institute of engineering mechanics, kit diploma thesis daniel tameling dipl.-ing. stephan...
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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
Institute of Engineering Mechanics, KIT
• Introduction
• Mathematical background
• Algorithms
• Comparing the algorithms
• Conclusion
D. Tameling KIT Karlsruhe Institute of Technology
19th September 20112
IntroductionOutline
Introduction
Mathematicalbackground
Algorithms
Comparing thealgorithms
Conclusion
Institute of Engineering Mechanics, KIT
D. Tameling KIT Karlsruhe Institute of Technology
19th September 20113
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 20114
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 20116
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 20118
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 20119
Algorithms
Method 3Method 2Method 1
Initialization
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 201111
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 201112
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 201113
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 201114
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 201115
Conclusion
Method 1 Method 2 Method 3
Stability o o o
Number of Active Set Searches o o o
Number of Newton steps - + +
Speed - + o
Result - + o
Outline
Introduction
Mathematicalbackground
Algorithms
Comparing thealgorithms
Conclusion
Institute of Engineering Mechanics, KIT
D. Tameling KIT Karlsruhe Institute of Technology
19th September 201116
Thank youfor your attention!
Outline
Introduction
Mathematicalbackground
Algorithms
Comparing thealgorithms
Conclusion
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