A New BEM for Modeling and Simulation of Laser Generated Ultrasound Waves in 3T Fractional Nonlinear Generalized Micropolar Poro-Thermoelastic FGA Structures

Abstract

In this chapter, we introduce a new theory called acoustic wave propagation of three-temperature fractional nonlinear generalized micropolar poro-thermoelasticity and we propose a new boundary element technique for modeling and simulation of laser-generated ultrasonic wave propagation problems of functionally graded anisotropic (FGA) structures which are linked with the proposed theory. Since it is very difficult to solve general acoustic problems of this theory analytically, we need to develop and use new computational modeling techniques. So, we propose a new boundary element technique for solving such problems. The numerical results are shown graphically to depict the effects of three temperatures on the thermal stress waves propagation. The validity, accuracy, and efficiency of our proposed theory and the technique are examined and demonstrated by comparing the obtained outcomes with those previously reported in the literature as special cases of our general study.

1. Introduction

The fractional calculus has recently been widely used to describe anomalous diffusion instead of classical diffusion, where the standard time derivative is replaced by fractional time derivative. Indeed, fractional calculus has important applications in electronics, wave propagation, nanotechnology, control theory, electricity, heat conduction modeling and identification, signal and image processing, biochemistry, biology, viscoelasticity, hereditary solid mechanics, and fluid dynamics.

Physically, according to the medium where the waves are transmitted, there are three wave types which are classified as mechanical waves, electromagnetic waves, and matter waves. Mechanical waves can travel through any medium with speed depending on elasticity and inertia and cannot travel through a vacuum. Electromagnetic waves can travel through a vacuum and do not need a medium to travel like X-ray, microwaves, ultraviolet waves, and radio waves. Matter waves are also called De Broglie waves that have wave-particle duality property. There are two mechanisms that have been proposed to explain wave generation, a first mechanism at high energy density, which leads to forces that generate ultrasound, and a second mechanism at low energy density, which generates elastic waves according to irradiation of laser pulses onto a material. The interaction between laser light and a metal surface led to great progress to develop theoretical models to describe the experimental data [1]. Scruby et al. [2] proved that the thermoelastic area source had been reduced to a surface point-source. This point-source ignores the optical absorption, the heat source thermal diffusion, and the limited side dimensions of the source. Based on point-source representation, Rose [3] introduced Surface Center of Expansion (SCOE) models which predict the major features of ultrasound waves generated by laser. Doyle [4] established that the existence of the metal precursor is due to subsurface sources which arise from thermal diffusion. According to McDonald [5], Spicer [6] used the generalized thermoelasticity theory to introduce a real circular laser source model taking into consideration spatial-temporal laser pulse design and thermal diffusion effect. The mathematical foundations of three-temperature were laid for nonlinear generalized thermoelasticity theory by Fahmy [7, 8, 9, 10, 11, 12]. Fahmy [7] introduced a new boundary element strategy for modeling and simulation of three-temperature nonlinear generalized micropolar-magneto-thermoelastic wave propagation problems in FGA structures. Fahmy [8] proposed a boundary element formulation for three-temperature thermal stresses in anisotropic circular cylindrical plate structures. Fahmy [9] developed a boundary element model to describe the three-temperature fractional-order heat transfer in magneto-thermoelastic functionally graded anisotropic structures. Fahmy [10] introduced a boundary element formulation for modeling and optimization of micropolar thermoviscoelastic problems. Fahmy [11] discussed modeling and optimization of photo-thermoelastic stresses in three-temperature anisotropic semiconductor structures. Fahmy [12] proposed a new boundary element algorithm for nonlinear modeling and simulation of three-temperature anisotropic generalized micropolar piezothermoelasticity with memory-dependent derivative. This chapter differs from the references mentioned above, because it constructs a new acoustic wave propagation theory and allows the effective, efficient, and simple solution to the considered complex problems related with the proposed theory.

Recently, research on nonlinear generalized micropolar thermoelastic wave propagation problems has become very popular due to its practical applications in various fields such as astronautics, oceanology, aeronautics, narrow-band and broad-band systems, fiber-optic communication, fluid mechanics, automobile industries, aircraft, space vehicles, materials science, geophysics, petroleum and mineral prospecting, geomechanics, earthquake engineering, plasma physics, nuclear reactors, high-energy particle accelerators, and other industrial applications. Due to computational difficulties in solving nonlinear generalized micropolar poro-thermoelastic problems analytically, many numerical techniques have been developed and implemented for solving such problems [13, 14]. The boundary element method (BEM) [15, 16, 17, 18, 19, 20, 21, 22] has been recognized as an attractive alternative numerical method to domain methods [23, 24, 25, 26] like finite difference method (FDM), finite element method (FEM), and finite volume method (FVM) in engineering applications. The superior feature of BEM over domain methods is that it only needs to discretize the boundary, which often leads to fewer elements and easier to use. In the boundary element method (BEM) formulation, boundary integral equations involving singular integrands, the proper treatment of the singular integration has become essential in terms of numerical accuracy and efficiency of BEM. Also, some domain integrals may appear representing body forces, nonlinear effects, etc. Through our BEM solution, several approaches have been used to transform domain integrals into equivalent boundary integrals, so that the final boundary element formulation solution involves only the boundary integrals. The boundary element formulation of the current general study has been derived by using the weighted residual method [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]. In engineering applications, both FEM and BEM are based on the weighted residual methods with the same approximation procedure based on interpolation functions over each element to approximate the state variables distribution. Both methods differ in choosing the weighting functions. FEM as a domain method needs discretization of the whole domain, which usually leads to large systems of equations. This advantage of BEM over FEM has significant importance for modeling and simulation of thermal stress wave propagation problems which can be implemented using BEM with little cost and less input data. The solutions by BEM, like boundary thermal stress wave problems, are more accurate than by FEM, especially near the place of stress concentration. This feature is very important for our proposed theory and the technique of solving its related problems.

In this chapter, we introduce a novel theory called acoustic wave propagation of three-temperature fractional nonlinear generalized micropolar poro-thermoelasticity and we propose a new boundary element technique for modeling and simulation of laser-generated ultrasonic wave propagation problems of functionally graded anisotropic (FGA) structures which are linked with the proposed theory. Since it is very difficult to solve general acoustic problems of this theory analytically and we need to develop and use new computational modeling techniques. So, we propose a new boundary element technique for solving such problems. The numerical results are shown graphically to depict the effects of three temperatures on the propagation of thermal stresses waves. Since there are no available data for comparison with our proposed technique results, so, we replace the radiative heat conduction equations with heat conduction as a special case from our present general study. In the special case under consideration, the BEM results have been compared graphically with the FDM and FEM in the heat conduction and radiative heat conductions cases; it can be noticed that the BEM results are in a good agreement with the FDM and FEM results and thus demonstrate the validity and accuracy of our proposed theory and the technique used to solve its general problems.

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 wave propagation problems in three-temperature nonlinear generalized micropolar poro-thermoelastic FGA structures and their applications. Section 2 describes the BEM modeling of the new theory and introduces the partial differential equations that govern its related problems. Section 3 outlines BEM simulation of temperature field. Section 4 discusses BEM simulation of micropolar porothermoelastic field to obtain the three temperatures thermal stress wave propagation. Section 5 presents the new numerical results that describe the thermal stress wave propagation under the effect of three-temperature on the FGA structures.

2. BEM modeling of the problem

We consider an anisotropic micropolar porous smart structure in a rectangular Cartesian system $\left(x_1,,,x_2,,,x_3\right)$ shown in Figure 1, with a configuration $R$ bounded by a closed surface $S$ , and $S_i\left(i=\mathrm{1,2,3,4,5,6}\right)$ denotes subsets of $S$ such that $S_1+S_2=g_3+S_4=S_5+S_6$ . The governing equations for modeling of fractional three-temperature nonlinear generalized micropolar poro-thermoelastic problems of functionally graded anisotropic structures (FGA) can be expressed as [7].

Figure 1.

Geometry of the FGA structure.

$${\sigma }_{\mathit{ij},j}+\rho F_i=\rho {\ddot{u}}_i+{\mathit{\varphi \rho }}_{\mathcal{F}}{\ddot{v}}_i$$ E1

$$m_{\mathit{ij},j}+{\epsilon }_{\mathit{ijk}}{\sigma }_{\mathit{jk}}+\rho M_i=\mathit{J\rho }{\ddot{\omega }}_i$$ E2

$$\dot{\zeta }+q_{i,i}=\mathbb{C}$$ E3

where

$${\sigma }_{\mathit{ij}}={\left(z+1\right)}^m\left[C_{\mathit{\text{ijkl}}}\mathit{\aleph e}{\delta }_{\mathit{ij}}-A{\delta }_{\mathit{ij}}p+\stackrel{⌣}{\alpha }\left(u_{j,i}-{\epsilon }_{\mathit{ijk}}{\omega }_k\right)-{\beta }_{\mathit{ij}}{T}_{\alpha }\right]$$ E4

$$C_{\mathit{\text{ijkl}}}=C_{\mathit{\text{klij}}}=C_{\mathit{\text{jikl}}},{\beta }_{\mathit{ij}}={\beta }_{\mathit{ji}}$$ E5

$$m_{\mathit{ij}}={\left(x+1\right)}^m\left[\alpha {\omega }_{k,k}{\delta }_{\mathit{ij}}+\overline{\alpha }{\omega }_{i,j}+\overline{\overline{\alpha }}={\omega }_{j,i}\right]$$ E6

$$\zeta ={\left(x+1\right)}^m\left[{\mathit{Au}}_{k,k}+\frac{{\varphi }^2}{R}p\right]$$ E7

$$q_i={\left(x+1\right)}^m\left[-\overline{k}\left(p_{,i}+{\rho }_{\mathcal{F}}{\ddot{u}}_i+\frac{{\rho }_0+{\mathit{\varphi \rho }}_{\mathcal{F}}}{\varphi }{\ddot{v}}_i\right)\right]$$ E8

$${\mathrm{ϵ}}_{\mathit{ij}}={\epsilon }_{\mathit{ij}}-{\epsilon }_{\mathit{ijk}}{\left(\chi +1\right)}^m\left(r_k-{\omega }_k\right)$$ E9

$${\epsilon }_{\mathit{ij}}=\frac{1}{2}\left(u_{i,j}1+u_{j,i}\right)$$ E10

$$r_i=\frac{1}{2}{\epsilon }_{\mathit{ikl}}{u}_{l,k}$$ E11

The time-fractional order three-temperature radiative heat conduction equations can be written as

$$D_{\tau }^a{T}_{\alpha }\left(r,\tau \right)=\xi \text{∇}\left[{\mathbb{K}}_{\alpha }\text{∇}{T}_{\alpha }\left(r,\tau \right)\right]+\xi \overline{\mathbb{W}}\left(r,\tau \right),\xi =\frac{1}{c_{\alpha }{\mathit{\rho \delta }}_1}$$ E12

where

$$\overline{\mathbb{W}}\left(r,\tau \right)=\left\{\begin{array}{l}\rho {\mathbb{W}}_{\mathit{ei}}\left(T_e-T_i\right)+\rho {\overline{\overline{\mathbb{W}}}}_{\mathit{er}}\left(T_e-T_p\right)+\overline{\overline{\mathbb{W}}},\alpha =e,{\delta }_1=1\\ -\rho {\mathbb{W}}_{\mathit{ei}}\left(T_e-T_i\right)+\overline{\overline{\mathbb{W}}},\alpha =i,{\delta }_1=1\\ -\rho {\mathbb{W}}_{\mathit{er}}\left(T_e-T_p\right)+\overline{\overline{\mathbb{W}}},\alpha =p,{\delta }_1=T_p^3\end{array}\right.$$ E13

$$\begin{array}{l}\overline{\overline{\mathbb{W}}}\left(r,\tau \right)=-{\delta }_{2n}{\mathbb{K}}_{\alpha }{\dot{T}}_{\alpha ,\mathit{ij}}+{\beta }_{\mathit{ij}}{T}_{\alpha 0}\left[\AA {\delta }_{1n}{\dot{u}}_{i,j}+\left({\tau }_0+{\delta }_{2n}\right){\ddot{u}}_{i,j}\right]\\ + \rho c_{\alpha }\left[\left({\tau }_0+{\delta }_{1n^{T_2}}+{\delta }_{2n}\right){\ddot{T}}_{\alpha }\right]\end{array}$$ E14

and

$${\mathbb{W}}_{\mathit{ei}}=\rho {\mathbb{A}}_{\mathit{ei}}{T}_e^{-2/3},{\mathbb{W}}_{\mathit{er}}=\rho {\mathbb{A}}_{\mathit{er}}{T}_e^{-1/2},{\mathbb{K}}_{\alpha }={\mathbb{A}}_{\alpha }{T}_{\alpha }^{5/2},\alpha =e,i,{\mathbb{K}}_p={\mathbb{A}}_p{T}_p^{3+\mathbb{B}}$$ E15

The total energy is

$$P=P_e+P_i+P_p,P_e=c_e{T}_e,P_i=c_i{T}_i,P_p=\frac{1}{4}{c}_p{T}_p^4$$ E16

where we considered that $\theta =T_e+T_i+T_r,$ $T_e,T_i$ , and $T_r$ are temperature functions of electron, ion, and photon, respectively, ${\mathbb{K}}_e,{\mathbb{K}}_i$ , and ${\mathbb{K}}_r$ are conductive coefficients of electron, ion, and photon, respectively, and $\rho$ is the material density which is constant inside each subdomain.

3. BEM simulation for temperature field

In this section, we are interested in using a boundary element method for modeling the nonlinear time-dependent two dimensions three temperature (2D-3T) radiation heat equations coupled with electron, ion, and phonon temperatures.

According to finite difference scheme of Caputo at times $\left(f+1\right)\mathrm{\Delta }\tau$ and $f\mathrm{\Delta }\tau$ , we obtain [52].

$$D_{\tau }^a{T}_{\alpha }^{f+1}+D_{\tau }^a{T}_{\alpha }^f\approx \sum _{j=0}^k{W}_{a,j}\left(T_{\alpha }^{f+1-j}\left(r\right)-T_{\alpha }^{f-j}\left(r\right)\right)$$ E17

where

$$W_{a,0}=\frac{{\left(\mathrm{\Delta }\tau \right)}^{-a}}{\mathrm{\Gamma }\left(2-a\right)},W_{a,j}=W_{a,0}\left({\left(j+1\right)}^{1-a}-{\left(j-1\right)}^{1-a}\right)$$ E18

Based on Eq. (17), the fractional order heat Eq. (12) can be replaced by the following system

$$\begin{array}{l}{W}_{a,0}{T}_{\alpha }^{f+1}\left(r\right)-{\mathbb{K}}_{\alpha }\left(x\right)T_{\alpha ,\mathit{II}}^{f+1}\left(r\right)-{\mathbb{K}}_{{\alpha }_{,I}},\left(x\right)T_{\alpha ,I}^{f+1}\left(r\right)\\ =W_{a,0}{T}_{\alpha }^f\left(r\right)-{\mathbb{K}}_{\alpha }\left(x\right)T_{\alpha ,\mathit{II}}^f\left(r\right)\\ -{\mathbb{K}}_{{\alpha }_{,I}},\left(x\right)T_{{\alpha }_{,J}]}^f\left(r\right)-\sum _{j=1}^f{W}_{a,j}\left(T_{\alpha }^{f+1-j}\left(r\right)-T_{\alpha }^{f-j}\left(r\right)\right)+{\overline{\mathbb{W}}}_m^{f+1}\left(x,\tau \right)\\ +{\overline{\mathbb{W}}}_m^f\left(x,\tau \right)\end{array}$$ E19

where, $j=1,2,\dots ,F,f=\mathrm{0,1,2},\dots ,F$ .

Now, according to Fahmy [9] and using the fundamental solution that satisfies the system (19), the boundary integral equations corresponding to (12) without internal heat sources can be written as

$${\mathit{CT}}_{\alpha }={\int }_s\left[T_{\alpha }{q}^{\ast }-T_{\alpha }^{\ast }q\right]\mathit{dS}-{\int }_R\frac{{\mathbb{K}}_{\alpha }}{D}\frac{\partial T_{\alpha }^{\ast }}{\partial \tau }{T}_{\alpha }\mathit{dR}$$ E20

Now, to transform the domain integral in (20) into the boundary, we assume that the time-temperature derivative can be approximated by using a series of known functions $f^j\left(r\right)$ and unknown coefficients $a^j\left(\tau \right)$ as

$$\frac{\partial T_{\alpha }}{\partial \tau }\cong \sum _{j=1}^N{f}^j{\left(r\right)}^j{a}^j\left(\tau \right)$$ E21

We assume that ${\widehat{T}}_{\alpha }^j$ is a solution of

$${\nabla }^2{\widehat{T}}_{\alpha }^j=f^j$$ E22

Thus, Eq. (20) can be written as

$$\begin{array}{l}{\mathit{CT}}_{\alpha }={\int }_S\left[T_{\alpha }{q}^{\ast }-T_{\alpha }^{\ast }q\right]\mathit{dS}\\ +\sum _{j=1}^N{a}^j\left(\tau \right)D^{-1}\left(C{\widehat{T}}_{\alpha }^j-{\int }_S\left[T_{\alpha }^j{q}^{\ast }-{\widehat{q}}^j{T}_{\alpha }^{\ast }\right]\mathit{dS}\right)\end{array}$$ E23

where

$${\widehat{q}}^j=-{\mathbb{K}}_{\alpha }\frac{\partial {\widehat{T}}_{\alpha }^j}{\partial n}$$ E24

and

$$a^j\left(\tau \right)=\sum _{i=1}^N{f}_{\mathit{ji}}^{-1}\frac{\partial T\left(r_i,\tau \right)}{\partial \tau }$$ E25

In which, the entries of ${\mathit{f}}_{\mathrm{ji}}^{-1}$ are the coefficients of ${\mathit{F}}^{-1}$ with matrix $\mathit{F}$ defined as

$${\left\{F\right\}}_{\mathit{ji}}=f^j\left(r_i\right)$$ E26

Using the standard boundary element discretization scheme [28], for Eq. (23) and using Eq. (25), we get

$$C{\dot{T}}_{\alpha }+{\mathit{HT}}_{\alpha }=GQ$$ E27

where the matrices $H$ and $G$ are depending on current time step, boundary geometry, and material properties.

The diffusion matrix can be defined as

$$C=-\left[H{\widehat{T}}_{\alpha }-G\widehat{Q}\right]F^{-1}{D}^{-1}$$ E28

with

$${\left\{\widehat{T}\right\}}_{\mathit{ij}}={\widehat{T}}^j\left(x_i\right)$$ E29

$${\left\{\widehat{Q}\right\}}_{\mathit{ij}}={\widehat{q}}^j\left({\chi }_i\right)$$ E30

In order to solve Eq. (27) numerically, the functions $T_{\alpha }$ and $\mathrm{q}$ are interpolated as

$$T_{\alpha }=\left(1-\theta \right)T_{\alpha }^m+\theta T_{\alpha }^{m+1}$$ E31

$$q=\left(1-\theta \right)q^m+\theta q^{m+1}$$ E32

The time derivative of the temperature can be written as

$${\dot{T}}_{\alpha }=\frac{{\mathit{dT}}_{\alpha }}{\mathit{d\theta }}\frac{\mathit{d\theta }}{\mathit{d\tau }}=\frac{T_{\alpha }^{m+1}-T_{\alpha }^m}{{\tau }^{m+1}-{\tau }^m}=\frac{T_{\alpha }^{m+1}-T_{\alpha }^m}{\mathrm{\Delta }{\tau }^m},\theta =\frac{\tau -{\tau }^m}{{\tau }^{m+1}-{\tau }^m},0\le \theta \le 1$$ E33

By substituting from Eqs. (31)–(33) into (27), we obtain

$$\left(\frac{c}{\mathrm{\Delta }{\tau }^m}+\mathit{\theta H}\right)T_{\alpha }^{m+1}-\theta {\mathit{GQ}}^{m+1}=\left(\frac{c}{\mathrm{\Delta }{\tau }^m}-\left(1-\theta \right)H\right)T_{\alpha }^m+\left(1-\theta \right){\mathit{GQ}}^m$$ E34

which can be written as follows [10].

$$\mathbb{a}X=\mathbb{b}$$ E35

where $\mathbb{a}$ is an unknown matrix, while $\mathbf{X}$ and $\mathbb{b}$ are known matrices.

The explicit staggered predictor-corrector procedure based on communication-avoiding Arnoldi (CA-Arnoldi) method [53] due to its numerical stability, convergence, and performance [7] has been implemented for obtaining the temperature field in terms of predicted displacement field which will be explained in the next section.

4. BEM simulation for micropolar poro-thermoelastic fields

By implementing the weighted residual method, the governing Eqs. (1)–(3) can be written as

$${\int }_R\left({\sigma }_{\mathit{ij},j}+U_i\right)u_i^{\ast }\mathit{dR}=0$$ E36

$${\int }_R\left(m_{\mathit{ij},j}+{\epsilon }_{\mathit{ijk}}{\sigma }_{\mathit{jk}}+V_i\right){\omega }_i^{\ast }\mathit{dR}=0$$ E37

$${\int }_R\left(q_i+{\dot{\zeta }}_i-{\mathbb{C}}_i\right)p_i^{\ast }\mathit{dR}=0$$ E38

in which

$$U_i={\phi }_{{\mathit{ij}}_{J}j}+\rho F_i-\rho {\ddot{u}}_i-{\mathit{\varphi \rho }}_{\mathcal{F}}{\ddot{v}}_i$$ E39

$$V_i=\rho \left(M_i-J{\ddot{\omega }}_i\right)$$ E40

where $u_i^{\ast },{\omega }_i^{\ast }$ and $p_i^{\ast }$ are weighting functions, $u_i,{\omega }_i$ , and $p_i$ are approximate solutions as shown in Eqs. (4)–(11)

The boundary conditions are

$$u_i={\overline{u}}_i\mathrm{on}{S}_1$$ E41

$${\lambda }_i={\sigma }_{\mathit{ii}}{n}_i={\overline{\lambda }}_i\mathrm{on}{S}_2$$ E42

$${\omega }_i={\overline{\omega }}_i\mathrm{on}{S}_3$$ E43

$${\mu }_i=m_{\mathit{ij}}{n}_j={\overline{\mu }}_i\mathrm{on}{S}_4$$ E44

$$p=\overline{p}\mathrm{on}{S}_5$$ E45

$$L=\frac{\partial p}{\partial n}=\overline{L}\mathrm{on}{S}_6$$ E46

By integrating by parts the first term of Eqs. (36)–(38), we obtain

$$-{\int }_R{\sigma }_{\mathit{ij}}{u}_{i,j}^{\ast }\mathit{dR}+{\int }_R{U}_i{u}_i^{\ast }\mathit{dR}=-{\int }_{S_2}{\lambda }_i{u}_i^{\ast }\mathit{dS}$$ E47

$$\begin{array}{l}-{\int }_R{m}_{\mathit{ij}}{\omega }_{i,j}^{\ast }\mathit{dR}+{\int }_R{\epsilon }_{\mathit{ijk}}{\sigma }_{\mathit{jk}}{\omega }_i^{\ast }\mathit{dR}+{\int }_R{V}_i{\omega }_i^{\ast }\mathit{dR}\\ =-{\int }_{S_4}{\mu }_i{\omega }_i^{\ast }\mathit{dS}\end{array}$$ E48

$$-{\int }_{R}q{\mathrm{p}}_{i,i}^{\ast }\mathit{dR}+{\int }_R{\dot{\zeta }}_i{\mathrm{p}}_i^{\ast }\mathit{dR}-{\int }_R{\mathbb{C}}_i{\mathrm{p}}_i^{\ast }\mathit{dR}=-{\int }_{S_6}{L}_i{\mathrm{p}}_i^{\ast }\mathit{dS}$$ E49

which according to Huang and Liang [54] can be expressed as

$$\begin{array}{l}-{\int }_R{\sigma }_{\mathit{ij},j}{u}_i^{\ast }\mathit{dR}+{\int }_R\left(m_{\mathit{ij},j}+{\epsilon }_{\mathit{ijk}}{\sigma }_{\mathit{jk}}\right){\omega }_i^{\ast }\mathit{dR}+{\int }_R{U}_i{u}_i^{\ast }\mathit{dR}\\ +{\int }_R{V}_i{\omega }_i^{\ast }\mathit{dR}-{\int }_{R}q{\mathrm{p}}_{i,i}^{\ast }\mathit{dR}+{\int }_R{\dot{\zeta }}_i{\mathrm{p}}_i^{\ast }\mathit{dR}-{\int }_R{\mathbb{C}}_i{\mathrm{p}}_i^{\ast }\mathit{dR}\\ ={\int }_{S_2}\left({\lambda }_i-{\overline{\lambda }}_i\right)u_i^{\ast }\mathit{dS}+{\int }_{S_1}\left({\overline{u}}_i-u_i\right){\lambda }_i^{\ast }\mathit{dS}\\ +{\int }_{S_4}\left({\mu }_i-{\overline{\mu }}_i\right){\omega }_i^{\ast }\mathit{dS}+{\int }_{S_3}\left({\overline{\omega }}_i-{\omega }_i\right){\mu }_i^{\ast }\mathit{dS}\\ +{\int }_{S_6}\left(L_i-{\overline{L}}_i\right){\mathrm{p}}_i^{\ast }\mathit{dS}+{\int }_{S_5}\left({\overline{\mathrm{p}}}_i-{\mathrm{p}}_i\right)L_i^{\ast }\mathit{dS}\end{array}$$ E50

Using integration by parts for the left-hand side of (50), we have

$$\begin{array}{l}-{\int }_R{\sigma }_{\mathit{ij}}{\epsilon }_{\mathit{ij}}^{\ast }\mathit{dR}-{\int }_R{m}_{\mathit{ij},j}{\omega }_{i,j}^{\ast }\mathit{dR}+{\int }_R{U}_i{u}_i^{\ast }\mathit{dR}\\ +{\int }_R{V}_i{\omega }_i^{\ast }\mathit{dR}-{\int }_{R}q{\mathrm{p}}_{i,i}^{\ast }\mathit{dR}+{\int }_R{\dot{\zeta }}_i{\mathrm{p}}_i^{\ast }\mathit{dR}-{\int }_R{\mathbb{C}}_i{\mathrm{p}}_i^{\ast }\mathit{dR}\\ =-{\int }_{S_2}{\overline{\lambda }}_i{u}_i^{\ast }\mathit{dS}-{\int }_{S_1}{\lambda }_i{u}_i^{\ast }\mathit{dS}\\ +{\int }_{S_1}\left({\overline{u}}_i-u_i\right){\lambda }_i^{\ast }\mathit{dS}-{\int }_{S_4}{\overline{\mu }}_i{\omega }_i^{\ast }\mathit{dS}-{\int }_{S_3}{\mathit{\mu \omega }}_i^{\ast }\mathit{dS}\\ +{\int }_{S_3}\left({\overline{\omega }}_i-{\omega }_i\right){\mu }_i^{\ast }\mathit{dS}-{\int }_{S_6}{\overline{L}}_i{\mathrm{p}}_i^{\ast }\mathit{dS}-{\int }_{S_6}{L}_i{\mathrm{p}}_i^{\ast }\mathit{dS}\\ +{\int }_{S_5}\left({\overline{\mathrm{p}}}_i-{\mathrm{p}}_i\right)L_i^{\ast }\mathit{dS}\end{array}$$ E51

By using the following elastic stress and couple stress (see Eringen [55])

$$\begin{array}{l}{\sigma }_{\mathit{ij}}={\mathbb{A}}_{\mathit{\text{ijkl}}}{\epsilon }_{\mathit{kl}},m_{\mathit{ij}}={\mathbb{B}}_{\mathit{\text{ijkl}}}{\omega }_{k,l} \text{where} {\mathbb{A}}_{\mathit{\text{ijkl}}}={\mathbb{A}}_{\mathit{\text{klij}}} \text{and} {\mathbb{B}}_{\mathit{\text{ijkl}}}={\mathbb{B}}_{\mathit{\text{klij}}}\end{array}$$ E52

Hence, Eq. (51) can be rewritten as

$$\begin{array}{l}-{\int }_R{\sigma }_{\mathit{ij}}^{\ast }{\epsilon }_{\mathit{ij}}\mathit{dR}-{\int }_R{m}_{\mathit{ij},j}^{\ast }{\omega }_{i,j}\mathit{dR}+{\int }_R{U}_i{u}_i^{\ast }\mathit{dR}\\ +{\int }_R{V}_i{\omega }_i^{\ast }\mathit{dR}-{\int }_{R}q{\mathrm{p}}_{i,i}^{\ast }\mathit{dR}+{\int }_R{\dot{\zeta }}_i{\mathrm{p}}_i^{\ast }\mathit{dR}-{\int }_R{\mathbb{C}}_i{\mathrm{p}}_i^{\ast }\mathit{dR}\\ =-{\int }_{S_2}{\overline{\lambda }}_i{u}_i^{\ast }\mathit{dS}-{\int }_{S_1}{\lambda }_i{u}_i^{\ast }\mathit{dS}\\ +{\int }_{S_1}\left({\overline{u}}_i-u_i\right){\lambda }_i^{\ast }\mathit{dS}-{\int }_{S_4}{\overline{\mu }}_i{\omega }_i^{\ast }\mathit{dS}-{\int }_{S_3}{\mu }_i{\omega }_i^{\ast }\mathit{dS}\\ +{\int }_{S_3}\left({\overline{\omega }}_i-{\omega }_i\right){\mu }_i^{\ast }\mathit{dS}-{\int }_{S_6}{\overline{L}}_i{\mathrm{p}}_i^{\ast }\mathit{dS}-{\int }_{S_6}{L}_i{\mathrm{p}}_i^{\ast }\mathit{dS}\\ +{\int }_{S_5}\left({\overline{\mathrm{p}}}_i-{\mathrm{p}}_i\right)L_i^{\ast }\mathit{dS}\end{array}$$ E53

Applying the integration by parts for the left-hand side of Eq. (53), we get

$$\begin{array}{l}{\int }_R{\sigma }_{\mathit{ij},j}^{\ast }{u}_i\mathit{dR}+{\int }_R\left(m_{\mathit{ij},j}^{\ast }+{\epsilon }_{\mathit{ijk}}{\sigma }_{\mathit{jk}}^{\ast }\right){\omega }_i\mathit{dR}\\ =-{\int }_S{u}_i^{\ast }{\lambda }_i\mathit{dS}-{\int }_S{\omega }_i^{\ast }{\mu }_i\mathit{dS}-{\int }_S{\mathrm{p}}_i^{\ast }{L}_i\mathit{dS}+{\int }_S{\lambda }_i^{\ast }{u}_i\mathit{dS}\\ +{\int }_S{\mu }_i^{\ast }{\omega }_i\mathit{dS}+{\int }_S{L}_i^{\ast }{\mathrm{p}}_i\mathit{dS}\end{array}$$ E54

The weighting functions for $U_i={\Delta }^n$ and ${\mathrm{V}}_{\mathrm{i}}=0$ along the unit vector direction $e_l$ are as follows:

$${\sigma }_{\mathit{lj},j}^{\ast }+{\mathrm{\Delta }}^n{e}_l=0$$ E55

$$m_{\mathit{ij},j}^{\ast }+{\epsilon }_{\mathit{ijk}}{\sigma }_{\mathit{jk}}^{\ast }=0$$ E56

The analytical fundamental solution of Dragos [56] can be written as

$$\begin{array}{l}{u}_i^{\ast }=u_{\mathit{li}}^{\ast }{e}_{l,}{\omega }_i^{\ast }={\omega }_{\mathit{li}}^{\ast }{e}_{l,}{p}_i^{\ast }=p_{\mathit{li}}^{\ast }{e}_l,{\lambda }_i^{\ast }={\lambda }_{\mathit{li}}^{\ast }{e}_l,\\ {\mu }_i^{\ast }={\mu }_{\mathit{li}}^{\ast }{e}_{l,}{L}_i^{\ast }=L_{\mathit{li}}^{\ast }{e}_l\end{array}$$ E57

The obtained weighting functions for a point load $U_{\mathrm{i}}=0$ and $V_i={\Delta }^n$ along the unit vector direction ${\mathrm{e}}_1$ were next used as follows:

$${\sigma }_{\mathit{ij},j}^{\ast \ast }=0$$ E58

$$m_{\mathit{lj},j}^{\ast \ast }+{\epsilon }_{\mathit{ljk}}{\sigma }_{\mathit{jk}}^{\ast \ast }+{\mathrm{\Delta }}^n{e}_l=0$$ E59

According to Dragos [56], the fundamental solution can be expressed as

$$\begin{array}{l}{u}_i^{\ast }=u_{\mathit{ii}}^{\ast \ast }{e}_l,{\omega }_i^{\ast }={\omega }_{\mathit{li}}^{\ast }{e}_l,p_i^{\ast }=p_{\mathit{li}}^{\ast \ast }{e}_l,{\lambda }_i^{\ast }={\lambda }_{\mathit{li}}^{\ast \ast }{e}_l,\\ {\mu }_i^{\ast }={\mu }_{\mathit{li}}^{\ast }{e}_l,L_i^{\ast }=L_{\mathit{li}}^{\ast \ast }{e}_l\end{array}$$ E60

Using the weighting functions of (57) and (60) into (54), we obtain

$$\begin{array}{l}{C}_{\mathit{li}}^n{u}_i^n=-{\int }_S{\lambda }_{\mathit{li}}^{\ast }{u}_i\mathit{dS}-{\int }_S{\mu }_{\mathit{li}}^{\ast }{\omega }_i\mathit{dS}-{\int }_S{\mathrm{L}}_{\mathit{li}}^{\ast }{p}_i\mathit{dS}+{\int }_S{u}_{\mathit{li}}^{\ast }{\lambda }_i\mathit{dS}\\ +{\int }_S{\omega }_{\mathit{li}}^{\ast }{\mu }_i\mathit{dS}+{\int }_S{p}_{\mathit{li}}^{\ast }{\mathrm{L}}_i\mathit{dS}\end{array}$$ E61

$$\begin{array}{l}{C}_{\mathit{li}}^n{\omega }_i^n=-{\int }_S{\lambda }_{\mathit{li}}^{\ast \ast }{u}_i\mathit{dS}-{\int }_S{\mu }_{\mathit{li}}^{\ast \ast }{\omega }_i\mathit{dS}-{\int }_S{\mathrm{L}}_{\mathit{li}}^{\ast \ast }{p}_i\mathit{dS}+{\int }_S{u}_{\mathit{li}}^{\ast \ast }{\lambda }_i\mathit{dS}\\ +{\int }_S{\omega }_{\mathit{li}}^{\ast \ast }{\mu }_i\mathit{dS}+{\int }_S{p}_{\mathit{li}}^{\ast \ast }{\mathrm{L}}_i\mathit{dS}\end{array}$$ E62

Thus, we can write

$$C^n{\mathbb{q}}^n=-{\int }_S{\mathbb{p}}^{\ast }\mathbb{q}\mathit{dS}+{\int }_S{\mathbb{q}}^{\ast }{\mathbb{p}}^{\mathit{dS}+}{\int }_S{\mathbb{a}}^{\ast }p\mathit{dS}+{\int }_S{\mathbb{b}}^{\ast }\frac{\partial p}{\partial n}\mathit{dS}$$ E63

where

$$\begin{array}{l}{C}^n=\left[\begin{array}{ll}{C}_{11}& C_{12}\\ C_{21}& C_{22}\end{array}\right],{\mathbb{q}}^{\ast }=\left[\begin{array}{ccc}{u}_{11}^{\ast }& u_{12}^{\ast }& {\omega }_{13}^{\ast }\\ u_{21}^{\ast }& u_{22}^{\ast }& {\omega }_{23}^{\ast }\\ u_{31}^{\ast }& u_{32}^{\ast }& {\omega }_{33}^{\ast }\end{array}\right],{\mathbb{p}}^{\ast }=\left[\begin{array}{ccc}{\lambda }_{11}^{\ast }& {\lambda }_{12}^{\ast }& {\lambda }_{13}^{\ast }\\ {\lambda }_{21}^{\ast }& {\lambda }_{22}^{\ast }& {\mu }_{23}^{\ast }\\ {\lambda }_{31}^{\ast }& {\lambda }_{32}^{\ast }& {\mu }_{33}^{\ast }\end{array}\right]\end{array}$$ E64

$$\begin{array}{l}\mathbb{q}=\left[\begin{array}{l}{u}_1\\ u_2\\ {\omega }_3\end{array}\right],\mathbb{p}=\left[\begin{array}{l}{\lambda }_1\\ {\lambda }_2\\ {\mu }_3\end{array}\right],{\mathbb{a}}^{\ast }=\left[\begin{array}{l}{\mathbb{a}}_1^{\ast }\\ {\mathbb{a}}_2^{\ast }\\ 0\end{array}\right],{\mathbb{b}}^{\ast }=\left[\begin{array}{l}{\mathbb{b}}_1^{\ast }\\ {\mathbb{b}}_2^{\ast }\\ 0\end{array}\right]\end{array}$$

In order to obtain the numerical solution of (63), we define the following functions

$$\mathbb{q}=\psi {\mathbb{q}}^j,\mathrm{p}=\psi {\mathbb{p}}^j,p={\psi }_0{p}^j,\frac{\partial p}{\partial n}={\psi }_0{\left(\frac{\partial p}{\partial n}\right)}^j$$ E65

substituting above functions into (63) and discretizing the boundary, we obtain

$$\begin{array}{l}{C}^n{\mathrm{q}}^n=\sum _{j=1}^{N_e}\left[-{\int }_{{\mathrm{\Gamma }}_j}{\mathrm{p}}^{\ast }\mathit{\psi d}\mathrm{\Gamma }\right]{\mathrm{q}}^j+\sum _{j=1}^{N_e}\left[{\int }_{{\mathrm{\Gamma }}_j}{\mathrm{q}}^{\ast }\mathit{\psi d}\mathrm{\Gamma }\right]{\mathrm{p}}^j\\ +\sum _{j=1}^{N_e}\left[{\int }_{{\mathrm{\Gamma }}_j}{\mathrm{a}}^{\ast }{\psi }_{0}d\mathrm{\Gamma }\right]p^j+\sum _{j=1}^{N_e}\left[-{\int }_{{\mathrm{\Gamma }}_j}{\mathrm{b}}^{\ast }{\psi }_{0}d\mathrm{\Gamma }\right]{\left(\frac{\partial p}{\partial n}\right)}^j\end{array}$$ E66

Equation after integration can be written as

$$C^i{\mathrm{q}}^i=-\sum _{j=1}^{N_e}{\widehat{\mathbb{H}}}^{\mathit{ij}}{\mathrm{q}}^j+\sum _{j=1}^{N_e}{\widehat{\mathrm{G}}}^{\mathit{ij}}{\mathrm{p}}^j+\sum _{j=1}^{N_e}{\widehat{\mathrm{a}}}^{\mathit{ij}}{p}^j+\sum _{j=1}^{N_e}{\widehat{\mathrm{b}}}^{\mathit{ij}}{\left(\frac{\partial p}{\partial n}\right)}^j$$ E67

By using the following representation

$${\mathbb{H}}^{\mathit{ij}}=\left\{\begin{array}{ll}{\widehat{\mathbb{H}}}^{\mathit{ij}}& \mathit{\text{if}}i\ne j\\ {\widehat{\mathbb{H}}}^{\mathit{ij}}+C^i& \mathit{\text{if}}i=j\end{array}\right.$$ E68

Thus, we can write (67) as follows

$$\sum _{j=1}^{N_e}{\mathbb{H}}^{\mathit{ij}}{\mathrm{q}}^j=\sum _{j=1}^{N_e}{\widehat{\mathrm{G}}}^{\mathit{ij}}{\mathrm{p}}^j+\sum _{j=1}^{N_e}{\widehat{\mathrm{a}}}^{\mathit{ij}}{p}^j+\sum _{j=1}^{N_e}{\widehat{\mathrm{b}}}^{\mathit{ij}}{\left(\frac{\partial p}{\partial n}\right)}^j$$ E69

The global matrix system equation for all $\mathit{i}$ nodes can be written as follows

$$\mathbb{HQ}=\mathbb{G}\mathbb{P}+\mathbb{ai}+\mathbb{bj}$$ E70

the vector $\mathbb{Q}$ represents all the values of displacements and microrotations, the vector $\mathbb{P}$ represents all the tractions and couple stress vector, the vector $\mathbb{i}$ represents all the values of pore pressure, and the vector $\mathbb{j}$ represents all the values of pore pressure gradients before applying boundary conditions.

Substituting the boundary conditions into (70), we obtain the following system of equations

$$\mathbb{AX}=\mathbb{B}$$ E71

where $\mathbb{A}$ is an unknown matrix, while $\mathbb{X}$ and $\mathbb{B}$ are known matrices.

Now, an explicit staggered predictor-corrector procedure based on communication-avoiding Arnoldi (CA-Arnoldi) method has been implemented in (71) for obtaining the corrected displacement. Then we can get the temperature field from (35).

5. Numerical results and discussion

In order to show the numerical results of this study, we consider a monoclinic graphite-epoxy as an anisotropic micropolar poro-thermoelastic material which has the following physical constants.

The elasticity tensor is expressed as

$$C_{\mathit{\text{pjkl}}}=\left[\begin{array}{cccccc}430.1& 130.4& 18.2& 0& 0& 201.3\\ 130.4& 116.7& 21.0& 0& 0& 70.1\\ 18.2& 21.0& 73.6& 0& 0& 2.4\\ 0& 0& 0& 19.8& -8.0& 0\\ 0& 0& 0& -8.0& 29.1& 0\\ 201.3& 70.1& 2.4& 0& 0& 147.3\end{array}\right]\mathrm{GPa}$$ E72

The mechanical temperature coefficient is

$${\beta }_{\mathit{pj}}=\left[\begin{array}{ccc}1.01& 2.00& 0\\ 2.00& 1.48& 0\\ 0& 0& 7.52\end{array}\right]\cdot {10}^6\frac{\mathrm{N}}{{\mathrm{km}}^2}$$ E73

The thermal conductivity tensor is

$$k_{\mathit{pj}}=\left[\begin{array}{lll}5.2& 0& 0\\ 0& 7.6& 0\\ 0& 0& 38.3\end{array}\right]\mathrm{W}/\mathrm{Km}$$ E74

Mass density $\rho =7820\mathrm{kg}/{\mathrm{m}}^3$ and heat capacity $c=461\mathrm{J}/\mathrm{kg}\mathrm{K}$ .

The proposed technique that has been utilized in the present chapter can be applicable to a wide variety of wave propagation of fractional nonlinear generalized micropolar poro-thermoelastic FGA structures problems related with the proposed theory.

The influence of three-temperature on the propagation of thermal stress waves plays a very important role during the simulation process. According to Fahmy [7], who compared and implemented communication-avoiding GMRES (CA-GMRES) of Saad and Schultz [57], communication-avoiding Arnoldi (CA-Arnoldi) of the Arnoldi [58] and communication-avoiding Lanczos (CA-Lanczos) of Lanczos [59] for solving the dense nonsymmetric algebraic system of linear equations arising from the BEM. So, the efficiency of the proposed technique has been developed using the communication-avoiding Arnoldi (CA-Arnoldi) solver to reduce the iterations number and CPU time, where the BEM discretization is employed 1280 quadrilateral elements, with 3964° of freedom (DOF).

Now, in order to assess the impact of three temperatures on the thermal stress waves, the numerical outcomes are completed and delineated graphically for electron, ion, and phonon temperatures.

Figures 2–4 show the propagation of the thermal stress ${\sigma }_{11}$ , ${\sigma }_{12}$ , and ${\sigma }_{22}$ waves along $x$ -axis for the three temperatures $T_e$ , $T_i$ , and $T_p$ and total temperature $T$ . It was noted from these figures that the three temperatures have significant effects on the thermal stress waves along $\mathrm{x}$ -axis through the thickness of the FGA structure.

Figure 2.

Propagation of the thermal stress ${\mathbf{\sigma }}_{\mathbf{11}}$ waves along $\mathit{x}$ -axis for the three temperatures $T_e$ , $T_i$ , $T_p$ and total temperature $T$ .

Figure 3.

Propagation of the thermal stress ${\mathbf{\sigma }}_{\mathbf{12}}$ waves along $\mathit{x}$ -axis for the three temperatures $T_e$ , $T_i$ , $T_p$ and total temperature $T$ .

Figure 4.

Propagation of the thermal stress ${\mathbf{\sigma }}_{\mathbf{22}}$ waves along $\mathit{x}$ -axis for the three temperatures $T_e$ , $T_i$ , $T_p$ and total temperature $T$ .

Since there are no available results for our considered problem. So, some literatures may be considered as special cases from our considered complex problem. For comparison purposes with the special cases of other methods treated by other authors, we only considered one-dimensional numerical results of the considered problem. In the special case under consideration, the BEM results have been plotted in Figures 5 and 6 with the results of finite difference method (FDM) and finite element method (FEM) in the two cases, namely, three-temperature (3T) theory and one-temperature (1T) theory.

Figure 5.

Propagation of the thermal stress ${\mathbf{\sigma }}_{\mathbf{11}}$ waves along $\mathit{x}$ -axis for 3T theory and different methods.

Figure 6.

Propagation of the thermal stress ${\mathbf{\sigma }}_{\mathbf{11}}$ waves along $\mathit{x}$ -axis for 1T theory and different methods.

Figure 5 shows a comparison of the propagation of the thermal stress ${\sigma }_{11}$ waves for the BEM results of three-temperature (3T) radiative heat conduction theory for the BEM results with those obtained using the FDM of Pazera and Jędrysiak [60] and FEM of Xiong and Tian [61], where we replaced the 1T heat conduction theory of their work by 3T radiative heat conduction theory of our work to obtain the results. It can be noticed that the BEM results are found to agree very well with the FDM and FEM results.

Figure 6 shows a comparison of the propagation of the thermal stress ${\sigma }_{11}$ waves for the BEM results of one-temperature (1T) heat conduction theory with those obtained using FDM of Pazera and Jędrysiak [60], FEM1 of Xiong and Tian [61], and FEM2 of COMSOL multiphysics software version 5.1, where we replaced 3T radiative heat conduction theory of our work by the 1T heat conduction theory of their work to obtain the results. It can be noticed that the BEM results are found to agree very well with the FDM, FEM1, and FEM2 results and thus demonstrate the validity and accuracy of our proposed theory and the technique used to solve its general problems.

6. Conclusion

The main purpose of this chapter is to introduce a novel theory called acoustic wave propagation of three-temperature fractional nonlinear generalized micropolar poro-thermoelasticity and we propose a new boundary element technique for modeling and simulation of ultrafast laser-induced thermal stress waves propagation problems in 3T nonlinear generalized micropolar poro-thermoelastic FGA structures which are linked with the proposed theory. By discretizing only, the boundary of the domain using BEM, where the unknowns on the domain boundary are expressed as functions depend only on the domain boundary values. Since it is very difficult to solve general acoustic problems of this theory analytically and we need to develop and use new computational modeling techniques. So, we propose a new boundary element technique for solving such problems. The numerical results are shown graphically to depict the effects of three temperatures on the thermal stress waves. Because there are no available results for comparison with the results of our proposed technique, we replace the three-temperature radiative heat conduction with one-temperature heat conduction as a special case from our present general study of three-temperature nonlinear generalized micropolar poro-thermoelasticity. In the special case under consideration, the BEM results have been compared graphically with the FDM and FEM in the two cases, namely three-temperature (3T) theory and one-temperature (1T) theory; it can be noticed that the BEM results are in a good agreement with the FDM and FEM results and thus demonstrate the validity and accuracy of our proposed theory and the technique used to solve its general problems. The numerical simulations are often faster and cheaper than experiments, and they are easily cross-platform, reproducible, relocatable, and customizable. So, the validation of the numerical simulation is of paramount importance. In this work, we implemented the explicit staggered predictor-corrector procedure based on communication-avoiding Arnoldi (CA-Arnoldi) solver due to its numerical stability, convergence, and performance as in Fahmy [10] to demonstrate the efficiency of the proposed technique. Thus, the numerical results of our proposed technique demonstrate the validity, accuracy, and efficiency of our proposed technique.

Nowadays, the knowledge of thermal stress wave propagation in three-temperature nonlinear generalized micropolar poro-thermoelastic problems associated with the ultrafast laser pulse proposed theory can be utilized by mechanical engineers in ceramic production applications and designing of boiler tubes and heat exchangers. As well as for chemists to observe the chemical reaction phenomena such as bond formation and bond breaking.