A coupled isotropic elasto-plastic damage model based on incremental minimization principles

Transcription

1 A coupled isotropic elasto-plastic damage model based on incremental minimization principles Olaf Kintzel*, Jörn Mosler GKSS Research Centre Geesthacht Materials Mechanics Simulation of Solids and Structures 1 st International Conference on Material Modeling Dortmund, Germany September 15-17, 2009 PAGE 1

2 Outline Motivation A novel ductile-brittle isotropic damage model Variational principle Incremental minimization principles Numerical example Uniaxial tensile test Conclusion / Outlook PAGE 2

3 Application fields of Aluminium 2024 Light-weight metallic materials in automobile or aeroplane structures In particular Al2024-alloy Fuselage of A380 Frames of motorcycles Valves or tubes High strength-to-weight ratio Increased fracture toughness PAGE 3

4 Microstructure of Al2024-plate (100mm thick) Manufacturing process of Al2024 plate Complex microstructure T L S 2000 m Different material properties in S-, L- and T-direction Material anisotropy and different plastic flow behavior (ductile/brittle) PAGE 4

5 Experimental observations - SEM stereo fractography - Number of cycles brittle vs. ductile Monotonic Cyclic Stress, MPa S-orientation T-orientation L-orientation 0 0,00 0,05 0,10 0,15 0,20 0,25 Material anisotropy brittle vs. ductile L-direction S-direction Strain, - PAGE 5

6 Motivation for current analysis Challenge: Modeling of ductile/brittle damage for Al2024-plate Simulate S-, L- and T-directions separately using isotropic damage variable Formulate analytical model as variational principle Robust numerical implementation (Numerical tools from Math-library) Variational principle straightforward for associative evolution law Not straightforward for non-associative evolution laws PAGE 6

7 Introduction to the isotropic damage model Ductile damage (Lemaitre 1990) Elastic and plastic parts used in the definition of Lemaitre dissipated PAGE 7 elasticitiy before yield limit

8 Introduction to the isotropic damage model Novel damage indicator function for brittle damage Damage growth independent of plastic evolution dissipated PAGE 8

9 Introduction to the isotropic damage model Strain equivalence principle (Chaboche & Lemaitre 1990) Fictive undamaged state Strains are equal Stiffnesses (effective stresses) are increased PAGE 9

10 A novel ductile-brittle isotropic damage model Helmholtz energy Plastic yield function PAGE 10

11 A novel ductile-brittle isotropic damage model Helmholtz energy Damage indicator function for quasi-brittle damage PAGE 11

12 A novel ductile-brittle isotropic damage model Modeling non-linear hardening with Armstrong-Frederick rules (Armstrong & Frederick 1966) Plastic and damage dissipation potentials Non-associative evolution equations Decomposition of the energy release rate into ductile/brittle PAGE 12

13 A novel ductile-brittle isotropic damage model Evolution laws considering generalized standard materials Plastic part Substitution: PAGE 13

14 A novel ductile-brittle isotropic damage model Brittle damage part Damage evolution law PAGE 14

15 A novel ductile-brittle isotropic damage model Hardening Integration by backward-euler First-order accurate Evolution rules integrated pointwise (at each Gauss point) Equations usually solved by iterative procedures Finding roots of residua E.g. Newton-solvers, Predictor-corrector steps, Return-map PAGE 15

16 A variational re-formulation An alternative solution technique Variational principle (locally) Considering dissipative processes within an extended variational principle Dislocation structures (Ortiz & Repetto 1999) Associative anisotropic finite strain plasticity (Miehe & Lambrecht 2001)... and many others Twinning in Magnesium (Mosler & Homayonifar 2009) Non-associative finite strain plasticity (Mosler 2009) PAGE 16

17 A variational re-formulation Physical state at minimal energy Considering internal variables Incremental energy minimization PAGE 17

18 Incremental energy minimization Plasticity with linear hardening Yield function homogeneous of order h=1 implicitly (like magic!) PAGE 18

19 Incremental energy minimization Plasticity with non-linear hardening (Mosler 2009) Backward-Euler integration Yield function not fulfilled exactly Cause: Non-associative terms First-order accurate Yield function fulfilled in the limit Consistent! PAGE 19

20 Incremental energy minimization Plasticity coupled with damage (Lemaitre) (linear hardening) Damage evolution law (backward-euler integration) Consistent? Variate incremental potential and test Euler-Lagrange equations PAGE 20

21 Incremental energy minimization Plasticity coupled with damage (Lemaitre) (linear hardening) Variation of Helmholtz energy: keep in mind: Variation of damage part: Variation: PAGE 21

22 Incremental energy minimization Plasticity coupled with damage (Lemaitre) (non-linear hardening) Variation: First-order accurate (non-associative terms), consistent for PAGE 22

23 Incremental energy minimization Plasticity coupled with damage (brittle) (linear hardening) Variation of Helmholtz-energy (damage part): Variation: PAGE 23

24 Incremental energy minimization Plasticity coupled with damage (brittle) (non-linear hardening) Variation: First-order accurate (non-associative terms), consistent for PAGE 24

25 Incremental energy minimization Fully coupled model (ductile/brittle) Possible only for linear damage evolution laws (for Lemaitre M=2) Variation: PAGE 25

26 Incremental energy minimization For non-linear damage evolution or more refined decompositions, e.g.: Multi-step methods Solving elasto-plasticity first (no coupling with damage) Solving brittle damage Finally damage evolution PAGE 26

27 Numerical example Uniaxial tensile test Material data: S 1 =0.001 S 1 = H d =20 H d =60 H d = Displacement u Displacement u 1 Ductile damage Brittle damage PAGE 27

28 Numerical example Uniaxial tensile test Material data: D 0.4 ductile brittle Sum Sum Decrease of Cauchy stress/cycle Damage evolution PAGE 28

29 Conclusion / Outlook Novel isotropic ductile/brittle damage model Variational principle for ductile damage (Lemaitre) First-order Variational principle for brittle damage First-order Future: Compare energy equivalence principle Anisotropic damage variable (structural tensor) PAGE 29

30 Acknowledgement Thanks go to: Shehzad Khan for experimental stuff Andriy Vyshnevskyy for bridging the gap Prof. Jörn Mosler for many discussions Malek Homayonifar for being such a nice guy Thank you for your attention PAGE 30