1. Institute of Engineering Mechanics, KIT Diploma Thesis Daniel Tameling Dipl.-Ing. Stephan Wulfinghoff Prof. Dr.-Ing. Thomas Böhlke Chair for Continuum Mechanics Institute of Engineering Mechanics Algorithms for nonlocal material laws in a gradient-theory of single-crystal plasticity TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAAA Outline Introduction Mathematical background Algorithms Comparing the algorithms Conclusion D. Tameling KIT Karlsruhe Institute of Technology 19 th September 2011 1

2. Institute of Engineering Mechanics, KIT Introduction Mathematical background Algorithms Comparing the algorithms Conclusion D. Tameling KIT Karlsruhe Institute of Technology 19 th September 2011 2 Introduction Outline Introduction Mathematical background Algorithms Comparing the algorithms Conclusion

3. Institute of Engineering Mechanics, KIT D. Tameling KIT Karlsruhe Institute of Technology 19 th September 2011 3 Motivation At dimensions smaller than approx. 10 µm there is a size dependency of plasticity Fleck et al. (1994) d1d1 d 2 <d 1 Not predicted by conventional theory Nonlinear variational formulation Finite-Element-Method with Newton‘s-method Active Set Search Gradient-theory related to dislocations especially at inhomogeneous deformation like torsion Possible solution: Outline Introduction Mathematical background Algorithms Comparing the algorithms Conclusion

4. Institute of Engineering Mechanics, KIT Kinematics of a single-crystal Decomposition of deformation gradient Single-crystal with small deformations D. Tameling KIT Karlsruhe Institute of Technology 19 th September 2011 4 rotation + lattice deformation plastic shearing One active slip-system: slip-parameter slip direction slip normal Schmid tensor: Elastic part of the displacenent gradient Gurtin, Needleman (2005) Outline Introduction Mathematical background Algorithms Comparing the algorithms Conclusion

5. Institute of Engineering Mechanics, KIT 5 D. Tameling KIT Karlsruhe Institute of Technology 19 th September 2011 Motivation Nye’s dislocation tensor After plastic deformation Reference placement Single-crystal Continuum Burgers-vektor: dislocation density Stokes’ theorem Nye (1953) Outline Introduction Mathematical background Algorithms Comparing the algorithms Conclusion

6. Institute of Engineering Mechanics, KIT D. Tameling KIT Karlsruhe Institute of Technology 19 th September 2011 6 Helmholtz free energy hardening modulus Hardening part: Elastic part: Total free energy: with Nye’s dislocation tensor Dislocation part: constant stiffness tensor Outline Introduction Mathematical background Algorithms Comparing the algorithms Conclusion

7. Institute of Engineering Mechanics, KIT D. Tameling KIT Karlsruhe Institute of Technology 19 th September 2011 Implementation Nonlinear variational formulation Newton’s-method Linearization Principle of virtual power Solution? Nonlinear finite-element-method System of linear equations Which nodes are active plastic? Active Set Search Equations for slip-parameter in inactive nodes are removed from the system of linear equations 7 Outline Introduction Mathematical background Algorithms Comparing the algorithms Conclusion

8. Institute of Engineering Mechanics, KIT D. Tameling KIT Karlsruhe Institute of Technology 19 th September 2011 8 System of linear equations Active Set Search constraints due to plasticity passive node active node becomes passive when becomes active when Active Set Search: Different ways of combining Active Set Search and Newton‘s method symmetric + positive-definite System of linear equations: Active Set: Set of all active nodes Miehe, Schröder (2001) Scope of the Diploma Thesis: Outline Introduction Mathematical background Algorithms Comparing the algorithms Conclusion

9. Institute of Engineering Mechanics, KIT Yes No D. Tameling KIT Karlsruhe Institute of Technology 19 th September 2011 9 Algorithms Method 3Method 2Method 1 Initialization Find exact solution Constraints violated? Change Active Set Solution found Yes No Initialization One Newton step Constraints violated? Change Active Set continue with old solution Solution accurate? YesNo Solution found Yes No Initialization One Newton step Constraints violated? Change Active Set continue with new solution Solution accurate? YesNo Solution found Outline Introduction Mathematical background Algorithms Comparing the algorithms Conclusion

10. Institute of Engineering Mechanics, KIT Grids 11x11x6 =726 nodes 26x26x14 =9464 nodes 10 D. Tameling KIT Karlsruhe Institute of Technology 19 th September 2011 Simulation Boundary conditions lower surface fixed upper surface is moved slip-parameter is zero on the entire boundary coarse grid fine grid Outline Introduction Mathematical background Algorithms Comparing the algorithms Conclusion

11. Institute of Engineering Mechanics, KIT D. Tameling KIT Karlsruhe Institute of Technology 19 th September 2011 11 Simulation Reference placement Simulations:u max =0,03µm with 10 time steps and fine grid u max =0,3µm with 4 und 10 time steps and coarse and fine grid Outline Introduction Mathematical background Algorithms Comparing the algorithms Conclusion

12. Institute of Engineering Mechanics, KIT D. Tameling KIT Karlsruhe Institute of Technology 19 th September 2011 12 Results 10 time steps, xz-plane displacement with scale factor 100 displacement with scale factor 20 Outline Introduction Mathematical background Algorithms Comparing the algorithms Conclusion

13. Institute of Engineering Mechanics, KIT D. Tameling KIT Karlsruhe Institute of Technology 19 th September 2011 13 Comparing the algorithms Number of Newton steps determines time consumption of a method Method 1Method 2Method 3 Sum of the number of all Newton steps from all simulations Method 1 is the slowest Outline Introduction Mathematical background Algorithms Comparing the algorithms Conclusion

14. Institute of Engineering Mechanics, KIT Method 2 it is only 51 times done instead of 101 times at Method 3 Why is the number of Newton steps determining the time consumption? D. Tameling KIT Karlsruhe Institute of Technology 19 th September 2011 14 Comparing the algorithms Method 1Method 2Method 3 Method 2 is the fastest Setting the system of linear equations up is very expensive This is for Method 2 only necessary if the Active Set is not changed Sum of the number of all changes of the Active Set from all simulations Outline Introduction Mathematical background Algorithms Comparing the algorithms Conclusion

15. Institute of Engineering Mechanics, KIT D. Tameling KIT Karlsruhe Institute of Technology 19 th September 2011 15 Conclusion Method 1Method 2Method 3 Stability ooo Number of Active Set Searches ooo Number of Newton steps - ++ Speed - +o Result - +o Outline Introduction Mathematical background Algorithms Comparing the algorithms Conclusion

16. Institute of Engineering Mechanics, KIT D. Tameling KIT Karlsruhe Institute of Technology 19 th September 2011 16 Thank you for your attention! Outline Introduction Mathematical background Algorithms Comparing the algorithms Conclusion