## 2 Collisions and the Coefficient of Restitution (\(CoR\), \(e\))  
### Theoretical Background  
Cereal-grain impacts--whether between two kernels or between a single kernel and a flat surface--are short-duration events in which contact forces rise and fall in microseconds, transferring momentum and altering velocity. The process is continuous but can be divided into compression and restitution stages [8,9].  
During compression, immediately after first contact (Figure 2a), equal and opposite forces resist the relative motion [10,11]. For a normal impact of a grain of mass \(m\) against a rigid plate, the analysis simplifies because the plate's mass is effectively infinite (\(M\)\(\rightarrow\)\(\infty\)), and its velocity remains zero [9]. After the instant of maximum compression, restitution begins (Figure 2b), and the contact force falls to zero as the grain returns--fully or partially--to its original shape [12]. Elastic collisions show almost immediate recovery, whereas elastic-plastic ones leave permanent deformation or delayed rebound [9,13, 14, 15, 16].  
Figure 1: Wheat grain [7].  
The impact obeys momentum conservation and is characterised by the coefficient of restitution (CoR, \(e\)). With the convention that \(u\) denotes pre-impact velocity and \(v\) post-impact velocity, conservation of linear momentum for two bodies is as follows:  
\[m_{1}\overset{\rightarrow}{u_{1}}+m_{2}\overset{\rightarrow}{u_{2}}=\ m_{1} \overset{\rightarrow}{v_{1}}+\ m_{2}\overset{\rightarrow}{v_{2}}, \tag{1}\]  
where  
* \(m_{1}\), \(m_{2}\)--masses of the bodies;
* \(\overset{\rightarrow}{u_{1}}\), \(\overset{\rightarrow}{u_{2}}\)--initial velocity vectors;
* \(\overset{\rightarrow}{v_{1}}\), \(\overset{\rightarrow}{v_{2}}\)--final velocity vectors.  
The coefficient of restitution \((CoR,\ e)\) quantifies elasticity as the ratio of the relative normal separation (final) speed to the relative normal approach (initial) speed. For a grain impacting a flat plate,  
\[CoR=e=\frac{|v_{n}|}{|u_{n}|}. \tag{2}\]  
where \(u_{n}\) and \(v_{n}\) are the grain's velocity components normal to the surface before and after impact, respectively.  
Collision regimes can be classified by the value of the coefficient of restitution \((CoR,\ e)\):  
* \(e=1\)--perfectly elastic; no kinetic energy is lost;
* \(0<e<1\)--partially elastic; some energy is dissipated, but the bodies separate;
* \(e=0\)--perfectly inelastic; the bodies stick together, and all translational kinetic energy is lost.  
Normal impact against an immovable plate  
For a normal collision of a particle of mass \(m\) with a rigid plate of effectively infinite mass \((M\rightarrow\infty)\), the plate velocity is zero before and after impact \((u_{2}=v_{2}\ =0)\). Momentum conservation therefore reduces to the particle alone, and its post-impact velocity is  
\[v_{1}=-eu_{1}. \tag{3}\]  
Accordingly, the following are true:  
* \(e=1\)--the particle rebounds with unchanged speed;
* \(e=0\)--the particle comes to rest on the plate;
* \(0<e<1\)--the particle rebounds with reduced speed.  
Elasticity versus plasticity of wheat kernels  
Elasticity is quantified directly by \(CoR\): values near 1 indicate an almost elastic response, whereas lower values signal energy loss through plastic deformation. Wheat kernels that absorb more energy (e.g., those with high moisture content or pre-damage) display the following:  
Figure 2: Contact phases in a grain–plate collision: (**a**) initial contact; (**b**) deformation until relative motion ceases; (**c**) end of contact [7].  
* Lower _CoR_;
* Longer contact times, because energy absorption prolongs the unloading phase.  
The mechanical properties therefore shape the force-time history (Figure 3). Elastic impacts exhibit symmetric rise and fall, while elastic-plastic and plastic impacts show asymmetry: compression lasts longer than restitution, and the total contact duration increases.