Background Element-free elastoplastic solid for nonsmooth Method Solver multidomain dynamics Elastic results Plastic method Plastic results John Nordberg john.nordberg@umu.se UMIT Research Lab - Ume˚ a University August 27, 2015 john.nordberg@umu.se, August 27, 2015 (1 : 14)
Background Method Solver Elastic results Plastic method Plastic results john.nordberg@umu.se, August 27, 2015 (2 : 14)
Approaches Background Method Solver Elastic results Plastic method Multibody system dynamics Solid mechanics Plastic results q − G T λ = f ρ ¨ u − ∇ · σ = ρb M ¨ σ = CE ελ + g ( q ) = 0 john.nordberg@umu.se, August 27, 2015 (3 : 14)
Background Method Solver Elastic results Plastic method Plastic results john.nordberg@umu.se, August 27, 2015 (4 : 14)
Displacement u ( x ) q ( x ) = x + u ( x ) J ≡ ∇ x q ( x ) = I + ∇ x u Background Green-Lagrange strain tensor Method Solver E ( x ) = 1 = 1 � � � � J T J − I ∇ x u + ∇ T x u + ∇ T Elastic results x u ∇ x u 2 2 Plastic method Plastic results john.nordberg@umu.se, August 27, 2015 (5 : 14)
Moving least squares (MLS) approximation of the displacement field u ( x ) Np � Ψ j ( x ) u j u α ( x ) = α Background j Method � � Ψ j ( x ) = p γ ( x ) A − 1 γτ ( x ) p τ ( x j ) W x − x j , h Solver Elastic results Np Plastic method � � � x − x j , h p γ ( x j ) p τ ( x j ) A γτ ( x ) = W Plastic results j 1, x , y , z , yz , xz , xy , x 2 , y 2 , z 2 � T � p ( x ) = john.nordberg@umu.se, August 27, 2015 (6 : 14)
Constraint energy U = 1 2 g T ε − 1 g Constitutive relation using Voigt notation Background σ = CE Method Solver λ + 2 µ 0 0 0 λ λ Elastic results λ λ + 2 µ λ 0 0 0 Plastic method λ + 2 µ 0 0 0 λ λ C = Plastic results 0 0 0 µ 0 0 0 0 0 0 0 µ 0 0 0 0 0 µ Strain energy U = 1 2 Ev 0 CE Elasticity strain tensor constraint and regularisation - 6D ε = ( v 0 C ) − 1 g i ( q ) = E i Jacobian where K τη ≡ ∇ τ u η αβ ( q ) = ∂g i = ∂g i ∂E γ ∂K τη G i α α ∂q β ∂E γ ∂K τη ∂q β john.nordberg@umu.se, August 27, 2015 (7 : 14)
Multibody dynamics - numerical solver Linearized varational time stepper SPOOK (Lacoursi` ere [2, 3]) Background q n + 1 = q n + h ˙ q n + 1 Method Solver Elastic results − ¯ Plastic method − G T G T ˙ M ˙ q n + hf n M q n + 1 Plastic results − 4 G Σ 0 λ = h Υg + ΥG ˙ q n ¯ ¯ ¯ 0 G Σ λ ω n � �� � � �� � � �� � − r H z regularization and stabilization matrices Σ = 4 � ε 1 ε 2 � ε = ( v 0 C ) − 1 h 2 diag , , . . . 1 + 4 τ 1 1 + 4 τ 2 h h Σ = 1 ¯ h diag ( γ 1 , γ 2 , . . . ) � 1 1 � Υ = diag , , . . . 1 + 4 τ 1 1 + 4 τ 2 h h john.nordberg@umu.se, August 27, 2015 (8 : 14)
Nonsmooth MBD - numerical solver Background Method Including frictional contacts, impacts, joint ant motor limits Solver Elastic results lead to limits and complementarity conditions on the solution Plastic method Plastic results variables Hz + r = w + − w − 0 � w + ⊥ z − l � 0 0 � w − ⊥ u − z � 0 The problem transforms from linear system to a mixed linear complementarity problem (MLCP) john.nordberg@umu.se, August 27, 2015 (9 : 14)
Background Method Solver 0.15 Elastic results hydrostatic compression ← → uniaxial stretch Plastic method 0.1 Plastic results 0.05 σ /c 0 analytic solution −0.05 simulation np = 11 3 simulation np = 6 3 −0.1 −0.1 −0.05 0 0.05 0.1 0.15 λ − 1 john.nordberg@umu.se, August 27, 2015 (10 : 14)
Elastoplastic terrain model Elastic and plastic strain components E = E e + E p Plastic flow rule d E p = d λ p ∂Φ ∂σ when yield Φ ( σ ) > 0 Background Method Solver Elastic results √ J 2 Plastic method Plastic results ɸ ( I 1, J 2 ) = 0 DP ɸ ( I 1 , J 2, κ ) = 0 C ɸ < 0 ɸ ( I 1, J 2 ) = 0 T - κ - κ 0 C T -I 1 ( κ ) I 1 I 1 Capped Drucker-Prager plasticity model (Dolarevic [1]) I 1 � I t Φ t ( I 1 , J 2 ) 1 I t 1 � I 1 � I c Φ ( σ , κ ) = (1) Φ e ( I 1 , J 2 ) 1 ( κ ) Φ c ( I 1 , J 2 , κ ( tr E p )) I 1 � I c 1 ( κ ) john.nordberg@umu.se, August 27, 2015 (11 : 14)
Background Method Solver Elastic results Plastic method Plastic results john.nordberg@umu.se, August 27, 2015 (12 : 14)
References Background Method Solver Elastic results Plastic method Plastic results [1] S. Dolarevic et al. A modifed three-surface elasto-plastic cap model and its numerical implementation. Comput. Struct., 85(7-8):419-430, April 2007. [2] C. Lacoursi` ere, M. Linde, SPOOK: a variational time-stepping scheme for rigid multibody systems subject to dry frictional contacts, submitted (2013). [3] C. Lacoursi` ere, Ghosts and Machines: Regularized Variational Methods for Interactive Simulations of Multibodies with Dry Frictional Contacts, PhD thesis, Ume˚ aUniversity, Sweden, (2007) john.nordberg@umu.se, August 27, 2015 (13 : 14)
Elastoplastic terrain model Drucker-Prager for cohesive soil Background Method J 2 + η ( φ ) I 1 � Φ e ( I 1 , J 2 ) = 3 − ξ ( φ ) c (2) Solver Elastic results Plastic method Tension cap Plastic results Φ t ( I 1 , J 2 ) = ( I 1 − T + R t ) 2 + J 2 − R 2 (3) t Compression cap Φ c ( I 1 , J 2 , κ ( tr ε p )) = ( I 1 − a ( κ )) 2 + J 2 − b ( κ ) 2 (4) R 2 Cap variables - for compressive hardening 1 + tr ( ǫ p ) κ = κ 0 + 1 � � D ln (5) W where κ 0 is the initial position of the compression cap (Dolarevic [1]). john.nordberg@umu.se, August 27, 2015 (14 : 14)
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