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 lowtemperature 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 threetemperature 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-ratedependent 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) [24–26]. The boundary element method (BEM)

A New Computerized Boundary Element Model for Three-Temperature Nonlinear Generalized… DOI: http://dx.doi.org/10.5772/intechopen.90053

has been performed successfully for solving various engineering, scientific and mathematical applications due to its simplicity, efficiency, and ease of implementation [27–46].

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 threetemperature 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 ð Þ r, θ, z for the circular cylindrical plate structure (Figure 1) within the region R which bounded by boundary 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 <sup>d</sup><sup>θ</sup> <sup>¼</sup> <sup>1</sup> r .

Figure 1. Geometry of circular cylindrical plate structure.

Figure 2. Pressure distribution over the structure's entire surface.

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

$$
\sigma\_{pj\circ} = \mathbf{0} \tag{1}
$$

where

$$
\sigma\_{p\circ} = \mathcal{C}\_{pjkl} u\_{k\downarrow} - \beta\_{pj} T\_a(r, \tau) \tag{2}
$$

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

$$\mathcal{L}\_{\varepsilon} \frac{\partial T\_{\varepsilon}(r, \tau)}{\partial \tau} - \frac{1}{\rho} \nabla [\mathbb{K}\_{\varepsilon} \nabla T\_{\varepsilon}(r, \tau)] = -\mathbb{W}\_{\varepsilon i}(T\_{\varepsilon} - T\_i) - \mathbb{W}\_{\varepsilon p} \left(T\_{\varepsilon} - T\_p\right) \tag{3}$$

$$\varepsilon\_i \frac{\partial T\_i(r, \tau)}{\partial \tau} - \frac{1}{\rho} \nabla [\mathbb{K}\_i \nabla T\_i(r, \tau)] = \mathbb{W}\_{\acute{e}i}(T\_e - T\_i) \tag{4}$$

$$\frac{4}{\rho}c\_p T\_p^3 \frac{\partial T\_p(r, \tau)}{\partial \tau} - \frac{1}{\rho} \nabla \left[ \mathbb{K}\_p \nabla T\_p(r, \tau) \right] = \mathbb{W}\_{ep} \left( T\_\epsilon - T\_p \right) \tag{5}$$

where Te, Ti, Tp , ce,ci,cp and e, i, <sup>p</sup> are respectively temperatures, specific heat capacities and conductive coefficients of electron, ion and phonon. A New Computerized Boundary Element Model for Three-Temperature Nonlinear Generalized… DOI: http://dx.doi.org/10.5772/intechopen.90053

The total temperature

$$T = T\_\epsilon + T\_i + T\_p \tag{6}$$

## 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

$$\nabla \left[ \left( \delta\_{\mathbf{l}\dot{}} \mathbb{K}\_a + \delta\_{\mathbf{\dot{\beta}}} \mathbb{K}\_a^\* \right) \nabla T\_a(r, \tau) \right] - \overline{\mathcal{W}}(r, \tau) = c\_a \rho \delta\_{\mathbf{l}} \delta\_{\mathbf{l}\dot{}} \frac{\partial T\_a(r, \tau)}{\partial \tau} \tag{7}$$

where

$$\overline{\mathbb{W}}(r,\varepsilon) = \begin{cases} \rho \,\,\mathbb{W}\_{\varepsilon i}(T\_{\varepsilon} - T\_i) + \rho \,\,\mathbb{W}\_{\sigma}(T\_{\varepsilon} - T\_p) + \overline{\overline{\mathbb{W}}}, a = \varepsilon, \delta\_1 = 1 \\ -\rho \,\,\mathbb{W}\_{\varepsilon i}(T\_{\varepsilon} - T\_i) + \overline{\overline{\mathbb{W}}}, \quad a = i, \delta\_1 = 1 \\ -\rho \,\,\mathbb{W}\_{\sigma}(T\_{\varepsilon} - T\_p) + \overline{\overline{\mathbb{W}}}, \quad a = p, \delta\_1 = \frac{4}{\rho} T\_p^3 \end{cases} \tag{8}$$
 
$$\varepsilon\_{\varepsilon} \le \varepsilon\_{\varepsilon} \le \varepsilon\_{\varepsilon} + \varepsilon\_{\varepsilon} \le \varepsilon\_{\varepsilon} \quad \text{for} \quad \varepsilon\_{\varepsilon} + \varepsilon\_{\varepsilon} \le \varepsilon\_{\varepsilon} + \varepsilon\_{\varepsilon} \quad \varepsilon\_{\varepsilon} \le \varepsilon\_{\varepsilon} \quad \varepsilon\_{\varepsilon} \le \varepsilon\_{\varepsilon} \tag{9}$$

$$\begin{aligned} \overline{\mathcal{W}}(r,\tau) &= -\delta\_{\overline{2}\overline{}}\mathbb{K}\_{a}\dot{T}\_{a,ab} + \beta\_{ab}T\_{a0}\left[\dot{\mathcal{A}}\delta\_{\overline{1}\overline{}}\dot{u}\_{a,b} + \left(\tau\_{0} + \delta\_{\overline{2}\overline{}}\right)\ddot{u}\_{a,b}\right] + \rho c\_{a}\left[\left(\tau\_{0} + \delta\_{\overline{2}\overline{}}\tau\_{2} + \delta\_{\overline{2}\overline{}}\right)\ddot{T}\_{a}\right] \\ &- \frac{T\_{a,a}}{r}, \quad \frac{T\_{a,a}}{r} = T\_{a,\mu a} \end{aligned}$$

$$\left(\mathfrak{g}\right)$$

$$\mathbb{V}\_{\text{ci}} = \rho \mathbb{A}\_{\text{ci}} T\_{\text{e}}^{-2/3}, \mathbb{W}\_{\text{cr}} = \rho \mathbb{A}\_{\text{cr}} T\_{\text{e}}^{-1/2}, \mathbb{K}\_{a} = \mathbb{A}\_{a} T\_{a}^{5/2}, a = \mathfrak{e}, \text{i}, \mathbb{K}\_{p} = \mathbb{A}\_{p} T\_{p}^{3+\mathbb{B}} \tag{10}$$

The total energy can be written as follows

$$P = P\_\ell + P\_i + P\_p,\\ P\_\ell = c\_\ell T\_\ell,\\ P\_i = c\_i T\_i,\\ P\_p = \frac{1}{\rho} c\_p T\_p^4 \tag{11}$$

By applying the following conditions

$$T\_a(\mathbf{x}, \mathbf{y}, \mathbf{0}) = T\_a^0(\mathbf{x}, \mathbf{y}) = \mathbf{g}\_1(\mathbf{x}, \mathbf{r}) \tag{12}$$

$$\left. \mathbb{K}\_a \frac{\partial T\_a}{\partial n} \right|\_{\Gamma\_1} = 0, a = e, i, T\_r \vert\_{\Gamma\_1} = \mathbb{g}\_2(\infty, \mathfrak{r}) \tag{13}$$

$$\mathbb{K}\_a \frac{\partial T\_a}{\partial n} \bigg|\_{\Gamma\_2} = 0, a = e, i, p \tag{14}$$

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

$$D\nabla^2 T\_a + \frac{\partial T\_a^\*}{\partial n} = -\delta(r - p\_i)\delta(\tau - r) \tag{15}$$

where <sup>D</sup> <sup>¼</sup> <sup>α</sup> <sup>ρ</sup><sup>c</sup> and pi are singular points.

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

$$\text{C}T\_a = \frac{D}{\mathbb{K}\_a} \int\_O^\tau \int\_S \left[T\_a q^\* - T\_a^\* q\right] d\text{S}d\tau + \frac{D}{\mathbb{K}\_a} \int\_O^\tau \int\_R b T\_a^\* d\text{R}d\tau + \int\_R T\_a^i T\_a^\* \Big|\_{\tau=0} dR \tag{16}$$

which can be expressed as

$$\text{CT}\_{a} = \int\_{S} \left[ T\_{a} q^{\*} - T\_{a}^{\*} q \right] d\mathbf{S} - \int\_{R} \frac{\mathbb{K}\_{a}}{D} \frac{\partial T\_{a}^{\*}}{\partial \tau} T\_{a} dR \tag{17}$$

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

$$\frac{\partial T\_a}{\partial \boldsymbol{\pi}} \cong \sum\_{j=1}^N f^j(\boldsymbol{r})^j \boldsymbol{a}^j(\boldsymbol{\pi}) \tag{18}$$

where f <sup>j</sup> ð Þ<sup>r</sup> and <sup>a</sup><sup>j</sup> ð Þτ are known functions and unknown coefficients, respectively.

We assume that <sup>T</sup>^ <sup>j</sup> <sup>α</sup> is a solution of

$$\nabla^2 \hat{T}\_a^{\hat{J}} = f^{\hat{J}} \tag{19}$$

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

$$\mathbf{CT}\_{a} = \int\_{\mathcal{S}} \left[ T\_{a} q^{\*} - T\_{a}^{\*} q \right] d\mathbf{S} + \sum\_{j=1}^{N} a^{j}(\mathbf{r}) D^{-1} \left( \mathbf{C} \hat{T}\_{a}^{j} - \int\_{\mathcal{S}} \left[ T\_{a}^{j} q^{\*} - \dot{q}^{j} T\_{a}^{\*} \right] d\mathbf{S} \right) \tag{20}$$

where

$$\hat{q}^j = -\mathbb{K}\_a \frac{\partial \hat{T}\_a^j}{\partial n} \tag{21}$$

$$\mathfrak{a}^j(\mathfrak{r}) = \sum\_{i=1}^N f\_{ji}^{-1} \frac{\partial T(r\_i, \mathfrak{r})}{\partial \mathfrak{r}} \tag{22}$$

$$\{F\}\_{j\!i} = f^j(r\_i) \tag{23}$$

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

$$C\dot{T}\_a + HT\_a = GQ \tag{24}$$

where

$$\mathbf{C} = -\left[H\hat{T}\_a - \mathbf{G}\hat{Q}\right] \mathbf{F}^{-1} \mathbf{D}^{-1} \tag{25}$$

and

$$\left\{\hat{T}\right\}\_{\vec{\eta}} = \hat{T}^{\vec{f}}(\mathbf{x}\_{i}) \tag{26}$$

$$\left\{\hat{Q}\right\}\_{\vec{\eta}} = \hat{q}^{\dagger}(\mathbf{x}\_{i})\tag{27}$$

For solving (24) numerically, the functions q, T<sup>α</sup> and its derivative with time can be written as

$$q = (\mathbb{1} - \Theta)q^m + \Theta \, q^{m+1}, 0 \le \Theta \le \mathbb{1} \tag{28}$$

A New Computerized Boundary Element Model for Three-Temperature Nonlinear Generalized… DOI: http://dx.doi.org/10.5772/intechopen.90053

$$T\_a = (\mathbf{1} - \Theta)T\_a^m + \theta T\_a^{m+1}, \mathbf{0} \le \Theta = \frac{\boldsymbol{\tau} - \boldsymbol{\tau}^m}{\boldsymbol{\tau}^{m+1} - \boldsymbol{\tau}^m} \le \mathbf{1} \tag{29}$$

$$\dot{T}\_a = \frac{dT\_a}{d\Theta} \frac{d\Theta}{d\tau} = \frac{T\_a^{m+1} - T\_a^m}{\tau^{m+1} - \tau^m} = \frac{T\_a^{m+1} - T\_a^m}{\Delta \tau^m} \tag{30}$$

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

$$\left(\frac{\mathbf{C}}{\Delta \tau^{m}} + \Theta H\right) T\_{a}^{m+1} - \Theta G Q^{m+1} = \left(\frac{\mathbf{C}}{\Delta \tau^{m}} - (\mathbf{1} - \Theta)H\right) T\_{a}^{m} + (\mathbf{1} - \Theta)G Q^{m} \tag{31}$$

By applying the initial and boundary conditions, we obtain

$$\mathbf{a} \mathbf{X} = \mathbf{b} \tag{32}$$

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