Box Shearing

According to findings from Chialvo et al. [15] Φc plays such an important role in governing the rhe-ology in each of the three flow regimes. However, this task is made difficult by fluctuations of pressure measurements in time t. There is a propensity for assemblies near Φc to form and break force chains intermittently during the shearing process, resulting in stress fluctuations of several orders of magnitude as seen in Figure 2.6(a). Though fluctuations occur at all volume fractions, their size relative to the mean is markedly large near the critical point. In Figure 2.6(b) the standard deviationσP ≡

q

hp²(t)i − hp(t)i² of the pressure, when scaled by the time-averaged pressure p, exhibits a spike centered slightly under Φc. The use of Φc as delimiter for the rheological model is important to have reliable stress data measured in inertial and quasi static regime.

Simulations were performed using an unscaled system and a scaled system according to previously obtained relation for stiffness and damping. For all simulations static friction used was 0.1, rolling friction 0.0, coefficient of restitution 0.75 and density 2500 kg/m³. Results of pressure for different parcel sizes are depicted in Figure 2.8. In the y-axis pressure is presented in a dimensionless form asP D_p/k, where Dp is the parcel diameter, and in the x-axis is depicted the dimensionless parcel diameter. Pressure is defined asP =^(σxxσ_3V^yyσzz).

Figure 2.7: DEM model of the shear box simulation. Lees-Edwards boundary conditions impose a steady shear motion on the particles and were used where indicated

Results of shear stress for different parcel sizes are depicted in Figure 2.9. Shear stress (τ) is defined as the stress component pointing in the shearing direction and acting on the surface normal to the gradient direction (the other components are much smaller).

The stress analysis is now extended to the Hertz non linear contact model. Cohesive models used in further studies are derived for Hertz and its behaviour using previous scaling rules for CGM modelling is now addressed. The difference between the linear model and Hertz model is how stiffness and damping coefficients are calculated. In Hertz model stiffness is calculated as in Equation (2.21). Since particle overlap scales with the particle radius, the stiffness condition is satisfied andkn scales with the particle radius. Damping is given by Equation (2.22) and S_n parameter is given by Equation (2.25). S_n scales with R multiplied bym^∗ (which scales withR³). From Equation (2.22) can be seen that damping scales withR² and satisfies the damping condition. Hertz contact model can be considered scale independent and no further adjustment in stiffness and damping parameters are necessary.

Figure 2.8: Dimensionless pressure in the formP D_p/k is depicted for different parcels diameters. Data corresponds to CGM scaled (blue points) and unscaled system (red points) using linear model for the inertial regime. Yellow points correspond to static regime. Predictive theory for inertial and quasi-static regime are also depicted as continuous line and details are found in [15].

Figure 2.9: Dimensionless shear stress in the form τ Dp/k is depicted for different parcels diameters.

Data corresponds to CGM scaled (blue points) and unscaled system (red points) using linear model for the inertial regime. Yellow points correspond to quasi-static regime. Predictive theory for inertial and quasi-static regime are also depicted as continuous line and details are found in [15].

Differently of the linear model where the stiffness is calculated directly from the material parameters, for the Hertz model there is a dependency of the stiffness with respect to the overlap. Stiffness increases with the overlap and in this case is not possible to define a fixed value as we did for the linear model.

However in the inertial regime the pressure is independent of the particle stiffness as shown in the work of Chialvo et al. [15]. One should expect pressure data obtained with Hertz and linear model to collapse in the same curve in the inertial regime.

In the quasi-static regime there is a dependency of stresses with respect to the stiffness and one should expected that pressure increases with the increase of the stiffness [15]. Although curves may not collapse (unless both models have equivalent stiffness) is expected a similar profile in the Hertz and scaled linear model curves. Same shear rate and volume fraction used for linear model were also used for the simula-tions using Hertz contact model.

Results of the shear box simulation for dimensionless pressure (P Dp/k) for the inertial and quasi-static flow regime are depicted in Figure 2.10. Hertz data is compared to data obtained for the linear model in the quasi static regime. In the inertial regime the stiffness in the Hertz model is varied and curves collapse in the the same values obtained with the linear model.

Results of the shear box simulation for dimensionless shear stress (τ D_p/k) for the inertial and quasi-static flow regime are depicted in Figure 2.11. Hertz model data is compared to data obtained for the linear model in the quasi static regime. In the inertial regime the stiffness in the Hertz model is varied and curves collapse in the the same values obtained with the linear model.

Similar stresses were obtained for Hertz-Mindlin and the linear contact models. In coarse graining is desired that stresses remain invariant with the increase in parcels size. In the inertial regime results show a significant pressure increase with the parcel size. Therefore more analysis and a relaxation model to correct this pressure would be necessary. For quasi-static regime it was demonstrated that the stresses re-mained almost constant with parcel sizes. A slight increase is observed for correlations ofDp/Dprim>10 when dimensionless shear rate are > 10⁻². This could be explained by the higher dimensionless shear rate for such parcels that results in flow regime where both inertial and quasi static flow merge in one curve and pressure increases linearly as shown by [15]. Numerical simulations presented in this thesis fit in dense quasi-static regime definition.

The same shear test was applied for cohesive material but it failed to obtain correct data. On the one hand for high cohesive forces particles stick together and this change their translational velocity.

This leads to differences in the module of the velocity in x-direction with respect xy-plane in y=0. In this scenario Lees-Edwards boundary conditions are not valid anymore. It is also not possible to hold the assumption that different parcel sizes have the same translational velocity. On the other hand with low cohesive forces the shear forces present in the flow easily separate the particles and the behavior is identical as Hertz contact law. A description of coarse graining for different cohesion models and a secondary validation test is given in the following section.

Figure 2.10: Pressure (dimensionless) for different parcels diameters. Linear model scaled and Hertz model are depicted for the inertial flow (φ= 0.62) with a virtually constant pressure with the increase in parcel size. Linear (scaled and unscaled) and Hertz models are depicted also for inertial regime (φ= 0.55).

No dependency w.r.t. stiffness is depicted in this region as already shown in [15].

Figure 2.11: Shear stresses (dimensionless) for different parcels diameters. Linear model scaled and Hertz model are depicted for the inertial flow (φ= 0.62) with virtually constant shear stresses with the increase in parcel size. Linear (scaled and unscaled) and Hertz models are depicted also for inertial regime (φ= 0.55). No dependency w.r.t. stiffness is depicted in this region as already shown in [15].