## 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.

### Keywords

- boundary element model
- three-temperature radiative heat conduction
- nonlinear generalized thermoelasticity
- thermal stresses
- anisotropic circular cylindrical plate structures

## 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

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

where

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

where

The total temperature

## 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

where

The total energy can be written as follows

By applying the following conditions

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

where

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

which can be expressed as

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

where

We assume that

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

where

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

where

and

For solving (24) numerically, the functions

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

By applying the initial and boundary conditions, we obtain

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]

where

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

where

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

where

The general solution of (39) can be obtained as

where

and the particular solution can be determined as

Thus, Eq. (41) can be written as

By implementing the following boundary conditions.

we can write the unknown

where

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

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

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

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

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

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

which roots

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 *r*. It was shown from this figure that the three temperatures are different and they may have great effects on the connected fields.

Figures 5 **–** 7 show the distributions of the thermal stresses *r* for the three temperatures

Figure 8 shows the distributions of the thermal stresses

## 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

stress-temperature coefficients

Kronecker delta

strain tensor

thermodynamic temperature

tractions

magnetic permeability

viscoelastic relaxation time

weights of control points

density

force stress tensor

specific heat capacity

constant elastic moduli

piezoelectric tensor

mass force vector

conductive coefficients

bending moment

total energy of unit mass

temperature functions

displacement vector

general solution

electron-ion energy coefficient

electron-photon energy coefficient