# Numerical Study of Characteristic Equations of Thermoelastic Waves Propagating in a Uniaxial Prestressed Isotropic Plate

Authors: Shih-Ming Hsu; Ching-Chung Yin
Journal: ISRN Applied Mathematics
 (2011)

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## Abstract

Thermoelastic waves propagating in an isotropic thin plate exerted by a uniaxial tensile stress are represented in this work. Characteristic equation of guided thermoelastic waves is formulated based on the theory of acoustoelasticity and classical thermoelasticity. Curve-tracing method for complex root finding is used to determine the attenuation, which is the imaginary part of the complex-value wavenumber. It is found that each plate mode of thermoelastic wave propagating in an isotropic plate with or without prestress has a minimum attenuation at a specific frequency except the A0 mode. These modes are called the Lamé modes, which are the volume resonances in the thickness direction and propagate along the plate with the least energy dissipation. Frequency spectra of the phase velocity dispersion and attenuation of thermoelastic waves propagating along various orientations in the uniaxial prestressed thin plate have further been discussed.


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## Body

## 1. Introduction

Determination of residual stresses in products is a major issue in most manufacturing industries. Both laser-induced ultrasound (LIU) [1, 2] and photoacoustics (PA) [3, 4] are the special techniques in “photothermal (PT) science” and have received intensive attention in nondestructive measurement of the residual stresses. The former determines the phase velocities from the far-field response generated by the pulsed laser, but the latter acquires the near-field response excited by the intensity-modulated CW laser. Both methods are correlative to the responses induced by “thermal acoustic waves.” Their dynamical behaviors are described based on the theory of thermoelasticity.Research on the theory of dynamical thermoelasticity which includes the displacement and temperature coupled fields was studied and established well in a book [5]. However, for describing the thermal transmission that propagates by means of wave with finite speed, several modified theories for the hyperbolic-type energy equation have been proposed. Differing from the classical theory of thermoelasticity, these theories are called the “generalized thermoelasticity” and classified as the followings: L-S theories [6] with one relaxation time, G-L theory [7] with two relaxation times, and three types of G-N theory [8–10] involved with the energy dissipation.On the other hand, the thermoelastic waves or laser-induced ultrasonic waves have become rather important in recent decades. Generalized thermoelastic wave propagation in the homogeneous transversely isotropic and anisotropic media has been analyzed [11, 12]. Afterward, based on four assumptions of the classical L-S, G-L, and G-N theories, Sharma et al. [13] investigated the thermoelastic wave propagation in a homogeneous isotropic plate, and showed the wavenumber spectra of the symmetric and antisymmetric modes for the insulated and isothermal boundary conditions. Verma and Hasebe  [14, 15]  and Verma  [16] studied the wave propagation in the infinite homogeneous isotropic and orthotropic plates as well as the anisotropic layered media using the generalized thermoelasticity with relaxation times. Salnikov and Scott [17] developed the asymptotic models of long-wave low-frequency and short-wave approximations to analyze the dispersion relations of thermoelastic waves in an infinite homogeneous isotropic plate subject to either isothermal or thermally insulated traction-free boundary conditions.This work presents an emerging method that unifies the advantages of LIU and PA without loss of generality for anisotropic inspection, which is of significance in the residual stress measurement. The classical theory of thermoelasticity [5] and the acoustoelasticity [18] are employed to model the thermoelastic waves propagating in a copper foil under distinct uniaxial stress state. The characteristic equations of phase velocity dispersion and attenuation are derived, and the spectra for each plate mode are determined numerically. Except for the fundamental antisymmetric (A0) mode, each thermoelastic plate mode has a unique characteristic in its attenuation spectrum, in which a minimum value occurs at a specific frequency. guided thermoelastic waves induced at these specific frequencies can propagate farther away because of smaller attenuation.


## 2. Theoretical Formulation

### 2.1. Constitutive Relations

The thermoelastic effect in a stressed flat plate is formulated within the framework of the natural, initial, and final states, shown in Figure1, originally proposed in [18] for the formulation of acoustoelasticity. Assume that there is no temperature change between the natural and initial states, that is, Θ0=Θi. The stress component TIJ and the increment of entropy Ξ satisfy the following constitutive relations: (2.1a)TIJ=c̅IJKLuK,L+λ̅IJΔΘ,(2.1b)Ξ=λ̅KLuK,L+α̅ΔΘ,where I,J,K,L=1,2,3. The physical field variables uK,L and ΔΘ indicate the strain changes and temperature rise caused by the external disturbance applied to the initial state. The material constants c̅IJKL, λ̅IJ, and α̅ denote the effective elastic constants, thermoelastic coupling coefficients, and thermal constant influenced by the initial strains uK,Li, respectively. They are defined by(2.2a)c̅IJKL=(1-eNNi)cIJKL+cIJKLMN′uM,Ni+cOJKLuI,Oi+cIOKLuJ,0i+cIJOLuK,Oi+cIJKOuL,Oi,
(2.2b)λ̅IJ=(1-eNNi)λIJ+λOJuI,Oi+λIOuJ,Oi,
(2.2c)α̅=(1-eNNi)α,where α≡ρ0CE/Θ0 and eNNi≡u1,1i+u2,2i+u3,3i denotes the cubic dilatation. The terms cIJKL, c′IJKLMN, λIJ, α, CE, ρ0, and Θ0 are the second- and third-order elastic constants, thermoelastic coupling coefficients, thermal constant, heat capacity, mass density, and temperature measured in the natural state. The relation between the initial strains uK,Li and the initial (residual) stresses TIJi is given by  TIJi=cIJKLuK,Li.Figure 1
Coordinate systems of the natural, initial or prestressed, and final states.


### 2.2. Governing Equations

The elastic wave propagation in a medium under residual stress must satisfy the equations of motion in the initial state of the form(2.3)(TJI+TJKiuI,K),J+ρibI=ρiüI,
where ρi is the mass density in the initial state and is defined by ρi=(1-eNNi)ρ0. The term ρibI is the body force applied to the initial state. Further, the balance of entropy and Fourier heat transfer equation in the initial state are of the form
(2.4a)-qJ,J+ρih=ΘiΞ̇,(2.4b)qI=-k̅IJΔΘ,J,where Θi(=Θ0), ρih, and qI are the temperature, distributed body heat source, and surface heat flux in the initial state, respectively. k̅IJ is the effective thermal conductivity influenced by the initial strains uK,Li and can be written as(2.5)k̅IJ=(1-eNNi)kIJ+kOJuI,Oi+kIOuJ,Oi,
where kIJ is the thermal conductivity measured in the natural state. Substituting (2.1a), (2.1b) into (2.3), (2.4a), (2.4b) subsequently yields the partial differential equations of thermoelastic waves as follows:
(2.6)(c̅IJKL+δIKTJLi)uK,JL-λ̅IJΔΘ,J+ρibI=ρiüI,k̅JLΔΘ,JL-Θiλ̅KLu̇K,L+ρih=Θiα̅ΔΘ̇.


### 2.3. Thin Isotropic Plate Subjected to a Uniaxial Prestress in the X1-Direction

In this study, an isotropic thin plate in the natural state is considered. Referring the schematic diagram shown in Figure2, the residual stresses TIJi are assumed to be homogeneous. Only normal stress is applied in the X1-direction, and shear stress components vanish, that is,  T22i=T33i=TIJi=0 (I≠J). According to the relation TIJi=cIJKLuK,Li, the initial strains uK,Li and the associated cubic dilatation eNNi of the isotropic plate are given as follows:
(2.7)u1,1i=(c11+c12)T11i(c11-c12)(c11+2c12),u2,2i=u3,3i=-c12T11i(c11-c12)(c11+2c12),u2,3i=u3,2i=u1,3i=u3,1i=u1,2i=u2,1i=0,eNNi=T11ic11+2c12.
The existing effective material constants c̅PQ, λ̅P, and k̅P (P,Q=1,2,…,6) in the initial state appeared in (2.2a), (2.2b), and (2.5) are derived as follows:(2.8){c̅11,c̅22,c̅33}T=[(1-eNNi)c11+c112′eNNi]{1,1,1}T+(4c11+c111′-c112′){u1,1i,u2,2i,u3,3i}T,{c̅23,c̅13,c̅12}T=[(1-eNNi)c12+(2c12+c112′)eNNi]{1,1,1}T-(2c12+c112′-c123′){u1,1i,u2,2i,u3,3i}T,{c̅44,c̅55,c̅66}T=[(1-eNNi)c44+(2c44+c155′)eNNi]{1,1,1}T-(2c44+c155′-c144′){u1,1i,u2,2i,u3,3i}T,{λ̅1,λ̅2,λ̅3}T=(1-eNNi)λ{1,1,1}T+2λ{u1,1i,u2,2i,u3,3i}T,{k̅1,k̅2,k̅3}T=(1-eNNi)k{1,1,1}T+2k{u1,1i,u2,2i,u3,3i}T.
According to (2.7) and (2.8), the special relations c̅22=c̅33, c̅13=c̅12, c̅55=c̅66, λ̅2=λ̅3, and k̅2=k̅3 are obtained. Rewriting (2.6) leads to the thermoelastic governing equations involving the effect of uniaxial prestress T11i as follows:
(2.9)(c̅11T11iu1,11+c̅66u1,22+c̅55u1,33)+(c̅12+c̅66)u2,12+(c̅13+c̅55)u3,13-λ̅1ΔΘ,1+ρib1=ρiü1,(c̅66T11iu2,11+c̅22u2,22+c̅44u2,33)+(c̅12+c̅66)u1,12+(c̅23+c̅44)u3,23-λ̅2ΔΘ,2+ρib2=ρiü2,(c̅55T11iu3,11+c̅44u3,22+c̅33u3,33)+(c̅13+c̅55)u1,13+(c̅23+c̅44)u2,23-λ̅3ΔΘ,3+ρib3=ρiü3,(k̅1ΔΘ,11+k̅2ΔΘ,22+k̅3ΔΘ,33)-(Θiλ̅1u̇1,1+Θiλ̅2u̇2,2+Θiλ̅3u̇3,3)+ρih=Θiα̅ΔΘ̇,
where c̅11T11i=c̅11+T11i, c̅55T11i=c̅55+T11i, and c̅66T11i=c̅66+T11i.Figure 2
Schematic diagram of a copper foil prestressed in theX1-direction. The unit vector n denotes the propagating direction in the X1X2-plane of guided thermoelastic waves.


### 2.4. Thermoelastic Waves Propagating along Any Direction from theX1-Axis

The schematic diagram of a thin plate exerted by normal stress in theX1-direction with the Cartesian coordinates is shown in Figure 2, in which the X3-axis is the thickness direction and the X1- and X2-axes both extend to infinity. This plate occupies the region -h/2≤X3≤h/2, where h is the thickness. In the absence of both ρibI and ρih in (2.9), all field quantities are assumed to have a plane wave harmonic function ei(ξJXJ+ζX3-ωt) based on the partial wave expansion (PWE) method. Then, the solutions of uI(I=1,2,3) and ΔΘ are represented as(2.10)uI=AIeiζX3ei(ξJXJ-ωt),ΔΘ=A4eiζX3ei(ξJXJ-ωt),
where AI and A4 indicate the amplitudes of uI and ΔΘ, respectively. The term ξJ (J=1,2) denotes the wave vector of guided wave in the X1X2-plane and ζ denotes the angular wavenumber in the thickness (X3-) direction. The components of wave vector ξJ are defined by ξ1=ξcosθ and ξ2=ξsinθ, where θ denotes the included angle of the arrowhead n and the X1-axis as shown in Figure 2. The parameters ξ and ω are the angular wavenumber and angular frequency defined as 2π times the wavenumber k and the frequency f, which are the reciprocal of the wavelength and period, respectively.Substitution of (2.10) into (2.9) is followed by the thermoelastic Christoffel equations for the coupled P, SV, SH, and thermal waves in the matrix form:(2.11)[ζ2a11+c11c12ζb13c14c21ζ2a22+c22ζb23c24ζb31ζb32ζ2a33+c33ζb34c41c42ζb43ζ2a44+c44]{A1A2A3A4}={0000},
where the unknown terms are defined as follows:(2.12)a11=c̅55,c11=ξ12c̅11T11i+ξ22c̅66-ω2ρi,a22=c̅44,c22=ξ12c̅66T11i+ξ22c̅22-ω2ρi,a33=c̅33,c33=ξ12c̅55T11i+ξ22c̅44-ω2ρi,a44=k̅3,c44=ξ12k̅1+ξ22k̅2-iωΘiα̅,c12=c21=ξ1ξ2(c̅12+c̅66),c14=iξ1λ̅1,c41=ωΘiξ1λ̅1,b13=b31=ξ1(c̅13+c̅55),c24=iξ2λ̅2,c42=ωΘiξ2λ̅2,b23=b32=ξ2(c̅23+c̅44),b34=iλ̅3,b43=ωΘiλ̅3.
According to the existence of a nontrivial solution in (2.11), the determinate vanishes and the eigenvalues ±ζk (k=1,2,3,4) must satisfy the quartic equation of ζ2 as follows:
(2.13)a8ζ8+a6ζ6+a4ζ4+a2ζ2+a0=0,
where the coefficients a8,a6,a4,a2, and a0 can be obtained and sorted by employing the software MAPLE for symbolic computation. The quartic equation (2.13) can be directly solved by quartic formula and eight complex roots ±ζk (k=1,2,3,4) are subsequently obtained. For the definiteness of a single-value function, a constraint for root ζk is selected as Im(ζk)≥0 to avoid the exponential increasing. The symbol “+” corresponds to the downgoing waves propagating along the positive X3-direction. On the other hand, the symbol “−” denotes those upgoing waves traveling toward the negative X3-direction. The eigenvector components (A1±,A2±,A3±,A4±)(k) with respect to ±ζk satisfy the proportional relation,(2.14)(A1±,A2±,A3±,A4±)(k)=(p1k±,p2k±,p3k±,p4k±)×Ck±,
where Ck± denotes the unknown amplitudes and the proportional factors (p1k±,p2k±,p3k±,p4k±) are the determinants of corresponding submatrices (or minors) in (2.11) of the form
(2.15)p1k+=ζk[c14b43(ζk2a22+c22)+c12b23(ζk2a44+c44)+c24c42b13-b13(ζk2a22+c22)(ζk2a44+c44)-c14c42b23-c12c24b43],p2k+=ζk[c24b43(ζk2a11+c11)+c21b13(ζk2a44+c44)+c14c41b23-b23(ζk2a11+c11)(ζk2a44+c44)-c24c41b13-c21c14b43],p3k+=(ζk2a11+c11)(ζk2a22+c22)(ζk2a44+c44)+c12c24c41+c14c42c21-c24c42(ζk2a11+c11)-c14c41(ζk2a22+c22)-c12c21(ζk2a44+c44),p4k+=ζk[c42b23(ζk2a11+c11)+c41b13(ζk2a22+c22)+c12c21b43-b43(ζk2a11+c11)(ζk2a22+c22)-c42c21b13-c41c12b23],p1k-=-p1k+,p2k-=-p2k+,p3k-=+p3k+,p4k-=-p4k+.
Applying (2.1a) and (2.4b), the solutions of traction tI (≡T3I, I=1,2,3) and heat influx qin (≡-q3) along the positive X3-direction can be represented in the form of(2.16)tI=T3I=BIeiζX3ei(ξJXJ-ωt),qin=-q3=B4eiζX3ei(ξJXJ-ωt),
where BI and B4 indicate the amplitudes of T3I and -q3, respectively. Following the above procedure in a similar manner, the components (B1±,B2±,B3±,B4±)(k) with respect to ±ζk also satisfy the proportional relation,(2.17)(B1±,B2±,B3±,B4±)(k)=(q1k±,q2k±,q3k±,q4k±)×Ck±,
where the proportional factors (q1k±,q2k±,q3k±,q4k±) are given by(2.18)q1k+=ic̅55(ζkp1k++ξ1p3k+),q2k+=ic̅44(ζkp2k++ξ2p3k+),q3k+=i(c̅13ξ1p1k++c̅23ξ2p2k++c̅33ζkp3k++iλ̅3p4k+),q4k+=ik̅3ζkp4k+,q1k-=+q1k+,q2k-=+q2k+,q3k-=-q3k+,q4k-=+q4k+.
Hence, applying the matrix technique [19] used in analysis for the layered structure, a combination of (2.10) and (2.16) yields(2.19){U(XJ,X3,t)T(XJ,X3,t)}=[P+P-Q+Q-][D+(X3)00D-(X3)]{C+C-}ei(ξJXJ-ωt),
where U={u1,u2,u3,ΔΘ}T and T={T31,T32,T33,-q3}T. The submatrices P± and Q± are the 4×4 matrices with elements pik± and qik± as given in (2.15) and (2.18), respectively. The diagonal matrix D±(X3) has a rank 4 with plane wave forms e±iζkX3. The uncertain 4×1 vector C± with elements Ck± can be determined by the boundary conditions.


### 2.5. Characteristic Equation of Guided Thermoelastic Waves

Thermoelastic waves propagating in a thin plate are dispersive because of the geometric constraints on the upper and bottom boundaries. In addition, they are dissipative due to transformation between strain and thermal energies. Assume that the upper and bottom surfaces (X3=±h/2) are both traction-free and adiabatic conditions, that is,  T3I=q3=0 (I=1,2,3). Applying the above boundary conditions to (2.19), the characteristic equations for the symmetric and antisymmetric modes of thermoelastic waves, indicated by the superscript “S” and “A”, are derived in the form(2.20)ΩS,A(ω,ξ)orΩS,A(f,k)≡det⁡(q11S,Aq12S,Aq13S,Aq14S,Aq21S,Aq22S,Aq23S,Aq24S,Aq31S,Aq32S,Aq33S,Aq34S,Aq41S,Aq42S,Aq43S,Aq44S,A),
where the elements of vector {q1kS,A,q2kS,A,q3kS,A,q4kS,A}T (k=1,2,3,4) are defined as follows:(2.21)qikS=sin(ζkh/2)qik+(i=1,2,4),q3kS=cos⁡(ζkh/2)q3k+,qikA=cos⁡(ζkh/2)qik+(i=1,2,4),q3kA=sin(ζkh/2)q3k+.In this study, due to energy dissipation of the propagating guided wave in the plate, the imaginary part of the complex-value wave number plays an important role. It is defined ask=kr+iki=kr(1+iγν/2π), where ki and γν are the attenuation per unit distance (Np/mm) and per wavelength (Np/ν), respectively. The characteristic equation ΩS,A(f,k) given in (2.19) can be represented as ΩS,A(f,kr,ki) or ΩS,A(f,kr,γν). Moreover, because of the difficulty in obtaining an exact zero solution to ΩS,A(f,kr,ki)=0 numerically. Instead, minimizing |ΩS,A(f,kr,ki)| is the common method for finding an approximate solution. Lowe [19] developed an effective method (also called the curve tracing algorithm) to find the complex roots of the characteristic equation for acoustic guided waves. Figures 3(a) and 3(b) show the dispersion and attenuation spectra for real wavenumber kr and imaginary wavenumber ki with respect to frequency f. There exist many roots of (f,ki) with respect to one point on the kr-axis. Besides, the MATLAB v6.5 subroutine fminbnd [20] which is an algorithm by the golden section search and parabolic interpolation and the subroutine amoeba [21] based on the downhill simplex method are frequently used in the numerical computation.Frequency spectra of (a) the dispersion (real wavenumberkr) and (b) the attenuation (imaginary wavenumber ki) for the symmetric modes (red lines) and antisymmetric modes (black lines) for the thermoelastic waves propagating in an isotropic copper foil under the natural state (without any prestress).
(a)(b)


## 2.1. Constitutive Relations

The thermoelastic effect in a stressed flat plate is formulated within the framework of the natural, initial, and final states, shown in Figure1, originally proposed in [18] for the formulation of acoustoelasticity. Assume that there is no temperature change between the natural and initial states, that is, Θ0=Θi. The stress component TIJ and the increment of entropy Ξ satisfy the following constitutive relations: (2.1a)TIJ=c̅IJKLuK,L+λ̅IJΔΘ,(2.1b)Ξ=λ̅KLuK,L+α̅ΔΘ,where I,J,K,L=1,2,3. The physical field variables uK,L and ΔΘ indicate the strain changes and temperature rise caused by the external disturbance applied to the initial state. The material constants c̅IJKL, λ̅IJ, and α̅ denote the effective elastic constants, thermoelastic coupling coefficients, and thermal constant influenced by the initial strains uK,Li, respectively. They are defined by(2.2a)c̅IJKL=(1-eNNi)cIJKL+cIJKLMN′uM,Ni+cOJKLuI,Oi+cIOKLuJ,0i+cIJOLuK,Oi+cIJKOuL,Oi,
(2.2b)λ̅IJ=(1-eNNi)λIJ+λOJuI,Oi+λIOuJ,Oi,
(2.2c)α̅=(1-eNNi)α,where α≡ρ0CE/Θ0 and eNNi≡u1,1i+u2,2i+u3,3i denotes the cubic dilatation. The terms cIJKL, c′IJKLMN, λIJ, α, CE, ρ0, and Θ0 are the second- and third-order elastic constants, thermoelastic coupling coefficients, thermal constant, heat capacity, mass density, and temperature measured in the natural state. The relation between the initial strains uK,Li and the initial (residual) stresses TIJi is given by  TIJi=cIJKLuK,Li.Figure 1
Coordinate systems of the natural, initial or prestressed, and final states.


## 2.2. Governing Equations

The elastic wave propagation in a medium under residual stress must satisfy the equations of motion in the initial state of the form(2.3)(TJI+TJKiuI,K),J+ρibI=ρiüI,
where ρi is the mass density in the initial state and is defined by ρi=(1-eNNi)ρ0. The term ρibI is the body force applied to the initial state. Further, the balance of entropy and Fourier heat transfer equation in the initial state are of the form
(2.4a)-qJ,J+ρih=ΘiΞ̇,(2.4b)qI=-k̅IJΔΘ,J,where Θi(=Θ0), ρih, and qI are the temperature, distributed body heat source, and surface heat flux in the initial state, respectively. k̅IJ is the effective thermal conductivity influenced by the initial strains uK,Li and can be written as(2.5)k̅IJ=(1-eNNi)kIJ+kOJuI,Oi+kIOuJ,Oi,
where kIJ is the thermal conductivity measured in the natural state. Substituting (2.1a), (2.1b) into (2.3), (2.4a), (2.4b) subsequently yields the partial differential equations of thermoelastic waves as follows:
(2.6)(c̅IJKL+δIKTJLi)uK,JL-λ̅IJΔΘ,J+ρibI=ρiüI,k̅JLΔΘ,JL-Θiλ̅KLu̇K,L+ρih=Θiα̅ΔΘ̇.


## 2.3. Thin Isotropic Plate Subjected to a Uniaxial Prestress in the X1-Direction

In this study, an isotropic thin plate in the natural state is considered. Referring the schematic diagram shown in Figure2, the residual stresses TIJi are assumed to be homogeneous. Only normal stress is applied in the X1-direction, and shear stress components vanish, that is,  T22i=T33i=TIJi=0 (I≠J). According to the relation TIJi=cIJKLuK,Li, the initial strains uK,Li and the associated cubic dilatation eNNi of the isotropic plate are given as follows:
(2.7)u1,1i=(c11+c12)T11i(c11-c12)(c11+2c12),u2,2i=u3,3i=-c12T11i(c11-c12)(c11+2c12),u2,3i=u3,2i=u1,3i=u3,1i=u1,2i=u2,1i=0,eNNi=T11ic11+2c12.
The existing effective material constants c̅PQ, λ̅P, and k̅P (P,Q=1,2,…,6) in the initial state appeared in (2.2a), (2.2b), and (2.5) are derived as follows:(2.8){c̅11,c̅22,c̅33}T=[(1-eNNi)c11+c112′eNNi]{1,1,1}T+(4c11+c111′-c112′){u1,1i,u2,2i,u3,3i}T,{c̅23,c̅13,c̅12}T=[(1-eNNi)c12+(2c12+c112′)eNNi]{1,1,1}T-(2c12+c112′-c123′){u1,1i,u2,2i,u3,3i}T,{c̅44,c̅55,c̅66}T=[(1-eNNi)c44+(2c44+c155′)eNNi]{1,1,1}T-(2c44+c155′-c144′){u1,1i,u2,2i,u3,3i}T,{λ̅1,λ̅2,λ̅3}T=(1-eNNi)λ{1,1,1}T+2λ{u1,1i,u2,2i,u3,3i}T,{k̅1,k̅2,k̅3}T=(1-eNNi)k{1,1,1}T+2k{u1,1i,u2,2i,u3,3i}T.
According to (2.7) and (2.8), the special relations c̅22=c̅33, c̅13=c̅12, c̅55=c̅66, λ̅2=λ̅3, and k̅2=k̅3 are obtained. Rewriting (2.6) leads to the thermoelastic governing equations involving the effect of uniaxial prestress T11i as follows:
(2.9)(c̅11T11iu1,11+c̅66u1,22+c̅55u1,33)+(c̅12+c̅66)u2,12+(c̅13+c̅55)u3,13-λ̅1ΔΘ,1+ρib1=ρiü1,(c̅66T11iu2,11+c̅22u2,22+c̅44u2,33)+(c̅12+c̅66)u1,12+(c̅23+c̅44)u3,23-λ̅2ΔΘ,2+ρib2=ρiü2,(c̅55T11iu3,11+c̅44u3,22+c̅33u3,33)+(c̅13+c̅55)u1,13+(c̅23+c̅44)u2,23-λ̅3ΔΘ,3+ρib3=ρiü3,(k̅1ΔΘ,11+k̅2ΔΘ,22+k̅3ΔΘ,33)-(Θiλ̅1u̇1,1+Θiλ̅2u̇2,2+Θiλ̅3u̇3,3)+ρih=Θiα̅ΔΘ̇,
where c̅11T11i=c̅11+T11i, c̅55T11i=c̅55+T11i, and c̅66T11i=c̅66+T11i.Figure 2
Schematic diagram of a copper foil prestressed in theX1-direction. The unit vector n denotes the propagating direction in the X1X2-plane of guided thermoelastic waves.


## 2.4. Thermoelastic Waves Propagating along Any Direction from theX1-Axis

The schematic diagram of a thin plate exerted by normal stress in theX1-direction with the Cartesian coordinates is shown in Figure 2, in which the X3-axis is the thickness direction and the X1- and X2-axes both extend to infinity. This plate occupies the region -h/2≤X3≤h/2, where h is the thickness. In the absence of both ρibI and ρih in (2.9), all field quantities are assumed to have a plane wave harmonic function ei(ξJXJ+ζX3-ωt) based on the partial wave expansion (PWE) method. Then, the solutions of uI(I=1,2,3) and ΔΘ are represented as(2.10)uI=AIeiζX3ei(ξJXJ-ωt),ΔΘ=A4eiζX3ei(ξJXJ-ωt),
where AI and A4 indicate the amplitudes of uI and ΔΘ, respectively. The term ξJ (J=1,2) denotes the wave vector of guided wave in the X1X2-plane and ζ denotes the angular wavenumber in the thickness (X3-) direction. The components of wave vector ξJ are defined by ξ1=ξcosθ and ξ2=ξsinθ, where θ denotes the included angle of the arrowhead n and the X1-axis as shown in Figure 2. The parameters ξ and ω are the angular wavenumber and angular frequency defined as 2π times the wavenumber k and the frequency f, which are the reciprocal of the wavelength and period, respectively.Substitution of (2.10) into (2.9) is followed by the thermoelastic Christoffel equations for the coupled P, SV, SH, and thermal waves in the matrix form:(2.11)[ζ2a11+c11c12ζb13c14c21ζ2a22+c22ζb23c24ζb31ζb32ζ2a33+c33ζb34c41c42ζb43ζ2a44+c44]{A1A2A3A4}={0000},
where the unknown terms are defined as follows:(2.12)a11=c̅55,c11=ξ12c̅11T11i+ξ22c̅66-ω2ρi,a22=c̅44,c22=ξ12c̅66T11i+ξ22c̅22-ω2ρi,a33=c̅33,c33=ξ12c̅55T11i+ξ22c̅44-ω2ρi,a44=k̅3,c44=ξ12k̅1+ξ22k̅2-iωΘiα̅,c12=c21=ξ1ξ2(c̅12+c̅66),c14=iξ1λ̅1,c41=ωΘiξ1λ̅1,b13=b31=ξ1(c̅13+c̅55),c24=iξ2λ̅2,c42=ωΘiξ2λ̅2,b23=b32=ξ2(c̅23+c̅44),b34=iλ̅3,b43=ωΘiλ̅3.
According to the existence of a nontrivial solution in (2.11), the determinate vanishes and the eigenvalues ±ζk (k=1,2,3,4) must satisfy the quartic equation of ζ2 as follows:
(2.13)a8ζ8+a6ζ6+a4ζ4+a2ζ2+a0=0,
where the coefficients a8,a6,a4,a2, and a0 can be obtained and sorted by employing the software MAPLE for symbolic computation. The quartic equation (2.13) can be directly solved by quartic formula and eight complex roots ±ζk (k=1,2,3,4) are subsequently obtained. For the definiteness of a single-value function, a constraint for root ζk is selected as Im(ζk)≥0 to avoid the exponential increasing. The symbol “+” corresponds to the downgoing waves propagating along the positive X3-direction. On the other hand, the symbol “−” denotes those upgoing waves traveling toward the negative X3-direction. The eigenvector components (A1±,A2±,A3±,A4±)(k) with respect to ±ζk satisfy the proportional relation,(2.14)(A1±,A2±,A3±,A4±)(k)=(p1k±,p2k±,p3k±,p4k±)×Ck±,
where Ck± denotes the unknown amplitudes and the proportional factors (p1k±,p2k±,p3k±,p4k±) are the determinants of corresponding submatrices (or minors) in (2.11) of the form
(2.15)p1k+=ζk[c14b43(ζk2a22+c22)+c12b23(ζk2a44+c44)+c24c42b13-b13(ζk2a22+c22)(ζk2a44+c44)-c14c42b23-c12c24b43],p2k+=ζk[c24b43(ζk2a11+c11)+c21b13(ζk2a44+c44)+c14c41b23-b23(ζk2a11+c11)(ζk2a44+c44)-c24c41b13-c21c14b43],p3k+=(ζk2a11+c11)(ζk2a22+c22)(ζk2a44+c44)+c12c24c41+c14c42c21-c24c42(ζk2a11+c11)-c14c41(ζk2a22+c22)-c12c21(ζk2a44+c44),p4k+=ζk[c42b23(ζk2a11+c11)+c41b13(ζk2a22+c22)+c12c21b43-b43(ζk2a11+c11)(ζk2a22+c22)-c42c21b13-c41c12b23],p1k-=-p1k+,p2k-=-p2k+,p3k-=+p3k+,p4k-=-p4k+.
Applying (2.1a) and (2.4b), the solutions of traction tI (≡T3I, I=1,2,3) and heat influx qin (≡-q3) along the positive X3-direction can be represented in the form of(2.16)tI=T3I=BIeiζX3ei(ξJXJ-ωt),qin=-q3=B4eiζX3ei(ξJXJ-ωt),
where BI and B4 indicate the amplitudes of T3I and -q3, respectively. Following the above procedure in a similar manner, the components (B1±,B2±,B3±,B4±)(k) with respect to ±ζk also satisfy the proportional relation,(2.17)(B1±,B2±,B3±,B4±)(k)=(q1k±,q2k±,q3k±,q4k±)×Ck±,
where the proportional factors (q1k±,q2k±,q3k±,q4k±) are given by(2.18)q1k+=ic̅55(ζkp1k++ξ1p3k+),q2k+=ic̅44(ζkp2k++ξ2p3k+),q3k+=i(c̅13ξ1p1k++c̅23ξ2p2k++c̅33ζkp3k++iλ̅3p4k+),q4k+=ik̅3ζkp4k+,q1k-=+q1k+,q2k-=+q2k+,q3k-=-q3k+,q4k-=+q4k+.
Hence, applying the matrix technique [19] used in analysis for the layered structure, a combination of (2.10) and (2.16) yields(2.19){U(XJ,X3,t)T(XJ,X3,t)}=[P+P-Q+Q-][D+(X3)00D-(X3)]{C+C-}ei(ξJXJ-ωt),
where U={u1,u2,u3,ΔΘ}T and T={T31,T32,T33,-q3}T. The submatrices P± and Q± are the 4×4 matrices with elements pik± and qik± as given in (2.15) and (2.18), respectively. The diagonal matrix D±(X3) has a rank 4 with plane wave forms e±iζkX3. The uncertain 4×1 vector C± with elements Ck± can be determined by the boundary conditions.


## 2.5. Characteristic Equation of Guided Thermoelastic Waves

Thermoelastic waves propagating in a thin plate are dispersive because of the geometric constraints on the upper and bottom boundaries. In addition, they are dissipative due to transformation between strain and thermal energies. Assume that the upper and bottom surfaces (X3=±h/2) are both traction-free and adiabatic conditions, that is,  T3I=q3=0 (I=1,2,3). Applying the above boundary conditions to (2.19), the characteristic equations for the symmetric and antisymmetric modes of thermoelastic waves, indicated by the superscript “S” and “A”, are derived in the form(2.20)ΩS,A(ω,ξ)orΩS,A(f,k)≡det⁡(q11S,Aq12S,Aq13S,Aq14S,Aq21S,Aq22S,Aq23S,Aq24S,Aq31S,Aq32S,Aq33S,Aq34S,Aq41S,Aq42S,Aq43S,Aq44S,A),
where the elements of vector {q1kS,A,q2kS,A,q3kS,A,q4kS,A}T (k=1,2,3,4) are defined as follows:(2.21)qikS=sin(ζkh/2)qik+(i=1,2,4),q3kS=cos⁡(ζkh/2)q3k+,qikA=cos⁡(ζkh/2)qik+(i=1,2,4),q3kA=sin(ζkh/2)q3k+.In this study, due to energy dissipation of the propagating guided wave in the plate, the imaginary part of the complex-value wave number plays an important role. It is defined ask=kr+iki=kr(1+iγν/2π), where ki and γν are the attenuation per unit distance (Np/mm) and per wavelength (Np/ν), respectively. The characteristic equation ΩS,A(f,k) given in (2.19) can be represented as ΩS,A(f,kr,ki) or ΩS,A(f,kr,γν). Moreover, because of the difficulty in obtaining an exact zero solution to ΩS,A(f,kr,ki)=0 numerically. Instead, minimizing |ΩS,A(f,kr,ki)| is the common method for finding an approximate solution. Lowe [19] developed an effective method (also called the curve tracing algorithm) to find the complex roots of the characteristic equation for acoustic guided waves. Figures 3(a) and 3(b) show the dispersion and attenuation spectra for real wavenumber kr and imaginary wavenumber ki with respect to frequency f. There exist many roots of (f,ki) with respect to one point on the kr-axis. Besides, the MATLAB v6.5 subroutine fminbnd [20] which is an algorithm by the golden section search and parabolic interpolation and the subroutine amoeba [21] based on the downhill simplex method are frequently used in the numerical computation.Frequency spectra of (a) the dispersion (real wavenumberkr) and (b) the attenuation (imaginary wavenumber ki) for the symmetric modes (red lines) and antisymmetric modes (black lines) for the thermoelastic waves propagating in an isotropic copper foil under the natural state (without any prestress).
(a)(b)


## 3. Numerical Results and Discussion

In this study, a thin copper foil is considered as an example. The material properties [22, 23] used in numerical computation are expressed in the units of “mg”, “mm”, “μs”, and “k°K” and listed in the first column of Table 1. Furthermore, the effective material properties under two initial states with two uniaxial prestresses 0.02c44 and 0.04c44 applied in the X1-direction are assembled in the second and third columns of Table 1, respectively. Obviously, according to the relations c̅22=c̅33≈c̅23+2c̅44, λ̅2=λ̅3, and k̅2=k̅3 in the X2X3-plane, the material properties in two distinct initial states reveal the characteristic of nearly transversely isotropic. Material properties in the plane perpendicular to the direction of applied prestress T11i are quasi-isotropic.Table 1
Material properties of the copper foil under the natural state and two initial states with two uniaxial prestresses0.02c44 and 0.04c44 applied in the X1-direction.
Natural statePrestressT11i=0.02c44PrestressT11i=0.04c44Thickness(mm)h=0.1h=0.1h=0.1Mass density (mg/mm3)ρ0=8.93ρi=8.913ρi=8.895Elastic constants(mg/mm·μs2)c11=188.1c12=108.9c44=39.6c111′=-1894 c112′=-754c123′=56 c144′=-401c155′=-287 c456′=57c̅11=183.377c̅22=c̅33=187.343c̅23=111.926c̅12=c̅13=106.094c̅44=37.704c̅55=c̅66=39.636c̅11=178.655c̅22=c̅33=186.587c̅23=114.952c̅12=c̅13=103.288c̅44=35.807c̅55=c̅66=39.671Temperature(k°K)Θ0=0.300Θi=0.300Θi=0.300Thermal constant(mg/mm·μs2·k°K2)α=(ρ0CE)/Θ0=11.129α̅=11.107α̅=11.085Thermoelastic coupling coeff. (mg/mm·μs2·k°K)λ=(c11+2c12)γ=6.697λ̅1=6.782λ̅2=λ̅3=6.648λ̅1=6.867λ̅2=λ̅3=6.599Thermal conductivity (mg/mm·μs2·k°K)k=0.398×10-3k̅1=0.403×10-3k̅2=k̅3=0.395×10-3k̅1=0.408×10-3k̅2=k̅3=0.392×10-3CE=3.74 is the heat capacity (mg/μs2·k°K), and γ=0.0165 is the thermal expansion coefficient (1/k°K).The speeds of the longitudinal (L0) and transverse (S0) waves, and Lamé mode (Lame0), and the Rayleigh wave (R0) in the natural state are, respectively, defined by(3.1a)cL0=c11/ρ0,
(3.1b)cS0=c44/ρ0,
(3.1c)cLame0=2cS0,
(3.1d)cR0=0.87+1.12ν1+νcS0,where Poison’s ratio ν=0.367 for the copper foil. The formulas given in (3.1c) and (3.1d) can be referred to the books by Graff [24] and Achenbach [25], respectively. In the absence of the thermoelastic coupling effect, the speeds of the longitudinal (L) and transverse (S) waves along the X1-, X2-, and X3-directions in the initial (prestressed) state are defined as follows:(3.2)cL1[100]=c̅11T11i/ρi,cS1[001]=c̅55T11i/ρi,cS1[010]=c̅66T11i/ρi,(3.3)cL2[010]=c̅22/ρi,cS2[001]=c̅44/ρi,cS2[100]=c̅66/ρi,(3.4)cL3[001]=c̅33/ρi,cS3[100]=c̅55/ρi,cS3[010]=c̅44/ρi,
where the superscript “[IJK]” denotes the polarized direction of plane wave. Next, their corresponding differences of speeds between the natural and initial states are defined by(3.5)ΔcL1[100]=cL1[100]-cL0,ΔcS1[001]=cS1[001]-cS0,ΔcS1[010]=cS1[010]-cS0,(3.6)ΔcL2[010]=cL2[010]-cL0,ΔcS2[001]=cS2[001]-cS0,ΔcS2[100]=cS2[100]-cS0,(3.7)ΔcL3[001]=cL3[001]-cL0,ΔcS3[100]=cS3[100]-cS0,ΔcS3[010]=cS3[010]-cS0.
The speeds of the Lamé modes along the X1- and X2-directions and their corresponding differences are given by(3.8)cLame1=2cS3[100],ΔcLame1=2ΔcS3[100],(3.9)cLame2=2cS3[010],ΔcLame2=2ΔcS3[010],
and they are related to the speeds of transverse (S) waves along the X3-direction according to the thickness resonance behavior of the Lamé mode. Equation (3.9) can provid an exact solution the Lamé modes because of the nearly isotropic property in the X2X3-plane for whereas (3.8) only provided the trend of speed change for the Lamé modes owing to the orthorhombic property in the X1X3-plane. Substituting the material properties listed in Table 1 into (3.1a)–(3.1d) to (3.9), the resultant data for the natural states and two initial states, that is, two prestresses: 0.02c44 and 0.04c44 applyied in the X1-direction, are assembled in Table 2.Table 2
Collected data of the wave speeds under the natural state and two initial states with two uniaxial prestresses0.02c44 and 0.04c44 applied in the X1-direction.
Wave speed (mm/μs)Longitudinal waveShear waveCorresponding differences of wave speed due to (3.5)–(3.7)Lamé modeNatural statecL0=4.590cS0=2.106——cLame0=2.978cL1[100]=4.546cS1[001]=2.130ΔcL1[100]=-0.044ΔcS1[001]=0.024cLame1=2.982cS1[010]=2.130ΔcS1[010]=0.024ΔcLame1=0.0004PrestressT11i=0.02c44cL2[010]=4.585cS2[001]=2.056ΔcL2[010]=-0.005ΔcS2[001]=-0.050cLame2=2.908cS2[100]=2.109ΔcS2[100]=0.003ΔcLame2=-0.0707cL3[001]=4.585cS3[100]=2.109ΔcL3[001]=-0.005ΔcS3[100]=0.003cS3[010]=2.056ΔcS3[010]=-0.050cL1[100]=4.502cS1[001]=2.154ΔcL1[100]=-0.088ΔcS1[001]=0.048cLame1=2.986cS1[010]=2.154ΔcS1[010]=0.048ΔcLame1=0.0008PrestressT11i=0.04c44cL2[010]=4.580cS2[001]=2.006ΔcL2[010]=-0.010ΔcS2[001]=-0.100cLame2=2.838cS2[100]=2.112ΔcS2[100]=0.006ΔcLame2=-0.1414cL3[001]=4.580cS3[100]=2.112ΔcL3[001]=-0.010ΔcS3[100]=0.006cS3[010]=2.006ΔcS3[010]=-0.100According to the results via the complex root finding, the frequency spectra of the dispersion (real wavenumberkr) and attenuation (imaginary wavenumber ki) for the symmetric modes (red lines) and antisymmetric modes (black lines) of a thermoelastic wave propagating in the copper foil in the natural state (without any prestress) are shown in Figures 3(a) and 3(b). The frequency spectra of the phase velocity cph and semilogarithmic plot of attenuation ki are also shown in Figures 4(a) and 4(b), where the phase velocity cph is defined as f/kr. In Figure 4(a), the four additional blue dashed lines, labeled by "cL0," "cS0," "cLame0," and "cR0" indicate the phase velocities of values 4.590, 2.106, 2.978, and 1.972 mm/μs due to (3.1a)–(3.1d). As frequency increases, both phase velocities of the A0 and S0 modes converge to a constant value corresponding to the Rayleigh wave speed cR0=1.972 mm/μs.Frequency spectra of (a) the phase velocitycph and (b) the semilogarithmic plot of attenuation ki for the symmetric modes (red lines) and antisymmetric modes (black lines) for the thermoelastic waves propagating in an isotropic copper foil under the natural state (without any prestress).
(a)(b)In Figure4(b), the attenuation spectrum of each mode, except the A0 mode, has a close-to-zero minimum at a specific frequency, which is called the “Lamé mode” [24]. The Lamé modes travel at a specific wave speed cLame0 and can be indicated using a dimensionless parameter krh=n+1/2 for the symmetric modes Sn(n=0,1,2,…) and krh=m for the antisymmetric modes Am(m=1,2,3,…). Reflecting on the “cLame0”-labeled blue dashed line shown in Figure 4(a), the specific frequencies, that is,  f=(cLame0/h)·(n+1/2) for the Sn modes and f=(cLame0/h)·m for the Am modes, can be also observed. The Lamé modes represent the volume resonances in the thickness direction and propagate along the plate with the close-to-zero attenuation. This means that the thermoelastic waves can propagate farther away without energy dissipation. Moreover, the attenuations of the A0 and S0 modes merge together at the frequency range higher than 40 MHz. The convergent value is about 0.5×10-3 mm−1 at 80 MHz.Let the copper foil be exerted by a tensile stress in theX1-direction. Two cases of applied stresses 0.02c44 and 0.04c44 are considered. Figures 5(a) and 5(b) show the frequency spectra of phase velocity dispersion and semilogarithmic attenuation for the symmetric modes (red lines) and antisymmetric modes (black lines) of thermoelastic waves propagating along the X1-direction. Figures 6(a) and 6(b) show those of thermoelastic waves propagating along the X2-direction. Comparison of Figures 5(a) and 6(a) indicates that the shift of each dispersion curve depends on the bulk wave speed differences between the natural and stressed states. According to Table 1, the changes of bulk wave speeds ΔcL1[100] and ΔcS1[001] in the X1-direction due to the tensile prestress T11i=0.02c44 are −0.044 and 0.024 mm/μs, respectively. As shown in Figure 5(a), the resulting phase velocities of plate waves increase over a broad frequency range but decrease in the vicinity of longitudinal wave speed cL0=4.590 mm/μs. Similarly, the phase velocities of the plate waves along the X2-direction decrease in accordance with the changes of bulk wave speeds ΔcL2[010]=−0.005 and ΔcS2[001]=−0.050 mm/μs. Figure 6(a) shows that the dispersion curves of higher modes change distinguishably in the region of higher frequency and wavenumber.Frequency spectra of (a) the phase velocitycph  and (b) the semilogarithmic plot of attenuation ki for the symmetric modes (red lines) and antisymmetric modes (black lines) for the thermoelastic waves propagating along the X1-direction in a copper foil under the natural state (without any prestress) and two initial states with two uniaxial prestresses 0.02c44 and 0.04c44 applied in the X1-direction.
(a)(b)Frequency spectra of (a) the phase velocitycph and (b) the semilogarithmic plot of attenuation ki for the symmetric modes (red lines) and antisymmetric modes (black lines) of a thermoelastic wave propagating along the X2-direction in the copper foil under the natural state (without any prestress) and two initial states with two uniaxial prestresses 0.02c44 and 0.04c44 applied in the X1-direction.
(a)(b)On the other hand, as shown in Figures5(b) and 6(b), the attenuation spectrum of each mode has a minimum at a specific frequency. These specific modes represent the volume resonances in the thickness direction and propagate along the plate with the least energy dissipation. Figure 5(b) shows that the minimum attenuation of each mode, except the A0 mode, increases as the tensile prestress in the same direction as wave propagation increases. This phenomenon is caused by the uniaxial prestress T11i and leads to the orthorhombic symmetry in the X1X3-plane, for example, the values c̅11, c̅33, c̅13, c̅55, λ̅1, λ̅3, k̅1, and k̅3 shown in Table 1. Next, Figure 6(b) shows that the specific frequencies of the Lamé modes are reduced if a tensile prestress is applied in the orientation perpendicular to the direction of wave propagation. The reductions of these specific frequencies can result from the changes of isotropic property in the X2X3-plane, that is, the specific relations c̅22=c̅33≈c̅23+2c̅44, λ̅2=λ̅3, and k̅2=k̅3 shown in Table 1. Therefore, the feature of close-to-zero attenuation of the Lamé modes is directly correlated with the isotropic material property in the sagittal plane of propagating thermoelastic guided wave.In the previous illustrations, the horizontally polarized motion (SH wave) has been decoupled from the thermoelastic waves propagating along theX1- and X2-directions. Figures 7(a) and 7(b) show the frequency spectra of phase velocity dispersion and semilogarithmic attenuation of thermoelastic waves propagating along the direction inclined at 45° to the X1-axis in a copper foil which is tensilely prestressed by 0.02c44 in the X1-direction. Owing to that the off-axis traveling wave motions are neither polarized in the sagittal plane nor horizontally polarized, and the P, SV, SH, and thermal waves are coupled. As shown in Figure 7(b), the “even” numbered antisymmetric modes (solid black lines), except the A0 mode, and the “odd” numbered symmetric modes (solid red lines) possess the Lamé modes. These modes have the minimum attenuation at some specific frequencies, which is similar to the feature previously mentioned. However, the Lamé mode vanishes from the “odd” numbered antisymmetric modes (dash-dotted black lines) and the “even” numbered symmetric modes (dash-dotted red lines) due to the participation of the horizontally polarized motions, that is, SH waves. This phenomenon represents that the energy of thermoelastic guided wave will dissipate into the region out of the sagittal plane during propagating along the orientation between the X1- and X2-directions. The special regions neighboring the intersecting dispersion curves represent coupling motion of different modes. Therefore, it is of difficulty to interpret the mode-converted response generated at those regions using LIU or PA technique.Frequency spectra of (a) the phase velocitycph and (b) the semilogarithmic plot of attenuation ki for the symmetric modes (red lines) and antisymmetric modes (black lines) for the thermoelastic waves propagating at 45° orientation to the X1-axis in a copper foil under the initial state with uniaxial prestress 0.02c44 applied in the X1-direction.
(a)(b)


## 4. Conclusion

In this paper, a copper foil exerted by a uniaxial tensile prestress in theX1-direction is considered as an example. Applying the curve-tracing method for the root finding of complex wavenumber, the numerical evidence indicates that the response of thermoelastic waves can characterize the uniaxial residual stresses in the plate-like structures through the frequency spectra of phase velocity dispersion and wavenumber attenuation, especially in the direction of wave propagation parallel or perpendicular to the loading direction. Except for the A0 mode, the attenuation spectra of thermoelastic waves have steep descents at the specific frequencies where their unique minima occur. The attenuation increases with increasing tensile prestress in the same direction as wave propagation. If the prestress orientation is perpendicular to the direction of thermoelastic wave propagation, the reductions in these specific frequencies of Lamé modes are proportional to the magnitudes of applied stress. Along the perpendicular direction, the phase velocities apparently decrease as the prestress increases. Furthermore, the isotropic material property in the sagittal plane of propagating thermoelastic guided wave can affect the appearance of close-to-zero attenuation.



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*Source: 803562-2011-07-18.xml*