Kinematic and static assumptions for homogenization in micromechanics of granular materials

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1 Kinemati and stati assumptions for homogenization in miromehanis of granular materials Overview N.P. Kruyt Department of Mehanial Engineering University of Twente, The Netherlands L. Rothenburg Department of Civil Engineering University of Waterloo, Canada Miromehanis of granular materials Homogenization Variational priniples Constrution of ompatible and euilibrated fields Loal adjustment fields Comparison with Disrete Element Simulations Conlusions

2 Miromehanis of granular materials Objetive: relation marosopi and mirosopi harateristis Marosopi level Mirosopi level stress ontat fore strain ontat relative displaement Contat uantities Partile + U p t p l p n p + U Partile p Contat point p U U p Kinemati and stati assumptions for homogenization in granular materials 2

3 Euilibrium onditions 0 f p Compatibility onditions 0 S RS Network onneting partile entres Kinemati and stati assumptions for homogenization in granular materials 3

4 Euilibrium onditions 0 f p Compatibility onditions 0 S RS Network onneting polygon entres Network onneting partile entres Kinemati and stati assumptions for homogenization in granular materials 4

5 Miromehanial stress and strain University of Twente σ ε I -- f l S S -- h S S -- l h S S Polygon S h RS g RS Polygon R Uniform fields Kinemati form ε l (uniform strain) Stati form f σ h (uniform stress) Kinemati and stati assumptions for homogenization in granular materials 5

6 Homogenization Marosopi level Stress Constitutive relation Strain Averaging Homogenization Averaging Fore Constitutive relation Mirosopi level (ontats) Relative displaement Examples of homogenisation ε l ζε l f σ h f ξσ h Kinemati and stati assumptions for homogenization in granular materials 6

7 Contat onstitutive relation University of Twente No partile rotation Elasti onstitutive relation No frition Bonded ontats k t f n n f S f t k t t C f Kinemati and stati assumptions for homogenization in granular materials 7

8 Energy priniples Minimum potential energy priniple { }, { f } ompatible, not euilibrated { }, { f } ompatible, euilibrated U U * With presribed strain upper bound to moduli Minimum omplementary energy priniple { f }, { } euilibrated, not ompatible { f }, { } euilibrated, ompatible Û Û * With presribed stress lower bound to moduli Kinemati and stati assumptions for homogenization in granular materials 8

9 Compatible and euilibrated fields Compatible form p U U p (partile based) Euilibrated form f RS Φ S Φ R (polygon based) B b A p C D a d f p f pa + f pb + f p + f pd [ Φ D Φ A ] Φ A Φ B + [ ] + [ Φ B Φ C ] + [ Φ C Φ D ] 0 Kinemati and stati assumptions for homogenization in granular materials 9

10 Loal adjustment fields University of Twente Assume: overall behaviour uniform field + loal adjustment Kinemati formulation Presribed 2 3 p Free Presribed U p U U m ( X p ; ε) + u p U m ( X ; ε) p Reuire (approximate) euilibrium 0 f p S p ε p u { S p ε l p } l p u p Stati formulation Presribed Φ R Φ S Φ m ( X R ; σ) + φ R Φ m ( X S ; σ) f RS σ h RS φ R S 2 S 3 R Presribed S Free Reuire (approximate) ompatibility 0 S S RS C RS R φ { C RS σ h RS } S Kinemati and stati assumptions for homogenization in granular materials 0

11 Results from Disrete Element simulations Loose system Dense system G LAF bounds Uniform strain Simulation G Uniform stress k t Uniform stress k t.2 Relative displaements 2.5 Fores ζ Simulation, tangential Bound, tangential Bound, normal ξ 2.5 Bound, normal Simulation, normal Simulation, tangential Simulation, normal 0 Bound, tangential n t k t ζ n ( ε l ) n ζ t ( ε l ) t f n f t k t ξ n ( σ ) n h ξ t ( σ ) t h Kinemati and stati assumptions for homogenization in granular materials

12 Conlusions onstrution of euilibrated ontat fore fields uniform field homogenization leads to upper and lower bounds for moduli uniform strain assumption is only orret for dense systems homogenization based on mean field and flutuation that is determined by loal environment tight bounds for moduli good preditions of relative displaement and fore fields at ontats Kinemati and stati assumptions for homogenization in granular materials 2