Open access peer-reviewed chapter - ONLINE FIRST

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

By Mohamed Abdelsabour Fahmy

Submitted: April 3rd 2020Reviewed: July 14th 2020Published: August 14th 2020

DOI: 10.5772/intechopen.93376

Downloaded: 21

Abstract

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

Keywords

  • boundary element method
  • modeling and simulation
  • laser ultrasonics
  • three-temperature
  • fractional-order
  • nonlinear generalized micropolar poro-thermoelasticity
  • functionally graded anisotropic structures

1. Introduction

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

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

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

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

A brief summary of the chapter is as follows: Section 1 introduces the background and provides the readers with the necessary information to books and articles for a better understanding of wave propagation problems in three-temperature nonlinear generalized micropolar poro-thermoelastic FGA structures and their applications. Section 2 describes the BEM modeling of the new theory and introduces the partial differential equations that govern its related problems. Section 3 outlines BEM simulation of temperature field. Section 4 discusses BEM simulation of micropolar porothermoelastic field to obtain the three temperatures thermal stress wave propagation. Section 5 presents the new numerical results that describe the thermal stress wave propagation under the effect of three-temperature on the FGA structures.

2. BEM modeling of the problem

We consider an anisotropic micropolar porous smart structure in a rectangular Cartesian system x1,x2,x3shown in Figure 1, with a configuration Rbounded by a closed surface S, and Sii=1,2,3,4,5,6denotes subsets of Ssuch that S1+S2=g3+S4=S5+S6. The governing equations for modeling of fractional three-temperature nonlinear generalized micropolar poro-thermoelastic problems of functionally graded anisotropic structures (FGA) can be expressed as [7].

Figure 1.

Geometry of the FGA structure.

σij,j+ρFi=ρu¨i+ϕρFv¨iE1
mij,j+εijkσjk+ρMi=ω¨iE2
ζ̇+qi,i=CE3

where

σij=z+1mCijklℵeδijAδijp+αuj,iεijkωkβijTαE4
Cijkl=Cklij=Cjikl,βij=βjiE5
mij=x+1mαωk,kδij+α¯ωi,j+α¯¯=ωj,iE6
ζ=x+1mAuk,k+ϕ2RpE7
qi=x+1mk¯p,i+ρFu¨i+ρ0+ϕρFϕv¨iE8
ϵij=εijεijkχ+1mrkωkE9
εij=12ui,j1+uj,iE10
ri=12εiklul,kE11

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

DτaTαrτ=ξKαTαrτ+ξW¯rτ,ξ=1cαρδ1E12

where

W¯rτ=ρWeiTeTi+ρW¯¯erTeTp+W¯¯,α=e,δ1=1ρWeiTeTi+W¯¯,α=i,δ1=1ρWerTeTp+W¯¯,α=p,δ1=Tp3E13
W¯¯rτ=δ2nKαṪα,ij+βijTα0Åδ1nu̇i,j+τ0+δ2nu¨i,j+ ρcατ0+δ1nT2+δ2nT¨αE14

and

Wei=ρAeiTe2/3,Wer=ρAerTe1/2,Kα=AαTα5/2,α=e,i,Kp=ApTp3+BE15

The total energy is

P=Pe+Pi+Pp,Pe=ceTe,Pi=ciTi,Pp=14cpTp4E16

where we considered that θ=Te+Ti+Tr,Te,Ti, and Trare temperature functions of electron, ion, and photon, respectively, Ke,Ki, and Krare conductive coefficients of electron, ion, and photon, respectively, and ρis the material density which is constant inside each subdomain.

3. BEM simulation for temperature field

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

According to finite difference scheme of Caputo at times f+1Δτand fΔτ, we obtain [52].

DτaTαf+1+DτaTαfj=0kWa,jTαf+1jrTαfjrE17

where

Wa,0=ΔτaΓ2a,Wa,j=Wa,0j+11aj11aE18

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

Wa,0Tαf+1rKαxTα,IIf+1rKα,I,xTα,If+1r=Wa,0TαfrKαxTα,IIfrKα,I,xTα,J]frj=1fWa,jTαf+1jrTαfjr+W¯mf+1xτ+W¯mfxτE19

where, j=1,2,,F,f=0,1,2,,F.

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

CTα=sTαqTαqdSRKαDTατTαdRE20

Now, to transform the domain integral in (20) into the boundary, we assume that the time-temperature derivative can be approximated by using a series of known functions fjrand unknown coefficients ajτas

Tατj=1NfjrjajτE21

We assume that T̂αjis a solution of

2T̂αj=fjE22

Thus, Eq. (20) can be written as

CTα=STαqTαqdS+j=1NajτD1CT̂αjSTαjqq̂jTαdSE23

where

q̂j=KαT̂αjnE24

and

ajτ=i=1Nfji1TriττE25

In which, the entries of fji1are the coefficients of F1with matrix Fdefined as

Fji=fjriE26

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

CṪα+HTα=GQE27

where the matrices Hand Gare depending on current time step, boundary geometry, and material properties.

The diffusion matrix can be defined as

C=HT̂αGQ̂F1D1E28

with

T̂ij=T̂jxiE29
Q̂ij=q̂jχiE30

In order to solve Eq. (27) numerically, the functions Tαand qare interpolated as

Tα=1θTαm+θTαm+1E31
q=1θqm+θqm+1E32

The time derivative of the temperature can be written as

Ṫα=dTα=Tαm+1Tαmτm+1τm=Tαm+1TαmΔτm,θ=ττmτm+1τm,0θ1E33

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

cΔτm+θHTαm+1θGQm+1=cΔτm1θHTαm+1θGQmE34

which can be written as follows [10].

aX=bE35

where ais an unknown matrix, while Xand bare known matrices.

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

4. BEM simulation for micropolar poro-thermoelastic fields

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

Rσij,j+UiuidR=0E36
Rmij,j+εijkσjk+ViωidR=0E37
Rqi+ζ̇iCipidR=0E38

in which

Ui=φijJj+ρFiρu¨iϕρFv¨iE39
Vi=ρMiJω¨iE40

where ui,ωiand piare weighting functions, ui,ωi, and piare approximate solutions as shown in Eqs. (4)(11)

The boundary conditions are

ui=u¯ionS1E41
λi=σiini=λ¯ionS2E42
ωi=ω¯ionS3E43
μi=mijnj=μ¯ionS4E44
p=p¯onS5E45
L=pn=L¯onS6E46

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

Rσijui,jdR+RUiuidR=S2λiuidSE47
Rmijωi,jdR+RεijkσjkωidR+RViωidR=S4μiωidSE48
Rqpi,idR+Rζ̇ipidRRCipidR=S6LipidSE49

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

Rσij,juidR+Rmij,j+εijkσjkωidR+RUiuidR+RViωidRRqpi,idR+Rζ̇ipidRRCipidR=S2λiλ¯iuidS+S1u¯iuiλidS+S4μiμ¯iωidS+S3ω¯iωiμidS+S6LiL¯ipidS+S5p¯ipiLidSE50

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

RσijεijdRRmij,jωi,jdR+RUiuidR+RViωidRRqpi,idR+Rζ̇ipidRRCipidR=S2λ¯iuidSS1λiuidS+S1u¯iuiλidSS4μ¯iωidSS3μωidS+S3ω¯iωiμidSS6L¯ipidSS6LipidS+S5p¯ipiLidSE51

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

σij=Aijklεkl,mij=Bijklωk,l where Aijkl=Aklij and Bijkl=BklijE52

Hence, Eq. (51) can be rewritten as

RσijεijdRRmij,jωi,jdR+RUiuidR+RViωidRRqpi,idR+Rζ̇ipidRRCipidR=S2λ¯iuidSS1λiuidS+S1u¯iuiλidSS4μ¯iωidSS3μiωidS+S3ω¯iωiμidSS6L¯ipidSS6LipidS+S5p¯ipiLidSE53

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

Rσij,juidR+Rmij,j+εijkσjkωidR=SuiλidSSωiμidSSpiLidS+SλiuidS+SμiωidS+SLipidSE54

The weighting functions for Ui=Δnand Vi=0along the unit vector direction elare as follows:

σlj,j+Δnel=0E55
mij,j+εijkσjk=0E56

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

ui=uliel,ωi=ωliel,pi=pliel,λi=λliel,μi=μliel,Li=LlielE57

The obtained weighting functions for a point load Ui=0and Vi=Δnalong the unit vector direction e1were next used as follows:

σij,j=0E58
mlj,j+εljkσjk+Δnel=0E59

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

ui=uiiel,ωi=ωliel,pi=pliel,λi=λliel,μi=μliel,Li=LlielE60

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

Clinuin=SλliuidSSμliωidSSLlipidS+SuliλidS+SωliμidS+SpliLidSE61
Clinωin=SλliuidSSμliωidSSLlipidS+SuliλidS+SωliμidS+SpliLidSE62

Thus, we can write

Cnqn=SpqdS+SqpdS+SapdS+SbpndSE63

where

Cn=C11C12C21C22,q=u11u12ω13u21u22ω23u31u32ω33,p=λ11λ12λ13λ21λ22μ23λ31λ32μ33E64
q=u1u2ω3,p=λ1λ2μ3,a=a1a20,b=b1b20

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

q=ψqj,p=ψpj,p=ψ0pj,pn=ψ0pnjE65

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

Cnqn=j=1NeΓjpψdΓqj+j=1NeΓjqψdΓpj+j=1NeΓjaψ0dΓpj+j=1NeΓjbψ0dΓpnjE66

Equation after integration can be written as

Ciqi=j=1NeĤijqj+j=1NeĜijpj+j=1Neâijpj+j=1Neb̂ijpnjE67

By using the following representation

Hij=ĤijifijĤij+Ciifi=jE68

Thus, we can write (67) as follows

j=1NeHijqj=j=1NeĜijpj+j=1Neâijpj+j=1Neb̂ijpnjE69

The global matrix system equation for all inodes can be written as follows

HQ=GP+ai+bjE70

the vector Qrepresents all the values of displacements and microrotations, the vector Prepresents all the tractions and couple stress vector, the vector irepresents all the values of pore pressure, and the vector jrepresents all the values of pore pressure gradients before applying boundary conditions.

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

AX=BE71

where Ais an unknown matrix, while Xand Bare known matrices.

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

5. Numerical results and discussion

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

The elasticity tensor is expressed as

Cpjkl=430.1130.418.200201.3130.4116.721.00070.118.221.073.6002.400019.88.000008.029.10201.370.12.400147.3GPaE72

The mechanical temperature coefficient is

βpj=1.012.0002.001.480007.52106Nkm2E73

The thermal conductivity tensor is

kpj=5.20007.600038.3W/KmE74

Mass density ρ=7820kg/m3and heat capacity c=461J/kgK.

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

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

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

Figures 24 show the propagation of the thermal stress σ11, σ12, and σ22waves along x-axis for the three temperatures Te, Ti, and Tpand total temperature T. It was noted from these figures that the three temperatures have significant effects on the thermal stress waves along x-axis through the thickness of the FGA structure.

Figure 2.

Propagation of the thermal stress σ 11 waves along x -axis for the three temperatures T e , T i , T p and total temperature T .

Figure 3.

Propagation of the thermal stress σ 12 waves along x -axis for the three temperatures T e , T i , T p and total temperature T .

Figure 4.

Propagation of the thermal stress σ 22 waves along x -axis for the three temperatures T e , T i , T p and total temperature T .

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

Figure 5.

Propagation of the thermal stress σ 11 waves along x -axis for 3T theory and different methods.

Figure 6.

Propagation of the thermal stress σ 11 waves along x -axis for 1T theory and different methods.

Figure 5 shows a comparison of the propagation of the thermal stress σ11waves for the BEM results of three-temperature (3T) radiative heat conduction theory for the BEM results with those obtained using the FDM of Pazera and Jędrysiak [60] and FEM of Xiong and Tian [61], where we replaced the 1T heat conduction theory of their work by 3T radiative heat conduction theory of our work to obtain the results. It can be noticed that the BEM results are found to agree very well with the FDM and FEM results.

Figure 6 shows a comparison of the propagation of the thermal stress σ11waves for the BEM results of one-temperature (1T) heat conduction theory with those obtained using FDM of Pazera and Jędrysiak [60], FEM1 of Xiong and Tian [61], and FEM2 of COMSOL multiphysics software version 5.1, where we replaced 3T radiative heat conduction theory of our work by the 1T heat conduction theory of their work to obtain the results. It can be noticed that the BEM results are found to agree very well with the FDM, FEM1, and FEM2 results and thus demonstrate the validity and accuracy of our proposed theory and the technique used to solve its general problems.

6. Conclusion

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

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

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Mohamed Abdelsabour Fahmy (August 14th 2020). A New BEM for Modeling and Simulation of Laser Generated Ultrasound Waves in 3T Fractional Nonlinear Generalized Micropolar Poro-Thermoelastic FGA Structures [Online First], IntechOpen, DOI: 10.5772/intechopen.93376. Available from:

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