# The Construction of Equivalent Particle Element Models for Conditioned Sandy Pebble
## 4 Physical Model between Particles
### Contact Constitutive Relation  
A schematic diagram of contact vectors in the H-M model can be found in Figure 9. In the model, the calculation of normal elastic force \(F_{\mathrm{nij}}e\) is based on Hertzian contact theory [29], while the tangential elastic force \(F_{\mathrm{fij}}e\) was derived by [PERSON] and [PERSON] [21; 30]. As for the energy dissipation in the contact process, [PERSON] et al. [31] proposed a nonlinear normal damping force \(F_{\mathrm{nij}}d\) and tangential damping force \(F_{\mathrm{fij}}d\). Hence, the contact action exerted on particle \(i\) by particle \(j\) can be expressed as follows  
\[\left\{\begin{array}{c}F_{\mathrm{nij}}e=\frac{4}{3}E^{*}\sqrt{R^{*}u_{ \mathrm{n}}}m^{\frac{3}{2}}\\ F_{\mathrm{nij}}e=-k_{\mathrm{t}}u_{\mathrm{t}}\\ F_{\mathrm{nij}}d=-2\sqrt{\frac{5}{6}}\beta\sqrt{m^{*}k_{\mathrm{n}}}^{*}v_{ \mathrm{n}}\mathrm{{}^{rel}}\\ F_{\mathrm{tij}}d=-2\sqrt{\frac{5}{6}}\beta\sqrt{m^{*}k_{\mathrm{t}}}v_{ \mathrm{t}}\mathrm{{}^{rel}}\end{array}\right. \tag{1}\]  
where, \(u_{\mathrm{n}}\) and \(u_{\mathrm{t}}\) denote the normal and tangential overlap of particle \(i\) and particle \(j\), respectively; \(k_{\mathrm{t}}\) is shear stiffness, \(k_{\mathrm{t}}=8C^{*}\sqrt{R^{*}u_{\mathrm{n}}}\); \(k_{\mathrm{n}}\)\({}^{*}\) denotes equivalent normal stiffness, \(k_{\mathrm{n}}{}^{*}=2E^{*}\sqrt{R^{*}u_{\mathrm{n}}}\); \(E^{*}\), \(R^{*}\), \(m^{*}\), and \(G^{*}\) denote [PERSON]'s modulus, radius, mass, and shear modulus of particles in contact, respectively; \(\beta\) denotes damping coefficient, which is associated to the restitution coefficient \(e\) by  
Figure 8: Physical contact models of conditioned soil.  
\(\beta=\frac{\ln e}{\sqrt{\ln^{2}e+\pi^{2}}}\); and \(v_{\rm n}\)rel and \(v_{\rm t}\)rel denote the normal and tangential components of relative speed of two particles, respectively.  
The tangential force obeys Coulomb law of friction [32], that is to say if \(\left|\overset{\rightarrow}{F}_{\rm uj}e+\overset{\rightarrow}{F}_{\rm uj}d \right|>F_{\rm u}=\mu_{\rm s}\left|\overset{\rightarrow}{F}_{\rm nij}\right|\) (\(F_{\rm u}\) is friction force and \(\mu_{\rm s}\) denotes the coefficient of sliding friction), then slip occurs. While the rolling friction is accounted for by applying directional constant torque model [33], namely \(T_{\rm u}=-\mu_{\rm r}\left|\overset{\rightarrow}{F}_{\rm nij}\right|r_{i} \overset{\rightarrow}{\omega}_{i}\), where \(\mu_{\rm r}\) is the coefficient of rolling friction, \(r_{i}\) is the distance from the mass center of particle \(i\) to the contact point, \(\omega_{i}\) denotes the unit vector of angular velocity.  
To describe adhesive action, JKR theory enriched the Hertz-Mindlin model with normal cohesion force, and the normal elastic force \(F_{\rm nij}e\) is revised to [34]  
\[F_{\rm nij}\overset{\rm JKR}{}=-4\sqrt{\pi kE^{*}}\alpha^{\frac{3}{2}}+\frac{4 E^{*}}{3R^{*}}\alpha^{3} \tag{2}\]  
where, \(k\) denotes surface energy per unit of contact area, measured in \(\rm I/m^{2}\); the relationship between coefficient \(\alpha\) and normal overlap \(u_{\rm n}\) can be expressed as \(u_{\rm n}=\frac{a^{2}}{R^{*}}-\sqrt{\frac{4\pi k\alpha}{E^{*}}}\).