Diploma Thesis Daniel Tameling Dipl.- Ing . Stephan Wulfinghoff Prof. Dr.- Ing . Thomas Böhlke

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

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

3. Motivation • Outline • Introduction • Mathematical • background • Algorithms • Comparing the • algorithms • Conclusion Fleck et al. (1994) At dimensions smaller than approx. 10 µm there is a size dependency of plasticity especially at inhomogeneous deformation like torsion d2 <d1 d1 Not predicted by conventional theory Possible solution: Finite-Element-Method with Newton‘s-method Active Set Search Gradient-theory related to dislocations Nonlinear variational formulation D. Tameling KIT Karlsruhe Institute of Technology 19th September 2011

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

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

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

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

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

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

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

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

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

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

14. Comparing the algorithms • Outline • Introduction • Mathematical • background • Algorithms • Comparing the • algorithms • Conclusion Why is the number of Newton steps determining the time consumption? 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 Method 1 Method 2 Method 3 Method 2 it is only 51 times done instead of 101 times at Method 3 Method 2 is the fastest D. Tameling KIT Karlsruhe Institute of Technology 19th September 2011

15. Conclusion • Outline • Introduction • Mathematical • background • Algorithms • Comparing the • algorithms • Conclusion D. Tameling KIT Karlsruhe Institute of Technology 19th September 2011

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