stress intensity factor as a function of nondimensional time (Fourier number) and crack length, for various values of the dimensionless Biot number. The temperature distribution and the thermal stresses in the uncracked strip are also included. The time lag, which occurs between the time at which the stress on the surface of the strip is a maximum and the time when a maximum occurs in the stress intensity factor, is clearly shown to be a function of the Biot number for any given ratio of crack length to strip thickness. A result of particular interest is the degree with which the maximum stress intensity factor decreases, as a function of crack length, for decreasing values of the Biot number.     

SOLUTION TO INVERSE HEAT CONDUCTION PROBLEM IN NANOSCALE USING SEQUENTIAL METHOD

An inverse heat conduction problem for nanoscale structure is studied. The conduction phenomenon is modeled using the Boltzmann transport equation. Phonon-mediated heat conduction in one dimension is considered. One boundary is exposed to an unknown temperature and the other boundary, where temperature observation takes place, is subject to a known boundary condition. A sequential scheme with constant function specification is employed for inverse estimation of the unknown temperature. Sample results are presented and discussed.     

AN INTERNAL PENNY-SHAPED CRACK IN AN INFINITE THERMOELASTIC SOLID

In this work, we solve a dynamical problem for an infinite thermoelastic solid with an internal penny-shaped crack, which is subjected to prescribed temperature and stress distributions. The problem is solved using the Laplace and Hankel transforms. The boundary conditions of the problem give a set of four dual integral equations. The operators of fractional calculus are used to transform the dual integral equations into a Fredholm integral equation of the second kind, which is solved numerically. The inverse Hankel and Laplace transforms are obtained using a numerical technique. Numerical results for the temperature, stress, and displacement distributions, as well as for the stress intensity factor, are shown graphically.     

DEFORMATION OF AN INFINITELY LONG,DC CABLE WITH TEMPERATURE-DEPENDENT ELECTRIC CONDUCTIVITY OF THE INSULATION

We investigate the deformation of an infinitely long, circular cylindrical electric conductor carrying a uniform axial current, for the case when the electric conductivity of the coating is temperature dependent. This model conforms with the real situation for many of the existing modern dc cables with polyethylene coating. The distributions of temperature, magnetic field, stresses, and displacements in the cable are obtained and discussed under a thermal radiation condition at the boundary of the cable. In particular, it appears that there is a critical temperature for the ambient medium to the cable, above which no solution for the steady heat problem can exist and the thermal equilibrium of the cable is no longer fulfilled. A formula for the calculation of this critical value is given, which may turn out to be of practical importance for a reliable design of the cable. The obtained results also show that the electric conductivity of the coating strictly decreases as one moves from the core to the boundary of the cable in conformity with the behavior of temperature. Some numerical results are presented.     

An efficient BEM formulation for three-dimensional steady-state heat conduction analysis of composites

In this work, an efficient boundary element formulation has been presented for three-dimensional steady-state heat conduction analysis of fiber reinforced composites. The cylindrical shaped fibers in the three-dimensional composite matrix are represented by a system of curvilinear line elements with a prescribed diameter which facilitates efficient analysis and modeling together with the reduction in dimensionality of the problem. The variations in the temperature and flux fields in the circumferential direction of the fiber are represented in terms of a trigonometric shape function together with a linear or quadratic variation in the longitudinal direction. The resulting integrals are then treated semi-analytically which reduces the computational task significantly. The computational effort is further minimized by analytically substituting the fiber equations into the boundary integral equation of the material matrix with hole, resulting in a modified boundary integral equation of the composite matrix. An efficient assembly process of the resulting system equations is demonstrated together with several numerical examples to validate the proposed formulation. An example of application is also included.     

NUMERICAL INVESTIGATION OF THE BLADE COOLING EFFECT GENERATED BY MULTIPLE JETS ISSUING AT AN ANGLE INTO AN INCOMPRESSIBLE HORIZONTAL CROSSFLOW

The thermal effect of cool jets issuing into an incompressible hot crossflow at an angle over a turbine blade is the subject. Numerical solutions for 12 multiple flow arrangements with two flow ratios, two different hole spacings, and four different jet issuing angles document strong to moderate secondary vortex structures spanning normal to the direction of the jet. This fully three-dimensional flowfield strongly influences the cooling performance of the hole-blade system. For a generalized body-fitted three-dimensional finite volume model, computational results suggest an optimum hole spacing and low issuing angle for maximum cooling efficiency.     

REWETTING OF AN INFINITE SLAB WITH BOUNDARY HEAT FLUX

This paper deals with a numerical solution of the two-dimensional quasi-static conduction equation, governing conduction controlled rewetting of an infinitely long slab with one side flooded and the other side subjected to a constant heat flux. The solution gives the quench front temperature as a function of various model parameters such as Peclet number, Biot number, and dimensionless boundary heat flux. Also, the critical boundary heat flux is obtained by setting the Peclet number equal to zero, which gives the minimum heat flux required to prevent the hot surface being rewetted.     

THERMAL AND MECHANICAL INSTABILITY OF AN IMPERFECT SHALLOW SPHERICAL CAP

In this article, the thermal and mechanical buckling loads of a cap of a shallow spherical shell of isotropic material and geometrically imperfect shell are considered. The equilibrium and stability equations are based on Donnell-Mushtari-Velasov (DMV) theory and are derived using the variational method. The Sander's nonlinear strain-displacement relations are used. The shell is under external pressure for mechanical loading and uniform temperature rise and radial temperature difference for thermal loadings. A simply supported boundary condition is assumed. The solutions for thermal and mechanical buckling loads are obtained using the stability equations and the Galerkin method. One-term approximation for the middle-plane shell displacement is considered. The expressions for the thermal and mechanical buckling loads are obtained analytically and are given by closed-form solutions.     

AN INVERSE SOURCE PROBLEM IN RADIATIVE TRANSFER FOR SPHERICAL MEDIA

Abstract

A method is presented to identify the source term in a one-dimensional, absorbing, emitting, scattering spherical medium from the knowledge of the exit radiation intensities. The inverse radiation problem is formulated as an optimization problem. The sensitivity problem and the gradient equation are derived. The conjugate gradient method is used for its solution. Although the source term is a function of the space variable, only radiation intensities exiting the outer boundary are required. Both data with and without measurement errors are used as input to identify the source term. The study shows that the estimation of the source term is more sensitive to increases in measurement errors as the optical thickness increases.     

Heat transfer in a reaction–diffusion system with a moving heat source

A general formulation is presented for a moving boundary problem in which heat is generated at the boundary due to an exothermic reaction involving a species which diffuses into a dispersed phase from an external medium of finite volume. The speed of the moving boundary is prescribed based on the solution of the mass diffusion problem and an analysis is presented of the thermal dynamics of the system. The set of equations describing heat transport leads to a Green’s function type problem with time dependent boundary conditions and the Galerkin finite element method is employed to develop a numerical solution. Transformations are introduced to freeze the moving boundary and partition the domain for ease of computation, and an iterative scheme is defined to satisfy the heat flux jump boundary condition and match the temperature field across the moving boundary. The numerical results are used to set the limits of applicability of an analytical perturbation solution. Essential aspects of thermal dynamics in the system are described and parametric regions resulting in a local temperature hot spot are delineated. Computed contour plots describing thermal evolution are presented for different combinations of parameter values. These may be of utility in the prediction of thermal development, for control and avoidance of hot spot formation, and in physical parameter estimation.