Institute of Engineering Mechanics, KIT Diploma Thesis Daniel Tameling Dipl.-Ing. Stephan Wulfinghoff Prof. Dr.-Ing. Thomas Böhlke Chair for Continuum.

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 20111


• Introduction

• Mathematical background

• Algorithms

• Comparing the algorithms

• Conclusion



IntroductionOutline

Introduction


Algorithms


Conclusion




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


Algorithms


Conclusion


Kinematics of a single-crystal

Decomposition of deformation gradient

Single-crystalwith small deformations



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


Algorithms


Conclusion


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


Algorithms


Conclusion




Helmholtz free energy

hardening modulus

Hardening part:

Elastic part:

Total free energy:

with Nye’s dislocation tensor

Dislocation part:

constant

stiffness tensor

Outline

Introduction


Algorithms


Conclusion



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


Algorithms


Conclusion




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


Algorithms


Conclusion


YesNo



Algorithms

Method 3Method 2Method 1

Initialization

Findexact

solution

Constraintsviolated?

Change Active Set

Solutionfound

YesNo

Initialization

OneNewton

step


Change Active

Setcontinuewith oldsolution

Solutionaccurate?

YesNo

Solutionfound

YesNo

Initialization

OneNewton

step


Change Active

Setcontinuewith newsolution

Solutionaccurate?

YesNo

Solutionfound

Outline

Introduction


Algorithms


Conclusion


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


Algorithms


Conclusion




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


Algorithms


Conclusion




Results10 time steps, xz-plane

displacement withscale factor 100

displacement withscale factor 20

Outline

Introduction


Algorithms


Conclusion




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


Algorithms


Conclusion


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

Why is the numberof Newton stepsdetermining the

time consumption?



Comparing the algorithms


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


Algorithms


Conclusion




Conclusion


Stability o o o

Number of Active Set Searches o o o

Number of Newton steps - + +

Speed - + o

Result - + o

Outline

Introduction


Algorithms


Conclusion




Thank youfor your attention!

Outline

Introduction


Algorithms


Conclusion

Institute of Engineering Mechanics, KIT Diploma Thesis Daniel Tameling Dipl.-Ing. Stephan Wulfinghoff Prof. Dr.-Ing. Thomas Böhlke Chair for Continuum.

Documents

algorithms conclusiond

diploma thesis daniel

nonlocal material laws

thomas bhlkechair

texpoint manual

stephan wulfinghoffprof