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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 A_{0} 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.

Determination of residual stresses in products is a major issue in most manufacturing industries. Both laser-induced ultrasound (LIU) [

Research on the theory of dynamical thermoelasticity which includes the displacement and temperature coupled fields was studied and established well in a book [

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 [

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 [_{0}) 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.

The thermoelastic effect in a stressed flat plate is formulated within the framework of the natural, initial, and final states, shown in Figure

Coordinate systems of the natural, initial or prestressed, and final states.

The elastic wave propagation in a medium under residual stress must satisfy the equations of motion in the initial state of the form

In this study, an isotropic thin plate in the natural state is considered. Referring the schematic diagram shown in Figure

Schematic diagram of a copper foil prestressed in the

The schematic diagram of a thin plate exerted by normal stress in the

Substitution of (

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 (

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

Frequency spectra of (a) the dispersion (real wavenumber

In this study, a thin copper foil is considered as an example. The material properties [

Material properties of the copper foil under the natural state and two initial states with two uniaxial prestresses

Natural state | Prestress | Prestress | |
---|---|---|---|

Thickness | |||

Mass density^{3}) | |||

Elastic constants^{2}) | |||

Temperature | |||

Thermal constant^{2}^{2}) | |||

Thermoelastic coupling coeff. (mg/mm^{2} | |||

Thermal conductivity (mg/mm^{2} |

^{2}

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_{1}-, _{2}-, and _{3}-directions in the initial (prestressed) state are defined as follows:

Collected data of the wave speeds under the natural state and two initial states with two uniaxial prestresses

Wave speed (mm/ | Longitudinal wave | Shear wave |
Corresponding differences of wave speed due to ( | Lamé mode | |

Natural state | — | — | |||

Prestress | |||||

Prestress | |||||

According to the results via the complex root finding, the frequency spectra of the dispersion (real wavenumber _{0} and S_{0} modes converge to a constant value corresponding to the Rayleigh wave speed

Frequency spectra of (a) the phase velocity

In Figure _{0} mode, has a close-to-zero minimum at a specific frequency, which is called the “Lamé mode” [_{n}_{m}_{n}_{m}_{0} and S_{0} modes merge together at the frequency range higher than 40 MHz. The convergent value is about ^{−1} at 80 MHz.

Let the copper foil be exerted by a tensile stress in the

Frequency spectra of (a) the phase velocity

Frequency spectra of (a) the phase velocity

On the other hand, as shown in Figures _{0} mode, increases as the tensile prestress in the same direction as wave propagation increases. This phenomenon is caused by the uniaxial prestress

In the previous illustrations, the horizontally polarized motion (SH wave) has been decoupled from the thermoelastic waves propagating along the _{0} 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

Frequency spectra of (a) the phase velocity

In this paper, a copper foil exerted by a uniaxial tensile prestress in the _{0} 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.

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