1. Introduction

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

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

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

A brief summary of the paper is as follows: Section 1 introduces the background and provides the readers with the necessary information to books and articles for a better understanding of wave propagation problems in three-temperature nonlinear generalized magneto-thermoelastic ISMFGA structures and their applications. Section 2 describes the formulation of the new theory and introduces the partial differential equations that govern its related problems. Section 3 outlines continuity and initial and boundary conditions of the considered problem. Section 4 discusses the implementation of the new BEM and its implementation for solving the governing equations of the problem to obtain the three temperatures and displacement fields. Section 5 presents the new numerical results that describe the displacement waves and three-temperature waves under the effect of initial stress on the homogeneous and functionally graded structures.

2. Formulation of the problem

Consider a multilayered structure with n functionally graded layers in the 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 ith layer is denoted by h and hi, respectively. The considered multilayered structure which occupies the region R=xyz:0<x<h0<y<b0<z<a has been placed in a primary magnetic field H0 acting in the direction of the y-axis.

Figure 1.
Geometry of the FGA structure.

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

σab,b+τab,b−Γab=ρix+1mu¨aiE1

σab=χ+1mCabfgiuf,gi−βabiTi−T0+τ1ṪiE2

τab=μix+1mh˜aHb+h˜bHa−δbah˜fHfE3

Γab=Pix+1m∂uai∂xb−∂ubi∂xaE4

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

∇δ1jKαi∗+δ2jKαi∇Tαirτ−W¯rτ=cαiρiδ1δ1j∂Tαirτ∂τE5

where

W¯rτ=ρiWeITei−TIi+ρiWerTei−Tpi+W¯¯,α=e,δ1=1−ρiWeITei−Tpi+W¯¯,α=I,δ1=1−ρiWerTei−Tpi+W¯¯,α=p,δ1=Tp3E6

in which

W¯¯rτ=−δ2jKαiṪα,ab+βabTα0τ0+δ2ju¨a,b+ρicαiτ0+δ1jτ2+δ2jT¨α−QxτE7

and

WeI=ρiAeITe−2/3,Wer=ρiAerTe−1/2,Kα=AαTα5/2,α=e,I,Kp=ApTp3+BE8

The total energy of unit mass can be described by

P=Pe+PI+Pp,Pe=cαeTei,PI=cαITli,Pp=14cαpTp4iE9

where σab, τab, and uki are the mechanical stress tensor, Maxwell’s electromagnetic stress tensor, and displacement vector, respectively; Tα0i is the reference temperature; Tαi is the temperature; Cabfgi and βabi are, respectively, the constant elastic moduli and stress-temperature coefficients of the anisotropic medium; μi, h˜, Pi, ρi, and cαi are the magnetic permeability, perturbed magnetic field, initial stress in the ith layer, density, and specific heat capacity, respectively; τ is the time; τ0, τ1, and τ2 are the relaxation times; i=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 τab,b=μ0iϵabfJbHf is the a-component of the Lorentz force and Jτ=J0ττ32eττ3 is the temporal profile of a non-Gaussian laser pulse, J0 is the total energy intensity, and Qxτ=1−Rx0exax0Jτ,a=1,2,3 is the heat source intensity.

According to Fahmy [57], we notice that there are two special cases of the Green and Naghdi theory of type III; when Kαi→0, the equations of GN III theory are reduced to the GN theory type II, and when Kαi∗→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:

Tαixzτx=hi=Tαi+1(xztτ)x=hiE10

qixzτx=hi=qi+1(xzτ)x=hiE11

ufixzτx=hi=ufi+1xzτx=hiE12

t¯aixzτx=hi=t¯ai+1xzτx=hiE13

where n is the total number of layers, t¯a are the tractions, which are defined by t¯a=σabnb, and i=1,2,…,n−1.

The remaining initial and boundary conditions for the current study are

ufixz0=u̇fixz0=0forxz∈R∪CE14

ufixzτ=Ψfxzτforxz∈C3E15

t¯aixzτ=Φfxzτforxz∈C4,τ>0,E16

Tαixz0=Tαixz0=0forxz∈R∪CE17

Tαixyτ=f¯xyτforxy∈C1,τ>0E18

qixzτ=h¯xzτforxz∈C2,τ>0E19

where Ψf, Φf,f, and h¯ are suitably prescribed functions and C=C1∪C2=C3∪C4, C1∩C2=C3∩C4=Ø.

4. BEM numerical implementation

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

Lgbufi=ρiüai−DaTαi−Pi∂ubi∂xa−∂uai∂xb=fgbE20

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:

Lgbufi=fgb,E21

LabTαi=fabE22

where the operators Lgb, fgb, Lab, and fab are as follows:

Lgb=Dabf∂∂xb+Daf+ΛDa1f,Lab=δ2jKαi∗∇E23

fgb=ρiu¨ai−DaTαi−Pi∂ubi∂xa−∂uai∂xbE24

fab=∇δ1jKαi∇+ρicαiδ1δ1jx+1mṪαi+W¯rτE25

where

Dabf=Cabfgε,ε=∂∂xg,Daf=μH02∂∂xa+δa1Λ∂∂xf,Da=−βabi∂∂xb+δb1Λ+τ1∂∂xb+Λ∂∂τ,Λ=mx+1.

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

∫RLgbufi−fgbudai∗dR=0E26

Now, the fundamental solution udfi∗ and traction vectors tdai∗ and tai can be written as follows:

Lgbuafi∗=−δadδxξE27

tdai∗=Cabfgudf,gi∗nbE28

tai=t¯aix+1m=Cabfguf,gi−βabiTαi+τ1TαinbE29

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:

udiξ=∫Cudai∗tai−tdai∗uai+udai∗βabiTαinbdC−∫Rfgbudai∗dRE30

The fundamental solution Ti∗ can be defined as

LabTi∗=−δxξE31

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

∫RLabTαiTαi∗−LabTαi∗TαidR=∫Cqi∗Tαi−qiTαi∗dCE32

where

qi=−KαiTα.binaE33

qi∗=−KαiTα,bi∗naE34

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

Tαiξ=∫Cqi∗Tαi−qiTαi∗dC−∫RfabTαi∗dRE35

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

[ udi(ξ)Tαi(ξ) ]=∫C{ −[ tdai∗−uaai∗βabnb0−qi∗ ] [ uaiTαi ]+[ udai∗00−Tαi∗ ][ τaiqi ] }dC−∫R[ udai∗00−Tαi∗ ][ fgb−fab ]dR E36

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

UAi=uaia=A=1,2,3TαiA=4E37

TαAi=taia=A=1,2,3qiA=4E38

UDAi∗=udai∗d=D=1,2,3;a=A=1,2,30d=D=1,2,3;A=40D=4;a=A=1,2,3−Tαi∗D=4;A=4E39

T˜αDAi∗=taai∗d=D=1,2,3;a=A=1,2,3−u˜di∗d=D=1,2,3;A=40D=4;a=A=1,2,3−qi∗D=4;A=4E40

u˜di∗=udai∗βafinfE41

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

UDiξ=∫CUDAi∗TαAi−T˜αDAiUAidC−∫RUDAi∗SAdRE42

The vector SA can be split as follows

SA=SA0+SAT+SAu+SAṪ+SAT¨+SAu¨E43

where

SA0=0A=1,2,31−Rx0e−xax0JτA=4E44

SAT=ωAFUFiwithωAF=−DaA=1,2,3;F=4∇δ2jKαi∗∇otherwiseE45

SAu=ψUFiwithψ=Pi∂∂xb−∂∂xaA=1,2,3;F=1,2,3,0A=4;F=4E46

SAṪ=ΓAFU̇FiwithΓAF=−βabiτ1∂∂xb+Λ∂∂τA=4;F=4ρicαiδ1δ1jotherwiseE47

SAT¨=δAFU¨FiwithδAF=0A=4;F=4ρicαiτ0+δ1jτ2+δ2jotherwiseE48

SAu¨=õU¨Fiwithõ=ρiA=1,2,3,F=1,2,3,βabiTα0iτ0+δ2iA=4;F=4E49

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

[ SA ]=−[ 0−1−Rx0e(−xax0)J(τ) ]+{ [ −DaTαi∇(δ2jKα i ∗)∇Tαi ] }+[ Pi(ub,ai0−ua,bi)0 ]+[ −βabiτ1(∂∂xb+Λ)T˙αiρicαiδ1δ1jT˙αi ]+ρicαi[ (τ0+δ1jτ2+δ2j) ][ 0T¨αi ]+[ ρiu¨aiβabiTα0i(τ0+δ2j)u¨f,gi ]E50

To transform the domain integral in (42) to the boundary, we approximate the source vector S_A by a series of given tensor functions fAEq and unknown coefficients αEq as follows:

SA≈∑q=1EfAEqαEqE51

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

UDξ=∫CUDAi∗TαAi−T˜αDAi∗UAidC−∑q=1N∫RUDAi∗fAEqdRαEqE52

By implementing the WRM to the following equations.

Lgbufeiq=faeqE53

LabTαiq=fpjq_,E54

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

udeiqξ=∫Cudai∗taeiq−tdai∗uaeiqdC−∫Rudai∗faeqdRE55

Tαiqξ=∫Cqi∗Tαiq−qiqTαi∗dC−∫RfqTαi∗dRE56

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

UDEiqξ=∫CUDAi∗TαAEiq−TαDAi∗UAEiqdC−∫RUDAi∗fAEiqdRE57

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

UDiξ=∫CUDAi∗TαAi−T⌣αDAi∗UAidC+∑q=1EUDEiqξ+∫CTαDAi∗UAEiq−UDAi∗TαAEiqdCαEqE58

To calculate interior stresses, (58) is differentiated with respect to ξl as follows:

∂UDiξ∂ξl=−∫CUDA,li∗TαA,li−T⌣αDA,li∗UAidC+∑q=1E∂UDEiqξ∂ξl−∫CTαDA,li∗UαAEiq−UDA,li∗TαAEiqdCαEqE59

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

ζ⌣U−ηTα=ζU⌣−η℘⌣αE60

The generalized displacements and velocities are approximated in terms of a series of known tensor functions fFDq and unknown coefficients γDq and γ˜Dq:

UFi≈∑q=1NfFDqxγDqE61

where

fFDq=ffdqf=F=1,2,3;d=D=1,2,3fqF=4;D=40otherwiseE62

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

UF,gi≈∑q=1NfFD,gqxγKqE63

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

SAT=∑q=1NSAFfFD,gqγDqE64

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

S⌣=Jα¯,Ui=J′γ,E65

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

S⌣Tαi=BTγE66

SAu=ψ¯UiE67

S⌣Ṫαl=Γ¯AFU̇iE68

S⌣T¨α̇l=δ¯AFU¨iE69

S⌣u¨=õ¯U¨iE70

where ψ¯, Γ¯AF, δ¯AF, and õ¯ are assembled using the submatrices ψ'ΓAF, δAF, and õ, respectively.

Solving the system (65) for α and γ yields

α¯=J−1S⌣,γ=J'−1UiE71

Now, the coefficients α can be expressed in terms of nodal values of the unknown displacements Uⁱ, velocities U̇i, and accelerations U¨i as follows:

α¯=J−1(S⌣0+BTJ'−1+ψ¯]Ui+Γ¯AFU̇i+õ¯+δ¯AFU¨iE72

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:

M⏞U¨i+Γ⏞U̇i+K⏞Ui=Q⏞iE73

X⏞T¨αi+A⏞Ṫαi+B⏞Tαi=Z⏞U¨i+R⏞E74

where V=η℘⌣−ζU⌣J−1, M⏞=Võ¯+δ¯AF, Γ⏞=VΓ¯AF, K⏞=−ζ⌣+VBTJ'−1+ψ¯,Q⏞i=−ηT⌣+VS⌣0, X⏞=−ρicix+1m, A⏞=kabi∂∂xa∂∂xb, B⏞=kabi∗∂∂xa∂∂xb, Z⏞=βabiT0_,R⏞=−1−Rx0exax0Jτ

where V, M⏞, Γ⏞, K⏞, A⏞, and B⏞ represent the volume, mass, damping, stiffness, capacity, and conductivity matrices, respectively, and U¨i,U̇i,Ui,Ti, and Q⏞i represent the acceleration, velocity, displacement, temperature, and external force vectors, respectively.

In many applications, the coupling term Z⏞U¨n+1i that appears in the heat conduction equation and which is induced by the effect of the strain rate is negligible.

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

M⏞U¨n+1i+Γ⏞U̇n+1i+K⏞Un+1i=Q⏞n+1ipE75

X⏞T¨αn+1i+A⏞Ṫαn+1i+B⏞Tαn+1i=Z⏞U¨n+1i+R⏞E76

where Q⏞n+1ip=ηTαn+1ip+VS⌣0 and Tαn+1ip is the predicted temperature.

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

Un+1i=U̇ni+Δτ2U¨n+1i+U¨ni=U̇ni+Δτ2U¨ni+M⏞−1Q⏞n+1ip−Γ⏞U̇n+1i−K⏞Un+1iE77

Un+1i=Uni+Δτ2U̇n+1i+U̇ni=Uni+ΔτU̇ni+Δτ24U¨ni+M⏞−1Q˜n+1ip−Γ⏞U̇n+1i−K⏞Un+1iE78

From Eq. (77) we have

U̇n+1i=ϒ¯−1U̇ni+Δτ2U¨ni+M⏞−1Q⏞n+1ip−K⏞Un+1iE79

where ϒ¯=I+Δτ2M⏞−1Γ⏞_.

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

Un+1i=Uni+ΔτU˙ni+Δτ24 [ U¨ni+M⏞−1 ( ℚ˜n+1ip−Γ⏞ϒ¯−1[ U˙ni+Δτ2 ][ U¨ni+M⏞ −1(ℚ⏞n+1ip−K⏞Un+1i) ]−K⏞Un+1i ) ]E80

Substituting U̇n+1i from Eq. (79) into Eq. (75), we obtain

U¨n+1i=M⏞−1 [ ℚ⏞n+1ip−Γ⏞ [ ϒ¯−1 [ U˙ni+Δτ2[ U¨ni+M⏞−1(ℚ⏞n+1ip−K⏞Un+1i) ] ] ]−K⏞Un+1i ]E81

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

Ṫαn+1i=Ṫni+Δτ2T¨αn+1i+T¨αni=Ṫαni+Δτ2X⏞−1Z⏞U¨n+1i+R⏞−A⏞Ṫαn+1iA−B⏞Tαn+1i+T¨αniE82

Tαn+1i=Tαni+Δτ2Ṫαn+1i+Ṫαni=Tαni+ΔτṪαni+Δτ24T¨αni+X⏞−1Z⏞U¨n+1i+R⏞−A⏞Ṫαn+1i−B⏞Tαn+1iE83

From Eq. (82) we get

Ṫαn+1i=ϒ−1Ṫαni+Δτ2X⏞−1Z⏞U¨n+1i+R⏞−B⏞Tαn+1i+T¨αniE84

where ϒ=I+12A⏞ΔτX⏞−1

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

Tα(n+1)i=Tαni+ΔτT˙αni+Δτ24( T¨αni+X⏞−1 [ ℤ⏞U¨n+1i+ℝ⏞−A⏞( Y−1[ T˙αni ] +Δτ2(X⏞−1[ ℤ⏞U¨n+1i+ℝ⏞−B⏞Tα(n+1)i ]+T¨αni) )−B˜Tα(n+1)i ] )E85

Substituting Ṫn+1i from Eq. (84) into Eq. (76), we obtain

T¨α(n+1)iA=X⏞−1 [ ℤ⏞U¨n+1i+ℝ⏞−A⏞( ϒ−1 [ T˙αni+Δτ2 (X⏞−1[ ℤ⏞U¨n+1i+ℝ⏞−B⏞Tα(n+1)i ]+T¨α(n+1)i) ] )−*B⏞Tα(n+1)i ]E86

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

The first step is to predict the propagation of the displacement wave field: Un+1ip=Uni. The second step is to substitute U̇n+1i and U¨n+1i 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 U̇n+1i, U¨n+1i, Ṫαn+1i, and T¨αn+1i from Eqs. (79), (81), (82), and (86), respectively.

5. Numerical results and discussion

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

The elasticity tensor is expressed as

Cpjkl=430.1130.418.200201.3130.4116.721.00070.118.221.073.6002.400019.8−8.00000−8.029.10201.370.12.400147.3GPaE87

The mechanical temperature coefficient is

βpj=1.012.0002.001.480007.52⋅106NKm2E88

The thermal conductivity tensor is

kpj=5.20007.600038.3W/KmE89

Mass density ρ=7820kg/m3 and heat capacity c=461J/kgK.

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 TT=Te+Ti+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.

Figure 2.
Propagation of the temperature T_e, T_i, T_p and T waves along the x-axis.

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

Figure 3.
Propagation of the displacement u1 waves along the x-axis.

Figure 4.
Propagation of the displacement u₂ waves along the x-axis.

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

Figure 5.
Propagation of the displacement u₁ waves along the x-axis.

Figure 6.
Propagation of the displacement u₂ waves along the x-axis.

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

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

Figure 8.
Propagation of the displacement u₁ waves along the x-axis for BEM, FDM, and FEM.

6. Conclusion

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

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

Zhenhua Dan

Chapters authored

Related collaborators

Jolanta Kumirska

Mirko X. Weinhold

Małgorzata Czerwicka

Zbigniew Kaczyński

Anna Bychowska

Hiroshi Tamura

Rangasamy Jayakumar

M. Prabaharan

Tetsuya Furuike

Shantikumar V. Nair