Modeling and simulation of mechanical systems with contact and friction within the nonsmooth contact dynamics framework. Possible applications in geosciences. Modeling and simulation of mechanical systems with contact and friction within the nonsmooth contact dynamics framework. Possible applications in geosciences. Seminario del Departamento de Ingenieria Matematica. Universidad de Concepcion Vincent Acary November 12, 2015 – 1/55
Modeling and simulation of mechanical systems with contact and friction within the nonsmooth contact dynamics framework. Possible applications in geosciences. Motivations Nonsmooth modeling of mechanical systems Numerical methods for the simulation Applications in geosciences and geotechnical engineering – 2/55
Modeling and simulation of mechanical systems with contact and friction within the nonsmooth contact dynamics framework. Possible applications in geosciences. Motivations Motivations ◮ Simulation of the mechanical behavior (statics and dynamics) of large collection of bodies in interaction through: ◮ contact and impact, ◮ Coulomb dry friction, ◮ cohesive interfaces with damage and plasticity. ◮ Nonsmooth mechanics modeling framework: ◮ dedicated time–integration schemes, ◮ numerical optimization solvers for SOCCP. ◮ Applications in geosciences. ◮ granular flows, rock avalanches, ◮ fracture processes, ◮ rock stability, ◮ friction instabilities (seismic waves). Motivations – 3/55
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source : http://caceresnatural.blogspot.cl/
Modeling and simulation of mechanical systems with contact and friction within the nonsmooth contact dynamics framework. Possible applications in geosciences. Motivations Nonsmooth modeling of mechanical systems Motivations – 7/55
Modeling and simulation of mechanical systems with contact and friction within the nonsmooth contact dynamics framework. Possible applications in geosciences. Nonsmooth modeling of mechanical systems Smooth multibody dynamics Equations of motion M ( q ) dv dt + F ( t , q , v ) = 0 , (1) v = ˙ q R n , R n , q ( t 0 ) = q 0 ∈ I v ( t 0 ) = v 0 ∈ I where ◮ F ( t , q , v ) = N ( q , v ) + F int ( t , q , v ) − F ext ( t ) Nonsmooth modeling of mechanical systems – 8/55
Modeling and simulation of mechanical systems with contact and friction within the nonsmooth contact dynamics framework. Possible applications in geosciences. Nonsmooth modeling of mechanical systems Unilateral contact and impact Body B ◮ Unilateral contact (Signorini condition) C B 0 � g N ( q ) ⊥ R N � 0 (2) N g N C A T 2 Complementarity condition T 1 ◮ Local relative velocity at contact � U N � Body A = G T ( q ) v U = (3) R N U T ◮ Impact Law (Newton Impact law) U + N = − e U − (4) N e is the coefficient of restitution. g N Nonsmooth modeling of mechanical systems – 9/55
Modeling and simulation of mechanical systems with contact and friction within the nonsmooth contact dynamics framework. Possible applications in geosciences. 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 ◦ If U T = 0 then R ∈ K If U T � = 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 || � µ | R N | } is the Coulomb friction cone Maximum dissipation principle in the tangent plane [Moreau, 1974]. R T ∈ D ( µ R N ) − U T max T R T (6) Nonsmooth modeling of mechanical systems – 10/55
Modeling and simulation of mechanical systems with contact and friction within the nonsmooth contact dynamics framework. Possible applications in geosciences. 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 – 11/55
Modeling and simulation of mechanical systems with contact and friction within the nonsmooth contact dynamics framework. Possible applications in geosciences. 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 � 3 Additional Energy � (BCED) 4 stored by viscosity (b) Rate dependent law (viscosity) Nonsmooth modeling of mechanical systems – 11/55
Modeling and simulation of mechanical systems with contact and friction within the nonsmooth contact dynamics framework. Possible applications in geosciences. Nonsmooth modeling of mechanical systems Why a nonsmooth modeling rather a smooth one ? Tribological reasons ◮ complexity of the behavior of the interface/interphase : elasticity, viscosity, damage, plasticity, wear, ... ◮ parameters are difficult to identify and to measure ◮ multi–scale problems: high stiffness coefficients, uncertainties on parameters. smoothing techniques and regularized models Regularization enables to use of standard PDE and/or ODE solvers, BUT ◮ the regularization parameters are in general not physical ◮ the results are highly sensible the regularization parameters ◮ the numerical tools are inefficient: stiff ODES, numerical instabilities. ◮ the intrinsic set–valuedness of the model is not well-represented (sticking state). ◮ The quasi–static process needs also unrealistic viscosity regularization. Nonsmooth modeling of mechanical systems – 12/55
Modeling and simulation of mechanical systems with contact and friction within the nonsmooth contact dynamics framework. Possible applications in geosciences. Nonsmooth modeling of mechanical systems Nonsmooth Lagrangian Dynamics Fundamental assumptions. ◮ The velocity v = ˙ q is assumed to of Bounded Variations (B.V) and right–continuous v + = ˙ q + (7) ◮ q is an absolutely continuous function such that � t v + ( t ) dt q ( t ) = q ( t 0 ) + (8) t 0 ◮ The acceleration (¨ q in the usual sense) is hence a differential measure dv associated with v such that � dv = v + ( b ) − v + ( a ) dv (( a , b ]) = (9) ( a , b ] Nonsmooth modeling of mechanical systems – 13/55
Modeling and simulation of mechanical systems with contact and friction within the nonsmooth contact dynamics framework. Possible applications in geosciences. Nonsmooth modeling of mechanical systems Nonsmooth Lagrangian Dynamics Definition 1 (Nonsmooth Lagrangian Dynamics) M ( q ) dv + F ( t , q , v + ) dt = di (10) v + = ˙ q + where di is the reaction measure and dt is the Lebesgue measure. Remarks ◮ The nonsmooth Dynamics contains the impact equations and the smooth evolution in a single equation. ◮ The formulation allows one to take into account very complex behaviors, especially, finite accumulation (Zeno-state). ◮ This formulation is sound from a mathematical Analysis point of view. References [Schatzman, 1973, 1978, Moreau, 1983, 1988] Nonsmooth modeling of mechanical systems – 14/55
Modeling and simulation of mechanical systems with contact and friction within the nonsmooth contact dynamics framework. Possible applications in geosciences. Nonsmooth modeling of mechanical systems Nonsmooth Lagrangian Dynamics Measures Decomposition (for dummies) � dv = ( v + − v − ) d ν + γ dt + dv s (11) di = f dt + p d ν + di s where ◮ γ = ¨ q is the acceleration defined in the usual sense. ◮ f is the Lebesgue measurable force, ◮ v + − v − is the difference between the right continuous and the left continuous functions associated with the B.V. function v = ˙ q , ◮ d ν is a purely atomic measure concentrated at the time t i of discontinuities of v , i.e. where ( v + − v − ) � = 0,i.e. d ν = � i δ t i ◮ p is the purely atomic impact percussions such that pd ν = � i p i δ t i ◮ dv S and di S are singular measures with the respect to dt + d η . Nonsmooth modeling of mechanical systems – 15/55
Modeling and simulation of mechanical systems with contact and friction within the nonsmooth contact dynamics framework. Possible applications in geosciences. 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, (12) or M ( q ( t i ))( v + ( t i ) − v − ( t i )) = p i , (13) Smooth Dynamics between impacts M ( q ) γ dt + F ( t , q , v ) dt = fdt (14) or M ( q ) γ + + F ( t , q , v + ) f + = [ dt − a . e . ] (15) Nonsmooth modeling of mechanical systems – 16/55
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