TECLAB Ding Xia shaw@teclab.cn

abstract：The Ultrasonic Resonance Spectroscopy (RUS) method obtains all elastic constants of solid material samples by analyzing the ultrasonic resonance frequency. This article provides a brief description of RUS technology and introduces its application in the research of bulk metallic glass (BMG) materials.

keyword：RUS elastic constant measurement ultrasonic resonance spectrum BMG

Resonant ultrasound spectroscopy is a technique that can be used to determine elastic constants of solids It measures the resonant spectrum of mechanical resonance for a sample of known geometry, dimensions and mass. The article gives a brief description of basic principles in RUSpec, introduces the elastic constant measurement using RUSpec technology for bulk metallic glass (BMG) materials.

【 Key word 】 Resonant ultrasound spectroscopy, RUS, Elastic constant measurement，BMG

introduce

The use of resonance in solid materials to determine their elastic modulus has always been a topic of great interest to condensed matter physicists, engineers, and materials scientists. The mechanical resonance spectrum response of solid materials depends on their shape, density, elastic modulus, and dissipation. For solid materials with low dissipation, their resonance frequency peak is very sharp (Q=, where f is the resonance frequency and Δ f is the full width at half maximum). When the Q value is large, the elastic hysteresis is small, and the elastic modulus is very clear, which means that there is a linear and unique relationship between stress and strain. However, the relationship between resonance frequency and modulus cannot be easily obtained and requires complex calculations based on finite element analysis. For objects with simple geometric shapes, such as parallelepipeds (RPs), cylinders, and spheres, the Lagrangian method can be used for analysis. With the continuous development of computing technology, the complex analysis of mechanical resonance can be performed by computers. This makes ultrasound resonance spectroscopy (RUS) technology widely applicable. For example, using the first personal computer produced by IBM, RUS analysis required nearly 2 hours of iteration, 10 seconds on the first Cary supercomputer, and only 0.8 seconds on current 2GHz laptop personal computers.

Ultrasonic resonance spectrum analysis is a technique used to measure all elastic constants of solids, which is fundamentally different from other ultrasonic techniques. At present, it has been widely used in various material research, and many foreign articles have studied and commented on the development of this technology, complex calculation technology details, and transducer design. However, there is no complete description of this technology in China. Through the introduction and experimental case analysis of this technology, this article hopes to provide a reference for domestic researchers.

Figure 1. Ultrasonic resonance spectrum

theoretical basis

According to Hooke's law, the stress-strain relationship can be expressed as, where c represents a one-dimensional elastic constant. For three-dimensional elastic solids, displacement becomes a three-dimensional vector, and strain is defined as Hooke's law expressed as:

(1)

Meanwhile, Newton's second law can be expressed as:

(2)

To determine the vibration mode, it is assumed that the sample surface satisfies the stress free boundary condition:

(3)

In the formula, n_jRepresents the unit vector of surface normals. Due to the properties of tensors, the relationship between particle displacement and wave direction is quite complex. If the time function is cos (2 π ft), based on the shape of the sample, the stress free boundary conditions, and the preset elastic constant, the natural frequency f can be obtained through equations 1-3_c.

For RUS technology, we need to measure the resonance frequency f based on it_mInvert to obtain the elastic constant C_ijkl。In most cases, the measured resonance frequency f_mThe number will be greater than the calculated natural frequency f of the sample_c. Therefore, it is necessary to match the two through calculation and analysis. Calculating f_cSet a set of "estimated" initial elastic constants, and then use the least squares method based on the Laplace iteration scheme to achieve the best match between the two:

(4)

In the formula, W_iThe representative weight factor is usually set to 1, but when the measured resonance peak is uncertain, the weight value can be increased according to the situation. N should be large enough to ensure accurate results, generally only 15 for isotropic samples, while at least 40 is required for orthorhombic symmetric crystal materials.

After achieving optimal matching, the independent elastic constants C of the sample can be obtained_ijkl。Taking isotropic materials (2 independent elastic constants, C11 and C44) as an example, there is a relationship between the elastic constants and the various moduli of the material as follows:

In the formula, E represents Young's modulus, B represents bulk modulus of elasticity, s represents Poisson's ratio, and G represents shear modulus.

test method

When conducting RUS measurements, the sample is gently placed on top of the transducer to meet stress free boundary conditions, and this method does not require a coupling agent. For general samples, RUS supports the tested sample through three piezoelectric transducers. One of the transducers is used to generate an elastic wave with a constant amplitude but frequency variation, while the other two transducers are used to measure the resonant frequency, as shown in Figure 2a; For small samples, only one send and one receive is needed, as shown in Figure 2b (cylindrical small sample) and Figure 2c (parallelepiped small sample). After frequency scanning, record a series of resonance peaks. The position of the peak can be used to determine the frequency f.

Figure 3 RUSPEC Ultrasonic Resonance Spectroscopy Material Characterization System

When measuring the modulus of a sample using RUS, the sample should be hard (with a Q value of several hundred or more, and a diamond Q value of 1.5x10)⁶）Uniform and regular in appearance. To achieve high accuracy, the appearance of the sample generally requires that the parallelism of the opposing surfaces be better than 0.05%, the axial angle perpendicular to any plane should be better than 0.05%, and the smoothness of the surface should be better than 0.01% of the shortest dimension of that plane. In the test results, the root mean square error of the matching should be less than 0.8%, generally less than 0.2%, and sometimes up to 0.03%.

In certain situations, it is necessary to extend high-temperature equipment to measure the modulus values of materials in extreme environments. During high-temperature RUS testing, the sample is placed between high-temperature buffer rods (usually made of silicon carbide) in a high-temperature furnace, and the signal is transmitted to the transducer outside the furnace through the high-temperature buffer rods. As shown in Figure 4.

Figure 4: A device and its schematic diagram for high-temperature RUS testing

Low temperature RUS measurement can be performed using a device similar to a casing. Place both the sample and the transducer in a non-metallic casing (for easy measurement in a magnetic field environment), stabilize the sample between the transducers with a slight load (about 1g), and place the entire casing in a low-temperature environment for measurement. As shown in Figure 5:

Figure 4: A device and its schematic diagram for low-temperature RUS testing

For very small and fragile samples (such as parallelepiped with a side length of about 0.03-1px), PVDF piezoelectric film can be used as a transducer to make a testing platform as shown in Figure 7 for testing:

RUS is used for BMG modulus measurement

The various modulus values of materials in the field of bulk metallic glass research (BMG) are also a hot research topic. Many researchers internationally use ultrasonic resonance spectroscopy technology to measure the modulus of materials with limited dimensions.

The research on bulk metallic glass (BMG) in this article uses pure metal blocks with a purity higher than 99.9% as the starting material for preparing alloys. The metal blocks prepared according to their composition are placed in a water-cooled Cu crucible in a vacuum arc furnace, and melted into a mother alloy ingot by arc melting in a high-purity (99.999%) Ar atmosphere with Ti gas collection. Then, the mother alloy ingot is repeatedly melted 5-6 times to ensure uniformity in composition. The mother alloy is then remelted in a water-cooled Cu pool in a small vacuum arc furnace, and the molten metal is sucked into the Cu mold cavity at the bottom of the Cu pool to obtain a round rod sample. The sample blank is cut, precision processed, polished, and polished to make its shape regular and meet the requirements. Test conditions. Table 1 shows the modulus values of 2mm diameter ternary (Zr Cu Al and Hf Ni Al) BMG materials measured using ultrasonic resonance spectroscopy analysis technology. The tested sample was placed between two flat top transducers, and the RUSPEC system was used to measure the two independent elastic constants C11 and C44 of the sample. Then, the system automatically obtained the shear modulus G, bulk modulus B, and Poisson's ratio v.

Table 1

Table 2 shows various modulus values of 2mm diameter BMG materials measured using ultrasonic resonance spectroscopy technology, including Young's modulus E, shear modulus, bulk modulus B, Poisson's ratio, and critical diameter D_cGlass transition temperature T_g.

Table 3 shows the modulus results of 3mm diameter BMG material measured using ultrasonic resonance spectroscopy technology: Young's modulus Y, shear modulus G, bulk modulus B, Poisson's ratio, mass density, critical diameter D_cGlass transition temperature T_g.

Table 3

RUS is used for heat-resistant hard glass (Pyrex glass), 7075 aluminum, 4140 steel, high-purity Al₂O₃Modulus measurement

In this study, each test sample (isotropic) was matched with the initial 40 resonance peaks (two independent elastic constants, Young's modulus E, and shear modulus G). Each sample underwent two independent measurements: with a resolution of 0.03KHz and a sweep frequency range from 0-200KHz. The matching results are evaluated using the root mean square error between the measured peak values and the calculated peak values. Generally speaking, the initial 2 to 3 frequency peaks cannot be matched well, especially for very thin samples (aspect ratio less than 0.04), where the resonance spectrum peaks shift towards the lower frequency range, and such frequencies will be excluded during the matching process.

The samples used for testing include Pyrex glass, 7075 aluminum, 4140 steel, and high-purity Al₂O₃The table below lists the materials, dimensions, shapes, and weights of all samples. The dimensional measurement accuracy is 0.001mm, and the weight measurement accuracy is 0.0001g. All aluminum samples are from the same aluminum rod (cut and polished). The same steel sample is also from the same steel rod, and the heat-resistant glass sample is cut and drilled from two glass plates (with original thicknesses of 3.28 and 6.53 mm), Al₂O₃The samples were made by the manufacturer based on the parameters in the table without any special treatment. The average roughness and parallelism of each sample were measured by corresponding instruments. The processed samples were additionally polished to obtain sharp edges without chamfering or grinding, in order to avoid significant impact on the amplitude of the elastic constant during measurement. Additionally, due to Al₂O₃The bulk density of Al varies significantly between samples when calculated based on mass and size, so each Al₂O₃The density of the samples was ultimately measured separately using the water immersion method.

Experimental results:

The meanings of each symbol in the results:

conclusion

The ultrasonic resonance spectrum analysis technology is currently the method with the highest absolute accuracy in material modulus measurement. Its most unique feature is that this technology can quickly, synchronously, and accurately measure all modulus values on small samples, and the results have high reproducibility. At present, many well-known domestic and foreign research institutions have widely used this technology for material research. Especially in the field of BMG bulk amorphous metallic glass research, more and more researchers are using RUS technology to measure the elastic modulus and independent elastic constants of limited samples of various sizes.

reference:

1. A. Migliori, J.L. Sarrao, William M. Visscher, T.M. Bell, Ming Lei, Z. Fisk ', and R.G. Leisure, "Resonant ultrasound spectroscopic techniques for measurement of the elastic module of solids," Physica B 183 (1993) 1-24
2. Julian MaynardRESONANT ULTRASOUND SPECTROSCOPYPHYSICS TODAY
3. M. Radovic, E. Lara Curzio, L. Riester《Comparison of different experimental techniques for determination of elastic properties of solids》 Materials Science and Engineering A368 (2004) 56–70
4. Albert Migliori，J. D. MaynardImplementation of a modern resonant ultrasound spectroscopy system for the measurement of the elastic module of small solid speciesREVIEW OF SCIENTIFIC INSTRUMENTS 76, 1 2005
5. Li Zhang, Yong Qiang Cheng, A-Jing Cao, Jian Xu, Evan Ma《Bulk metallic glasses with large plasticity: Composition design from the structural perspective》Acta Materialia 57 (2009) 1154-1164
6. Peng Jia, Zhen Dong Zhu, Evan Ma, and Jian Xu, "Notch toughness of Cu based bulk metallic glasses," Scripta Materialia 61 (2009) 137-140
7. Li Zhang, Ling Ling Shi, Jian Xu, "Hf Cu Ni Al bulk metallic glasses: Optimization of glass forming ability and plasticity" Journal of Non Crystalline Solids 355 (2009) 1005-1007