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Research Papers

Resonant Ultrasound Spectroscopy: Sensitivity Analysis for Isotropic Materials and Anisotropic Materials With Cubic Symmetry

[+] Author and Article Information
Farhad Farzbod

Mem. ASME
Department of Mechanical Engineering,
University of Mississippi,
201A Carrier Hall,
University, MS 38677
e-mail: farzbod@olemiss.edu

Onome E. Scott-Emuakpor

Mem. ASME
Aerospace Systems Directorate (AFRL/RQTI),
1950 Fifth Street, Bldg. 18D,
Wright-Patterson AFB, OH 45433

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received October 13, 2017; final manuscript received September 20, 2018; published online November 14, 2018. Assoc. Editor: Miao Yu. This work is in part a work of the U.S. Government. ASME disclaims all interest in the U.S. Government's contributions.

J. Vib. Acoust 141(2), 021010 (Nov 14, 2018) (10 pages) Paper No: VIB-17-1457; doi: 10.1115/1.4041593 History: Received October 13, 2017; Revised September 20, 2018

Resonant ultrasound spectroscopy (RUS) is an experimental method to measure elastic and anelastic properties of materials. The RUS experiment is conducted by exciting a specimen with a simple geometry and measuring resonant frequencies. From the resonant behaviors, both elastic and anelastic properties of the sample material can be extracted. This paper investigates the sensitivities of measured resonant frequencies to changes in elastic constants for an isotropic material and anisotropic material with cubic symmetry. Also under investigation is whether different specimen geometries increase the sensitivity of RUS; in other words, a path for optimizing the reliability of RUS data is explored.

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Figures

Grahic Jump Location
Fig. 1

First, 200th, and 500th resonant frequencies for a sample with the dimensions of 50.29 × 30.32 × 25.0 mm and the density of 8940 kg/m3 are calculated for various values of C11 and C12 while C44 was set to 75 GPa

Grahic Jump Location
Fig. 2

Mode shapes for samples with cubic symmetry. C44 was set to be 75 GPa while C11 and C12 vary along the line of C11–C12 = constant. As can be seen, the mode shapes for these different elastic constants are very similar even at high resonant frequencies, with the exception of some minor change in the order of modes.

Grahic Jump Location
Fig. 3

Root-mean-square (RMS) difference between resonant frequencies of sample with various C11 and C12 versus a sample with C11 and C12 of 90 and 30 GPa. In all cases, the C44, density, and the dimensions are set to be 75 GPa, 8940 kg/m3, and 50.29 × 30.32 × 25.0 mm.

Grahic Jump Location
Fig. 4

Relative difference between resonant frequencies of two samples with (C11, C12) = (118, 50) and (153, 90) when they are compared to the ones generated by (C11, C12) = (135, 70)

Grahic Jump Location
Fig. 5

Lines of constant resonant frequencies vary for different frequencies when we keep C12 constant and change C11 and C44. This allows us to pin point C11 and C44 when we use couple of frequencies in the error analysis.

Grahic Jump Location
Fig. 6

Resonant frequencies versus C11 and C12 for an isotropic sample (C44 = (C11–C12)/2) with the dimensions of 50.29 × 30.32 × 25.0 mm and the density of 8940 kg/m3. Similar to the sample with cubic crystal symmetry, resonant frequency changes by minuscule amount on the lines of C11–C12 = constant.

Grahic Jump Location
Fig. 7

RMS difference between resonant frequencies of isotropic samples with various C11 and C12 versus a sample with C11 and C12 of 90 and 30 GPa. In all cases, the C44 = (C11–C12)/2 and the Poisson's ratios are indicated by dotted lines.

Grahic Jump Location
Fig. 8

Resonant frequencies versus C11 and C44 for an isotropic sample; as is evident, C44 can be found with high certainty while resonant frequencies does not define C11 with high certainty

Grahic Jump Location
Fig. 9

Hessian eigenvector corresponding to the small curvature direction is depicted at each point along with the vectors orthogonal to the gradient. These vectors are calculated for the 20th resonant frequency of the sample with cubic crystal symmetry. On the right, a zoom out of the image is depicted.

Grahic Jump Location
Fig. 10

RMS difference between resonant frequencies of samples with cubic crystal symmetries, C44 = 75 GPa and various elastic constants (C11, C12) when compared to the center value of (C11, C12) = (170,120). The sample assumed to be 50 times longer or shorter in each principal direction.

Grahic Jump Location
Fig. 11

Samples with three different geometries: parallelepiped, cylindrical, and quarter cylindrical samples. Here, the tenth resonant mode is depicted for these samples.

Grahic Jump Location
Fig. 12

First, 20th, and 100th resonant frequencies for the quarter cylindrical sample. As is evident, the same pattern of miniscule variations in frequency is observed along the approximate lines of C11–C12 = constant.

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