239:
damage accumulated during a thermal shock treatment can be determined for refractory materials. This can be an advantage in understanding the physical properties of certain materials. Finally, the technique can be used to check the quality of systems. In this case, a reference piece is required to obtain a reference frequency spectrum. Engine blocks for example can be tested by tapping them and comparing the recorded signal with a pre-recorded signal of a reference engine block. By using simple cluster analysis algorithms or principal component analysis, sample's pattern recognition is also achievable with a set of pre-recorded signals.
1631:(also called "Mixed numerical experimental method"). The non destructive Resonalyser procedure allows a fast and accurate simultaneous identification of the 4 Engineering constants E1, E2, G12 and v12 for orthotropic materials. For the identification of the four orthotropic material constants, the first three natural frequencies of a rectangular test plate with constant thickness and the first natural frequency of two test beams with rectangular cross section must be measured. One test beam is cut along the longitudinal direction 1, the other one cut along the transversal direction 2 (see Figure on the right).
1663:
1653:
1658:
67:. To optimize the results a microphone or a laser vibrometer can be used as there is no contact between the test-piece and the sensor. Laser vibrometers are preferred to measure signals in vacuum. Afterwards, the acquired vibration signal in the time domain is converted to the frequency domain by a
1595:
The figure above shows some examples of common orthotropic materials: layered uni-directionally reinforced composites with fiber directions parallel to the plate edges, layered bi-directionally reinforced composites, short fiber reinforced composites with preference directions (like wooden particle
83:
Different resonant frequencies can be excited dependent on the position of the support wires, the mechanical impulse and the microphone. The two most important resonant frequencies are the flexural which is controlled by the Young's modulus of the sample and the torsional which is controlled by the
181:
Material damping or internal friction is characterized by the decay of the vibration amplitude of the sample in free vibration as the logarithmic decrement. The damping behaviour originates from anelastic processes occurring in a strained solid i.e. thermoelastic damping, magnetic damping, viscous
1647:
This ratio yields a so-called "Poisson plate". The interesting property of a Freely suspended
Poisson plate is that the modal shapes that are associated with the 3 first resonance frequencies are fixed: the first resonance frequency is associated with a torsional modal shape, the second resonance
1612:
require two tensile, bending of IET tests, one on a beam cut along the 1-direction and one on a beam cut along the 2-direction. Major and minor
Poisson's ratios can be identified if also the transverse strains are measured during the tensile tests. The identification of the in-plane shear modulus
238:
tool to study the transitions as function of time and temperature. A detailed insight into the material crystal structure can be obtained by studying the elastic and damping properties. For example, the interaction of dislocations and point defects in carbon steels are studied. Also the material
224:
The most important parameters to define the measurement uncertainty are the mass and dimensions of the sample. Therefore, each parameter has to be measured (and prepared) to a level of accuracy of 0.1%. Especially, the sample thickness is most critical (third power in the equation for Young's
1971:
Germany, GHI/RWTH-Aachen, Aachen, Germany, Institute of
Mineral Engineering – Department of Ceramics and Refractory Materials, Aachen (2015-01-01). "Estimation of Damage in Refractory Materials after Progressive Thermal Shocks with Resonant Frequency Damping Analysis".
1617:
1671:
The question is now how to extract the orthotropic
Engineering constants from the frequencies measured with IET on the beams and Poisson plate. This problem can be solved by an inverse method (also called" Mixed numerical/experimental method") based on a
793:
233:
The impulse excitation technique can be used in a wide range of applications. Nowadays, IET equipment can perform measurements between −50 °C and 1700 °C in different atmospheres (air, inert, vacuum). IET is mostly used in research and as
1533:
351:
207:
and impulse excitation technique) provide an advantage over static methods because the measurements are relatively quick and simple and involve small elastic strains. Therefore, IET is very suitable for porous and brittle materials like
194:
Considering the importance of elastic properties for design and engineering applications, a number of experimental techniques are developed and these can be classified into 2 groups; static and dynamic methods. Statics methods (like the
164:
is a measure in which a material tends to expand in directions perpendicular to the direction of compression. After measuring the Young's modulus and the shear modulus, dedicated software determines the
Poisson's ratio using
1804:
Roebben, G.; Bollen, B.; Brebels, A.; Van
Humbeeck, J.; Van Der Biest, O. (1997-12-01). "Impulse excitation apparatus to measure resonant frequencies, elastic moduli, and internal friction at room and high temperature".
1346:
942:
1529:
when the elastic properties are symmetric with respect to a rectangular
Cartesian system of axes. In case of a two dimensional state of stress, like in thin sheets, the stress-strain relations for orthotropic material
47:
of predefined shapes like rectangular bars, cylindrical rods and disc shaped samples. The measurements can be performed at room temperature or at elevated temperatures (up to 1700 °C) under different atmospheres.
120:. To minimize the damping of the test-piece, it has to be supported at the nodes where the vibration amplitude is zero. The test-piece is mechanically excited at one of the anti-nodes to cause maximum vibration.
1445:
1139:
152:. To minimize the damping of the test-piece, it has to be supported at the center of both axis. The mechanical excitation has to be performed in one corner in order to twist the beam rather than flexing it.
1010:
87:
For predefined shapes like rectangular bars, discs, rods and grinding wheels, dedicated software calculates the sample's elastic properties using the sample dimensions, weight and resonant frequency (ASTM
414:
1234:
533:
112:
mode. This induced vibration is also referred as the out-of-plane vibration mode. The in-plane vibration will be excited by turning the sample 90° on the axis parallel to its length. The
1590:
1679:
In an inverse method, the material properties in the finite element model are updated in such a way that the computed resonance frequencies match the measured resonance frequencies.
544:
1848:
Roebben, G; Basu, B; Vleugels, J; Van
Humbeeck, J; Van der Biest, O (2000-09-28). "The innovative impulse excitation technique for high-temperature mechanical spectroscopy".
2015:"IETeasy: An open source and low-cost instrument for impulse excitation technique, applied to materials classification by acoustical and mechanical properties assessment"
1877:"IETeasy: An open source and low-cost instrument for impulse excitation technique, applied to materials classification by acoustical and mechanical properties assessment"
2137:
1999:
260:
1668:
So, without the necessity to do an investigation to the nature of the modal shapes, the IET on a
Poisson plate reveals the vibrational behaviour of a Poisson plate.
2192:
Material
Identification Using Mixed Numerical Experimental Methods : Proceedings of the EUROMECH Colloquium held in Kerkrade, the Netherlands, 7-9 April 1997
1480:
v. For isotropic materials the relation between strains and stresses in any point of flat sheets is given by the flexibility matrix in the following expression:
1673:
1485:
27:) is a non-destructive material characterization technique to determine the elastic properties and internal friction of a material of interest. It measures the
1518:. The orientation of the axes 1 and 2 in the above figure is arbitrary. This means that the values for E, G and v are the same in any material direction.
1770:
ISO 12680-1:2005 - Methods of test for refractory products -- Part 1: Determination of dynamic Young's modulus (MOE) by impulse excitation of vibration
2063:
Non-destructive testing : proceedings of the First Joint Belgian-Hellenic Conference on Non-Destructive Testing, Patras, Greece, 22-23 May 1995
1641:
1261:
124:
92:
2199:
863:
1376:
216:. The technique can also be easily modified for high temperature experiments and only a small amount of material needs to be available.
1684:
1070:
1760:
ASTM E1876 - 15 Standard Test Method for Dynamic Youngs Modulus, Shear Modulus, and Poissons Ratio by Impulse Excitation of Vibration
2070:
186:, vacancies, ...) can contribute to an increase in the internal friction between the vibrating defects and the neighboring regions.
1926:"Impulse Excitation Internal Friction Study of Dislocation and Point Defect Interactions in Ultra-Low Carbon Bake-Hardenable Steel"
1746:
the knowledge of the modal shapes of a Poisson plate can be used to generate very good starting values using a virtual field method
1676:(FE) computer model of the Poisson plate. A FE model allows computing resonance frequencies for a given set of material properties
1248:
51:
The measurement principle is based on tapping the sample with a small projectile and recording the induced vibration signal with a
1634:
The Young's modulus of the test beams can be found using the bending IET formula for test beams with a rectangular cross section.
1749:
and the first 3 natural frequencies of a Poisson plate are sensitive for variations of all the orthotropic Engineering constants.
1721:· The computed frequencies in the FE model of the Poisson plate must be sensitive for variations of all the material parameters
1648:
frequency is associated with a saddle modal shape and the third resonance frequency is associated with a breathing modal shape.
1244:
The induced vibration signal (in the time domain) is fitted as a sum of exponentially damped sinusoidal functions according to:
72:
2229:
953:
71:. Dedicated software will determine the resonant frequency with high accuracy to calculate the elastic properties based on the
362:
1729:
if only the Poisson's ratio v12 and the in-plane shear modulus G12 are taken as variable parameters in the FE-model, the
1187:
2244:
1769:
1725:
In the case the Young's moduli (obtained by IET) are fixed (as non variable parameters) in the inverse method procedure
1718:· The starting values must be close enough to the final solution to avoid a local minimum (instead of a global minimum)
2239:
2094:
474:
1780:
DIN EN 843-2:2007 Advanced technical ceramics - Mechanical properties of monolithic ceramics at room temperature"
2234:
196:
60:
788:{\displaystyle R=\left\left-0.060\left({\frac {b}{L}}\right)^{\frac {3}{2}}\left({\frac {b}{t}}-1\right)^{2}}
204:
203:) are based on direct measurements of stresses and strains during mechanical tests. Dynamic methods (like
68:
1993:
1181:
If the Young's modulus and shear modulus are known, the Poisson's ratio can be calculated according to:
28:
2151:
Lauwagie, Tom; Sol, Hugo; Roebben, Gert; Heylen, Ward; Shi, Yinming; Van der Biest, Omer (2003-10-01).
2152:
1814:
1526:
52:
1484:
225:
modulus). In that case, an overall accuracy of 1% can be obtained practically in most applications.
1779:
209:
141:
128:
2088:
1953:
1563:
1547:
1477:
1469:
346:{\displaystyle E=0.9465\left({\frac {mf_{f}^{2}}{b}}\right)\left({\frac {L^{3}}{t^{3}}}\right)T}
161:
117:
40:
32:
2153:"Mixed numerical–experimental identification of elastic properties of orthotropic metal plates"
2205:
2195:
2172:
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1945:
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1640:
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123:
113:
91:
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1981:
1937:
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1857:
1822:
2013:
Massara, Nazareno; Boccaleri, Enrico; Milanesio, Marco; Lopresti, Mattia (1 October 2021).
1875:
Massara, Nazareno; Boccaleri, Enrico; Milanesio, Marco; Lopresti, Mattia (1 October 2021).
235:
200:
1637:
The ratio Width/Length of the test plate must be cut according to the following formula:
1521:
More complex material behaviour like orthotropic material behaviour can be identified by
1818:
1743:
a FE of a plate can be made very accurate by selecting a sufficiently fine element grid,
2065:. Hemelrijck, Danny van., Anastassopoulos, Athanassios. Rotterdam: A.A. Balkema. 1996.
2039:
2014:
1901:
1876:
166:
2168:
1861:
1852:. Intern. Conf. Internal Friction and Ultrasonic Attentuation in Solids (ICIFUAS-12).
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1957:
1596:
boards), plastics with preference orientation, rolled metal sheets, and much more...
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1473:
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64:
36:
2124:"Validation_of_the_Resonalyser_method_an_inverse_method_for_material_identification"
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1925:
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IET yields very accurate resonance frequencies, even with non-expert equipment,
2030:
1941:
1892:
213:
56:
2209:
2176:
1949:
1834:
2080:
1683:
1461:
1341:{\displaystyle x\left(t\right)=\sum Ae^{-kt}\sin \left(2\pi ft+\phi \right)}
2048:
1985:
1910:
1759:
1698:· Are the parameters converging to the correct physical solution?
170:
44:
1532:
1247:
937:{\displaystyle E=1.6067\left({\frac {L^{3}}{d^{4}}}\right)mf_{f}^{2}T'}
182:
damping, defect damping, ... For example, different materials defects (
109:
96:
2108:
1826:
1695:· The need of good starting values for the material properties
1440:{\displaystyle Q^{-1}={\frac {\Delta W}{2\pi W}}={\frac {k}{\pi f}}}
140:
The second figure gives an example of a test-piece vibrating in the
1604:
Standard methods for the identification of the two Young's moduli E
108:
The first figure gives an example of a test-piece vibrating in the
1246:
122:
116:
of this flexural vibration mode is characteristic for the dynamic
90:
1134:{\displaystyle G=16\left({\frac {L}{\pi d^{2}}}\right)mf_{t}^{2}}
1924:
Jung, Il-Chan; Kang, Deok-Gu; Cooman, Bruno C. De (2013-11-26).
1252:
1573:
is symmetric. The minor Poisson's ratio can hence be found if E
2122:
T. Lauwagie, H. Sol, G. Roebben, W. Heylen and Y. Shi (2002).
1639:
1615:
1005:{\displaystyle T'=1+4.939\left({\frac {d}{L}}\right)^{2}}
409:{\displaystyle T=1+6.585\left({\frac {t}{L}}\right)^{2}}
2109:"Theoretical Background of the Resonalyser Procedure"
1715:· The IET measurements must be sufficiently accurate
1570:
is the minor Poisson's ratio. The flexibility matrix
1468:
using the above described empirical formulas for the
1379:
1264:
1190:
1073:
956:
866:
547:
477:
365:
263:
2194:. Oomens, C. W. J. Dordrecht: Springer Netherlands.
1229:{\displaystyle \nu =\left({\frac {E}{2G}}\right)-1}
461:
The correction factor can only be used if L/t ≥ 20!
1498:are normal strains in the 1- and 2-direction and Υ
1439:
1340:
1228:
1133:
1004:
936:
787:
527:
408:
345:
1452:Extended IET applications: the Resonalyser Method
173:materials according to the different standards.
1613:requires an additional in plane shearing test.
1600:Extended IET for orthotropic material behaviour
1457:Isotropic versus orthotropic material behaviour
1712:· The FE-model must be sufficiently accurate
528:{\displaystyle G={\frac {4Lmf_{t}^{2}}{bt}}R}
8:
2136:: CS1 maint: multiple names: authors list (
1998:: CS1 maint: multiple names: authors list (
148:of this vibration is characteristic for the
1930:Metallurgical and Materials Transactions A
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1974:Journal of Ceramic Science and Technology
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1054:The correction factor can only be used if
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84:shear modulus for isotropic materials.
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2086:
1991:
1627:" is an extension of the IET using an
7:
1464:elastic properties can be found by
1733:satisfies all above requirements.
1399:
14:
1701:· Is the solution unique?
1807:Review of Scientific Instruments
1682:
1661:
1656:
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1531:
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16:Method to characterize materials
1850:Journal of Alloys and Compounds
1550:in the 1- and 2-direction and G
1447:with W the energy of the system
1:
2169:10.1016/S0963-8695(03)00048-3
1862:10.1016/S0925-8388(00)00966-X
1510:are the normal stresses and τ
169:which can only be applied to
1708:to obtain good results are:
799:Note that we assume that b≥t
127:Test-piece vibrating in the
21:impulse excitation technique
177:Internal friction / Damping
69:fast Fourier transformation
2261:
1692:with inverse methods are:
190:Dynamic vs. static methods
31:in order to calculate the
2157:NDT & E International
2031:10.1016/j.ohx.2021.e00231
1942:10.1007/s11661-013-2122-z
1893:10.1016/j.ohx.2021.e00231
1364:the logarithmic decrement
855:
243:Experimental correlations
220:Accuracy and uncertainty
95:Test-piece vibrating in
1525:. A material is called
1523:extended IET procedures
1154:the torsional frequency
816:the torsional frequency
252:
205:ultrasound spectroscopy
197:four-point bending test
2230:Nondestructive testing
2093:: CS1 maint: others (
1986:10.4416/jcst2015-00080
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1620:
1502:is the shear strain. σ
1441:
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1032:the flexural frequency
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938:
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436:the flexural frequency
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347:
132:
100:
1731:Resonalyser procedure
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1625:Resonalyser procedure
1619:
1564:major Poisson's ratio
1490:In this expression, ε
1442:
1358:the natural frequency
1343:
1250:
1231:
1136:
1050:the correction factor
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846:the correction factor
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457:the correction factor
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73:classical beam theory
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53:piezoelectric sensor
29:resonant frequencies
2245:Continuum mechanics
1819:1997RScI...68.4511R
1369:damping parameter Q
1240:Damping coefficient
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1017:the Young's modulus
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421:the Young's modulus
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1371:can be defined as:
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806:the shear modulus
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79:Elastic properties
2240:Materials science
2201:978-94-009-1471-1
1827:10.1063/1.1148422
1813:(12): 4511–4515.
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1936:(4): 1962–1978.
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1470:Young's modulus
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162:Poisson's ratio
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156:Poisson's ratio
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33:Young's modulus
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1629:inverse method
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1349:
1348:
1336:
1332:
1329:
1326:
1323:
1320:
1317:
1313:
1309:
1306:
1301:
1298:
1295:
1291:
1287:
1284:
1281:
1277:
1274:
1271:
1267:
1241:
1238:
1237:
1236:
1225:
1222:
1218:
1212:
1209:
1205:
1200:
1196:
1193:
1178:
1175:
1174:
1173:
1169:
1167:
1163:
1161:
1157:
1155:
1150:
1148:
1142:
1141:
1128:
1123:
1119:
1115:
1111:
1103:
1099:
1095:
1091:
1086:
1082:
1079:
1076:
1064:
1061:
1060:
1059:
1051:
1045:
1041:
1039:
1035:
1033:
1028:
1026:
1024:
1020:
1018:
1014:
1012:
999:
994:
989:
986:
981:
976:
973:
970:
967:
963:
960:
945:
944:
932:
929:
923:
918:
914:
910:
906:
899:
895:
889:
885:
879:
875:
872:
869:
857:
854:
852:
849:
848:
847:
841:
837:
835:
831:
829:
825:
823:
819:
817:
812:
810:
801:
800:
796:
795:
782:
777:
773:
770:
765:
762:
756:
748:
745:
739:
734:
731:
726:
721:
718:
714:
706:
702:
695:
691:
687:
681:
678:
674:
669:
662:
655:
652:
645:
642:
637:
633:
628:
623:
620:
616:
610:
607:
602:
599:
596:
589:
584:
579:
576:
571:
566:
563:
557:
553:
550:
536:
535:
524:
518:
515:
508:
503:
499:
495:
492:
489:
483:
480:
468:
465:
464:
463:
458:
452:
448:
446:
442:
440:
437:
432:
430:
428:
424:
422:
418:
416:
403:
398:
393:
390:
385:
380:
377:
374:
371:
368:
354:
353:
342:
338:
331:
327:
321:
317:
311:
306:
301:
295:
290:
286:
282:
276:
272:
269:
266:
254:
251:
249:
246:
244:
241:
230:
227:
221:
218:
191:
188:
178:
175:
157:
154:
137:
134:
105:
102:
80:
77:
15:
13:
10:
9:
6:
4:
3:
2:
2257:
2246:
2243:
2241:
2238:
2236:
2233:
2231:
2228:
2227:
2225:
2211:
2207:
2203:
2197:
2193:
2186:
2183:
2178:
2174:
2170:
2166:
2162:
2158:
2154:
2147:
2144:
2139:
2133:
2125:
2118:
2115:
2110:
2104:
2101:
2096:
2090:
2082:
2078:
2074:
2072:90-5410-595-X
2068:
2064:
2058:
2055:
2050:
2046:
2041:
2036:
2032:
2028:
2024:
2020:
2016:
2009:
2006:
2001:
1995:
1987:
1983:
1979:
1975:
1967:
1964:
1959:
1955:
1951:
1947:
1943:
1939:
1935:
1931:
1927:
1920:
1917:
1912:
1908:
1903:
1898:
1894:
1890:
1886:
1882:
1878:
1871:
1868:
1863:
1859:
1855:
1851:
1844:
1841:
1836:
1832:
1828:
1824:
1820:
1816:
1812:
1808:
1800:
1797:
1790:
1785:
1781:
1778:
1775:
1771:
1768:
1765:
1761:
1758:
1757:
1753:
1748:
1745:
1742:
1739:
1738:
1737:
1734:
1732:
1728:
1720:
1717:
1714:
1711:
1710:
1709:
1707:
1702:
1699:
1696:
1693:
1691:
1687:
1685:
1680:
1677:
1675:
1669:
1664:
1659:
1654:
1649:
1642:
1638:
1635:
1632:
1630:
1626:
1618:
1614:
1599:
1597:
1593:
1591:
1586:
1572:
1565:
1557:
1556:shear modulus
1549:
1536:
1534:
1528:
1524:
1519:
1517:
1488:
1486:
1481:
1479:
1475:
1474:shear modulus
1471:
1467:
1463:
1456:
1451:
1431:
1428:
1424:
1419:
1413:
1410:
1407:
1402:
1393:
1388:
1385:
1381:
1373:
1370:
1366:
1360:
1354:
1353:
1352:
1334:
1330:
1327:
1324:
1321:
1318:
1315:
1311:
1307:
1304:
1299:
1296:
1293:
1289:
1285:
1282:
1279:
1275:
1272:
1269:
1265:
1258:
1257:
1254:
1249:
1245:
1239:
1223:
1220:
1216:
1210:
1207:
1203:
1198:
1194:
1191:
1184:
1183:
1182:
1177:Poisson ratio
1176:
1168:
1162:
1156:
1147:
1146:
1145:
1126:
1121:
1117:
1113:
1109:
1101:
1097:
1093:
1089:
1084:
1080:
1077:
1074:
1067:
1066:
1063:Shear modulus
1062:
1058:
1055:
1052:
1049:
1046:
1040:
1034:
1025:
1019:
1013:
997:
992:
987:
984:
979:
974:
971:
968:
965:
961:
958:
950:
949:
948:
930:
927:
921:
916:
912:
908:
904:
897:
893:
887:
883:
877:
873:
870:
867:
860:
859:
850:
845:
842:
840:the thickness
836:
830:
824:
818:
809:
808:
807:
805:
798:
797:
780:
775:
771:
768:
763:
760:
754:
746:
743:
737:
732:
729:
724:
719:
716:
712:
704:
700:
693:
689:
685:
679:
676:
672:
667:
660:
653:
650:
643:
640:
635:
631:
626:
621:
618:
614:
608:
605:
600:
597:
594:
587:
582:
577:
574:
569:
564:
561:
555:
551:
548:
541:
540:
539:
522:
516:
513:
506:
501:
497:
493:
490:
487:
481:
478:
471:
470:
467:Shear modulus
466:
462:
459:
456:
453:
451:the thickness
447:
441:
438:
429:
423:
417:
401:
396:
391:
388:
383:
378:
375:
372:
369:
366:
359:
358:
357:
340:
336:
329:
325:
319:
315:
309:
304:
299:
293:
288:
284:
280:
274:
270:
267:
264:
257:
256:
247:
242:
240:
237:
228:
226:
219:
217:
215:
211:
206:
202:
198:
189:
187:
185:
176:
174:
172:
168:
163:
153:
151:
150:shear modulus
147:
143:
135:
130:
125:
121:
119:
115:
111:
103:
98:
93:
89:
85:
78:
76:
74:
70:
66:
65:accelerometer
62:
58:
54:
49:
46:
42:
38:
37:shear modulus
34:
30:
26:
22:
2191:
2185:
2160:
2156:
2146:
2117:
2103:
2062:
2057:
2022:
2018:
2008:
1994:cite journal
1977:
1973:
1966:
1933:
1929:
1919:
1884:
1880:
1870:
1853:
1849:
1843:
1810:
1806:
1799:
1783:
1773:
1764:www.astm.org
1763:
1735:
1730:
1726:
1724:
1706:requirements
1705:
1703:
1700:
1697:
1694:
1689:
1688:
1681:
1678:
1670:
1667:
1646:
1636:
1633:
1628:
1624:
1622:
1603:
1594:
1587:
1571:
1537:
1522:
1520:
1516:shear stress
1489:
1482:
1465:
1460:
1368:
1350:
1243:
1180:
1166:the diameter
1143:
1056:
1053:
1047:
1038:the diameter
946:
843:
803:
802:
537:
460:
454:
355:
232:
229:Applications
223:
214:refractories
193:
184:dislocations
180:
159:
139:
136:Torsion mode
107:
104:Flexure mode
86:
82:
50:
24:
20:
18:
1585:are known.
1527:orthotropic
439:b the width
167:Hooke's law
2224:Categories
2025:: e00231.
1887:: e00231.
1791:References
1172:the length
1044:the length
834:the length
445:the length
144:mode. The
88:E1876-15).
57:microphone
2210:851370715
2177:0963-8695
2089:cite book
2019:HardwareX
1958:136855858
1950:1073-5623
1881:HardwareX
1835:0034-6748
1754:Standards
1462:Isotropic
1429:π
1411:π
1400:Δ
1386:−
1331:ϕ
1319:π
1308:
1294:−
1283:∑
1221:−
1192:ν
1094:π
1057:L/d ≥ 20!
828:the width
769:−
717:−
636:π
622:−
598:−
171:isotropic
2132:cite web
2081:35306088
2049:35607698
1911:35607698
1736:Indeed,
1690:Problems
1546:are the
1160:the mass
1023:the mass
962:′
931:′
822:the mass
427:the mass
210:ceramics
2040:9123443
1902:9123443
1815:Bibcode
1562:is the
1530:become:
1514:is the
1472:E, the
1251:Damped
686:0.00851
142:torsion
129:torsion
110:flexure
97:flexure
2208:
2198:
2175:
2079:
2069:
2047:
2037:
1956:
1948:
1909:
1899:
1833:
1476:G and
1362:δ = kt
874:1.6067
271:0.9465
1980:(2).
1954:S2CID
1623:The "
1608:and E
1581:and v
1566:and v
1542:and E
1506:and σ
1494:and ε
1351:with
1144:with
975:4.939
947:with
720:0.060
627:1.991
601:2.521
538:with
379:6.585
356:with
2206:OCLC
2196:ISBN
2173:ISSN
2138:link
2095:link
2077:OCLC
2067:ISBN
2045:PMID
2000:link
1946:ISSN
1907:PMID
1831:ISSN
1704:The
1253:sine
212:and
199:and
160:The
131:mode
99:mode
43:and
19:The
2165:doi
2035:PMC
2027:doi
1982:doi
1938:doi
1897:PMC
1889:doi
1858:doi
1854:310
1823:doi
1774:ISO
1727:and
1577:, E
1558:. v
1466:IET
1305:sin
63:or
25:IET
2226::
2204:.
2171:.
2161:36
2159:.
2155:.
2134:}}
2130:{{
2091:}}
2087:{{
2075:.
2043:.
2033:.
2023:10
2021:.
2017:.
1996:}}
1992:{{
1976:.
1952:.
1944:.
1934:45
1932:.
1928:.
1905:.
1895:.
1885:10
1883:.
1879:.
1829:.
1821:.
1811:68
1809:.
1782:.
1772:.
1762:.
1583:12
1568:21
1560:12
1552:12
1512:12
1500:12
1081:16
1048:T'
75:.
59:,
55:,
39:,
35:,
2212:.
2179:.
2167::
2140:)
2126:.
2111:.
2097:)
2083:.
2051:.
2029::
2002:)
1988:.
1984::
1978:7
1960:.
1940::
1913:.
1891::
1864:.
1860::
1837:.
1825::
1817::
1786:.
1776:.
1766:.
1610:2
1606:1
1579:2
1575:1
1544:2
1540:1
1538:E
1508:2
1504:1
1496:2
1492:1
1432:f
1425:k
1420:=
1414:W
1408:2
1403:W
1394:=
1389:1
1382:Q
1356:f
1335:)
1328:+
1325:t
1322:f
1316:2
1312:(
1300:t
1297:k
1290:e
1286:A
1280:=
1276:)
1273:t
1270:(
1266:x
1224:1
1217:)
1211:G
1208:2
1204:E
1199:(
1195:=
1170:L
1164:d
1158:m
1151:t
1149:f
1127:2
1122:t
1118:f
1114:m
1110:)
1102:2
1098:d
1090:L
1085:(
1078:=
1075:G
1042:L
1036:d
1029:f
1027:f
1021:m
1015:E
998:2
993:)
988:L
985:d
980:(
972:+
969:1
966:=
959:T
928:T
922:2
917:f
913:f
909:m
905:)
898:4
894:d
888:3
884:L
878:(
871:=
868:E
844:R
838:t
832:L
826:b
820:m
813:t
811:f
804:G
781:2
776:)
772:1
764:t
761:b
755:(
747:2
744:3
738:)
733:L
730:b
725:(
713:]
705:2
701:L
694:2
690:b
680:+
677:1
673:[
668:]
661:)
654:1
651:+
644:t
641:b
632:e
619:1
615:(
609:b
606:t
595:4
588:2
583:)
578:t
575:b
570:(
565:+
562:1
556:[
552:=
549:R
523:R
517:t
514:b
507:2
502:t
498:f
494:m
491:L
488:4
482:=
479:G
455:T
449:t
443:L
433:f
431:f
425:m
419:E
402:2
397:)
392:L
389:t
384:(
376:+
373:1
370:=
367:T
341:T
337:)
330:3
326:t
320:3
316:L
310:(
305:)
300:b
294:2
289:f
285:f
281:m
275:(
268:=
265:E
23:(
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