A New Computerized Boundary Element Model for Three-Temperature Nonlinear Generalized Thermoelastic Stresses in Anisotropic Circular Cylindrical Plate Structures

Abstract

In this chapter, we propose a new theory called nonlinear generalized thermoelasticity involving three temperatures. Because of strong nonlinearity of the proposed theory, therefore, it is much more difficult to develop analytical solution for solving problems related with the proposed theory. So, we propose a new computerized boundary element model for the solution of such problems and obtaining the three-temperature nonlinear generalized thermoelastic stresses in anisotropic circular cylindrical plate structures problems which are related with the proposed theory, where we used two-dimensional three temperature nonlinear radiative heat conduction equations coupled with electron, ion and phonon temperatures. The numerical results of the current study show the temperatures effects on the thermal stresses. Also, these numerical results demonstrate the validity and accuracy of our proposed model.

1. Introduction

The spiral formed tube which has been used in water transmission pipelines [1, 2] is the most common structural application of a cylindrical shell. Spiral formed pipes were initially constructed by riveting together appropriately bent plates [3] until advances in welding technology allowed for efficient tandem arc welding [1]. Recently, increasing attention has been devoted to the study of spiral welded tubes due to its many applications in water, gas and oil pipelines under both low and high pressure [4] as well as for foundation piles and primary load-bearing members in Combi-walls [5]. Spiral welded tubes provide certain benefits over traditional longitudinal and butt-welded tubes. In particular, continuous or very long tubular members may be constructed efficiently from compact coils of metal strip, eliminating the need for costly transport of long tubular members. The coil material is usually manufactured to very tight tolerances which results in a tube with consistent wall thickness [6]. Further, they exhibit a superior fatigue performance compared to longitudinal seam welded tubes [7]. They also exhibit a comparable resistance to crack growth propagation in ductile materials [8]. However, spiral welded tubes are not suitable for offshore and deep-water applications, because their diameter and wall thickness are limited to nearly 3 m and 30 mm, respectively [9] which generally makes them unsuitable for offshore and deep-water applications [10].

In recent years, great attention has been directed towards the study of generalized thermoelastic interactions in anisotropic thermoelastic models due to its many applications in physics, geophysics, astronautics, aeronautics, earthquake engineering, military technologies, plasma, robotics, mining engineering, accelerators, nuclear reactors, nuclear plants, soil dynamics, automobile industries, high-energy particle accelerators and other science and engineering applications. The main notion of photons, which are particles of light energy, has been introduced by Albert Einstein in 1905. It is difficult to interpret why temperature depends on the specific heat of the crystalline solids. So, the original notion of phonons, which are particles of heat, has also introduced by Albert Einstein in 1907 to explain this phenomenon. Our three-temperature study is essential for a wide range of low-temperature applications, such as pool and basin heating, unglazed and uninsulated flat-plate organic collectors, cold storage warehouses, outdoor applications in extreme low temperatures, cryogenic gas processing plants and frozen food processing facilities. Also, our three-temperature study is very important high temperature applications such as turbine blades, piston engine valves, turbo charger components, microwave devices, laser diodes, RF power amplifiers, tubes of steam power plant, recuperators in the metallurgical and glass industries. The proposed boundary element method (BEM) can be easily implemented for solving nonlinear generalized thermoelasticity problems. Through the present paper, the three-temperature concept introduced for the first time in the field of nonlinear generalized thermoelasticity. Duhamel [11] and Neumann [12] developed the classical thermo-elasticity (CTE) theory and obtained the strain-temperature gradients equations in an elastic body, but their theory has the following two shortcomings: First, the heat conduction equation is predicting infinite speeds of propagation. Second, the heat conduction equation does not contain elastic terms. Biot [13] developed the classical coupled thermo-elasticity (CCTE) theory to overcome the first shortcoming in CTE. Then, several generalized theories based on a modified Fourier’s law predict finite propagation speed of thermal waves such as extended thermo-elasticity (ETE) theory of Lord and Shulman (L-S) [14], temperature-rate-dependent thermo-elasticity (TRDTE) theory of Green and Lindsay (G-L) [15] and three linear generalized thermoelasticity models of Green and Naghdi (G-N) [16, 17], where Type I discusses the heat conduction theory based on Fourier’s law, type II describes the thermoelasticity theory without energy dissipation (TEWOED), and type III discusses the thermoelasticity theory with energy dissipation (TEWED). Due to the computational difficulties, inherent in solving nonlinear generalized thermoelastic problems [18], for such problems, it is very difficult to obtain the analytical solution in a general case. Instead of analytical methods, many numerical methods were developed for solving such problems approximately including the finite difference method (FDM) [19, 20], discontinuous Galerkin method (DGM) [21], finite element method (FEM) [22, 23] and boundary element method (BEM) [24, 25, 26]. The boundary element method (BEM) has been performed successfully for solving various engineering, scientific and mathematical applications due to its simplicity, efficiency, and ease of implementation [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46].

The main aim of the present chapter is to propose a new theory called nonlinear generalized thermoelasticity involving three-temperature. A new boundary element model was proposed for solving nonlinear generalized thermoelastic problems in anisotropic circular cylindrical plate structures which are associated with the proposed theory, where we used two-dimensional three-temperature (2D-3T) nonlinear time-dependent radiative heat conduction equations coupled with electron, ion and photon temperatures in the formulation of such problems. The numerical results are presented graphically to show the effects of electron, ion and photon temperatures on the thermal stress components. The validity and accuracy of our proposed BEM model were confirmed by comparing our BEM obtained results with the corresponding results of finite element method (FEM).

A brief summary of the chapter is as follows: Section 1 outlines the background and provides the readers with the necessary information to books and articles for a better understanding of mechanical behaviour of anisotropic circular cylindrical plate structures and their applications. Section 2 describes the formulation of the new theory and its related problems. Section 3 discusses the implementation of the new BEM for solving the three-temperature heat conduction equations, to obtain the temperature fields. Section 4 studies the development of new BEM and its implementation for solving the equilibrium equation based on the three-temperature fields. Section 5 presents the new numerical results that describe the temperatures effects on the thermal stresses generated in anisotropic circular cylindrical plate structures.

2. Formulation of the problem

We consider a cylindrical coordinate system $\left(r,\theta ,z\right)$ for the circular cylindrical plate structure ( Figure 1 ) within the region $\mathrm{R}$ which bounded by boundary $\mathrm{S}$ . Pressure distribution over the structure’s entire surface has been shown in Figure 2 . Geometry of meridional cross section of the considered structure has been shown in Figure 3 , where $\mathit{d\theta }=\frac{1}{r}$ .

Figure 1.

Geometry of circular cylindrical plate structure.

Figure 2.

Pressure distribution over the structure’s entire surface.

Figure 3.

Geometry of meridional cross section of considered structure.

The equilibrium equations for anisotropic plate structures can be written as follows

$${\sigma }_{\mathit{pj},j}=0$$ E1

where

$${\sigma }_{\mathit{pj}}=C_{\mathit{\text{pjkl}}}{u}_{k,l}-{\beta }_{\mathit{pj}}{T}_{\alpha }\left(r,\tau \right)$$ E2

Three radiative heat conduction equations coupled with electron, ion and phonon temperatures can be written as follows

$$c_e\frac{\partial T_e\left(r,\tau \right)}{\partial \tau }-\frac{1}{\rho }\nabla \left[{\mathbb{K}}_e\nabla T_e\left(r,\tau \right)\right]=-{\mathbb{W}}_{\mathit{ei}}\left(T_e-T_i\right)-{\mathbb{W}}_{\mathit{ep}}\left(T_e-T_p\right)$$ E3

$$c_i\frac{\partial T_i\left(r,\tau \right)}{\partial \tau }-\frac{1}{\rho }\nabla \left[{\mathbb{K}}_i\nabla T_i\left(r,\tau \right)\right]={\mathbb{W}}_{\mathit{ei}}\left(T_e-T_i\right)$$ E4

$$\frac{4}{\rho }{c}_p{T}_p^3\frac{\partial T_p\left(r,\tau \right)}{\partial \tau }-\frac{1}{\rho }\nabla \left[{\mathbb{K}}_p\nabla T_p\left(r,\tau \right)\right]={\mathbb{W}}_{\mathit{ep}}\left(T_e-T_p\right)$$ E5

where $\left(T_e,T_i,T_p\right),\left(c_e,c_i,c_p\right)$ and $\left({\mathbb{K}}_e,{\mathbb{K}}_i,{\mathbb{K}}_p\right)$ are respectively temperatures, specific heat capacities and conductive coefficients of electron, ion and phonon.

The total temperature

$$T=T_e+T_i+T_p$$ E6

3. BEM solution for three-temperature field

The nonlinear time-dependent two dimensions three temperature (2D-3T) radiative heat conduction Eqs. (3)–(5) coupled by electron, ion and phonon temperatures can be written as

$$\nabla \left[\left({\delta }_{1j}{\mathbb{K}}_{\alpha }+{\delta }_{2j}{\mathbb{K}}_{\alpha }^{\ast }\right)\nabla T_{\alpha }\left(r,\tau \right)\right]-\overline{\mathbb{W}}\left(r,\tau \right)=c_{\alpha }\rho {\delta }_1{\delta }_{1j}\frac{\partial T_{\alpha }\left(r,\tau \right)}{\partial \tau }$$ E7

where

$$\overline{\mathbb{W}}\left(r,\tau \right)=\left\{\begin{array}{c}\rho {\mathbb{W}}_{\mathit{ei}}\left(T_e-T_i\right)+\rho {\mathbb{W}}_{\mathit{er}}\left(T_e-T_p\right)+\overline{\overline{\mathbb{W}}},\alpha =e,{\delta }_1=1\\ -\rho {\mathbb{W}}_{\mathit{ei}}\left(T_e-T_i\right)+\overline{\overline{\mathbb{W}}},\alpha =i,{\delta }_1=1\\ -\rho {\mathbb{W}}_{\mathit{er}}\left(T_e-T_p\right)+\overline{\overline{\mathbb{W}}},\alpha =p,{\delta }_1=\frac{4}{\rho }{T}_p^3\end{array}\right.$$ E8

$$\begin{array}{ll}\overline{\overline{\mathbb{W}}}\left(r\text{,}\tau \right)=& -{\delta }_{2j}{\mathbb{K}}_{\alpha }{\dot{T}}_{\alpha ,ab}+{\beta }_{ab}{T}_{\alpha 0}\left[\AA {\delta }_{1j}{\dot{u}}_{a,b}+\left({\tau }_0+{\delta }_{2j}\right){\ddot{u}}_{a,b}\right]+\rho c_{\alpha }\left[\left({\tau }_0+{\delta }_{1j}{\tau }_2+{\delta }_{2j}\right){\ddot{T}}_{\alpha }\right]\\ & -\frac{T_{\alpha ,a}}{r},\frac{T_{\alpha ,a}}{r}=T_{\alpha ,aa}\end{array}$$ E9

$${\mathbb{W}}_{\mathit{ei}}=\rho {\mathbb{A}}_{\mathit{ei}}{T}_e^{-2/3},{\mathbb{W}}_{\mathit{er}}=\rho {\mathbb{A}}_{\mathit{er}}{T}_e^{-1/2},{\mathbb{K}}_{\alpha }={\mathbb{A}}_{\alpha }{T}_{\alpha }^{5/2},\alpha =e,i,{\mathbb{K}}_p={\mathbb{A}}_p{T}_p^{3+\mathbb{B}}$$ E10

The total energy can be written as follows

$$P=P_e+P_i+P_p,P_e=c_e{T}_e,P_i=c_i{T}_i,P_p=\frac{1}{\rho }{c}_p{T}_p^4$$ E11

By applying the following conditions

$$T_{\alpha }\left(x,y,0\right)=T_{\alpha }^0\left(x,y\right)=g_1\left(x,\tau \right)$$ E12

$${\left.{\mathbb{K}}_{\alpha }\frac{\partial T_{\alpha }}{\partial n}\right|}_{{\mathrm{\Gamma }}_1}=0,\alpha =e,i,{\left.T_r\right|}_{{\mathrm{\Gamma }}_1}=g_2\left(x,\tau \right)$$ E13

$${\left.{\mathbb{K}}_{\alpha }\frac{\partial T_{\alpha }}{\partial n}\right|}_{{\mathrm{\Gamma }}_2}=0,\alpha =e,i,p$$ E14

By using the fundamental solution that satisfies the following Eq. [46]

$$D{\nabla }^2{T}_{\alpha }+\frac{\partial T_{\alpha }^{\ast }}{\partial n}=-\delta \left(r-p_i\right)\delta \left(\tau -r\right)$$ E15

where $D=\frac{{\mathbb{K}}_{\alpha }}{\mathit{\rho c}}$ and $p_i$ are singular points.

The corresponding dual reciprocity boundary integral equation can be written as [46]

$$C{T}_{\alpha }=\frac{D}{{\mathbb{K}}_{\alpha }}{\int }_O^{\tau }{\int }_S\left[T_{\alpha }{q}^{\ast }-T_{\alpha }^{\ast }q\right]\mathit{\text{dSdτ}}+\frac{D}{{\mathbb{K}}_{\alpha }}{\int }_O^{\tau }{\int }_{R}b{T}_{\alpha }^{\ast }\mathit{\text{dRdτ}}+{\int }_R{\left.T_{\alpha }^i{T}_{\alpha }^{\ast }\right|}_{\tau =0}\mathit{dR}$$ E16

which can be expressed as

$$C{T}_{\alpha }={\int }_S\left[T_{\alpha }{q}^{\ast }-T_{\alpha }^{\ast }q\right]\mathit{dS}-{\int }_R\frac{{\mathbb{K}}_{\alpha }}{D}\frac{\partial T_{\alpha }^{\ast }}{\partial \tau }{T}_{\alpha }\mathit{dR}$$ E17

In order to transform the domain integral into the boundary, we assume that

$$\frac{\partial T_{\alpha }}{\partial \tau }\cong \sum _{j=1}^N{f}^j{\left(r\right)}^j{a}^j\left(\tau \right)$$ E18

where $f^j\left(r\right)$ and $a^j\left(\tau \right)$ are known functions and unknown coefficients, respectively.

We assume that ${\widehat{T}}_{\alpha }^j$ is a solution of

$${\nabla }^2{\widehat{T}}_{\alpha }^j=f^j$$ E19

Thus, from (17) we can write the following boundary integral equation

$$C{T}_{\alpha }={\int }_S\left[T_{\alpha }{q}^{\ast }-T_{\alpha }^{\ast }q\right]\mathit{dS}+\sum _{j=1}^N{a}^j\left(\tau \right)D^{-1}\left(C{\widehat{T}}_{\alpha }^j-{\int }_S\left[T_{\alpha }^j{q}^{\ast }-{\widehat{q}}^j{T}_{\alpha }^{\ast }\right]\mathit{dS}\right)$$ E20

where

$${\widehat{q}}^j=-{\mathbb{K}}_{\alpha }\frac{\partial {\widehat{T}}_{\alpha }^j}{\partial n}$$ E21

$$a^j\left(\tau \right)=\sum _{i=1}^N{f}_{\mathit{ji}}^{-1}\frac{\partial T\left(r_i,\tau \right)}{\partial \tau }$$ E22

$${\left\{F\right\}}_{\mathit{ji}}=f^j\left(r_i\right)$$ E23

By using (20) and (22), we obtain

$$C{\dot{T}}_{\alpha }+H{T}_{\alpha }=\mathit{GQ}$$ E24

where

$$C=-\left[H{\widehat{T}}_{\alpha }-G\widehat{Q}\right]F^{-1}{D}^{-1}$$ E25

and

$${\left\{\widehat{T}\right\}}_{\mathit{ij}}={\widehat{T}}^j\left(x_i\right)$$ E26

$${\left\{\widehat{Q}\right\}}_{\mathit{ij}}={\widehat{q}}^j\left(x_i\right)$$ E27

For solving (24) numerically, the functions $q$ , $T_{\alpha }$ and its derivative with time can be written as

$$q=\left(1-\mathrm{\Theta }\right)q^m+\mathrm{\Theta }{q}^{m+1},0\le \mathrm{\Theta }\le 1$$ E28

$$T_{\alpha }=\left(1-\mathrm{\Theta }\right)T_{\alpha }^m+\theta T_{\alpha }^{m+1},0\le \mathrm{\Theta }=\frac{\tau -{\tau }^m}{{\tau }^{m+1}-{\tau }^m}\le 1$$ E29

$${\dot{T}}_{\alpha }=\frac{d{T}_{\alpha }}{d\mathrm{\Theta }}\frac{d\mathrm{\Theta }}{\mathit{d\tau }}=\frac{T_{\alpha }^{m+1}-T_{\alpha }^m}{{\tau }^{m+1}-{\tau }^m}=\frac{T_{\alpha }^{m+1}-T_{\alpha }^m}{∆{\tau }^m}$$ E30

By substituting from Eqs. (28)–(30) into Eq. (24), we obtain

$$\left(\frac{C}{∆{\tau }^m}+\mathrm{\Theta }H\right)T_{\alpha }^{m+1}-\mathrm{\Theta }G{Q}^{m+1}=\left(\frac{C}{∆{\tau }^m}-\left(1-\mathrm{\Theta }\right)H\right)T_{\alpha }^m+\left(1-\mathrm{\Theta }\right)G{Q}^m$$ E31

By applying the initial and boundary conditions, we obtain

$$\mathbb{a}\mathrm{X}=\mathbb{b}$$ E32

This system yields the temperature in terms of the displacement field.

4. BEM solution for displacement field

The equilibrium Eqs. (1) for anisotropic plate structures can be written as follows [47]

$$C_{\mathit{\text{ijkl}}}\frac{d^{4}w}{d{x}^4}-T\frac{d^{2}w}{d{x}^2}=p+\frac{T_1}{r}$$ E33

where

$$T_2=-\frac{\mathit{pr}}{2}$$ E34

$$T_1=C_{\mathit{\text{ijkl}}}{T}_2-C_{\mathit{\text{ijkl}}}h\frac{w}{r}$$ E35

By using (34) and (35), we can write (33) in the following form

$$C_{\mathit{\text{ijkl}}}\frac{d^{4}w}{d{x}^4}+\frac{\mathit{pr}}{2}\frac{d^{2}w}{d{x}^2}+\frac{C_{\mathit{\text{ijkl}}}h}{r^2}w=p\left(1-\frac{C_{\mathit{\text{ijkl}}}}{2}\right)$$ E36

where

$$\mathrm{A}=\frac{C_{\mathit{\text{ijkl}}}h}{r^2}$$ E37

$$\mathrm{B}=p\left(1-\frac{C_{\mathit{\text{ijkl}}}}{2}\right)$$ E38

By using Eqs. (37) and (38), we can write (36) as follows

$$C_{\mathit{\text{ijkl}}}\frac{d^{4}w}{d{x}^4}-T\frac{d^{2}w}{d{x}^2}+\mathrm{A}w=\mathrm{B}$$ E39

where

$$\beta =\frac{T_2}{2\sqrt{C_{\mathit{\text{ijkl}}}k}},0<{\beta }^2<1$$ E40

The general solution of (39) can be obtained as

$$w\left(x\right)=C_1\mathit{\text{chδx}}cos\mathit{\gamma x}+C_2\mathit{\text{chδx}}sin\mathit{\gamma x}+C_3\mathit{\text{shδx}}cos\mathit{\gamma x}+C_4\mathit{\text{shδx}}sin\mathit{\gamma x}+w_{\text{part}}$$ E41

where

$$\delta =\alpha \sqrt{1+\beta };\gamma =\alpha \sqrt{1-\beta };\alpha =\sqrt[4]{\frac{k}{4C_{\mathit{\text{ijkl}}}}},\beta =\frac{T_2}{2\sqrt{C_{\mathit{\text{ijkl}}}k}}$$ E42

and the particular solution can be determined as $p=\text{constant}$ as follows

$$w_{\text{part}}=\frac{p{r}^2}{C_{\mathit{\text{ijkl}}}h}\left(1-\frac{C_{\mathit{\text{ijkl}}}}{2}\right)$$ E43

Thus, Eq. (41) can be written as

$$w\left(x\right)=\frac{p{r}^2}{C_{\mathit{\text{ijkl}}}h}\left(1-\frac{C_{\mathit{\text{ijkl}}}}{2}\right)+C_1\mathit{\text{chδx}}cos\mathit{\gamma x}+C_4\mathit{\text{shδx}}sin\mathit{\gamma x}$$ E44

By implementing the following boundary conditions.

$$\mathrm{at}x=\pm \frac{l}{2}\frac{\mathit{dw}}{\mathit{dx}}=0$$ E45

$$\mathrm{at}x=\frac{l}{2}w=\frac{2p{r}^2}{C_{\mathit{\text{ijkl}}}h}\frac{d^{3}w}{d{x}^3}$$ E46

we can write the unknown $C_1$ and $C_4$ as follows

$$C_1=-\frac{2p{r}^2}{C_{\mathit{\text{ijkl}}}h}\left(1-\frac{C_{\mathit{\text{ijkl}}}}{2}\right)\frac{u_1\mathit{ch}{u}_{1}sin{u}_2+u_2\mathit{sh}{u}_{1}cos{u}_2}{u_2\mathit{sh}2u_1+u_{1}sin2u_2}{\epsilon }_1$$ E47

$$C_4=-\frac{2p{r}^2}{C_{\mathit{\text{ijkl}}}h}\left(1-\frac{C_{\mathit{\text{ijkl}}}}{2}\right)\frac{u_2\mathit{ch}{u}_{1}sin{u}_2-u_1\mathit{sh}{u}_{1}cos{u}_2}{u_2\mathit{sh}2u_1+u_{1}sin2u_2}{\epsilon }_1$$ E48

where

$${\epsilon }_1=\frac{1}{1+\frac{\mathit{lh}}{A}{A}_1\left(u_1,u_2\right)}$$ E49

$$A_1\left(u_1,u_2\right)=\sqrt{1-{\beta }^2}\frac{\mathit{ch}2u_1-cos2u_2}{u_2\mathit{sh}2u_1+u_{1}sin2u_2}$$ E50

$$u_1=\frac{\mathit{\delta l}}{2}=u\sqrt{1+\beta },u_2=\frac{\mathit{\gamma l}}{2}=u\sqrt{1+\beta },u=0.6425\frac{1}{\sqrt{\mathit{rh}}}$$ E51

If we neglected the longitudinal forces influence on the bending of the circular cylindrical shell, we can write (39) in the following form

$$C_{\mathit{\text{ijkl}}}{w}^{\mathit{IV}}+\mathit{kw}=q$$ E52

Now, the approximate solution has been reduced for solving problem of bending single span beam with the following compliance

$$k_{\mathit{II}}=\frac{2r^2}{C_{\mathit{\text{ijkl}}}A}$$ E53

The deflection of the considered shell in the cross section and reference section, respectively, is as follows

$$w\left(0\right)=\frac{p{r}^2}{C_{\mathit{\text{ijkl}}}h}\left(1-\frac{v}{2}\right)\left[1-\frac{{\varnothing }_1\left(u\right)}{1+B_1}\right]$$ E54

$$w\left(\frac{l}{2}\right)=\frac{p{r}^2}{C_{\mathit{\text{ijkl}}}h}\left(1-\frac{v}{2}\right)\frac{B_1}{1+B_1}$$ E55

Also, the bending moment in the cross section and reference section, respectively, is as follows

$$M_1\left(0\right)=-\frac{p{l}^2}{24}\left(1-\frac{v}{2}\right)\left[1-\frac{{\chi }_1\left(u\right)}{1+B_1}\right]$$ E56

$$M_1\left(\frac{l}{2}\right)=\frac{p{l}^2}{12}\left(1-\frac{v}{2}\right)\frac{{\chi }_2\left(u\right)}{1+B_1}$$ E57

The Cauchy model with two-bed scheme can be described as follows

$$D{v}^{\mathit{IY}}\left(x\right)+\frac{\mathit{pr}}{2}{v}^{\prime \prime }\left(x\right)+\frac{C_{\mathit{\text{ijkl}}}h}{r^2}v\left(x\right)=p\left(1-\frac{\mu }{2}\right)$$ E58

$$v\left(0\right);\phi \left(0\right)=v^{\prime }\left(0\right)$$ E59

$$M\left(0\right)=-D{v}^{\prime \prime }\left(0\right)-\mathit{Tv}\left(0\right)$$ E60

$$Q\left(0\right)=-D{v}^{\prime \prime \prime }\left(0\right)-T{v}^{\prime }\left(0\right)$$ E61

where the characteristic equation of (58) can be defined as

$$C_{\mathit{\text{ijkl}}}{k}^4+\frac{\mathit{pr}}{2}{k}^2+\frac{C_{\mathit{\text{ijkl}}}h}{r^2}=0,k^2=t$$ E62

$$C_{\mathit{\text{ijkl}}}{t}^2+\frac{\mathit{pr}}{2}t+\frac{C_{\mathit{\text{ijkl}}}h}{r^2}=0$$ E63

which roots

$$k_{1,2,3,4}=\pm \sqrt{\frac{-\frac{\mathit{pr}}{2}\pm \sqrt{{\left(\frac{\mathit{pr}}{2}\right)}^2-4{C_{\mathit{\text{ijkl}}}}^2\frac{h}{r^2}}}{2C_{\mathit{\text{ijkl}}}}}$$ E64

$$t_{1,2}=\frac{-\frac{\mathit{pr}}{2}\pm \sqrt{{\left(\frac{\mathit{pr}}{2}\right)}^2-4{C_{\mathit{\text{ijkl}}}}^2\frac{h}{r^2}}}{2C_{\mathit{\text{ijkl}}}}$$ E65

The systems (32) and (58) can be solved by using the algorithm of Fahmy [35] to obtain the three temperatures and displacements components. Then we can compute thermal stresses distributions along radial distance r. we refer the reader to recent references [48, 49, 50, 51] for details of boundary element technique.

5. Numerical results and discussion

The BEM that has been used in the current chapter can be applicable to a wide variety of plate structures problems associated with the proposed theory of three temperatures nonlinear generalized thermoelasticity. In order to evaluate temperatures effects on the thermal stresses, the numerical results are carried out and depicted graphically for electron, ion and phonon temperatures.

Figure 4 shows the distributions of the three temperatures $T_e,T_i,T_p$ and total temperature $T\left(T=T_e+T_i+T_p\right)$ along the radial distance r. It was shown from this figure that the three temperatures are different and they may have great effects on the connected fields.

Figure 4.

Variation of the temperatures T_e, T_i, T_p and T along the radial distance r.

Figures 5 – 7 show the distributions of the thermal stresses ${\sigma }_{11},{\sigma }_{12}\text{and}{\sigma }_{22}$ respectively, with the radial distance r for the three temperatures $T_e,T_i,T_p$ and total temperature. It was noticed from these figures that the three temperatures have great effects on the thermal stresses.

Figure 5.

Variation of the thermal stress σ₁₁ with radial distance r.

Figure 6.

Variation of the thermal stress σ₁₂ with radial distance r.

Figure 7.

Variation of the thermal stress σ₂₂ with radial distance r.

Figure 8 shows the distributions of the thermal stresses ${\mathrm{\sigma }}_{11},{\mathrm{\sigma }}_{12},{\mathrm{\sigma }}_{22}$ and total temperature $\mathrm{T}$ with the radial distance r for BEM results and finite element method (FEM) results of COMSOL Multiphysics software version 5.4 to demonstrate the validity and accuracy of our proposed model based on replacing heat conduction with three-temperature heat conduction.

Figure 8.

Thermal stresses and total temperature variations with r.

6. Conclusion

The main objective of this chapter is to propose a new theory called nonlinear generalized thermoelasticity involving three-temperature and new BEM model for the solution of problems which are associated with the proposed nonlinear theory, where we used the three-temperature radiative heat conduction equations coupled with electron, ion and phonon temperatures to describe the thermal stresses in anisotropic circular cylindrical plate structures. It can be concluded from numerical results of our proposed model that the generalized theories of thermoelasticity can be connected with the three-temperature radiative heat conduction to describe the deformation of anisotropic circular cylindrical plate structures. The validity and accuracy of the proposed model was examined and confirmed by comparing the obtained results with those known previously. Because there are no available data to confirm the validity and accuracy of our results, we replace the three-temperature radiative heat conduction results with one-temperature heat conduction results as a special case from results of our current general model for circular cylindrical plate structures. In the special case under consideration, the results obtained with the BEM have been compared graphically with the FEM results of COMSOL Multiphysics software version 5.4. Excellent agreement is obtained between BEM results and FEM results. Understanding the behaviour of the three-temperature thermal stresses in anisotropic circular cylindrical plate structures should be a key for extending the application of these behaviors to a wide range of structures. The numerical results for our general model which is associated with our proposed theory may provide interesting information for computer scientists and engineers, geotechnical and geothermal engineers, researchers who will industrialize the thermoelastic devices using additive manufacturing and the materials designers and developers, etc.

Nomenclature

${\beta }_{\mathit{ij}}$

stress-temperature coefficients

${\delta }_{\mathit{ij}}$

Kronecker delta $\left(i,j=1,2\right)$

${\epsilon }_{\mathit{ij}}$

strain tensor

$\theta$

thermodynamic temperature

$\lambda$

tractions

${\mu }_0$

magnetic permeability

${\vartheta }_0$

viscoelastic relaxation time

$\varpi$

weights of control points

$\rho$

density

${\sigma }_{\mathit{ij}}$

force stress tensor

$c$

specific heat capacity

$C_{\mathit{\text{ijkl}}}$

constant elastic moduli

${\mathbb{e}}_{\mathit{lij}}$

piezoelectric tensor

$F_i$

mass force vector

${\mathbb{K}}_{\alpha }$

conductive coefficients

$M_1$

bending moment

$P$

total energy of unit mass

$T_{\alpha }$

temperature functions

$u_i$

displacement vector

$w\left(x\right)$

general solution

${\mathbb{W}}_{\mathrm{ei}}$

electron-ion energy coefficient

${\mathbb{W}}_{\mathrm{ep}}$

electron-photon energy coefficient