Francisco J. Nieves

Estimation of dynamic elastic constants from the amplitude and velocity of Rayleigh waves

In this paper a method is proposed to characterize the elasticity of isotropic linear materials from the generation and detection of an acoustic surface wave. For the calculation of the elastic constants, it is sufficient that only one of the faces of the sample be accessible. The methodology is based on both the measurement of the Rayleigh wave velocity and on the determination of the normal to longitudinal amplitude ratio calculated from the normal and longitudinal components of the displacement of a point. The detection of two consecutive surface wave pulses using a single experimental setup permits the determination of the elastic constants. The method is applied to calculate Youngs modulus and Poissons ratio of an aluminum sample as well as their systematic uncertainties. The results obtained give a relative uncertainty for Youngs modulus on the order of the sixth part of that calculated for Poissons ratio.

Estimation of the elastic constants of a cylinder with a length equal to its diameter

A simple and noncomputational intensive method is proposed for elastically characterizing an isotropic material. The Poisson’s ratio and the shear modulus are determined from the axisymmetric vibrations of a cylinder with a length equal to its diameter. These vibrations are excited by means of a wide spectrum impact. The optical system used allows the simultaneous detection of several vibration modes. The out-of-plane displacement is detected by speckle interferometry. From the resulting vibration displacement spectrum, the two lowest frequencies are obtained, both corresponding to axisymmetric modes—specifically, the first symmetric mode and the first antisymmetric mode. The values of both dynamic elastic constants are obtained by comparing previously computed nondimensional natural frequencies and the measured frequencies. The method requires only one experiment and needs no electronic computation. The results obtained for an aluminum cylinder are coherent and confirm the appropriateness of the method. ...

Measurement of the dynamic elastic constants of short isotropic cylinders

Abstract A method based on a single test is proposed to characterize the elasticity of an isotropic homogeneous material in the shape of a cylinder of any slenderness (length–diameter) ratio. Firstly, the Rayleigh–Ritz method is used to determine the natural frequencies of the cylinders vibrating axisymmetrically. The study is focused on cylindrical samples with diameter and length of similar magnitude so that the shear modulus and the Poisson ratio can be calculated simultaneously. Subsequently, the theoretical results for cylinders of slenderness ratio between 0.1 and 3 are analyzed in order to obtain the data required to determine the elastic constants from one of the two lowest measured natural frequencies and their quotient. The analysis of the results demonstrates that any slenderness ratio is useful in the calculation of the elastic constants, although in some cases the third natural frequency should be used. Furthermore, the influence of the length–diameter quotient on the sensitivity of the method is analyzed by evaluating the systematic uncertainties for both dynamic elastic constants. Finally, the method is experimentally tested by characterizing two steel cylinders with slenderness ratios 0.1 and 1, respectively. The results demonstrate that uncertainties for both Poisson ratio and the shear modulus are smaller when the slenderness ratio is 1.

Precise and direct determination of the elastic constants of a cylinder with a length equal to its diameter

A cylinder made of a homogeneous isotropic material and with a length equal to its diameter is excited by axial percussion. Its axial displacement is detected by speckle interferometry. The two lowest frequencies, the first symmetric mode and first antisymmetric mode, are compared with the corresponding nondimensional natural frequencies, calculated to six significant figures using the Ritz method for 51 values of Poisson’s ratio. The values of both dynamic elastic constants (shear modulus and Poisson’s ratio) are found based on the quotient of the measured frequencies by using a table that is given. The method requires only one experiment and a simple calculator. The origin of the uncertainties is analyzed to improve the method’s precision. The systematic uncertainty of the results shows relative values of 0.23% for shear modulus and 0.59% for Poisson’s ratio. The values of the elastic constants calculated for a stainless steel test piece are compared with the values obtained by other methods.

On the natural frequencies of short cylinders and the universal point. Direct determination of the shear modulus

This work presents a study of the relation between the lowest nondimensional natural frequencies of a short, free cylinder vibrating in axisymmetric modes, its slenderness ratio, and its Poisson’s ratio. Ritz’s method applied to the study of the symmetric vibration of cylinders confirms that all curves, which show the dependence of frequency versus slenderness, pass through a point, called universal, independently of Poisson’s ratio. The lowest universal frequency is 2.6036 and corresponds to a cylinder whose quotient of its length and its diameter is 0.853 22. The rules leading to the identification of the first symmetric mode are inferred from the numerical results. A cylinder with universal slenderness ratio is set into free vibration by applying an axial impact. The lowest axisymmetric natural frequencies are obtained from measurement of the axial displacement by speckle interferometry. A simple arithmetical operation permits calculation of the shear modulus from the value of the first symmetric frequency, the diameter of the cylinder, and its density. The quotient of frequencies of two similar cylinders is studied as a function of their elastic properties and diameters.

Magnetic field homogeneity of a conical coaxial coil pair

An analytical study of the magnetic field created by a double-conical conducting sheet is presented. The analysis is based on the expansion of the magnetic field in terms of Legendre polynomials. It is demonstrated analytically that the angle of the conical surface that produces a nearly homogeneous magnetic field coincides with that of a pair of loops that fulfills the Helmholtz condition. From the results obtained, we propose an electric circuit formed by pairs of isolated conducting loops tightly wound around a pair of conical surfaces, calculating numerically the magnetic field produced by this system and its heterogeneity. An experimental setup of the proposed circuit was constructed and its magnetic field was measured. The results were compared with those obtained by numerical calculation, finding a good agreement. The numerical results demonstrate a significant improvement in homogeneity in the field of the proposed pair of conical coils compared with that achieved with a simple pair of Helmholtz loops or with a double solenoid. Moreover, a new design of a double pair of conical coils based on Braunbeks four loops is also proposed to achieve greater homogeneity. Regarding homogeneity, the rating of the analyzed configurations from best to worst is as follows: (1) double pair of conical coils, (2) pair of conical coils, (3) Braunbeks four loops, (4) Helmholtz pair, and (5) solenoid pair.

Straightforward estimation of the elastic constants of an isotropic cube excited by a single percussion

Ritzs 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 Poissons 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, Poissons ratio and the shear modulus are calculated by means of elementary arithmetical operations.

Direct measurement of dynamic elastic constants of a disc

In two preceding papers [F. J. Nieves, F. Gascon, and A. Bayon, J. Acoust. Soc. Am. 104, 176–180 (1998); J. Acoust. Soc. Am. (to be published)], the possibility of measuring the two dynamic elastic constants by excitation of axisymmetric vibration of an isotropic cylinder, whose length L equals its diameter D, and by means of the detection of the two lowest natural frequencies, one symmetric and another antisymmetric, has been demonstrated. The variational method and the Ritz procedure with polynomies are followed to calculate numerically the two lowest nondimensional frequencies for a disc with L/D=0.1 and the result is antisymmetric. The quotient between both nondimensional frequencies is a function only of the Poisson ratio and it is single‐valuated. Then the measurement of these frequencies allows the calculation of that elastic constant. The quotient dependence on the Poisson ratio is strong and almost linear, which implies a much better sensibility than it was for L/D=1. Moreover, one of the calcula...