2. Formulation of the problem

instead of pure copper, to overcome the loss of strength due to the softening of the

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.

Mechanics of Functionally Graded Materials and Structures

sensors, heat exchanger, fire retardant doors, etc.

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

The carbon nanotubes (CNT) in FGM have new applications such as reinforced

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,

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],

There was widespread interest in functionally graded materials, which has developed a lot of analytical methods for analysis of elasticity [25–32] and

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

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

copper matrix.

30

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 H0 .

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

$$
\sigma\_{p\,\,\,\,\,\ell} + \tau\_{p\,\,\,\,\,\ell} = \rho(\chi + 1)^m \ddot{u}\_k \tag{1}
$$

$$
\sigma\_{p\uparrow} = (\mathfrak{x} + \mathfrak{1})^m \left[ \mathcal{C}\_{p\,fkl} u\_{k,l} \text{-} \beta\_{p\,f} T\_a(r, \mathfrak{r}) \right] \tag{2}
$$

$$
\pi\_{pf} = \mu (\varkappa + 1)^m \Big( h\_p H\_f + H\_l h\_p \text{-} \delta\_{pf} (h\_k H\_k) \Big) \tag{3}
$$

where , , uk, Cpjkl (Cpjkl ¼ Cklpj ¼ Ckljp), ( ), μ and hp 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

Figure 1. Geometry of the FGA structure.

$$D\_{\mathbf{r}}^{\alpha}T\_{\alpha}(r,\tau) = \xi \nabla[\mathbb{K}\_{\alpha}\,\nabla T\_{\alpha}(r,\tau)] + \xi \overline{\mathbb{W}}(r,\tau), \xi = \frac{1}{c\_{\alpha}\rho\delta\_{1}}\tag{4}$$

Now, let us consider and discretize the time interval 0½ � ; F into F þ 1 equal time steps, where , Let be the solution at time . Assuming that the time derivative of temperature within the

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

denotes the Caputo fractional time derivative of order a defined by [75].

By using a finite difference scheme of Caputo fractional time derivative of order

According to Eq. (18), the fractional order heat Eq. (4) can be replaced by the

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

ð16Þ

ð17Þ

ð18Þ

ð19Þ

ð20Þ

ð21Þ

ð22Þ

time interval can be approximated by.

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

a (17) at times and , we obtain:

Where

following system

33

where

$$
\overline{\mathbb{W}}(\mathbf{x}, \mathbf{y}, \mathbf{r}) = \begin{cases}
\ \begin{array}{l}
\ \begin{array}{l}
\ \begin{array}{l}
\ \rho
\end{array} \ \begin{array}{l}
\ \begin{array}{l}
\ \begin{array}{l}
\ \begin{array}{l}
\ \end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{aligned} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{aligned} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{aligned} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{array} \ \begin{array}{l}
\end{$$

and

$$\mathbb{V}\mathbb{W}\_{\nu i} = \rho \mathbb{A}\_{\nu i} T\_{\rho}^{\cdot 2/3}, \mathbb{W}\_{\text{ep}} = \rho \mathbb{A}\_{\text{ep}} T\_{\text{e}}^{\cdot 1/2} \mathbb{K}\_{\text{r}} = \mathbb{A}\_{\text{r}} T\_{\text{e}}^{5/2}, \alpha = \text{e}\_{\text{r}} \mathbb{i}\_{\mathbb{Y}} \mathbb{K}\_{\text{p}} = \mathbb{A}\_{\text{p}} T\_{\text{p}}^{3 + \mathbb{B}} \tag{6}$$

The total energy per unit mass can be expressed as follows

$$P = P\_{\text{e}} + P\_{l} + P\_{p\text{\prime}} \\ P\_{\text{e}} = c\_{\text{e}}T\_{\text{e}}, \\ P\_{l} = c\_{l}T\_{l\text{\prime}} \\ P\_{p} = \frac{1}{\rho}c\_{p}T\_{p}^{4} \tag{7}$$

where are conductive coefficients, are temperature functions, are isochore specific-heat coefficients, ρ is the density, τ is the time. In which, , , B, Aei, Aep are constant inside each subdomain, Wei and Wep are electron-ion coefficient and electron–phonon coefficient, respectively.

Initial and boundary conditions can be written as

$$T\_a(r,0) = T\_a^0(r) = g\_1(r,\tau),\tag{8}$$

$$
\dot{u}\_k(r,0) = \dot{u}\_k(r,0) = 0 \text{ for } r \in R \cup \mathcal{C}(r,\tau), \tag{9}
$$

$$\left. \mathbb{K}\_{\alpha} \frac{\partial T\_{\alpha}}{\partial n} \right|\_{\Gamma\_1} = 0, \alpha = e, i, T\_r \vert\_{\Gamma\_1} = g\_2(r, r) \tag{10}$$

$$
\mu\_k(r,\tau) = \Psi\_k(r,\tau) \text{ for } \quad r \in \mathcal{L}\_3 \tag{11}
$$

$$\text{tr}\_k(r,\tau) = \delta\_k(r,\tau) \quad \text{for} \quad r \in \mathcal{C}\_{4\prime}\mathcal{C} = \mathcal{C}\_3 \cup \mathcal{C}\_{4\prime}\mathcal{C}\_3 \cap \mathcal{C}\_4 = \emptyset \tag{12}$$

$$\left. \mathbb{K}\_{\alpha} \frac{\partial T\_{\alpha}}{\partial n} \right|\_{\Gamma\_2} = 0, \alpha = e, i, p \tag{13}$$

$$T(r,\tau) = \mathbf{H}(r,\tau) \quad \text{for} \quad r \in \mathcal{L}\_1, \qquad \tau > 0 \tag{14}$$

$$q(r,\tau) = h(r,\tau) \quad \text{for} \quad r \in \mathcal{C}\_2, \qquad \tau > 0,\\ \mathcal{C} = \mathcal{C}\_1 \cup \mathcal{C}\_2, \mathcal{C}\_1 \cap \mathcal{C}\_2 = \emptyset \qquad (15)$$

## 3. 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.

Boundary Element Model for Nonlinear Fractional-Order Heat Transfer in Magneto… DOI: http://dx.doi.org/10.5772/intechopen.88255

Now, let us consider and discretize the time interval 0½ � ; F into F þ 1 equal time steps, where , Let be the solution at time . Assuming that the time derivative of temperature within the time interval can be approximated by.

$$\dot{T}\_a(r,\tau) = \frac{T\_a^{f+1}(r) \cdot T\_a^f(r)}{\Delta \tau} + O(\Delta \tau) \tag{16}$$

denotes the Caputo fractional time derivative of order a defined by [75].

$$D\_{\mathbf{r}}^{a}T\_{a}(r,\mathbf{r}) = \frac{1}{\Gamma(1\cdot a)} \int\_{0}^{\tau} \frac{\partial T\_{a}(r,\mathbf{s})}{\partial \mathbf{s}} \frac{d\mathbf{s}}{(\tau\cdot\mathbf{s})^{a}}, \ 0 < a < 1\tag{17}$$

By using a finite difference scheme of Caputo fractional time derivative of order a (17) at times and , we obtain:

$$D\_{\mathbf{r}}^{a}T\_{a}^{f+1} + D\_{\mathbf{r}}^{a}T\_{a}^{f} \approx \sum\_{l=0}^{k} \mathcal{W}\_{a,j} \left(T\_{a}^{f+1-l}(r) \cdot T\_{a}^{f\cdot j}(r)\right), \{f = 1, 2, \dots, F\} \tag{18}$$

Where

ð4Þ

ð5Þ

<sup>p</sup> (6)

ð7Þ

ð8Þ

ð9Þ

ð10Þ

ð11Þ

ð12Þ

ð13Þ

ð14Þ

ð15Þ

, , <sup>K</sup><sup>p</sup> <sup>¼</sup> <sup>A</sup>pT<sup>3</sup>þ<sup>B</sup>

are isochore specific-heat coefficients, ρ is the density, τ is the time.

The total energy per unit mass can be expressed as follows

Mechanics of Functionally Graded Materials and Structures

Initial and boundary conditions can be written as

3. BEM numerical implementation for temperature field

a boundary element method.

32

This section outlines the solution of 2D nonlinear time-dependent three temperatures (electron, ion and phonon) radiation diffusion equations using

where are conductive coefficients, are temperature functions,

In which, , , B, Aei, Aep are constant inside each subdomain, Wei and Wep are electron-ion coefficient and electron–phonon coefficient, respectively.

where

and

$$\mathcal{W}\_{a,0} = \frac{(\Delta x)^{-a}}{\Gamma(2 \cdot a)} \tag{19}$$

$$\mathcal{W}\_{a,f} = \mathcal{W}\_{a,0}((f+1)^{\mathfrak{l}\cdot\mathfrak{a}} \cdot (f\text{-}1)^{\mathfrak{l}\cdot\mathfrak{a}}), f = 1, \mathbb{Z}, \dots, \mathbb{F} \tag{20}$$

According to Eq. (18), the fractional order heat Eq. (4) can be replaced by the following system

$$W\_{a,0}T\_a^{f+1}(r) \cdot \mathbb{K}\_a(\mathbf{x})T\_{a,ll}^{f+1}(r) \cdot \mathbb{K}\_{a,l}(\mathbf{x})T\_{a,l}^{f+1}(r)$$

$$= W\_{a,0}T\_a^{f}(r) \cdot \mathbb{K}\_r(\mathbf{x})T\_{a,m}^{f}(r) \cdot \mathbb{K}\_r(\mathbf{x})T\_a^{f}(r) \tag{21}$$

$$\cdot \sum\_{j=1}^{f} W\_{a,j} \{T\_a^{f+1-f}(r) \cdot T\_a^{f+j}(r)\} + \overline{W}\_m^{f+1}(\mathbf{x},\tau) + \overline{W}\_m^f(\mathbf{x},\tau), f = 0,1,2,\ldots,F$$

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

$$\text{C}T\_{a} = \frac{D}{\mathbb{K}\_{a}} \int\_{0}^{\pi} \int\_{S} \left[ T\_{a} q^{\*} \cdot T\_{a}^{\*} q \right] dS \, d\tau + \frac{D}{\mathbb{K}\_{a}} \int\_{0}^{\pi} \int\_{R} \left. b \left. T\_{a}^{\*} \right| dR \, d\tau + \int\_{R} \left. T\_{a}^{l} T\_{a}^{\*} \right|\_{\tau=0} dR \,\tag{22}$$

which can be written in the absence of internal heat sources as follows

$$CT\_a = \int\_S \left[ T\_a q^\* \cdot T\_a^\* q \right] \, dS \cdot \int\_R \, \frac{\mathbb{K}\_a}{D} \, \frac{\partial T\_a^\*}{\partial \tau} T\_a \, dR \tag{23}$$

Time temperature derivative can be written as

$$\frac{\partial T\_a}{\partial \boldsymbol{\pi}} \cong \sum\_{f=1}^N f^f(\boldsymbol{r})^f a^f(\boldsymbol{\pi}) \tag{24}$$

with

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

step.

displacement.

where

35

T^ � �

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

<sup>Q</sup>^ n o

By differentiating Eq. (34) with respect to time we get

The substitution of Eqs. (34)–(36) into Eq. (30) leads to

This system yields the temperature, that can be used to solve (1) for the

when the temperatures are known, the displacement can be computed by

<sup>p</sup> as the weight function and applying the

4. BEM numerical implementation for displacement field

Based on Eqs. (2) and (3), Eq. (1) can be rewritten as

weighted residual method, Eq. (39) can be reexpressed as

solving (39) using BEM. By choosing u<sup>∗</sup>

By using initial and boundary conditions, we get

ij <sup>¼</sup> <sup>T</sup>^<sup>j</sup>

ij <sup>¼</sup> <sup>q</sup>^<sup>j</sup>

In order to solve Eq. (30) numerically the functions and q are interpolated as

where determines the practical time τ in the current time

ð Þ xi (32)

ð Þ xi (33)

ð34Þ

ð35Þ

ð36Þ

ð37Þ

ð38Þ

ð39Þ

ð40Þ

where f j ð Þr are known functions and are unknown coefficients. We suppose that is a solution of

$$\nabla^2 \hat{\mathbb{T}}\_a^{\dot{j}} = f^{\dot{j}} \tag{25}$$

Then, Eq. (23) yields the following boundary integral equation

$$\mathcal{C}\ T = \int\_{\mathcal{S}} \left[ T\_a q^\* \cdot T\_a^\* q \right] dS + \sum\_{j=1}^{N} a^j(\mathfrak{r}) \mathcal{D}^{-1} \left( \mathcal{C} \mathcal{T}\_a^j \cdot \int\_{\mathcal{S}} \left[ T\_a^j q^\* \cdot \mathfrak{l}^j T\_a^\* \right] dS \right) \qquad (26)$$

where

$$\mathfrak{q}^f = \text{-} \mathbb{K}\_{\alpha} \frac{\partial \mathfrak{T}\_{\alpha}^f}{\partial n} \tag{27}$$

and

$$a^{\dagger}(\mathbf{r}) = \sum\_{l=1}^{N} f\_{ll}^{\mathbf{i}^1} \frac{\partial T(r\_{l\bullet}\mathbf{r})}{\partial \mathbf{r}} \tag{28}$$

In which the entries of f �1 ji are the coefficients of F�<sup>1</sup> with matrix F defined as [76].

$$\{\mathbf{F}\}\_{\mathcal{U}} = f^{\mathcal{J}}(\mathbf{r}\_{l}) \tag{29}$$

Using the standard boundary element discretization scheme for Eq. (26) and using Eq. (28), we have

$$\mathcal{C}\,\dot{T}\_{\sigma} + H\,T\_{\sigma} = \mathcal{G}\,\mathcal{Q} \tag{30}$$

The diffusion matrix can be defined as

$$\mathcal{C} = -[H \, \hat{T}\_a \, \text{-} \, \hat{Q}] F^{\text{-1}} D^{\text{-1}} \tag{31}$$

Boundary Element Model for Nonlinear Fractional-Order Heat Transfer in Magneto… DOI: http://dx.doi.org/10.5772/intechopen.88255

with

which can be written in the absence of internal heat sources as follows

ð Þr are known functions and are unknown coefficients.

Then, Eq. (23) yields the following boundary integral equation

Time temperature derivative can be written as

Mechanics of Functionally Graded Materials and Structures

We suppose that is a solution of

where f j

where

and

[76].

34

In which the entries of f

using Eq. (28), we have

�1

The diffusion matrix can be defined as

ji are the coefficients of F�<sup>1</sup> with matrix F defined as

Using the standard boundary element discretization scheme for Eq. (26) and

ð23Þ

ð24Þ

ð25Þ

ð26Þ

ð27Þ

ð28Þ

ð29Þ

ð30Þ

ð31Þ

$$\{\hat{\mathbf{T}}\}\_{\mathbf{ij}} = \hat{\mathbf{T}}^{\mathbf{j}}(\mathbf{x}\_{\mathbf{i}}) \tag{32}$$

$$\left\{\hat{\mathbb{Q}}\right\}\_{\mathfrak{j}} = \hat{\mathbf{q}}^{\dagger}(\mathbf{x}\_{\mathfrak{i}}) \tag{33}$$

In order to solve Eq. (30) numerically the functions and q are interpolated as

$$T\_a = \begin{pmatrix} \mathbf{1} - \boldsymbol{\theta} \end{pmatrix} T\_a^m + \boldsymbol{\theta} \ \boldsymbol{T}\_a^{m+1} \tag{34}$$

$$q = (1 - \theta)q^m + \theta \, q^{m+1} \tag{35}$$

where determines the practical time τ in the current time step.

By differentiating Eq. (34) with respect to time we get

$$\dot{T}\_{\alpha} = \frac{dT\_{\alpha}}{d\theta} \frac{d\theta}{d\tau} = \frac{T\_{\alpha}^{m+1} \cdot T\_{\alpha}^{m}}{\tau^{m+1} \cdot \tau^{m}} = \frac{T\_{\alpha}^{m+1} \cdot T\_{\alpha}^{m}}{\Delta \tau^{m}} \tag{36}$$

The substitution of Eqs. (34)–(36) into Eq. (30) leads to

$$\left(\frac{\mathcal{C}}{\Delta \tau^m} + \theta H\right) T\_a^{m+1} \cdot \theta G Q^{m+1} = \left(\frac{\mathcal{C}}{\Delta \tau^m} \cdot (1 - \theta) H\right) T\_a^m + (1 - \theta) G Q^m \qquad (37)$$

By using initial and boundary conditions, we get

$$\mathbf{a}\mathbf{X} = \mathbf{h} \tag{38}$$

This system yields the temperature, that can be used to solve (1) for the displacement.
