Straightforward estimation of the elastic constants of

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Nieves et al.: JASA Express Letters

Published Online 8 October 2009

Straightforward estimation of the elastic constants of an isotropic cube excited by a single percussion F. J. Nieves and F. Gascón Departamento de Física Aplicada II, E.T.S. Arquitectura, Universidad de Sevilla, Reina Mercedes 2, 41012 Sevilla, Spain [email protected], [email protected]

A. Bayón and F. Salazar Departamento de Física Aplicada, E.T.S.I. Minas, Universidad Politécnica de Madrid, Ríos Rosas 21, 28003 Madrid, Spain [email protected], [email protected]

Abstract: Ritz’s method is applied to calculate accurate values of the lowest non-dimensional natural frequencies of a freely vibrating isotropic cube. The dependence of such frequencies and their quotients on Poisson’s ratio is established. Vibration of a cube caused by percussion is detected at a point by a laser interferometer. With the help of the tables and graphs provided and with the values of the first lowest frequencies obtained experimentally in a single test, Poisson’s ratio and the shear modulus are calculated by means of elementary arithmetical operations. © 2009 Acoustical Society of America PACS numbers: 43.40.At, 43.40.Dx, 43.20.Ks [JM] Date Received: April 30, 2009 Date Accepted: September 11, 2009

1. Introduction As an antecedent of the work presented here, it is possible to mention the calculation of the dynamic elastic constants from the axisymmetric vibration of a cylinder whose length is equal to its diameter.1 The measurement of two or three natural frequencies of vibration of the cylinder, originated by an axial percussion, enables the calculation of Poisson’s ratio and the shear modulus. A general analytical solution for the free vibration problem of a thick plate does not exist. An approximate solution can be obtained by Ritz’s, finite element method (FEM), and other methods. In this paper the Ritz method is employed due to its advantages with respect to the FEM. Ritz’s technique utilizes global basis functions, which are more accurate per degree of freedom than FEM. In addition, Ritz’s method allows breaking the total vibration problem into smaller problems.2 A solution for the amplitude of vibration of rectangular parallelepipeds was proposed in the form of power series of the coordinates.2,3 Ritz’s method has been optimized4 in two-dimensional studies with the aim of simplifying calculations by means of an automatic search of the maximum exponents of the series, which gives a good convergence. A three-dimensional calculation of the free vibration frequencies of an isotropic cube is here performed by the optimized Ritz method. It is demonstrated that the values of the five lowest natural frequencies enable the immediate calculation of the dynamic elastic constants. 2. Calculation of the natural frequencies of a free cube by the optimized Ritz method For the sake of simplicity, the following non-dimensional frequency is used: ⍀ ⬅ ␲fL冑␳/G,

共1兲

where f is the ordinary frequency measured in hertz, L is the length of each side of the cube, G its shear modulus, and ␳ its density.

EL140 J. Acoust. Soc. Am. 126 共5兲, November 2009

© 2009 Acoustical Society of America

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Table 1. The five lowest, non-dimensional, different, non-null frequencies for a cube for six different values of Poisson’s ratio and their modes. Poisson’s ratio

0 0.1 0.2 0.3 0.4 0.499

⍀1

⍀2

⍀3

⍀4

⍀5

1.427 418 EV1 1.427 646 EV1 1.427 879 EV1 1.428 087 EV1 1.428 234 EV1 1.428 406 EV1

1.866 441 EX1 + EX2 1.890 008 EX1 1.908 425 EX1 1.923 045 EX1 1.934 846 EX1 1.944 319 EX1

1.936 756 OX1 1.945 570 OX1 1.951 381 OX1 1.955 635 OX1 1.958 959 OX1 1.961 633 OX1

2.169 072 OX2 2.009 933 EX2 2.151 210 EX2 2.221 445 OD1 2.221 445 OD1 2.221 444 OD1

2.221 442 OD1 + OD2 + OD3 2.221 689 OD1 2.221 445 OD1 2.286 637 EX2 2.412 044 EX2 2.511 337 OX2

The Mindlin–Lamé modes5 have an analytical solution; the lowest non-dimensional frequency is given by ⍀m = ␲/冑2.

共2兲

Let us suppose a harmonic solution for the displacements. Polynomials of monomials formed by products of powers of the coordinates are chosen for the amplitudes,1 Ui = 兺pqrAipqrxp1xq2xr3 (i = 1, 2, and 3). In applying Ritz’s method, an optimization procedure4 that improves the calculation is here generalized to three dimensions. The method involves initiating the calculation with a first stage in which very low maximum exponents are taken, and each exponent is then sequentially increased by one and the resulting frequencies are calculated and compared. The calculation is based on the hypothesis that the first five non-null lowest frequencies are sufficient to determine the elastic constants of the cube. The reason why five frequencies are considered suitable was gathered from Fig. 1 of Ref. 6, in which it can be seen that some of the modes corresponding to those five frequencies have different behaviors with respect to Poisson’s ratio. A criterion to decide which group of exponents is optimal in each stage could be the following: The best set is that for which the sum of the first five frequencies is minimum. Using symmetry arguments of the displacement-field components it is concluded7 that a single rectangular parallelepiped can vibrate in only eight basic forms or different groups of vibration. In the particular case of a cube, it can vibrate freely in the form defined by the four groups OD (dilatation), EV (torsion), EX (bending), and OX (shear), in agreement with the nomenclature of Heyliger et al.8 Because of this reduction in modes, a cubic shaped sample is employed to make use of its high symmetry and simplify the calculation. The values of Poisson’s ratio used in the calculations are 0.0, 0.1, 0.2, 0.3, 0.4, and 0.499. This reduced number of values of v, six, is chosen in order to avoid excessive calculations and extensive tables of results, but it allows interpolation, and therefore Poisson’s ratio may be calculated. The natural frequencies of the four indicated groups of vibration, numerically calculated and expressed to six decimal places, were shown in four basic tables. Table 1 summarizes the four basic tables. The lowest natural frequency appearing in Table 1 corresponds to a torsion mode (mode EV1). Another identifiable frequency in the table is that of lower Mindlin, OD1, which, according to Eq. (2), is ␲ / 冑 2. In Table 1 none of its values differs from the theoretical value by more than 0.01%. From the calculated frequencies, Fig. 1 is drawn, which represents the smallest ⍀ against v for the lowest modes. Figure 1 has actually been drawn from each basic table by means

J. Acoust. Soc. Am. 126 共5兲, November 2009

Nieves et al.: Elastic constants by a single percussion EL141


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Ω

OX2

2.4 2.2

c3

2.0 1.8


c5

EX2 OD1

c4

OX1 EX1

c2 c1

1.6 EV1

1.4 1.2 1.0 0.0

0.1

0.2

0.3

0.4

0.5

ν

Fig. 1. The lowest computed non-dimensional frequencies ⍀ versus Poisson’s ratio for a cube. The labels indicate the group of modes to which they belong.

of the parabolic fit of each of the columns ⍀i through the polynomial expression ⍀i = ai + biv + ci␯2. In this figure it was observed that no curves corresponding to ⍀ superior to the sixth intersected with any of the six lower frequencies. None of the ⍀ frequencies is a decreasing function of v. This figure is better, for low frequencies, than that published by Demarest,6 at least for the reason that it includes values of Poisson’s ratio from 0 to very near to 0.5. Since ⍀OD1 is independent of v, and ⍀1 ⬅ ⍀EV1 has such a small variation with v, the ratio ⍀OD1 / ⍀1 ⬅ ⍀OD1 / ⍀EV1 also has a small variation. In effect, from Table 1, the interval of possible values of ⍀OD1 / ⍀EV1 is deduced to be from 1.5552 to 1.5563 and may be written as ⍀OD1 = 共1.5557± 0.0006兲⍀EV1. A similarly small interval results with the modes ⍀OD1 and ⍀OX1, whose quotient is in the interval ⍀OD1 / ⍀OX1 = 1.1397± 0.0073. 3. Calculation of the elastic constants of a cube From the experimental natural frequencies, the quotients of all the pairs of the first five lowest frequencies fi / fj are calculated. From the definition of ⍀, Eq. (1), it is deduced that the quotient ⍀i / ⍀j = fi / fj only depends on v. In Table 2, the numerical results for such quotients are listed. Figure 2 shows the quotients of lowest frequencies versus ␯ and has been drawn from the parabolic fit and by applying the equalities f i ⍀ i a i + b i␯ + c i␯ 2 ⬅ = . f j ⍀ j a j + b j␯ + c j␯ 2

共3兲

Table 2. The ten quotients between the five lowest frequencies for a cube, numerically calculated. Poisson’s

⍀2 / ⍀1

⍀3 / ⍀1

⍀4 / ⍀1

⍀5 / ⍀1

⍀3 / ⍀2

⍀4 / ⍀2

⍀5 / ⍀2

⍀4 / ⍀3

⍀5 / ⍀3

⍀5 / ⍀4

0 0.1 0.2 0.3 0.4 0.499

1.3076 1.3239 1.3365 1.3466 1.3547 1.3612

1.3568 1.3628 1.3666 1.3694 1.3716 1.3733

1.5196 1.4079 1.5066 1.5555 1.5554 1.5552

1.5563 1.5562 1.5558 1.6012 1.6888 1.7581

1.0377 1.0294 1.0225 1.0169 1.0125 1.0089

1.1621 1.0635 1.1272 1.1552 1.1481 1.1425

1.1902 1.1755 1.1640 1.1891 1.2466 1.2916

1.1200 1.0331 1.1024 1.1359 1.1340 1.1324

1.1470 1.1419 1.1384 1.1693 1.2313 1.2802

1.0241 1.1054 1.0326 1.0293 1.0858 1.1305


Nieves et al.: Elastic constants by a single percussion


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Ωi/Ωj f5/f1 1.7

1.6 f4/f1 1.5

1.4

f3/f1 f2/f1

1.3

f5/f2 f5/f3

1.2

f4/f2 f4/f3

1.1

f5/f4 f3/f2

1.0 0.0

0.1

0.2

0.3

0.4

ν

0.5

Fig. 2. The ten ratios ⍀i / ⍀ j of the lowest frequencies versus Poisson’s ratio for a cube.

The knowledge of the quotients obtained in the laboratory for the lowest frequencies fi / fj allows Poisson’s ratio to be deduced from Table 2, or from Fig. 2, or from the equation of second degree in v, Eq. (3). Observe that the calculation of v is independent of the properties L, ␳, and G. However, it is necessary to be sure, by means of a detailed study, if ␯ is a single-valued function for some of the quotients of frequencies. The following should be borne in mind. (a) (b) (c) (d)

Quotient ⍀3 / ⍀2, for a typical value of v = 0.3, is 1.0169, and therefore its relative difference is of the order of 2%. For this reason, frequencies f3 and f2 must be measured with high precision to be considered useful. Although the frequencies of Mindlin modes do not depend on v, the quotients ⍀i / ⍀Mindlin ⬅ ⍀i / ⍀OD have the advantage that the denominator is known with great precision as is desired and all the quotients are increasing functions of ␯. To ensure that the error in the estimation of Poisson’s ratio is small, it is necessary for the absolute values of slopes of the curves in this Fig. 2 to be great for all the values of Poisson’s ratio. Poisson’s ratio v is a single-valued function of the quotients ⍀i / ⍀j, only in certain intervals. Therefore, the knowledge of the first five lowest frequencies and their quotients does not assure, a priori, deduction of Poisson’s ratio.

However, once the experimental frequencies are measured and ordered in increasing order, it is possible to determine Poisson’s ratio by means of the following steps. (1) Ten quotients fi / fj are calculated by always dividing the highest by the lowest, which guarantees quotients greater than the unit. (2) Each of these quotients is marked on the vertical axis of Fig. 2. A horizontal straight line is drawn from each of these marks. (3) The points where the horizontal straight lines cross the curves of Fig. 2, or at least the curves of greater slope, are marked, and, therefore, a set of crosses is obtained. Note that each horizontal line generally intersects the curves several times.





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The vertical line drawn through ␯ of the cube crosses about ten times the curves of Fig. 2. Reciprocally, as a result of the measurement of the frequencies, and the accurate numerical results, ten crosses on the same vertical are obtained, and such a vertical line in turn intersects the horizontal axis at a point corresponding to the value of Poisson’s ratio. Since the measures of the frequencies are not perfect and neither are the calculated fit curves, the crosses are on a broken line around the vertical line through the expected value of v. Therefore a graphical interpolation with a vertical straight line provides an average value for v. Given Poisson’s ratio, an experimental natural frequency fi and its ith order spectrum, either Table 1 or Fig. 1, provide the value of the corresponding ⍀i. With this value of ⍀i, measured length L of the edge of the cube, its mass, and calculated density ␳, the application of the definition of ⍀, Eq. (1), gives the value of the shear module. A practical guide for the estimation of elastic constants can be constructed to avoid the main difficulties of the proposed method: the multiple-valued relation of v with the quotient of experimental frequencies, and the high uncertainty that may appear in certain quotients. To this end the values of Poisson’s ratio have been numerically calculated corresponding to the five intersections ci that appear in Fig. 1: vc1 = 0.0020, vc2 = 0.0512, vc3 = 0.0626, vc4 = 0.2511, and vc5 = 0.4783. The values of the quotients of frequencies, at the crossings that are of more interest here, are 关⍀2 / ⍀1兴␯c1 = 1.3080, 关⍀3 / ⍀1兴␯c2 = 1.3599, and 关⍀4 / ⍀1兴␯c3 = 1.3717. Some rules leading to the identification of the interval of existence of ␯ are proposed: First, the lowest frequency always corresponds to the mode EV1, and the non-dimensional frequency ⍀1 may be assigned to f1. Second, application of the ratio given in Sec. 2, ⍀OD1 / ⍀EV1 = 共1.5557± 0.0006兲, may give the frequency fOD1 of the first Mindlin mode. Third, if, from the spectrum fi, it is deduced that fOD1 is the fourth frequency, then Fig. 2 implies that the material under study has Poisson’s coefficient v ⱖ vc4. With this an interval of existence of v is defined, which allows discernment to be made between assignable multiple values ␯ for a certain ratio fi / fj in Fig. 2. Fourth, if, for instance, it is deduced that fOD1 is the fifth frequency, then vc3 ⱕ ␯ ⬍ ␯c4. With this interval of existence, ␯ is determined. Note that between each pair of neighboring crossings of the curves in Fig. 2, v is a single-valued function of the quotients. The calculations of v and of G may be made graphically by placing Fig. 1 on Fig. 2 so that their axes are aligned. In effect, knowledge of the point of crossing of a quotient ⍀i / ⍀j with a curve of Fig. 2 is enough to draw a descending vertical line to find v. The same vertical, but in its ascending sense cuts the line of one of the modes drawn in Fig. 1 at a point from which a horizontal line leads to ⍀i and Eq. (1) gives G. To increase accuracy an easy analytic procedure may be followed by applying Eq. (3) to an adequate quotient and by solving the second degree equation to obtain v. 4. Experimental arrangement and experimental results The cube under study is of commercial stainless steel, annealed, and of dimensions a = 50.05 mm, b = 50.04 mm, and c = 50.03 mm: an almost perfect cube. The density of the cube is ␳ = 7836 kg/ m3. The elastic constants have been obtained from measurements of the P and S wave velocities in three perpendicular directions and with two polarizations. The values calculated for v belong in the interval (0.2866, 0.2881), and their average value is v = 0.2873. The values calculated for G are in the interval (77.55, 77.97), and their average value is G = 77.76 GPa. Measurement of the natural frequencies of vibration is made with a laser interferometer I-O.1 The interferometer detects displacements in the order of magnitude of 1 nm, and the bandwidth of detection is of the order of tens of MHz. The fast Fourier transform (FFT) of the out-of-plane displacement component enables us to obtain the natural frequencies of vibration of the cube. The sample is supported on a rubber. The location of the rubber does not affect the frequencies measured. Advantages of this method with respect to resonant ultrasound spectroscopy (RUS) is that it is a non-contact technique, the vibration spectrum is obtained in one single test, and it does not need an iterative process to obtain the frequencies with which Poisson’s ratio and the shear modulus are directly determined.





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4.1 Impact and detection at two corners In a first experiment one percussion is applied near to a corner and the consequent vibration is detected at the diagonally opposite corner. The five lowest frequencies are f1 = 28 625 Hz, f2 = 38 450 Hz, f3 = 39 125 Hz, f4 = 44 475 Hz, and f5 = 45 450 Hz. If all the lowest frequencies have been detected, the lowest from among them must correspond to a torsion, i.e., f1 = ft. The quotients of the first five lowest frequencies obtained experimentally are ⍀2 / ⍀1 = 1.3430, ⍀3 / ⍀1 = 1.3668, ⍀4 / ⍀1 = 1.5537, ⍀5 / ⍀1 = 1.5878, ⍀3 / ⍀2 = 1.0176, ⍀4 / ⍀2 = 1.1567, ⍀5 / ⍀2 = 1.1821, ⍀4 / ⍀3 = 1.1367, ⍀5 / ⍀3 = 1.1617, and ⍀5 / ⍀4 = 1.0219. These quotients of frequencies can be compared with those of Table 2, be substituted in Eq. (3), and be marked in Fig. 2. The quotient ⍀OD1 / ⍀EV1 = 1.5557± 0.0006, obtained previously by numerical methods, does not appear in the series of experimental quotients. The nearest is ⍀4 / ⍀1 whose difference with respect to the theoretical value is 0.13%, which can be due to small errors in the measurement and geometry of the sample. However, the quotient ⍀OD1 / ⍀OX1 = 1.1397± 0.0073 appears among the experimental quotients 共⍀4 / ⍀3兲. Therefore the fourth frequency of the spectrum corresponds to the mode OD1. Let us see that the methodology to find v is still correct and can be applied to the sample studied in spite of the small errors in the spectrum of frequencies. Since it has been deduced that fOD1 is the fourth frequency, the material under study has Poisson’s ratio v ⱖ ␯4cs = 0.2511. This leads to the formation of an interval of existence for v, which in turn enables the determination of the value of v for a certain ratio fi / fj with Table 2, Fig. 2, or Eq. (3). The quotient of greatest slope in Fig. 2 is f5 / f1. For the experiment carried out ⍀5 / ⍀1 = 1.5878; this quotient should be in the column of ⍀5 / ⍀1 in Table 2 in the interval from lower value 共⍀5 / ⍀1兲l = 1.5558 and the higher value 共⍀5 / ⍀1兲h = 1.6012. By a linear interpolation, ␯ = 0.2705 is obtained. Another independent consequence is deduced: A simultaneous interpolation in Tables 1 and 2 gives directly ⍀k from the double linear interpolation formulas ⍀k − ⍀kl ⍀i/⍀j − 共⍀i/⍀j兲l = , ⍀kh − ⍀kl 共⍀i/⍀j兲h − 共⍀i/⍀j兲l

共4兲

and then G may be calculated from ⍀k and fk. In this way ⍀1 or ⍀5 is found. For example, the table gives ⍀1 = 1.428 025. Application of Eq. (1) with the pair ⍀1 and f1 gives G = 77.81 GPa, whose difference with the value calculated from the velocity measurements is 0.06%. Note that either of the elastic constants can be calculated independently of the other. In the interval of existence of v, the most suitable quotients due to their greatest slope are, in order of better to worse, f5 / f1, f5 / f4, f5 / f3, and f5 / f2.The application of Eq. (3) gives the respective values of Poisson’s ratio 0.2863, 0.2884, 0.2896, and 0.2899. This set has an average value v = 0.2870, whose difference with that calculated from the P and S wave velocities is 0.10%. As in the zone of existence of v, Fig. 1, the arrangement of the modes in order of increasing frequencies is EV1, EX1 , . . ., the correspondence of fi with ⍀i is known, and Eq. (1) may be applied. The resulting value of G for frequency f4 (mode OD1) is 77.62 GPa. 4.2 Perpendicular impact and detection at the centers of opposite faces For this kind of excitation and detection and by symmetry considerations, it is deduced that the detectable modes must be in the OD or EX mode groups. The detected frequencies are f1 = 44 475 Hz, f2 = 45 450 Hz, . . .. Quotient f2 / f1 = 1.0219 may correspond to the quotient ⍀5 / ⍀4 according to Fig. 2 and Table 2. Therefore the first three lowest frequencies have not been detected. Application of Eq. (3) to this pair of detected frequencies gives v = 0.2884 whose difference with the value obtained from the velocity measurements is 0.38%.





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If the pair f1, ⍀4 is used to calculate G, then application of Eq. (1) gives directly G = 77.62 MPa, which leads to a difference of 0.18% with respect to the value calculated from the velocity measurements. Observe that either of the elastic constants can be calculated independently of the other. The appearance in the spectrum of the Mindlin frequencies, whose ⍀ is independent of ␯, and of the second frequency of bending 共EX2兲 has facilitated an accurate and rapid calculation of the two elastic constants, as was foreseen. 4.3 Systematic uncertainty The systematic uncertainty U of the measurement is estimated. Let us suppose resolution to be the only source of uncertainty. The frequencies are measured directly in the experiments carried out; its systematic uncertainty is Uf = 25 Hz. For representative values in this paper: v = 0.3, f = 40 kHz, and the relative uncertainty of Poisson’s ratio for the quotient ⍀EX2 / ⍀OD1 is U␯ / ␯ = 兩⳵␯ / ⳵共⍀i / ⍀j兲兩U⍀i/⍀j / ␯ ⬇ 4%. For the quotient ⍀EX2 / ⍀OD1, Uv / ␯ = 0.1%. Note the importance of an adequate quotient fi / fj to accurately calculate ␯. From Eq. (1) the relative systematic uncertainty of the shear modulus gives UG / G = 2Uf / f + Um / m + UL / L + 2U⍀ / ⍀ = 12.5⫻ 10−4 + 3.48⫻ 10−4 + 2.00⫻ 10−4 + 0.01⫻ 10−4 = 0.2%, where Um = 10−4 kg, UL = 10−5 m, U⍀ = 10−6, and ⍀ = 2 have been taken. The most important source of error of the shear modulus is the uncertainty of frequency.

References and links 1

F. J. Nieves, F. Gascón, and A. Bayón, “Estimation of the elastic constants of a cylinder with a length equal to its diameter,” J. Acoust. Soc. Am. 104, 176–180 (1998). 2 P. Heyliger, P. Ugander, and H. Ledbetter, “Anisotropic elastic constants: Measurement by impact resonance,” J. Mater. Civ. Eng. 13, 356–362 (2001). 3 R. Holland, “Resonant properties of rectangular piezoelectric ceramic parallelepipeds,” J. Acoust. Soc. Am. 43, 988–997 (1968). 4 F. J. Nieves, A. Bayón, and F. Gascón, “Optimization of the Ritz method to calculate axisymmetric natural vibration frequencies of cylinders,” J. Sound Vib. 311, 588–596 (2008). 5 T. Lee, R. S. Lakes, and A. Lal, “Resonant ultrasound spectroscopy measurement of mechanical damping: Comparison with broadband viscoelastic spectroscopy,” Rev. Sci. Instrum. 71, 2855–2861 (2000). 6 H. H. Demarest, “Cube-resonance method to determines the elastic constants of solids,” J. Acoust. Soc. Am. 49, 768–775 (1971). 7 I. Ohno, “Free vibration of a rectangular parallelepiped crystal and its application to determination of elastic constants of orthorhombic crystals,” J. Phys. Earth 24, 355–379 (1976). 8 P. R. Heyliger, H. Ledbetter, S. Kin, and I. Reimanis, “Elastic constants of layers in isotropic laminates,” J. Acoust. Soc. Am. 114, 2618–2625 (2003).


