The problem of detecting and characterizing cracks in solids is a major research area involving contributions from mathematics, physics, mechanics, and materials science.</p

Fortunko, Christopher

Heyliger, Paul

Kim, Sudook

Ledbetter, Hassel

Pei, Kuang-Chih

Digital Repository @ Iowa State University (ISU)

ARTIFICIAL CRACK IN STEEL: AN ULTRASONIC-RESONANCE-SPECTROSCOPY AND MODELING STUDY Hassel Ledbetter Materials Science and Engineering Laboratory National Institute of Standards and Technology Boulder, Colorado 80303 Paul Heyliger and Kuang-Chih Pei Civil Engineering Department Colorado State University Fort Collins, Colorado 80523 Sudook Kim and Christopher Fortunko Materials Science and Engineering Laboratory National Institute of Standards and Technology Boulder, Colorado 80303 INTRODUCTION The problem of detecting and characterizing cracks in solids is a major research area involving contributions from mathematics, physics, mechanics, and materials science. In the present study we approached this problem from both measurement and modeling viewpoints. In the measurements, we used ultrasonic-resonance spectroscopy to determine the macroscopic-vibration-frequency shifts caused by an artificial (electromachined) crack. The modeling used a finite-element method to approximate the macroscopic vibration frequencies of a cracked solid. As a check, we used the Ritz method to calculate the frequencies of the noncracked solid. MEASUREMENTS Specimen As a specimen material, we chose an AISI-304 steel polycrystal, which showed an Archimedes-method mass density of 7.898 g/cm3 • We prepared a rectangular-parallelepiped specimen, approximately 0.5 x 1.0 x 1.4 cm. Longitudinal- wave and shear-wave pulse-echo sound-velocity measurements verified the material's elastic isotropy within 1% and gave the following elastic constants: Young modulus 199.6 GPa, shear modulus 77.4 GPa, bulk modulus 158.2 GPa, Poisson ratio 0.290. *Contribution of NISTi not subject to copyright. Rt!IIiew of Progress in Quantitalive Nondestructive Evaluation. Vol. 14 Edited by D.O. Thompson and D.E. Chimenti. Plenum Press. New Yark. 1995 2019 L . :-:, :: -:-::- :- ... - ~ -. .. " : • .. " .'.. .......... • ...... ,e ..... ~ I ........ _ ...... '. -.- ,,-.. .. ... ' . -. ... ... .- .. . . .,.. . ... . : . .. ." '.. '. ..... ' . . " , ." .',. -. '. . .. : .. ,,' .... ' .. ,. I' " • • . •.•• I '.' . _,I. ..., • • '. • .. "<III • ~ • :.' !. . I . • ".r ~. '. • • I • '. .:' ... #. • I" _:'.,4 ............. . /1. .. " . • : •• 1 .. -:.1 '. '. : I '. . :.: .'. I .. ': .. I .'. : : ~I T H x --------........ -. .. Figure 1. Diagram of specimen: polycrystalline - austenitic-steel parallelepiped with electromachined O.3-mm-wide notch. We measured the specimen's macroscopic vibration frequencies in the nonnotched condition and in three notched conditions, with progressively deepened notches approximately 1, 3, and 5-mm deep. Figure 1 shows a schematic diagram of the notched specimen, the electromachined notch being O.3-mm wide. Measurements Several authors [1-3] reviewed the ultrasonic-spectroscopy method, and one of our recent studies [4] gives the particulars of our measurement system. Without liquid couplant, the specimen contacts two piezoelectric transducers at two of the specimen's diagonally opposite corners, effectively at two points. One transducer introduces a continuous-frequency sinusoidal longitudinal displacement. The other detects the specimen's resonance frequencies manifested as displacement-versus-frequency maxima. Figure 2 shows an example spectrum. MODELING Most of the computational analysis was completed using standard three-dimensional finite-element analysis using 8-noded linear brick elements and a code developed by the authors. The details of this formulation for the equations of periodic motion for three-dimensional elasticity can be found in any standard reference on the finite-element method (for example [5]). The eventual result requires solution of the 2020 0.25 0.20 ~ W 0.15 C ~ I-::::i fl. ~ 0.10 <C 0.05 ., , ;; 1 1 li 1 L i A II i i 1 0.00 0 50 100 150 200 250 300 FREQUENCY (kHz) Figure 2. Ultrasonic-resonance spectrum of specimen containing 5-mm-deep notch. standard eigenvalue problem defined by the mass and stiffness matrices [M] and [K] as [K] {x} = A [M] {x}. Here lambda denotes the square of the frequency and {x} the corresponding eigenvector that describes the mode shape of vibration. (1) As a check of the accuracy of the finite-element results, the Ritz method [6] was used to study the nonflawed specimen. This method uses power series to approximate the three dis-placement components. Because this method is secondary in the present study, the details of this approach are omitted except to note that approximation functions up to X 7y 7 z 7 were used. The flawed specimen is symmetric about the x-z and y-z planes, and this characteristic was exploited to reduce the domain of the problem to one quadrant of the original specimen. The mesh configuration was formed by a 4 x 5 x 8 discretization in the short, middle, and long dimensions of the specimen. The notch or void was modeled explicitly, with elements in this region removed from the mesh and appropriate traction-free boundary conditions imposed along these surfaces. Four different sets (I-IV) of boundary conditions were imposed to form the complete spectrum for the specimen. These are described by noting which displacements (u,v,w) were specified as zero for each of the four cases. Along the x-z plane, these are I(O,v,O), II(O,v,O) III(u,O,w), and IV(u,O,w). This implies, for example, that for the first set of conditions, u and ware specified to be zero and v is unknown. The four corresponding displacements on the surface along the y-z plane are I(u,O,O), II(O,v,w), III(O,v,w), and IV(u,O,O). 2021 Completing the analyses and combining the results yields the specimen's total vibration spectrum. RESULTS For the nonflawed specimen, Table I shows for the first fifteen resonance modes three sets of results: Ritz model, finite-element model, measurements. The table shows three principal features. First, the Ritz model agrees with measurement within about 1%. Second, the finite-element model gives expectedly higher predictions, by about 2-6%. For present purposes, we found this agreement satisfactorYi and we made no attempt to improve further the finite-element calculations. Third, one predicted mode, number 8, was not observed. Our experience shows that such resonance modes can usually be extracted from the background. But in this case, we did not search. For the largest-flaw case (5-mm), Table II shows the finite-element-model results together with measurements. Figure 3 shows for the first fifteen vibration modes the predicted-versus-measured frequency shifts. DISCUSSION Table I verifies that one can use either a Ritz model or a finite-element model to predict the macroscopic vibration frequencies of a regular-shape solid, in this case a rectangular parallelepiped. Our calculations suggest that a finer finite-element mesh would give results agreeing more Table I. Calculated and measured resonance-frequencies for first fifteen modes, nonnotched case, units kHz. Mode Ritz Finite Element Measurement 1 80.622 81. 813 80.933 2 98.256 102.154 98.063 3 135.427 138.550 134.531 4 158.940 164.127 158.509 5 167.229 170.473 165.830 6 168.408 174.091 168.719 7 174.080 175.625 173.429 8 184.887 192.683 - --9 208.479 220.615 209.202 10 219.163 223.412 221.187 11 221.004 226.707 222.608 12 232.487 244.403 232.132 13 257.182 262.275 255.416 14 259.705 265.625 260.197 15 259.883 270.159 262.849 2022 I Table II. For the 5-mm-deep-notch case, calculated and measured resonance frequencies, units kHz. FINITE ELEMENT I MEASUREMENT No Percent No Percent Mode Flaw Flaw Difference Flaw Flaw Difference 1 81.813 47.789 -67.69 80.933 45.043 -79.68 2 102.154 53.864 -89.65 98.063 51.097 -91. 91 3 138.550 73.412 -88.73 134.531 68.871 -95.33 4 164.127 116.236 -41.20 158.509 110.221 -43.81 5 170.473 129.894 -31. 24 165.830 121.128 -36.90 6 174.091 153.881 -13.13 168.719 148.578 -13.56 7 175.625 156.211 -12.43 173.429 151.212 -14.69 8 192.683 172.808 -11.50 - -- 167.190 ---9 220.615 192.500 -14.61 209.202 185.740 -12.63 10 223.412 213.323 - 4.73 221.187 208.037 - 6.32 11 226.707 213.681 - 6.10 222.608 208.224 - 6.91 12 244.403 226.509 - 7.90 232.132 213.068 - 8.95 13 262.275 247.328 - 6.04 255.416 243.653 - 4.83 14 265.625 249.147 - 6.61 260.197 244.349 - 6.49 15 270.159 265.640 - 1. 70 262.849 259.366 - 1. 34 closely with the Ritz model, which agree closely with observation. I Table II and Figure 3 verify that a finite-element model of a cracked (notched) regular-shape solid gives results in good agreement with observation. Again, the model-observation agreement should improve with a finer finite-element mesh. For the studied specimen, the frequency shifts are large, approaching 100% for some modes in the 5mm-notch case! Resonance-peak positions can be determined within about 1 Hz. Taking typical resonance-peak frequency as 100,000 Hz, we see that (in principle) ultrasonic resonance spectroscopy can detect a very small crack (notch). Despite the occurrence of a deep crack, halfway across the specimen, no new resonance peaks appeared. Nor did the model predict new peaks. This feature simplifies greatly both the model-measurement comparison and tracking the resonance-peak frequency shifts. A question arises concerning what happens if we reduce the notch width to approximate more closely a real crack. We did not yet do such calculations, but we guess the effect would be small. For a thinner crack, we expect essentially the same vibrational spectrum as for the 0.3-mm-thick crack that we studied. 2023 100 0 0 80 0 C Q) E .!!1 w , 60 2 c u::: hi OJ c 0 co 40 ..c 0 0 ~ a 20 FLA W SIZE: 5 mm o o 20 40 60 80 100 % Change: Measurement Figure 3. Predicted-versus-measured resonance-frequency shifts. Often, acoustic studies on cracked materials report reductions in either sound velocities or in elastic constants Cij • Unlike previous studies [4,7] where we used ultrasonic-resonance spectroscopy to determine a material's elastic constants by measuring the resonance frequencies fn and doing an inverse calculation to get the Cij , such an approach fails in the present case because the crack destroys the material (specimen) symmetry. If cracks were smaller and occluded within the specimen, then the inverse calculation for the reduced Cij should succeed. Another acoustic property that shows sensitivity to crack formation is the internal friction Q-l, which relates simply to the resonance-peak width: We shall discuss Q-l in a future, fuller description of the present study. CONCLUSIONS This study led to two principal conclusions: (2 ) 1. Ultrasonic-resonance spectroscopy provides a powerful tool for detecting and characterizing a crack in a regular-shape solid. 2024 2. Although this study focused on a relatively large crack (0.5 x 0.5 cm), the large resonance-peak frequency shifts suggest the possibility to detect cracks smaller by many orders of magnitude. ACKNOWLEDGMENT P. Heyliger acknowledges receipt of a summer research grant from NIST-Boulder. REFERENCES 1. E. Schreiber, o. Anderson, N. Soga, Elastic Constants and Their Measurement (McGraw-Hill, New York, 1973), Chap. 5. 2. J. Maynard, M. McKenna, A. Migliori, W. Visscher, in Ultrasonics of High-Tc and Other Unconventional Superconductors (Academic, Boston, 1992), p. 381. 3. o. Anderson, J. Acoust. Soc. Am. 91, 2245 (1992). 4. H. Ledbetter, C. Fortunko, P. Heyliger, J. Appl. Phys, forthcoming. 5. J. Reddy, An Introduction to the Finite-Element Method (McGraw-Hill, New York, 1993). 6. I. Ohno, J. Phys. Earth 23, 335 (1976). 7. H. Ledbetter, S. Kim, C. Fortunko, P. Heyliger, Int. J. Thermophys., forthcoming. 2025 

Artificial Crack in Steel: An Ultrasonic-Resonance-Spectrscopy and Modeling Study

https://dr.lib.iastate.edu/handle/20.500.12876/60524

Artificial Crack in Steel: An Ultrasonic-Resonance-Spectrscopy and Modeling Study

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