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

Institute of Engineering Mechanics, KIT

Diploma Thesis

Daniel Tameling

Dipl.-Ing. Stephan WulfinghoffProf. Dr.-Ing. Thomas Böhlke

Chair for Continuum MechanicsInstitute of Engineering Mechanics

Algorithms for nonlocal material laws in a gradient-theory of single-crystal plasticityOutline

Introduction

Mathematicalbackground

Algorithms

Comparing thealgorithms

Conclusion

D. Tameling KIT Karlsruhe Institute of Technology 19th September 2011

1

Institute of Engineering Mechanics, KIT

• Introduction

• Mathematical background

• Algorithms

• Comparing the algorithms

• Conclusion

D. Tameling KIT Karlsruhe Institute of Technology 19th September 2011

2

IntroductionOutline

Introduction

Mathematicalbackground

Algorithms

Comparing thealgorithms

Conclusion

Institute of Engineering Mechanics, KIT

D. Tameling KIT Karlsruhe Institute of Technology 19th September 2011

3

Motivation

At dimensions smaller than approx. 10 µmthere is a size dependency of plasticity

Fleck et al. (1994)

d1

d2 <d1

Not predicted byconventional theory

Nonlinearvariationalformulation

Finite-Element-Methodwith

Newton‘s-methodActive Set Search

Gradient-theoryrelated to

dislocations

especially at inhomogeneous deformation like torsion

Possible solution:

Outline

Introduction

Mathematicalbackground

Algorithms

Comparing thealgorithms

Conclusion

Institute of Engineering Mechanics, KIT

Kinematics of a single-crystal

Decomposition of deformation gradient

Single-crystalwith small deformations

D. Tameling KIT Karlsruhe Institute of Technology 19th September 2011

4

rotation + lattice deformation plastic shearing

One active slip-system:

slip-parameter

slip direction

slip normal

Schmid tensor:

Elastic part of thedisplacenent gradient

Gurtin, Needleman (2005)

Outline

Introduction

Mathematicalbackground

Algorithms

Comparing thealgorithms

Conclusion

Institute of Engineering Mechanics, KIT

5D. Tameling KIT Karlsruhe Institute of Technology 19th September 2011

Motivation Nye’s dislocation tensor

After plasticdeformation

Referenceplacement

Single-crystal

Continuum

Burgers-vektor:

dislocation densityStokes’theorem

Nye (1953)

Outline

Introduction

Mathematicalbackground

Algorithms

Comparing thealgorithms

Conclusion

Institute of Engineering Mechanics, KIT

D. Tameling KIT Karlsruhe Institute of Technology 19th 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

Mathematicalbackground

Algorithms

Comparing thealgorithms

Conclusion

Institute of Engineering Mechanics, KIT

D. Tameling KIT Karlsruhe Institute of Technology 19th September 2011

Implementation

Nonlinear variational formulation

Newton’s-method

Linearization

Principle of virtual power

Solution?

Nonlinear finite-element-method

System oflinear equations

Which nodes areactive plastic?

Active Set Search

Equations for slip-parameterin inactive nodes are removed

from the system of linear equations

7

Outline

Introduction

Mathematicalbackground

Algorithms

Comparing thealgorithms

Conclusion

Institute of Engineering Mechanics, KIT

D. Tameling KIT Karlsruhe Institute of Technology 19th September 2011

8

System of linear equationsActive Set Search

constraints due to plasticity

passive node active node

becomes passivewhen

becomes activewhen

Active Set Search:

Different ways of combiningActive 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

Mathematicalbackground

Algorithms

Comparing thealgorithms

Conclusion

Institute of Engineering Mechanics, KIT

YesNo

D. Tameling KIT Karlsruhe Institute of Technology 19th September 2011

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Algorithms

Method 3Method 2Method 1Initialization

Findexact

solution

Constraintsviolated?

Change Active Set

Solutionfound

YesNo

Initialization

OneNewton

step

Constraintsviolated?

Change Active

Setcontinuewith oldsolution

Solutionaccurate?

YesNo

Solutionfound

YesNo

Initialization

OneNewton

step

Constraintsviolated?

Change Active

Setcontinuewith newsolution

Solutionaccurate?

YesNo

Solutionfound

Outline

Introduction

Mathematicalbackground

Algorithms

Comparing thealgorithms

Conclusion

Institute of Engineering Mechanics, KIT

Grids11x11x6

=726 nodes26x26x14

=9464 nodes

10D. Tameling KIT Karlsruhe Institute of Technology 19th September 2011

Simulation

Boundary conditionslower surface fixed

upper surface is moved

slip-parameteris zero on the

entire boundary

coarse grid fine grid

Outline

Introduction

Mathematicalbackground

Algorithms

Comparing thealgorithms

Conclusion

Institute of Engineering Mechanics, KIT

D. Tameling KIT Karlsruhe Institute of Technology 19th September 2011

11

Simulation

Reference placement

Simulations: umax=0,03µm with 10 time steps and fine gridumax=0,3µm with 4 und 10 time steps and coarse and fine grid

Outline

Introduction

Mathematicalbackground

Algorithms

Comparing thealgorithms

Conclusion

Institute of Engineering Mechanics, KIT

D. Tameling KIT Karlsruhe Institute of Technology 19th September 2011

12

Results10 time steps, xz-plane

displacement withscale factor 100

displacement withscale factor 20

Outline

Introduction

Mathematicalbackground

Algorithms

Comparing thealgorithms

Conclusion

Institute of Engineering Mechanics, KIT

D. Tameling KIT Karlsruhe Institute of Technology 19th September 2011

13

Comparing the algorithms

Number of Newton stepsdetermines time consumption

of a method

Method 1 Method 2 Method 3

Sum of thenumber of allNewton steps

from allsimulations

0

20

40

60

80

100

120

140

160

Method 1 is the slowest

Outline

Introduction

Mathematicalbackground

Algorithms

Comparing thealgorithms

Conclusion

Institute of Engineering Mechanics, KIT

Method 2 it is only 51 times doneinstead of 101 times at Method 3

Why is the numberof Newton stepsdetermining the

time consumption?

D. Tameling KIT Karlsruhe Institute of Technology 19th September 2011

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Comparing the algorithms

Method 1 Method 2 Method 3

Method 2 is the fastest

0

5

10

15

20

25

30

35

40

45

50

Setting thesystem of linearequations up isvery expensive

This is for Method 2only necessary if

the Active Setis not changed

Sum of thenumber of all

changes of theActive Setfrom all

simulations

Outline

Introduction

Mathematicalbackground

Algorithms

Comparing thealgorithms

Conclusion

Institute of Engineering Mechanics, KIT

D. Tameling KIT Karlsruhe Institute of Technology 19th September 2011

15

Conclusion

Method 1 Method 2 Method 3

Stability o o oNumber of Active

Set Searches o o oNumber of

Newton steps - + +Speed - + oResult - + o

Outline

Introduction

Mathematicalbackground

Algorithms

Comparing thealgorithms

Conclusion

Institute of Engineering Mechanics, KIT

D. Tameling KIT Karlsruhe Institute of Technology 19th September 2011

16

Thank youfor your attention!

Outline

Introduction

Mathematicalbackground

Algorithms

Comparing thealgorithms

Conclusion