Presentation on theme: "L. Rothenburg Department of Civil Engineering"— Presentation transcript:
1 Strength, dilatancy, energy and dissipation in quasi-static deformation of granular materials L. RothenburgDepartment of Civil EngineeringUniversity of Waterloo, CanadaN.P. KruytDepartment of Mechanical EngineeringUniversity of Twente, The Netherlands
2 Objective Relation interparticle friction and macroscopic behaviour: shear strengthdilatancyenergydissipation
3 DEM simulations two-dimensional biaxial tests various friction coefficients mperiodic boundary conditions50,000 disksdense and loose initial assemblyInvariants:
4 Contact constitutive relation Special cases:m ® 0m ® µmktknElastic behaviour at contact in these special cases.
5 Shear strength and dilatancy Increasing mQualitative 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.
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 DissipationWork inputChange in energyDissipation
13 Dissipation function Work input dissipated: Strength-dilatancy: Resulting dissipation function:
14 ConclusionsFor all m (including m®0 and m®µ), macroscopic frictional behaviourMacroscopic dissipation even for m®0 and m®µStrength-dilatancy relationConstant ratio of ‘tangential’ over ‘normal’ energy beyond initial rangeAll work input is dissipated beyond initial rangeStrength-dilatancy relation; geometrical effect?
15 Discussion Origin of macroscopic frictional behaviour: Interparticle friction: NOContact disruption and creation: POSSIBLYOrigin of dissipation if m®0 or m® µ:Dynamics & viscous dissipationMicroscopic 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 Thank you for your attention!Questions and comments?Presentation can be downloaded from:Thank you all for your attention. I welcome questions and comments.