Thermal Stresses—Advanced Theory and Applications

The basic laws of thermoelasticity, similar to those of the theory of elasticity, include the equations of motion, the compatibility equations, and the constitutive law. This chapter begins with the derivation of the basic laws of linear thermoelasticity, where the linear strain–displacement relations are obtained, following the general discussion of the Green and Almansi nonlinear strain tensors. The necessity and sufficiency of the compatibility conditions for the simply and multiply connected regions are presented for three-dimensional conditions. These conditions are then reduced to the two-dimensional case called Michell conditions. The classical general and simple thermoelastic plane strain and plane stress formulations are presented.

A new presentation of the thermodynamic principles for solid elastic continuum is given. The first and the second laws of thermodynamics in variational form are stated, and the variational principle of thermodynamics in terms of entropy follows. The principle of thermoelasticity linearization is discussed and the classical, coupled, as well as the generalized (with second sound effect) theories are derived using the linearization technique. A unique generalized formulation, considering Lord–Shulman, Green–Lindsay, and Green–Naghdi models, for the heterogeneous anisotropic material is presented, where the formulation is properly reduced to those of isotropic material. The uniqueness theorem and the variational form of the generalized thermoelasticity are derived, and the exposition of the Maxwell reciprocity theorem concludes the chapter.

Not all types of temperature changes in a solid continuum result in creation of thermal stresses. The chapter begins with the discussion of the condition on what type of temperature distribution causes thermal stresses. The analogy of temperature gradient with body forces is stated. Then the theoretical analysis of thermal stress problems is presented in three main classical coordinate systems, that is, the rectangular Cartesian coordinates, the cylindrical coordinates, and the spherical coordinates. In discussing the analytical methods of solution, the Airy stress function, Boussinesq function, the displacement potential, the Michell function, and the Papkovich functions are defined and the general solution in three-coordinate systems are given in terms of these functions.

The ability of obtaining the temperature distribution in an elastic continuum through the solutions of the heat conduction equation is an essential tool for the analysis of thermal stress problems. The analytical methods of solution of heat conduction problems may be classified as the differential, the lumped, and the integral formulations. This chapter presents the analytical methods of solution of heat conduction problems based on the differential and lumped formulations. Similar to the analysis given in Chap. 3, this chapter presents the method of analysis of heat conduction in three main classical coordinate systems, namely the rectangular Cartesian coordinates, the cylindrical coordinates, and the spherical coordinates. The temperature distributions in each coordinate system for one-, two-, and three-dimensional problems for steady-state and transient conditions are obtained. The temperature distributions in a number of structures of functionally graded materials are also presented. The basic mathematical techniques of solution are discussed. The extent of the heat conduction analysis given in this chapter should be adequate for further applications to various thermal stress problems.

As an application of the theory of thermoelasticity, thermal stress analysis of beams based on the elementary beam theory is the objective of this chapter. It begins with the derivation of formulas for axial thermal stresses and thermal lateral deflections in beams, and the associated boundary conditions are stated, see [1, 2]. The discussion on transient thermal stresses is presented, and the analysis of beams with internal heat generation follows. The formulas for thermal stresses in a bimetallic beam are discussed. The analysis of beams of functionally graded materials under steady-state and transient temperature distributions is presented, and analysis of thermal stresses in curved beams concludes the chapter.

Thick cylinders, spheres, and disks are components of many structural systems. Due to their capacity to withstand high pressures, radial loads, and radial temperature gradients, the problem of thermal stress calculations is an important design issue. This chapter presents the method to calculate thermal stresses in such structural members which are made either of homogeneous/isotropic materials or of functionally graded materials. The latter ones, classified as new materials, are mainly designed to withstand high temperatures and high temperature gradients, and they may be designed in such a way that the applied loads, mechanical or thermal, produce a uniform stress distribution across their radial direction. Functionally graded materials exhibit the unique design features, where by selection of proper grading profiles, stress distribution within the element may be optimized.

Piping systems, such as those installed in refineries, are transmitters of high-pressure fluids or gases at high temperatures, and failure of such systems may cause catastrophic damage. Advanced design codes have been developed to provide safety instructions for designers. Piping systems are initially installed at reference temperature. At working conditions, under elevated temperatures, the end expansion and contraction forces and bending moments are created. The resulting thermal expansions, due to the imposed constraints, may produce large stresses which, if not properly taken care of, may cause failure. This chapter presents a simple method of analysis of piping systems. The main scope of the chapter is to provide an analytical method to calculate the end expansion and contraction forces and bending moments. In a number of examples, it is shown how the end forces and moments may be made smaller. The chapter then continues to present the stiffness method of piping system. While the method of elastic center is restricted to only one branch of piping system and the pipe elements must be parallel to the coordinate axes, the stiffness method has none of these limitations and handles a piping system with any arbitrary number of branches and the pipe elements may have any arbitrary directions and not necessarily parallel to the coordinate axes.

A structure under thermal shock load, when the period of shock is of the same order of magnitude as the lowest natural frequency of the structure, should be analyzed through the coupled form of the energy and thermoelasticity equations. Analytical solutions of this class of problems are mathematically complex and are limited to those of an infinite body or a half-space, where the boundary conditions are simple. This chapter begins with the analytical solutions of a number of classical problems of coupled thermoelasticity for an infinite body, a half-space, and a layer. Coupled thermoelasticity problem for a thick cylinder is discussed when the inertia terms are ignored. The generalized thermoelasticity problems for a layer, based on the Green–Naghdi, Green–Lindsay, and the Lord–Shulman models are discussed, when the analytical solution in the space domain is found. The solution in the time domain is obtained by numerical inversion of Laplace transforms. The generalized thermoelasticity of thick cylinders and spheres, in a unified form, is discussed, and problems are solved analytically in the space domain, while the inversion of Laplace transforms are carried out by numerical methods.

Because the analytical solutions to coupled and generalized thermoelasticity problems are mathematically complicated, the numerical methods, such as the finite and the boundary element methods, have become powerful means of analysis. This chapter presents a new treatment of the finite and the boundary element methods for this class of problems. The finite element method based on the Galerkin technique is employed in order to model the general form of the coupled equations, and the application is then expanded to the two- and one-dimensional cases. The generalized thermoelasticity problems for a functionally graded layer, a thick sphere, a disk, and a beam are discussed using the Galerkin finite element technique. To show the strong rate of convergence of the Galerkin-based finite element, a problem for a radially symmetric loaded disk with three types of shape functions, linear, quadratic, and cubic, is solved. It is shown that the linear solution rapidly converges to that of the cubic solution. When the temperature change compared to the reference temperature may not be ignored, the heat conduction becomes nonlinear. The problem of thermally nonlinear generalized thermoelasticity of a layer based on the Lord–Shulman model is presented in this section, and it is indicated that how and when this assumption is essential to be used. The chapter concludes with the boundary element formulation for the generalized thermoelasticity. A unique principal solution satisfying both the thermoelasticity and the coupled energy equations is employed to obtain the boundary element formulation.

Under the combination of elevated temperature and mechanical loads, structural members tend to creep. Basic laws of creep are presented, and the effect of temperature changes in the constitutive law of creep is discussed. The rheological models of two important engineering concepts of stress, namely the load- and the deformation-controlled stresses, are presented. The chapter concludes with the description of numerical techniques of solutions to creep problems. The nature of pure thermal stresses as the deformation-controlled stresses is presented in the example problems. It is shown that pure thermal stresses relax as the time advances.