F. Gascón

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.

Experimental results on bulk waves and Rayleigh waves in slate

This paper describes experimental results on velocity measurements of bulk and Rayleigh waves in a highly anisotropic natural medium (slate). Bulk waves velocities are measured using a standard ultrasonic pulse transmission setup. The Rayleigh waves are excited using the wedge technique and recorded by an interferometer which detects both the in‐plane and out‐of‐plane components of the displacement they produce in a point of a sample surface. The experiments constitute part of a study of the propagation of surface seismic waves in anisotropic geological structures using scaled‐down models. The elastic behavior of a block of slate whose faces are traversed by these surface waves is studied, and it is verified that this rock shows a good fit to a physical model of a solid with an axis of symmetry of revolution.

Determination of the elastic constants of isotropic solids by optical heterodyne interferometry

An experimental method based on the application of speckle interferometry is proposed in order to determine the dynamic elastic constants of an isotropic material. A broad frequency spectrum impact is applied to a cylindrical sample and, when this vibrates freely, a heterodyne laser interferometer is used to detect the displacement of a point of the material. Once the sample is suitably excited, the displacement component of interest, either in‐plane or out‐of‐plane, is detected in order to determine its frequency content. A spectral analysis allows the determination of the longitudinal, transverse, and torsional natural frequencies. From this the elastic constants are calculated. The measurement precision is checked experimentally using the same method by means of a third detection.

Measurement of longitudinal vibrations in a slender rod by optical heterodyne interferometry

Abstract The vibration of a slender rod subject to a longitudinal harmonic perturbation without physical contact and near the first resonance frequency is experimentally analyzed. The rod is excited by a magnetic field which produces an harmonic force acting on a small ferromagnetic disc glued at the end of the rod. The detection technique applied is based on the optical heterodyne speckle interferometry, which permits both the normal-to-surface displacement and the tangential displacement to be detected sequentially. The out-of-plane technique is used to estimate the first eigenfrequency. By means of the in-plane technique the amplitude and phase for different frequencies near resonance are measured and the displacements of 17 equidistant points on the lateral surface of the rod are plotted as a function of time. The comparison between the experimental results and the prediction of the classical theory shows a good agreement, making apparent the usefulness of the speckle interferometry for the vibration analysis. In addition, Youngs modulus is calculated and Poissons ratio is estimated from the measurement of both the out-of-plane and in-plane displacements of two opposite lateral points of the rod.

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.

Elastic characterization of isotropic materials by a single test based on the experimental determination of natural frequencies using laser interferometry

An experimental method is proposed for the elastic characterization of an isotropic material based upon the vibration natural frequencies recorded by means of a single assembly and experiment. Speckle heterodyne optic interferometry is applied to detect the tangential component of the vibration at a point. The sample is a slender rod excited through a broad-spectrum percussion, where the vibration recorded is a superposition of the transverse and torsional natural modes. Spectral analysis of the vibration allows the identification of the transverse and torsional natural frequencies. This is followed by calculation of the elastic moduli.

Simulation of Rough Surfaces and Analysis of Roughness by MATLAB

The simulation of physical phenomena in science and engineering has become an important tool because it allows studying a wide range of real problems. On the other hand, it allows resolving problems that, because of its difficulty, it would be not possible to solve by analytical methods. Moreover, simulation is fast and versatile since it permits to vary parameters of the problem easily, allowing analyzing the effect of the modification of them in the response of the system examined. Simulation requires programming, for which there are many different languages. Each of them has a particular internal structure that distinguishes it from others. Therefore, depending on the problem to be study, it may be advisable to use a specific programming language. In the scientific-technical context MATLAB has been increasingly used by the great advantages that it offers. For example, the instructions are interpreted and not compiled, the user to enter commands interactively. The data processing is flexible. They can be read and stored in two different formats, ASCII and MATLAB format. ASCII has the advantage that the data and results may be used for other programs. However, MATLAB format may be faster. On the other hand, many functions and libraries of MATLAB are written MATLAB language, enabling the user access to the source files. It is possible to execute instructions of the operating system without exiting the program. Moreover, this language is portable in platforms as Windows or Apple, commonly employed by the researcher. From the point of view of numerical calculation, the use of matrices as basic elements makes it efficient and easy to employ, being also possible to perform graphics of curves and surfaces. Finally, the operations can be performed with simple and intuitive expressions similar to those used in science and engineering. MATLAB has been used for many applications in general physics, mathematics, optics, electronics, chemistry, biology, medicine, and artificial intelligence, among others. Now we want to employ MATLAB to simulate an optical procedure to measure surface roughness. Thus, the aim of this paper is the determination of the roughness of a surface from the analysis of the speckle pattern obtained in the far field, when the object is illuminated with a monochromatic beam perpendicularly to its surface.

Maximum diameter of the read-out laser beam in double-exposure speckle photography

A simple model is proposed for a speckle pattern produced by a diffuse surface and intercepted by the plane of a photographic plate. In this model, all the spots have the same form but are randomly distributed. During the translation, rotation, and strain of the diffuse surface, it is assumed that the shape of each spot remains the same. The transmittance of the processed plate that would be produced when exposed before and after the displacement of the diffuse surface, is calculated. An equation is obtained for the intensity of the diffracted light when the plate is illuminated with a circular interrogation beam. The lower limit that the read-out laser beam diameter should have so that the Young fringes are visible is calculated and a criterion is proposed. The theory is checked with laboratory experiments.