Strength, dilatancy, energy and dissipation in quasi-static deformation of granular materials L. Rothenburg Department of Civil Engineering University of Waterloo, Canada leoroth@utwaterloo.ca N.P. Kruyt Department of Mechanical Engineering University of Twente, The Netherlands n.p.kruyt@utwente.nl

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

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

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

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

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

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

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

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.

Energy

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.

Dissipation Work input Change in energy Dissipation

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

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?

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 53 2005] 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.

Questions and comments Thank you for your attention! Questions and comments? Presentation can be downloaded from: www.ts.wb.utwente.nl/kruyt/pg2005.pps Thank you all for your attention. I welcome questions and comments.