# On the Measurement of Particle Contact Curvature and Young's Modulus Using X-ray \(\mu\)CT
## 2 Young's Modulus Measurement  
Before proceeding, the methods to determine [PERSON]'s modulus of a single particle are first briefed in the literature as a prelude to interpret the interlink between contact curvature and [PERSON]'s modulus. The first method is using single particle compression test to determine the elastic modulus. The force-displacement curve of a particle subject to uniaxial compression is employed to observe the elastic and elastic-plastic deformation until rupture. A typical deformation curve is shown in Figure 1. At near-zero stress and strain, the stress-strain curve is linear and theoretically follows the Hertz contact model until the yield point, which tops the boundary of elastic deformation. After that,the slope of the force-displacement curve decreases with the embarkation on the elastic-plastic deformation. The elastic modulus of a particle in a diametric compression can be determined following the equation below:  
\[F=\frac{1}{3}E^{*}\sqrt{d}\Delta^{3/2} \tag{1}\]  
where \(F\) is the applied force; \(d\) and \(\Delta\) are the particle diameter and total displacement, respectively; \(E^{*}\) is the effective modulus of elastic modulus of the particle and it further gives:  
\[E^{*}=\frac{E}{1-v^{2}} \tag{2}\]  
where \(E\) and \(v\) are the Young's modulus and Poisson's ratio of the particle, respectively. Examples of determining Young's modulus by this method can be extensively found in [1; 19]. Alternatively, nanoindentation where the diamond tip is placed against the particle surface by generating minute indentation depth can be adopted to determine the elastic modulus of a particle. In a complete cycle of indentation test shown in Figure 2, the force-displacement is composed of loading and unloading stages. The data from the unloading stage are used for analysis as only the elastic displacement is recovered. As a result, this method is not applicable to particles where plasticity reverses during unloading. However, studies from finite element simulations indicate that reverse plastic deformation is negligible over the unloading stage [16]. Note that the most commonly used tip is the Berkovich indenter having a three-sided pyramid with a half-angle of 65.3\({}^{\circ}\). The superiority of Berkovich indenter over other indenters such as the four-sided Vickers indenter is the easier generation of a sharp tip, resulting in more attainable results at lower loading conditions [20]. The equation to compute the elastic modulus by nanoindentation can be expressed:  
\[E_{r}=\frac{\sqrt{\pi}}{2}\frac{S}{\sqrt{A}} \tag{3}\]  
Figure 1: A typical force-displacement curve to determine particle elastic modulus. where \(E_{r}\) is the reduced Young's modulus; \(S\) and \(A\) denote the unloading stiffness and the projected area of indentation, respectively. However, it has been noted that the indentation method is not suitable for particles that are not sufficiently flat. The unsuitability of this method results from the slipping and bending of indenter when the particle is not reasonably flat in the contact region. As a result, a vertical compression test of the irregularly shaped particle between two rigid platens is proposed with the aid from the 3D laser scanning technique. This improved method to measure the Young's modulus was developed by capturing the three-dimensional surface geometry of individual granule. A pair of orthogonal planes intersecting at the contact point is shown in Figure 3, where the curvature of the mutually perpendicular planes can be computed from the scanned data. The measured Young's modulus of a corn granule based on the proposed method was found to differ significantly from the indentation test because of the imperfect roundness. Figure 3: Digital profile of a corn granule with orthogonal XZ and YZ planes (modified from [18]). Figure 2: A typical loading curve from nanoindentation to measure Young’s modulus. (Modified from [14,20]).