1. François Chevoir, Jean-Noël Roux Laboratoire Navier (LCPC, ENPC, CNRS) DENSE GRANULAR MATERIALS Microscopic origins of macroscopic behaviour GdR CHANT Ecole Nationale des Ponts et Chaussées Cité Descartes, Nov. 2007

2. GRANULAR MATERIALS Variety of material nature, properties, uses… Assemblies of solid objects interacting at their contacts

3. OUTLINE 1. Macroscopic aspects: known properties and phenomena, simple modelling schemes 2. Microscopic models for grains and contacts 3. Some connections between 1 and 2 for solidlike granular assemblies 4. Dense granular flows

4. Macroscopic behaviour of granular materials. Phenomenological description Internal friction and dilatancy Density and structure of packings Constitutive law Elasticity Strain localization

5. INTERNAL FRICTION On any plane cut through the material, shear component  of stress vector .n limited as Internal friction angle A slope inclined at angle cannot be stable With z = normal to free surface, x = along slope, assuming all derivatives vanish except w.r.t. z, use:

6. DILATANCY Deviatoric and volumetric strains couple Dense assemblies have to dilate to get sheared (grains packs in flexible membrane under vacuum are rigid) In simple shear define dilatancy angle Classically invoked picture: 2 nd row moves up, slope angle 

7. See other document (pdf) for more on macroscopic behavior

8. Contact law: normal elastic force Normal contact force between smooth objects: Hertz law d h  d = effective diameter (sphere diameter if identical)  h = normal deflection of contact  E,  Young modulus and Poisson ratio  Valid for smooth objects   radius of contact region prop. to

9. Contact law: tangential elasticity and friction Tangential ‘elastic’ force is history-dependent,to be dealt with incrementally Stiffness at zero tangential relative displacement : Coulomb condition: Apply condition locally for force density on surface History-dependent slip and no-slip regions within contact surface with

10. History dependence of contact law: Even without any sliding zone within contact surface ! (Elata and Berryman, Mechanics of Materials, 24, 229-240 1996)

11. Simplified contact law. Thermodynamic consistence Keep = u T -independent ? increase in stored elastic energy in tangential elasticity in case of a receding normal relative motion Rescale F T when F N is reduced ( proportionnally to stiffness)

12. Further simplification: linear unilateral elasticity Use constantand Keep correct order of magnitude for normal deflection h or Assume rigid, undeformable contacts and penalize impenetrability constraints. Motivation: Definition and use of limit of rigid contacts ?

13. Contact forces in rolling and pivoting. Objectivity Tangential elastic/frictional forces are not determined by current position/orientation of grains. Evolution for arbitrary motions ? Rolling = rotation about axis in tangential plane Pivoting = rotation about normal axis Possible rule for tangential force: 1. Follow rolling motion of normal vector n 2. Rotate about n with average angular velocity of the 2 solids Ensures objectivity. (e.g. tangential force should follow global rigid body motion) (Most often, rule 1 is adopted – without discussion - but not rule 2 !) u T constant

14. Viscous part of contact law. Damping, restitution. Add viscous terms in contact forces, opposing relative motion at contact point : With linear elasticity, a pair in contact is a damped harmonic oscillator If not overdamped, restitution coefficients e N,T relate to ratio  N,T of damping coefficient to critical value (e.g. ) as In Hertzian case, similar definition possible with tangent stiffnesses K N,T Ad-hoc parameters, little physical justification Coulomb condition: includes viscous terms ? Yes  forbid traction. No  viscous fluid ?

15. Remarks on contact laws Strong stress concentration in contacts (worse for angular contacts) In experimental practice, only measured or controlled with relatively large, regular-shaped bodies Dissipative part of contact law ??? Tempting to use rigid contacts… But other approach is needed then … and what about macroscopic elasticity ? Necessary anyway to systematically assess parameter influence

16. See other document (pdf) for part III, about micro/macro connections for solidlike granular materials