What Is Thermoelasticity?

Thermoelasticity is also called thermoelasticity. A branch of solid mechanics, it mainly studies the problems of stress and deformation in the elastic range caused by the non-uniform temperature field caused by heat.

Chinese name: Thermoelasticity

Foreign name: Thermoelasticity

A Brief History of the Development of Thermoelasticity

The foundation of a brief history of thermoelasticity was laid down by JMC Duhamel and FE Neumann as early as the first half of the 19th century. Duhamel established the foundation of thermoelastic theory in 1838, obtained a set of equations, and used it to solve the thermal stress problem of axisymmetric temperature distribution cylinders and centrally symmetric temperature distribution spheres. Neumann proceeded from certain assumptions and obtained the same equation in 1841. Since the 20th century, due to the development of industry, the importance of thermal stress has gradually been recognized, so many articles about thermal stress have appeared. However, an in-depth and extensive study of this science was after World War II. In high-tech aircraft, rockets, missiles, thermonuclear reactors and other cutting-edge technology fields, the problem of thermal stress is particularly prominent. At that time, many scientific workers engaged in research in this area, which promoted the development of thermoelasticity.

In recent years, people have been interested in the problem of coupling and propagation of thermoelastic waves. Great progress has also been made in the study of thermal stresses in anisotropic bodies, composite materials, and fractures. In addition, research on nonlinear thermoelasticity theory, electromagnetic thermoelasticity theory, and thermoelasticity of piezoelectric crystals is also developing.

Research content of thermoelasticity

It mainly studies the stress and deformation of the object in the elastic range due to the non-uniform temperature field caused by heating. Thermoelasticity is a generalization of elasticity. It considers the influence of temperature on the basis of elasticity and adds a strain due to temperature changes to the stress-strain relationship. In the process of establishing thermoelasticity theory, the heat conduction equation and the first and second laws of thermodynamics need to be used.

When the object is heated, the parts of the object will expand outward due to the temperature rise. If every part of the object can expand freely, there will be no stress despite strain. If each part of the object cannot expand freely (the object is uniformly heated but subject to a certain constraint or the object is not uniformly heated and the object is continuous), the various parts will cause stress due to mutual constraints. This stress is called temperature stress or thermal stress. In addition, the elastic modulus of a material (see the mechanical properties of the material) decreases with increasing temperature.

According to the relationship between temperature and stress with time, it can be divided into steady thermal stress and unsteady thermal stress. According to the relationship between temperature and deformation, it can be divided into coupled thermoelasticity and uncoupled thermoelasticity.

Major Problems in Thermoelasticity

Steady thermal stress in thermoelasticity

Steady thermal stress is a thermal stress caused by a steady temperature field. "Stability" means that temperature and stress are independent of time. When the transient temperature change approaches zero and the temperature distribution reaches a stable state, the temperature distribution can be obtained from the heat conduction equation and temperature boundary conditions; and the elasticity equation including the temperature term can be used to obtain the displacement and stress. At present, the research on stationary thermal stress is mainly focused on the following aspects: Two-dimensional thermal stress, that is, plane stress and plane strain, such as thick-walled tubes, cylinders, circular plates, annular plates, and half-planes. Thermal stress issues. Thermal stress problems of rotating bodies, infinite bodies or semi-infinite bodies in an axisymmetric temperature field, for example, thermal stress problems caused by a point heat source or a thermal dipole on the surface of an infinite body or a semi-infinite body. The thermal bending and thermal wrinkling of the plate and shell are similar to the problems of bending and wrinkling of the plate and shell at normal temperature, except that the temperature term is reduced to a considerable external load term. The thermal stress of inclusions in infinite plates, infinite bodies or semi-infinite bodies.

The problem of unsteady thermal stress in thermoelasticity

Unsteady thermal stress is a thermal stress caused by an unsteady temperature field. "Unsteady" means that temperature or stress changes over time. In principle, the unsteady thermal stress problem is no longer a static problem, but a dynamic problem. However, in general, the temperature changes slowly, and the influence of acceleration can be ignored. The movement is regarded as a series of equilibrium states, and the thermal stress at that time is calculated according to the current temperature distribution at each moment. This treatment method is called quasi-static treatment of unsteady thermal stress. The difference between the unsteady thermal stress problem and the steady thermal stress problem lies only in the solution of the heat conduction equation. According to quasi-static treatment, there are many problems, such as unsteady thermal stress problems of cylinders and spheres, quasi-steady thermal stress problems with periodic temperature field changes, and quasi-steady thermal stress problems caused by moving heat sources. The problem of dynamic thermal stress must consider the impact of acceleration. The problems in this area are: thermal shock, that is, the thermal stress caused by the sudden heating of the object; transient heat source, this kind of problem is generally not quasi-static treatment, but must be considered Influence of inertia term; thermoelastic vibration problems, such as vibration problems caused by sudden heating of thin rods or thin plates; thermoelastic wave propagation problems.

Thermoelasticity coupled thermoelasticity

Coupling thermoelasticity is the most common problem in thermoelasticity. It considers the interaction between temperature and deformation, that is, not only temperature will cause deformation, but deformation will also generate or consume energy, which affects temperature. Thus, there is an additional term that includes strain in the heat conduction equation, called the coupling term of temperature field and strain field. The heat conduction equation and thermoelastic equation are no longer independent, and must be simultaneous to solve temperature, displacement, and stress. But it is more difficult to solve the coupled thermoelastic problem. The corresponding theory is called coupled thermoelastic theory.

Thermoelasticity Non-Coupling Thermoelasticity

In practical applications, the coupling term can often be ignored, so the heat conduction equation becomes a common heat conduction equation. In this way, the temperature distribution can be obtained from the heat conduction equation first, and then the displacement and stress can be solved from the thermoelastic equation. The corresponding thermoelastic theory is called uncoupled thermoelastic theory.

For some issues, the role of coupling terms needs to be considered. For example, in the propagation of waves, due to the dissipation of thermal energy, thermoelastic coupling plays a more important role in damping the waves. The effects of coupling terms must also be considered in stress or strain discontinuities and thermal shock issues.

Thermoelastic theory

Linear thermoelasticity theory is a relatively mature theory. The theory assumes that objects are slightly disturbed in the equilibrium state. The deviation of all physical quantities from the corresponding quantities in the equilibrium state is very small. The products are negligible. The equations established under these simplified assumptions are all linear equations.

The thermoelastic potential method proposed by JN Goodall in 1937 is a widely used analytical method. This solution has nothing to do with whether the temperature field is unsteady and can also be applied to dynamic problems. Green's function method, integral transformation method, and elastic mechanics complex function method in two-dimensional problems are also used, and many results have been obtained. In numerical methods, due to the rapid development of computers, the finite element method has become one of the most effective tools for solving practical problems in engineering.