**1. Introduction**

Smart materials, which are also called intelligent materials, are engineered materials that have the ability to respond to the changes that occur around them in a controlled fashion by external stimuli, such as stress, heat, light, ultraviolet, moisture, chemical compounds, mechanical strength, and electric and magnetic fields. We can simply define smart materials as materials which adapt themselves as per required condition. The history of the discovery of these materials dates back to the 1880s when Jacques and Pierre Curie noticed a phenomenon that pressure generates electrification around a number of minerals such as quartz and tourmaline, and this phenomenon is called piezoelectric effect, so the piezoelectric materials are the oldest type of smart materials, which are utilized extensively in the fabrication of various devices such as transducers, sensors, actuators, surface acoustic wave

devices, frequency control, etc. There are a lot of smart material types like piezoelectric materials, thermochromic pigments, shape memory alloys, magnetostrictive, shape memory polymers, hydrogels, electroactive polymers and bi-component fibers, etc.

*dv x*‐*a*

*DOI: http://dx.doi.org/10.5772/intechopen.92829*

Laurent defined the integration of arbitrary order *v*>0 as

*xf x*ð Þ¼ *cD<sup>m</sup>*‐*<sup>ρ</sup> <sup>x</sup> f x*ð Þ¼ *dm*

*cDv*

*Dα*

<sup>∗</sup> *f t*ðÞ¼ <sup>1</sup>

Γð Þ *m* � *α*

*Dζ af*ð Þ¼ *τ*

*Dζ*

kernel functions are 1, 1½ � � ð Þ *<sup>τ</sup>*‐*<sup>ξ</sup>* and 1 � *<sup>τ</sup>*‐*<sup>ξ</sup>*

*<sup>ω</sup>f*ð Þ¼ *<sup>τ</sup>* <sup>1</sup> *ω* ð*τ*

*<sup>D</sup>ωf*ð Þ¼ *<sup>τ</sup>* <sup>1</sup>

the integration does not depend on the past time (local operator).

where *cD<sup>v</sup>*

order *α*>0 as

where *f*

expressed as

*f* 0

**95**

*dx<sup>v</sup>* ¼ �ð Þ<sup>1</sup> *<sup>v</sup> <sup>Γ</sup>*ð Þ *<sup>a</sup>* <sup>þ</sup> *<sup>v</sup>*

*A Novel MDD-Based BEM Model for Transient 3T Nonlinear Thermal Stresses in FGA Smart…*

By using Cauchy's integral formula for complex valued analytical functions,

1 Γð Þ*ρ*

*dx<sup>m</sup>*

Cauchy presented the following fractional order derivative:

ð*t* 0 *f* ð Þ *<sup>m</sup>* ð Þ*<sup>τ</sup>*

ð*τ a*

*f* ð Þ *α* <sup>þ</sup> ¼ ð *Γ*ð Þ *a*

ð*x c* ð Þ *<sup>x</sup>*‐*<sup>t</sup> <sup>ρ</sup>*‐<sup>1</sup>

*<sup>x</sup>* denotes differentiation of order *v* of the function *f* along the *x*-axis.

*<sup>f</sup>*ð Þ*<sup>τ</sup>* ð Þ *<sup>t</sup>* � *<sup>τ</sup>* �*α*�<sup>1</sup> Γð Þ �*a*

In 1967, the Italian mathematician Caputo presented his fractional derivative of

Diethelm [45] has suggested the Caputo derivative to be in the following form:

*<sup>K</sup>ζ*ð Þ *<sup>τ</sup>*‐*<sup>ξ</sup> <sup>f</sup>*

*<sup>K</sup>ζ*ð Þ¼ *<sup>τ</sup>*‐*<sup>ξ</sup>* ð Þ *<sup>τ</sup>*‐*<sup>ξ</sup> <sup>m</sup>*‐*ζ*‐<sup>1</sup>

Wang and Li [46] have introduced a memory-dependent derivative (MDD)

*<sup>K</sup>ζ*ð Þ *<sup>τ</sup>*‐*<sup>ξ</sup> <sup>f</sup>*

*<sup>K</sup>*ð Þ *<sup>τ</sup>*‐*<sup>ξ</sup> <sup>f</sup>* 0

*τ*‐*ω*

*ω* ð*τ*

Based on several practical applications, the memory effect needs weight <sup>0</sup>≤*K*ð Þ *<sup>τ</sup>*‐*<sup>ξ</sup>* <sup>&</sup>lt; 1 for *<sup>ξ</sup>*∈½ � *<sup>τ</sup>*‐*ω*, *<sup>τ</sup>* , so the MDD magnitude *<sup>D</sup>ωf*ð Þ*<sup>τ</sup>* is usually smaller than

ð Þ*<sup>τ</sup>* , where the time delay ð Þ *<sup>ω</sup>*<sup>&</sup>gt; <sup>0</sup> and the kernel function (0 <sup>≤</sup>*K*ð Þ *<sup>τ</sup>*‐*<sup>ξ</sup>* <sup>≤</sup>1 for *<sup>ξ</sup>*∈½ � *<sup>τ</sup>*‐*ξ*, *<sup>τ</sup>* ) can be chosen arbitrarily on the delayed interval ½ � *<sup>τ</sup>*‐*ω*, *<sup>τ</sup>* , the practical

*ω* � �*<sup>p</sup>*

monotonically increasing with *<sup>K</sup>* <sup>¼</sup> 0 for the past time *<sup>τ</sup>*‐*<sup>ξ</sup>* and *<sup>K</sup>* <sup>¼</sup> 1 for the present time τ. The main feature of MDD is that the real-time functional value depends also on the past time <sup>½</sup>*τ*‐*ξ*‐*τ*½. So, *<sup>D</sup><sup>ω</sup>* depends on the past time (nonlocal operator), while

*, p* <sup>¼</sup> <sup>1</sup>

where the first-order ð Þ *ζ* ¼ 1 of MDD for a differentiable function *f*ð Þ*τ* can be

*τ*‐*ω*

(*m*) is the *<sup>m</sup>*th order derivative and *<sup>m</sup>* is an integer such that *<sup>m</sup>*‐1<*<sup>ζ</sup>* <sup>≤</sup> *<sup>m</sup>*

*x*‐*a*‐*<sup>v</sup>* (2)

*dτ* (4)

ð Þ *<sup>m</sup>* ð Þ*<sup>ξ</sup> <sup>d</sup><sup>ξ</sup>* (6)

<sup>Γ</sup>ð Þ *<sup>m</sup>*‐*<sup>ζ</sup>* (7)

ð Þ *<sup>m</sup>* ð Þ*<sup>ξ</sup> <sup>d</sup><sup>ξ</sup>* (8)

ð Þ*ξ dξ* (9)

<sup>4</sup> , 1, 2, etc. These functions are

*f t*ð Þ*dt* � �, 0 <sup>&</sup>lt;*ρ*≤1 (3)

ð Þ *<sup>t</sup>* � *<sup>τ</sup> <sup>α</sup>*þ1�*<sup>m</sup> <sup>d</sup>τ*, *<sup>m</sup>* � <sup>1</sup><sup>&</sup>lt; *<sup>α</sup>*<sup>&</sup>lt; *<sup>m</sup>*, *<sup>α</sup>* <sup>&</sup>lt;0 (5)

Anisotropic smart structures (ASSs) are getting great attention of researchers due to their applications in textile, aerospace, mass transit, marine, automotive, computers and other electronic industries, consumer goods applications, mechanical and civil engineering, infertility treatment, micropumps, medical equipment applications, ultrasonic micromotors, microvalves and photovoltaics, rotating machinery applications, and much more [1–12].

The classical thermoelasticity (CTE) theory of Duhamel [13] and Neumann [14] has two shortcomings based on parabolic heat conduction equation of this theory: the first does not involve any elastic terms, while the second has infinite propagation speeds of thermoelastic waves. In order to overcome the first shortcoming, Biot [15] proposed the classical coupled thermoelasticity (CCTE). But CTE and CCTE have the second shortcoming. So, several generalized thermoelasticity theories have been developed to overcome the second shortcoming of CTE. Among of these theories are Lord and Shulman (LS) [16], Green and Lindsay (GL) [17], and Green and Naghdi [18, 19] theories of thermoelasticity with and without energy dissipation, dual-phase-lag thermoelasticity (DPLTE) [20, 21] and three-phase-lag thermoelasticity (TPLTE) [22]. Although thermoelastic phenomena in the majority of practical applications are adequately modeled with the classical Fourier heat conduction equation, there are an important number of problems that require consideration of nonlinear heat conduction equation. It is appropriate in these cases to apply the nonlinear generalized theory of thermoelasticity; great attention has been paid to investigate the nonlinear generalized thermoelastic problems by using numerical methods [23–34]. Fahmy [35–39] introduced the mathematical foundations of three-temperature (3T) field to thermoelasticity.

The fractional calculus is the mathematical branch that used to study the theory and applications of derivatives and integrals of arbitrary non-integer order. This branch has emerged in recent years as an effective tool for modeling and simulation of various engineering and industrial applications [40, 41]. Due to the nonlocal nature of fractional order operators, they are useful for describing the memory and hereditary properties of various materials and processes. Also, the fractional calculus has drawn wide attention from the researchers of various countries in recent years due to its applications in solid mechanics, fluid dynamics, viscoelasticity, heat conduction modeling and identification, biology, food engineering, econophysics, biophysics, biochemistry, electrochemistry, electrical engineering, finance and control theory, robotics and control theory, signal and image processing, electronics, electric circuits, wave propagation, nanotechnology, etc. [42–44].

Several mathematics researchers have contributed to the history of fractional calculus, where Euler mentioned interpolating between integral orders of a derivative in 1730. Then, Laplace defined a fractional derivative by means of an integral in 1812.

Lacroix presented the first formula for the fractional order derivative appeared in 1819, where he introduced the nth derivative of the function *<sup>y</sup>* <sup>¼</sup> *xm* as follows:

$$\frac{d^n}{d\mathbf{x}^n} = \frac{\Gamma(m+1)}{\Gamma(m\cdot n+1)} \mathbf{x}^{m\cdot n} \tag{1}$$

Liouville supposed that *dv dx<sup>v</sup> eax* ð Þ¼ *<sup>a</sup>veax* for *<sup>v</sup>*>0 to get the following fractional order derivative:

*A Novel MDD-Based BEM Model for Transient 3T Nonlinear Thermal Stresses in FGA Smart… DOI: http://dx.doi.org/10.5772/intechopen.92829*

$$\frac{d^v \mathcal{X}^{\cdot^a}}{d\mathbf{x}^v} = (-1)^v \frac{\Gamma(a+v)}{\Gamma(a)} \mathcal{X}^{\cdot^{a\cdot v}} \tag{2}$$

By using Cauchy's integral formula for complex valued analytical functions, Laurent defined the integration of arbitrary order *v*>0 as

$$\_cD\_x^{\nu}f(\mathbf{x}) = \_cD\_x^{m \cdot \rho}f(\mathbf{x}) = \frac{d^m}{d\mathbf{x}^m} \left[ \frac{1}{\Gamma(\rho)} \int\_c^{\infty} (\mathbf{x} \cdot \mathbf{t})^{\rho \cdot 1} f(t) dt \right], 0 < \rho \le 1 \tag{3}$$

where *cD<sup>v</sup> <sup>x</sup>* denotes differentiation of order *v* of the function *f* along the *x*-axis. Cauchy presented the following fractional order derivative:

$$f\_{+}^{(a)} = \left[ f(\tau) \frac{(t-\tau)^{-a-1}}{\Gamma(-a)} d\tau \right] \tag{4}$$

In 1967, the Italian mathematician Caputo presented his fractional derivative of order *α*>0 as

$$D\_{\ast}^{a}f(t) = \frac{1}{\Gamma(m-a)} \int\_{0}^{t} \frac{f^{(m)}(\tau)}{(t-\tau)^{a+1-m}} d\tau, m-1 < a < m, \quad a < 0 \tag{5}$$

Diethelm [45] has suggested the Caputo derivative to be in the following form:

$$D\_{\mathfrak{a}}^{\zeta}f(\mathfrak{r}) = \int\_{a}^{\mathfrak{r}} K\_{\zeta}(\mathfrak{r}\cdot\mathfrak{k}) f^{(m)}(\xi) \,d\xi \tag{6}$$

where *f* (*m*) is the *<sup>m</sup>*th order derivative and *<sup>m</sup>* is an integer such that *<sup>m</sup>*‐1<*<sup>ζ</sup>* <sup>≤</sup> *<sup>m</sup>*

$$K\_{\zeta}(\mathfrak{r}\cdot\mathfrak{F}) = \frac{\left(\mathfrak{r}\cdot\mathfrak{f}\right)^{m\cdot\zeta\cdot 1}}{\Gamma(m\cdot\zeta)}\tag{7}$$

Wang and Li [46] have introduced a memory-dependent derivative (MDD)

$$D\_{\alpha}^{\zeta}f(\boldsymbol{\tau}) = \frac{1}{\alpha} \int\_{\boldsymbol{\tau} \cdot \boldsymbol{\omega}}^{\boldsymbol{\tau}} K\_{\zeta}(\boldsymbol{\tau} \cdot \boldsymbol{\xi}) f^{(m)}(\boldsymbol{\xi}) \, d\boldsymbol{\xi} \tag{8}$$

where the first-order ð Þ *ζ* ¼ 1 of MDD for a differentiable function *f*ð Þ*τ* can be expressed as

$$D\_{\alpha}f(\boldsymbol{\pi}) = \frac{1}{\alpha} \int\_{\boldsymbol{\pi} \cdot \alpha \boldsymbol{\nu}}^{\boldsymbol{\pi}} K(\boldsymbol{\pi} \cdot \boldsymbol{\xi}) f'(\boldsymbol{\xi}) \, d\boldsymbol{\xi} \tag{9}$$

Based on several practical applications, the memory effect needs weight <sup>0</sup>≤*K*ð Þ *<sup>τ</sup>*‐*<sup>ξ</sup>* <sup>&</sup>lt; 1 for *<sup>ξ</sup>*∈½ � *<sup>τ</sup>*‐*ω*, *<sup>τ</sup>* , so the MDD magnitude *<sup>D</sup>ωf*ð Þ*<sup>τ</sup>* is usually smaller than *f* 0 ð Þ*<sup>τ</sup>* , where the time delay ð Þ *<sup>ω</sup>*<sup>&</sup>gt; <sup>0</sup> and the kernel function (0 <sup>≤</sup>*K*ð Þ *<sup>τ</sup>*‐*<sup>ξ</sup>* <sup>≤</sup>1 for *<sup>ξ</sup>*∈½ � *<sup>τ</sup>*‐*ξ*, *<sup>τ</sup>* ) can be chosen arbitrarily on the delayed interval ½ � *<sup>τ</sup>*‐*ω*, *<sup>τ</sup>* , the practical kernel functions are 1, 1½ � � ð Þ *<sup>τ</sup>*‐*<sup>ξ</sup>* and 1 � *<sup>τ</sup>*‐*<sup>ξ</sup> ω* � �*<sup>p</sup> , p* <sup>¼</sup> <sup>1</sup> <sup>4</sup> , 1, 2, etc. These functions are monotonically increasing with *<sup>K</sup>* <sup>¼</sup> 0 for the past time *<sup>τ</sup>*‐*<sup>ξ</sup>* and *<sup>K</sup>* <sup>¼</sup> 1 for the present time τ. The main feature of MDD is that the real-time functional value depends also on the past time <sup>½</sup>*τ*‐*ξ*‐*τ*½. So, *<sup>D</sup><sup>ω</sup>* depends on the past time (nonlocal operator), while the integration does not depend on the past time (local operator).

devices, frequency control, etc. There are a lot of smart material types like piezoelectric materials, thermochromic pigments, shape memory alloys,

bi-component fibers, etc.

*Advanced Functional Materials*

ogy, etc. [42–44].

integral in 1812.

order derivative:

**94**

Liouville supposed that *dv*

machinery applications, and much more [1–12].

tions of three-temperature (3T) field to thermoelasticity.

magnetostrictive, shape memory polymers, hydrogels, electroactive polymers and

Anisotropic smart structures (ASSs) are getting great attention of researchers due to their applications in textile, aerospace, mass transit, marine, automotive, computers and other electronic industries, consumer goods applications, mechanical and civil engineering, infertility treatment, micropumps, medical equipment applications, ultrasonic micromotors, microvalves and photovoltaics, rotating

The classical thermoelasticity (CTE) theory of Duhamel [13] and Neumann [14] has two shortcomings based on parabolic heat conduction equation of this theory: the first does not involve any elastic terms, while the second has infinite propagation speeds of thermoelastic waves. In order to overcome the first shortcoming, Biot [15] proposed the classical coupled thermoelasticity (CCTE). But CTE and CCTE have the second shortcoming. So, several generalized thermoelasticity theories have been developed to overcome the second shortcoming of CTE. Among of these theories are Lord and Shulman (LS) [16], Green and Lindsay (GL) [17], and Green and Naghdi [18, 19] theories of thermoelasticity with and without energy dissipation, dual-phase-lag thermoelasticity (DPLTE) [20, 21] and three-phase-lag

thermoelasticity (TPLTE) [22]. Although thermoelastic phenomena in the majority of practical applications are adequately modeled with the classical Fourier heat conduction equation, there are an important number of problems that require consideration of nonlinear heat conduction equation. It is appropriate in these cases to apply the nonlinear generalized theory of thermoelasticity; great attention has been paid to investigate the nonlinear generalized thermoelastic problems by using numerical methods [23–34]. Fahmy [35–39] introduced the mathematical founda-

The fractional calculus is the mathematical branch that used to study the theory and applications of derivatives and integrals of arbitrary non-integer order. This branch has emerged in recent years as an effective tool for modeling and simulation of various engineering and industrial applications [40, 41]. Due to the nonlocal nature of fractional order operators, they are useful for describing the memory and hereditary properties of various materials and processes. Also, the fractional calculus has drawn wide attention from the researchers of various countries in recent years due to its applications in solid mechanics, fluid dynamics, viscoelasticity, heat conduction modeling and identification, biology, food engineering, econophysics, biophysics, biochemistry, electrochemistry, electrical engineering, finance and control theory, robotics and control theory, signal and image processing, electronics, electric circuits, wave propagation, nanotechnol-

Several mathematics researchers have contributed to the history of fractional

Lacroix presented the first formula for the fractional order derivative appeared in 1819, where he introduced the nth derivative of the function *<sup>y</sup>* <sup>¼</sup> *xm* as follows:

*xm*‐*<sup>n</sup>* (1)

*dx<sup>v</sup> eax* ð Þ¼ *<sup>a</sup>veax* for *<sup>v</sup>*>0 to get the following fractional

calculus, where Euler mentioned interpolating between integral orders of a derivative in 1730. Then, Laplace defined a fractional derivative by means of an

> *dx<sup>n</sup>* <sup>¼</sup> *<sup>Γ</sup>*ð Þ *<sup>m</sup>* <sup>þ</sup> <sup>1</sup> *<sup>Γ</sup>*ð Þ *<sup>m</sup>*‐*<sup>n</sup>* <sup>þ</sup> <sup>1</sup>

*dn*

As a special case *<sup>K</sup>*ð Þ� *<sup>τ</sup>*‐*<sup>ξ</sup>* 1, we have

$$\mathbf{D}\_{\alpha}\mathbf{f}(\tau) = \frac{\mathbf{1}}{\alpha} \int\_{\tau-\alpha}^{\tau} \mathbf{f}'(\xi)d\xi = \frac{\mathbf{f}(\tau) - \mathbf{f}(\tau-\alpha)}{\alpha} \to \mathbf{f}'(\tau) \tag{10}$$

The above equation shows that the common derivative *<sup>d</sup> <sup>d</sup><sup>τ</sup>* is the limit of *D<sup>ω</sup>* as *ω* ! 0. That is,

$$D\_{\alpha}f(\tau) \le \left| \frac{\partial f}{\partial \tau} \right| = \lim\_{\alpha \to 0} \frac{f(\tau + \alpha) - f(\tau)}{\alpha} \tag{11}$$

The governing equations for the transient three-temperature nonlinear thermal stresses problems of FGA smart structures with memory-dependent derivatives can

*A Novel MDD-Based BEM Model for Transient 3T Nonlinear Thermal Stresses in FGA Smart…*

*<sup>σ</sup>ij* <sup>¼</sup> ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup> Cijkleδij*‐*βab <sup>T</sup>α*‐*Tα*<sup>0</sup> <sup>þ</sup> *<sup>τ</sup>*1*T*\_ *<sup>α</sup>*

*<sup>ε</sup>ij* <sup>¼</sup> <sup>1</sup>

where *σij*, *Fi*, *εij*, *εijk*, *ui*, and *ρ* are the force stress tensor, mass force vector, strain

∇½ �¼� *Ke*∇*Te*ð Þ *r; τ Wei*ð Þ� *Te* � *Ti Wep Te* � *Tp*

<sup>∇</sup> *Kp*∇*Tp*ð Þ *<sup>r</sup>; <sup>τ</sup>* <sup>¼</sup> *Wep Te* � *Tp*

are specific heat capacities, conductive coefficients,

*Cijkl Cijkl* <sup>¼</sup> *Cklij* <sup>¼</sup> *Cjikl* is the constant elastic moduli, e is the dilatation, *<sup>β</sup>ij <sup>β</sup>ij* <sup>¼</sup> *<sup>β</sup>ji*

are the stress-temperature coefficients, *Di* is the electric displacement, *eijk* is the piezoelectric tensor, fik is the permittivity tensor, and *Ek* is the electric field vector. The two-dimensional three-temperature (2D-3T) radiative heat conduction

where *e*, *i*, ∧ *p* denote electron, ion, and phonon, respectively; *ce*,*ci*,*cp*

*Di* <sup>¼</sup> ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup> eijkεjk* <sup>þ</sup> <sup>f</sup>*ikEk*

tensor, alternate tensor, displacement vector, and density, respectively,

*σij*,*<sup>j</sup>* þ *ρFi* ¼ *π*̈u*<sup>i</sup>* (12) *Di*,*<sup>i</sup>* ¼ 0 (13)

(14)

*ui*,*<sup>j</sup>* þ *u <sup>j</sup>*,*<sup>i</sup>*

∇½ �¼ *Ki*∇*Ti*ð Þ *r; τ Wei*ð Þ *Te* � *Ti* (17)

, (15)

(16)

(18)

,

2

be written as [35].

*Computational domain of the considered smart structure.*

*DOI: http://dx.doi.org/10.5772/intechopen.92829*

**Figure 1.**

where

equations can be expressed as

*ci*

*cpT*<sup>3</sup> *p* *<sup>∂</sup>Ti*ð Þ *<sup>r</sup>; <sup>τ</sup> <sup>∂</sup><sup>τ</sup>* � <sup>1</sup> *ρ*

*<sup>∂</sup>Tp*ð Þ *<sup>r</sup>; <sup>τ</sup> <sup>∂</sup><sup>τ</sup>* � <sup>1</sup> *ρ*

*<sup>∂</sup>Te*ð Þ *<sup>r</sup>; <sup>τ</sup> <sup>∂</sup><sup>τ</sup>* � <sup>1</sup> *ρ*

, and *Te*, *Ti*, *Tp*

*ce*

*Ke*, *Ki*, *Kp*

**97**

Now, the boundary element method (BEM) [47–80] is widely adopted for solving several engineering problems due to its easy implementation. In the BEM, only the boundary of the domain needs to be discretized, so it has a major advantage over other methods requiring full domain discretization [81–87] such as finite difference method (FDM), finite element method (FEM), and finite volume method (FVM) in engineering applications. This advantage of BEM over domain methods has significant importance for modeling of nonlinear generalized thermoelastic problems which can be implemented using BEM with little cost and less input data. Previously scientists have proven that FEM covers more engineering applications than BEM which is more efficient for infinite domain problems. But currently BEM scientists have changed their thinking and vision on BEM, where the BEM researchers developed the BEM technique for solving inhomogeneous and nonlinear problems involving infinite and semi-infinite domains by using a lot of software like FastBEM and ExaFMM.

The main objective of this chapter is to introduce a novel memory-dependent derivative model for solving transient three-temperature nonlinear thermal stress problems in functionally graded anisotropic (FGA) smart structures. The governing equations of the considered model are nonlinear and very difficult if not impossible to solve analytically. Therefore, we develop a new efficient boundary element technique for solving such equations. Numerical results show the effects of MDD on the three-temperature distributions and the influence of MDD and anisotropy on the nonlinear thermal stresses of FGA smart structures. Also, numerical results demonstrate the validity and accuracy of the proposed model.

A brief summary of the chapter is as follows: Section 1 introduces the background and provides the readers with the necessary information to books and articles for a better understanding of smart material problems, memory-dependent derivative history, and their applications. Section 2 describes the physical modeling of memory-dependent derivative problems of three-temperature nonlinear thermal stresses in FGA structures. Section 3 outlines the BEM implementation for obtaining the temperature field of the considered problem. Section 4 outlines the BEM implementation for obtaining the dispacement field of the considered problem. Section 5 introduces computing performance of the proposed model. Section 6 presents the new numerical results that describe the effects of memory-dependent derivative and anisotropy on the problem's field variations. Lastly, Section 7 outlines the significant findings of this chapter.

#### **2. Formulation of the problem**

With reference to a Cartesian system xð Þ 1, x2, x3 with a configuration R bounded by a closed surface S as shown in **Figure 1**.

*A Novel MDD-Based BEM Model for Transient 3T Nonlinear Thermal Stresses in FGA Smart… DOI: http://dx.doi.org/10.5772/intechopen.92829*

**Figure 1.** *Computational domain of the considered smart structure.*

The governing equations for the transient three-temperature nonlinear thermal stresses problems of FGA smart structures with memory-dependent derivatives can be written as [35].

$$
\sigma\_{\vec{\eta},\vec{j}} + \rho F\_i = \ddot{\pi} \mathbf{u}\_i \tag{12}
$$

$$D\_{i,i} = \mathbf{0} \tag{13}$$

where

As a special case *<sup>K</sup>*ð Þ� *<sup>τ</sup>*‐*<sup>ξ</sup>* 1, we have

*Advanced Functional Materials*

Dωfð Þ¼ τ

*ω* ! 0. That is,

1 ω ðτ

*Dωf*ð Þ*τ* ≤

demonstrate the validity and accuracy of the proposed model.

significant findings of this chapter.

**2. Formulation of the problem**

**96**

by a closed surface S as shown in **Figure 1**.

A brief summary of the chapter is as follows: Section 1 introduces the background and provides the readers with the necessary information to books and articles for a better understanding of smart material problems, memory-dependent derivative history, and their applications. Section 2 describes the physical modeling of memory-dependent derivative problems of three-temperature nonlinear thermal stresses in FGA structures. Section 3 outlines the BEM implementation for obtaining the temperature field of the considered problem. Section 4 outlines the BEM implementation for obtaining the dispacement field of the considered problem. Section 5 introduces computing performance of the proposed model. Section 6 presents the new numerical results that describe the effects of memory-dependent derivative and anisotropy on the problem's field variations. Lastly, Section 7 outlines the

With reference to a Cartesian system xð Þ 1, x2, x3 with a configuration R bounded

τ�ω f 0

The above equation shows that the common derivative *<sup>d</sup>*

*∂f ∂τ* � � � �

� � � � <sup>¼</sup> lim*<sup>ω</sup>*!<sup>0</sup>

ð Þ<sup>ξ</sup> <sup>d</sup><sup>ξ</sup> <sup>¼</sup> <sup>f</sup>ð Þ� <sup>τ</sup> <sup>f</sup>ð Þ <sup>τ</sup> � <sup>ω</sup>

Now, the boundary element method (BEM) [47–80] is widely adopted for solving several engineering problems due to its easy implementation. In the BEM, only the boundary of the domain needs to be discretized, so it has a major advantage over other methods requiring full domain discretization [81–87] such as finite difference method (FDM), finite element method (FEM), and finite volume method (FVM) in engineering applications. This advantage of BEM over domain methods has significant importance for modeling of nonlinear generalized thermoelastic problems which can be implemented using BEM with little cost and less input data. Previously scientists have proven that FEM covers more engineering applications than BEM which is more efficient for infinite domain problems. But currently BEM scientists have changed their thinking and vision on BEM, where the BEM researchers developed the BEM technique for solving inhomogeneous and nonlinear problems involving infinite and semi-infinite domains by using a lot of software like FastBEM and ExaFMM. The main objective of this chapter is to introduce a novel memory-dependent derivative model for solving transient three-temperature nonlinear thermal stress problems in functionally graded anisotropic (FGA) smart structures. The governing equations of the considered model are nonlinear and very difficult if not impossible to solve analytically. Therefore, we develop a new efficient boundary element technique for solving such equations. Numerical results show the effects of MDD on the three-temperature distributions and the influence of MDD and anisotropy on the nonlinear thermal stresses of FGA smart structures. Also, numerical results

<sup>ω</sup> ! <sup>f</sup>

*f*ð Þ� *τ* þ *ω f*ð Þ*τ*

0

ð Þτ (10)

*<sup>d</sup><sup>τ</sup>* is the limit of *D<sup>ω</sup>* as

*<sup>ω</sup>* (11)

$$
\sigma\_{\vec{\eta}} = (\infty + \mathbf{1})^m \left[ \mathbf{C}\_{\vec{\eta}kl} e \delta\_{\vec{\eta}} \cdot \beta\_{ab} \left( T\_a \mathbf{-} T\_{a0} + \tau\_1 \dot{T}\_a \right) \right] \tag{14}
$$

$$D\_i = \left(\mathbf{x} + \mathbf{1}\right)^m \left[\mathbf{e}\_{ijk}\mathbf{e}\_{jk} + \mathbf{f}\_{ik}\mathbf{E}\_k\right] \mathbf{e}\_{ij} = \frac{\mathbf{1}}{2}\left(\mathbf{u}\_{ij} + \mathbf{u}\_{j,i}\right),\tag{15}$$

where *σij*, *Fi*, *εij*, *εijk*, *ui*, and *ρ* are the force stress tensor, mass force vector, strain tensor, alternate tensor, displacement vector, and density, respectively, *Cijkl Cijkl* <sup>¼</sup> *Cklij* <sup>¼</sup> *Cjikl* is the constant elastic moduli, e is the dilatation, *<sup>β</sup>ij <sup>β</sup>ij* <sup>¼</sup> *<sup>β</sup>ji* are the stress-temperature coefficients, *Di* is the electric displacement, *eijk* is the piezoelectric tensor, fik is the permittivity tensor, and *Ek* is the electric field vector.

The two-dimensional three-temperature (2D-3T) radiative heat conduction equations can be expressed as

$$\mathcal{L}\_{\varepsilon} \frac{\partial T\_{\varepsilon}(r,\tau)}{\partial \tau} - \frac{1}{\rho} \nabla [K\_{\varepsilon} \nabla T\_{\varepsilon}(r,\tau)] = -\mathcal{W}\_{\varepsilon i} (T\_{\varepsilon} - T\_i) - \mathcal{W}\_{\varepsilon p} \left(T\_{\varepsilon} - T\_p\right) \tag{16}$$

$$
\sigma\_i \frac{\partial T\_i(r, \tau)}{\partial \tau} - \frac{1}{\rho} \nabla [K\_i \nabla T\_i(r, \tau)] = \mathcal{W}\_{\acute{e}i}(T\_e - T\_i) \tag{17}
$$

$$
\varepsilon\_p T\_p^3 \frac{\partial T\_p(r, \tau)}{\partial \tau} - \frac{1}{\rho} \nabla \left[ K\_p \nabla T\_p(r, \tau) \right] = W\_{cp} \left( T\_e - T\_p \right) \tag{18}
$$

where *e*, *i*, ∧ *p* denote electron, ion, and phonon, respectively; *ce*,*ci*,*cp* , *Ke*, *Ki*, *Kp* , and *Te*, *Ti*, *Tp* are specific heat capacities, conductive coefficients, and temperature functions, respectively; *Wei* is the electron-ion coefficient; and *Wep* is the electron-phonon coefficient.

#### **3. BEM solution of temperature field**

This section concerns using a boundary element method to solve the temperature model.

The above 2D-3T radiative heat conduction Eqs. (16)-(18) can be expressed in the context of nonlinear thermal stresses of FGA smart structures as in [36].

$$\nabla \left[ \left( \delta\_{\mathbf{\dot{\mathcal{H}}}} K\_a + \delta\_{\mathbf{\dot{\mathcal{H}}}} K\_a^\* \right) \nabla T\_a(r, \tau) \right] - \dot{\mathcal{W}}(r, \tau) = c\_a \rho \delta\_{\mathbf{\dot{\mathcal{H}}}} \delta\_{\mathbf{\dot{\mathcal{H}}}} D\_{a a} T\_a(r, \tau) \tag{19}$$

which can be written in the following form:

$$L\_{ab}T\_a(r,\tau) = f\_{ab} \tag{20}$$

*<sup>α</sup> ∂T<sup>α</sup> ∂n* � � � � Γ1

*DOI: http://dx.doi.org/10.5772/intechopen.92829*

By using the fundamental solutions *T*<sup>∗</sup>

*<sup>T</sup>α<sup>q</sup>* <sup>∗</sup> ‐*T*<sup>∗</sup> *<sup>α</sup> <sup>q</sup>* � �*dS d<sup>τ</sup>* <sup>þ</sup>

*CT<sup>α</sup>* ¼

mate the temperature time derivative as

*\_ j*

*<sup>T</sup>α<sup>q</sup>* <sup>∗</sup> ‐*<sup>T</sup>* <sup>∗</sup>

*<sup>α</sup> <sup>q</sup>* � �*dS* <sup>þ</sup><sup>X</sup>

ð *S*

ð Þ*<sup>r</sup>* are known functions and *<sup>a</sup> <sup>j</sup>*

*<sup>α</sup>* is a solution of

equation:

*CT<sup>α</sup>* <sup>¼</sup> *<sup>D</sup> <sup>α</sup>* ð*τ O* ð *S*

where *f <sup>j</sup>*

*CT<sup>α</sup>* ¼

where

and

where *f*

**99**

�1

We assume that *T*

ð *S* *<sup>α</sup> ∂T<sup>α</sup> ∂n* � � � � Γ2

¼ 0, *α* ¼ *e*, *i*, *Tp*

*A Novel MDD-Based BEM Model for Transient 3T Nonlinear Thermal Stresses in FGA Smart…*

*LabT*<sup>∗</sup>

Now, by implementing the technique of Fahmy [35], we can write (19) as

*D <sup>α</sup>* ð*τ O* ð *R bT* <sup>∗</sup>

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

*∂T<sup>α</sup> <sup>∂</sup><sup>τ</sup>* ffi <sup>X</sup> *N*

*<sup>T</sup>α<sup>q</sup>* <sup>∗</sup> � *<sup>T</sup>*<sup>∗</sup> *<sup>α</sup> <sup>q</sup>* � �*dS*‐ � �

<sup>Γ</sup><sup>1</sup> ¼ *g*2ð Þ *x*, *τ* (29)

¼ 0, *α* ¼ *e*, *i*, *p* (30)

*<sup>α</sup>* that satisfies the following differential

*<sup>α</sup>* ¼ *f ab* (31)

ð *R Ti <sup>α</sup>T* <sup>∗</sup> *α* � �

*<sup>τ</sup>*¼<sup>0</sup>*dR* (32)

*<sup>∂</sup><sup>τ</sup> <sup>T</sup><sup>α</sup> dR* (33)

ð Þ*τ* (34)

*<sup>α</sup> dR dτ* þ

*∂T*<sup>∗</sup> *α*

ð Þ*τ* are unknown coefficients.

*<sup>α</sup>* <sup>¼</sup> *<sup>f</sup> <sup>j</sup>* (35)

*T*∗ *α* � �*dS* � � (36)

*<sup>∂</sup><sup>τ</sup>* (38)

ð Þ *ri* (39)

(37)

ð *R <sup>α</sup> D*

In order to transform the domain integral in (33) to the boundary, we approxi-

*i*¼1 *f j* ð Þ*r j a j*

∇2 *T \_ j*

ð Þ*<sup>τ</sup> <sup>D</sup>*�<sup>1</sup> *CT*

*∂T \_ j α ∂n*

*<sup>∂</sup>T<sup>α</sup> ri* ð Þ , *<sup>τ</sup>*

^*<sup>q</sup> <sup>j</sup>* <sup>¼</sup> ‐*<sup>α</sup>*

*\_ j α*‐ ð *S T j <sup>α</sup><sup>q</sup>* <sup>∗</sup> ‐^*<sup>q</sup> <sup>j</sup>*

Then, Eq. (33) leads to the following boundary integral equation

*N*

*i*¼1 *a j*

*a j*

ð Þ¼ *<sup>τ</sup>* <sup>X</sup> *N*

*i*¼1 *f* ‐1 *ji*

*ji* are the coefficients of *F*�<sup>1</sup> which are defined as [58].

f g*<sup>F</sup> ji* <sup>¼</sup> *<sup>f</sup> <sup>j</sup>*

where

$$L\_{ab} = \nabla \left[ \left( \delta\_{\sharp j} K\_a + \delta\_{\sharp j} K\_a^\* \right) \nabla \right] \tag{21}$$

$$f\_{ab} = \acute{W}(r, \pi) + \acute{W}(r, \pi) \tag{22}$$

where

$$\dot{\mathcal{W}}(r,\varepsilon) = \begin{cases} \rho \, \mathcal{W}\_{\varepsilon i} \left( T\_{\varepsilon} - T\_{i} \right) + \rho \, \mathcal{W}\_{\varepsilon r} \left( T\_{\varepsilon} - T\_{p} \right) + \dot{\mathcal{W}}, a = \varepsilon, \delta\_{1} = 1 \\ -\rho \, \mathcal{W}\_{\varepsilon i} \left( T\_{\varepsilon} - T\_{i} \right) + \dot{\mathcal{W}}, & a = i, \delta\_{1} = 1 \\ -\rho \, \mathcal{W}\_{\varepsilon r} \left( T\_{\varepsilon} - T\_{p} \right) + \dot{\mathcal{W}}, & a = p, \delta\_{1} = T\_{p}^{3} \end{cases} \tag{23}$$
 
$$\dot{\mathcal{W}}(r,\varepsilon) = \mathcal{E}(r,\varepsilon) \quad \delta\_{2} \neq \varepsilon^{\prime} \quad \delta\_{1} \left( \varepsilon^{2} T\_{\varepsilon}^{2} \left( \varepsilon^{-2} \right) \right) J^{\varepsilon}$$

$$
\begin{split}
\dot{\mathcal{W}}(r,\tau) &= F(r,\tau) - \frac{\delta\_{\mathcal{G}}K\_{a}}{\alpha\_{a}} \int\_{\tau-\alpha\_{a}}^{\tau} K(\tau-\xi) \frac{\partial}{\partial\xi} \Big(\nabla^{2}T\_{a}(r,\tau)\big) d\xi \\ &+ \frac{\rho \mathcal{C}\_{a}\delta\_{\mathcal{I}}\delta\_{\mathcal{I}}}{\alpha\_{a}} \int\_{\tau-\alpha\_{a}}^{\tau} K(\tau-\xi) \frac{\partial}{\partial\xi} (T\_{a}(r,\tau)) d\xi \\ &+ \frac{\rho \mathcal{C}\_{a}(\tau\_{0}+\delta\_{\mathcal{I}}\tau\_{2}+\delta\_{\mathcal{I}})}{\alpha\_{a}} \int\_{\tau-\alpha\_{a}}^{\tau} K(\tau-\xi) \frac{\partial^{2}}{\partial\xi^{2}} (T\_{a}(\mathbf{r},\tau)) d\xi \end{split} \tag{24}
$$

$$F(r,\tau) = \beta\_{ab} T\_{a0} \left[ \rhd \delta\_{1\dot{j}} \acute{u}\_{a,b} + (\tau\_0 + \delta\_{2\dot{j}}) \acute{u}\_{a,b} \right] \tag{25}$$

and

$$\mathbf{W}\_{\text{ci}} = \rho \mathbf{A}\_{\text{ci}} \mathbf{T}\_{\text{e}}^{-2/3}, \mathbf{W}\_{\text{cr}} = \rho \mathbf{A}\_{\text{cr}} \mathbf{T}\_{\text{e}}^{-1/2}, \mathbf{K}\_{a} = \mathbf{A}\_{a} \mathbf{T}\_{a}^{5/2}, a = e, \mathbf{i}, \mathbf{K}\_{p} = \mathbf{A}\_{p} \mathbf{T}\_{p}^{3+B} \tag{26}$$

where *δij*,ð Þ *i*, *j* ¼ 1, 2 , *ωα*ð Þ 0 ð Þ *α* ¼ *e*, *i* ∧ *p* , and *K*ð Þ *τ* � *ξ* are the Kronecker delta, delay times, and kernel function, respectively.

The total energy can be expressed as

$$P = P\_{\epsilon} + P\_{i} + P\_{p'} \\ P\_{\epsilon} = c\_{\epsilon}T\_{\epsilon}, \\ P\_{i} = c\_{i}T\_{i}, \\ P\_{p} = \frac{1}{4}c\_{p}T\_{p}^{4} \tag{27}$$

Initial and boundary conditions can be expressed as

$$T\_a(\mathbf{x}, \mathbf{y}, \mathbf{0}) = T\_a^0(\mathbf{x}, \mathbf{y}) = \mathbf{g}\_1(\mathbf{x}, \mathbf{r})\tag{28}$$

*A Novel MDD-Based BEM Model for Transient 3T Nonlinear Thermal Stresses in FGA Smart… DOI: http://dx.doi.org/10.5772/intechopen.92829*

$$\left. \mathbb{K}\_a \frac{\partial T\_a}{\partial n} \right|\_{\Gamma\_1} = 0, a = e, i, \left. T\_p \right|\_{\Gamma\_1} = \mathbf{g}\_2(\mathbf{x}, \mathbf{r}) \tag{29}$$

$$\mathbb{K}\_a \frac{\partial T\_a}{\partial n} \bigg|\_{\Gamma\_2} = \mathbf{0}, a = e, i, p \tag{30}$$

By using the fundamental solutions *T*<sup>∗</sup> *<sup>α</sup>* that satisfies the following differential equation:

$$L\_{ab}T\_a^\* = f\_{ab} \tag{31}$$

Now, by implementing the technique of Fahmy [35], we can write (19) as

$$\text{CT}\_{a} = \frac{D}{\mathbb{K}\_{a}} \int\_{O}^{\tau} \int\_{S} \left[ T\_{a} q^{\*} \cdot T\_{a}^{\*} q \right] d\mathbf{S} d\tau + \frac{D}{\mathbb{K}\_{a}} \int\_{O}^{\tau} \int\_{R} b \left. T\_{a}^{\*} \right|\_{} d\mathbf{R} d\tau + \int\_{R} \left. T\_{a}^{i} T\_{a}^{\*} \right|\_{\tau=0} d\mathbf{R} \tag{32}$$

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

$$\text{CT}\_{a} = \int\_{\mathcal{S}} \left[ T\_{a} q^{\*} - T\_{a}^{\*} q \right] d\mathbf{S} \cdot \int\_{R} \frac{\mathbb{K}\_{a}}{D} \frac{\partial T\_{a}^{\*}}{\partial \boldsymbol{\pi}} T\_{a} d\mathbf{R} \tag{33}$$

In order to transform the domain integral in (33) to the boundary, we approximate the temperature time derivative as

$$\frac{\partial T\_a}{\partial \mathbf{r}} \cong \sum\_{i=1}^N f^j(r)^j a^j(\mathbf{r}) \tag{34}$$

where *f <sup>j</sup>* ð Þ*<sup>r</sup>* are known functions and *<sup>a</sup> <sup>j</sup>* ð Þ*τ* are unknown coefficients. We assume that *T \_ j <sup>α</sup>* is a solution of

$$\nabla^2 \hat{T}\_a^j = f^j \tag{35}$$

Then, Eq. (33) leads to the following boundary integral equation

$$\mathbf{C}\,T\_{a} = \int\_{S} \left[T\_{a}q^{\*}\,\prescript{\*}{}{T}\_{a}^{\*}q\right]d\mathbf{S} + \sum\_{i=1}^{N}a^{j}(\tau)D^{-1}\left(\mathbf{C}\overset{\frown}{T}\_{a^{\*}}^{j}\int\_{S}\left[T\_{a}^{j}q^{\*}\cdot\dot{q}^{j}T\_{a}^{\*}\right]d\mathbf{S}\right) \tag{36}$$

where

and temperature functions, respectively; *Wei* is the electron-ion coefficient; and

This section concerns using a boundary element method to solve the tempera-

The above 2D-3T radiative heat conduction Eqs. (16)-(18) can be expressed in

� �∇*Tα*ð Þ *<sup>r</sup>*, *<sup>τ</sup>* � � � *W r* ´ ð Þ¼ , *<sup>τ</sup> <sup>c</sup>αρδ*1*δ*1*jDωαTα*ð Þ *<sup>r</sup>*, *<sup>τ</sup>* (19)

*LabTα*ð Þ¼ *r*, *τ f ab* (20)

� �∇ � � (21)

*<sup>f</sup> ab* <sup>¼</sup> *W r* ´ ð Þþ , *<sup>τ</sup> W r* ´ ð Þ , *<sup>τ</sup>* (22)

� � <sup>þ</sup> *<sup>W</sup>*´ , *<sup>α</sup>* <sup>¼</sup> *<sup>e</sup>*, *<sup>δ</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup>

*<sup>T</sup>α*ð Þ *<sup>r</sup>*, *<sup>τ</sup>* � �*d<sup>ξ</sup>*

� �*u*´*<sup>a</sup>*,*<sup>b</sup>*

*<sup>α</sup>* , *<sup>α</sup>* <sup>¼</sup> *<sup>e</sup>*, *<sup>i</sup>*, *Kp* <sup>¼</sup> *ApT*<sup>3</sup>þ*<sup>B</sup>*

4 *cpT*<sup>4</sup>

*<sup>α</sup>*ð Þ¼ *x*, *y g*1ð Þ *x*, *τ* (28)

*p*

*<sup>∂</sup>ξ*<sup>2</sup> ð Þ *<sup>T</sup>α*ð Þ r, *<sup>τ</sup> <sup>d</sup><sup>ξ</sup>* (24)

(23)

(25)

*<sup>p</sup>* (26)

*<sup>p</sup>* (27)

*α*

the context of nonlinear thermal stresses of FGA smart structures as in [36].

*Lab* <sup>¼</sup> <sup>∇</sup> *<sup>δ</sup>*1*jK<sup>α</sup>* <sup>þ</sup> *<sup>δ</sup>*2*jK*<sup>∗</sup>

�*ρWei*ð Þþ *Te* � *Ti <sup>W</sup>*´ , *<sup>α</sup>* <sup>¼</sup> *<sup>i</sup>*, *<sup>δ</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup>

*<sup>K</sup>*ð Þ *<sup>τ</sup>* � *<sup>ξ</sup> <sup>∂</sup>*

*F r*ð Þ¼ , *τ βabT<sup>α</sup>*<sup>0</sup> Å *δ*1*ju*´*<sup>a</sup>*,*<sup>b</sup>* þ *τ*<sup>0</sup> þ *δ*2*<sup>j</sup>*

� � <sup>þ</sup> *<sup>W</sup>*´ , *<sup>α</sup>* <sup>¼</sup> *<sup>p</sup>*, *<sup>δ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>T</sup>*<sup>3</sup>

*<sup>∂</sup><sup>ξ</sup>* <sup>∇</sup><sup>2</sup>

ð Þ *Tα*ð Þ *r*, *τ dξ*

*<sup>K</sup>*ð Þ *<sup>τ</sup>* � *<sup>ξ</sup> <sup>∂</sup>*<sup>2</sup>

h i

*<sup>K</sup>*ð Þ *<sup>τ</sup>* � *<sup>ξ</sup> <sup>∂</sup>*

*∂ξ*

τ�*ωα*

where *δij*,ð Þ *i*, *j* ¼ 1, 2 , *ωα*ð Þ 0 ð Þ *α* ¼ *e*, *i* ∧ *p* , and *K*ð Þ *τ* � *ξ* are the Kronecker delta,

*<sup>P</sup>* <sup>¼</sup> *Pe* <sup>þ</sup> *Pi* <sup>þ</sup> *Pp*0*Pe* <sup>¼</sup> *ceTe*, *Pi* <sup>¼</sup> *ciTi*, *Pp* <sup>¼</sup> <sup>1</sup>

ðτ

*ρWei*ð Þþ *Te* � *Ti ρWer Te* � *Tp*

ð*τ*

*τ*�*ωα*

�*ρWer Te* � *Tp*

*ωα*

ð*τ*

<sup>þ</sup> *<sup>ρ</sup>C<sup>α</sup> <sup>τ</sup>*<sup>0</sup> <sup>þ</sup> *<sup>δ</sup>*1*<sup>j</sup>τ*<sup>2</sup> <sup>þ</sup> *<sup>δ</sup>*2*<sup>j</sup>* � � *ωα*

*Wei* <sup>¼</sup> *<sup>ρ</sup>AeiT*�2*=*<sup>3</sup> *<sup>e</sup>* ,*Wer* <sup>¼</sup> *<sup>ρ</sup>AerT*�1*=*<sup>2</sup> *<sup>e</sup>* ,*K<sup>α</sup>* <sup>¼</sup> *<sup>A</sup>αT*<sup>5</sup>*=*<sup>2</sup>

Initial and boundary conditions can be expressed as

*<sup>T</sup>α*ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>*, 0 *<sup>T</sup>*<sup>0</sup>

delay times, and kernel function, respectively. The total energy can be expressed as

*τ*�*ωα*

*Wep* is the electron-phonon coefficient.

*Advanced Functional Materials*

**3. BEM solution of temperature field**

*α*

which can be written in the following form:

<sup>∇</sup> *<sup>δ</sup>*1*jK<sup>α</sup>* <sup>þ</sup> *<sup>δ</sup>*2*jK* <sup>∗</sup>

ture model.

where

where

and

**98**

*W r* ´ ð Þ¼ , *<sup>τ</sup>*

8 >><

>>:

*W r* ´ ð Þ¼ , *<sup>τ</sup> F r*ð Þ� , *<sup>τ</sup> <sup>δ</sup>*2*jK<sup>α</sup>*

<sup>þ</sup> *<sup>ρ</sup>Cαδ*1*δ*1*<sup>j</sup> ωα*

$$
\hat{q}^{\circ} = \text{-} \mathbb{K}\_a \frac{\partial \hat{T}\_a^{\circ}}{\partial n} \tag{37}
$$

and

$$a^j(\boldsymbol{\pi}) = \sum\_{i=1}^{N} f\_{ji}^1 \frac{\partial T\_a(r\_i, \boldsymbol{\pi})}{\partial \boldsymbol{\pi}} \tag{38}$$

where *f* �1 *ji* are the coefficients of *F*�<sup>1</sup> which are defined as [58].

$$\{F\}\_{\vec{\jmath}i} = f^{\vec{\jmath}}(r\_i) \tag{39}$$

By discretizing Eq. (36) and using Eq. (38), we get [35].

$$\mathbf{C}\dot{T}\_a + HT\_a = \mathbf{G}\,\mathbf{Q} \tag{40}$$

*Ui* <sup>¼</sup> *<sup>ρ</sup>Fi*‐*ρu*€*i*, (51)

*ui* ¼ *ui on* S1 (52) *λ<sup>i</sup>* ¼ *σijn <sup>j</sup>* ¼ *λ<sup>i</sup> on* S2 (53)

Φ ¼ Φ *on* S5 (54)

*<sup>∂</sup><sup>n</sup>* <sup>¼</sup> *Q on* S6 (55)

*<sup>i</sup> dS* (56)

*<sup>i</sup> dS* (57)

ð *S*1

*<sup>i</sup> dS*

*<sup>i</sup> dS*

ð Þ *ui* � *ui <sup>λ</sup>* <sup>∗</sup>

*<sup>i</sup> dS*

(58)

*<sup>i</sup> dS*

(59)

*<sup>i</sup>* are weighting functions and *ui* and Φ*<sup>i</sup>* are approximate

*A Novel MDD-Based BEM Model for Transient 3T Nonlinear Thermal Stresses in FGA Smart…*

*<sup>i</sup> dR* ¼ �<sup>ð</sup>

*S*6 *Qi*Φ<sup>∗</sup>

ð *S*2

þ ð *S*6

þ ð *S*5

> *S*2 *λiu*<sup>∗</sup> *<sup>i</sup> dS* �

þ ð *S*1

� ð *S*5 *Qi* Φ<sup>∗</sup>

*<sup>i</sup>*,*<sup>i</sup> dR* ¼ �<sup>ð</sup>

*S*2 *λ<sup>i</sup> u*<sup>∗</sup>

*λ<sup>i</sup>* � *λ<sup>i</sup>* � �*u*<sup>∗</sup>

> *Qi* � *Qi* � �Φ<sup>∗</sup>

Φ*<sup>i</sup>* � Φ*<sup>i</sup>* � �*Q* <sup>∗</sup>

ð Þ *ui* � *ui <sup>λ</sup>* <sup>∗</sup>

*<sup>i</sup> dS* þ

*σij* ¼ *ijkl εkl*, (60)

*ijkl* ¼ *klij* (61)

*<sup>i</sup> dS* þ

ð *S*1 *λiu*<sup>∗</sup> *<sup>i</sup> dS*�

*<sup>i</sup> dS* �

ð *S*5 ð *S*6 *Qi* Φ<sup>∗</sup> *<sup>i</sup> dS*

Φ*<sup>i</sup>* � Φ*<sup>i</sup>* � �*Q* <sup>∗</sup>

where *u*<sup>∗</sup>

solutions.

expressed as

*<sup>i</sup> dR* þ

ð *R Ui u*<sup>∗</sup>

ð *R Ui u*<sup>∗</sup>

Hence, Eq. (59) can be rewritten as

� ð *R σij*,*<sup>j</sup> u*<sup>∗</sup>

� ð *R σij ε* <sup>∗</sup> *ij dR* þ

where

**101**

*<sup>i</sup>* and Φ<sup>∗</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.92829*

Now, we assume the following boundary conditions:

� ð *R σij u*<sup>∗</sup> *<sup>Q</sup>* <sup>¼</sup> <sup>∂</sup><sup>Φ</sup>

*<sup>i</sup>*,*<sup>j</sup> dR* þ

ð *R D*Φ<sup>∗</sup>

By integrating by parts for the left-hand side of (58), we get

Based on Eringen [89], the elastic stress can be expressed as

ð *R D*Φ<sup>∗</sup>

� ð *R D*Φ<sup>∗</sup>

*<sup>i</sup> dR* �

*<sup>i</sup> dR* �

By integration by parts for the first term of Eqs. (49) and (50), we have

*<sup>i</sup>*,*<sup>i</sup> dR* ¼ �<sup>ð</sup>

*<sup>i</sup>*,*<sup>i</sup> dR* ¼

Based on Huang and Liang [88], the boundary integral equation can be

ð *R Ui u*<sup>∗</sup>

where Q is the heat flux vector and H and G are matrices. The diffusion matrix can be defined as

$$\mathbf{C} = -\left[ H\hat{T}\_a - G\hat{Q} \right] \mathbf{F}^{-1} \mathbf{D}^{-1} \tag{41}$$

where

$$\left\{\hat{T}\right\}\_{\vec{\eta}} = \hat{T}^{\vec{\jmath}}(\mathbf{x}\_{i})\tag{42}$$

$$\left\{\hat{Q}\right\}\_{\vec{\eta}} = \hat{q}^{\dagger}(\mathbf{x}\_{i})\tag{43}$$

To solve numerically Eq. (41), the functions *T<sup>α</sup>* and q were interpolated as

$$T\_a = \left(\mathbf{1} - \theta\right) T\_a^m + \theta \, T\_a^{m+1} \tag{44}$$

$$q = (1 - \theta)q^m + \theta q^{m+1} \tag{45}$$

where 0 <sup>≤</sup><sup>0</sup> <sup>¼</sup> *<sup>τ</sup>*�*τ<sup>m</sup> <sup>τ</sup>m*þ1‐*τ<sup>m</sup>* <sup>≤</sup>1 determines the practical time *<sup>τ</sup>* of the current time step. By time differentiation of Eq. (44), we obtain

$$\dot{T}\_a = \frac{dT\_a}{d\theta} \frac{d\theta}{d\tau} = \frac{T\_a^{m+1} \cdot T\_a^m}{\tau^{m+1} \cdot \tau^m} = \frac{T\_a^{m+1} \cdot T\_a^m}{\Delta \tau^m} \tag{46}$$

By substitution from (44)–(46) into (40), we get

$$\left(\frac{\mathbf{C}}{\Delta \tau^{m}} + \theta H\right) T\_{a}^{m+1} - \theta G Q^{m+1} = \left(\frac{\mathbf{C}}{\Delta \tau^{m}} \cdot (\mathbf{1} - \theta) H\right) T\_{a}^{m} + (\mathbf{1} - \theta) G Q^{m} \tag{47}$$

By considering the initial and boundary conditions, we can write the following system of equations

$$\mathbf{a}X = \mathbf{b} \tag{48}$$

We apply an explicit staggered algorithm to solve the system (48) and obtain the temperature in terms of the displacement field.

#### **4. BEM solution of displacement field**

By using the weighted residual method, we can write (12) and (13) in the following form:

$$\int\_{R} \left(\sigma\_{\vec{\eta}\vec{j}} + U\_i\right) u\_i^\* \, dR = 0 \tag{49}$$

$$\int\_{R} (D\_{,i}) \, \Phi\_{i}^{\*} \, dR = \mathbf{0} \tag{50}$$

where

*A Novel MDD-Based BEM Model for Transient 3T Nonlinear Thermal Stresses in FGA Smart… DOI: http://dx.doi.org/10.5772/intechopen.92829*

$$U\_i = \rho F\_{i^\*} \rho \ddot{u}\_i,\tag{51}$$

where *u*<sup>∗</sup> *<sup>i</sup>* and Φ<sup>∗</sup> *<sup>i</sup>* are weighting functions and *ui* and Φ*<sup>i</sup>* are approximate solutions.

Now, we assume the following boundary conditions:

$$
u\_i = \overline{u}\_i \quad on \text{ } \mathbb{S}\_1 \tag{52}$$

$$
\lambda\_i = \sigma\_{i\bar{\jmath}} n\_{\bar{\jmath}} = \overline{\lambda}\_i \quad \text{on } \mathbb{S}\_2 \tag{53}
$$

$$
\Phi = \overline{\Phi} \quad \text{on } \mathbb{S}\_{\mathbb{S}} \tag{54}
$$

$$Q = \frac{\partial \Phi}{\partial n} = \overline{Q} \quad \text{on } \mathbb{S}\_6 \tag{55}$$

By integration by parts for the first term of Eqs. (49) and (50), we have

$$-\int\_{R} \sigma\_{\vec{\eta}} u\_{i\vec{j}}^{\*} \, dR + \int\_{R} U\_{i} u\_{i}^{\*} \, dR = -\int\_{S\_{2}} \lambda\_{i} u\_{i}^{\*} \, dS \tag{56}$$

$$-\int\_{R} D\Phi\_{i,i}^{\*} \, dR = -\int\_{S\_{6}} Q\_{i} \Phi\_{i}^{\*} \, d\mathcal{S} \tag{57}$$

Based on Huang and Liang [88], the boundary integral equation can be expressed as

$$\begin{aligned} -\int\_{R} \sigma\_{ij;j} \boldsymbol{u}\_{i}^{\*} \, d\boldsymbol{R} + \int\_{R} U\_{i} \boldsymbol{u}\_{i}^{\*} \, d\boldsymbol{R} - \int\_{R} D \, \boldsymbol{\Phi}\_{i,i}^{\*} \, d\boldsymbol{R} &= \int\_{S\_{2}} \left(\boldsymbol{\lambda}\_{i} - \overline{\boldsymbol{\lambda}}\_{i}\right) \boldsymbol{u}\_{i}^{\*} \, d\boldsymbol{S} + \int\_{S\_{1}} \left(\overline{\boldsymbol{u}\_{i}} - \boldsymbol{u}\_{i}\right) \boldsymbol{\lambda}\_{i}^{\*} \, d\boldsymbol{S} \\ &+ \int\_{S\_{6}} \left(\boldsymbol{Q}\_{i} - \overline{\boldsymbol{Q}\_{i}}\right) \boldsymbol{\Phi}\_{i}^{\*} \, d\boldsymbol{S} \\ &+ \int\_{S\_{9}} \left(\overline{\boldsymbol{\Phi}\_{i}} - \boldsymbol{\Phi}\_{i}\right) \boldsymbol{Q}\_{i}^{\*} \, d\boldsymbol{S} \end{aligned} \tag{58}$$

By integrating by parts for the left-hand side of (58), we get

$$\begin{aligned} -\int\_{R} \sigma\_{\vec{\eta}} \boldsymbol{e}\_{\vec{\eta}}^{\*} \, d\mathcal{R} + \int\_{R} U\_{i} \boldsymbol{u}\_{i}^{\*} \, d\mathcal{R} - \int\_{R} D \, \Phi\_{i,i}^{\*} \, d\mathcal{R} &= -\int\_{\mathcal{S}\_{2}} \overline{\lambda}\_{i} \boldsymbol{u}\_{i}^{\*} \, d\mathcal{S} - \int\_{\mathcal{S}\_{1}} \lambda\_{i} \boldsymbol{u}\_{i}^{\*} \, d\mathcal{S} \\ &+ \int\_{\mathcal{S}\_{1}} (\overline{\boldsymbol{u}\_{i}} - \boldsymbol{u}\_{i}) \, \lambda\_{i}^{\*} \, dS - \int\_{\mathcal{S}\_{6}} \overline{\boldsymbol{Q}\_{i}} \Phi\_{i}^{\*} \, d\mathcal{S} \\ &- \int\_{\mathcal{S}\_{5}} \boldsymbol{Q}\_{i} \, \Phi\_{i}^{\*} \, d\mathcal{S} + \int\_{\mathcal{S}\_{6}} \left( \overline{\boldsymbol{\Phi}\_{i}} - \boldsymbol{\Phi}\_{i} \right) \boldsymbol{Q}\_{i}^{\*} \, d\mathcal{S} \, \end{aligned} \tag{59}$$

Based on Eringen [89], the elastic stress can be expressed as

$$
\sigma\_{ij} = \mathbb{A}\_{ijkl}\,\varepsilon\_{kl},\tag{60}
$$

where

$$
\mathbb{A}\_{ijkl} = \mathbb{A}\_{klij} \tag{61}
$$

Hence, Eq. (59) can be rewritten as

By discretizing Eq. (36) and using Eq. (38), we get [35].

where Q is the heat flux vector and H and G are matrices.

*<sup>C</sup>* ¼ � *H T\_*

*T* n o*\_*

*Q* n o*\_*

*<sup>T</sup><sup>α</sup>* <sup>¼</sup> ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>T</sup><sup>m</sup>*

*dθ*

*<sup>α</sup>* � *GQ\_* h i

*ij* <sup>¼</sup> *<sup>T</sup> \_ j*

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

To solve numerically Eq. (41), the functions *T<sup>α</sup>* and q were interpolated as

*<sup>d</sup><sup>τ</sup>* <sup>¼</sup> *<sup>T</sup><sup>m</sup>*þ<sup>1</sup> *<sup>α</sup>* ‐*T<sup>m</sup>*

*F*�<sup>1</sup>

The diffusion matrix can be defined as

*Advanced Functional Materials*

By time differentiation of Eq. (44), we obtain

*<sup>T</sup>*\_ *<sup>α</sup>* <sup>¼</sup> *dT<sup>α</sup> dθ*

By substitution from (44)–(46) into (40), we get

temperature in terms of the displacement field.

**4. BEM solution of displacement field**

*<sup>T</sup><sup>m</sup>*þ<sup>1</sup> *<sup>α</sup>* � *<sup>θ</sup>GQ<sup>m</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>C</sup>*

ð *R*

> ð *R*

where

where 0 <sup>≤</sup><sup>0</sup> <sup>¼</sup> *<sup>τ</sup>*�*τ<sup>m</sup>*

*C* <sup>Δ</sup>*τ<sup>m</sup>* <sup>þ</sup> *<sup>θ</sup><sup>H</sup>* � �

system of equations

following form:

where

**100**

*CT*\_ *<sup>α</sup>* <sup>þ</sup> *H T<sup>α</sup>* <sup>¼</sup> *GQ* (40)

*D*�<sup>1</sup> (41)

ð Þ *xi* (42)

ð Þ *xi* (43)

*<sup>α</sup>* <sup>þ</sup> *<sup>θ</sup>T<sup>m</sup>*þ<sup>1</sup> *<sup>α</sup>* (44)

*<sup>q</sup>* <sup>¼</sup> ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>q</sup><sup>m</sup>* <sup>þ</sup> *<sup>θ</sup> <sup>q</sup><sup>m</sup>*þ<sup>1</sup> (45)

*α*

a*X* ¼ b (48)

*<sup>i</sup> dR* ¼ 0 (49)

*<sup>i</sup> dR* ¼ 0 (50)

*T<sup>m</sup>*

<sup>Δ</sup>*τ<sup>m</sup>* (46)

*<sup>α</sup>* <sup>þ</sup> ð Þ <sup>1</sup> � *<sup>θ</sup> GQ<sup>m</sup>* (47)

*<sup>τ</sup>m*þ1‐*τ<sup>m</sup>* <sup>≤</sup>1 determines the practical time *<sup>τ</sup>* of the current time step.

*α <sup>τ</sup><sup>m</sup>*þ<sup>1</sup>‐*τ<sup>m</sup>* <sup>¼</sup> *<sup>T</sup><sup>m</sup>*þ<sup>1</sup> *<sup>α</sup>* ‐*T<sup>m</sup>*

<sup>Δ</sup>*τ<sup>m</sup>* ‐ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>H</sup>* � �

By considering the initial and boundary conditions, we can write the following

We apply an explicit staggered algorithm to solve the system (48) and obtain the

By using the weighted residual method, we can write (12) and (13) in the

*σij*,*<sup>j</sup>* þ *Ui* � �*u*<sup>∗</sup>

ð Þ *<sup>D</sup>*,*<sup>i</sup>* <sup>Φ</sup><sup>∗</sup>

$$\begin{aligned} -\int\_{R} \boldsymbol{\sigma}^{\dot{y}\ast} \, \boldsymbol{\varepsilon}\_{\ddot{y}\cdot} d\boldsymbol{R} + \int\_{R} U\_{i} \boldsymbol{u}\_{i}^{\*} \, d\boldsymbol{R} - \int\_{R} D \, \boldsymbol{\Phi}\_{i,i}^{\*} d\boldsymbol{R} &= -\int\_{\mathcal{S}\_{2}} \overline{\boldsymbol{\lambda}}\_{i} \boldsymbol{u}\_{i}^{\*} \, d\boldsymbol{S} - \int\_{\mathcal{S}\_{1}} \boldsymbol{\lambda}\_{i} \boldsymbol{u}\_{i}^{\*} \, d\boldsymbol{S} \\ &+ \int\_{\mathcal{S}\_{1}} (\overline{\boldsymbol{u}\_{i}} - \boldsymbol{u}\_{i}) \, \boldsymbol{\lambda}\_{i}^{\*} \, d\boldsymbol{S} - \int\_{\mathcal{S}\_{6}} \overline{\boldsymbol{Q}\_{i}} \, \boldsymbol{\Phi}\_{i}^{\*} \, d\boldsymbol{S} \\ &- \int\_{\mathcal{S}\_{5}} \boldsymbol{Q}\_{i} \, \boldsymbol{\Phi}\_{i}^{\*} \, d\boldsymbol{S} + \int\_{\mathcal{S}\_{6}} \left( \overline{\boldsymbol{\Phi}\_{i}} - \boldsymbol{\Phi}\_{i} \right) \, \boldsymbol{Q}\_{i}^{\*} \, d\boldsymbol{S} \, \end{aligned} \tag{62}$$

By integration by parts again, we obtain

$$\int\_{R} \sigma\_{\vec{\eta},i}^{\*} \mu\_{i} dR = -\int\_{S} u\_{i}^{\*} \, \lambda\_{i} d\mathbf{S} - \int\_{S} \Phi\_{i}^{\*} \, \mathcal{Q}\_{i} d\mathbf{S} + \int\_{S} \lambda\_{i}^{\*} \, u\_{i} d\mathbf{S} + \int\_{S} \mathcal{Q}\_{i}^{\*} \, \Phi\_{i} d\mathbf{S} \tag{63}$$

The weighting functions of *Ui* <sup>¼</sup> <sup>Δ</sup>*<sup>n</sup>* and *Vi* <sup>¼</sup> 0 along e1 can be obtained as follows:

$$
\sigma^\*\_{\mathfrak{z}/\mathfrak{z}} + \Delta^n \mathfrak{e}\_1 = \mathbf{0} \tag{64}
$$

In order to solve (70) numerically, we suppose the following definitions:

*A Novel MDD-Based BEM Model for Transient 3T Nonlinear Thermal Stresses in FGA Smart…*

, <sup>Φ</sup> <sup>¼</sup> *<sup>ψ</sup>*<sup>0</sup> <sup>Φ</sup>*<sup>j</sup>*

q<sup>∗</sup> *ψ d*Γ " #

Substituting from (72) into (70) and discretizing the boundary, we obtain

<sup>p</sup> *<sup>j</sup>* <sup>þ</sup><sup>X</sup> *Ne*

*ij* <sup>¼</sup> ^ *ij if i* 6¼ *<sup>j</sup>* ^ *ij*

> <sup>p</sup> *<sup>j</sup>* <sup>þ</sup><sup>X</sup> *Ne*

*j*¼1 ^*ij*

where is the displacement vector, is the traction vector, Θ is the electric

Substituting the boundary conditions into (77), we obtain the following system

We apply an explicit staggered algorithm to solve the system (78) and obtain the

1.From Eq. (48) we obtain the temperature field in terms of the displacement

2.We predict the displacement field and solve the resulted equation for the

3.We correct the displacement field using the computed temperature field for

*j*¼1 ^*ij*

<sup>þ</sup> *Ci if i* <sup>¼</sup> *<sup>j</sup>*

<sup>Φ</sup> *<sup>j</sup>* <sup>þ</sup><sup>X</sup> *Ne*

*j*¼1

¼ þ Θ þ (77)

¼ (78)

^*ij* <sup>∂</sup><sup>Φ</sup> *∂n* � �*<sup>j</sup>*

ð Γ *j* , ∂Φ *<sup>∂</sup><sup>n</sup>* <sup>¼</sup> *<sup>ψ</sup>*<sup>0</sup>

> <sup>p</sup> *<sup>j</sup>* <sup>þ</sup><sup>X</sup> *Ne*

<sup>Φ</sup>*<sup>i</sup>* <sup>þ</sup><sup>X</sup> *Ne*

*j*¼1

^*ij* <sup>∂</sup><sup>Φ</sup> *∂n* � � *<sup>j</sup>*

*j*¼1

∂Φ *∂n* � � *<sup>j</sup>*

> ð Γ *j*

d<sup>∗</sup> *ψ*<sup>0</sup> *d*Γ " #

Φ*j*

(72)

(73)

(74)

(75)

(76)

<sup>q</sup> <sup>¼</sup> *<sup>ψ</sup>* <sup>q</sup> *<sup>j</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.92829*

p<sup>∗</sup> *ψ d*Γ " #

> f <sup>∗</sup> *ψ*<sup>0</sup> *d*Γ " # <sup>∂</sup><sup>Φ</sup>

Equation after integration can be written as

*<sup>C</sup><sup>n</sup>* <sup>q</sup>*<sup>n</sup>* <sup>¼</sup> <sup>X</sup>

*Ne*

*j*¼1 � ð Γ *j*

þ<sup>X</sup> *Ne*

*Ci*

of equations:

field.

Eq. (78).

**103**

temperature field.

*j*¼1

ð Γ *j*

<sup>q</sup>*<sup>i</sup>* ¼ �<sup>X</sup> *Ne*

*j*¼1 ^ *ij*

Thus, we can write (74) as follows:

*ij* <sup>q</sup> *<sup>j</sup>* <sup>¼</sup> <sup>X</sup>

temperature and displacement fields as follows:

*Ne*

*j*¼1 ^ *ij*

potential vector, and is the electric potential gradient vector.

The global matrix equation for all **i** nodes can be expressed as

X *Ne*

*j*¼1

By using the following representation:

, p <sup>¼</sup> *<sup>ψ</sup>* <sup>p</sup> *<sup>j</sup>*

<sup>q</sup> *<sup>j</sup>* <sup>þ</sup><sup>X</sup> *Ne*

> *∂n* � � *<sup>j</sup>*

<sup>q</sup> *<sup>j</sup>* <sup>þ</sup><sup>X</sup> *Ne*

*j*¼1 ^ *ij*

(

*j*¼1

According to Dragos [90], the fundamental solution can be written as

$$
\mu\_i^\* = \mu\_{\text{li}}^\* e\_1,\\
\Phi\_i^\* = \Phi\_{\text{li}}^\* e\_1,\\
\lambda\_i^\* = \lambda\_{\text{li}}^\* e\_1,\\
\mathbf{Q}\_i^\* = Q\_{\text{li}}^\* e\_1 \tag{65}
$$

The weighting functions of *Ui* <sup>¼</sup> 0 and *Vi* <sup>¼</sup> <sup>Δ</sup>*<sup>n</sup>* along e1 can be written as follows:

$$
\sigma^\*\_{\vec{\eta}, \vec{y}} = \mathbf{0} \tag{66}
$$

Based on Dragos [90], the fundamental solution can be obtained analytically as

$$u\_i^\* = u\_{1i}^{\*\ast} \ e\_1,\\ \Phi\_i^\* = \Phi\_{1i}^{\*\ast} \ e\_1,\\ \lambda\_i^\* = \lambda\_{1i}^{\*\ast} \ e\_1,\\ \mathbf{Q}\_i^\* = \mathbf{Q}\_{1i}^{\*\ast} \ e\_1 \tag{67}$$

By using the weighting functions of (65) and (67) into (63), we have

$$\mathbf{C}\_{\mathbf{i}l}^{n}u\_{i}^{n} = -\int\_{\mathcal{S}} \boldsymbol{\lambda}\_{\mathbf{i}l}^{\*} \boldsymbol{u}\_{i} d\mathbf{S} - \int\_{\mathcal{S}} \mathbf{Q}\_{\mathbf{i}l}^{\*} \boldsymbol{\Phi}\_{l} d\mathbf{S} + \int\_{\mathcal{S}} \boldsymbol{u}\_{\mathbf{i}l}^{\*} \boldsymbol{\lambda}\_{l} d\mathbf{S} + \int\_{\mathcal{S}} \boldsymbol{\Phi}\_{\mathbf{i}l}^{\*} \boldsymbol{Q}\_{l} d\mathbf{S} \tag{68}$$

$$\mathbf{C}\_{1i}^{\mathbf{u}}\boldsymbol{\alpha}\_{i}^{\mathbf{u}} = \cdot \left[\boldsymbol{\lambda}\_{1i}^{\*\;\ast} \,\,\boldsymbol{u}\_{i}\,\mathrm{d}\mathbf{S} - \int\_{\mathcal{S}} \mathbf{Q}\_{1i}^{\*\;\ast} \,\,\boldsymbol{\Phi}\_{i}\mathrm{d}\mathbf{S} + \int\_{\mathcal{S}} \boldsymbol{u}\_{1i}^{\*\;\ast} \,\,\boldsymbol{\lambda}\_{i}\mathrm{d}\mathbf{S} + \int\_{\mathcal{S}} \boldsymbol{\Phi}\_{1i}^{\*\;\ast} \,\,\mathbf{Q}\_{i}\mathrm{d}\mathbf{S}\right] \tag{69}$$

Thus, we can write

$$\mathbf{C}^{n}\mathbf{q}^{n} = -\int\_{\mathcal{S}} \mathbf{p}^{\*} \cdot \mathbf{q} \, d\mathbf{S} + \int\_{\mathcal{S}} \mathbf{q}^{\*} \cdot \mathbf{p} \, d\mathbf{S} + \int\_{\mathcal{S}} \mathbf{d}^{\*} \cdot \boldsymbol{\Phi} \, ds + \int\_{\mathcal{S}} \mathbf{f}^{\*} \cdot \frac{\partial \boldsymbol{\Phi}}{\partial n} d\mathbf{S} \tag{70}$$

where

$$\mathbf{C}^{\mathbf{u}} = \begin{bmatrix} \mathbf{C}\_{11} & \mathbf{C}\_{12} \\ \mathbf{C}\_{21} & \mathbf{C}\_{22} \end{bmatrix}, \mathbf{q}^{\ast} = \begin{bmatrix} u\_{11}^{\ast} & u\_{12}^{\ast} & \mathbf{0} \\ u\_{21}^{\ast} & u\_{22}^{\ast} & \mathbf{0} \\ u\_{31}^{\ast \ast} & u\_{32}^{\ast \ast} & \mathbf{0} \end{bmatrix}, \mathbf{p}^{\ast} = \begin{bmatrix} \lambda\_{11}^{\ast} & \lambda\_{12}^{\ast} & \mathbf{0} \\ \lambda\_{21}^{\ast} & \lambda\_{22}^{\ast} & \mathbf{0} \\ \lambda\_{31}^{\ast \ast} & \lambda\_{32}^{\ast \ast} & \mathbf{0} \end{bmatrix}, \mathbf{q} = \begin{bmatrix} u\_{1} \\ u\_{2} \\ u\_{3} \end{bmatrix},$$

$$\mathbf{p} = \begin{bmatrix} \lambda\_{1} \\ \lambda \\ \mu\_{3} \end{bmatrix}, \mathbf{d}^{\ast} = \begin{bmatrix} \mathbf{d}\_{1}^{\ast} \\ \mathbf{d}\_{2}^{\ast} \\ \mathbf{0} \end{bmatrix}, \mathbf{f}^{\ast} = \begin{bmatrix} \mathbf{f}\_{1}^{\ast} \\ \mathbf{f}\_{2}^{\ast} \\ \mathbf{0} \end{bmatrix} \tag{71}$$

*A Novel MDD-Based BEM Model for Transient 3T Nonlinear Thermal Stresses in FGA Smart… DOI: http://dx.doi.org/10.5772/intechopen.92829*

In order to solve (70) numerically, we suppose the following definitions:

$$\mathbf{q} = \boldsymbol{\nu} \cdot \mathbf{q}^j, \mathbf{p} = \boldsymbol{\nu} \cdot \mathbf{p}^j, \boldsymbol{\Phi} = \boldsymbol{\nu}\_0 \cdot \boldsymbol{\Phi}^j, \frac{\partial \boldsymbol{\Phi}}{\partial \boldsymbol{n}} = \boldsymbol{\nu}\_0 \left(\frac{\partial \boldsymbol{\Phi}}{\partial \boldsymbol{n}}\right)^j \tag{72}$$

Substituting from (72) into (70) and discretizing the boundary, we obtain

$$\begin{aligned} \mathbf{C}^{\mathbf{u}} \mathbf{q}^{\mathbf{u}} &= \sum\_{j=1}^{N\_{\epsilon}} \left[ -\int\_{\Gamma\_{j}} \mathbf{p}^{\*} \, \boldsymbol{\Psi} \, d\Gamma \right] \mathbf{q}^{j} + \sum\_{j=1}^{N\_{\epsilon}} \left[ \int\_{\Gamma\_{j}} \mathbf{q}^{\*} \, \boldsymbol{\Psi} \, d\Gamma \right] \mathbf{p}^{j} + \sum\_{j=1}^{N\_{\epsilon}} \left[ \int\_{\Gamma\_{j}} \mathbf{d}^{\*} \, \boldsymbol{\Psi} \, d\Gamma \right] \boldsymbol{\Phi}^{j} \\ &+ \sum\_{j=1}^{N\_{\epsilon}} \left[ \int\_{\Gamma\_{j}} \mathbf{f}^{\*} \, \boldsymbol{\Psi} \, d\Gamma \right] \left( \frac{\partial \boldsymbol{\Phi}}{\partial \boldsymbol{n}} \right)^{j} \end{aligned} \tag{73}$$

Equation after integration can be written as

$$\mathbf{C}^{\dot{\mathbf{q}}}\mathbf{q}^{\dot{\mathbf{i}}} = -\sum\_{j=1}^{N\_{\epsilon}} \hat{\mathbf{H}}^{\ddot{\mathbf{j}}\prime} \mathbf{q}^{j} + \sum\_{j=1}^{N\_{\epsilon}} \hat{\mathbf{G}}^{\ddot{\mathbf{j}}\prime} \mathbf{p}^{j} + \sum\_{j=1}^{N\_{\epsilon}} \hat{\mathbf{D}}^{\ddot{\mathbf{j}}\prime} \mathbf{o}^{j} + \sum\_{j=1}^{N\_{\epsilon}} \hat{\mathbf{F}}^{\ddot{\mathbf{j}}\prime} \left(\frac{\partial \Phi}{\partial n}\right)^{j} \tag{74}$$

By using the following representation:

$$\mathbb{H}^{\sharp} = \begin{cases} \mathbb{H}^{\sharp} & \text{if } i \neq j \\ \mathbb{H}^{\sharp} + \mathbb{C}^{i} & \text{if } i = j \end{cases} \tag{75}$$

Thus, we can write (74) as follows:

$$\sum\_{j=1}^{N\_\epsilon} \mathbb{H}^{\vec{\jmath}} \mathbf{q}^j = \sum\_{j=1}^{N\_\epsilon} \hat{\mathbb{G}}^{\vec{\jmath}} \mathbf{p}^j + \sum\_{j=1}^{N\_\epsilon} \hat{\mathbb{D}}^{\vec{\jmath}} \boldsymbol{\Phi}^j + \sum\_{j=1}^{N\_\epsilon} \hat{\mathbb{F}}^{\vec{\jmath}} \left(\frac{\partial \boldsymbol{\Phi}}{\partial \boldsymbol{n}}\right)^j \tag{76}$$

The global matrix equation for all **i** nodes can be expressed as

$$\mathbb{HPQ} = \mathbb{GP} + \mathbb{DP} + \mathbb{RP} \tag{77}$$

where is the displacement vector, is the traction vector, Θ is the electric potential vector, and is the electric potential gradient vector.

Substituting the boundary conditions into (77), we obtain the following system of equations:

$$\mathsf{VX} = \mathsf{B} \tag{78}$$

We apply an explicit staggered algorithm to solve the system (78) and obtain the temperature and displacement fields as follows:


� ð *R*

> ð *R σ* ∗ *ij*,*i*

follows:

follows:

*<sup>σ</sup>ij* <sup>∗</sup> *<sup>ε</sup>ij dR* <sup>þ</sup>

*Advanced Functional Materials*

ð *R Ui u*<sup>∗</sup>

*<sup>i</sup> dR* �

By integration by parts again, we obtain

ð *S u*∗ *<sup>i</sup> λ<sup>i</sup> dS* �

*uidR* ¼ �

*u*∗ *<sup>i</sup>* <sup>¼</sup> *<sup>u</sup>*<sup>∗</sup>

*u*∗ *<sup>i</sup>* <sup>¼</sup> *<sup>u</sup>*∗ ∗

*Cn* 1*i un <sup>i</sup>* ¼ � ð *S λ* ∗ <sup>1</sup>*<sup>i</sup> ui dS* �

Thus, we can write

*<sup>C</sup><sup>n</sup>*q*<sup>n</sup>* ¼ �

ð *S*

, q<sup>∗</sup> <sup>¼</sup>

p ¼

*Cn* 1*i ωn <sup>i</sup>* <sup>¼</sup> ‐ ð *S λ* ∗ ∗ <sup>1</sup>*<sup>i</sup> ui dS* �

where

**102**

*<sup>C</sup><sup>n</sup>* <sup>¼</sup> *<sup>C</sup>*<sup>11</sup> *<sup>C</sup>*<sup>12</sup> *C*<sup>21</sup> *C*<sup>22</sup> � � ð *R D*Φ<sup>∗</sup>

> ð *S* Φ<sup>∗</sup>

> > *σ* ∗

*<sup>i</sup>* <sup>¼</sup> <sup>Φ</sup><sup>∗</sup>

*<sup>i</sup>* <sup>¼</sup> <sup>Φ</sup>∗ ∗

ð *S Q* <sup>∗</sup>

ð *S Q* ∗ ∗

<sup>p</sup><sup>∗</sup> <sup>q</sup>*dS* <sup>þ</sup>

*u*∗ <sup>11</sup> *u*<sup>∗</sup>

2 6 4

*u*∗ <sup>21</sup> *u*<sup>∗</sup>

*u*∗ ∗ <sup>31</sup> *u*∗ ∗

> *λ*1 *λ μ*3

3 7 <sup>5</sup>, d<sup>∗</sup> <sup>¼</sup>

2 6 4

<sup>1</sup>*<sup>i</sup> e*1, Φ<sup>∗</sup>

<sup>1</sup>*<sup>i</sup> e*1, Φ<sup>∗</sup>

According to Dragos [90], the fundamental solution can be written as

<sup>1</sup>*<sup>i</sup> e*1, *λ* <sup>∗</sup>

The weighting functions of *Ui* <sup>¼</sup> 0 and *Vi* <sup>¼</sup> <sup>Δ</sup>*<sup>n</sup>* along e1 can be written as

*σ* ∗

<sup>1</sup>*<sup>i</sup> e*1, *λ* <sup>∗</sup>

<sup>1</sup>*<sup>i</sup>* Φ*<sup>i</sup> dS* þ

<sup>1</sup>*<sup>i</sup>* Φ*<sup>i</sup> dS* þ

<sup>q</sup><sup>∗</sup> <sup>p</sup>*dS* <sup>þ</sup>

3 7 <sup>5</sup>, p<sup>∗</sup> <sup>¼</sup>

d∗ 1 d∗ 2 0

3 7 <sup>5</sup>, f <sup>∗</sup> <sup>¼</sup>

2 6 4

<sup>12</sup> 0

<sup>22</sup> 0

<sup>32</sup> 0

ð *S*

By using the weighting functions of (65) and (67) into (63), we have

Based on Dragos [90], the fundamental solution can be obtained analytically as

*<sup>i</sup>* <sup>¼</sup> *<sup>λ</sup>* <sup>∗</sup>

*<sup>i</sup>* <sup>¼</sup> *<sup>λ</sup>* ∗ ∗

ð *S u*∗ <sup>1</sup>*<sup>i</sup> λ<sup>i</sup> dS* þ

ð *S u*∗ ∗ <sup>1</sup>*<sup>i</sup> λ<sup>i</sup> dS* þ

> ð *S*

<sup>d</sup><sup>∗</sup> <sup>Φ</sup>*ds* <sup>þ</sup>

*λ* ∗ <sup>11</sup> *λ* <sup>∗</sup>

2 6 4

*λ* ∗ <sup>21</sup> *λ* <sup>∗</sup>

*λ* ∗ ∗ <sup>31</sup> *λ* ∗ ∗

2 6 4 f ∗ 1 f ∗ 2 0

3 7

<sup>1</sup>*<sup>i</sup> e*1, Q <sup>∗</sup>

<sup>1</sup>*<sup>i</sup> e*1, Q <sup>∗</sup>

*<sup>i</sup>*,*<sup>i</sup> dR* ¼ �

*<sup>i</sup> Qi dS* þ

The weighting functions of *Ui* <sup>¼</sup> <sup>Δ</sup>*<sup>n</sup>* and *Vi* <sup>¼</sup> 0 along e1 can be obtained as

ð *S*2 *λ<sup>i</sup> u*<sup>∗</sup> *<sup>i</sup> dS* �

þ ð *S*1

� ð *S*5 *Qi* Φ<sup>∗</sup>

> ð *S λ* ∗ *<sup>i</sup> ui dS* þ

ð *S*1 *λ<sup>i</sup> u*<sup>∗</sup> *<sup>i</sup> dS*

*<sup>i</sup> dS* �

ð *S*5 ð *S*6 *Qi* Φ<sup>∗</sup> *<sup>i</sup> dS*

Φ*<sup>i</sup>* � Φ*<sup>i</sup>* � �*Q* <sup>∗</sup>

*<sup>i</sup>* Φ*<sup>i</sup> dS* (63)

<sup>1</sup>*ie*<sup>1</sup> (65)

<sup>1</sup>*<sup>i</sup> e*<sup>1</sup> (67)

<sup>1</sup>*<sup>i</sup> Qi dS* (68)

<sup>1</sup>*<sup>i</sup> Qi dS* (69)

*dS* (70)

*u*1 *u*2 *ω*3 3 7 5,

2 6 4 *<sup>i</sup> dS*

(62)

ð Þ *ui* � *ui <sup>λ</sup>* <sup>∗</sup>

*<sup>i</sup> dS* þ

ð *S Q* <sup>∗</sup>

<sup>1</sup>*j*,*<sup>j</sup>* <sup>þ</sup> <sup>Δ</sup>*ne*<sup>1</sup> <sup>¼</sup> <sup>0</sup> (64)

*<sup>i</sup>* <sup>¼</sup> *<sup>Q</sup>* <sup>∗</sup>

*ij*,*<sup>j</sup>* ¼ 0 (66)

*<sup>i</sup>* <sup>¼</sup> *<sup>Q</sup>* ∗ ∗

ð *S* Φ<sup>∗</sup>

> ð *S* Φ∗ ∗

> > ð *S* <sup>f</sup> <sup>∗</sup> <sup>∂</sup><sup>Φ</sup> *∂n*

> > > <sup>12</sup> 0

3 7 <sup>5</sup>, q <sup>¼</sup>

<sup>5</sup> (71)

<sup>22</sup> 0

<sup>32</sup> 0

#### *Advanced Functional Materials*

An explicit staggered algorithm based on communication-avoiding Arnoldi as described in Hoemmen [91] is very suitable for efficient implementation in Matlab (R2019a) with the aim of specifically improving its performance for the solution of the resulting linear algebraic systems.

and

*DOI: http://dx.doi.org/10.5772/intechopen.92829*

**Figure 3.**

**105**

*CA-GMRES iteration algorithm.*

Q*<sup>k</sup>* ¼ Q0, Q1, … , Q*k*�<sup>1</sup> ½ � (80)

The generalized minimal residual (GMRES) method of Saad and Schultz [92] is a Krylov subspace method for solving nonsymmetric linear systems. The CA-GMRES algorithm is based on Arnoldi (s, t) and equivalent to standard GMRES in exact arithmetic. Also, the GMRES or CA-GMRES are convergent at the same rate for problems, but Hoemmen [91] proved that CA-GMRES algorithm shown in **Figure 3** converges for the s-step basis lengths and restart lengths used for obtaining maximum performance. Lanczos method can be considered as a special case of Arnoldi method for symmetric and real case of A or Hermitian and complex case of A. Symmetric Lanczos which is also called Lanczos is different from nonsymmetric Lanczos. We implemented a communication-avoiding version of symmetric Lanczos (CA-Lanczos) for solving symmetric positive definite (SPD) eigenvalue problems. Also, we implement CA-Lanczos iteration algorithm shown in **Figure 4**, which is also called Lanczos (s, t), where s is the s-step basis length and t is the outer iterations number before restart. This algorithm is based on using rank revealingtall skinny QR-block Gram-Schmidt (RR-TSQR-BGS) orthogonalization method

*A Novel MDD-Based BEM Model for Transient 3T Nonlinear Thermal Stresses in FGA Smart…*
