Transient Thermal Stresses in a Sphere by Local Heating

1974 ◽  
Vol 41 (4) ◽  
pp. 930-934 ◽  
Author(s):  
J. B. Cheung ◽  
T. S. Chen ◽  
K. Thirumalai
Keyword(s):  
Thermal Stresses ◽  
Numerical Solutions ◽  
Integral Transformation ◽  
Separation Of Variables ◽  
Local Heating ◽  
Local Surface ◽  
Displacement Potential ◽  
Heated Zone ◽  
Localized Heating ◽  
Transient Thermal

The problem of transient thermal stresses in a solid, elastic, homogeneous, and isotropic sphere is solved for uniform and nonuniform, local surface heating. The temperature solutions are obtained by using separation of variables and integral transformation. The corresponding thermal stresses are derived by superposing a particular displacement potential function on Boussinesq solutions. Numerical solutions for two particular cases of localized heating of a typical brittle spherical solid have been obtained and presented. The results indicate a tensile stress concentration in the interior of the solid below the heated zone.

scholarly journals Transient Thermoelastic Analysis of Pressurized Rotating Disks Subjected to Arbitrary Boundary and Initial Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Mohammad Zamani Nejad ◽  
Azam Afshin
Keyword(s):  
Thermal Stresses ◽  
Initial Conditions ◽  
Exact Analytical Solution ◽  
Separation Of Variables ◽  
Outer Radius ◽  
Rotating Disks ◽  
Arbitrary Boundary ◽  
Inner Radius ◽  
Transient Thermal

This paper focuses on exact analytical solution of transient thermoelastic behaviors of rotating pressurized disks subjected to arbitrary boundary and initial conditions. The pressure, inner radius, and outer radius are considered constant. The basic thermoelasticity theory under generalized assumptions is used to solve the thermoelastic problem. Using the method of the separation of variables, the relations of temperature and transient thermal stresses in the radial direction are obtained. In the case study, the disk is considered under heat flux. Some useful discussions and numerical examples are presented. The analytical results were compared with those of the finite element method and good agreement was found. The relations obtained in this paper can be applied to any arbitrary boundary and initial conditions.

Transient Two-Dimensional Heat Conduction Problem With Partial Heating Near Corners

2016 ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
James V. Beck ◽  
Donald E. Amos
Keyword(s):  
Heat Conduction ◽  
Numerical Integration ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Thermal Protection ◽  
Heat Conduction Problem ◽  
Separation Of Variables ◽  
Two Dimensional ◽  
Rectangular Region ◽  
Long Distance

This paper provides a solution for two-dimensional heating over a rectangular region on a homogeneous plate. It has application to verification of numerical conduction codes as well as direct application for heating and cooling of electronic equipment. Additionally, it can be applied as a direct solution for the inverse heat conduction problem, most notably used in thermal protection systems for re-entry vehicles. The solutions used in this work are generated using Green’s functions. Two approaches are used which provide solutions for either semi-infinite plates or finite plates with isothermal conditions which are located a long distance from the heating. The methods are both efficient numerically and have extreme accuracy, which can be used to provide additional solution verification. The solutions have components that are shown to have physical significance. The extremely precise nature of analytical solutions allows them to be used as prime standards for their respective transient conduction cases. This extreme precision also allows an accurate calculation of heat flux by finite differences between two points of very close proximity which would not be possible with numerical solutions. This is particularly useful near heated surfaces and near corners. Similarly, sensitivity coefficients for parameter estimation problems can be calculated with extreme precision using this same technique. Another contribution of these solutions is the insight that they can bring. Important dimensionless groups are identified and their influence can be more readily seen than with numerical results. For linear problems, basic heating elements on plates, for example, can be solved to aid in understanding more complex cases. Furthermore these basic solutions can be superimposed both in time and space to obtain solutions for numerous other problems. This paper provides an analytical two-dimensional, transient solution for heating over a rectangular region on a homogeneous square plate. Several methods are available for the solution of such problems. One of the most common is the separation of variables (SOV) method. In the standard implementation of the SOV method, convergence can be slow and accuracy lacking. Another method of generating a solution to this problem makes use of time-partitioning which can produce accurate results. However, numerical integration may be required in these cases, which, in some ways, negates the advantages offered by the analytical solutions. The method given herein requires no numerical integration; it also exhibits exponential series convergence and can provide excellent accuracy. The procedure involves the derivation of previously-unknown simpler forms for the summations, in some cases by virtue of the use of algebraic components. Also, a mathematical identity given in this paper can be used for a variety of related problems.

Axisymmetric Thermal Stresses of a Piezoelectric Poroelastic Circular Solid Plate

2021 ◽  
Vol 143 (5) ◽  
Author(s):  
Ali Abjadi ◽  
Mohsen Jabbari ◽  
Ahmad Reza Khorshidvand
Keyword(s):  
Cadmium Selenide ◽  
Material Properties ◽  
Thermal Stresses ◽  
Separation Of Variables ◽  
Electrical Potential ◽  
Material Symmetry ◽  
Exponential Functions ◽  
Cylindrical Coordinates ◽  
Solid Plate ◽  
Electrical Loads

Abstract This paper presents the steady-state thermoelasticity solution for a circular solid plate is made of an undrained porous piezoelectric hexagonal material symmetry of class 6 mm. The porosities of the plate vary through the thickness; thus, material properties, except Poisson's ratio, are assumed as exponential functions of axial variable z in cylindrical coordinates. Having axisymmetric general form, external thermal and electrical loads are acted on the plate and the piezothermoelastic behavior of the plate is investigated. Using a full analytical method based on Bessel Fourier's series and separation of variables, the governing partial differential equations are derived. A formulation is given for the displacements, electric potential, thermal stresses, and electric displacements resulting from prescribed the general form of thermal, mechanical, and electric boundary conditions. Finally, the application of the derived formulas is illustrated by an example for a cadmium selenide solid, the results of which are presented graphically. Also, the effects of material property indexes, the porosity, and Skempton coefficients are discussed on the displacements, thermal stresses, electrical potential function, and electric displacements.

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