**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 [1–3]. 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 [10–14]. 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) [18–31], and other developed techniques [32–36]. The boundary element method [37–67] is actualized effectively for tackling a few designing and logical applications because of its straightforwardness, precision, and simplicity of

*A New BEM for Modeling of Acoustic Wave Propagation in Three-Temperature Nonlinear…*

*DOI: http://dx.doi.org/10.5772/intechopen.92784*

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

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

Consider a multilayered structure with *n* functionally graded layers in the *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

which occupies the region *R* ¼ f g ð Þ *x*, *y*, *z* : 0< *x*< *h*, 0< *y*<*b*, 0 <*z*<*a* has been placed in a primary magnetic field *H*<sup>0</sup> acting in the direction of the *y*-axis. 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

*<sup>σ</sup>ab*,*<sup>b</sup>* <sup>þ</sup> *<sup>τ</sup>ab*,*<sup>b</sup>* � <sup>Γ</sup>*ab* <sup>¼</sup> *<sup>ρ</sup><sup>i</sup>*

, respectively. The considered multilayered structure

*<sup>a</sup>* (1)

ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *mu*€*<sup>i</sup>*

homogeneous and functionally graded structures.

**2. Formulation of the problem**

*i*th layer is denoted by *h* and *h<sup>i</sup>*

in the following form:

**131**

execution.

FEM results.

#### *A New BEM for Modeling of Acoustic Wave Propagation in Three-Temperature Nonlinear… DOI: http://dx.doi.org/10.5772/intechopen.92784*

The mathematical foundations of three-temperature thermoelasticity were defined for the first time by Fahmy [10–14]. 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) [18–31], and other developed techniques [32–36]. The boundary element method [37–67] 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 *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<sup>i</sup>* , respectively. The considered multilayered structure which occupies the region *R* ¼ f g ð Þ *x*, *y*, *z* : 0< *x*< *h*, 0< *y*<*b*, 0 <*z*<*a* has been placed in a primary magnetic field *H*<sup>0</sup> acting in the direction of the *y*-axis.

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:

$$
\sigma\_{ab,b} + \pi\_{ab,b} - \Gamma\_{ab} = \rho^i (\mathfrak{x} + \mathfrak{1})^m \ddot{u}\_a^i \tag{1}
$$

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 [1–3]. 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.

*Noise and Environment*

**130**

$$\sigma\_{ab} = (\chi + \mathbf{1})^m \left[ \mathbf{C}\_{ab\text{fg}}^i u\_{f\text{g}}^i - \beta\_{ab}^i \left( T^i - T\_0 + \tau\_1 \dot{T}^i \right) \right] \tag{2}$$

$$\pi\_{ab} = \mu^i (\mathfrak{x} + \mathfrak{1})^m \left( \tilde{h}\_a H\_b + \tilde{h}\_b H\_a - \delta\_{ba} \left( \tilde{h}\_f H\_f \right) \right) \tag{3}$$

$$
\Gamma\_{ab} = P^i (\varkappa + 1)^m \left( \frac{\partial u^i\_a}{\partial \varkappa\_b} - \frac{\partial u^i\_b}{\partial \varkappa\_a} \right) \tag{4}
$$

The total energy of unit mass can be described by

*DOI: http://dx.doi.org/10.5772/intechopen.92784*

*<sup>P</sup>* <sup>¼</sup> *Pe* <sup>þ</sup> *PI* <sup>þ</sup> *Pp*, *Pe* <sup>¼</sup> *<sup>c</sup>αeT<sup>i</sup>*

netic stress tensor, and displacement vector, respectively; *T<sup>i</sup>*

*<sup>x</sup>*<sup>0</sup> *e xa x*0 *<sup>J</sup>*ð Þ*<sup>τ</sup>*

*<sup>α</sup>* is the temperature; *C<sup>i</sup>*

where *σab*, *τab*, and *ui*

the current study that *<sup>τ</sup>ab*,*<sup>b</sup>* <sup>¼</sup> *<sup>μ</sup><sup>i</sup>*

energy intensity, and *Q x*ð Þ¼ , *<sup>τ</sup>* <sup>1</sup>�*<sup>R</sup>*

and Naghdi theory of type III; when *<sup>i</sup>*

reduced to the GN theory type II, and when *<sup>i</sup>* <sup>∗</sup>

**3. Continuity and initial and boundary conditions**

displacement, and traction can be expressed as follows:

*Ti*

*qi*

*ui*

*t i <sup>a</sup>*ð Þ *x*, *z*, *τ*

*<sup>f</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>z</sup>*, 0 *<sup>u</sup>*\_ *<sup>i</sup>*

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

*ui*

*ta* ¼ *σabnb*, and *i* ¼ 1, 2, … , *n* � 1.

*ui*

*t i*

*Ti*

*Ti*

**133**

*qi*

*<sup>f</sup>*ð Þ *x*, *z*, *τ*

theory are reduced to the GN theory type I.

*τ*2 3 *e τ*

temperature; *T<sup>i</sup>*

, and *c<sup>i</sup>*

and *<sup>J</sup>*ð Þ¼ *<sup>τ</sup> <sup>J</sup>*0*<sup>τ</sup>*

*Pi* , *ρ<sup>i</sup>* *<sup>e</sup>*, *PI* <sup>¼</sup> *<sup>c</sup>αIT<sup>i</sup>*

*<sup>α</sup>* are the magnetic permeability, perturbed magnetic field, initial stress

*<sup>τ</sup>*<sup>3</sup> is the temporal profile of a non-Gaussian laser pulse, *J*<sup>0</sup> is the total

*abfg* and *β<sup>i</sup>*

elastic moduli and stress-temperature coefficients of the anisotropic medium; *μ<sup>i</sup>*

*A New BEM for Modeling of Acoustic Wave Propagation in Three-Temperature Nonlinear…*

in the *i*th layer, density, and specific heat capacity, respectively; *τ* is the time; *τ*0, *τ*1, and *τ*<sup>2</sup> are the relaxation times; *i* ¼ 1, 2, … , *n* � 1 represents the parameters in a multilayered structure; and *m* is a dimensionless constant. Also, we considered in

According to Fahmy [57], we notice that there are two special cases of the Green

The continuity conditions along interfaces for the temperature, heat flux,

*<sup>α</sup>*ð Þ *<sup>x</sup>*, *<sup>z</sup>*, *<sup>τ</sup> <sup>x</sup>*¼*h<sup>i</sup>* <sup>¼</sup> *<sup>T</sup>*ð Þ *<sup>i</sup>*þ<sup>1</sup> *<sup>α</sup>* <sup>ð</sup>*x*, *ztτ*Þj

ð Þ *<sup>x</sup>*, *<sup>z</sup>*, *<sup>τ</sup> <sup>x</sup>*¼*h<sup>i</sup>* <sup>¼</sup> *<sup>q</sup>*ð Þ *<sup>i</sup>*þ<sup>1</sup> <sup>ð</sup>*x*, *<sup>z</sup>*, *<sup>τ</sup>*Þj

*<sup>x</sup>*¼*h<sup>i</sup>* <sup>¼</sup> *<sup>u</sup>*ð Þ *<sup>i</sup>*þ<sup>1</sup>

ð Þ *i*þ1

where *n* is the total number of layers, *ta* are the tractions, which are defined by

The remaining initial and boundary conditions for the current study are

 

 *<sup>x</sup>*¼*h<sup>i</sup>* <sup>¼</sup> *<sup>t</sup>* *l* , *Pp* <sup>¼</sup> <sup>1</sup> 4 *cαpT*<sup>4</sup>*<sup>i</sup>*

<sup>0</sup>ϵ*abfJbH <sup>f</sup>* is the *a*-component of the Lorentz force

*<sup>k</sup>* are the mechanical stress tensor, Maxwell's electromag-

*<sup>p</sup>* (9)

, ~ *h*,

*<sup>α</sup>*<sup>0</sup> is the reference

*ab* are, respectively, the constant

, *a* ¼ 1, 2, 3 is the heat source intensity.

*<sup>α</sup>* ! 0, the equations of the GN III

*<sup>α</sup>* ! 0, the equations of GN III theory are

*<sup>x</sup>*¼*h<sup>i</sup>*

*<sup>x</sup>*¼*h<sup>i</sup>*

(10)

(11)

*<sup>f</sup>*ð Þ¼ *x*, *z*, 0 0 for ð Þ *x*, *z* ∈*R* ∪*C* (14)

*<sup>α</sup>*ð Þ¼ *x*, *z*, 0 0 for ð Þ *x*, *z* ∈ *R* ∪*C* (17)

*<sup>f</sup>*ð Þ¼ *x*, *z*, *τ* Ψ *<sup>f</sup>*ð Þ *x*, *z*, *τ* for ð Þ *x*, *z* ∈*C*<sup>3</sup> (15)

*<sup>a</sup>*ð Þ¼ *x*, *z*, *τ* Φ *<sup>f</sup>*ð Þ *x*, *z*, *τ* for ð Þ *x*, *z* ∈*C*4, *τ* >0, (16)

*<sup>α</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>τ</sup> f x*, y, *<sup>τ</sup>* for ð Þ *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup>*C*1, *<sup>τ</sup>* <sup>&</sup>gt;<sup>0</sup> (18)

ð Þ¼ *x*, *z*, *τ h x*ð Þ , *z*, *τ* for ð Þ *x*, *z* ∈*C*2, *τ* >0 (19)

*<sup>f</sup>* ð Þ *<sup>x</sup>*, *<sup>z</sup>*, *<sup>τ</sup> <sup>x</sup>*¼*h<sup>i</sup>* (12)

*<sup>a</sup>* ð Þ *<sup>x</sup>*, *<sup>z</sup>*, *<sup>τ</sup> <sup>x</sup>*¼*h<sup>i</sup>* (13)

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

$$\nabla \left[ \left( \delta\_{\mathbf{l}\dot{\jmath}} \mathbb{K}\_a^{i\*} + \delta\_{\dot{\jmath}\dot{\jmath}} \mathbb{K}\_a^i \right) \nabla T\_a^i(r, \tau) \right] - \overline{\mathbf{W}}(r, \tau) = c\_a^i \rho^i \delta\_{\mathbf{l}} \delta\_{\mathbf{l}\dot{\jmath}} \frac{\partial T\_a^i(r, \tau)}{\partial \tau} \tag{5}$$

where

$$
\overline{\mathbb{W}}(\boldsymbol{r},\boldsymbol{\pi}) = \begin{cases}
\rho^{i} \mathbb{W}\_{el} \left( T\_{\epsilon}^{i} - T\_{1}^{i} \right) + \rho^{i} \mathbb{W}\_{er} \left( T\_{\epsilon}^{i} - T\_{p}^{i} \right) + \overline{\overline{\mathbb{W}}}, a = \boldsymbol{e}, & \delta\_{1} = 1 \\
\end{cases} \tag{6}
$$

in which

$$\overline{\mathbf{W}}(\tau,\tau) = -\delta\_{\vec{2}\vec{\ell}} \mathbb{K}\_{a}^{i} \dot{T}\_{a,ab} + \beta\_{ab} T\_{a0} \left[ \left( \tau\_{0} + \delta\_{\vec{2}} \right) \ddot{u}\_{a,b} \right] + \rho^{i} \mathbf{c}\_{a}^{i} \left[ \left( \tau\_{0} + \delta\_{\vec{1}} \tau\_{2} + \delta\_{\vec{2}} \right) \ddot{T}\_{a} \right] - Q(\mathbf{x},\tau) \tag{7}$$

and

$$\begin{aligned} \mathbb{W}\_{el} &= \rho^i \mathbb{A}\_{el} T\_\epsilon^{-2/3}, \mathbb{W}\_{er} = \rho^i \mathbb{A}\_{er} T\_\epsilon^{-1/2}, \mathbb{K}\_a = \mathbb{A}\_a T\_a^{5/2}, a = e, I, \\ \mathbb{K}\_p &= \mathbb{A}\_p T\_p^{3+\mathbb{B}} \end{aligned} \tag{8}$$

**Figure 1.** *Geometry of the FGA structure.*

*A New BEM for Modeling of Acoustic Wave Propagation in Three-Temperature Nonlinear… DOI: http://dx.doi.org/10.5772/intechopen.92784*

The total energy of unit mass can be described by

*<sup>σ</sup>ab* <sup>¼</sup> ð Þ *<sup>χ</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup> Ci*

*<sup>τ</sup>ab* <sup>¼</sup> *<sup>μ</sup><sup>i</sup>*

*<sup>α</sup>* <sup>þ</sup> *<sup>δ</sup>*2*<sup>j</sup><sup>i</sup>*

*<sup>e</sup>* � *<sup>T</sup><sup>i</sup> I* � � <sup>þ</sup> *<sup>ρ</sup><sup>i</sup>er <sup>T</sup><sup>i</sup>*

*<sup>e</sup>* � *<sup>T</sup><sup>i</sup> p* � �

*<sup>e</sup>* � *<sup>T</sup><sup>i</sup> p* � �

*<sup>α</sup>T*\_ *<sup>α</sup>*,*ab* <sup>þ</sup> *<sup>β</sup>abT<sup>α</sup>*<sup>0</sup> *<sup>τ</sup>*<sup>0</sup> <sup>þ</sup> *<sup>δ</sup>*2*<sup>j</sup>*

*eIT*�2*=*<sup>3</sup> *<sup>e</sup>* ,*er* <sup>¼</sup> *<sup>ρ</sup><sup>i</sup>*

� �*u*€*<sup>a</sup>*,*<sup>b</sup>* � � <sup>þ</sup> *<sup>ρ</sup><sup>i</sup>*

*eI T<sup>i</sup>*

*er T<sup>i</sup>*

� �∇*T<sup>i</sup>*

*α*

*<sup>α</sup>*ð Þ *<sup>r</sup>*, *<sup>τ</sup>* � � � ð Þ¼ *<sup>r</sup>*, *<sup>τ</sup> <sup>c</sup>*

expressed as follows:

*Noise and Environment*

where

ð Þ¼ *r*, *τ*

in which

<sup>W</sup>ð Þ¼� *<sup>r</sup>*, *<sup>τ</sup> <sup>δ</sup>*2*<sup>j</sup><sup>i</sup>*

and

**Figure 1.**

**132**

*Geometry of the FGA structure.*

∇ *δ*1*<sup>j</sup><sup>i</sup>* <sup>∗</sup>

*ρi eI T<sup>i</sup>*

8 >>>><

>>>>:

�*ρ<sup>i</sup>*

�*ρ<sup>i</sup>*

*eI* <sup>¼</sup> *<sup>ρ</sup><sup>i</sup>*

*<sup>p</sup>* <sup>¼</sup> *pT*<sup>3</sup>þ *p*

*abfgui*

ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup>* <sup>~</sup>

<sup>Γ</sup>*ab* <sup>¼</sup> *<sup>P</sup><sup>i</sup>*

*<sup>f</sup>*,*<sup>g</sup>* � *<sup>β</sup><sup>i</sup>*

*haHb* <sup>þ</sup> <sup>~</sup>

ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup> <sup>∂</sup>u<sup>i</sup>*

According to Fahmy [10], the *2D-3 T radiative heat conduction equations* can be

*<sup>e</sup>* � *<sup>T</sup><sup>i</sup> p* � �

*ab <sup>T</sup><sup>i</sup>* � *<sup>T</sup>*<sup>0</sup> <sup>þ</sup> *<sup>τ</sup>*1*T*\_ *<sup>i</sup>* h i � �

*hbHa* � *<sup>δ</sup>ba* <sup>~</sup>

� *∂ui b ∂xa* � �

> *i αρi δ*1*δ*1*<sup>j</sup> ∂T<sup>i</sup> <sup>α</sup>*ð Þ *r*, *τ*

þ , *α* ¼ *I*, *δ*<sup>1</sup> ¼ 1

<sup>þ</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>

*c i*

*erT*�1*=*<sup>2</sup> *<sup>e</sup>* , *<sup>α</sup>* <sup>¼</sup> *αT*<sup>5</sup>*=*<sup>2</sup>

þ , *α* ¼ *e*, *δ*<sup>1</sup> ¼ 1

*<sup>α</sup> τ*<sup>0</sup> þ *δ*1*<sup>j</sup>τ*<sup>2</sup> þ *δ*2*<sup>j</sup>* � �*T*€*<sup>α</sup>*

� � � *Q x*ð Þ , *<sup>τ</sup>*

*<sup>α</sup>* , *α* ¼ *e*,*I*,

� � � �

*a ∂xb* *h fH <sup>f</sup>*

(2)

(3)

(4)

*p*

(6)

(7)

(8)

*<sup>∂</sup><sup>τ</sup>* (5)

$$P = P\_e + P\_I + P\_p,\ P\_e = c\_{at}T\_e^i,\ P\_I = c\_{al}T\_l^i,\ P\_p = \frac{1}{4}c\_{ap}T\_p^{4i} \tag{9}$$

where *σab*, *τab*, and *ui <sup>k</sup>* are the mechanical stress tensor, Maxwell's electromagnetic stress tensor, and displacement vector, respectively; *T<sup>i</sup> <sup>α</sup>*<sup>0</sup> is the reference temperature; *T<sup>i</sup> <sup>α</sup>* is the temperature; *C<sup>i</sup> abfg* and *β<sup>i</sup> ab* are, respectively, the constant elastic moduli and stress-temperature coefficients of the anisotropic medium; *μ<sup>i</sup>* , ~ *h*, *Pi* , *ρ<sup>i</sup>* , and *c<sup>i</sup> <sup>α</sup>* are the magnetic permeability, perturbed magnetic field, initial stress in the *i*th layer, density, and specific heat capacity, respectively; *τ* is the time; *τ*0, *τ*1, and *τ*<sup>2</sup> are the relaxation times; *i* ¼ 1, 2, … , *n* � 1 represents the parameters in a multilayered structure; and *m* is a dimensionless constant. Also, we considered in the current study that *<sup>τ</sup>ab*,*<sup>b</sup>* <sup>¼</sup> *<sup>μ</sup><sup>i</sup>* <sup>0</sup>ϵ*abfJbH <sup>f</sup>* is the *a*-component of the Lorentz force and *<sup>J</sup>*ð Þ¼ *<sup>τ</sup> <sup>J</sup>*0*<sup>τ</sup> τ*2 3 *e τ <sup>τ</sup>*<sup>3</sup> is the temporal profile of a non-Gaussian laser pulse, *J*<sup>0</sup> is the total

energy intensity, and *Q x*ð Þ¼ , *<sup>τ</sup>* <sup>1</sup>�*<sup>R</sup> <sup>x</sup>*<sup>0</sup> *e xa x*0 *<sup>J</sup>*ð Þ*<sup>τ</sup>* , *a* ¼ 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 *<sup>i</sup> <sup>α</sup>* ! 0, the equations of GN III theory are reduced to the GN theory type II, and when *<sup>i</sup>* <sup>∗</sup> *<sup>α</sup>* ! 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:

$$\left.T\_a^i(\varkappa, z, \mathfrak{r})\right|\_{\mathfrak{x} = \mathfrak{h}^i} = \left.T\_a^{(i+1)}(\varkappa, z\_t \mathfrak{r})\right|\_{\mathfrak{x} = \mathfrak{h}^i} \tag{10}$$

$$q^i(\mathbf{x}, z, \mathfrak{r})|\_{\mathfrak{x} = \mathfrak{h}^i} = q^{(i+1)}(\mathfrak{x}, z, \mathfrak{r})|\_{\mathfrak{x} = \mathfrak{h}^i} \tag{11}$$

$$\left.u\_f^i(\mathbf{x}, \mathbf{z}, \mathbf{z})\right|\_{\mathbf{x}=h^i} = \left.u\_f^{(i+1)}(\mathbf{x}, \mathbf{z}, \mathbf{z})\right|\_{\mathbf{x}=h^i} \tag{12}$$

$$\overline{t}\_a^i(\infty, z, \mathfrak{r})\Big|\_{\mathfrak{x}=\mathfrak{h}^i} = \overline{t}\_a^{(i+1)}(\infty, \mathfrak{z}, \mathfrak{r})\_{\mathfrak{x}=\mathfrak{h}^i} \tag{13}$$

where *n* is the total number of layers, *ta* are the tractions, which are defined by *ta* ¼ *σabnb*, and *i* ¼ 1, 2, … , *n* � 1.

The remaining initial and boundary conditions for the current study are

$$u^i\_f(\mathbf{x}, z, \mathbf{0}) = \dot{u}^i\_f(\mathbf{x}, z, \mathbf{0}) = \mathbf{0} \quad \text{for } (\mathbf{x}, z) \in \mathbb{R} \cup \mathbb{C} \tag{14}$$

$$u\_f^i(\mathbf{x}, z, \tau) = \Psi\_f(\mathbf{x}, z, \tau) \text{ for } (\mathbf{x}, z) \in \mathbb{C}\_3 \tag{15}$$

$$\vec{t}\_a(\varkappa, z, \tau) = \Phi\_f(\varkappa, z, \tau) \text{ for } (\varkappa, z) \in \mathbb{C}\_4, \tau > 0,\tag{16}$$

$$T\_a^i(\mathbf{x}, \mathbf{z}, \mathbf{0}) = T\_a^i(\mathbf{x}, \mathbf{z}, \mathbf{0}) = \mathbf{0} \text{ for } (\mathbf{x}, \mathbf{z}) \in \mathbb{R} \cup \mathbb{C} \tag{17}$$

$$T\_a^\sharp(\mathbf{x}, \mathbf{y}, \boldsymbol{\tau}) = \overline{f}(\mathbf{x}, \mathbf{y}, \boldsymbol{\tau}) \text{ for } (\mathbf{x}, \mathbf{y}) \in \mathcal{C}\_1, \quad \boldsymbol{\tau} > \mathbf{0} \tag{18}$$

$$q^i(\mathbf{x}, \mathbf{z}, \tau) = \overline{h}(\mathbf{x}, \mathbf{z}, \tau) \text{ for } (\mathbf{x}, \mathbf{z}) \in \mathcal{C}\_2, \tau > \mathbf{0} \tag{19}$$

where Ψ *<sup>f</sup>* , Φ *<sup>f</sup>* , *f*, and *h* are suitably prescribed functions and *C* ¼ *C*<sup>1</sup> ∪*C*<sup>2</sup> ¼ *C*<sup>3</sup> ∪*C*4, *C*<sup>1</sup> ∩*C*<sup>2</sup> ¼ *C*<sup>3</sup> ∩*C*<sup>4</sup> ¼ Ø.

## **4. BEM numerical implementation**

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

$$L\_{gb}u\_f^i = \rho^i \ddot{u}\_a^i - \left(D\_a T\_a^i - P^i \left(\frac{\partial u\_b^i}{\partial \mathbf{x}\_a} - \frac{\partial u\_a^i}{\partial \mathbf{x}\_b}\right)\right) = f\_{gb} \tag{20}$$

where the inertia term *ρu*€*a*, the temperature gradient *DaT*, and the initial stress term are treated as the body forces.

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

$$L\_{\mathfrak{g}^b} u^i\_f = f\_{\mathfrak{g}^b},\tag{21}$$

*ui <sup>d</sup>*ð Þ¼ *ξ*

The fundamental solution *T<sup>i</sup>* <sup>∗</sup>

ð

*LabT<sup>i</sup> αTi* ∗ *<sup>α</sup>* � *LabT<sup>i</sup>* <sup>∗</sup>

> *Ti <sup>α</sup>*ð Þ¼ *ξ*

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

� *t i* ∗ *da* �*ui* <sup>∗</sup>

> *ui* <sup>∗</sup> *da* 0 <sup>0</sup> �*T<sup>i</sup>* <sup>∗</sup> *α*

ð

*qi* ∗ *Ti <sup>α</sup>* � *<sup>q</sup><sup>i</sup> Ti* ∗ *α* � �*dC* �

*aa βabnb*

*C*

( " # *ui*

" # *<sup>f</sup> gb*

*Ui*

T*i <sup>α</sup><sup>A</sup>* <sup>¼</sup> *<sup>t</sup>*

*ui* <sup>∗</sup>

8 >>><

>>>:

8 >>><

>>>:

�*T<sup>i</sup>* <sup>∗</sup>

�*u*~*<sup>i</sup>* <sup>∗</sup>

*t i* ∗

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

(

*Ti*

*i*

(

<sup>0</sup> �*q<sup>i</sup>* <sup>∗</sup>

*R*

where

*ui <sup>d</sup>*ð Þ*ξ Ti <sup>α</sup>*ð Þ*ξ*

" #

form as follows:

**135**

¼ ð

*C*

� ð

*R*

*U<sup>i</sup>* <sup>∗</sup> *DA* ¼

*T*~*i* ∗ *<sup>α</sup>DA* ¼

representation formula:

ð

*DOI: http://dx.doi.org/10.5772/intechopen.92784*

*ui* <sup>∗</sup> *da t i <sup>a</sup>* � *t i* ∗ *daui*

*<sup>a</sup>* <sup>þ</sup> *ui* <sup>∗</sup> *da β<sup>i</sup> abT<sup>i</sup> <sup>α</sup>nb*

*A New BEM for Modeling of Acoustic Wave Propagation in Three-Temperature Nonlinear…*

� �*dC* �

can be defined as

ð

*C qi* ∗ *Ti <sup>α</sup>* � *<sup>q</sup><sup>i</sup> Ti* ∗ *α* � �*dC* (32)

*LabT<sup>i</sup>* <sup>∗</sup>

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

*<sup>α</sup> T<sup>i</sup> α*

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

*qi* <sup>∗</sup> ¼ �*<sup>i</sup>*

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

*αTi*

*a Ti α*

þ

*<sup>a</sup> a* ¼ *A* ¼ 1, 2, 3

*<sup>a</sup> a* ¼ *A* ¼ 1, 2, 3

*da d* ¼ *D* ¼ 1, 2, 3; *a* ¼ *A* ¼ 1, 2, 3

*aa d* ¼ *D* ¼ 1, 2, 3; *a* ¼ *A* ¼ 1, 2, 3

*<sup>d</sup> d* ¼ *D* ¼ 1, 2, 3; *A* ¼ 4 0 *D* ¼ 4; *a* ¼ *A* ¼ 1, 2, 3

" #

The generalized thermoelastic vectors can be expressed in contracted notation

*<sup>α</sup> A* ¼ 4

*<sup>q</sup><sup>i</sup> <sup>A</sup>* <sup>¼</sup> <sup>4</sup>

0 *d* ¼ *D* ¼ 1, 2, 3; *A* ¼ 4 0 *D* ¼ 4; *a* ¼ *A* ¼ 1, 2, 3

*<sup>α</sup> D* ¼ 4; *A* ¼ 4

�*q<sup>i</sup>* <sup>∗</sup> *<sup>D</sup>* <sup>¼</sup> 4; *<sup>A</sup>* <sup>¼</sup> <sup>4</sup>

*αT<sup>i</sup>* <sup>∗</sup>

ð

*<sup>f</sup> abT<sup>i</sup>* <sup>∗</sup>

" # *τ<sup>i</sup>*

�*<sup>f</sup> ab* " #*dR* (36)

*R*

*ui* <sup>∗</sup> *da* 0 <sup>0</sup> �*T<sup>i</sup>* <sup>∗</sup> *α*

� �*dR* <sup>¼</sup>

ð

*f gbui* <sup>∗</sup>

*dadR* (30)

*R*

¼ �*δ*ð Þ *x*, *ξ* (31)

*<sup>α</sup>:bna* (33)

*<sup>α</sup>*,*bna* (34)

*<sup>α</sup> dR* (35)

*dC*

(37)

(38)

(39)

(40)

*a qi* " #)

*C*

$$L\_{ab}T^i\_{\;\;a} = f\_{\;\;ab} \tag{22}$$

where the operators *Lgb*, *f gb*, *Lab*, and *f ab* are as follows:

$$L\_{\rm gb} = D\_{\rm abf} \frac{\partial}{\partial \mathbf{x}\_b} + D\_{\rm af} + \Lambda D\_{\rm aff}, \qquad L\_{\rm ab} = \left(\delta\_{\mathbf{\dot{\beta}}} \mathbb{K}\_a^{i \ast} \right) \nabla \tag{23}$$

$$f\_{gb} = \rho^i \ddot{u}\_a^i - \left(D\_a T\_a^i - P^i \left(\frac{\partial u\_b^i}{\partial \mathbf{x}\_a} - \frac{\partial u\_a^i}{\partial \mathbf{x}\_b}\right)\right) \tag{24}$$

$$f\_{\phantom{a}ab} = \nabla \left(\delta\_{\mathbb{I}\dot{\jmath}} \mathbb{K}\_a^i \right) \nabla + \rho^i c\_a^i \delta\_{\mathbb{I}\dot{\vartheta}} \delta\_{\mathbb{I}\dot{\jmath}} (\varkappa + \mathbf{1})^m \dot{T}\_a^{\dot{\jmath}} + \overline{\mathbb{W}}(r, \tau) \tag{25}$$

where

$$\begin{split} D\_{\mathrm{abf}} &= \mathcal{C}\_{\mathrm{abfg}} \varepsilon, \varepsilon = \frac{\partial}{\partial \mathbf{x}\_g}, D\_{\mathrm{af}} = \mu H\_0^2 \left( \frac{\partial}{\partial \mathbf{x}\_a} + \delta\_{\mathrm{al}1} \Lambda \right) \frac{\partial}{\partial \mathbf{x}\_f}, \\ D\_a &= -\beta\_{\mathrm{ab}}^i \left( \frac{\partial}{\partial \mathbf{x}\_b} + \delta\_{\mathrm{b1}} \Lambda + \tau\_1 \left( \frac{\partial}{\partial \mathbf{x}\_b} + \Lambda \right) \frac{\partial}{\partial \tau} \right), \quad \Lambda = \frac{m}{\varkappa + 1}. \end{split}$$

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

$$\int\_{R} \left( L\_{gb} u\_f^i - f\_{gb} \right) u\_{da}^{i\*} dR = 0 \tag{26}$$

Now, the fundamental solution *ui* <sup>∗</sup> *df* and traction vectors *t i* ∗ *da* and *t i <sup>a</sup>* can be written as follows:

$$L\_{gb}u\_{af}^{i\*} = -\delta\_{ad}\delta(\mathfrak{x}, \mathfrak{f})\tag{27}$$

$$\mathbf{t}\_{da}^{i\*} = \mathbf{C}\_{ab \text{fg}} \boldsymbol{\mu}\_{\text{df} \text{ g}}^{i\*} \boldsymbol{n}\_{b} \tag{28}$$

$$t\_a^i = \frac{\overline{t}\_a^i}{\left(\varkappa + 1\right)^m} = \left(C\_{abf\mathfrak{g}}\mu\_{f\mathfrak{g}}^i - \beta\_{ab}^i \left(T\_a^i + \tau\_1 T\_a^i\right)\right) n\_b \tag{29}$$

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:

*A New BEM for Modeling of Acoustic Wave Propagation in Three-Temperature Nonlinear… DOI: http://dx.doi.org/10.5772/intechopen.92784*

$$u\_d^i(\xi) = \int\_C (u\_{da}^{i\*} t\_a^i - t\_{da}^{i\*} u\_a^i + u\_{da}^{i\*} \beta\_{ab}^i T\_a^i n\_b) \, d\mathcal{C} - \int\_R f\_{gb} u\_{da}^{i\*} \, dR \tag{30}$$

The fundamental solution *T<sup>i</sup>* <sup>∗</sup> can be defined as

$$L\_{ab}T^{i^\*} = -\delta(\mathfrak{x}, \mathfrak{xi})\tag{31}$$

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

$$\int\_{R} (L\_{ab}T\_a^i T\_a^{i^\*} - L\_{ab}T\_a^{i^\*}T\_a^i)dR = \int\_{C} (q^{i^\*}T\_a^i - q^i T\_a^{i^\*})dC \tag{32}$$

where

where Ψ *<sup>f</sup>* , Φ *<sup>f</sup>* , *f*, and *h* are suitably prescribed functions and *C* ¼ *C*<sup>1</sup> ∪*C*<sup>2</sup> ¼

*<sup>α</sup>* � *Pi <sup>∂</sup>u<sup>i</sup>*

where the inertia term *ρu*€*a*, the temperature gradient *DaT*, and the initial stress

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

*Lgbui*

*LabT<sup>i</sup>*

� � � �

<sup>þ</sup> *Daf* <sup>þ</sup> <sup>Λ</sup>*Da*1*<sup>f</sup>* , *Lab* <sup>¼</sup> *<sup>δ</sup>*2*<sup>j</sup><sup>i</sup>* <sup>∗</sup>

� � � �

*<sup>α</sup>δ*1*δ*1*<sup>j</sup>*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *mT*\_ *<sup>i</sup>*

*∂ ∂xa*

*b ∂xa* � *∂ui a ∂xb*

þ *δ<sup>a</sup>*1Λ � � *∂*

*∂τ*

*<sup>α</sup>* � *Pi <sup>∂</sup>ui*

0

*ui* <sup>∗</sup>

*<sup>f</sup>*,*<sup>g</sup>* � *<sup>β</sup><sup>i</sup>*

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 represen-

*ab T<sup>i</sup>*

*df* and traction vectors *t*

*∂ ∂xb* þ Λ � � *∂*

The differential equation (21) can be solved using the weighted residual method

*b ∂xa* � *∂ui a ∂xb*

*<sup>f</sup>* ¼ *f gb*, (21)

*<sup>α</sup>* ¼ *f ab* (22)

*α*

*∂x <sup>f</sup>* ,

, <sup>Λ</sup> <sup>¼</sup> *<sup>m</sup>*

*dadR* ¼ 0 (26)

*df*,*gnb* (28)

*i* ∗ *da* and *t i*

*af* ¼ �*δadδ*ð Þ *x*, *ξ* (27)

*<sup>α</sup>* <sup>þ</sup> *<sup>τ</sup>*1*T<sup>i</sup> α* � � � � *nb* (29)

*x* þ 1 *:*

� �∇ (23)

*<sup>α</sup>* þ ð Þ *r*, *τ* (25)

(24)

*<sup>a</sup>* can be written

¼ *f gb* (20)

*C*<sup>3</sup> ∪*C*4, *C*<sup>1</sup> ∩*C*<sup>2</sup> ¼ *C*<sup>3</sup> ∩*C*<sup>4</sup> ¼ Ø.

*Noise and Environment*

**4. BEM numerical implementation**

*Lgbui*

*Lgb* ¼ *Dabf*

where

as follows:

tation formula:

**134**

term are treated as the body forces.

*<sup>f</sup>* <sup>¼</sup> *<sup>ρ</sup><sup>i</sup> üi*

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

where the operators *Lgb*, *f gb*, *Lab*, and *f ab* are as follows:

*u*€*i*

*α* � �<sup>∇</sup> <sup>þ</sup> *<sup>ρ</sup><sup>i</sup>*

*∂xg*

þ δb1Λ þ *τ*<sup>1</sup>

*Lgbu<sup>i</sup>*

*Lgbu<sup>i</sup>* <sup>∗</sup>

*t i* ∗

ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup>* <sup>¼</sup> *Cabfgui*

*<sup>a</sup>* � *DaT<sup>i</sup>*

*c i*

, *Daf* <sup>¼</sup> *<sup>μ</sup>H*<sup>2</sup>

� �

*<sup>f</sup>* � *f gb* � �

*da* <sup>¼</sup> *Cabfgui* <sup>∗</sup>

*∂ ∂xb*

*<sup>f</sup> gb* <sup>¼</sup> *<sup>ρ</sup><sup>i</sup>*

*<sup>f</sup> ab* <sup>¼</sup> <sup>∇</sup> *<sup>δ</sup>*1*<sup>j</sup><sup>i</sup>*

*Dabf* <sup>¼</sup> *Cabfgε*, *<sup>ε</sup>* <sup>¼</sup> *<sup>∂</sup>*

*∂ ∂xb*

*ab*

Now, the fundamental solution *ui* <sup>∗</sup>

*t i <sup>a</sup>* <sup>¼</sup> *<sup>t</sup>*

(WRM) to obtain the following integral equation:

ð *R*

*i a*

*Da* ¼ �*β<sup>i</sup>*

*<sup>a</sup>* � *DaT<sup>i</sup>*

$$q^i = -\mathbb{K}\_a^i T\_{a.b}^i n\_a \tag{33}$$

$$q^{i\*} = -\mathbb{K}\_a^i T\_{a,b}^{i^\*} \mathfrak{n}\_a \tag{34}$$

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

$$T\_a^i(\xi) = \int\_C (q^{i^\*} T\_a^i - q^i T\_a^{i^\*}) d\mathcal{C} - \int\_R f\_{ab} T\_a^{i^\*} d\mathcal{R} \tag{35}$$

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

$$
\begin{split}
\begin{bmatrix} u\_d^i(\xi) \\ T\_a^i(\xi) \end{bmatrix} &= \int\_C \left\{ -\begin{bmatrix} t\_{da}^{i\*} & -u\_{aa}^{i\*}\beta\_{ab}u\_b \\ 0 & -q^{i\*} \end{bmatrix} \begin{bmatrix} u\_a^i \\ T\_a^i \end{bmatrix} + \begin{bmatrix} u\_{da}^{i\*} & 0 \\ 0 & -T\_a^{i\*} \end{bmatrix} \begin{bmatrix} \tau\_a^i \\ q^i \end{bmatrix} \right\} dR \\ &- \int\_{\tilde{R}} \begin{bmatrix} u\_{da}^{i\*} & 0 \\ 0 & -T\_a^{i\*} \end{bmatrix} \begin{bmatrix} f\_{gb}^b \\ -f\_{ab} \end{bmatrix} dR \end{split} \tag{36}
$$

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

$$U\_A^i = \begin{cases} u\_a^i & a = A = 1,2,3 \\ T\_a^i & A = 4 \end{cases} \tag{37}$$

$$\mathbf{T}\_{aA}^{i} = \begin{cases} t\_a^i & a=A=1,2,3\\ q^i & A=4 \end{cases} \tag{38}$$

$$U\_{DA}^{i\*} = \begin{cases} u\_{da}^{i\*} & d=D=1,2,3; a=A=1,2,3\\ 0 & d=D=1,2,3; A=4\\ 0 & D=4; a=A=1,2,3\\ -T\_{-}^{i\*} & D=4; A=4 \end{cases} \tag{39}$$

$$\begin{aligned} -T\_a^{i\*} & \quad D = 4; A = 4\\ \bar{T}\_{a\Delta}^{i^\*} &= \begin{cases} t\_{a a}^{i\*} & d = D = 1, 2, 3; a = A = 1, 2, 3\\ -\bar{u}\_d^{i\*} & d = D = 1, 2, 3; A = 4\\ 0 & D = 4; a = A = 1, 2, 3 \end{cases} \end{aligned} \tag{40}$$

$$
\tilde{\boldsymbol{u}}\_d^{i\*} = \boldsymbol{u}\_{da}^{i\*} \boldsymbol{\beta}\_{\boldsymbol{a}f}^i \boldsymbol{n}\_f \tag{41}
$$

*SA* <sup>≈</sup> <sup>X</sup> *E*

*A New BEM for Modeling of Acoustic Wave Propagation in Three-Temperature Nonlinear…*

*q*¼1 *f q AEα<sup>q</sup>*

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

*αDAU<sup>i</sup> A* � �*d*<sup>C</sup> �<sup>X</sup>

> *Lgbuiq fe* ¼ *f q*

*LabTiq <sup>α</sup>* ¼ *f q*

*pj* , (54)

*<sup>α</sup>* � *<sup>q</sup>iqTi* <sup>∗</sup> *α* � � *dC* �

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

*αDAUiq AE* � �*dC* �

*<sup>α</sup>AE* � *<sup>T</sup><sup>i</sup>* <sup>∗</sup>

With the substitution of (57) into (52), the dual reciprocity representation

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

*N*

ð

*R U<sup>i</sup>* <sup>∗</sup> *DA f q AEdRα<sup>q</sup>*

*q*¼1

ð

*R ui* <sup>∗</sup> *da f q*

ð

*R f q T<sup>i</sup>* <sup>∗</sup>

ð

*R U<sup>i</sup>* <sup>∗</sup> *DA f iq*

*<sup>α</sup><sup>A</sup>* � *<sup>T</sup>*~*<sup>i</sup>* <sup>∗</sup>

following form:

single equation:

*Ui <sup>D</sup>*ð Þ¼ *ξ*

*∂U<sup>i</sup> <sup>D</sup>*ð Þ*ξ ∂ξl*

**137**

*UD*ð Þ¼ *ξ*

*DOI: http://dx.doi.org/10.5772/intechopen.92784*

ð

*U<sup>i</sup>* <sup>∗</sup> *DAT<sup>i</sup>*

By implementing the WRM to the following equations.

ð

*ui* <sup>∗</sup> *da t iq ae* � *t i* ∗ *dauiq ae* � �*dC* �

*qi* <sup>∗</sup> *Tiq*

*C*

ð

*C*

*U<sup>i</sup>* <sup>∗</sup> *DATiq*

formula of coupled thermoelasticity can be expressed as follows:

� �*d*<sup>C</sup>

ð

*T<sup>i</sup>* <sup>∗</sup> *αDAUiq*

To calculate interior stresses, (58) is differentiated with respect to *ξ<sup>l</sup>* as follows:

*AE* � *<sup>U</sup><sup>i</sup>* <sup>∗</sup>

*<sup>α</sup>AE* � *<sup>U</sup><sup>i</sup>* <sup>∗</sup>

*<sup>α</sup>AE* � �*dC*

*DA*,*l Tiq*

*DATiq <sup>α</sup>AE* � �*dC*

1 CA*αq*

> 1 CA*αq*

*C*

� �*d*<sup>C</sup>

� ð

Fahmy [44], we can write (58) in the following system of equations:

*C*

*T<sup>i</sup>* <sup>∗</sup> *αDA*,*l Uiq*

According to the dual reciprocity boundary integral equation procedure of

*<sup>α</sup><sup>A</sup>* � *T ^ i* ∗ *αDAU<sup>i</sup> A*

*Uiq DE*ð Þþ *ξ*

ð

*C*

C

*uiq de*ð Þ¼ *ξ*

*Tiq <sup>α</sup>* ð Þ¼ *ξ*

*Uiq DE*ð Þ¼ *ξ*

ð

*U<sup>i</sup>* <sup>∗</sup> *DAT<sup>i</sup>*

0

B@

*C*

¼ �<sup>ð</sup>

*C*

þ<sup>X</sup> *E*

*q*¼1

þ<sup>X</sup> *E*

*q*¼1

*U<sup>i</sup>* <sup>∗</sup> *DA*,*l Ti <sup>α</sup>A*,*<sup>l</sup>* � *T ^ i* <sup>∗</sup> *αDA*,*l Ui A*

0

B@

*∂Uiq DE*ð Þ*ξ ∂ξl*

*<sup>E</sup>* (51)

*ae* (53)

*aedR* (55)

*<sup>α</sup> dR* (56)

*AEdR* (57)

*<sup>E</sup>* (58)

*<sup>E</sup>* (59)

*<sup>E</sup>* (52)

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

$$\boldsymbol{U}\_{D}^{i}(\xi) = \int\_{C} \left( \boldsymbol{U}\_{DA}^{i\*} \boldsymbol{\Gamma}\_{aA}^{i} - \boldsymbol{\tilde{T}}\_{aDA}^{i} \boldsymbol{U}\_{A}^{i} \right) d\boldsymbol{C} - \int\_{R} \boldsymbol{U}\_{DA}^{i\*} \boldsymbol{S}\_{A} d\boldsymbol{R} \tag{42}$$

The vector *SA* can be split as follows

$$\mathbf{S}\_{A} = \mathbf{S}\_{A}^{0} + \mathbf{S}\_{A}^{T} + \mathbf{S}\_{A}^{u} + \mathbf{S}\_{A}^{T} + \mathbf{S}\_{A}^{T} + \mathbf{S}\_{A}^{\vec{u}} \tag{43}$$

where

$$S\_A^0 = \begin{cases} 0 & A = 1,2,3\\ \frac{1 - R}{\infty\_0} e^{\left(-\frac{x\mathfrak{a}}{x\_0}\right) l(\mathfrak{a})} & A = 4 \end{cases} \tag{44}$$

$$\mathcal{S}\_A^T = \text{o}\_{AF} U\_F^i \text{witho}\_{AF} = \begin{cases} -D\_a & A = 1, 2, 3; F = 4\\ \nabla \left( \delta\_{2\circ} \mathbb{K}\_a^{i^\*} \right) \nabla & \text{otherwise} \end{cases} \tag{45}$$

$$S\_A^t = \mathcal{y}U\_F^i \text{ with} \\ \mathcal{y} = \begin{cases} P^i \left( \frac{\partial}{\partial \mathbf{x}b} - \frac{\partial}{\partial \mathbf{x}\_d} \right) & A = 1,2,3; F = 1,2,3, \\ \mathbf{0} & A = 4; F = 4 \end{cases} \tag{46}$$

$$\mathbf{S}\_A^{\hat{T}} = \Gamma\_{AF} \dot{U}\_F^i \quad \text{with} \quad \Gamma\_{AF} = \begin{cases} -\rho\_{ab}^i \tau\_1 \left(\frac{\partial}{\partial \mathbf{x}\_b} + \Lambda\right) \frac{\partial}{\partial \tau} & \text{ $A = 4$ ;} F = 4 \\\ \rho^i \mathbf{c}\_a^i \delta\_1 \delta\_{\hat{l}} & \text{otherwise} \end{cases} \tag{47}$$

$$\boldsymbol{\mathcal{S}}\_{A}^{T} = \delta\_{\rm AF} \ddot{\boldsymbol{U}}\_{F}^{i} \quad \text{with} \quad \delta\_{\rm AF} = \begin{cases} \boldsymbol{0} & \boldsymbol{A} = \boldsymbol{4}; F = \boldsymbol{4} \\\boldsymbol{\rho}^{i} \boldsymbol{c}\_{a}^{i} \Big[ \left( \boldsymbol{\tau}\_{\rm 0} + \delta\_{\rm \dot{\boldsymbol{\mu}}} \boldsymbol{\tau}\_{2} + \delta\_{\rm \dot{\boldsymbol{\mu}}} \right) \Big] & \text{otherwise} \end{cases} \tag{48}$$

$$S\_A^{\bar{u}} = \tilde{o} \ddot{U}\_F^i \text{ with } \ \tilde{o} = \begin{cases} \rho^i & A = 1, 2, 3, F = 1, 2, 3, \\ \rho\_{ab}^i T\_{a0}^i (\tau\_0 + \delta\_{2i}) & A = 4; F = 4 \end{cases} \tag{49}$$

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

$$\begin{aligned} \left[S\_{A}\right] &= -\begin{bmatrix} 0 \\ -\frac{1-R}{\varkappa\_{0}} & \varepsilon \left(-\frac{\varkappa\_{a}}{\varkappa\_{0}}\right) \end{bmatrix} + \left\{ \begin{bmatrix} -D\_{a}\dot{T}^{i}\_{a} \\ \nabla\left(\delta\_{\hat{g}}\dot{\mathsf{R}}^{i^{\top}}\_{a}\right)\nabla\dot{T}^{i}\_{a} \end{bmatrix} \right\} + \begin{bmatrix} \dot{P}^{i}\Big{(}u^{i\_{0}}\_{b,a} - u^{i}\_{a,b}\Big{)} \\ \nabla\left(\delta\_{\hat{g}}\dot{\mathsf{R}}^{i^{\top}}\_{a}\right)\nabla\dot{T}^{i}\_{a} \end{bmatrix} \right\} \\ &+ \begin{bmatrix} -\dot{\rho}^{i}\_{a\dot{b}}\tau\_{1}\Big{(}\frac{\partial}{\partial\textbf{x}\_{b}} + \Lambda\Big{)}\dot{T}^{i}\_{a} \\ \dot{\rho}^{i}c^{i}\_{a}\delta\_{\hat{b}}\delta\_{\hat{g}}\dot{T}^{i}\_{a} \end{bmatrix} + \dot{\rho}^{i}c^{i}\_{a}\Big{[}\left(\tau\_{0} + \delta\_{\hat{y}}\tau\_{2} + \delta\_{\hat{y}}\right)\Big{]} \begin{bmatrix} 0 \\ \ddot{T}^{i}\_{a} \end{bmatrix} \\ &+ \begin{bmatrix} \dot{\rho}^{i}\_{a}\dot{T}^{i}\_{a} \\ \dot{\rho}^{i}\_{a}\dot{T}^{i}\_{a0}\big{(}\tau\_{0} + \delta\_{\hat{y}}\rangle\ddot{u}^{i}\_{f\mathcal{R}} \end{bmatrix} \end{aligned} \tag{50}$$

To transform the domain integral in (42) to the boundary, we approximate the source vector *SA* by a series of given tensor functions *f q AE* and unknown coefficients *αq <sup>E</sup>* as follows:

*A New BEM for Modeling of Acoustic Wave Propagation in Three-Temperature Nonlinear… DOI: http://dx.doi.org/10.5772/intechopen.92784*

$$S\_A \approx \sum\_{q=1}^{E} f\_{AE}^q a\_E^q \tag{51}$$

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

$$dU\_D(\xi) = \left[ \left( U\_{DA}^{i^\*} T\_{aA}^i - \tilde{T}\_{aDA}^{i^\*} U\_A^i \right) d\mathbb{C} - \sum\_{q=1}^N \left[ U\_{DA}^{i^\*} f\_{AB}^q d\mathcal{R} \alpha\_E^q \right] \tag{52}$$

By implementing the WRM to the following equations.

$$L\_{\rm gb}u^{iq}\_{fe} = f^q\_{ae} \tag{53}$$

$$L\_{ab}T^{iq}\_a = f^q\_{pj} \tag{54}$$

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

$$u\_{d\epsilon}^{iq}(\xi) = \int\_C (u\_{da}^{i\*}t\_{a\epsilon}^{iq} - t\_{da}^{i\*}u\_{a\epsilon}^{iq})d\mathcal{C} - \int\_R u\_{da}^{i\*}f\_{a\epsilon}^q d\mathcal{R} \tag{55}$$

$$T\_a^{iq}(\xi) = \int\_C (q^{i\*}T\_a^{iq} - q^{iq}T\_a^{i\*}) \, d\mathcal{C} - \int\_R f^q T\_a^{i\*} \, d\mathcal{R} \tag{56}$$

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

$$\mathcal{U}\_{DE}^{iq}(\xi) = \int\_{C} \left( \mathcal{U}\_{DA}^{i\*} \mathcal{T}\_{aAE}^{iq} - \mathcal{T}\_{aDA}^{i\*} \mathcal{U}\_{AE}^{iq} \right) d\mathcal{C} - \int\_{R} \mathcal{U}\_{DA}^{i\*} \mathcal{f}\_{AE}^{iq} d\mathcal{R} \tag{57}$$

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

$$\begin{aligned} U\_D^i(\xi) &= \int\_C \left( U\_{DA}^{i\*} T\_{aA}^i - \breve{T}\_{aDA}^{i\*} U\_A^i \right) d\mathbf{C} \\ &+ \sum\_{q=1}^E \left( U\_{DE}^{iq}(\xi) + \int\_C \left( T\_{aDA}^{i\*} U\_{AE}^{iq} - U\_{DA}^{i\*} T\_{aAE}^{iq} \right) d\mathbf{C} \right) d\mathbf{C} \end{aligned} \tag{58}$$

To calculate interior stresses, (58) is differentiated with respect to *ξ<sup>l</sup>* as follows:

$$\begin{split} \frac{\partial \mathbf{U}\_{D}^{i}(\xi)}{\partial \xi\_{l}} &= -\int\_{C} \left( \mathbf{U}\_{DA,l}^{i\*} \mathbf{T}\_{aA,l}^{i} - \overline{\mathbf{T}}\_{aDA,l}^{i^\*} \mathbf{U}\_{A}^{i} \right) d\mathbf{C} \\ &+ \sum\_{q=1}^{E} \left( \frac{\partial \mathbf{U}\_{DE}^{iq}(\xi)}{\partial \xi\_{l}} - \int\_{C} \left( \mathbf{T}\_{aDA,l}^{i\*} \mathbf{U}\_{aAE}^{iq} - \mathbf{U}\_{DA,l}^{i\*} \mathbf{T}\_{aAE}^{iq} \right) d\mathbf{C} \right) a\_{E}^{q} \end{split} \tag{59}$$

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

*u*~*<sup>i</sup>* <sup>∗</sup> *<sup>d</sup>* <sup>¼</sup> *ui* <sup>∗</sup> *da β<sup>i</sup>*

*<sup>α</sup><sup>A</sup>* � *<sup>T</sup>*~*<sup>i</sup>*

*<sup>A</sup>* <sup>þ</sup> *<sup>S</sup>*<sup>u</sup>

*<sup>e</sup>* �*xa x*0 � � *J*ð Þ*τ*

� �

*αDAU<sup>i</sup> A*

*<sup>A</sup>* <sup>þ</sup> *<sup>S</sup>T*\_

0 *A* ¼ 1, 2, 3

*<sup>F</sup>*withω*AF* <sup>¼</sup> �*Da <sup>A</sup>* <sup>¼</sup> 1, 2, 3; *<sup>F</sup>* <sup>¼</sup> <sup>4</sup> ∇ *δ*2*<sup>j</sup><sup>i</sup>* <sup>∗</sup> *α*

0 *A* ¼ 4; *F* ¼ 4

*<sup>F</sup>* with *<sup>δ</sup>AF* <sup>¼</sup> <sup>0</sup> *<sup>A</sup>* <sup>¼</sup> 4; *<sup>F</sup>* <sup>¼</sup> <sup>4</sup>

*<sup>F</sup>* with *<sup>õ</sup>* <sup>¼</sup> *<sup>ρ</sup><sup>i</sup> <sup>A</sup>* <sup>¼</sup> 1, 2, 3, *<sup>F</sup>* <sup>¼</sup> 1, 2, 3,

The thermoelastic representation formula (36) can also be written in matrix

<sup>5</sup> <sup>þ</sup> �*DaT<sup>i</sup>*

To transform the domain integral in (42) to the boundary, we approximate the

∇ *δ*2*<sup>j</sup><sup>i</sup>* <sup>∗</sup> *α* � �∇*T<sup>i</sup>*

*<sup>α</sup> τ*<sup>0</sup> þ *δ*1*<sup>j</sup>τ*<sup>2</sup> þ *δ*2*<sup>j</sup>*

*<sup>α</sup>*0ð Þ *τ*<sup>0</sup> þ *δ*2*<sup>i</sup> A* ¼ 4; *F* ¼ 4

*∂ ∂xb* þ Λ � � *∂*

*abτ*<sup>1</sup>

*dC* � ð

*<sup>A</sup>* <sup>þ</sup> *<sup>S</sup>T*€

*A* ¼ 4

� �∇ otherwise

*R U<sup>i</sup>* <sup>∗</sup>

*<sup>A</sup>* <sup>þ</sup> *<sup>S</sup>u*€

*A* ¼ 1, 2, 3; *F* ¼ 1, 2, 3,

*<sup>α</sup>δ*1*δ*1*<sup>j</sup>* otherwise

� � � � otherwise

*α*

( ) " #

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

*α*

*q*

þ

*T*€*i α* � �

*P<sup>i</sup> ui*<sup>0</sup>

*AE* and unknown coefficients

*<sup>b</sup>*,*<sup>a</sup>* � *<sup>u</sup><sup>i</sup> a*,*b* � �

" #

0

*<sup>∂</sup><sup>τ</sup> <sup>A</sup>* <sup>¼</sup> 4; *<sup>F</sup>* <sup>¼</sup> <sup>4</sup>

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

ð

*U<sup>i</sup>* <sup>∗</sup> *DA*T*<sup>i</sup>*

*C*

*SA* <sup>¼</sup> *<sup>S</sup>*<sup>0</sup>

8 ><

>:

*S*0 *<sup>A</sup>* ¼

*<sup>F</sup>* with*<sup>ψ</sup>* <sup>¼</sup> *<sup>P</sup><sup>i</sup> <sup>∂</sup>*

8 < :

*<sup>F</sup>* with <sup>Γ</sup>*AF* <sup>¼</sup> �*β<sup>i</sup>*

*<sup>A</sup>* <sup>þ</sup> *<sup>S</sup><sup>T</sup>*

1 � *R x*0

�

8 ><

>:

*ρi ci*

�

*βi abT<sup>i</sup>*

*J*ð Þ*τ*

3 7 <sup>5</sup> <sup>þ</sup> *<sup>ρ</sup><sup>i</sup> c i*

*T*\_ *i α*

*f*,*g*

source vector *SA* by a series of given tensor functions *f*

3

(

0

*∂ ∂xb* þ Λ � �

*<sup>α</sup>*<sup>0</sup> *τ*<sup>0</sup> þ *δ*2*<sup>j</sup>* � �*u*€*<sup>i</sup>*

" #

*<sup>e</sup>* � *xa x*0 � �

� <sup>1</sup> � *<sup>R</sup> x*0

*abτ*<sup>1</sup>

*ρi ci <sup>α</sup>δ*1*δ*1*jT*\_ *<sup>i</sup> α*

*<sup>∂</sup>xb* � *<sup>∂</sup> ∂xa* � �

> *ρi ci*

*Ui <sup>D</sup>*ð Þ¼ *ξ*

where

*Noise and Environment*

*ST*

*<sup>A</sup>* <sup>¼</sup> <sup>Γ</sup>*AFU*\_ *<sup>i</sup>*

*<sup>A</sup>* <sup>¼</sup> *<sup>δ</sup>AFU*€ *<sup>i</sup>*

*Su <sup>A</sup>* <sup>¼</sup> *<sup>ψ</sup>U<sup>i</sup>*

*ST*\_

*ST*€

*S*u€ *<sup>A</sup>* <sup>¼</sup> *õU*€ *<sup>i</sup>*

form as follows:

2 4

<sup>þ</sup> �*β<sup>i</sup>*

*ρi u*€*i a βi abT<sup>i</sup>*

2 6 4

þ

*<sup>E</sup>* as follows:

*αq*

**136**

½ �¼� *SA*

*<sup>A</sup>* <sup>¼</sup> <sup>ω</sup>*AFU<sup>i</sup>*

The vector *SA* can be split as follows

*af n <sup>f</sup>* (41)

*DASAdR* (42)

*<sup>A</sup>* (43)

(44)

(45)

(46)

(47)

(48)

(49)

(50)

$$
\breve{\zeta}U - \eta T\_a = \left(\zeta \breve{U} - \eta \breve{\wp}\right)a\tag{60}
$$

Now, the coefficients *α* can be expressed in terms of nodal values of the

*A New BEM for Modeling of Acoustic Wave Propagation in Three-Temperature Nonlinear…*

0�<sup>1</sup> <sup>þ</sup> *<sup>ψ</sup>*�*U<sup>i</sup>* <sup>þ</sup> <sup>Γ</sup>*AFU*\_ *<sup>i</sup>*

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

z}|{ <sup>¼</sup> *<sup>V</sup> <sup>õ</sup>* <sup>þ</sup> *<sup>δ</sup>AF* � �, <sup>Γ</sup>

*abT*0,

represent the acceleration, velocity, displacement, temperature, and external

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

*<sup>n</sup>*þ<sup>1</sup> þ *K* z}|{ *Ui*

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup>

z}|{ *Ti*

and *Tip*

 z}|{*ip*

*<sup>n</sup>*þ<sup>1</sup> � Γ z}|{ *U*\_ *i*

> ~ *ip <sup>n</sup>*þ<sup>1</sup> � Γ z}|{ *U*\_ *i*

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

z}|{ *U*€ *i*

¼ �*ηT ^* þ *VS ^*0 , X z}|{ ¼ �*ρ<sup>i</sup>*

, and B z}|{

conduction equation and which is induced by the effect of the strain rate is

, and accelerations *U*€ *<sup>i</sup>*

� � � (72)

*<sup>U</sup><sup>i</sup>* <sup>¼</sup> z}|{*<sup>i</sup>*

z}|{ <sup>¼</sup> *<sup>V</sup>*Γ*AF*,

z}|{ ¼ � <sup>1</sup>�*<sup>R</sup>*

*<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup>

z}|{ *U*€ *i*

z}|{*ip*

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> is the predicted temperature.

*<sup>n</sup>*þ<sup>1</sup> � *K* z}|{ *Ui n*þ1

� � � � (78)

*<sup>n</sup>*þ<sup>1</sup> � *K* z}|{ *Ui n*þ1

� � � � (77)

*<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup>

*ci*

represent the volume, mass, damping,

*<sup>n</sup>*þ<sup>1</sup> that appears in the heat

*<sup>x</sup>*<sup>0</sup> *e xa x*0 � �*<sup>J</sup>*ð Þ*<sup>τ</sup>*

ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup>*,

, *U*\_ *<sup>i</sup>* , *U<sup>i</sup>* , *T<sup>i</sup>* , and

*<sup>n</sup>*þ<sup>1</sup> (75)

z}|{ (76)

as follows:

z}|{ (74)

(73)

<sup>þ</sup> *<sup>õ</sup>* <sup>þ</sup> *<sup>δ</sup>AF* � �*U*€ *<sup>i</sup>*

, velocities *U*\_ *<sup>i</sup>*

þ B*TJ*

unknown displacements *U<sup>i</sup>*

where *V* ¼ *η*℘

*^*

*ab ∂ ∂xa ∂ ∂xb* , B z}|{ <sup>¼</sup> *<sup>k</sup><sup>i</sup>* <sup>∗</sup> *ab ∂ ∂xa ∂ ∂xb* , z}|{ <sup>¼</sup> *<sup>β</sup><sup>i</sup>*

where *V*, *M*

force vectors, respectively.

algebraic equations (DAEs):

X z}|{ *T*€*i*

z}|{*<sup>i</sup>*<sup>p</sup>

*<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>U</sup>*\_ *<sup>i</sup>*

<sup>¼</sup> *<sup>U</sup>*\_ *<sup>i</sup> <sup>n</sup>* þ Δ*τ* <sup>2</sup> *<sup>U</sup>*€ *<sup>i</sup>*

*<sup>n</sup>* þ Δ*τ* <sup>2</sup> *<sup>U</sup>*\_ *<sup>i</sup>*

*<sup>n</sup>* <sup>þ</sup> <sup>Δ</sup>*τU*\_ *<sup>i</sup>*

where

*Ui*

*<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>U</sup><sup>i</sup>*

<sup>¼</sup> *<sup>U</sup><sup>i</sup>*

*Ui*

**139**

*K* z}|{ ¼ �*<sup>ζ</sup>*

A z}|{ <sup>¼</sup> *<sup>k</sup><sup>i</sup>*

 z}|{*<sup>i</sup>*

negligible.

*α* ¼ *J*

�<sup>1</sup> *S ^*0

*DOI: http://dx.doi.org/10.5772/intechopen.92784*

substitution of Eq. (72) into Eq. (60) as follows:

X z}|{ *T*€*i <sup>α</sup>* þ A z}|{ *T*\_ *i <sup>α</sup>* þ B z}|{ *Ti <sup>α</sup>* ¼ z}|{ *U*€ *i* þ

�<sup>1</sup> <sup>þ</sup> *<sup>ψ</sup>* � �,

z}|{, *K* z}|{ , A z}|{

In many applications, the coupling term

*M* z}|{ *U*€ *i*

> z}|{ *T*\_ *i*

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> *VS*

*<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> *<sup>U</sup>*€ *<sup>i</sup> n*

� �

*<sup>n</sup>* þ *M* z}|{�<sup>1</sup>

> *U*€ *i <sup>n</sup>* þ *M* z}|{�<sup>1</sup>

*<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> *<sup>U</sup>*\_ *<sup>i</sup> n*

� �

*<sup>n</sup>* þ Δ*τ*<sup>2</sup> 4

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> þ A

*<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>η</sup>Tip*

*<sup>n</sup>* þ Δ*τ* <sup>2</sup> *<sup>U</sup>*€ *<sup>i</sup>*

*<sup>n</sup>*þ<sup>1</sup> þ Γ z}|{ *U*\_ *i*

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> þ B

*^*0

*^* � *ζU ^* � �*<sup>J</sup>*

<sup>þ</sup> *<sup>V</sup>* <sup>B</sup>*TJ*<sup>0</sup>

z}|{, <sup>Γ</sup>

*M* z}|{ *U*€ *i* þ Γ z}|{ *U*\_ *i* þ *K* z}|{

�1 , *M*

z}|{*<sup>i</sup>*

stiffness, capacity, and conductivity matrices, respectively, and *U*€ *<sup>i</sup>*

The generalized displacements and velocities are approximated in terms of a series of known tensor functions *f q FD* and unknown coefficients *γ q <sup>D</sup>* and ~*γ q D*:

$$U\_F^i \approx \sum\_{q=1}^N f\_{FD}^q(\mathbf{x}) \chi\_D^q \tag{61}$$

where

$$f\_{FD}^q = \begin{cases} f\_{fd}^q & f = F = 1,2,3; d = D = 1,2,3 \\ f^q & F = 4; D = 4 \\ 0 & \text{otherwise} \end{cases} \tag{62}$$

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

$$U\_{F, \mathfrak{g}}^i \approx \sum\_{q=1}^N f\_{FD, \mathfrak{g}}^q(\mathfrak{x}) \chi\_K^q \tag{63}$$

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

$$\mathbf{S}\_A^T = \sum\_{q=1}^N \mathbf{S}\_{AF} f\_{FD, \mathbf{g}}^q \mathbf{y}\_D^q \tag{64}$$

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

$$
\breve{S} = J\overline{a}, \quad U^{i} = J^{\prime}\gamma,\tag{65}
$$

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

$$
\overline{\mathcal{S}}^{T^i\_a} = \mathcal{B}^T \mathcal{Y} \tag{66}
$$

$$\mathbf{S}\_A^u = \overline{\boldsymbol{\varphi}} \mathbf{U}^i \tag{67}$$

$$
\stackrel{\smile}{\dot{\bf S}}^{\dot{\mathcal{I}}^{\dot{\mathcal{I}}}\_{a}} = \overline{\Gamma}\_{AF} \dot{\mathcal{U}}^{\dot{\mathcal{I}}} \tag{68}
$$

$$
\bar{\mathbf{S}}^{\bar{T}^l\_{\dot{a}}} = \overline{\delta}\_{AF} \ddot{\mathbf{U}}^i \tag{69}
$$

$$
\stackrel{\smile}{\ddot{S}}^{\ddot{\mu}} = \overline{\sigma}\ddot{U}^i \tag{70}
$$

where *ψ*, Γ*AF*, *δAF*, and *õ* are assembled using the submatrices ½ � *ψ* <sup>0</sup> ½ � Γ*AF* , ½ � *δAF* , and ½ � *õ* , respectively.

Solving the system (65) for *α* and *γ* yields

$$\overline{a} = \mathcal{J}^{-1}\overline{\mathcal{S}}, \quad \gamma = \mathcal{J}^{-1}\mathcal{U}^{i} \tag{71}$$

*A New BEM for Modeling of Acoustic Wave Propagation in Three-Temperature Nonlinear… DOI: http://dx.doi.org/10.5772/intechopen.92784*

Now, the coefficients *α* can be expressed in terms of nodal values of the unknown displacements *U<sup>i</sup>* , velocities *U*\_ *<sup>i</sup>* , and accelerations *U*€ *<sup>i</sup>* as follows:

$$\overline{a} = \boldsymbol{J}^{-1} \left( \stackrel{\cdot}{\mathcal{S}}^{0} + \left( \mathcal{S}^{T} \boldsymbol{J}^{\prime -1} + \overline{\boldsymbol{\nu}} \right) \boldsymbol{U}^{i} + \overline{\boldsymbol{\Gamma}}\_{AF} \dot{\boldsymbol{U}}^{i} + \left( \overline{\boldsymbol{\sigma}} + \overline{\boldsymbol{\delta}}\_{AF} \right) \ddot{\boldsymbol{U}}^{i} \right) \tag{72}$$

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:

$$
\widehat{\Gamma^i \ddot{U}^i} + \widehat{\Gamma^i \ddot{U}^i} + \widehat{K^i U^i} = \widehat{\mathbb{Q}^i} \tag{73}
$$

$$
\widehat{\mathbf{X}}^i \ddot{T}\_a^i + \widehat{\mathbf{A}}^i \ddot{T}\_a^i + \widehat{\mathbf{B}}^i T\_a^i = \widehat{\mathbb{Z}}^i \ddot{U}^i + \widehat{\mathbb{R}}^i \tag{74}
$$

$$\begin{aligned} \text{where } V &= \left(\eta\breve{\wp} - \zeta\breve{U}\right)l^{-1}, \widehat{\widetilde{M}} = V\left(\overline{\widetilde{o}} + \overline{\delta}\_{\text{AF}}\right), \widehat{\Gamma} = V\overline{\Gamma}\_{\text{AF}},\\ \widehat{\widetilde{K}} &= -\breve{\zeta} + V\left(\mathcal{B}^{T}l^{-1} + \overline{\eta}\right), \widehat{\mathbb{Q}^{i}} = -\eta\breve{T} + V\breve{\mathcal{S}}^{0}, \widehat{\mathbf{X}} = -\rho^{i}c^{i}(\varkappa + 1)^{m},\\ \widehat{\mathbf{A}} &= k^{i}\_{ab}\frac{\partial}{\partial x\_{a}}\frac{\partial}{\partial x\_{b}}, \widehat{\mathbf{B}} = k^{i}\_{ab}\frac{\partial}{\partial x\_{a}}\frac{\partial}{\partial x\_{b}}, \widehat{\mathbb{Z}} = \beta^{i}\_{ab}T\_{0}, \widehat{\mathbb{R}} = -\frac{1-\mathcal{R}}{x\_{0}}e^{\left(\frac{x\_{a}}{x\_{0}}\right)l(\mathbf{r})} \end{aligned}$$

where *V*, *M* z}|{, <sup>Γ</sup> z}|{, *K* z}|{ , A z}|{ , and B z}|{ represent the volume, mass, damping, stiffness, capacity, and conductivity matrices, respectively, and *U*€ *<sup>i</sup>* , *U*\_ *<sup>i</sup>* , *U<sup>i</sup>* , *T<sup>i</sup>* , and z}|{*<sup>i</sup>* represent the acceleration, velocity, displacement, temperature, and external force vectors, respectively.

In many applications, the coupling term z}|{ *U*€ *i <sup>n</sup>*þ<sup>1</sup> 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 differentialalgebraic equations (DAEs):

$$
\widehat{\phantom{M}M}\ddot{\boldsymbol{U}}\_{n+1}^{i} + \widehat{\phantom{\Gamma}\boldsymbol{U}}\dot{\boldsymbol{U}}\_{n+1}^{i} + \widehat{\phantom{K}M}\boldsymbol{U}\_{n+1}^{i} = \widehat{\phantom{\mathbb{Q}}\boldsymbol{U}}\_{n+1}^{ip} \tag{75}
$$

$$\widehat{\mathbf{X}}^{i}\overleftarrow{\mathbf{T}}\_{a(n+1)}^{i} + \widehat{\mathbf{A}}^{i}\overleftarrow{\mathbf{T}}\_{a(n+1)}^{i} + \widehat{\mathbf{B}}^{i}T\_{a(n+1)}^{i} = \widehat{\mathbf{Z}}^{i}\overleftarrow{\mathbf{U}}\_{n+1}^{i} + \widehat{\mathbf{R}} \tag{76}$$

where z}|{*<sup>i</sup>*<sup>p</sup> *<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>η</sup>Tip <sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> *VS ^*0 and *Tip <sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> is the predicted temperature. Integrating Eq. (73) with the use of trapezoidal rule and Eq. (75), we obtain

$$\begin{split} U\_{n+1}^{i} &= \dot{U}\_{n}^{i} + \frac{\Delta \tau}{2} \left( \ddot{U}\_{n+1}^{i} + \ddot{U}\_{n}^{i} \right) \\ &= \dot{U}\_{n}^{i} + \frac{\Delta \tau}{2} \left[ \ddot{U}\_{n}^{i} + \widetilde{\mathcal{M}}^{-1} \left( \widetilde{\mathbb{Q}}\_{n+1} - \widetilde{\Gamma}^{\circ} \dot{U}\_{n+1}^{i} - \widetilde{\mathcal{K}}^{\circ} \dot{U}\_{n+1}^{i} \right) \right] \end{split} \tag{77}$$

$$\begin{split} U\_{n+1}^{i} &= U\_{n}^{i} + \frac{\Delta \tau}{2} \left( \dot{U}\_{n+1}^{i} + \dot{U}\_{n}^{i} \right) \\ &= U\_{n}^{i} + \Delta \tau \dot{U}\_{n}^{i} + \frac{\Delta \tau^{2}}{4} \left[ \ddot{U}\_{n}^{i} + \widetilde{\mathcal{M}}^{-1} \left( \dot{\mathbb{Q}}\_{n+1}^{ip} - \widetilde{\Gamma}^{\cdot} \dot{U}\_{n+1}^{i} - \widetilde{\mathcal{K}}^{\cdot} \dot{U}\_{n+1}^{i} \right) \right] \tag{78}$$

*ζ ^*

> *f q*

8 >><

>>:

*f*

series of known tensor functions *f*

*f q FD* ¼

to Eqs. (51) and (61), we have

½ � *õ* , respectively.

**138**

where

*Noise and Environment*

*U* � *ηT<sup>α</sup>* ¼ *ζU*

*q*

*Ui <sup>F</sup>* <sup>≈</sup> <sup>X</sup> *N*

The generalized displacements and velocities are approximated in terms of a

*q*¼1 *f q FD*ð Þ *x γ q*

*<sup>q</sup> <sup>F</sup>* <sup>¼</sup> 4; *<sup>D</sup>* <sup>¼</sup> <sup>4</sup>

The gradients of the generalized displacements and velocities can also be

*q*¼1 *f q FD*,*<sup>g</sup>* ð Þ *x γ q*

*q*¼1

*SAF f q FD*,*<sup>g</sup> γ q*

By implementing the point collocation procedure introduced by Gaul et al. [43]

0

<sup>¼</sup> *<sup>J</sup>α*, *<sup>U</sup><sup>i</sup>* <sup>¼</sup> *<sup>J</sup>*

Similarly, the implementation of the point collocation procedure to Eqs. (64),

*S ^T<sup>i</sup> α*

*Su*

*S ^T*\_ *<sup>l</sup> α*

*S ^T*€*<sup>l</sup> α*\_

*α* ¼ *J* �1 *S ^*

*S ^u*€

where *ψ*, Γ*AF*, *δAF*, and *õ* are assembled using the submatrices ½ � *ψ* <sup>0</sup> ½ � Γ*AF* , ½ � *δAF* , and

, *γ* ¼ *J*

0 otherwise

approximated in terms of the derivatives of tensor functions as follows:

*<sup>F</sup>*,*<sup>g</sup>* <sup>≈</sup> <sup>X</sup> *N*

*Ui*

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

*ST <sup>A</sup>* <sup>¼</sup> <sup>X</sup> *N*

*S ^*

(46), (47), (48), and (49) leads to the following equations:

Solving the system (65) for *α* and *γ* yields

*^* � *η*℘ *^* � �

*FD* and unknown coefficients *γ*

*fd f* ¼ *F* ¼ 1, 2, 3; *d* ¼ *D* ¼ 1, 2, 3

*α* (60)

*q <sup>D</sup>* and ~*γ q D*:

*<sup>D</sup>* (61)

*<sup>K</sup>* (63)

*<sup>D</sup>* (64)

*γ*, (65)

¼ B*<sup>T</sup><sup>γ</sup>* (66)

*<sup>A</sup>* <sup>¼</sup> *<sup>ψ</sup>U<sup>i</sup>* (67)

<sup>¼</sup> <sup>Γ</sup>*AFU*\_ *<sup>i</sup>* (68)

<sup>¼</sup> *<sup>δ</sup>AFU*€ *<sup>i</sup>* (69)

<sup>¼</sup> *õU*€ *<sup>i</sup>* (70)

0�<sup>1</sup> *U<sup>i</sup>* (71)

(62)

From Eq. (77) we have

$$\dot{\boldsymbol{U}}\_{n+1}^{i} = \overline{\mathbf{Y}}^{-1} \left[ \dot{\boldsymbol{U}}\_{n}^{i} + \frac{\Delta \tau}{2} \left[ \ddot{\boldsymbol{U}}\_{n}^{i} + \widehat{\boldsymbol{\mathcal{M}}}^{-1} \left( \widehat{\boldsymbol{\mathbb{Q}}}\_{n+1}^{ip} - \widehat{\boldsymbol{\mathcal{K}}}^{i} \boldsymbol{U}\_{n+1}^{i} \right) \right] \right] \tag{79}$$

Substituting *T*\_ *<sup>i</sup>*

 z}|{ *U*€ *i*

(85) is as follows:

*<sup>n</sup>*þ1, *<sup>T</sup>*\_ *<sup>i</sup>*

physical constants [57]:

*Cpjkl* ¼

*Uip <sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>U</sup><sup>i</sup>*

*U*\_ *i <sup>n</sup>*þ1, *<sup>U</sup>*€ *<sup>i</sup>*

**141**

*<sup>T</sup>*€ *<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> *<sup>i</sup>*

<sup>A</sup> <sup>¼</sup> <sup>X</sup> z}|{�<sup>1</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.92784*

*<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup>

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> , and *<sup>T</sup>*€*<sup>i</sup>*

**5. Numerical results and discussion**

The elasticity tensor is expressed as

The mechanical temperature coefficient is

*βpj* ¼

The thermal conductivity tensor is

2 6 4

*kpj* ¼

2 6 4

Mass density *<sup>ρ</sup>* <sup>¼</sup> 7820 kg*=*m<sup>3</sup> and heat capacity *<sup>c</sup>* <sup>¼</sup> 461 J*=*kg K.

this paper was to assess the impact of three temperatures on the acoustic

*<sup>n</sup>*þ<sup>1</sup> from Eq. (84) into Eq. (76), we obtain

*A New BEM for Modeling of Acoustic Wave Propagation in Three-Temperature Nonlinear…*

*<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup>

z}|{ � <sup>A</sup> z}|{

<sup>þ</sup> *<sup>T</sup>*€*<sup>i</sup> α*ð Þ *n*þ1

Now, a displacement-predicted staggered procedure for the solution of (80) and

The first step is to predict the propagation of the displacement wave field:

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

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

> *:*1 130*:*4 18*:*2 0 0 201*:*3 *:*4 116*:*7 21*:*0 0 0 70*:*1 *:*2 21*:*0 73*:*60 0 2*:*4 0 0 0 19*:*8 �8*:*0 0 0 00 �8*:*0 29*:*1 0 *:*3 70*:*1 2*:*4 0 0 147*:*3

> > 1*:*01 2*:*00 0 2*:*00 1*:*48 0 0 07*:*52

> > > 5*:*20 0 0 7*:*6 0 0 0 38*:*3

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

3 7 <sup>5</sup> � <sup>10</sup><sup>6</sup> <sup>N</sup>

3 7 5 <sup>ϒ</sup>�<sup>1</sup> *<sup>T</sup>*\_ *<sup>i</sup>*

� � B z}|{ *Ti α*ð Þ *n*þ1 �

�z}|{�<sup>1</sup> � � �

���

*<sup>n</sup>*þ<sup>1</sup> and *<sup>U</sup>*€ *<sup>i</sup>*

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> from Eqs. (79), (81), (82), and (86), respectively.

*<sup>α</sup>n* þ Δ*τ* 2

GPa (87)

Km<sup>2</sup> (88)

W*=*Km (89)

X

*<sup>n</sup>*þ<sup>1</sup> from Eqs. (77) and (75),

(86)

 z}|{ *U*€ *i*

z}|{ � <sup>B</sup> z}|{ *Ti α*ð Þ *n*þ1

*<sup>n</sup>*. The second step is to substitute *<sup>U</sup>*\_ *<sup>i</sup>*

� �

where <sup>ϒ</sup> <sup>¼</sup> *<sup>I</sup>* <sup>þ</sup> <sup>Δ</sup>*<sup>τ</sup>* <sup>2</sup> *M* z}|{�<sup>1</sup> Γ z}|{ � �. Substituting from Eq. (79) into Eq. (78), we derive

$$\begin{split} \boldsymbol{U}\_{n+1}^{i} &= \boldsymbol{U}\_{n}^{i} + \Delta \boldsymbol{r} \dot{\boldsymbol{U}}\_{n}^{i} + \frac{\Delta \tau^{2}}{4} \left[ \ddot{\boldsymbol{U}}\_{n}^{i} + \widetilde{\boldsymbol{M}}^{n-1} \left( \ddot{\boldsymbol{Q}}\_{n+1}^{ip} - \widetilde{\boldsymbol{\Gamma}}^{\cdot} \ddot{\boldsymbol{\Gamma}}^{\cdot -1} \right. \\ & \left. \left[ \dot{\boldsymbol{U}}\_{n}^{i} + \frac{\Delta \tau}{2} \right] \left[ \ddot{\boldsymbol{U}}\_{n}^{i} + \widetilde{\boldsymbol{M}}^{\cdot -1} \left( \widetilde{\boldsymbol{\Psi}}^{\cdot \, \, \, n+1} - \widetilde{\boldsymbol{K}}^{\cdot \, \, \,} \boldsymbol{U}\_{n+1}^{i} \right) \right] - \widetilde{\boldsymbol{K}}^{\cdot \, \, \, i} \boldsymbol{U}\_{n+1}^{i} \right) \right] \end{split} \tag{80}$$

Substituting *U*\_ *<sup>i</sup> <sup>n</sup>*þ<sup>1</sup> from Eq. (79) into Eq. (75), we obtain

$$\ddot{\boldsymbol{U}}\_{n+1}^{\boldsymbol{i}} = \widehat{\boldsymbol{M}}^{\boldsymbol{i}-1} \left[ \widehat{\boldsymbol{\he{Q}}}\_{n+1}^{\boldsymbol{i}\boldsymbol{p}} - \widehat{\boldsymbol{\Gamma}}^{\boldsymbol{i}} \left[ \widehat{\boldsymbol{Y}}^{\boldsymbol{i}-1} \Big[ \dot{\boldsymbol{U}}\_{n}^{\boldsymbol{i}} + \frac{\Delta \boldsymbol{\pi}}{2} \Big[ \ddot{\boldsymbol{U}}\_{n}^{\boldsymbol{i}} + \widehat{\boldsymbol{\boldsymbol{M}}}^{\boldsymbol{i}-1} \Big( \widehat{\boldsymbol{\he{Q}}}\_{n+1}^{\boldsymbol{i}\boldsymbol{p}} - \widehat{\boldsymbol{\boldsymbol{K}}}^{\boldsymbol{i}} \boldsymbol{U}\_{n+1}^{\boldsymbol{i}} \Big] \Big] \right] - \widehat{\boldsymbol{\mathcal{K}}}^{\boldsymbol{i}} \boldsymbol{U}\_{n+1}^{\boldsymbol{i}} \right] \tag{81}$$

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

$$\begin{split} \dot{T}\_{a(n+1)}^{j} &= \dot{T}\_{n}^{j} + \frac{\Delta \tau}{2} \Big( \ddot{T}\_{a(n+1)}^{j} + \dddot{T}\_{m}^{j} \Big) \\ &= \dot{T}\_{an}^{j} + \frac{\Delta \tau}{2} \Big( \widetilde{\mathbf{X}}^{-1} \Big[ \widetilde{\mathbf{Z}}^{-1} \ddot{U}\_{n+1}^{j} + \widetilde{\mathbf{R}}^{-} - \widetilde{\mathbf{A}}^{-} \dot{T}\_{a(n+1)}^{j} - \widetilde{\mathbf{B}}^{-} \dot{T}\_{a(n+1)}^{j} \Big] + \ddot{T}\_{am}^{j} \Big) \\ \\ \dot{T}\_{a(n+1)}^{i} &= T\_{an}^{i} + \frac{\Delta \tau}{2} \Big( \dot{T}\_{a(n+1)}^{j} + \ddot{T}\_{am}^{j} \Big) \\ &= \dot{T}\_{an}^{i} + \Delta \tau \dot{T}\_{am}^{j} \\ &+ \frac{\Delta \tau^{2}}{4} \Big( \ddot{T}\_{an}^{i} + \widetilde{\mathbf{X}}^{-1} \Big[ \widetilde{\mathbf{Z}}^{-1} \dddot{U}\_{n+1}^{i} + \widetilde{\mathbf{R}}^{-} - \widetilde{\mathbf{A}}^{-} \dot{T}\_{a(n+1)}^{j} - \widetilde{\mathbf{B}}^{-} \dot{T}\_{a(n+1)}^{i} \Big] \Big) \end{split} \tag{83}$$

From Eq. (82) we get

$$\begin{aligned} \dot{T}\_{a(n+1)}^i &= \Upsilon^{-1} \left[ \dot{T}\_{an}^i + \frac{\Delta \tau}{2} \left( \widetilde{\mathbf{X}^{-1}} \left[ \widetilde{\mathbf{U}^i}\_{n+1} + \widetilde{\mathbf{R}^-} - \widetilde{\mathbf{B}^-} \mathbf{T}\_{a(n+1)}^i \right] + \ddot{T}\_{an}^i \right) \right] \text{ (84)}\\ &\text{where } \Upsilon = \left( I + \frac{1}{2} \widetilde{\mathbf{A}^- \Delta \tau} \widetilde{\mathbf{X}^-} \right) \end{aligned} $$
 
$$\text{Substituting from Eq. (84) into Eq. (83), we have} $$

$$\begin{split} T^i\_{a(n+1)} &= T^i\_{an} + \Delta \tau \dot{T}^i\_{an} + \frac{\Delta \tau^2}{4} \left( \ddot{T}^i\_{an} + \widetilde{\mathbf{X}}^{-1} \Big[ \widetilde{\mathbb{Z}} \ \ddot{\mathbf{U}}^i\_{n+1} + \widetilde{\mathbb{R}} - \widetilde{\mathbf{A}}^i \left( \mathbf{Y}^{-1} \Big[ \dot{T}^i\_{an} \Big] \right) \right) \\ &+ \frac{\Delta \tau}{2} \Big( \widetilde{\mathbf{X}}^{-1} \Big[ \widetilde{\mathbb{Z}} \ \ddot{\mathbf{U}}^i\_{n+1} + \widetilde{\mathbb{R}} - \widetilde{\mathbf{B}}^i \left[ \mathbf{Y}^i\_{a(n+1)} \right] + \ddot{T}^i\_{an} \Big) \Big) - \ddot{\mathsf{B}} T^i\_{a(n+1)} \Big] \Big) \end{split} \tag{85}$$

*A New BEM for Modeling of Acoustic Wave Propagation in Three-Temperature Nonlinear… DOI: http://dx.doi.org/10.5772/intechopen.92784*

Substituting *T*\_ *<sup>i</sup> <sup>n</sup>*þ<sup>1</sup> from Eq. (84) into Eq. (76), we obtain

$$\ddot{T}\_{a(n+1)}\dot{\mathbf{r}}^{\text{A}} = \widehat{\mathbf{X}}^{-1} \Big[ \widehat{\mathbf{Z}}^{\text{I}} \ddot{\mathbf{U}}\_{n+1}^{i} + \widehat{\mathbf{R}}^{\text{I}} - \widehat{\mathbf{A}} \left( \mathbf{Y}^{-1} \Big[ \dot{T}\_{an}^{i} + \frac{\Delta \tau}{2} \Big( \widehat{\mathbf{X}}^{\text{A}} \Big)^{-1} \right)$$

$$\left[ \widehat{\mathbf{Z}}^{\text{I}} \ddot{\mathbf{U}}\_{n+1}^{i} + \widehat{\mathbf{R}}^{\text{I}} - \widehat{\mathbf{B}} \left. \mathbf{T}\_{a(n+1)}^{i} \right] + \ddot{T}\_{a(n+1)}^{i} \right) \Big] \Big) - \ast \widehat{\mathbf{B}}^{\text{I}} \ddot{T}\_{a(n+1)}^{i} \Big] \tag{86}$$

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: *Uip <sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>U</sup><sup>i</sup> <sup>n</sup>*. The second step is to substitute *<sup>U</sup>*\_ *<sup>i</sup> <sup>n</sup>*þ<sup>1</sup> and *<sup>U</sup>*€ *<sup>i</sup> <sup>n</sup>*þ<sup>1</sup> 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 *U*\_ *i <sup>n</sup>*þ1, *<sup>U</sup>*€ *<sup>i</sup> <sup>n</sup>*þ1, *<sup>T</sup>*\_ *<sup>i</sup> <sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> , and *<sup>T</sup>*€*<sup>i</sup> <sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> from Eqs. (79), (81), (82), and (86), respectively.

#### **5. Numerical results and discussion**

From Eq. (77) we have

*<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> <sup>ϒ</sup>�<sup>1</sup> *<sup>U</sup>*\_ *<sup>i</sup>*

*<sup>n</sup>* <sup>þ</sup> <sup>Δ</sup>*τU*\_ *<sup>i</sup>*

*n* þ Δ*τ* <sup>2</sup> *<sup>U</sup>*€ *<sup>i</sup>*

z}|{ � �

Γ

*U*€ *i <sup>n</sup>* þ *M* z}|{�<sup>1</sup>

<sup>ϒ</sup>�<sup>1</sup> *<sup>U</sup>*\_ *<sup>i</sup> <sup>n</sup>* þ Δ*τ* <sup>2</sup> *<sup>U</sup>*€ *<sup>i</sup>*

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> *<sup>T</sup>*€*<sup>i</sup>*

X z}|{�<sup>1</sup>

*αn*

*T*€*i <sup>α</sup><sup>n</sup>* þ X z}|{�<sup>1</sup>

*<sup>α</sup><sup>n</sup>* þ Δ*τ* 2

*<sup>α</sup><sup>n</sup>* <sup>þ</sup> <sup>Δ</sup>*τT*\_ *<sup>i</sup>*

 z}|{ *U*€ *i*

<sup>2</sup> A z}|{

� �

*αn*

*αn*

 z}|{ *U*€ *i*

 z}|{ *U*€ *i*

 z}|{ *U*€ *i*

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> *<sup>T</sup>*\_ *<sup>i</sup>*

X z}|{�<sup>1</sup>

Δ*τ* X � � z}|{�<sup>1</sup>

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

Δ*τ*<sup>2</sup> 4

*<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup>

*T*€*i <sup>α</sup><sup>n</sup>* þ X z}|{�<sup>1</sup>

� �

� ��

z}|{ � <sup>B</sup> z}|{ *Ti α*ð Þ *n*þ1

*<sup>α</sup><sup>n</sup>* þ

� �

z}|{�<sup>1</sup>

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

<sup>2</sup> *M* z}|{�<sup>1</sup>

*n* þ Δ*τ*<sup>2</sup> 4

> *U*€ *i <sup>n</sup>* þ *M*

*<sup>n</sup>* þ *M* z}|{�<sup>1</sup>

.

� �

z}|{*ip*

*<sup>n</sup>*þ<sup>1</sup> from Eq. (79) into Eq. (75), we obtain

 z}|{*ip*

� � � � � �

~ *ip <sup>n</sup>*þ<sup>1</sup> � Γ z}|{ ϒ� �<sup>1</sup>

� ��

� � � (80)

 z}|{*ip*

� � � � � � � �

� �

z}|{ � <sup>A</sup> z}|{ *T*\_ *i* A

*<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup>

*<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup>

� � � �

� �

z}|{ � <sup>A</sup> z}|{ *T*\_ *i*

z}|{ � <sup>B</sup> z}|{ *Ti α*ð Þ *n*þ1

� �

 z}|{ *U*€ *i*

> <sup>þ</sup> *<sup>T</sup>*€*<sup>i</sup> αn*

*<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup>

� h i � �

z}|{ � <sup>A</sup> z}|{

� <sup>B</sup>~*T<sup>i</sup>*

*α*ð Þ *n*þ1

� � � �

� �

*<sup>n</sup>*þ<sup>1</sup> � *K* z}|{ *Ui n*þ1

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> � B

z}|{ *Ti α*ð Þ *n*þ1

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> � B

z}|{ *Ti α*ð Þ *n*þ1

<sup>þ</sup> *<sup>T</sup>*€*<sup>i</sup> αn*

*<sup>n</sup>*þ<sup>1</sup> � *K* z}|{ *Ui n*þ1

*<sup>n</sup>* þ *M* z}|{�<sup>1</sup>

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

*<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup>

*<sup>n</sup>*þ<sup>1</sup> � *K* z}|{ *Ui n*þ1

> � *K* z}|{ *Ui n*þ1

(79)

� *K* z}|{ *Ui n*þ1

(81)

<sup>þ</sup> *<sup>T</sup>*€*<sup>i</sup> αn*

(82)

(83)

(84)

(85)

Y�<sup>1</sup> *T*\_ *<sup>i</sup> αn*

��

*U*\_ *i*

*Noise and Environment*

*<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>U</sup><sup>i</sup>*

*Ui*

*U*€ *i <sup>n</sup>*þ<sup>1</sup> ¼ *M* z}|{�<sup>1</sup>

*T*\_ *i*

*Ti*

*T*\_ *i*

*Ti*

**140**

þ Δ*τ* 2

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>T</sup>*\_ *<sup>i</sup>*

<sup>¼</sup> *<sup>T</sup>*\_ *<sup>i</sup> <sup>α</sup><sup>n</sup>* þ Δ*τ* 2

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>T</sup><sup>i</sup>*

<sup>¼</sup> *<sup>T</sup><sup>i</sup>*

þ Δ*τ*<sup>2</sup> 4

From Eq. (82) we get

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> <sup>ϒ</sup>�<sup>1</sup> *<sup>T</sup>*\_ *<sup>i</sup>*

where <sup>ϒ</sup> <sup>¼</sup> *<sup>I</sup>* <sup>þ</sup> <sup>1</sup>

X z}|{�<sup>1</sup>

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>T</sup><sup>i</sup>*

where <sup>ϒ</sup> <sup>¼</sup> *<sup>I</sup>* <sup>þ</sup> <sup>Δ</sup>*<sup>τ</sup>*

*U*\_ *i n* þ Δ*τ* 2 � �

> z}|{*ip*

*<sup>n</sup>* þ Δ*τ* 2 *T*€*i*

> *<sup>α</sup><sup>n</sup>* þ Δ*τ* 2 *T*\_ *i*

*<sup>α</sup><sup>n</sup>* <sup>þ</sup> <sup>Δ</sup>*τT*\_ *<sup>i</sup>*

*<sup>n</sup>*þ<sup>1</sup> � Γ z}|{

Substituting *U*\_ *<sup>i</sup>*

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

$$C\_{pjkl} = \begin{bmatrix} 430.1 & 130.4 & 18.2 & 0 & 0 & 201.3\\ 130.4 & 116.7 & 21.0 & 0 & 0 & 70.1\\ 18.2 & 21.0 & 73.6 & 0 & 0 & 2.4\\ 0 & 0 & 0 & 19.8 & -8.0 & 0\\ 0 & 0 & 0 & -8.0 & 29.1 & 0\\ 201.3 & 70.1 & 2.4 & 0 & 0 & 147.3 \end{bmatrix} \text{GPa} \tag{87}$$

The mechanical temperature coefficient is

$$
\beta\_{pj} = \begin{bmatrix}
\mathbf{1.01} & \mathbf{2.00} & \mathbf{0} \\
\mathbf{2.00} & \mathbf{1.48} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{7.52}
\end{bmatrix} \cdot \mathbf{10}^6 \frac{\mathbf{N}}{\mathbf{K} \mathbf{m}^2} \tag{88}
$$

The thermal conductivity tensor is

$$k\_{pj} = \begin{bmatrix} 5.2 & 0 & 0 \\ 0 & 7.6 & 0 \\ 0 & 0 & 38.3 \end{bmatrix} \text{W/Km} \tag{89}$$

Mass density *<sup>ρ</sup>* <sup>¼</sup> 7820 kg*=*m<sup>3</sup> and heat capacity *<sup>c</sup>* <sup>¼</sup> 461 J*=*kg 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 *Te*, *Ti*, and *Tp* and total temperature *T T* ¼ *Te* þ *Ti* þ *Tp* 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.

**Figures 3** and **4** show the displacement *u*<sup>1</sup> and *u*<sup>2</sup> acoustic waves propagation along *x*-axis for the three temperatures *Te*,*Ti*,*Tp* 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.

In order to evaluate the influence of the functionally graded parameter and initial stress on the propagation of the displacement waves *u*<sup>1</sup> and *u*<sup>2</sup> 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,

**Figure 4.**

**Figure 5.**

**Figure 6.**

**143**

*Propagation of the displacement* **u***<sup>2</sup> waves along the* **x***-axis.*

*DOI: http://dx.doi.org/10.5772/intechopen.92784*

*A New BEM for Modeling of Acoustic Wave Propagation in Three-Temperature Nonlinear…*

*Propagation of the displacement* **u***<sup>1</sup> waves along the* **x***-axis.*

*Propagation of the displacement* **u***<sup>2</sup> waves along the* **x***-axis.*

**Figure 2.** *Propagation of the temperature Te, Ti, Tp and T waves along the* **x***-axis.*

**Figure 3.** *Propagation of the displacement u*<sup>1</sup> *waves along the x-axis.*

*A New BEM for Modeling of Acoustic Wave Propagation in Three-Temperature Nonlinear… DOI: http://dx.doi.org/10.5772/intechopen.92784*

#### **Figure 4.**

displacement waves; the numerical outcomes are completed and delineated graph-

**Figure 2** shows the three temperatures *Te*, *Ti*, and *Tp* and total temperature

 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

**Figures 3** and **4** show the displacement *u*<sup>1</sup> and *u*<sup>2</sup> acoustic waves propagation along *x*-axis for the three temperatures *Te*,*Ti*,*Tp* and total temperature *T*. It was noticed from **Figures 3** and **4** that the three temperatures and total temperature

In order to evaluate the influence of the functionally graded parameter and initial stress on the propagation of the displacement waves *u*<sup>1</sup> and *u*<sup>2</sup> 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,

ically for electron, ion, phonon, and total temperatures.

have great effects on the acoustic displacement waves.

*Propagation of the temperature Te, Ti, Tp and T waves along the* **x***-axis.*

*Propagation of the displacement u*<sup>1</sup> *waves along the x-axis.*

*T T* ¼ *Te* þ *Ti* þ *Tp*

*Noise and Environment*

the connected fields.

**Figure 2.**

**Figure 3.**

**142**

*Propagation of the displacement* **u***<sup>2</sup> waves along the* **x***-axis.*

**Figure 5.** *Propagation of the displacement* **u***<sup>1</sup> waves along the* **x***-axis.*

**Figure 6.** *Propagation of the displacement* **u***<sup>2</sup> waves along the* **x***-axis.*

where A represents the numerical results for homogeneous ð Þ *m* ¼ 0 structures in the absence of initial stress (*P* ¼ 0), B represents the numerical results for functionally graded ð Þ *m* ¼ 0*:*5 structures in the absence of initial stress (*P* ¼ 0), C represents the numerical results for homogeneous ð Þ *m* ¼ 0 structures in the presence of initial stress (*P* ¼ 0*:*5), and D represents the numerical results for functionally graded ð Þ *m* ¼ 0*:*5 structures in the presence of initial stress (*P* ¼ 0*:*5). It can be seen from **Figures 5** and **6** that the effects of initial stress and functionally graded parameter are very pronounced.

Since there are no available results for the three-temperature thermoelastic problem, except for Fahmy's research [10–14]. For comparison purposes with the special cases of other methods treated by other authors, we only considered a onedimensional 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

*A New BEM for Modeling of Acoustic Wave Propagation in Three-Temperature Nonlinear…*

Propagation of displacements and temperature acoustic waves in threetemperature 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 onedimensional 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

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

FDM or FEM results.

*DOI: http://dx.doi.org/10.5772/intechopen.92784*

**6. Conclusion**

FEM results.

laser ultrasonics.

**145**

**Figure 7.** *Propagation of the temperature T waves along the* **x***-axis for BEM, FDM, and FEM.*

**Figure 8.** *Propagation of the displacement* **u***<sup>1</sup> waves along the* **x***-axis for BEM, FDM, and FEM.*

*A New BEM for Modeling of Acoustic Wave Propagation in Three-Temperature Nonlinear… DOI: http://dx.doi.org/10.5772/intechopen.92784*

Since there are no available results for the three-temperature thermoelastic problem, except for Fahmy's research [10–14]. For comparison purposes with the special cases of other methods treated by other authors, we only considered a onedimensional 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.

## **6. Conclusion**

where A represents the numerical results for homogeneous ð Þ *m* ¼ 0 structures in the absence of initial stress (*P* ¼ 0), B represents the numerical results for functionally graded ð Þ *m* ¼ 0*:*5 structures in the absence of initial stress (*P* ¼ 0), C represents the numerical results for homogeneous ð Þ *m* ¼ 0 structures in the presence of initial stress (*P* ¼ 0*:*5), and D represents the numerical results for functionally graded ð Þ *m* ¼ 0*:*5 structures in the presence of initial stress (*P* ¼ 0*:*5). It can be seen from **Figures 5** and **6** that the effects of initial stress and functionally graded

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

*Propagation of the displacement* **u***<sup>1</sup> waves along the* **x***-axis for BEM, FDM, and FEM.*

parameter are very pronounced.

*Noise and Environment*

**Figure 7.**

**Figure 8.**

**144**

Propagation of displacements and temperature acoustic waves in threetemperature 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 onedimensional 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.

*Noise and Environment*
