Discrete Element Modeling of a fault reveals that viscous rolling relaxation controls friction weakening

In this paper, the eﬀect of the rolling and damping values on the macro friction coeﬃcient have been investigated. The introduction of the viscous rolling relaxation induces a friction weakening. Hence, it has been highlighted that this element can act against the angular spring and decreases the sample stiﬀness. This inﬂuence must be considered to not overestimate the toughness of a fault.


Introduction
If the focus is made on the damage slip zone of a fault, it appears the rock mass is pulverized and can be considered as a granular material [1], [2].The global behavior observed at the macro scale is a result of interactions at the micro-scale.
Our work is motivated by previous experiments made on antigorite [23].The main goal is to assess the influence on global behavior of contact laws and parameters values.Even if different relevant outputs for dense granular flows are reviewed by the French research group Groupement de Recherche Milieux Divisés (GDR MiDi) [24], we are focused in this paper on the macro friction coefficient at steady state.
We are going to explore the influence of the rolling resistance model.Experimental results [25], [26], and numerical ones [27], [28], [29] have highlighted that grains rolling has a real impact on the sample behavior.A lot of rolling models have been formulated [30], [31].But it appears the most accurate is an elastic-plastic spring dashpot model [32], [33].From this formulation, a lot of investigations have been done about the angular spring parameter.For examples: (i) the rolling helps the formation of shear bands and decrease the sample strength [34], [35], [36], [37] or (ii) the stress-dilatancy curves are modified [38], [39], [40], [41].It appears that the rolling resistance comes from friction [42], [43] and roughness [44], [45], [46].Even if superquadric particles allow to approximate the shape, those simulations represent an enormous computational cost.Because of this fact, some geometric laws about the rolling friction have been developed [47], [48].Thanks to that simulations can keep using round particles with a rolling resistance (from an equivalent shape).
On other hand, the angular damping influence is not well known.Hence, it is not considered in most of the DEM simulations.It appears this parameter is used more for stability reasons [30], [32] than physical meaning.Nevertheless, Jiang has formulated a link between the rolling and the normal dampings [49].But it was under the assumption of a geometrically derived kinematical model.
We decided to realize in this paper a parametric study over the rolling and the damping parameters to understand better their influence on the macro friction coefficient of a shearing fault.

Theory and formulation
The Discrete Element Model (DEM) is an approach developed by Cundall & Strack [3] to simulate granular materials at the particles level.The foundation of this method is to consider inside the material the individual particles and their interactions explicitly.Newton's laws defined at equations 1 and 2 help to compute the motion of the grain.
where m is the particle mass, I the moment of inertia, g is the gravity acceleration, f the contact forces, M the contact moments, r the radius.
Considering two particles with radii R 1 and R 2 , the interaction between particles is computed only if the distance δ between grains verifies equation 3.
Once contact is detected between grains 1 and 2, interactions (force and moment) are computed from relative motions ∆u and ∆ω with equations 4 and 5.
(5) where u is the particle displacement, θ the angular displacement and ω the angular velocity of the grain.
The contact models between cohesionless particles obey the Hertz contact theory [50].Normal, tangential and rolling models are shown at figure 1 and formulated at equations 6 and 7. Some details about Hertz laws are given at equation 8.A lot of rolling models could be applied to our case but we decide to use an elastic-plastic spring-dashpot model because it is the most accurate choice [30].
Fig. 1 The contact between two particles obeys to normal, tangential and rolling elasticplastic spring-dashpot laws.
The normal force The tangentiel force The tangential displacement ∆ t is computed by integrating the relative tangential velocity at the contact point over time.

Numerical model
The simulation setup is illustrated at figure 2. The box is a 0, 004 m×0, 006 m× 0, 0024 m region.Faces x and z are under periodic conditions.The gravity is not considered because its effect stays negligible under the vertical pressure applied.The simulation, made by the open-source software LIGGGHTS [51], is in several steps illustrated at figure 3: 1.The box, bottom and top triangle plates are created.2. 2500 particles are generated following the distribution presented in table 1 equivalent to the one used by Morgan [13].3. Top plate applies vertical stress of 10 MPa by moving following the y axis.
This plate is free to move vertically to verify this confining and allow volume change.4. The sample is sheared by moving the bottom plate at the speed of 100 µm/s until 100% strain.This step is then repeated at the speed of 300 and 1000 µm/s.The different parameters needed for the DEM simulation are presented in table 2. We can notice the time step dt must verify the Rayleigh condition [50], [52], [53] defined at equation 9.
With every computing test, the main problem is the running time.The time step dt must be selected considering the number of particles, the computing power, the stability of the simulation and the time scale of the test.In our case, we are looking for a 10 2 seconds term [54].If we include the default value into equation 9 the time step is around 10 −8 second and the running time skyrockets.To answer this we can easily change the density ρ and the shear modulus G.We will see those parameters are included in two dimensionless numbers defined at the equation 10: the contact stiffness number κ [55], [56], [57] and the inertial number I [24], [56].
The contact stiffness number The inertial number (10) Where γ ′ = v shear /h is the shear rate, h is the height of the sample during the shear and d 50 the mean diameter.
It appears the constitutive law is sensitive to κ because grains are not rigid enough (κ ≤ 10 4 ) [55].By this fact, it becomes not possible to change the Young modulus Y (and so the shear modulus G).The inertial number I represents the behavior of the grains, which can be associated with solids, liquids or gases [58].This dimensionless parameter does not affect the constitutive law if the flow regime is at critical state (I ≤ 10 −3 ) [24], [56].In conclusion, the density ρ can be modified, if we stay under the condition I ≤ 10 −3 , to increase the time step and solve our computing problem.

Results and discussion
A parametric study has been done on the rolling friction coefficient µ r and the rolling viscous damping coefficient η r .As figure 4 shows, the macro friction coefficient is plotted following the shear strain.This coefficient µ is computed by considering µ = F y /F x , where F y (resp.F x ) is the component following the y-axis (resp.x-axis) of the force applied on the top plate.Because of the granular aspect, there is a lot of oscillation.To reduce this noise, at least 3 simulations are run by a set of parameters (µ r , η r ) and a mean curve is computed.Moreover, only the steady-state is considered and an average value is estimated.
The comparison of the macro friction coefficient with different parameters set is highlighted at figures 5, 6, 7. It appears there is an increase of the fault friction coefficient with the rolling resistance µ r until a critical point depending on the rolling damping η r .This reduction of the stiffness with the rolling damping is not easy to understand at the first point.The larger is the damping, the stiffer should be the system.Two main questions should be answered: why does the friction coefficient decrease with the rolling resistance if there is damping?Why is the reduction larger with the damping value?10, 11 help to understand the behavior.It shows the rotation of particle (in red) during four different cases.We can notice that the fewer rotations there are, the stiffer the system will be.It appears the number of rolling particle increases with the rolling resistance, see figures 8, 9 and 10.The decrease of the friction coefficient is explained by particles rolling.Moreover, it is shown at figures 9 and 11 that damping increases the number of rolling particles and so the friction coefficient is reduced.A focus on the model equations must be done at relation 11 to understand better those observations (the input rolling parameters are emphasized in red).First, it appears the increment of the spring ∆M k r depends on µ 2 r while the plastic limit µ r R * F n depends only on µ r .There is a square factor between those values.Thus, this plastic limit, and so grain rolling, is reached easier with a larger rolling resistance µ r for a same angular displacement θ r .
Concerning the damping, it avoids the variation of the angular position ( .θ r → 0) during the elastic phase.As we have seen before, the main part of the sample is at the plastic phase and particles roll.So, it is as the damping acts in opposition of the angular spring, keeping grain into the plastic phase.We can notice that we have decided in this paper to shut down the damping moment when the angular plastic limit is reached (see equation 11).
In this way, we can understand better the reduction of the friction coefficient with the rolling stiffness µ r if damping is active.We can notice there is no decrease but an increase of the friction coefficient in the case of no damping.In absence of this one, the angular spring can act normally.The larger is the rolling parameter, the stiffer is the global sample.Figures 12,13,14,15 highlight the shearing speed influence on the system.It is the same results as before but plotted in another way.It appears there is no speed effect visible in most simulations as the friction coefficient keeps the same value.It is not surprising that no speed effects are spotted because there are no other parameters except the damping parameter which depends on speed or time.A speed influence is nevertheless noticed for cases where the friction coefficient starts to decrease with rolling resistance (for example the case µ r = 0, 5 and η r = 0, 5 at figure 14).As shown at figure 9, few particles (in orange or in white) are still not rolling during this critical step.The damping value is not large enough to cancel the effect of the spring and few grains are in the elastic phase.The damping creates so in this case a speed influence.If the damping value is larger, we have seen particles tend to be all in the plastic phase.If it is lower, the damping is negligible or null.In both cases, the speed effect disappears.

Conclusion
In this paper, we have considered granular materials into a plane shear configuration to investigate the effect of the rolling resistance and damping on 1.In the no damping case, the sample stiffness increases with the rolling resistance.
2. The consideration of the rolling damping introduces a critical point.For a constant damping value, the sample stiffness increases the rolling parameter until this critical point is reached.Then, the stiffness starts to decrease until a residual value.Hence, the damping tend to act against the spring and grains roll.3.No visible speed effects have been highlighted except at critical point.For the same rolling resistance value : (i) When the damping parameter is not large enough, the angular spring is the main element and no speed dependency is spotted, (ii) when the damping parameter is too large, all grains are in the plastic phase (roll) and the residual value is reached and (iii) when the damping parameter is at critical value, there is no main element in the rolling model, speed dependency occurs.

Fig. 2
Fig.2The simulation box with triangle plates and periodic faces.

Fig. 3 Table 1
Fig.3The simulation is in multiple steps : creation of the box and particles, application of the normal force and shearing.

Fig. 5 v
Fig. 5 v shear = 100 µm/s Figures 8,9,10,11 help to understand the behavior.It shows the rotation of particle (in red) during four different cases.We can notice that the fewer rotations there are, the stiffer the system will be.It appears the number of rolling particle increases with the rolling resistance, see figures 8, 9 and 10.The decrease of the friction coefficient is explained by particles rolling.Moreover, it is shown at figures 9 and 11 that damping increases the number of rolling particles and so the friction coefficient is reduced.