Boundary Element Model for Nonlinear Fractional-Order Heat Transfer in Magneto-Thermoelastic FGA Structures Involving Three Temperatures

The principal objective of this chapter is to introduce a new fractional-order theory for functionally graded anisotropic (FGA) structures. This theory called nonlinear uncoupled magneto-thermoelasticity theory involving three temperatures. Because of strong nonlinearity, it is very difficult to solve the problems related to this theory analytically. Therefore, it is necessary to develop new numerical methods for solving such problems. So, we propose a new boundary element model for the solution of general and complex problems associated with this theory. The numerical results are presented graphically in order to display the effect of the graded parameter on the temperatures and displacements. The numerical results also confirm the validity and accuracy of our proposed model.


Introduction
Functionally graded material (FGM) is a special type of advanced inhomogeneous materials. Functionally graded structure is a mixture of two or more distinct materials (usually heat-resisting ceramic on the outside surface and fractureresisting metal on the inside surface) that have specified properties in specified direction of the structure to achieve a require function [1,2]. This feature enables obtaining structures with the best of both material's properties, and suitable for applications requiring high thermal resistance and high mechanical strength [3][4][5][6][7][8][9][10][11][12].
Functionally Graded Materials have been wide range of thermoelastic applications in several fields, for example, the water-cooling model of a fusion reactor divertor is one of the most widely used models in industrial design, which is consisting of a tungsten (W) and a copper (Cu), that subjected to a structural integrity issue due to thermal stresses resulted from thermal expansion mismatch between the bond materials. Recently, functionally graded tungsten (W)-copper (Cu) has been developed by using a precipitation-hardened copper alloy as matrix instead of pure copper, to overcome the loss of strength due to the softening of the copper matrix.
The carbon nanotubes (CNT) in FGM have new applications such as reinforced functionally graded piezoelectric actuators, reinforced functionally graded polyestercalcium phosphate materials for bone replacement, reinforced functionally graded tools and dies for reduce scrap, better wear resistance, better thermal management, and improved process productivity, reinforced metal matrix functionally graded composites used in mining, geothermal drilling, cutting tools, drills and machining of wear resistant materials. Also, they used as furnace liners and thermal shielding elements in microelectronics.
There are many areas of application for elastic and thermoelastic functionally graded materials, for example, industrial applications such as MRI scanner cryogenic tubes, eyeglass frames, musical instruments, pressure vessels, fuel tanks, cutting tool inserts, laptop cases, wind turbine blades, firefighting air bottles, drilling motor shaft, X-ray tables, helmets and aircraft structures. Automobiles applications such as combustion chambers, engine cylinder liners, leaf springs, diesel engine pistons, shock absorbers, flywheels, drive shafts and racing car brakes. Aerospace applications rocket nozzle, heat exchange panels, spacecraft truss structure, reflectors, solar panels, camera housing, turbine wheels and Space shuttle. Submarine applications such as propulsion shaft, cylindrical pressure hull, sonar domes, diving cylinders and composite piping system. Biotechnology applications such as functional gradient nanohydroxyapatite reinforced polyvinyl alcohol gel biocomposites. Defense applications such as armor plates and bullet-proof vests. High-temperature environment applications such as aerospace and space vehicles. Biomedical applications such as orthopedic applications for teeth and bone replacement. Energy applications such as energy conversion devices and as thermoelectric converter for energy conservation. They also provide thermal barrier and are used as protective coating on turbine blades in gas turbine engine. Marine applications such as parallelogram slabs in buildings and bridges, swept wings of aircrafts and ship hulls. Optoelectronic applications such as automobile engine components, cutting tool insert coating, nuclear reactor components, turbine blade, tribology, sensors, heat exchanger, fire retardant doors, etc.
According to continuous and smooth variation of FGM properties throughout in depth, there are many laws to describe the behavior of FGM such as index [ [73,74] and the references therein.
In the present paper, we propose new FGA structures theory and new boundary element technique for modeling problems of nonlinear uncoupled magnetothermoelasticity involving three temperatures. The boundary element method reduces the dimension of the problem, therefore, we obtain a reduction of numerical approximation, linear equations system and input data. Since there is strong nonlinearity in the proposed theory and its related problems. So, we develop new boundary element technique for modeling such problems. The numerical results are presented graphically through the thickness of the homogeneous and functionally graded structures to show the effect of graded parameter on the temperatures and displacements. The numerical results demonstrate the validity and accuracy of our proposed model.
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 magneto-thermoelastic FGA 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 nonlinear radiative heat conduction equation, to obtain the three temperature fields. Section 4 studies the development of new BEM and its implementation for solving the move equation based on the known three temperature fields, to obtain the displacement field. Section 5 presents the new numerical results that describe the through-thickness mechanical behaviour of homogeneous and functionally graded structures.

Formulation of the problem
We consider a Cartesian coordinate system for 2D structure (see Figure 1) which is functionally graded along the 0x direction, and considering z-axis is the direction of the effect of the constant magnetic field H 0 .
The fractional-order governing equations of three temperatures nonlinear uncoupled magneto-thermoelasticity in FGA structures can be written as follows [6]. where , , u k , C pjkl (C pjkl ¼ C klpj ¼ C kljp ), ( ), μ and h p are respectively, mechanic stress tensor, Maxwell stress tensor, displacement, constant elastic moduli, stress-temperature coefficients, magnetic permeability and perturbed magnetic field.
The nonlinear time-dependent two dimensions three temperature (2D-3 T) radiation diffusion equations coupled by electron, ion and phonon temperatures may be written as follows  , The total energy per unit mass can be expressed as follows where are conductive coefficients, are temperature functions, are isochore specific-heat coefficients, ρ is the density, τ is the time. In which, , , B, A ei , A ep are constant inside each subdomain, W ei and W ep are electron-ion coefficient and electron-phonon coefficient, respectively.
Initial and boundary conditions can be written as

BEM numerical implementation for temperature field
This section outlines the solution of 2D nonlinear time-dependent three temperatures (electron, ion and phonon) radiation diffusion equations using a boundary element method.
denotes the Caputo fractional time derivative of order a defined by [75].

ð17Þ
By using a finite difference scheme of Caputo fractional time derivative of order a (17) at times and , we obtain: Where ð19Þ ð20Þ According to Eq. (18), the fractional order heat Eq. (4) can be replaced by the following system ð21Þ According to Fahmy [60], and using the fundamental solution which satisfies the system (21), the boundary integral equations corresponding to nonlinear three temperature heat conduction-radiation equations can be written as ð22Þ which can be written in the absence of internal heat sources as follows In which the entries of f À1 ji are the coefficients of F À1 with matrix F defined as [76].

ð29Þ
Using the standard boundary element discretization scheme for Eq. (26) and using Eq. (28), we have The diffusion matrix can be defined as By using initial and boundary conditions, we get This system yields the temperature, that can be used to solve (1) for the displacement. The first term in (41) can be integrated partially using Gau β theory yields ð

BEM numerical implementation for displacement field
The last term in (42) can be integrated partially twice using Gau β theory yields ð Based on Eq. (43), Eq. (42) can be rewritten as ð (44) which can be written as The boundary tractions are By using the symmetry relation of elasticity tensor, we obtain The displacement particular solution may be defined as Differentiation of (54) leads to. The domain integral may be approximated as follows The use of (57) together with the dual reciprocity From (50), we can write By using (52), (59) and (60), we obtain According to Fahmy [9-11], the right-hand side integrals of (61) can be reexpressed as According to Fahmy [12], Guiggiani and Gigante [77] and Mantič [78] Eqs. (62) and (63) can respectively be expressed as By implementing the point collocation procedure and using (67) and (68) From (73) using (75) we have By considering the following known k and unknown u superscripts nodal vectors Hence (76) may be written as From the first row of (79), we can calculate the unknown fluxes as follows From the second row of (79) and using (80) we get Eq. (81) can be written at n þ 1 ð Þtime step as where In order to solve (83), The implicit backward finite difference scheme has been applied based on the Houbolt's algorithm and the following approximations In which.

ð88Þ ð89Þ
We implement the successive over-relaxation (SOR) of Golub and Van Loan [79] for solving (87) to obtain . Then, the unknown and can be obtained from (76) and (77), respectively. By using the procedure of Bathe [80], we obtain the traction vector t u nþ1 from (73).

Numerical results and discussion
The BEM that has been used in the current paper can be applicable to a wide variety of FGA structures problems associated with the proposed theory of three temperatures nonlinear uncoupled magneto-thermoelasticity. In order to evaluate the influence of graded parameter on the three temperatures and displacements, the numerical results are carried out and depicted graphically for homogeneous (m ¼ 0) and functionally graded (m ¼ 0:5 and 1:0) structures.   2-4 show the distributions of the three temperatures T e , T i and T p through the thickness coordinate Ox. It was shown from these figures that the three temperatures increase with increasing value of graded parameter m. Figures 5 and 6 show the distributions of the displacements u 1 and u 2 through the thickness coordinate Ox. It was noticed from these figures that the displacement components increase with increasing value of graded parameter m.    Figures 7 and 8 show the distributions of the displacements u 1 and u 2 with the time for boundary element method (BEM), finite difference method (FDM) and finite element method (FEM) to demonstrate the validity and accuracy of our proposed technique. It is noted from numerical results that the BEM obtained results are agree quite well with those obtained using the FDM of Pazera and Jędrysiak [81] and FEM of Xiong and Tian [82] results based on replacing heat conduction with three-temperature heat conduction.

Conclusion
The main aim of this article is to introduce a new fractional-order theory called nonlinear uncoupled magneto-thermoelasticity theory involving three temperatures for FGA structures and new boundary element technique for solving problems related to the proposed theory. Since the nonlinear three temperatures radiative heat conduction equation is independent of the displacement field, we first determine the temperature field using the BEM, then based on the known temperature field, the displacement field is obtained by solving the move equation using the BEM. It can be seen from the numerical results that the graded parameter had a significant effect on the temperatures and displacements through the thickness of the functionally graded structures. Since there are no available results for the considered problem. So, some literatures may be considered as special cases from the considered problem based on replacing the heat conduction by three temperatures radiative heat conduction. The numerical results demonstrate the validity and accuracy of our proposed model. From the proposed BEM technique that has been performed in the present paper, it is possible to conclude that the proposed BEM should be applicable to any FGM uncoupled magneto-thermoelastic problem with three-temperature. BEM is more efficient, accurate and easy to use than FDM or FEM, because it only needs to solve the unknowns on the boundaries and BEM users need only to deal with real geometry boundaries. Also, BEM is reducing the computational cost of its solver. The present numerical results for our complex problem may provide interesting information for computer scientists, designers of new FGM materials and researchers in FGM science as well as for those working on the development of new functionally graded structures.