2D Quasi-Static Accurate Solutions for Isotropic Thermoelastic Materials with Applications

0ermally 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. 0e general solution is simple in form and can be derived for arbitrary boundary problems subjected to a line heat load. 0is 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. 0e 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.


Introduction
ermal stress occurs when the temperature changes for structures [1]. e applications of thermoelastic mechanics have great significance in engineering. e stiffness and strength problems caused by thermal stress are indescribably important in some engineering structures which has been an important factor for the design of engineering devices in the field of aerospace, mechanical, civil, electronic engineering, and some microstructures [2][3][4][5][6].
ermal stresses are generated during high-temperature curing of the material, which induces tensile and shear stress in the structure. ese stresses may cause cracks during the heat conduction process, especially during the thermal shock process [5,6]. e thermodynamic analysis is very important to reveal the interaction mechanism between the thermal source and structures [7][8][9][10].
Many scholars have done excellent work in thermodynamics [11][12][13][14]. Goodier proposed the integration expression of thermoelastic equations in 1937, which laid the foundation for the study of thermodynamics [11]. Boley and Weiner explored the thermal stress problems in curve beams, rings, trusses, frames, and built-up structures [15]. In 1961, Mindlin derived the governing equations for thermal media from the fundamental laws of continuum mechanics and thermodynamics [16]. Nowinski proved the laws of uniqueness for the governing equations [17]. Hetnarski introduced the thermodynamic theory systematically [13]. erefore, it is true to use these classical elasticity theories to solve many basic thermoelastic problems. In recent years, many research studies on the thermal elastic problem can be found in the literature. Numerical method and analytical method are used to study the thermoelastic problems. e finite element method (FEM) and boundary element method (BEM) are often preferred to analyze the thermoelastic problems. For example, a thermal finite element model for the multicomponent system of high-speed metal cutting was presented [18]. e thermal stress and local damage of the composite were simulated by finite element and discrete element coupling methods [19], but it is hard to obtain the accurate solutions of the thermal elastic structure, especially near the heat source. e reason is that extremely high-gradient coupling field near the heat source requires a high-density grid, which usually causes ill-conditioned equations in the calculation. us, the calculation cannot converge to obtain an exact solution of the coupling field naturally by the traditional software platforms. e 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. e Laplace and Hankel transform techniques have been used for the transversely isotropic thermoelastic thin plate [20]. e method of Fourier transformation is used to obtain an analytical expansion for the temperature and thermal stresses [21]. In the context of generalized theory of thermoelasticity proposed by Green and Naghdi, the thermal stress and temperature fields are obtained by using Fourier transforms [22,23]. Lata and Parveen investigated the two-dimensional deformation of a homogeneous isotropic thermoelastic solid using the same method [24][25][26]. For quasi-static thermal conduction problems, three-dimensional transient heat conduction analysis was conducted by using the Laplace transformation method [27]. e thermoelasticity problem for the thick-walled cylinder subjected to transient thermal loading was analytically studied [28]. With the complexity of the structure, the integral matrix becomes larger and less efficient. And the integral transformation condition is very harsh in many cases which is difficult to get accurate results by inverse transformation. For using the complex variable method, Hasebe et al. presented thermal stresses and temperature distribution for an infinite plate with circular holes and cracks under uniform heat flux in the arbitrary direction [29]. Gao et al. presented an exact solution for the problem of an elliptical hole or a crack in a thermo-piezoelectric material [30]. Mohammad studied thermal stress analysis of the orthotropic plate containing a rectangular hole using the complex variable method [31], but complex function methods are usually only applicable to solve plane problems, which is very difficult to obtain a simple form. e 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. e 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 highprecision numerical methods, such as boundary element method [18,19]. For isotropic materials, some scholars gave the basic solution of the famous closed form [32][33][34]. Hou's team did some work on Green's function of thermal elastic structure based on the general solutions [35][36][37][38], but the thermal elastic analysis of the isotropic structure under dynamic heat load has not been seen. In this literature, we would introduce how to get a compact and complete thermoelastic general solution for quasi-static heat conduction problems.
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. irdly, 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. e fundamental solutions for the arbitrary distributed heat source are obtained by numerical integration using Green's function under the point heat source. e method proposed in this study presents an accurate and efficient tool to analyze the thermal elastic problems, which have great engineering application value.

Two-Dimensional Governing Equations for the Isotropic Thermoelastic Material
If all components are independent of coordinate y in Cartesian coordinates (x, y, z), one will obtain the so-called two-dimensional problem or plane problem in two-dimensional Cartesian coordinates (x, z). In this case, the corresponding constitutive equations of isotropic thermoelastic materials are in the form of where u and ω are the components of the mechanical displacement in xand z-directions, respectively, σ x , σ z , and τ zx are the components of stress, θ is the temperature increment, and α is the thermal modulus. λ and G are the Lame constant and shear modulus, which are in the following forms for the plane stress problem and plane strain problem, respectively. For the plane stress problem, . (12c) When i � 1, 2 and A i3 � 0, we can obtain the general solutions in the isothermal case. e solution is the case i � 3 considering the variable temperature. In this paper, the case i � 3 is analyzed.
By virtue of Almansi's theorem, the function f in equation (10) can be expressed in the following form: where f j satisfy the following harmonic equation: For the convenience of operation, let and then Based on equations (14a)- (15), one can find that the functions F j (j � 1, 2) in equation (16) still satisfy the following harmonic equation: Two general solutions are obtained in which θ � 0 when i � 1, 2. ese solutions are in fact identical to those without the thermal effect which are not discussed here. erefore, the solution should be taken in equation (12c) in which i � 3.
e following general solution is obtained: Let Based on equations (17) and (19), one can find that the functions ψ j (j � 1, 2) still satisfy the following harmonic equation: In combination with equations (18) and (20), the solutions are obtained: where Substituting general solution equation (21) into constitutive equation (1), the stress general solution can be simplified as follows: e two-dimensional quasi-static general solutions for isotropic thermoelastic materials are obtained in equations (21) and (23).
ese general solutions are expressed in terms of two functions ψ j (j � 1, 2) in equation (20). e suitable functions ψ j (j � 1, 2) with undetermined constants should be constructed according to the configuration of the structures and the form of loadings. e undetermined constants will be determined by the corresponding boundary conditions, compatible conditions, and other requests. In addition, it is important that the function ψ 1 satisfies the harmonic equation and the function ψ 2 satisfies the quasi-static heat conduction equation. As an example, the suitable functions ψ j (j � 1, 2) under a dynamic line heat source will be constructed based on the obtained general solutions. For the arbitrary heat sources in actual engineering problems, their stress fields can be obtained based on the fundamental solutions for the line heat source using the superposition principle.

Fundamental Solution for an Infinite Isotropic Thermoelastic Plane under a Pulse Line Heat Source
Consider an infinite isotropic thermoelastic thin plane in two-dimensional Cartesian coordinates (x, z) with the principle material direction parallel to the xand z-axis ( Figure 1). A pulse line heat source H(x, z, t) � Hδ(t) is applied at the origin (0, 0) on the xoz plane. Based on general solutions (21) and (23), the coupled thermoelastic field in the infinite thermoelastic plane is derived in this section. e following functions are introduced by the trial and error method: where A j (j � 1, 2) are arbitrary constants to be determined and Substituting function (24) into general solutions (21) and (23) yields When the mechanical equilibrium is considered (Figure 1), the following equation can be obtained: Substituting equation (26a) into (27)  − ρc e One useful integral is listed as follows: where erf(x) is the error function. It is defined as follows: Substituting equation (29) into (28) yields If the time tends to infinity, all the components are given as follows: e shear stress would be zero at infinity, and the following equation can be obtained: Substituting equation (31) into (33) yields Hence, Green's solution for a pulse line heat source in an infinite isotropic thermoelastic plane is determined by equation (34).

Results and Discussion
Based on the solutions derived above, we now present some numerical results. e following nondimensional components are used in the figures: , , where r 0 is a nonzero dimension and α 0 and T 0 are the thermal expansion coefficient and reference temperature, respectively. Equations (32a) and (32b) should be rewritten in the following nondimensional form: where δ is a nondimensional heat source given as follows: Here, let δ � 1 act at the point (0, 0), so the heat source H � r 2 0 T 0 ρc e . e material properties for steel are listed in Table 1.

ermoelastic Field Contours under the Pulse Line Heat
Source. Temperature and thermal-stress distribution contours of steel are given in Figures 2-5 when a pulse line heat source acts on the center of the plane. On the basis of the contours plotted, one can find all components are singular at the origin in which the line heat source is located. e components change gradually with time τ and then tend to be stable which do not change over time. It can be found that there is a great thermal impact near the heat source at the moment of the heat source action. So, the stresses are relatively large which change quite dramatically. With time going on, the stress values decrease gradually with the diffusion of the heat source. We can clearly see the trend of the temperature field and stress field, which helps engineers to predict the engineering structure stress state under the action of thermal load in advance.
(1) Figure 2 shows that the contours of temperature increment ϑ(θ) in the infinite plane are all normal circles, which is because the thermal diffusivity κ along the xand z-axis is equal for steel, respectively. In the initial stage of the heat source acting on the plane, the heat source is confined in a small region. And the temperature increment is very small which nearly approaches to be zero, far away from the heat source region. is also indicates that the temperature gradient is very large under the pulse heat source. A large stress field is easily generated near the operating point of the heat source which could cause  structural damage. With the passage of time, the gradients of temperature increment decrease gradually and tend to be in a stable state which clearly reveals the change process of the temperature field.
(2) It can be seen from Figures 3 and 4 that the radial component σ ξ (σ x ) and axial component σ ς (σ z ) of thermal stress have positive and negative values which vary with the heat source acting time. e gradients of negative contours for the axial stress σ ς (σ z ) are much larger than those of positive contours at the time τ ′ � 1. As time goes on, the zero contour moves outward, and the whole stress fields would be compressive stress. In the initial stage of the heat acting, the stresses rise and drop sharply because the heat energy is mainly concentrated near the heat source region.
(3) Figure 5 shows that there is a zero common tangent for shear stress τ ςξ (τ zx ) in an infinite steel plane. And the contours of shear stress τ ςξ (τ zx ) are symmetric with the zero contours. e shear stress τ ςξ (τ zx ) changes significantly between the interfaces of the zero contour at the beginning which means the stress failure of the materials is possible to occur at the time.

ermoelastic Field Contours under Arbitrary Distributed
Pulse Heat Source Load. 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 e region of the distributed pulse heat source is e contours of the stress components and temperature component for the distributed heat source are shown in Figures 6-9. Figures 6-9 show that the temperature and stress components tend to be stable gradually which is the same as the variation trend of that in Section 5.1.

Variation Characteristics of the ermal Stress Field Components.
e distributions of temperature and thermal stress at different positions of the plane with different times are given in Figures 10-13. Figure 10 depicts the variation of temperature distribution with t. It indicates the values of temperature increment ϑ have a sharp rise and then have a rapid decline when 0 < t < 20. It is noticed that the temperature values slightly decrease when t ≥ 50 and then tend       to be stable. Figures 11 and 12 exhibit the variations of radial and axial stress components with time t. e thermal stresses show compressive stress when they are stable. Figure 13 shows the variation of the shear stress with time t. It is noticed that the values of the shear stress τ ξζ become stable when t > 20. When 0 < t < 20, the shear stress values are negative and increase rapidly.

Conclusions
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. e 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. e 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. e 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.

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

Additional Points
Highlights. (1) A 2D quasi-static isotropic thermoelastic model for thermal fields was established. (2) Green's function is expressed in terms of concise elementary functions. (3) e accurate thermal stress is analyzed.

Conflicts of Interest
e authors declare that they have no conflicts of interest.