1. Strength, dilatancy, energy and dissipation in quasi-static deformation of granular materials
L. Rothenburg
Department of Civil Engineering
University of Waterloo, Canada
N.P. Kruyt
Department of Mechanical Engineering
University of Twente, The Netherlands

2. Objective Relation interparticle friction and macroscopic behaviour:
shear strength
dilatancy
energy
dissipation

3. DEM simulations two-dimensional biaxial tests
various friction coefficients m
periodic boundary conditions
50,000 disks
dense and loose initial assembly
Invariants:

4. Contact constitutive relation
Special cases:
m ® 0
m ® µ m kt kn
Elastic behaviour at contact in these special cases.

5. Shear strength and dilatancy
Increasing m
Qualitative identical behaviour in limit cases, so microscopic friction is not essential in determining macroscopic frictional behaviour

6. Plastic strains: unloading paths
Note of caution on interpretation: effect of micro-structural changes upon unloading (coordination number; contact anisotropy)

7. Dilatancy rate Formulated in plastic strains
Large range where a linear relation (with offset) is observed between shear strength and dilatancy rate.

8. Strength at peak & steady-state, dilatancy rate at peak strength

9. Strength-dilatancy relation
Note that result is for various values of interparticle friction coefficients. Note that (q/p) at steady state depends on mu, but alpha does not. Hence strength-dilatancy relation may be geometrical in nature.

10. Energy

11. Dissipation Work input Change in energy Dissipation
All dissipated energy is transferred out of the system as heat. Since free energy can not be determined from DEM simulation, the assumption is made that the free energy approximately equals the internal energy in the quasi-static case with ‘granular temperature’ -> 0.

12. Dissipation
Work input
Change in energy
Dissipation

13. Dissipation function Work input dissipated: Strength-dilatancy:
Resulting dissipation function:

14. Conclusions
For all m (including m®0 and m®µ), macroscopic frictional behaviour
Macroscopic dissipation even for m®0 and m®µ
Strength-dilatancy relation
Constant ratio of ‘tangential’ over ‘normal’ energy beyond initial range
All work input is dissipated beyond initial range
Strength-dilatancy relation; geometrical effect?

15. Discussion Origin of macroscopic frictional behaviour:
Interparticle friction: NO
Contact disruption and creation: POSSIBLY
Origin of dissipation if m®0 or m® µ:
Dynamics & viscous dissipation
Microscopic viscous dissipation Þ macroscopic plastic dissipation [Puglisi & Truskinovsky, JMPS ]
Dissipation occurs due to viscous damping during the fast dynamical processes that take place at the contact level during the transition of the system from one stable static state to the next after the disruption of contacts.

16. Questions and comments
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