# Negative Stiffness, Incompressibility, and Strain Localisation in Particulate Materials
## 2 Apparent Negative Stiffness Created by Rotation of Clusters of Particles  
Consider the Dyskin and Pasternak [1] rotation of a partially detached non-spherical particle in a granular material. Such a particle can represent either non-spherical grains or clusters of grains, even when the grains themselves are spherical (Figure 1). In Figure 1, \(T\) and \(P\) are the forces created by the shear and uniaxial compressive stress, \(d\) is the length of the line between the two contact points (cluster diameter), \(\alpha\) is the initial angle between this line and the vertical direction, and \(u\) is the shear displacement due to rolling. The moment equilibrium about \(O\) is expressed as  
\[T\sqrt{d^{2}-\left(d\sin\alpha-u\right)^{2}}=P(d\sin\alpha-u.\bigg{)} \tag{1}\]  
Normalisation of Equation (1) by \(d\sin\alpha\) (assuming the positions with \(\alpha\
eq 0\)) gives  
\[T\sqrt{\frac{1}{sina^{2}}-\left(1-\frac{u}{disna}\right)^{2}}=P\Big{(}1-\frac {u}{disna}\Big{)}. \tag{2}\]  
By denoting \(\xi=\frac{u}{disna}\), \(\delta=\frac{1}{sina}\), \(\alpha>0\), Equation (2) can be transferred to the following form:  
\[T\sqrt{\delta^{2}-\left(1-\xi\right)^{2}}=P(1-\xi). \tag{3}\]Examining Equation (1) shows that, as the rolling progresses, the term inside the square root on the left-hand side of the equation increases. On the right-hand side, the term inside the bracket decreases. If \(P\) is fixed, then the shear stress must be decreasing to keep the moments about \(O\) in equilibrium. The slope of the decrease, \(dT/d\xi=-\left[\delta^{2}+(1-\xi)^{2}\right]\left[\delta^{2}-(1-\xi)^{2}\right] ^{-3/2}<0\). Thus, the apparent negative stiffness is produced by rolling of a non-spherical cluster of particles in compression. This effect was also seen in PFC 2D simulations presented in [43]. In the case of loaded samples that exhibit negative stiffness, the stability of the sample with negative stiffness is ensured by the boundary conditions which represent an encompassing positive stiffness loading frame. In this case, the energy of the system consisting of the sample plus the loading frame is positive definite [1; 2; 43]. In order to illustrate the existence of rotating clusters, a 2D discrete element simulation of uniaxial compression test on a sample of particulate material was conducted. The virtual sample was created by randomly placing particles with radii between \(D_{min}\) and \(D_{max}=D_{r}D_{min}\), where \(D_{r}\) is the maximum to minimum disc diameter ratio, until the target porosity was reached. This was performed in four steps: (1) the initial unbonded compact virtual sample was built within four frictionless walls; (2) floaters, i.e., the particles with no contact with other particles, were deleted; (3) the particle bonds were installed, leading to the particles being \"glued\" together; (4) the initial frictionless walls were removed, and the sample was relaxed by performing the modelling such that the average displacement became less than \(10^{-6}\) of the average disc radius. This procedure produced a sample containing 3786 particles. The built-in linear parallel bond model was used as the contact bond model. This bond model represents bonds between particles that are able to break one-by-one, eventually separating particle clusters and leaving them free to rotate. It is assumed that the bond breakage takes place only when the force acting on it reaches the tensile strength of the bond. Parameters of the virtual sample are shown in Tables 1 and 2. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Parameters of the Sample**} \\ \hline \(H\) (m) & \(W\) (m) & \(D_{min}\) (m) & \(D_{r}\) & Porosity & \(\rho\) (kg/m\({}^{3}\)) \\ \hline
0.0634 & 0.0317 & 0.00055 & 1.66 & 0.16 & 2630 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the virtual sample. \(H\), \(W\), \(D_{min}\), \(D_{r}\), and \(\rho\) are the sample height, sample width, minimum disc diameter, maximum to minimum disc diameter ratio, and density of the particles.