Measurement of dynamic young s modulus by ultrasonic resonance with cylindrical rods and finite element modelling analysis

Transcription

1 PROCEEDINGS of the 22 nd International Congress on Acoustics Ultrasound: Paper ICA Measurement of dynamic young s modulus by ultrasonic resonance with cylindrical rods and finite element modelling analysis Martín Iofrida (a), JuanCarricondo (b), Augusto Bonelli Toro (c), Martín Gómez (d), Guido Ferrari (e) (a) Universidad Nacional de Tres de Febrero (UNTREF) Comisión Nacional de Energía Atómica (CNEA), Argentina, miofrida@gmail.com (b) Universidad Nacional de Tres de Febrero (UNTREF) Comisión Nacional de Energía Atómica (CNEA), Argentina, carricondojuan@gmail.com (b) (c) Universidad Nacional de Tres de Febrero (UNTREF) (d) Universidad Nacional de Tres de Febrero (UNTREF) Argentina, mgomez@cnea.gov.ar (e) Universidad Tecnológica Nacional Regional Delta (UTN FRD), Argentina, ferrari@cnea.gov.ar Abstract This work provides first measurements of a method to obtain the dynamic elastic modulus from the fundamental frequency of the longitudinal mode of a cylindrical rod, using a sonotrode (piezoelectric transducer with mechanical amplifier) as the signal source. The transducer can generate longitudinal mechanical waves from an electrical signal, in particular a sine sweep. To perform this measurement, the transducer is characterized with laboratory tests and simulation by finite element modelling software with COMSOL.This paper seeks to link this technique with the metallurgical industry and geology, collecting literature that justifies the need on developing faster, portable and inexpensive techniques to achieve measuring the dynamic Young ś modulus of materials with reasonable accuracy. Keywords: Elastic Dynamic Modulus, Ultrasound, Materials Sciences

2 Measurement of dynamic young s modulus by ultrasonic resonance with cylindrical rods and finite element modelling analysis 1 Introduction In this work, first measurements of dynamic Young s modulus are presented, asociated to an electromechanical transducer frequentrly used for ultrasonic fatigue testing. The interest in developing a method for obtaining the dynamic modulus is given by several factors. In the first instance, many materials are subject to cyclic stress, it is necessary to evaluate the elastic properties of the same under these conditions. The dynamic elastic modulus represents the aforementioned Young's Modulus (E) under dynamic conditions at high frequencies [1]. Furthermore, there is a wide variation between the static and dynamic values in nonmetallic materials. Mashinsky studied the physical causes of the difference between the elastic modulus and dynamic modulus rocks. The differences observed between both modules are observed at different frequencies and levels of applied strain. Various experiments show that the dynamic modulus is larger than the static, four to eight times. Both modules are controlled by visco elastic behavior and plastic micro rocks under stress and strain with less than the critical values. The differences observed between the static and dynamic modulus are caused by different contributions of inelastic mechanisms under quasi-static and dynamic function of the frequency and amplitude loads. To compare properly both modules must be evaluated under the same conditions of deformation and energy. Geological applications in situ would be useful [2].Obtaining rapidly (minutes) accurate values of Elastic Dynamic Modulus it is very useful to enhace accurracy in computational modeling. For ultrasound fatigue testing and other techniques where the resonance of a mechanical system is critical, the geometry of samples must be controlled finely [3]. The most common piezoelectric transducer is the Langevin transducer (Fig. 1 (left)), also known as sandwich transducer, consisting of a series of piezoelectric elements which are stacked between the electrodes and tensioned by a front and a rear mass. When signal is injected to the piezos in the transducer (constrained by front mass and rear mass), it generates an acoustic wave preferently in logitudinal front and rear directions. However, the rear mass is designed in a material with higher acoustic impedance (Za) than the front mass, transmitting the acoustic energy efficiently to the load [4].When a piezoelectric element is subjected to a sinusoidal electrical signal, this generates a vibration of very low amplitude. Often this vibration is not enough to be used in ultrasound applications. This can be solved taking advantage of the structural geometry of the parts to amplify this vibration.generally ultrasound transducers are designed taking advantage of three modes of fundamental vibration. Longitudinal (L) mode, torsional mode (T) and flexural mode (F) as shown in Fig. 1 (right) [5]. 2

3 Figure 1: Langevin Transducer (left); vibration modes (right). Spinner et al. conducted a comparison between experimental and theoretical relationships between Young's modulus and the fundamental frequency in the longitudinal mode in solid bars. In this paper empirical results that were used for the preparation of the above standards [6] they are.popovics performed a study based on vibration and wave propagation to improve methods for calculating the dynamic elastic modulus in building materials such as concrete and others. This and many other works can be found in the area of concrete, asphalt, etc. [7] 1.1 ASTM E : Standard Test Method for Dynamic Young's Modulus, Shear Modulus and Poisson's Ratio by Impulse Excitation of Vibration. This standard allows measuring the resonance frequency of specimens as specified geometry using a single elastic impulse momentum generated with a suitable tool. A transducer (eg. Accelerometer, microphone, etc.) senses the vibrations generated by the drive. The positions of source and receiver are positioned so as to receive the specific modes of vibration of interest [8]. 1.2 ASTM E : Standard Test Method for Dynamic Young's Modulus, Shear Modulus and Poisson's Ratio by Sonic Resonance This standard covers the determination of dynamic elastic properties of elastic materials. Specimens such materials have a characteristic resonance frequency determined by the elastic modulus, mass and geometry thereof. The dynamic elastic properties of such materials can be calculated if the geometry, mass and mechanical resonant frequency of a suitable specimen can be measured. In this standard the resonance frequency of bending modes [9] is used. The method is applicable specifically to elastic, homogeneous and isotropic materials. The method is nondestructive, voltages are applied to the span of minutes, minimizing the possibility of fracture, charging cycles of the order of 100 us allowing measuring at high temperatures also apply. To calculate the dynamic elastic modulus of a cylindrical rod, in the longitudinal basic mode resonance, the specific rule: =

4 K = Where,. = /!' =!"# $!%% & = '$## '(%# " $ "#!' '$! )* = '$! #+h = '$!!' = &$$# $!%% = + K = $'' $# $' $' $#+%!# '$## '(%# " COMSOL Multiphysics is a multipurpose platform for modeling and simulation based on physical problems through advanced numerical methods, such as finite element analysis (FEM), Finite Volume Method (FVM), Boundary Element Method (BEM), and Particle Tracing Method (PTM) which, COMSOL Multiphysics emphasizes using FEM [10]. 2 Experimental 2.1 Materials The material to analyze was a nickel-based alloy with high chromium content, of special design, with characteristics close to those of Inconel 690 (Nickel 58%, Chrome 27%), to evaluate fatigue for an industrial application. As mentioned above, in the ultrasonic fatigue testing samples with controlled geometries are necessary to resonate at a given frequency. The resonance is conditioned by the dynamic elastic modulus. In addition, these alloys are formed by crystal structures, their atoms are arranged. Therefore it is simpler assaying a lower degree of uncertainty in compounds or viscoelastic materials, where variations between Young's modulus and dynamic elastic modulus differ in higher proportions. From an essay by X-ray fluorescence (XRF) the percentage sample concentrations were obtained. Base resulting 52% nickel (Ni) and the main alloying 35% Chromium (Cr). Two cylindrical samples (Long and Short) were constructed from rectangular pieces of materials shown in Fig. 2. 4

5 Figure 2: Material pieces (Left); sample (right). To make geometric measures Vernier caliper was used with an error of 0.02 mm. Two cylinders constitute each sample,the body and a thread to connect the sample to the transducer. The volume of each piece and the whole sample was calculated with ec. 3 and error propagation was calculated. Dimensions obtained are shown in Table 1. V =! h 4 3 Table 1: Dimensions of samples. Dimension Long Sample Short Sample Short Cyl. Cylinder Thread Lenght (mm) Radius (mm) Volume (mm 3 ) Volume Error (mm 3 ) (0,3372%) (0,3474%) Subsequently, to characterize the material the specimens were weighed on a Mettler AE 240 in 4-digit precision mode 200 g, ie g error (0.1mg <4,16e-4%). Each specimen was weighed 3 times and the average was calculated. Density on material for each sample was calculated with eq. 4 with mass (m) and volume (V) and associated error. These values are given in Table 2. ρ = 4 4 Table 2: Properties of samples and material. Property Long Sample Short Sample Weight (g) Density (g/cm 3 ) Density Error (g/cm 3 ) 2.28e e-2 5

6 nd 22 International Congress on Acoustics, ICA 2016 st Acoustics for the 21 Century 2.2 Measurement Measurements equipment and wiring schematics is represented in Fig. 3. The signal source was an audio generator with fine frequency control which allowed a very accurate scanning (error less than 1%) across the spectrum from 1 khz to 100 khz. The generator voltage output was connected into a power amplifier and adjusted to 30V for the whole bandwidth. The sonotrode and a resistor 33ohm V - 1% were connected to the power output. Voltage amplitudes were measured in resonance on the resistor and the transducer. With an oscilloscope both signals were visualized respectively, relieving the frequency, phase shift and shape. The error in the measurement frequency is given by the fine sweep, which as mentioned is less than 1% and the oscilloscope whose measurement error is less than 0.002% to less than 100 khz. Phase shifts of voltage in the resistor and the sonotrode out of resonance and in resonance are shown in Fig. 4. Three measurements were performed, the sonotrode unloaded, and the sonotrode with the long and short sample connected. Figure 3: Measurement Schematics. Figure 4: OSC screens during measurements. Out of resonance (left); on resonance (right). 2.3 Computational Modeling Using the software COMSOL simulations for different Multiphysics isolated specimens and the test pieces were performed connected to the transducer. Then make a comparison with experimental results. The materials constituting each part were faithfully respected and all data obtained from the measurements were used to model the samples. A model of the sonotrode with the samples was obtained (Fig. 5). 6

7 Figure5:Sonotrode and sample model. 3 Results and Discussion In Table 3, simulated and measured resonance frequencies in samples and full system are compared. The frequency error between simulation and measurement for the test tubes, which are the values used to calculate the dynamic elastic modulus is very small, 1.23% for the specimen base and 0.16% for the short specimen. These are promising results, however it is important to study other alloys. Table 3: Simulated and measured frequencies for specific pieces. Comparison Piece Simulated Frequency (Hz) Measured Frequency (Hz) Diference (Hz) Error (%) Sample ,85 Short Sample ,53 Transducer ,42 Sonotrode ,13 Sonotrode + Sample ,26 Sonotrode + Short Sample ,40 To calculate the elastic dynamic modulus, the measured resonance frequencies of both samples were used. From the International Standard "ASTM Standard the dynamic elastic modulus of the material and it s associated error was calculated. Poisson s modulus () of nickel based alloy with high chromium content standard it is assumed ( = 0.305). Error propagation in this case is expressed in percentage using the values provided by the standards due to the exponents used in the equations, and assuming maximum error values for each of 7

8 the variables 0.01%, error value is obtained of 0.632%.The average of the values obtained can be expressed as the dynamic elastic modulus: GPa (error <0.632%). Table4: Measured Elastic Dynamic Moduli for both Samples. Dimensions Long Sample Short Sample Ed [GPa] 220, ,497 4 Conclusions Nickel-based alloy with high chromium content used for specific industrial use was selected as sample material. Cylindrical specimens compatible with the measurement system (sonotrode) were manufactured. These specimens were characterized in dimensions, weight and density, and the associated error in each case. Measurements were performed on the sonotrode, characterizing it s spectrum unloaded and loaded with two samples of different length. Modeling was performed by finite elements, evaluating the geometry and materials of the parts constituting the sonotrode and the specimens. Modeling generated a simulated spectrum between khz with the fundamental frequencies of longitudinal resonance of all parts. Comparison of spectra allowed to determine which of all the resonance frequencies measured corresponded to the samples. From ASTM International Standard, the dynamic elastic modulus was calculated, with the measured resonance frequencies and material characteristics. Twodynamic elastic modulus values were obtained: GPa and GPa; corresponding to measurements with the measuring cylinder with the cylindrical base and cut specimen respectively. Considering the average of measurements, a value of GPa and an error of 0.628% associated it is obtained. This value is within the range used for different nickel base alloys similar to those of the measured specimen. This paper presents promising results to begin the evaluation of this method as a fast, inexpensive and portable option for measuring the dynamic elastic modulus. Acknowledgments Comisión Nacional de Energía Atómica (CNEA).Departamento Química analítica, Departamento Materiales Grupo Difusión, Departamento de Proyectos Especiales Ciencias de la Tierra. Universidad Tecnológica Nacional - Facultad Regional Delta (UTN-FRD) Laboratorio de Emisión Acústica. Universidad Nacional de Tres de Febrero (UNTREF). 8

9 References [1] Callister, W.D., Introducción a la Ciencia e Ingeniería de los Materiales, 3ra edición, Editorial Revertré, Barcelona, (2007). [2] Mashinsky, E.I., Differences Between Static and Dynamic Elastic Moduli of Rocks: Physical Causes, Russian Gwology and Geophysics, 44, 9, , (1993). [3] Carricondo J., Ensayo de fatiga ultrasónica y modelado por elementos finitos de probetas de acero con alto contenido de carbón, UNTREF, (2015). [4] A. Cardoni, Characterising the Dynamic Response of UltrasonicCuttingDevices, PhD Thesis, University of Glasgow, UK, [5] Abramov O., High Intensity Ultrasonics: Theory and Industial Applications, Gordon and Breach Science Publishers (1998). [6] Spinner A., A comparison of Experimental and Theoretical Relations Between Young s Modulus and the flexural and longitudinal resonance frequencies of uniform bars, Journal of research on NIST,64A, 2, [7] Popovics, J. S., Study of static and Dynamic Modulus of Elasticity of concrete, ACI-CRC Report, Illinois (2008). [8] ASTM Standard Test Method for Dynamic Young s Modulus, Shear Modulus and Poisson s Ratio by Impulse Excitation of Vibration. [9] ASTM Standard Test Method for Dynamic Young s Modulus, Shear Modulus and Poisson s Ratio by Sonic Resonance. [10] Comsol Multiphysics, Introduction, Extracted