Element-free elastoplastic solid for nonsmooth Method Solver - - PowerPoint PPT Presentation

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Element-free elastoplastic solid for nonsmooth Method Solver - - PowerPoint PPT Presentation

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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

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Background Method Solver Elastic results Plastic method Plastic results

Element-free elastoplastic solid for nonsmooth multidomain dynamics

John Nordberg

john.nordberg@umu.se

UMIT Research Lab - Ume˚ a University

August 27, 2015

john.nordberg@umu.se, August 27, 2015 (1 : 14)

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Background Method Solver Elastic results Plastic method Plastic results john.nordberg@umu.se, August 27, 2015 (2 : 14)

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Background Method Solver Elastic results Plastic method Plastic results

Approaches

Solid mechanics Multibody system dynamics

ρ¨ u − ∇ · σ = ρb σ = CE M¨ q − GT λ = f ελ + g (q) = 0

john.nordberg@umu.se, August 27, 2015 (3 : 14)

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Background Method Solver Elastic results Plastic method Plastic results john.nordberg@umu.se, August 27, 2015 (4 : 14)

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Background Method Solver Elastic results Plastic method Plastic results

Displacement u(x) q(x) = x + u(x) J ≡ ∇xq(x) = I + ∇xu Green-Lagrange strain tensor E(x) = 1 2

  • JT J − I
  • = 1

2

  • ∇xu + ∇T

xu + ∇T xu∇xu

  • john.nordberg@umu.se, August 27, 2015

(5 : 14)

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Background Method Solver Elastic results Plastic method Plastic results

Moving least squares (MLS) approximation of the displacement field u (x)

uα(x) =

Np

  • j

Ψj(x)uj

α

Ψj(x) = pγ(x)A−1

γτ(x)pτ(xj)W

  • x − xj, h
  • Aγτ(x) =

Np

  • j

W

  • x − xj, h
  • pγ(xj)pτ(xj)

p(x) =

  • 1, x, y, z, yz, xz, xy, x2, y2, z2T

john.nordberg@umu.se, August 27, 2015 (6 : 14)

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Background Method Solver Elastic results Plastic method Plastic results

Constraint energy U = 1 2 gT ε−1g Constitutive relation using Voigt notation σ = CE C =        λ + 2µ λ λ λ λ + 2µ λ λ λ λ + 2µ µ µ µ        Strain energy U = 1 2 Ev0CE Elasticity strain tensor constraint and regularisation - 6D gi(q) = Ei ε = (v0C)−1 Jacobian where Kτη ≡ ∇τuη Gi

αβ(q) = ∂gi α

∂qβ = ∂gi

α

∂Eγ ∂Eγ ∂Kτη ∂Kτη ∂qβ

john.nordberg@umu.se, August 27, 2015 (7 : 14)

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Background Method Solver Elastic results Plastic method Plastic results

Multibody dynamics - numerical solver

Linearized varational time stepper SPOOK (Lacoursi` ere [2, 3]) qn+1 = qn + h ˙ qn+1   M −GT − ¯ GT G Σ ¯ G ¯ Σ  

  • H

  ˙ qn+1 λ ¯ λ  

  • z

=   M ˙ qn + hfn − 4

hΥg + ΥG ˙

qn ωn  

  • −r

regularization and stabilization matrices

Σ = 4 h2 diag

  • ε1

1 + 4 τ1

h

, ε2 1 + 4 τ2

h

, . . .

  • ¯

Σ = 1 h diag (γ1, γ2, . . . ) Υ = diag

  • 1

1 + 4 τ1

h

, 1 1 + 4 τ2

h

, . . .

  • ε = (v0C)−1

john.nordberg@umu.se, August 27, 2015 (8 : 14)

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Background Method Solver Elastic results Plastic method Plastic results

Nonsmooth MBD - numerical solver

Including frictional contacts, impacts, joint ant motor limits lead to limits and complementarity conditions on the solution 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)

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Background Method Solver Elastic results Plastic method Plastic results −0.1 −0.05 0.05 0.1 0.15 −0.1 −0.05 0.05 0.1 0.15 λ − 1 σ/c hydrostatic compression ← → uniaxial stretch analytic solution simulation np = 113 simulation np = 63 john.nordberg@umu.se, August 27, 2015 (10 : 14)

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Background Method Solver Elastic results Plastic method Plastic results

Elastoplastic terrain model

Elastic and plastic strain components E = Ee + Ep Plastic flow rule dEp = dλp ∂Φ

∂σ when yield Φ(σ) > 0

√J2

I1 I1

T

  • I1(κ)

C

  • κ
  • κ0

ɸ (I1, J2) = 0

DP

ɸ (I1, J2, κ) = 0

C

ɸ (I1, J2) = 0

T

ɸ< 0

Capped Drucker-Prager plasticity model (Dolarevic [1])

Φ (σ, κ) =      Φt (I1, J2) I1 It

1

Φe (I1, J2) It

1 I1 Ic 1 (κ)

Φc (I1, J2, κ (tr Ep)) I1 Ic

1 (κ)

(1)

john.nordberg@umu.se, August 27, 2015 (11 : 14)

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Background Method Solver Elastic results Plastic method Plastic results john.nordberg@umu.se, August 27, 2015 (12 : 14)

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Background Method Solver Elastic results Plastic method Plastic results

References

[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

  • f Multibodies with Dry Frictional Contacts, PhD thesis, Ume˚

aUniversity, Sweden, (2007) john.nordberg@umu.se, August 27, 2015 (13 : 14)

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Background Method Solver Elastic results Plastic method Plastic results

Elastoplastic terrain model

Drucker-Prager for cohesive soil Φe (I1, J2) =

  • J2 + η (φ) I1

3 − ξ (φ) c (2) Tension cap Φt (I1, J2) = (I1 − T + Rt)2 + J2 − R2

t

(3) Compression cap Φc (I1, J2, κ (tr εp)) = (I1 − a (κ))2 R2 + J2 − b (κ)2 (4) Cap variables - for compressive hardening κ = κ0 + 1 D ln

  • 1 + tr (ǫp)

W

  • (5)

where κ0 is the initial position of the compression cap (Dolarevic [1]).

john.nordberg@umu.se, August 27, 2015 (14 : 14)