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Determination of thickness and elastic constants of aluminum plates from full-field wavelength measurements of single-mode narrowband Lamb waves
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10.1121/1.2945707
/content/asa/journal/jasa/124/3/10.1121/1.2945707
http://aip.metastore.ingenta.com/content/asa/journal/jasa/124/3/10.1121/1.2945707

Figures

Image of FIG. 1.
FIG. 1.

Representation of the frequency spectrum of antisymmetric (continuous lines) and symmetric (dashed lines) Lamb modes in a plate having and .

Image of FIG. 2.
FIG. 2.

Representation of the sensitivity estimators against the frequency-thickness product in a plate with and . The continuous lines correspond to the antisymmetric modes and the dashed lines to the symmetric ones. (a) Sensitivity estimator for multiplied by . (b) Sensitivity estimator for multiplied by . (c) Sensitivity estimator for multiplied by squared. The advantage of using the factors and squared is that the represented functions do not depend on .

Image of FIG. 3.
FIG. 3.

Each curve , with , corresponds to the solution for the bulk velocities that satisfies Eq. (7) yielded by a particular experimental point when is fixed, calculated for a plate with and . (a) Mode at (continuous line) and mode at (dashed line). (b) Same mode as in (a) (continuous line) and mode at (dashed line).

Image of FIG. 4.
FIG. 4.

Representation of the slopes of the curves yielded by a generic experimental point when one parameter is fixed, against the frequency-thickness product, calculated for a plate with and . The continuous lines correspond to the antisymmetric modes and the dashed lines to the symmetric ones. (a) Slope of with respect to , divided by . (b) Slope of with respect to , divided by . (c) Slope of with respect to .

Image of FIG. 5.
FIG. 5.

Layout of the double-pulsed ESPI system.

Image of FIG. 6.
FIG. 6.

Out-of-plane displacement maps of several Lamb waves at a given instant measured with our double-pulsed ESPI system. (i) Instantaneous optical phase-change map, . (ii) Fast Fourier transform of the optical phase-change map indicating the position of the filter. (iii) Acoustic amplitude, . (iv) Acoustic phase, . (a) Mode at in an aluminum plate thick. The actual size of the field of view is . (b) Modes and at in an aluminum plate thick, being the amplitude of mode much larger than the amplitude of mode . The actual size of the field of view is . (c) Modes and with approximately the same amplitude at in an aluminum plate thick. The actual size of the field of view is .

Image of FIG. 7.
FIG. 7.

Out-of-plane displacement map at a given instant of a Lamb wave consisting of modes and at in an aluminum plate thick, measured with our double-pulsed ESPI system. The actual size of the field of view is . (i) Instantaneous optical phase-change map, . (ii) Fast Fourier transform of the optical phase-change map indicating the position of the filter. (iii) Acoustic amplitude, . (iv) Acoustic phase, . (a) A filter is applied in order to isolate mode . (b) A filter is applied in order to isolate mode . (c) A coarser filter is applied so that both modes and are recovered and the beating between them is detected.

Image of FIG. 8.
FIG. 8.

Histograms showing the dispersion of the calculated values of the parameters by means of the Monte Carlo method for the plate thick. (a) Longitudinal wave phase velocity . (b) Shear wave phase velocity . (c) Thickness of the plate .

Image of FIG. 9.
FIG. 9.

Experimental points together with the theoretical representation of the frequency spectra of Lamb modes for the calculated thickness and elastic constants. For the identification of each mode, see Fig. 1. (a) Plate thick. (b) Plate thick. (c) Plate thick. (d) Plate thick. (e) Plate thick.

Image of FIG. 10.
FIG. 10.

Complex representation of the out-of-plane displacement at a point of the plate surface when there are two modes and propagating along the plate.

Tables

Generic image for table
TABLE I.

Calculated values of the thickness , the longitudinal wave velocity , and the shear wave velocity obtained from minimizing with experimental points. From them, the values of , being Young’s modulus and the mass density, and Poisson’s ratio were calculated. The values of and obtained, respectively, with a micrometer and by means of the pulse-echo method are also given for reference purposes.

Generic image for table
TABLE II.

Calculated values of the thickness , the constant , and Poisson’s ratio obtained from minimizing with experimental points. The value of Young’s modulus is obtained as the product , being the mass density.

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/content/asa/journal/jasa/124/3/10.1121/1.2945707
2008-09-01
2014-04-25
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Determination of thickness and elastic constants of aluminum plates from full-field wavelength measurements of single-mode narrowband Lamb waves
http://aip.metastore.ingenta.com/content/asa/journal/jasa/124/3/10.1121/1.2945707
10.1121/1.2945707
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