# Measurement of Elastic Properties of Brittle Materials by Ultrasonic and Indentation Methods
## 2 Experiments and Methods  
Two 40 \(\times\) 40 \(\times\) 1.1 mm ITO glass (sample No. 1 and 2), one 10 \(\times\) 10 \(\times\) 2 mm Zr-Cu-Ni-Al ((Zr53 Cu30 Ni9 Al8)SiO.5) BMG (sample No. 3) [26] and one \(\O\) 12 \(\times\) 3.5 mm CM247 LC superalloy sample [21] (sample No. 4) were prepared for the measurement. The CM247 LC superalloy sample was cut from raw material ingot and polished down by 1.0 m aluminum-oxide powders to avoid any surface oxidation. The density measurement was performed under ASTM D792 (Standard Test Methods for Density and Specific Gravity of Plastics by Displacement) with relative controlled under 0.05%.  
The ultrasonic measurements are performed under ASTM standard D2845-08 [27]. The setup is as shown in Figure 1 with Parametrics 5073 PR pulse-receiver, GaGe 100 MHz sampling rate 12-bit A/D converter, Olympus M208-RM 0.125-inch 20 MHz longitudinal, and V157-RM 0.125-inch 5 MHz shear transducer. The errors of the A/D conversion are less than 10\({}^{-7}\) s in time and 0.2 mV in amplitude. The density is measured by the Archimedes principle from samples of proper sample size as listed in Table 1. The Young's modulus E, shear modulus G and Poisson ratio \(v\) are calculated as follows [6; 7]:  
\[\mathrm{E}=\frac{V_{L}^{2}p(1+v)(1-2v)}{1-v} \tag{1}\]  
\[\mathrm{G}=V_{\mathrm{T}}{}^{2}\rho \tag{2}\]  
\[v=\frac{1-2(V_{\mathrm{T}}/V_{\mathrm{L}})^{2}}{2-2(V_{\mathrm{T}}/V_{ \mathrm{L}})^{2}} \tag{3}\]where \(V_{T}\) is the transverse acoustic speed, \(V_{L}\) is the longitudinal acoustic speed which equals two times thickness divided by round trip transit time, \(\rho\) is the density. Attenuation coefficient \(\alpha\), is calculated as follows [5; 7; 8; 28]:  
\[\alpha=\frac{20\log(A_{0}/A_{1})}{2t} \tag{4}\]  
where \(A_{0}\) and \(A_{1}\) are the peak amplitude of the first and second echo signal respectively.  
The nanoindentation measurement is performed by MTS Nano Indenter XP system capable of Continual Stiffness Measurement (CSM) with software TestWorks as shown in Figure 2. The hardness \(H\) is defined as the maximum loading \(P_{max}\) divided by the normal contact projection area \(A(h_{c})\) i.e., \(H\) = \(P_{max}\)/\(A(h_{c})\), where \(h_{c}\) is the depth (displacement) of contact [10]. The relationship between initial unloading stiffness \(S\) and reduced Young's modulus \(E_{r}\) can be obtained by  
\[S=2\sqrt{\frac{A(h_{c})}{\pi}}E_{r} \tag{5}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Sample Number**} & \multirow{2}{*}{**Material**} & \multicolumn{6}{c}{**Sample Size and Density**} \\ \cline{3-8}  & & **Length** & **Width** & **Thickness** & **Air** & **Water** & **Density** \\  & & **(mm)** & **(mm)** & **(mm)** & **Weight (g)** & **Weight (g)** & **(g/mm\({}^{3}\))** \\ \hline No. 1 & ITO-glass & 39.37 & 39.37 & 1.11 & 4.11 & 2.42 & 2.42 \\ No. 2 & ITO-glass & 39.28 & 39.31 & 1.11 & 4.09 & 2.41 & 2.43 \\ No. 3 & BMG & 9.55 & 9.95 & 2.11 & 1.47 & 1.27 & 7.28 \\ No. 4 & CM247 LC & 12.20 & 12.20 & 3.49 & 3.54 & 3.13 & 8.56 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Size and density of each sample used in the measurements.  
Figure 1: Experimental setup for ultrasonic measurement.  
Figure 2: MTS Nano Indenter XP system.  
Metallurgical microscopic photos are also taken from the BMG and CM247 LC superalloy samples to explain how the acoustic property is affected by the peculiar microstructure.