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The nonsmooth contact dynamics method for the simulation of granular matter flows and fracture in mining applications The nonsmooth contact dynamics method for the simulation of granular matter flows and fracture in mining applications Seminar CMM. Universidad de Chile Vincent Acary August 27, 2015 – 1/49

The nonsmooth contact dynamics method for the simulation of granular matter flows and fracture in mining applications Motivations Nonsmooth modeling of mechanical systems Numerical methods for the simulation Applications in mining and geotechnical engineering – 2/49

The nonsmooth contact dynamics method for the simulation of granular matter flows and fracture in mining applications Motivations Motivations I Simulation of the mechanical behavior (statics and dynamics) of large collection of bodies in interaction through: I contact and impact, I Coulomb dry friction, I cohesive interfaces with damage and plasticity. I Nonsmooth mechanics modeling framework: I dedicated time–integration schemes, I numerical optimization solvers for SOCCP. I Applications in mining and geotechnical engineering. I granular flows, I fracture processes, I rock stability. Motivations – 3/49

The nonsmooth contact dynamics method for the simulation of granular matter flows and fracture in mining applications Motivations Nonsmooth modeling of mechanical systems Motivations – 5/49

The nonsmooth contact dynamics method for the simulation of granular matter flows and fracture in mining applications Nonsmooth modeling of mechanical systems Smooth multibody dynamics Equations of motion 8 M ( q ) dv > > dt + F ( t , q , v ) = 0 , > > > > > < (1) v = ˙ q > > > > > > > : R n , R n , q ( t 0 ) = q 0 2 I v ( t 0 ) = v 0 2 I where I F ( t , q , v ) = N ( q , v ) + F int ( t , q , v ) � F ext ( t ) Nonsmooth modeling of mechanical systems – 6/49

The nonsmooth contact dynamics method for the simulation of granular matter flows and fracture in mining applications Nonsmooth modeling of mechanical systems Unilateral contact and impact Body B I Unilateral contact (Signorini condition) C B 0 6 g N ( q ) ? R N > 0 (2) N g N C A T 2 Complementarity condition T 1 I Local relative velocity at contact  U N � Body A = G T ( q ) v U = (3) R N U T I Impact Law (Newton Impact law) U + N = � e U � (4) N e is the coe ffi cient of restitution. g N Nonsmooth modeling of mechanical systems – 7/49

The nonsmooth contact dynamics method for the simulation of granular matter flows and fracture in mining applications Nonsmooth modeling of mechanical systems Coulomb’s friction K � u T � ˆ u r N � ˆ u N Coulomb’s friction r N Coulomb’s friction says the following: r T � ˆ u T T 1 If g N ( q ) = 0 then: P � ˆ u � ˆ u N T 2 K � 8 If U T = 0 then R 2 K < If U T 6 = 0 then || R T ( t ) || = µ | R N | and there exists a scalar a > 0 : such that R T = � aU T (5) where K = { R , || R T || 6 µ | R N | } is the Coulomb friction cone Maximum dissipation principle in the tangent plane [Moreau, 1974]. R T 2 D ( µ R N ) � U T max T R T (6) Nonsmooth modeling of mechanical systems – 8/49

The nonsmooth contact dynamics method for the simulation of granular matter flows and fracture in mining applications Nonsmooth modeling of mechanical systems Coulomb’s friction as a Second–Order Cone Complementarity Problem (SOCCP) Let us introduce the modified velocity b U defined by U = [ U N + µ || U T || , U T ] T . b (7) This notation provides us with a synthetic form of the Coulomb friction as � b U 2 I N K ( R ) , (8) or K ⇤ 3 b U ? R 2 K . (9) where K ⇤ = { v 2 I R n | r T v > 0 , 8 r 2 K } is the dual cone. Nonsmooth modeling of mechanical systems – 9/49

The nonsmooth contact dynamics method for the simulation of granular matter flows and fracture in mining applications Nonsmooth modeling of mechanical systems Nonsmooth cohesive zone model R N 0 < β < 1 g C 0 � 2 1 � 1 β 2 c N Dissipated Energy � (OAB) 1 B by damage β = 1 Stored Energy � (OBC) 2 c N by the surface bond 1 A (a) Rate independent law Nonsmooth modeling of mechanical systems – 10/49

The nonsmooth contact dynamics method for the simulation of granular matter flows and fracture in mining applications Nonsmooth modeling of mechanical systems Nonsmooth cohesive zone model R N g C E 0 � 2 � 1 � 4 Dissipated Energy � (OAB) 1 by damage B Stored Energy � (OBC) 2 by the surface bond D 0 < β < 1 Dissipated Energy � (ABD) 3 by viscosity A β = 1 � Additional Energy 3 � (BCED) 4 stored by viscosity (b) Rate dependent law (viscosity) Nonsmooth modeling of mechanical systems – 10/49

The nonsmooth contact dynamics method for the simulation of granular matter flows and fracture in mining applications Nonsmooth modeling of mechanical systems Nonsmooth Lagrangian Dynamics Fundamental assumptions. I The velocity v = ˙ q is assumed to of Bounded Variations (B.V) and right–continuous v + = ˙ q + (10) I q is an absolutely continuous function such that Z t v + ( t ) dt q ( t ) = q ( t 0 ) + (11) t 0 I The acceleration (¨ q in the usual sense) is hence a di ff erential measure dv associated with v such that Z dv = v + ( b ) � v + ( a ) dv (( a , b ]) = (12) ( a , b ] Nonsmooth modeling of mechanical systems – 11/49

The nonsmooth contact dynamics method for the simulation of granular matter flows and fracture in mining applications Nonsmooth modeling of mechanical systems Nonsmooth Lagrangian Dynamics Definition 1 (Nonsmooth Lagrangian Dynamics) 8 M ( q ) dv + F ( t , q , v + ) dt = di > < (13) > : v + = ˙ q + where di is the reaction measure and dt is the Lebesgue measure. Remarks I The nonsmooth Dynamics contains the impact equations and the smooth evolution in a single equation. I The formulation allows one to take into account very complex behaviors, especially, finite accumulation (Zeno-state). I This formulation is sound from a mathematical Analysis point of view. References [Schatzman, 1973, 1978, Moreau, 1983, 1988] Nonsmooth modeling of mechanical systems – 12/49

The nonsmooth contact dynamics method for the simulation of granular matter flows and fracture in mining applications Nonsmooth modeling of mechanical systems Nonsmooth Lagrangian Dynamics Measures Decomposition (for dummies) ⇢ dv = ( v + � v � ) d ν + γ dt + dv s (14) di = f dt + p d ν + di s where I γ = ¨ q is the acceleration defined in the usual sense. I f is the Lebesgue measurable force, I v + � v � is the di ff erence between the right continuous and the left continuous functions associated with the B.V. function v = ˙ q , I d ν is a purely atomic measure concentrated at the time t i of discontinuities of v , i.e. where ( v + � v � ) 6 = 0,i.e. d ν = P i δ t i I p is the purely atomic impact percussions such that pd ν = P i p i δ t i I dv S and di S are singular measures with the respect to dt + d η . Nonsmooth modeling of mechanical systems – 13/49

The nonsmooth contact dynamics method for the simulation of granular matter flows and fracture in mining applications Nonsmooth modeling of mechanical systems Impact equations and Smooth Lagrangian dynamics Substituting the decomposition of measures into the nonsmooth Lagrangian Dynamics, one obtains Impact equations M ( q )( v + � v � ) d ν = pd ν , (15) or M ( q ( t i ))( v + ( t i ) � v � ( t i )) = p i , (16) Smooth Dynamics between impacts M ( q ) γ dt + F ( t , q , v ) dt = fdt (17) or M ( q ) γ + + F ( t , q , v + ) f + = [ dt � a . e . ] (18) Nonsmooth modeling of mechanical systems – 14/49

The nonsmooth contact dynamics method for the simulation of granular matter flows and fracture in mining applications Nonsmooth modeling of mechanical systems The Moreau’s sweeping process of second order Moreau [1983, 1988] A key stone of this formulation is the inclusion in terms of velocity. 8 M ( q ) dv + F ( t , q , v + ) dt = di = G ( q ) dI > > > > > > > > > v + = ˙ q + > < (19) > > U + = G T ( q ) v + > > > > > > > > : ) 0 6 U + + eU � ? dI > 0 g N ( q ) 6 0 = Comments R + ( g N ( q )) ( U + ) � dI 2 N T I (20) This formulation provides a common framework for the nonsmooth dynamics containing inelastic impacts without decomposition. ‹ Foundation of the time–stepping approaches. Nonsmooth modeling of mechanical systems – 15/49

The nonsmooth contact dynamics method for the simulation of granular matter flows and fracture in mining applications Nonsmooth modeling of mechanical systems Numerical methods for the simulation Nonsmooth modeling of mechanical systems – 16/49