1. 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 [3–12].

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.

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

Boundary Element Model for Nonlinear Fractional-Order Heat Transfer in Magneto…

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

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

The fractional-order governing equations of three temperatures nonlinear uncoupled magneto-thermoelasticity in FGA structures can be written as follows [6].

where , , uk, Cpjkl (Cpjkl ¼ Cklpj ¼ Ckljp), ( ), μ and hp are respectively, mechanic stress tensor, Maxwell stress tensor, displacement, constant

The nonlinear time-dependent two dimensions three temperature (2D-3 T) radiation diffusion equations coupled by electron, ion and phonon temperatures

elastic moduli, stress-temperature coefficients, magnetic permeability and

ð1Þ

ð2Þ

ð3Þ

proposed model.

and functionally graded structures.

DOI: http://dx.doi.org/10.5772/intechopen.88255

2. Formulation of the problem

perturbed magnetic field.

may be written as follows

Figure 1.

31

Geometry of the FGA structure.

direction of the effect of the constant magnetic field H0 .

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 [13], sigmoid law [14], exponential law [15] and power law [16–24].

There was widespread interest in functionally graded materials, which has developed a lot of analytical methods for analysis of elasticity [25–32] and thermoelasticity [33–53] problems, some of which have become dominant in scientific literature. For the numerical methods, the isogeometric finite element method (FEM) has been used by Valizadeh et al. [54] for static characteristics of FGM and by Bhardwaj et al. [55] for solving crack problem of FGM. Nowadays, the boundary element method is a simple, efficient and powerful numerical tool which provides an excellent alternative to the finite element method for the solution of FGM problems, Sladek et al. [56–58] have been developed BEM formulation for transient thermal problems in FGMs. Gao et al. [59] developed fracture analysis of functionally graded materials by a BEM. Fahmy [60–72] developed BEM to solve elastic, thermoelastic and biomechanic problems in anisotropic functionally graded structures. Further details on the BEM are given in [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 Model for Nonlinear Fractional-Order Heat Transfer in Magneto… DOI: http://dx.doi.org/10.5772/intechopen.88255

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.
