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Thermally induced stress is an important scientific problem in engineering applications. In this paper, an accurate and efficient method for the two-dimensional quasi-static thermal elastic problem is established to explore the thermal stress problem. First, the compact quasi-static two-dimensional general solution is derived in terms of simple potential functions. The general solution is simple in form and can be derived for arbitrary boundary problems subjected to a line heat load. This is completely new to the literature. Second, Green’s function solutions of an infinite plane under the line pulse heat source based on the general solutions are presented to analyze the thermal stress field. Lastly, numerical results are taken into account to study the temperature and stress field induced by the dynamic heat source load. The corresponding analysis can constitute to reveal the mechanism of thermal elastic problems and provide some guidance for experiments or engineering structural design in the future work.

Thermal stress occurs when the temperature changes for structures [

Many scholars have done excellent work in thermodynamics [

The main analytical methods are the integral transformation method, complex variable method, and classical general solution method. Integral transformation is a common method to solve differential equations. The Laplace and Hankel transform techniques have been used for the transversely isotropic thermoelastic thin plate [

The general solution method is a basic method to solve mechanical problems. It directly starts from the governing equations of mechanical models to find the general solution of partial differential equations and then combines the boundary conditions of specific problems to find the specific solution. The general solution under point loading is called Green’s function solution. Green’s function is an efficient tool for refined analysis of thermal stress of solid structures which can provide necessary preconditions to many high-precision numerical methods, such as boundary element method [

In this background, a harmonic function and a function which satisfies the quasi-static heat conduction equation for Green’s functions under the quasi-static line heat source will be constructed in this study. Firstly, we use the differential operator theory to derive three general solutions. Secondly, by virtue of Almansi’s theorem, the three general solutions can be transformed to the general solution which is expressed in terms of two harmonic functions and a function which satisfies the quasi-static heat conduction equation, respectively. Thirdly, the integral general solution expressed in one harmonic function and one function which satisfies the quasi-static heat conduction equation is obtained. As an example, the suitable functions for a quasi-static line heat source in the interior of the infinite steel plane are constructed, and the corresponding Green’s solutions are presented by virtue of the obtained general solutions. In this literature, all these harmonic functions are in the form of elementary functions, and all the thermal stress components are presented in a closed form at an arbitrary point in a two-phase infinite body. The fundamental solutions for the arbitrary distributed heat source are obtained by numerical integration using Green’s function under the point heat source. The method proposed in this study presents an accurate and efficient tool to analyze the thermal elastic problems, which have great engineering application value.

If all components are independent of coordinate

For the plane stress problem,

For the plane strain problem,

In the absence of body forces, the mechanical equilibrium equations are

The two-dimensional quasi-static heat conduction equation is

In this section, the quasi-static general solutions are derived using the governing equations

A combination of the results of equations (

Based on the differential operator theory, three general solutions of homogeneous differential equation (

The determinant of derivative operator matrix

When

By virtue of Almansi’s theorem, the function

For the convenience of operation, let

Based on equations (

Two general solutions are obtained in which

Let

Based on equations (

In combination with equations (

Substituting general solution equation (

The two-dimensional quasi-static general solutions for isotropic thermoelastic materials are obtained in equations (

Consider an infinite isotropic thermoelastic thin plane in two-dimensional Cartesian coordinates

An infinite isotropic thermoelastic plane applied by a pulse line heat source

The following functions are introduced by the trial and error method:

Substituting function (

When the mechanical equilibrium is considered (Figure

Substituting equation (

One useful integral is listed as follows:

Substituting equation (

If the time tends to infinity, all the components are given as follows:

The shear stress would be zero at infinity, and the following equation can be obtained:

Substituting equation (

Hence, Green’s solution for a pulse line heat source in an infinite isotropic thermoelastic plane is determined by equation (

Based on the solutions derived above, we now present some numerical results. The following nondimensional components are used in the figures:

Here, let

Material properties.

Materials | Young’s modulus | Poisson’s ratio | Density | Specific heat | Thermal expansion coefficient |
---|---|---|---|---|---|

Steel | 210.0 | 0.29 | 7.8 | 0.46 | 12.0 |

Temperature and thermal-stress distribution contours of steel are given in Figures

Figure

It can be seen from Figures

Figure

Temperature increment contour

Stress contour

Stress contour

Stress contour

In this section, we calculate the temperature and thermal stress distributions under different pulse-type heat source loads. By virtue of the superposition principle and numerical integration method, the coupled field under arbitrary distributed pulse heat source can be obtained easily. In this section, we consider one kind of heat source loaded on the plane.

For example, the dimensionless distributed pulse heat source loaded on the plane is in the form of

The region of the distributed pulse heat source is

The contours of the stress components and temperature component for the distributed heat source are shown in Figures

Temperature increment contour

Stress contour

Stress contour

Stress contour

The distributions of temperature and thermal stress at different positions of the plane with different times are given in Figures

Temperature distributions at different locations.

Radial component of thermal stress distributions at different locations.

Axial component of thermal stress distributions at different locations.

Shear stress component of thermal stress distributions at different locations.

In this paper, the compact quasi-static two-dimensional general solution for isotropic thermoelastic materials is presented by virtue of differential operator theory and Almansi’s theorem. The general solutions are expressed using simple explicit functions. Based on the general solutions, Green’s functions are obtained for a line heat source. Also, all Green’s components are expressed in simple functions, which are very convenient to use. The solution in this paper is obtained completely from the governing equation. No simplifying assumptions have been made. As an example, Green’s solutions for a pulse line heat source in the interior of an infinite steel plane are presented by virtue of the obtained general solution. Some numerical analyses are taken into account to study the distribution of the thermoelastic field under the action of the dynamic heat source. The numerical results of this study can give some theoretical basis for revealing the mechanism of the thermal elastic problem and provide some guidance for experiments or engineering structural design in the future work.

All the data generated or analyzed during this study are included within this article.

The authors declare that they have no conflicts of interest.

The authors thankfully acknowledge the financial support from the Guangzhou Science, Technology and Innovation Commission of China (no. 201904010204), the Hunan Province Natural Science Foundation of China (no. 2019JJ50633), the Department of Education of Guangdong Province of China (nos. 2018GKTSCX107, 2019GKTSCX031, and 2019GKTSCX032), and the Guangdong Province Natural Science Foundation of China (no. 2017A030310578).