A New BEM for Modeling of Acoustic Wave Propagation in Three-Temperature Nonlinear Generalized Magneto-Thermoelastic ISMFGA Structures Using Laser Ultrasonics

Abstract

The principal aim of this chapter is to introduce a new theory called acoustic wave propagation of three-temperature nonlinear generalized magneto-thermoelasticity, and we propose a new boundary element model for solving problems of initially stressed multilayered functionally graded anisotropic (ISMFGA) structures using laser ultrasonics, which connected with the proposed theory. Since there are no available analytical or numerical solutions for the considered nonlinear wave propagation problems in the literature, we propose a new boundary element modeling formulation for the solution of such problems. The numerical results are depicted graphically to show the propagation of three temperatures and displacement waves. The results also show the effects of initial stress and functionally graded material on the displacement waves and confirm the validity and accuracy of our proposed theory and solution technique.

1. Introduction

Physically, according to particle motion orientation and energy direction, there are three wave types, which are categorized as mechanical waves, electromagnetic waves, and matter waves. Mechanical waves are waves, which cannot travel through a vacuum and can travel through any medium at a wave speed, which depends on elasticity and inertia. There are three types of mechanical waves: longitudinal, transverse, and surface waves. Longitudinal waves occur when the movement of the particles is parallel to the energy motion like sound waves and pressure waves. Transverse waves appear when the movement of the particles is perpendicular to the energy motion like light waves, polarized waves, and electromagnetic waves. Surface waves happen when the movement of the particles is in a circular motion. These waves usually occur at interfaces like ocean waves and cup of water ripples. Electromagnetic waves are generated by a fusion of electric and magnetic fields. These waves travel through a vacuum and do not need a medium to travel like microwaves, X-ray, radio waves, and ultraviolet waves. The matter has a wave–particle duality property, where in 1905, Albert Einstein introduced a quantum mechanics theory stating that light has a dual nature; when the light is moving, it shows the wave properties, and when it is at rest, it shows the particle properties, where each light particle has an energy quantum called a photon. Sound is a pressure variation, where a condensation is an increased pressure region on a sound wave and a dilation is a decreased pressure region on a sound wave. Acoustics is the science of study related to the study of sound in gases, liquids, and solids including subjects such as vibration, sound, ultrasound, and infrasound and has grown to encompass the realm of ultrasonics and infrasonics in addition to the audio range, as the result of applications in oceanology, materials science, medicine, dentistry, communications, industrial processes, petroleum and mineral prospecting, music and voice synthesis, marine navigation, animal bioacoustics, and noise cancelation. There are two mechanisms that have been proposed to explain wave generation, which depend on the energy density of laser pulse, a first mechanism at high-energy density, where a thin layer of solid material melts, followed by a dissolution process where the particles fly off the surface, which leads to forces that generate ultrasound, and a second mechanism at low-energy density, where irradiation of laser pulses onto a material generates elastic waves due to the thermoelastic process of expansion of a surface at a high rate. Ultrasound generation with lasers offers a number of advantages over conventional generation with piezoelectric transducers. Since the ultrasound generation by a laser pulse in the thermoelastic range does not damage the material surface, it has several applications such as fiber-optic communication, narrow-band and broadband systems, the ability to work on hard to reach places, curved and rough surfaces, absolute beam energy measurements, and digital images having higher spatial resolution. The process of converting a laser source into an equivalent set of stress boundary conditions takes the largest share of the effort involved in modeling of laser-generated ultrasound, which is very useful in describing the features of a laser-generated ultrasonic in the thermoelastic system [1, 2, 3]. Due to the interaction between laser light and a metal surface, the generation of high-frequency acoustic pulses causes the laser irradiation of a metal surface. It led to great progress to develop theoretical models to describe the experimental data [4]. Scruby et al. [5] demonstrated that the thermoelastic area source has been reduced to a point-source influential on the surface. This source point ignores the optical absorption of laser energy into the bulk material and the thermal diffusion from the heat source. Moreover, it does not take into account the limited side dimensions of the source. Rose [6] introduced surface center of expansion (SCOE) based on point-source representation. The SCOE models predict the major features of laser-generated ultrasound waves and agree with experiments particularly well for highly focused Q-switched laser pulses. It fails to predict a precursor in ultrasonic waveforms on and near the epicenter. The precursor is a small sharp initial spike observed in metals signaling the arrival of the longitudinal wave. Doyle [7] established that the existence of the metal precursor is due to subsurface sources which arise from thermal diffusion, since the optical absorption depth is very small in comparison to the thermal diffusion length. According to McDonald [8], Spicer [9] used the generalized thermoelasticity theory to constitute a real model, taking into consideration spatial–temporal shape of the laser pulse and the effect of thermal diffusion.

The mathematical foundations of three-temperature thermoelasticity were defined for the first time by Fahmy [10, 11, 12, 13, 14]. Analytical solutions for the current nonlinear generalized thermoelastic problems which are associated with the proposed theory are very difficult to obtain, so many numerical methods were developed for solving such problems like finite difference method [15], discontinuous Galerkin method [16], finite element method (FEM) [17], boundary element method (BEM) [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], and other developed techniques [32, 33, 34, 35, 36]. The boundary element method [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67] is actualized effectively for tackling a few designing and logical applications because of its straightforwardness, precision, and simplicity of execution.

In the present chapter, we introduce a new acoustic wave propagation theory called three-temperature nonlinear generalized magneto-thermoelasticity, and we propose a new boundary element technique for modeling problems of initially stressed multilayered functionally graded anisotropic (ISMFGA) structures using laser ultrasonics, which connected with the proposed theory, where we used the three-temperature (3T) radiative heat conduction equations combined with electron, ion, and photon temperatures in the formulation of such problems. The numerical results are presented graphically to show the effects of three temperatures on the displacement wave propagation in the $x$-axis direction of ISMFGA structures. The numerical results also show the propagation of the displacement waves of homogenous and functionally graded structures under the effect of initial stress. The validity and accuracy of our proposed model was demonstrated by comparing our BEM results with the corresponding FDM and FEM results.

A brief summary of the paper 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 magneto-thermoelastic ISMFGA structures and their applications. Section 2 describes the formulation of the new theory and introduces the partial differential equations that govern its related problems. Section 3 outlines continuity and initial and boundary conditions of the considered problem. Section 4 discusses the implementation of the new BEM and its implementation for solving the governing equations of the problem to obtain the three temperatures and displacement fields. Section 5 presents the new numerical results that describe the displacement waves and three-temperature waves under the effect of initial stress on the homogeneous and functionally graded structures.

2. Formulation of the problem

Consider a multilayered structure with $n$ functionally graded layers in the $\mathit{yz}$-plane of a Cartesian coordinate. The $x$-axis is the common normal to all layers as shown in Figure 1. The thickness of the considered multilayered structure and the $i$th layer is denoted by $h$ and $h^i$, respectively. The considered multilayered structure which occupies the region $R=\left\{\left(x,y,z\right):0<x<h,0<y<b,0<z<a\right\}$ has been placed in a primary magnetic field $H_0$ acting in the direction of the $y$-axis.

Figure 1.

Geometry of the FGA structure.

According to the three-temperature theory, the governing equations of nonlinear generalized magneto-thermoelasticity in an initially stressed multilayered functionally graded anisotropic (ISMFGA) structure for the $i$th layer can be written in the following form:

$${\sigma }_{\mathit{ab},b}+{\tau }_{\mathit{ab},b}-{\mathrm{\Gamma }}_{\mathit{ab}}={\rho }^i{\left(x+1\right)}^m{\ddot{u}}_a^i$$E1

$${\sigma }_{\mathit{ab}}={\left(\chi +1\right)}^m\left[C_{\mathit{\text{abfg}}}^i{u}_{f,g}^i-{\beta }_{\mathit{ab}}^i\left(T^i-T_0+{\tau }_1{\dot{T}}^i\right)\right]$$E2

$${\tau }_{\mathit{ab}}={\mu }^i{\left(x+1\right)}^m\left({\tilde{h}}_a{H}_b+{\tilde{h}}_b{H}_a-{\delta }_{\mathit{ba}}\left({\tilde{h}}_f{H}_f\right)\right)$$E3

$${\mathrm{\Gamma }}_{\mathit{ab}}=P^i{\left(x+1\right)}^m\left(\frac{\partial u_a^i}{\partial x_b}-\frac{\partial u_b^i}{\partial x_a}\right)$$E4

According to Fahmy [10], the 2D-3 T radiative heat conduction equations can be expressed as follows:

$$\nabla \left[\left({\delta }_{1j}{\mathbb{K}}_{\alpha }^{i\ast }+{\delta }_{2j}{\mathbb{K}}_{\alpha }^i\right)\nabla T_{\alpha }^i\left(r,\tau \right)\right]-\overline{\mathbb{W}}\left(r,\tau \right)=c_{\alpha }^i{\rho }^i{\delta }_1{\delta }_{1j}\frac{\partial T_{\alpha }^i\left(r,\tau \right)}{\partial \tau }$$E5

where

$$\overline{\mathbb{W}}\left(r,\tau \right)=\left\{\begin{array}{ll}{\rho }^i{\mathbb{W}}_{\mathit{eI}}\left(T_e^i-T_I^i\right)+{\rho }^i{\mathbb{W}}_{\mathit{er}}\left(T_e^i-T_p^i\right)+\overline{\overline{\mathbb{W}}},\alpha =e,& {\delta }_1=1\\ -{\rho }^i{\mathbb{W}}_{\mathit{eI}}\left(T_e^i-T_p^i\right)+\overline{\overline{\mathbb{W}}},& \alpha =I,{\delta }_1=1\\ -{\rho }^i{\mathbb{W}}_{\mathit{er}}\left(T_e^i-T_p^i\right)+\overline{\overline{\mathbb{W}}},& \alpha =p,{\delta }_1=T_p^3\end{array}\right.$$E6

in which

$$\overline{\overline{\mathrm{W}}}\left(r,\tau \right)=-{\delta }_{2j}{\mathbb{K}}_{\alpha }^i{\dot{T}}_{\alpha ,\mathit{ab}}+{\beta }_{\mathit{ab}}{T}_{\alpha 0}\left[\left({\tau }_0+{\delta }_{2j}\right){\ddot{u}}_{a,b}\right]+{\rho }^i{c}_{\alpha }^i\left[\left({\tau }_0+{\delta }_{1j}{\tau }_2+{\delta }_{2j}\right){\ddot{T}}_{\alpha }\right]-Q\left(x,\tau \right)$$E7

and

$$\begin{array}{l}{\mathbb{W}}_{\mathit{eI}}={\rho }^i{\mathbb{A}}_{\mathit{eI}}{T}_e^{-2/3},{\mathbb{W}}_{\mathit{er}}={\rho }^i{\mathbb{A}}_{\mathit{er}}{T}_e^{-1/2},{\mathbb{K}}_{\alpha }={\mathbb{A}}_{\alpha }{T}_{\alpha }^{5/2},\alpha =e,I,\\ {\mathbb{K}}_p={\mathbb{A}}_p{T}_p^{3+\mathbb{B}}\end{array}$$E8

The total energy of unit mass can be described by

$$P=P_e+P_I+P_p,P_e=c_{\mathit{\alpha e}}{T}_{\mathit{e}}^i,P_I=c_{\mathit{\alpha I}}{T}_l^i,P_p=\frac{1}{4}{c}_{\mathit{\alpha p}}{T}_p^{4i}$$E9

where ${\sigma }_{\mathit{ab}}$, ${\tau }_{\mathit{ab}}$, and $u_k^i$ are the mechanical stress tensor, Maxwell’s electromagnetic stress tensor, and displacement vector, respectively; $T_{\alpha 0}^i$ is the reference temperature; $T_{\alpha }^i$ is the temperature; $C_{\mathit{\text{abfg}}}^i$ and ${\beta }_{\mathit{ab}}^i$ are, respectively, the constant elastic moduli and stress-temperature coefficients of the anisotropic medium; ${\mu }^i$, $\tilde{h}$, $P^i$, ${\rho }^i$, and $c_{\alpha }^i$ are the magnetic permeability, perturbed magnetic field, initial stress in the $i$th layer, density, and specific heat capacity, respectively; $\tau$ is the time; ${\tau }_0$, ${\tau }_1$, and ${\tau }_2$ are the relaxation times; $i=1,2,\dots ,n-1$ represents the parameters in a multilayered structure; and $m$ is a dimensionless constant. Also, we considered in the current study that ${\tau }_{\mathit{ab},b}={\mu }_0^i{\mathrm{ϵ}}_{\mathit{abf}}{J}_b{H}_f$ is the a-component of the Lorentz force and $J\left(\tau \right)=\frac{J_0\tau }{{\tau }_3^2}{e}^{\frac{\tau }{{\tau }_3}}$ is the temporal profile of a non-Gaussian laser pulse, $J_0$ is the total energy intensity, and $Q\left(x,\tau \right)=\frac{1-R}{x_0}{e}^{\left(\frac{x_a}{x_0}\right)J\left(\tau \right)},a=1,2,3$ is the heat source intensity.

According to Fahmy [57], we notice that there are two special cases of the Green and Naghdi theory of type III; when ${\mathbb{K}}_{\alpha }^i\to 0$, the equations of GN III theory are reduced to the GN theory type II, and when ${\mathbb{K}}_{\alpha }^{i\ast }\to 0$, the equations of the GN III theory are reduced to the GN theory type I.

3. Continuity and initial and boundary conditions

The continuity conditions along interfaces for the temperature, heat flux, displacement, and traction can be expressed as follows:

$$T_{\alpha }^i\left(x,z,\tau \right){\left|{}_{x=h^i}=T_{\alpha }^{\left(i+1\right)}(x,z_t\tau )\right|}_{x=h^i}$$E10

$$q^i\left(x,z,\tau \right){\left|{}_{x=h^i}=q^{\left(i+1\right)}(x,z,\tau )\right|}_{x=h^i}$$E11

$${\left.u_f^i\left(x,z,\tau \right)\right|}_{x=h^i}=u_f^{\left(i+1\right)}{\left(x,z,\tau \right)}_{x=h^i}$$E12

$${\left.{\overline{t}}_a^i\left(x,z,\tau \right)\right|}_{x=h^i}={\overline{t}}_a^{\left(i+1\right)}{\left(x,z,\tau \right)}_{x=h^i}$$E13

where $n$ is the total number of layers, ${\overline{t}}_a$ are the tractions, which are defined by ${\overline{t}}_a={\sigma }_{\mathit{ab}}{n}_b$, and $i=1,2,\dots ,n-1$.

The remaining initial and boundary conditions for the current study are

$$u_f^i\left(x,z,0\right)={\dot{u}}_f^i\left(x,z,0\right)=0\text{for}\left(x,z\right)\in R\cup C$$E14

$$u_f^i\left(x,z,\tau \right)={\mathrm{\Psi }}_f\left(x,z,\tau \right)\text{for}\left(x,z\right)\in C_3$$E15

$${\overline{t}}_a^i\left(x,z,\tau \right)={\mathrm{\Phi }}_f\left(x,z,\tau \right)\text{for}\left(x,z\right)\in C_4,\tau >0,$$E16

$$T_{\alpha }^i\left(x,z,0\right)=T_{\alpha }^i\left(x,z,0\right)=0\text{for}\left(x,z\right)\in R\cup C$$E17

$$T_{\alpha }^i\left(x,y,\tau \right)=\overline{f}\left(x,\mathrm{y},\tau \right)\text{for}\left(x,y\right)\in C_1,\tau >0$$E18

$$q^i\left(x,z,\tau \right)=\overline{h}\left(x,z,\tau \right)\text{for}\left(x,z\right)\in C_2,\tau >0$$E19

where ${\mathrm{\Psi }}_f$, ${\mathrm{\Phi }}_f,f$, and $\overline{h}$ are suitably prescribed functions and $C=C_1\cup C_2=C_3\cup C_4$, $C_1\cap C_2=C_3\cap C_4=\mathrm{Ø}$.

4. BEM numerical implementation

Making use of Eqs. (2)–(4), we can write (1) as follows:

$$L_{\mathit{gb}}{u}_f^i={\rho }^i{ü}_a^i-\left(D_a{T}_{\alpha }^i-P^i\left(\frac{\partial u_b^i}{\partial x_a}-\frac{\partial u_a^i}{\partial x_b}\right)\right)=f_{\mathit{gb}}$$E20

where the inertia term $\rho {\ddot{u}}_a$, the temperature gradient $D_{a}T$, and the initial stress term are treated as the body forces.

The field equations may be expressed in the operator form as follows:

$$L_{\mathit{gb}}{u}_f^i=f_{\mathit{gb}},$$E21

$$L_{\mathit{ab}}{T}_{\alpha }^i=f_{\mathit{ab}}$$E22

where the operators $L_{\mathit{gb}}$, $f_{\mathit{gb}}$, $L_{\mathit{ab}}$, and $f_{\mathit{ab}}$ are as follows:

$$L_{\mathit{gb}}=D_{\mathit{abf}}\frac{\partial }{\partial x_b}+D_{\mathit{af}}+\mathrm{\Lambda }{D}_{a1f},L_{\mathit{ab}}=\left({\delta }_{2j}{\mathbb{K}}_{\alpha }^{i\ast }\right)\nabla$$E23

$$f_{\mathit{gb}}={\rho }^i{\ddot{u}}_a^i-\left(D_a{T}_{\alpha }^i-P^i\left(\frac{\partial u_b^i}{\partial x_a}-\frac{\partial u_a^i}{\partial x_b}\right)\right)$$E24

$$f_{\mathit{ab}}=\nabla \left({\delta }_{1j}{\mathbb{K}}_{\alpha }^i\right)\nabla +{\rho }^i{c}_{\alpha }^i{\delta }_1{\delta }_{1j}{\left(x+1\right)}^m{\dot{T}}_{\alpha }^i+\overline{\mathbb{W}}\left(r,\tau \right)$$E25

where

$$\begin{array}{l}{D}_{\mathit{abf}}=C_{\mathit{\text{abfg}}}\epsilon ,\epsilon =\frac{\partial }{\partial x_g},D_{\mathit{af}}=\mu H_0^2\left(\frac{\partial }{\partial x_a}+{\delta }_{a1}\mathrm{\Lambda }\right)\frac{\partial }{\partial x_f},\\ D_a=-{\beta }_{\mathit{ab}}^i\left(\frac{\partial }{\partial x_b}+{\mathrm{\delta }}_{\mathrm{b}1}\mathrm{\Lambda }+{\tau }_1\left(\frac{\partial }{\partial x_b}+\mathrm{\Lambda }\right)\frac{\partial }{\partial \tau }\right),\mathrm{\Lambda }=\frac{m}{x+1}.\end{array}$$

The differential equation (21) can be solved using the weighted residual method (WRM) to obtain the following integral equation:

$${\int }_R\left(L_{\mathit{gb}}{u}_f^i-f_{\mathit{gb}}\right)u_{\mathit{da}}^{i\ast }\mathit{dR}=0$$E26

Now, the fundamental solution $u_{\mathit{df}}^{i\ast }$ and traction vectors $t_{\mathit{da}}^{i\ast }$ and $t_a^i$ can be written as follows:

$$L_{\mathit{gb}}{u}_{\mathit{af}}^{i\ast }=-{\delta }_{\mathit{ad}}\delta \left(x,\xi \right)$$E27

$$t_{\mathit{da}}^{i\ast }=C_{\mathit{\text{abfg}}}{u}_{\mathit{df},g}^{i\ast }{n}_b$$E28

$$t_a^i=\frac{{\overline{t}}_a^i}{{\left(x+1\right)}^m}=\left(C_{\mathit{\text{abfg}}}{u}_{f,g}^i-{\beta }_{\mathit{ab}}^i\left(T_{\alpha }^i+{\tau }_1{T}_{\alpha }^i\right)\right)n_b$$E29

Using integration by parts and sifting property of the Dirac distribution for (26), then using Eqs. (27) and (29), we can write the following elastic integral representation formula:

$$u_d^i\left(\xi \right)=\underset{C}{\int }\left(u_{\mathit{da}}^{i\ast }{t}_a^i-t_{\mathit{da}}^{i\ast }{u}_a^i+u_{\mathit{da}}^{i\ast }{\beta }_{\mathit{ab}}^i{T}_{\alpha }^i{n}_b\right)\mathit{dC}-\underset{R}{\int }{f}_{\mathit{gb}}{u}_{\mathit{da}}^{i\ast }\mathit{dR}$$E30

The fundamental solution $T^{i^{\ast }}$ can be defined as

$$L_{\mathit{ab}}{T}^{i^{\ast }}=-\delta \left(x,\xi \right)$$E31

By using WRM and integration by parts, we can write (23) as follows:

$$\underset{R}{\int }\left(L_{\mathit{ab}}{T}_{\alpha }^i{T}_{\alpha }^{i^{\ast }}-L_{\mathit{ab}}{T}_{\alpha }^{i^{\ast }}{T}_{\alpha }^i\right)\mathit{dR}=\underset{C}{\int }\left(q^{i^{\ast }}{T}_{\alpha }^i-q^i{T}_{\alpha }^{i^{\ast }}\right)\mathit{dC}$$E32

where

$$q^i=-{\mathbb{K}}_{\alpha }^i{T}_{\alpha .b}^i{n}_a$$E33

$$q^{i\ast }=-{\mathbb{K}}_{\alpha }^i{T}_{\alpha ,b}^{i^{\ast }}{n}_a$$E34

By the use of sifting property, we obtain from (32) the thermal integral representation formula:

$$T_{\alpha }^i\left(\xi \right)=\underset{C}{\int }\left(q^{i^{\ast }}{T}_{\alpha }^i-q^i{T}_{\alpha }^{i^{\ast }}\right)\mathit{dC}-\underset{R}{\int }{f}_{\mathit{ab}}{T}_{\alpha }^{i^{\ast }}\mathit{dR}$$E35

By combining (30) and (35), we have

$$\begin{array}{c}\left[\begin{array}{l}{u}_d^i\left(\xi \right)\hfill \\ T_{\alpha }^i\left(\xi \right)\hfill \end{array}\right]=\underset{C}{{\displaystyle \int }}\{-\left[\begin{array}{ll}{t}_{da}^{i\ast }\hfill & -u_{aa}^{i\ast }{\beta }_{ab}{n}_b\hfill \\ 0\hfill & -q^{i\ast }\hfill \end{array}\right]\left[\begin{array}{l}{u}_a^i\hfill \\ T_{\alpha }^i\hfill \end{array}\right]+\left[\begin{array}{ll}{u}_{da}^{i\ast }\hfill & 0\hfill \\ 0\hfill & -T_{\alpha }^{i\ast }\hfill \end{array}\right]\left[\begin{array}{l}{\tau }_a^i\hfill \\ q^i\hfill \end{array}\right]\}dC\\ -\underset{R}{{\displaystyle \int }}\left[\begin{array}{ll}{u}_{da}^{i\ast }\hfill & 0\hfill \\ 0\hfill & -T_{\alpha }^{i\ast }\hfill \end{array}\right]\left[\begin{array}{l}{f}_{gb}\hfill \\ -f_{ab}\hfill \end{array}\right]dR\end{array}$$E36

The generalized thermoelastic vectors can be expressed in contracted notation form as follows:

$$U_A^i=\left\{\begin{array}{ll}{u}_a^i& a=A=1,2,3\\ T_{\alpha }^i& A=4\end{array}\right.$$E37

$${\mathrm{T}}_{\mathit{\alpha A}}^i=\left\{\begin{array}{ll}{t}_a^i& a=A=1,2,3\\ q^i& A=4\end{array}\right.$$E38

$$U_{\mathit{DA}}^{i\ast }=\left\{\begin{array}{ll}{u}_{\mathit{da}}^{i\ast }& d=D=\mathrm{1,2,3};a=A=1,2,3\\ 0& d=D=\mathrm{1,2,3};A=4\\ 0& D=4;a=A=1,2,3\\ -T_{\alpha }^{i\ast }& D=4;A=4\end{array}\right.$$E39

$${\tilde{T}}_{\mathit{\alpha DA}}^{i^{\ast }}=\left\{\begin{array}{ll}{t}_{\mathit{aa}}^{i\ast }& d=D=\mathrm{1,2,3};a=A=1,2,3\\ -{\tilde{u}}_d^{i\ast }& d=D=\mathrm{1,2,3};A=4\\ 0& D=4;a=A=1,2,3\\ -q^{i\ast }& D=4;A=4\end{array}\right.$$E40

$${\tilde{u}}_d^{i\ast }=u_{\mathit{da}}^{i\ast }{\beta }_{\mathit{af}}^i{n}_f$$E41

Using the previous vectors, we can write (36) as

$$U_D^i\left(\xi \right)=\underset{C}{\int }\left(U_{\mathit{DA}}^{i\ast }{\mathrm{T}}_{\mathit{\alpha A}}^i-{\tilde{T}}_{\mathit{\alpha DA}}^i{U}_A^i\right)\mathit{dC}-\underset{R}{\int }{U}_{\mathit{DA}}^{i\ast }{S}_A\mathit{dR}$$E42

The vector $S_A$ can be split as follows

$$S_A=S_A^0+S_A^T+S_A^{\mathrm{u}}+S_A^{\dot{T}}+S_A^{\ddot{T}}+S_A^{\ddot{u}}$$E43

where

$$S_A^0=\left\{\begin{array}{ll}0& A=1,2,3\\ \frac{1-R}{x_0}{e}^{\left(-\frac{x_a}{x_0}\right)J\left(\tau \right)}& A=4\end{array}\right.$$E44

$$S_A^T={\mathrm{\omega }}_{\mathit{AF}}{U}_F^i\text{with}{\mathrm{\omega }}_{\mathit{AF}}=\left\{\begin{array}{cc}-D_a& A=\mathrm{1,2,3};F=4\\ \nabla \left({\delta }_{2j}{\mathbb{K}}_{\alpha }^{i^{\ast }}\right)\nabla & \text{otherwise}\end{array}\right.$$E45

$$S_A^u=\psi U_F^i\text{with}\psi =\left\{\begin{array}{ll}{P}^i\left(\frac{\partial }{\partial \mathit{xb}}-\frac{\partial }{\partial x_a}\right)& A=\mathrm{1,2,3};F=\mathrm{1,2,3},\\ 0& A=4;F=4\end{array}\right.$$E46

$$S_A^{\dot{T}}={\mathrm{\Gamma }}_{\mathit{AF}}{\dot{U}}_F^i\text{with}{\mathrm{\Gamma }}_{\mathit{AF}}=\left\{\begin{array}{ll}-{\beta }_{\mathit{ab}}^i{\tau }_1\left(\frac{\partial }{\partial x_b}+\mathrm{\Lambda }\right)\frac{\partial }{\partial \tau }& A=4;F=4\\ {\rho }^i{c}_{\alpha }^i{\delta }_1{\delta }_{1j}& \text{otherwise}\end{array}\right.$$E47

$$S_A^{\ddot{T}}={\delta }_{\mathit{AF}}{\ddot{U}}_F^i\text{with}{\delta }_{\mathit{AF}}=\left\{\begin{array}{ll}0& A=4;F=4\\ {\rho }^i{c}_{\alpha }^i\left[\left({\tau }_0+{\delta }_{1j}{\tau }_2+{\delta }_{2j}\right)\right]& \text{otherwise}\end{array}\right.$$E48

$$S_A^{\ddot{\mathrm{u}}}=õ{\ddot{U}}_F^i\text{with}õ=\left\{\begin{array}{ll}{\rho }^i& A=1,2,3,F=\mathrm{1,2,3},\\ {\beta }_{\mathit{ab}}^i{T}_{\alpha 0}^i\left({\tau }_0+{\delta }_{2i}\right)& A=4;F=4\end{array}\right.$$E49

The thermoelastic representation formula (36) can also be written in matrix form as follows:

$$\begin{array}{c}\left[S_A\right]=-\left[\begin{array}{ll}\hfill & 0\hfill \\ -\frac{1-R}{x_0}\hfill & e\left(-\frac{x_a}{x_0}\right)J\left(\tau \right)\hfill \end{array}\right]+\left\{\left[\begin{array}{c}-D_a{T}_{\alpha }^i\\ \nabla \left({\delta }_{2j}{\mathbb{K}}_{\alpha }^{i^{\ast }}\right)\nabla T_{\alpha }^i\end{array}\right]\right\}\\ +\left[\begin{array}{c}{P}^i\left(u_{b,a}^{i_0}-u_{a,b}^i\right)\\ 0\end{array}\right]+\left[\begin{array}{c}-{\beta }_{ab}^i{\tau }_1\left(\frac{\partial }{\partial x_b}+\text{Λ}\right){\dot{T}}_{\alpha }^i\\ {\rho }^i{c}_{\alpha }^i{\delta }_1{\delta }_{1j}{\dot{T}}_{\alpha }^i\end{array}\right]\\ +{\rho }^i{c}_{\alpha }^i\left[\left({\tau }_0+{\delta }_{1j}{\tau }_2+{\delta }_{2j}\right)\right]\left[\begin{array}{l}0\hfill \\ {\ddot{T}}_{\alpha }^i\hfill \end{array}\right]+\left[\begin{array}{l}{\rho }^i{\ddot{u}}_a^i\hfill \\ {\beta }_{ab}^i{T}_{\alpha 0}^i\left({\tau }_0+{\delta }_{2j}\right){\ddot{u}}_{f,g}^i\hfill \end{array}\right]\end{array}$$E50

To transform the domain integral in (42) to the boundary, we approximate the source vector S_A by a series of given tensor functions $f_{\mathit{AE}}^q$ and unknown coefficients ${\alpha }_E^q$ as follows:

$$S_A\approx \sum _{q=1}^E{f}_{\mathit{AE}}^q{\alpha }_E^q$$E51

Thus, the thermoelastic representation formula (42) can be written in the following form:

$$U_D\left(\xi \right)=\underset{\mathrm{C}}{\int }\left(U_{\mathit{DA}}^{i^{\ast }}{T}_{\mathit{\alpha A}}^i-{\tilde{T}}_{\mathit{\alpha DA}}^{i^{\ast }}{U}_A^i\right)d\mathrm{C}-\sum _{q=1}^N\underset{R}{\int }{U}_{\mathit{DA}}^{i^{\ast }}{f}_{\mathit{AE}}^q\mathit{dR}{\alpha }_E^q$$E52

By implementing the WRM to the following equations.

$$L_{\mathit{gb}}{u}_{\mathit{fe}}^{\mathit{iq}}=f_{\mathit{ae}}^q$$E53

$$L_{\mathit{ab}}{T}_{\alpha }^{\mathit{iq}}=f_{\mathit{pj}}^q$$_,E54

Then, the elastic and thermal representation formulae are given as follows [46]:

$$u_{\mathit{de}}^{\mathit{iq}}\left(\xi \right)=\underset{C}{\int }\left(u_{\mathit{da}}^{i\ast }{t}_{\mathit{ae}}^{\mathit{iq}}-t_{\mathit{da}}^{i\ast }{u}_{\mathit{ae}}^{\mathit{iq}}\right)\mathit{dC}-\underset{R}{\int }{u}_{\mathit{da}}^{i\ast }{f}_{\mathit{ae}}^q\mathit{dR}$$E55

$$T_{\alpha }^{\mathit{iq}}\left(\xi \right)=\underset{C}{\int }\left(q^{i\ast }{T}_{\alpha }^{\mathit{iq}}-q^{\mathit{iq}}{T}_{\alpha }^{i\ast }\right)\mathit{dC}-\underset{R}{\int }{f}^q{T}_{\alpha }^{i\ast }\mathit{dR}$$E56

The representation formulae (55) and (56) can be combined into the following single equation:

$$U_{\mathit{DE}}^{\mathit{iq}}\left(\xi \right)=\underset{C}{\int }\left(U_{\mathit{DA}}^{i\ast }{T}_{\mathit{\alpha AE}}^{\mathit{iq}}-T_{\mathit{\alpha DA}}^{i\ast }{U}_{\mathit{AE}}^{\mathit{iq}}\right)\mathit{dC}-\underset{R}{\int }{U}_{\mathit{DA}}^{i\ast }{f}_{\mathit{AE}}^{\mathit{iq}}\mathit{dR}$$E57

With the substitution of (57) into (52), the dual reciprocity representation formula of coupled thermoelasticity can be expressed as follows:

$$U_D^i\left(\xi \right)=\underset{C}{\int }\left(U_{\mathit{DA}}^{i\ast }{T}_{\mathit{\alpha A}}^i-{\stackrel{⌣}{T}}_{\mathit{\alpha DA}}^{i\ast }{U}_A^i\right)d\mathrm{C}+\sum _{q=1}^E\left(U_{\mathit{DE}}^{\mathit{iq}}\left(\xi \right)+\underset{C}{\int }\left(T_{\mathit{\alpha DA}}^{i\ast }{U}_{\mathit{AE}}^{\mathit{iq}}-U_{\mathit{DA}}^{i\ast }{T}_{\mathit{\alpha AE}}^{\mathit{iq}}\right)\mathit{dC}\right){\alpha }_E^q$$E58

To calculate interior stresses, (58) is differentiated with respect to ${\xi }_l$ as follows:

$$\frac{\partial U_D^i\left(\xi \right)}{\partial {\xi }_l}=-\underset{C}{\int }\left(U_{\mathit{DA},l}^{i\ast }{T}_{\mathit{\alpha A},l}^i-{\stackrel{⌣}{T}}_{\mathit{\alpha DA},l}^{i^{\ast }}{U}_A^i\right)d\mathrm{C}+\sum _{q=1}^E\left(\frac{\partial U_{\mathit{DE}}^{\mathit{iq}}\left(\xi \right)}{\partial {\xi }_l}-\underset{C}{\int }\left(T_{\mathit{\alpha DA},l}^{i\ast }{U}_{\mathit{\alpha AE}}^{\mathit{iq}}-U_{\mathit{DA},l}^{i\ast }{T}_{\mathit{\alpha AE}}^{\mathit{iq}}\right)\mathit{dC}\right){\alpha }_E^q$$E59

According to the dual reciprocity boundary integral equation procedure of Fahmy [44], we can write (58) in the following system of equations:

$$\stackrel{⌣}{\zeta }U-\eta T_{\alpha }=\left(\zeta \stackrel{⌣}{U}-\eta \stackrel{⌣}{\wp }\right)\alpha$$E60

The generalized displacements and velocities are approximated in terms of a series of known tensor functions $f_{\mathit{FD}}^q$ and unknown coefficients ${\gamma }_D^q$ and ${\tilde{\gamma }}_D^q$:

$$U_F^i\approx \sum _{q=1}^N{f}_{\mathit{FD}}^q\left(x\right){\gamma }_D^q$$E61

where

$$f_{\mathit{FD}}^q=\left\{\begin{array}{ll}{f}_{\mathit{fd}}^q& f=F=\mathrm{1,2,3};d=D=\mathrm{1,2,3}\\ f^q& F=4;D=4\\ 0& \text{otherwise}\end{array}\right.$$E62

The gradients of the generalized displacements and velocities can also be approximated in terms of the derivatives of tensor functions as follows:

$$U_{F,g}^i\approx \sum _{q=1}^N{f}_{\mathit{FD},g}^q\left(x\right){\gamma }_K^q$$E63

These approximations are substituted into Eq. (45) to obtain

$$S_A^T=\sum _{q=1}^N{S}_{\mathit{AF}}{f}_{\mathit{FD},g}^q{\gamma }_D^q$$E64

By implementing the point collocation procedure introduced by Gaul et al. [43] to Eqs. (51) and (61), we have

$$\stackrel{⌣}{S}=J\overline{\alpha },U^i=J^{\prime }\gamma ,$$E65

Similarly, the implementation of the point collocation procedure to Eqs. (64), (46), (47), (48), and (49) leads to the following equations:

$${\stackrel{⌣}{S}}^{T_{\alpha }^i}={\mathcal{B}}^T\gamma$$E66

$$S_A^u=\overline{\psi }{U}^i$$E67

$${\stackrel{⌣}{S}}^{{\dot{T}}_{\alpha }^l}={\overline{\mathrm{\Gamma }}}_{\mathit{AF}}{\dot{U}}^i$$E68

$${\stackrel{⌣}{S}}^{{\ddot{T}}_{\dot{\alpha }}^l}={\overline{\delta }}_{\mathit{AF}}{\ddot{U}}^i$$E69

$${\stackrel{⌣}{S}}^{\ddot{u}}=\overline{õ}{\ddot{U}}^i$$E70

where $\overline{\psi }$, ${\overline{\mathrm{\Gamma }}}_{\mathit{AF}}$, ${\overline{\delta }}_{\mathit{AF}}$, and $\overline{õ}$ are assembled using the submatrices ${\left[\psi \right]}_{\text{'}}\left[{\mathrm{\Gamma }}_{\mathit{AF}}\right]$, $\left[{\delta }_{\mathit{AF}}\right]$, and $\left[õ\right]$, respectively.

Solving the system (65) for $\alpha$ and $\gamma$ yields

$$\overline{\alpha }=J^{-1}\stackrel{⌣}{S},\gamma =J^{\text{'}-1}{U}^i$$E71

Now, the coefficients $\alpha$ can be expressed in terms of nodal values of the unknown displacements Uⁱ, velocities ${\dot{U}}^i$, and accelerations ${\ddot{U}}^i$ as follows:

$$\overline{\alpha }=J^{-1}({\stackrel{⌣}{S}}^0+\left({\mathcal{B}}^T{J}^{\text{'}-1}+\overline{\psi }]U^i+{\overline{\mathrm{\Gamma }}}_{\mathit{AF}}{\dot{U}}^i+\left(\overline{õ}+{\overline{\delta }}_{\mathit{AF}}\right){\ddot{U}}^i\right)$$E72

An implicit-implicit staggered algorithm for the integration of the governing equations was developed and implemented for use with the DRBEM for solving the governing equations which may now be written in a more convenient form after substitution of Eq. (72) into Eq. (60) as follows:

$$\overbrace{M}{\ddot{U}}^i+\overbrace{\mathrm{\Gamma }}{\dot{U}}^i+\overbrace{K}{U}^i={\overbrace{\mathbb{Q}}}^i$$E73

$$\overbrace{\mathrm{X}}{\ddot{T}}_{\alpha }^i+\overbrace{\mathrm{A}}{\dot{T}}_{\alpha }^i+\overbrace{\mathrm{B}}{T}_{\alpha }^i=\overbrace{\mathbb{Z}}{\ddot{U}}^i+\overbrace{\mathbb{R}}$$E74

where $V=\left(\eta \stackrel{⌣}{\wp }-\zeta \stackrel{⌣}{U}\right)J^{-1}$, $\overbrace{M}=V\left(\overline{õ}+{\overline{\delta }}_{\mathit{AF}}\right)$, $\overbrace{\mathrm{\Gamma }}=V{\overline{\mathrm{\Gamma }}}_{\mathit{AF}}$, $\overbrace{K}=-\stackrel{⌣}{\zeta }+V\left({\mathcal{B}}^{T}J{\text{'}}^{-1}+\overline{\psi }\right),$${\overbrace{\mathbb{Q}}}^i=-\eta \stackrel{⌣}{T}+V{\stackrel{⌣}{S}}^0$, $\overbrace{\mathrm{X}}=-{\rho }^i{c}^i{\left(x+1\right)}^m$, $\overbrace{\mathrm{A}}=k_{\mathit{ab}}^i\frac{\partial }{\partial x_a}\frac{\partial }{\partial x_b}$, $\overbrace{\mathrm{B}}=k_{\mathit{ab}}^{i^{\ast }}\frac{\partial }{\partial x_a}\frac{\partial }{\partial x_b}$, $\overbrace{\mathbb{Z}}={\beta }_{\mathit{ab}}^i{T}_0$_,$\overbrace{\mathbb{R}}=-\frac{1-R}{x_0}{e}^{\left(\frac{x_a}{x_0}\right)J\left(\tau \right)}$

where V, $\overbrace{M}$, $\overbrace{\mathrm{\Gamma }}$, $\overbrace{K}$, $\overbrace{\mathrm{A}}$, and $\overbrace{\mathrm{B}}$ represent the volume, mass, damping, stiffness, capacity, and conductivity matrices, respectively, and ${\ddot{U}}^i,{\dot{U}}^i,U^i,T^i$, and ${\overbrace{\mathbb{Q}}}^i$ represent the acceleration, velocity, displacement, temperature, and external force vectors, respectively.

In many applications, the coupling term $\overbrace{\mathbb{Z}}{\ddot{U}}_{n+1}^i$ that appears in the heat conduction equation and which is induced by the effect of the strain rate is negligible.

Hence, Eqs. (73) and (74) lead to the following coupled system of differential-algebraic equations (DAEs):

$$\overbrace{M}{\ddot{U}}_{n+1}^i+\overbrace{\mathrm{\Gamma }}{\dot{U}}_{n+1}^i+\overbrace{K}{U}_{n+1}^i={\overbrace{\mathbb{Q}}}_{n+1}^{\mathit{ip}}$$E75

$$\overbrace{\mathrm{X}}{\ddot{T}}_{\alpha \left(n+1\right)}^i+\overbrace{\mathrm{A}}{\dot{T}}_{\alpha \left(n+1\right)}^i+\overbrace{\mathrm{B}}{T}_{\alpha \left(n+1\right)}^i=\overbrace{\mathbb{Z}}{\ddot{U}}_{n+1}^i+\overbrace{\mathbb{R}}$$E76

where ${\overbrace{\mathbb{Q}}}_{n+1}^{i\mathrm{p}}=\eta T_{\alpha \left(n+1\right)}^{\mathit{ip}}+V{\stackrel{⌣}{S}}^0$ and $T_{\alpha \left(n+1\right)}^{\mathit{ip}}$ is the predicted temperature.

Integrating Eq. (73) with the use of trapezoidal rule and Eq. (75), we obtain

$$U_{n+1}^i={\dot{U}}_n^i+\frac{\mathrm{\Delta }\tau }{2}\left({\ddot{U}}_{n+1}^i+{\ddot{U}}_n^i\right)={\dot{U}}_n^i+\frac{\mathrm{\Delta }\tau }{2}\left[{\ddot{U}}_n^i+{\overbrace{M}}^{-1}\left({\overbrace{\mathbb{Q}}}_{n+1}^{\mathit{ip}}-\overbrace{\mathrm{\Gamma }}{\dot{U}}_{n+1}^i-\overbrace{K}{U}_{n+1}^i\right)\right]$$E77

$$U_{n+1}^i=U_n^i+\frac{\mathrm{\Delta }\tau }{2}\left({\dot{U}}_{n+1}^i+{\dot{U}}_n^i\right)=U_n^i+\mathrm{\Delta }\tau {\dot{U}}_n^i+\frac{\mathrm{\Delta }{\tau }^2}{4}\left[{\ddot{U}}_n^i+{\overbrace{M}}^{-1}\left({\tilde{\mathbb{Q}}}_{n+1}^{\mathit{ip}}-\overbrace{\mathrm{\Gamma }}{\dot{U}}_{n+1}^i-\overbrace{K}{U}_{n+1}^i\right)\right]$$E78

From Eq. (77) we have

$${\dot{U}}_{n+1}^i={\overline{\mathrm{\Upsilon }}}^{-1}\left[{\dot{U}}_n^i+\frac{\mathrm{\Delta }\tau }{2}\left[{\ddot{U}}_n^i+{\overbrace{M}}^{-1}\left({\overbrace{\mathbb{Q}}}_{n+1}^{\mathit{ip}}-\overbrace{K}{U}_{n+1}^i\right)\right]\right]$$E79

where $\overline{\mathrm{\Upsilon }}=\left(I+\frac{\mathrm{\Delta }\tau }{2}{\overbrace{M}}^{-1}\overbrace{\mathrm{\Gamma }}\right)$_.

Substituting from Eq. (79) into Eq. (78), we derive

$$\begin{array}{c}{U}_{n+1}^i=U_n^i+\text{Δ}\tau {\dot{U}}_n^i+\frac{\text{Δ}{\tau }^2}{4}[{\ddot{U}}_n^i+{\overbrace{M}}^{-1}({\tilde{\mathbb{Q}}}_{n+1}^{ip}-\overbrace{\text{Γ}}{\overline{\Upsilon }}^{-1}\\ \left[{\dot{U}}_n^i+\frac{\text{Δ}\tau }{2}\right]\left[{\ddot{U}}_n^i+\overbrace{M}{}^{-1}\left({\overbrace{\mathbb{Q}}}_{n+1}^{ip}-\overbrace{K}{U}_{n+1}^i\right)]-\overbrace{K}{U}_{n+1}^i)\right]\end{array}$$E80

Substituting ${\dot{U}}_{n+1}^i$ from Eq. (79) into Eq. (75), we obtain

$$\begin{array}{c}{\ddot{U}}_{n+1}^i={\overbrace{M}}^{-1}[{\overbrace{\mathbb{Q}}}_{n+1}^{ip}-\overbrace{\text{Γ}}[{\overline{\Upsilon }}^{-1}[{\dot{U}}_n^i+\frac{\text{Δ}\tau }{2}\left[{\ddot{U}}_n^i+{\overbrace{M}}^{-1}\left({\overbrace{\mathbb{Q}}}_{n+1}^{ip}-\overbrace{K}{U}_{n+1}^i\right)\right]]]\\ -\overbrace{K}{U}_{n+1}^i]\end{array}$$E81

Integrating the heat Eq. (74) using the trapezoidal rule and Eq. (76), we get

$$\begin{array}{l}{\dot{T}}_{\alpha \left(n+1\right)}^i={\dot{T}}_n^i+\frac{\mathrm{\Delta }\tau }{2}\left({\ddot{T}}_{\alpha \left(n+1\right)}^i+{\ddot{T}}_{\mathit{\alpha n}}^i\right)={\dot{T}}_{\mathit{\alpha n}}^i\\ +\frac{\mathrm{\Delta }\tau }{2}\left({\overbrace{\mathrm{X}}}^{-1}\left[\overbrace{\mathbb{Z}}{\ddot{U}}_{n+1}^i+\overbrace{\mathbb{R}}-\overbrace{\mathrm{A}}{\dot{T}}_{\alpha \left(n+1\right)}^{i^{\mathrm{A}}}-\overbrace{\mathrm{B}}{T}_{\alpha \left(n+1\right)}^i\right]+{\ddot{T}}_{\mathit{\alpha n}}^i\right)\end{array}$$E82

$$\begin{array}{l}{T}_{\alpha \left(n+1\right)}^i=T_{\mathit{\alpha n}}^i+\frac{\mathrm{\Delta }\tau }{2}\left({\dot{T}}_{\alpha \left(n+1\right)}^i+{\dot{T}}_{\mathit{\alpha n}}^i\right)=T_{\mathit{\alpha n}}^i+\mathrm{\Delta }\tau {\dot{T}}_{\mathit{\alpha n}}^i\\ +\frac{\mathrm{\Delta }{\tau }^2}{4}\left({\ddot{T}}_{\mathit{\alpha n}}^i+{\overbrace{\mathrm{X}}}^{-1}\left[\overbrace{\mathbb{Z}}{\ddot{U}}_{n+1}^i+\overbrace{\mathbb{R}}-\overbrace{\mathrm{A}}{\dot{T}}_{\alpha \left(n+1\right)}^i-\overbrace{\mathrm{B}}{T}_{\alpha \left(n+1\right)}^i\right]\right)\end{array}$$E83

From Eq. (82) we get

$${\dot{T}}_{\alpha \left(n+1\right)}^i={\mathrm{\Upsilon }}^{-1}\left[{\dot{T}}_{\mathit{\alpha n}}^i+\frac{\mathrm{\Delta }\tau }{2}\left({\overbrace{\mathrm{X}}}^{-1}\left[\overbrace{\mathbb{Z}}{\ddot{U}}_{n+1}^i+\overbrace{\mathbb{R}}-\overbrace{\mathrm{B}}{T}_{\alpha \left(n+1\right)}^i\right]+{\ddot{T}}_{\mathit{\alpha n}}^i\right)\right]$$E84

where $\mathrm{\Upsilon }=\left(I+\frac{1}{2}\overbrace{\mathrm{A}}\mathrm{\Delta }\tau {\overbrace{\mathrm{X}}}^{-1}\right)$

Substituting from Eq. (84) into Eq. (83), we have

$$\begin{array}{c}{T}_{\alpha \left(n+1\right)}^i=T_{\alpha n}^i+\text{Δ}\tau {\dot{T}}_{\alpha n}^i+\frac{\text{Δ}{\tau }^2}{4}({\ddot{T}}_{\alpha n}^i+{\overbrace{\text{X}}}^{-1}[\overbrace{\mathbb{Z}}{\ddot{U}}_{n+1}^i+\overbrace{ℝ}-\overbrace{\text{A}}({\text{Y}}^{-1}\left[{\dot{T}}_{\alpha n}^i\right]\\ +\frac{\text{Δ}\tau }{2}\left({\overbrace{\text{X}}}^{-1}\left[\overbrace{\mathbb{Z}}{\ddot{U}}_{n+1}^i+\overbrace{ℝ}-\overbrace{\text{B}}{T}_{\alpha \left(n+1\right)}^i\right]+{\ddot{T}}_{\alpha n}^i\right))-\tilde{\text{B}}{T}_{\alpha \left(n+1\right)}^i])\end{array}$$E85

Substituting ${\dot{T}}_{n+1}^i$ from Eq. (84) into Eq. (76), we obtain

$$\begin{array}{c}{\ddot{T}}_{\alpha \left(n+1\right)}{i}^{\text{A}}={\overbrace{\text{X}}}^{-1}[\overbrace{\mathbb{Z}}{\ddot{U}}_{n+1}^i+\overbrace{ℝ}-\overbrace{\text{A}}({\Upsilon }^{-1}[{\dot{T}}_{\alpha n}^i+\frac{\text{Δ}\tau }{2}\\ \left({\overbrace{\text{X}}}^{-1}\left[\overbrace{\mathbb{Z}}{\ddot{U}}_{n+1}^i+\overbrace{ℝ}-\overbrace{\text{B}}{T}_{\alpha \left(n+1\right)}^i\right]+{\ddot{T}}_{\alpha \left(n+1\right)}^i\right)])-*\overbrace{\text{B}}{T}_{\alpha \left(n+1\right)}^i]\end{array}$$E86

Now, a displacement-predicted staggered procedure for the solution of (80) and (85) is as follows:

The first step is to predict the propagation of the displacement wave field: $U_{n+1}^{\mathit{ip}}=U_n^i$. The second step is to substitute ${\dot{U}}_{n+1}^i$ and ${\ddot{U}}_{n+1}^i$ from Eqs. (77) and (75), respectively, in Eq. (85) and solve the resulting equation for the three-temperature wave fields. The third step is to correct the displacement wave propagation using the computed three-temperature fields for Eq. (80). The fourth step is to compute ${\dot{U}}_{n+1}^i$, ${\ddot{U}}_{n+1}^i$, ${\dot{T}}_{\alpha \left(n+1\right)}^i$, and ${\ddot{T}}_{\alpha \left(n+1\right)}^i$ from Eqs. (79), (81), (82), and (86), respectively.

5. Numerical results and discussion

In order to show the numerical results of this study, we consider a monoclinic graphite-epoxy as an anisotropic thermoelastic material which has the following physical constants [57]:

The elasticity tensor is expressed as

$$C_{\mathit{\text{pjkl}}}=\left[\begin{array}{cccccc}430.1& 130.4& 18.2& 0& 0& 201.3\\ 130.4& 116.7& 21.0& 0& 0& 70.1\\ 18.2& 21.0& 73.6& 0& 0& 2.4\\ 0& 0& 0& 19.8& -8.0& 0\\ 0& 0& 0& -8.0& 29.1& 0\\ 201.3& 70.1& 2.4& 0& 0& 147.3\end{array}\right]\mathrm{GPa}$$E87

The mechanical temperature coefficient is

$${\beta }_{\mathit{pj}}=\left[\begin{array}{lll}1.01& 2.00& 0\\ 2.00& 1.48& 0\\ 0& 0& 7.52\end{array}\right]\cdot {10}^6\frac{\mathrm{N}}{{\mathrm{Km}}^2}$$E88

The thermal conductivity tensor is

$$k_{\mathit{pj}}=\left[\begin{array}{lll}5.2& 0& 0\\ 0& 7.6& 0\\ 0& 0& 38.3\end{array}\right]\mathrm{W}/\mathrm{Km}$$E89

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 range of laser wave propagations in three-temperature nonlinear generalized thermoelastic problems of FGA structures. The main aim of this paper was to assess the impact of three temperatures on the acoustic displacement waves; the numerical outcomes are completed and delineated graphically for electron, ion, phonon, and total temperatures.

Figure 2 shows the three temperatures $T_e$, $T_i$, and $T_p$ and total temperature $T\left(T=T_e+T_i+T_p\right)$ wave propagation along the x-axis. It was shown from this figure that the three temperatures are different and they may have great effects on the connected fields.

Figure 2.

Propagation of the temperature T_e, T_i, T_p and T waves along the x-axis.

Figures 3 and 4 show the displacement u₁ and u₂ acoustic waves propagation along x-axis for the three temperatures T_e, T_i, T_p and total temperature T. It was noticed from Figures 3 and 4 that the three temperatures and total temperature have great effects on the acoustic displacement waves.

Figure 3.

Propagation of the displacement $u_1$ waves along the $\mathit{x}$-axis.

Figure 4.

Propagation of the displacement u₂ waves along the x-axis.

In order to evaluate the influence of the functionally graded parameter and initial stress on the propagation of the displacement waves u₁ and u₂ along the x-axis, the numerical results are presented graphically, as shown in Figures 5 and 6. These results are compared for different values of initial stress parameter and functionally graded parameter according to the following cases, A, B, C, and D, where A represents the numerical results for homogeneous $\left(m=0\right)$ structures in the absence of initial stress ($P=0$), B represents the numerical results for functionally graded $\left(m=0.5\right)$ structures in the absence of initial stress ($P=0$), C represents the numerical results for homogeneous $\left(m=0\right)$ structures in the presence of initial stress ($P=0.5$), and D represents the numerical results for functionally graded $\left(m=0.5\right)$ structures in the presence of initial stress ($P=0.5$). It can be seen from Figures 5 and 6 that the effects of initial stress and functionally graded parameter are very pronounced.

Figure 5.

Propagation of the displacement u₁ waves along the x-axis.

Figure 6.

Propagation of the displacement u₂ waves along the x-axis.

Since there are no available results for the three-temperature thermoelastic problem, except for Fahmy’s research [10, 11, 12, 13, 14]. For comparison purposes with the special cases of other methods treated by other authors, we only considered a one-dimensional special case of nonlinear generalized magneto-thermoelastic of anisotropic structure [11, 12] as a special case of the considered problem. In the special case under consideration, the temperature and displacement wave propagation results are plotted in Figures 7 and 8. The validity and accuracy of our proposed BEM technique was demonstrated by comparing graphically the BEM results for the considered problem with those obtained using the finite difference method (FDM) of Pazera and Jędrysiak [68] and finite element method (FEM) of Xiong and Tian [69] results based on replacing heat conduction with three-temperature heat conduction; it can be noticed that the BEM results are found to agree very well with the FDM or FEM results.

Figure 7.

Propagation of the temperature T waves along the x-axis for BEM, FDM, and FEM.

Figure 8.

Propagation of the displacement u₁ waves along the x-axis for BEM, FDM, and FEM.

6. Conclusion

Propagation of displacements and temperature acoustic waves in three-temperature nonlinear generalized magneto-thermoelastic ISMFGA structures has been studied, where we used the three-temperature nonlinear radiative heat conduction equations combined with electron, ion, and phonon temperatures. The BEM results of the considered model show the differences between electron, ion, phonon, and total temperature distributions within the ISMFGA structures. The effects of electron, ion, phonon, and total temperatures on the propagation of acoustic displacement waves have been investigated. Also, the effects of functionally graded parameter and initial stress on the propagation of acoustic displacement waves have been established. Since there are no available results for comparison, except for the one-temperature heat conduction problems, we considered the one-dimensional special case of our general model based on replacing three-temperature radiative heat conductions with one-temperature heat conduction for the verification and demonstration of the considered model results. In the considered special case, the BEM results have been compared graphically with the FDM and FEM, and it can be noticed that the BEM results are in excellent agreement with the FDM and FEM results.

Nowadays, knowledge and understanding of the propagation of acoustic waves of three-temperature nonlinear generalized magneto-thermoelasticity theory can be utilized by computer scientists and engineers, geotechnical and geothermal engineers, material science researchers and designers, and mechanical engineers for designing heat exchangers, semiconductor nanomaterials, and boiler tubes, as well as for chemists to observe the chemical reaction processes such as bond forming and bond breaking. In the application of three-temperature theories in advanced manufacturing technologies, with the development of soft machines and robotics in biomedical engineering and advanced manufacturing, acoustic displacement waves will be encountered more often where three-temperature nonlinear generalized magneto-thermoelasticity theory will turn out to be the best choice for thermomechanical analysis in the design and analysis of advanced ISMFGA structures using laser ultrasonics.