Table of Contents
ISRN Applied Mathematics
Volume 2011, Article ID 803562, 16 pages
http://dx.doi.org/10.5402/2011/803562
Research Article

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

Department of Mechanical Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan

Received 19 April 2011; Accepted 15 May 2011

Academic Editors: L.-Y. Li and M. Tabata

Copyright © 2011 Shih-Ming Hsu and Ching-Chung Yin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 [810] 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 Figure 1, 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: 𝑇𝐼𝐽=𝑐𝐼𝐽𝐾𝐿𝑢𝐾,𝐿+𝜆𝐼𝐽ΔΘ,(2.1a)Ξ=𝜆𝐾𝐿𝑢𝐾,𝐿+𝛼ΔΘ,(2.1b)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 𝑐𝐼𝐽𝐾𝐿, 𝜆𝐼𝐽, and 𝛼 denote the effective elastic constants, thermoelastic coupling coefficients, and thermal constant influenced by the initial strains 𝑢𝑖𝐾,𝐿, respectively. They are defined by𝑐𝐼𝐽𝐾𝐿=1𝑒𝑖𝑁𝑁𝑐𝐼𝐽𝐾𝐿+𝑐𝐼𝐽𝐾𝐿𝑀𝑁𝑢𝑖𝑀,𝑁+𝑐𝑂𝐽𝐾𝐿𝑢𝑖𝐼,𝑂+𝑐𝐼𝑂𝐾𝐿𝑢𝑖𝐽,0+𝑐𝐼𝐽𝑂𝐿𝑢𝑖𝐾,𝑂+𝑐𝐼𝐽𝐾𝑂𝑢𝑖𝐿,𝑂,(2.2a)𝜆𝐼𝐽=1𝑒𝑖𝑁𝑁𝜆𝐼𝐽+𝜆𝑂𝐽𝑢𝑖𝐼,𝑂+𝜆𝐼𝑂𝑢𝑖𝐽,𝑂,(2.2b)𝛼=1𝑒𝑖𝑁𝑁𝛼,(2.2c)where 𝛼𝜌0𝐶E/Θ0 and 𝑒𝑖𝑁𝑁𝑢𝑖1,1+𝑢𝑖2,2+𝑢𝑖3,3 denotes the cubic dilatation. The terms 𝑐𝐼𝐽𝐾𝐿, 𝑐𝐼𝐽𝐾𝐿𝑀𝑁, 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 𝑢𝑖𝐾,𝐿 and the initial (residual) stresses 𝑇𝑖𝐼𝐽 is given by 𝑇𝑖𝐼𝐽=𝑐𝐼𝐽𝐾𝐿𝑢𝑖𝐾,𝐿.

803562.fig.001
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 form𝑇𝐽𝐼+𝑇𝑖𝐽𝐾𝑢𝐼,𝐾,𝐽+𝜌𝑖𝑏𝐼=𝜌𝑖̈𝑢𝐼,(2.3) where 𝜌𝑖 is the mass density in the initial state and is defined by 𝜌𝑖=(1𝑒𝑖𝑁𝑁)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 form𝐪𝐽,𝐽+𝜌𝑖𝐡=Θ𝑖̇Ξ,(2.4a)𝐪𝐼=𝐤𝐼𝐽ΔΘ,𝐽,(2.4b)where Θ𝑖(=Θ0), 𝜌𝑖𝐡, and 𝐪𝐼 are the temperature, distributed body heat source, and surface heat flux in the initial state, respectively. 𝐤𝐼𝐽 is the effective thermal conductivity influenced by the initial strains 𝑢𝑖𝐾,𝐿 and can be written as𝐤𝐼𝐽=1𝑒𝑖𝑁𝑁𝐤𝐼𝐽+𝐤𝑂𝐽𝑢𝑖𝐼,𝑂+𝐤𝐼𝑂𝑢𝑖𝐽,𝑂,(2.5) 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: 𝑐𝐼𝐽𝐾𝐿+𝛿𝐼𝐾𝑇𝑖𝐽𝐿𝑢𝐾,𝐽𝐿𝜆𝐼𝐽ΔΘ,𝐽+𝜌𝑖𝑏𝐼=𝜌𝑖̈𝑢𝐼,𝐤𝐽𝐿ΔΘ,𝐽𝐿Θ𝑖𝜆𝐾𝐿̇𝑢𝐾,𝐿+𝜌𝑖𝐡=Θ𝑖̇𝛼ΔΘ.(2.6)

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 Figure 2, the residual stresses 𝑇𝑖𝐼𝐽 are assumed to be homogeneous. Only normal stress is applied in the 𝑋1-direction, and shear stress components vanish, that is, 𝑇𝑖22=𝑇𝑖33=𝑇𝑖𝐼𝐽=0 (𝐼𝐽). According to the relation 𝑇𝑖𝐼𝐽=𝑐𝐼𝐽𝐾𝐿𝑢𝑖𝐾,𝐿, the initial strains 𝑢𝑖𝐾,𝐿 and the associated cubic dilatation 𝑒𝑖𝑁𝑁 of the isotropic plate are given as follows: 𝑢𝑖1,1=𝑐11+𝑐12𝑇𝑖11𝑐11𝑐12𝑐11+2𝑐12,𝑢𝑖2,2=𝑢𝑖3,3=𝑐12𝑇𝑖11𝑐11𝑐12𝑐11+2𝑐12,𝑢𝑖2,3=𝑢𝑖3,2=𝑢𝑖1,3=𝑢𝑖3,1=𝑢𝑖1,2=𝑢𝑖2,1𝑒=0,𝑖𝑁𝑁=𝑇𝑖11𝑐11+2𝑐12.(2.7) The existing effective material constants 𝑐𝑃𝑄, 𝜆𝑃, and 𝐤𝑃 (𝑃,𝑄=1,2,,6) in the initial state appeared in (2.2a), (2.2b), and (2.5) are derived as follows:𝑐11,𝑐22,𝑐33T=1𝑒𝑖𝑁𝑁𝑐11+𝑐112𝑒𝑖𝑁𝑁{1,1,1}T+4𝑐11+𝑐111𝑐112𝑢𝑖1,1,𝑢𝑖2,2,𝑢𝑖3,3T,𝑐23,𝑐13,𝑐12T=1𝑒𝑖𝑁𝑁𝑐12+2𝑐12+𝑐112𝑒𝑖𝑁𝑁{1,1,1}T2𝑐12+𝑐112𝑐123𝑢𝑖1,1,𝑢𝑖2,2,𝑢𝑖3,3T,𝑐44,𝑐55,𝑐66T=1𝑒𝑖𝑁𝑁𝑐44+2𝑐44+𝑐155𝑒𝑖𝑁𝑁{1,1,1}T2𝑐44+𝑐155𝑐144𝑢𝑖1,1,𝑢𝑖2,2,𝑢𝑖3,3T,𝜆1,𝜆2,𝜆3T=1𝑒𝑖𝑁𝑁𝜆{1,1,1}T𝑢+2𝜆𝑖1,1,𝑢𝑖2,2,𝑢𝑖3,3T,𝐤1,𝐤2,𝐤3T=1𝑒𝑖𝑁𝑁𝐤{1,1,1}T𝑢+2𝐤𝑖1,1,𝑢𝑖2,2,𝑢𝑖3,3T.(2.8) According to (2.7) and (2.8), the special relations 𝑐22=𝑐33, 𝑐13=𝑐12, 𝑐55=𝑐66, 𝜆2=𝜆3, and 𝐤2=𝐤3 are obtained. Rewriting (2.6) leads to the thermoelastic governing equations involving the effect of uniaxial prestress 𝑇𝑖11 as follows: 𝑐𝑇𝑖1111𝑢1,11+𝑐66𝑢1,22+𝑐55𝑢1,33+𝑐12+𝑐66𝑢2,12+𝑐13+𝑐55𝑢3,13𝜆1ΔΘ,1+𝜌𝑖𝑏1=𝜌𝑖̈𝑢1,𝑐𝑇𝑖1166𝑢2,11+𝑐22𝑢2,22+𝑐44𝑢2,33+𝑐12+𝑐66𝑢1,12+𝑐23+𝑐44𝑢3,23𝜆2ΔΘ,2+𝜌𝑖𝑏2=𝜌𝑖̈𝑢2,𝑐𝑇𝑖1155𝑢3,11+𝑐44𝑢3,22+𝑐33𝑢3,33+𝑐13+𝑐55𝑢1,13+𝑐23+𝑐44𝑢2,23𝜆3ΔΘ,3+𝜌𝑖𝑏3=𝜌𝑖̈𝑢3,𝐤1ΔΘ,11+𝐤2ΔΘ,22+𝐤3ΔΘ,33Θ𝑖𝜆1̇𝑢1,1+Θ𝑖𝜆2̇𝑢2,2+Θ𝑖𝜆3̇𝑢3,3+𝜌𝑖𝐡=Θ𝑖̇𝛼ΔΘ,(2.9) where 𝑐𝑇𝑖1111=𝑐11+𝑇𝑖11, 𝑐𝑇𝑖1155=𝑐55+𝑇𝑖11, and 𝑐𝑇𝑖1166=𝑐66+𝑇𝑖11.

803562.fig.002
Figure 2: Schematic diagram of a copper foil prestressed in the 𝑋1-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 the 𝑋1-Axis

The schematic diagram of a thin plate exerted by normal stress in the 𝑋1-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 /2𝑋3/2, where 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+𝜁𝑋3𝜔𝑡) based on the partial wave expansion (PWE) method. Then, the solutions of 𝑢𝐼(𝐼=1,2,3) and ΔΘ are represented as𝑢𝐼=𝐴𝐼e𝑖𝜁𝑋3e𝑖(𝜉𝐽𝑋𝐽𝜔𝑡),ΔΘ=𝐴4e𝑖𝜁𝑋3e𝑖(𝜉𝐽𝑋𝐽𝜔𝑡),(2.10) 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:𝜁2𝐚11+𝐜11𝐜12𝜁𝐛13𝐜14𝐜21𝜁2𝐚22+𝐜22𝜁𝐛23𝐜24𝜁𝐛31𝜁𝐛32𝜁2𝐚33+𝐜33𝜁𝐛34𝐜41𝐜42𝜁𝐛43𝜁2𝐚44+𝐜44𝐴1𝐴2𝐴3𝐴4=0000,(2.11) where the unknown terms are defined as follows:𝐚11=𝑐55,𝐜11=𝜉21𝑐𝑇𝑖1111+𝜉22𝑐66𝜔2𝜌𝑖,𝐚22=𝑐44,𝐜22=𝜉21𝑐𝑇𝑖1166+𝜉22𝑐22𝜔2𝜌𝑖,𝐚33=𝑐33,𝐜33=𝜉21𝑐𝑇𝑖1155+𝜉22𝑐44𝜔2𝜌𝑖,𝐚44=𝐤3,𝐜44=𝜉21𝐤1+𝜉22𝐤2𝑖𝜔Θ𝑖𝐜𝛼,12=𝐜21=𝜉1𝜉2𝑐12+𝑐66,𝐜14=𝑖𝜉1𝜆1,𝐜41=𝜔Θ𝑖𝜉1𝜆1,𝐛13=𝐛31=𝜉1𝑐13+𝑐55,𝐜24=𝑖𝜉2𝜆2,𝐜42=𝜔Θ𝑖𝜉2𝜆2,𝐛23=𝐛32=𝜉2𝑐23+𝑐44,𝐛34=𝑖𝜆3,𝐛43=𝜔Θ𝑖𝜆3.(2.12) 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: 𝑎8𝜁8+𝑎6𝜁6+𝑎4𝜁4+𝑎2𝜁2+𝑎0=0,(2.13) 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 (𝐴±1,𝐴±2,𝐴±3,𝐴±4)(𝑘) with respect to ±𝜁𝑘 satisfy the proportional relation,𝐴±1,𝐴±2,𝐴±3,𝐴±4(𝑘)=𝑝±1𝑘,𝑝±2𝑘,𝑝±3𝑘,𝑝±4𝑘×𝐶±𝑘,(2.14) where 𝐶±𝑘 denotes the unknown amplitudes and the proportional factors (𝑝±1𝑘,𝑝±2𝑘,𝑝±3𝑘,𝑝±4𝑘) are the determinants of corresponding submatrices (or minors) in (2.11) of the form 𝑝+1𝑘=𝜁𝑘𝐜14𝐛43𝜁2𝑘𝐚22+𝐜22+𝐜12𝐛23𝜁2𝑘𝐚44+𝐜44+𝐜24𝐜42𝐛13𝐛13𝜁2𝑘𝐚22+𝐜22𝜁2𝑘𝐚44+𝐜44𝐜14𝐜42𝐛23𝐜12𝐜24𝐛43,𝑝+2𝑘=𝜁𝑘𝐜24𝐛43𝜁2𝑘𝐚11+𝐜11+𝐜21𝐛13𝜁2𝑘𝐚44+𝐜44+𝐜14𝐜41𝐛23𝐛23𝜁2𝑘𝐚11+𝐜11𝜁2𝑘𝐚44+𝐜44𝐜24𝐜41𝐛13𝐜21𝐜14𝐛43,𝑝+3𝑘=𝜁2𝑘𝐚11+𝐜11𝜁2𝑘𝐚22+𝐜22𝜁2𝑘𝐚44+𝐜44+𝐜12𝐜24𝐜41+𝐜14𝐜42𝐜21𝐜24𝐜42𝜁2𝑘𝐚11+𝐜11𝐜14𝐜41𝜁2𝑘𝐚22+𝐜22𝐜12𝐜21𝜁2𝑘𝐚44+𝐜44,𝑝+4𝑘=𝜁𝑘𝐜42𝐛23𝜁2𝑘𝐚11+𝐜11+𝐜41𝐛13𝜁2𝑘𝐚22+𝐜22+𝐜12𝐜21𝐛43𝐛43𝜁2𝑘𝐚11+𝐜11𝜁2𝑘𝐚22+𝐜22𝐜42𝐜21𝐛13𝐜41𝐜12𝐛23,𝑝1𝑘=𝑝+1𝑘,𝑝2𝑘=𝑝+2𝑘,𝑝3𝑘=+𝑝+3𝑘,𝑝4𝑘=𝑝+4𝑘.(2.15) 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 of𝑡𝐼=𝑇3𝐼=𝐵𝐼e𝑖𝜁𝑋3e𝑖(𝜉𝐽𝑋𝐽𝜔𝑡),𝐪in=𝐪3=𝐵4e𝑖𝜁𝑋3e𝑖(𝜉𝐽𝑋𝐽𝜔𝑡),(2.16) where 𝐵𝐼 and 𝐵4 indicate the amplitudes of 𝑇3𝐼 and 𝐪3, respectively. Following the above procedure in a similar manner, the components (𝐵±1,𝐵±2,𝐵±3,𝐵±4)(𝑘) with respect to ±𝜁𝑘 also satisfy the proportional relation,𝐵±1,𝐵±2,𝐵±3,𝐵±4(𝑘)=𝑞±1𝑘,𝑞±2𝑘,𝑞±3𝑘,𝑞±4𝑘×𝐶±𝑘,(2.17) where the proportional factors (𝑞±1𝑘,𝑞±2𝑘,𝑞±3𝑘,𝑞±4𝑘) are given by𝑞+1𝑘=𝑖𝑐55𝜁𝑘𝑝+1𝑘+𝜉1𝑝+3𝑘,𝑞+2𝑘=𝑖𝑐44𝜁𝑘𝑝+2𝑘+𝜉2𝑝+3𝑘,𝑞+3𝑘=𝑖𝑐13𝜉1𝑝+1𝑘+𝑐23𝜉2𝑝+2𝑘+𝑐33𝜁𝑘𝑝+3𝑘+𝑖𝜆3𝑝+4𝑘,𝑞+4𝑘=𝑖𝐤3𝜁𝑘𝑝+4𝑘,𝑞1𝑘=+𝑞+1𝑘,𝑞2𝑘=+𝑞+2𝑘,𝑞3𝑘=𝑞+3𝑘,𝑞4𝑘=+𝑞+4𝑘.(2.18) Hence, applying the matrix technique [19] used in analysis for the layered structure, a combination of (2.10) and (2.16) yields𝐔𝑋𝐽,𝑋3𝐓𝑋,𝑡𝐽,𝑋3=𝐏,𝑡+𝐏𝐐+𝐐𝐃+𝑋3𝟎𝟎𝐃𝑋3𝐂+𝐂e𝑖(𝜉𝐽𝑋𝐽𝜔𝑡),(2.19) 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 𝑝±𝑖𝑘 and 𝑞±𝑖𝑘 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 𝐶±𝑘 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=±/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 formΩS,AΩ(𝜔,𝜉)orS,A𝑞(𝑓,𝑘)detS,A11𝑞S,A12𝑞S,A13𝑞S,A14𝑞S,A21𝑞S,A22𝑞S,A23𝑞S,A24𝑞S,A31𝑞S,A32𝑞S,A33𝑞S,A34𝑞S,A41𝑞S,A42𝑞S,A43𝑞S,A44,(2.20) where the elements of vector {𝑞S,A1𝑘,𝑞S,A2𝑘,𝑞S,A3𝑘,𝑞S,A4𝑘}T (𝑘=1,2,3,4) are defined as follows:𝑞S𝑖𝑘𝜁=sin𝑘𝑞/2+𝑖𝑘(𝑖=1,2,4),𝑞S3𝑘𝜁=cos𝑘𝑞/2+3𝑘,𝑞A𝑖𝑘𝜁=cos𝑘𝑞/2+𝑖𝑘(𝑖=1,2,4),𝑞A3𝑘𝜁=sin𝑘𝑞/2+3𝑘.(2.21)

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 as 𝑘=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.

fig3
Figure 3: Frequency spectra of (a) the dispersion (real wavenumber 𝑘𝑟) 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).

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 “kK” 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 𝑐22=𝑐33𝑐23+2𝑐44, 𝜆2=𝜆3, and 𝐤2=𝐤3 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 𝑇𝑖11 are quasi-isotropic.

tab1
Table 1: Material properties of the copper foil under the natural state and two initial states with two uniaxial prestresses 0.02𝑐44 and 0.04𝑐44 applied in the 𝑋1-direction.

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 by𝑐L0=𝑐11/𝜌0,(3.1a)𝑐S0=𝑐44/𝜌0,(3.1b)𝑐Lame0=2𝑐S0,(3.1c)𝑐R0=0.87+1.12𝜈𝑐1+𝜈S0,(3.1d)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:𝑐[100]L1=𝑐𝑇𝑖1111/𝜌𝑖,𝑐[001]S1=𝑐𝑇𝑖1155/𝜌𝑖,𝑐[010]S1=𝑐𝑇𝑖1166/𝜌𝑖𝑐,(3.2)[010]L2=𝑐22/𝜌𝑖,𝑐[001]S2=𝑐44/𝜌𝑖,𝑐[100]S2=𝑐66/𝜌𝑖𝑐,(3.3)[001]L3=𝑐33/𝜌𝑖,𝑐[100]S3=𝑐55/𝜌𝑖,𝑐[010]S3=𝑐44/𝜌𝑖,(3.4) 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 byΔ𝑐[100]L1=𝑐[100]L1𝑐L0,Δ𝑐[001]S1=𝑐[001]S1𝑐S0,Δ𝑐[010]S1=𝑐[010]S1𝑐S0,(3.5)Δ𝑐[010]L2=𝑐[010]L2𝑐L0,Δ𝑐[001]S2=𝑐[001]S2𝑐S0,Δ𝑐[100]S2=𝑐[100]S2𝑐S0,(3.6)Δ𝑐[001]L3=𝑐[001]L3𝑐L0,Δ𝑐[100]S3=𝑐[100]S3𝑐S0,Δ𝑐[010]S3=𝑐[010]S3𝑐S0.(3.7) The speeds of the Lamé modes along the 𝑋1- and 𝑋2-directions and their corresponding differences are given by𝑐Lame1=2𝑐[100]S3,Δ𝑐Lame1=2Δ𝑐[100]S3,𝑐(3.8)Lame2=2𝑐[010]S3,Δ𝑐Lame2=2Δ𝑐[010]S3,(3.9) 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.

tab2
Table 2: Collected data of the wave speeds under the natural state and two initial states with two uniaxial prestresses 0.02𝑐44 and 0.04𝑐44 applied in the 𝑋1-direction.

According to the results via the complex root finding, the frequency spectra of the dispersion (real wavenumber 𝑘𝑟) 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.

fig4
Figure 4: Frequency spectra of (a) the phase velocity 𝑐ph 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).

In Figure 4(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 𝑘𝑟=𝑛+1/2 for the symmetric modes Sn(𝑛=0,1,2,) and 𝑘𝑟=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/)(𝑛+1/2) for the Sn modes and 𝑓=(𝑐Lame0/)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 0.5×103 mm−1 at 80 MHz.

Let the copper foil be exerted by a tensile stress in the 𝑋1-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 Δ𝑐[100]L1 and Δ𝑐[001]S1 in the 𝑋1-direction due to the tensile prestress 𝑇𝑖11=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 Δ𝑐[010]L2=−0.005 and Δ𝑐[001]S2=−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.

fig5
Figure 5: Frequency spectra of (a) the phase velocity 𝑐ph 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.
fig6
Figure 6: Frequency spectra of (a) the phase velocity 𝑐ph 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.

On the other hand, as shown in Figures 5(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 𝑇𝑖11 and leads to the orthorhombic symmetry in the 𝑋1𝑋3-plane, for example, the values 𝑐11, 𝑐33, 𝑐13, 𝑐55, 𝜆1, 𝜆3, 𝐤1, and 𝐤3 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 𝑐22=𝑐33𝑐23+2𝑐44, 𝜆2=𝜆3, and 𝐤2=𝐤3 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 the 𝑋1- 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.

fig7
Figure 7: Frequency spectra of (a) the phase velocity 𝑐ph 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.

4. Conclusion

In this paper, a copper foil exerted by a uniaxial tensile prestress in the 𝑋1-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.

Acknowledgment

This research was partially supported by National Science Council under Grant no. NSC 98-2221-E-009-007.

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