Continuum damage mechanics of geomaterials at finite strain

1. Continuum damage mechanics ofgeomaterials at finite strain MDU A. Karrech, Research Scientist, CSIRO K. Regenauer-Lieb, T. Poulet, P. Schaubs, Y, Zhang 29 September 2010

2. Outline 1 Background Motivation Current approach 2 Elasto-visco-plasticity at finite strain Multiplicative decomposition Constitutive relations 3 Damage mechanism Void growth under several control mechanisms The limit theory approximation 4 Validation / Application Validation of the large transformations model Damage of a notched plate and effects of pressure Chemo-thermo-hydro-mechanics (See Thomas Poulet) Damage down under (See Peter Schaubs) 5 Summary

3. Instabilities Large transformations to describe earth systems instabilities Damage at Finite Strain

4. Material Softening • The predicted forces for splitting continents apart are much higher then available from plate tectonics. • Time and length scales can’t be achieved in the laboratory. • Regenauer-Lieb et al 06, Nature Damage at Finite Strain

5. Outline 1 Background Motivation Current approach 2 Elasto-visco-plasticity at finite strain Multiplicative decomposition Constitutive relations 3 Damage mechanism Void growth under several control mechanisms The limit theory approximation 4 Validation / Application Validation of the large transformations model Damage of a notched plate and effects of pressure Chemo-thermo-hydro-mechanics (See Thomas Poulet) Damage down under (See Peter Schaubs) 5 Summary

6. Finite strain -- Review • Additive strain rate decomposition (similar to small deformations): Green Naghdi(65), Mandel (72) , Nemat-Nasser (81)... • Multiplicative gradient decomposition: Lee and Liu(67), Lee (69) • Numerical integration: Simo et al. (80s-94), Argyris and Doltsinis(80s), Miehe(90s) • Several inconsistencies (aberrant oscillations observed by Dienes (79) Simo and Pister (82), K. Regenauer-Lieb and H. Mulhaus (06)…) • Logarithmic corotational rates: Xiao, Buhrns Meyers (98-06) • Metallic materials: Lin, Brocks, Betten (02,04,06) • Formulation + numerical integration for geomaterials: current work

7. Finite strain – Basic concept • Small perturbations: • (+) Well understood + Easy integration • (-) Limitations in predicting instabilities Large transformations:

8. Finite strain – Oscillations How to formulate thermo-mechanical coupled models for frictional materials in finite strain How to overcome these spurious oscillations? Source of the figure: www.wikepidia.com

9. Decomposition The deformation gradient is: Hence, the multiplicative decomposition: We consider the measure of athermal strain:

10. Objective rates

11. Objective rates

12. Dissipation inequality

13. Helmholtz F. E. and dissipation

14. Helmholtz F. E. and dissipation

15. Principle of maximum dissipation

16. Outline 1 Background Motivation Current approach 2 Elasto-visco-plasticity at finite strain Multiplicative decomposition Constitutive relations 3 Damage mechanism Void growth under several control mechanisms The limit theory approximation 4 Validation / Application Validation of the large transformations model Damage of a notched plate and effects of pressure Chemo-thermo-hydro-mechanics (See Thomas Poulet) Damage down under (See Peter Schaubs) 5 Summary

17. Micro-scale model A. C. F. Cocks and M. F. Ashby, progress in materials science, 1982, Vol. 27, pp. 189 to 244

18. Comparison with other damage models For small f, Cocks and Ashby models coincide with the descriptions of Kachanov (58) and Lemaitre and chaboche (80s) The comparison highlights what we believe to be certain fundamental weaknesses of the continuum equations: first, the prediction that the damage-rate is finite even when there is no damage; second, the prediction that the damage-rate always accelerates with damage;

19. Current approach • Assumptions: • Vacancies within a given RVE are assumed to be within a spacing of min(2d, 2L), • (d and L are distances in the longitudinal and radial directions) • Voids are assumed to be of small size as compared to the • Voids are self-similar in terms of shape during the deformation process. • Upper limit (MARTIN, JMPS, 62)

20. Current approach After Integration (Karrech el al., ICAMEM Conference 2010) Similarly to Dahar et al (1996), we add a nucleation effect (no justification yet) Integration with respect to the thermodynamic force of damage:

21. Outline 1 Background Motivation Current approach 2 Elasto-visco-plasticity at finite strain Multiplicative decomposition Constitutive relations 3 Damage mechanism Void growth under several control mechanisms The limit theory approximation 4 Validation / Application Validation of the large transformations model Damage of a notched plate and effects of pressure Chemo-thermo-hydro-mechanics (See Thomas Poulet) Damage down under (See Peter Schaubs) 5 Summary

22. Axially loaded sample

23. Axially loaded sample

24. Simple Shear

25. Simple Shear in hyer-elasto-plasticity

26. Necking problem Good agreement between the experimental and numerical results

27. Triaxial test

28. Damage of a notched plate (Olivine)

29. Effect of pressure dependency

30. Effect of pressure dependency Courtesy of Arcady Dyskin, UWA

31. Outline 1 Background Motivation Current approach 2 Elasto-visco-plasticity at finite strain Multiplicative decomposition Constitutive relations 3 Damage mechanism Void growth under several control mechanisms The limit theory approximation 4 Validation / Application Validation of the large transformations model Damage of a notched plate and effects of pressure Chemo-thermo-hydro-mechanics (See Thomas Poulet) Damage down under (See Peter Schaubs) 5 Summary

32. Chemo-thermo-hydro-mechanics

33. Chemo-thermo-hydro-mechanics Permeability evolution with damage

34. Chemo-thermo-hydro-mechanics Fluid flow through damaged zones Preliminary chemistry

35. Invitation I invite you to talk to Thomas Poulet for more details about multi-physics Problems

36. Outline 1 Background Motivation Current approach 2 Elasto-visco-plasticity at finite strain Multiplicative decomposition Constitutive relations 3 Damage mechanism Void growth under several control mechanisms The limit theory approximation 4 Validation / Application Validation of the large transformations model Damage of a notched plate and effects of pressure Chemo-thermo-hydro-mechanics (See Thomas Poulet) Damage down under (See Peter Schaubs) 5 Summary

37. Damage & thermo-coupling • The Late Archaean Yilgarn Craton of Western Australia hosting orogenic gold deposits • Different loading scenarios

40. Invitation I invite you to talk to Peter Schaubs for more details about the field application

41. Outline 1 Background Motivation Current approach 2 Elasto-visco-plasticity at finite strain Multiplicative decomposition Constitutive relations 3 Damage mechanism Void growth under several control mechanisms The limit theory approximation 4 Validation / Application Validation of the large transformations model Damage of a notched plate and effects of pressure Chemo-thermo-hydro-mechanics (See Thomas Poulet) Damage down under (See Peter Schaubs) 5 Conclusions

42. Outline • Finite strain for geo-materials based on logarithmic strain measures and corotational rates. • Solution for the spurious oscillations • Continuum damage mechanics following based on approximate potential • Instabilities and localizations are accelerated in such circumstances • Multi-physics problems in the context of mining

43. Computational Geoscience Group Dr Ali Karrech Research Scientist @ CSIRO Adjunct Associate Professor @ UWA Phone: +61 8 64 36 86 96 Email: ali.karrech@csiro.au Web: www.csiro.au Thank you