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Weierstrass Institut Berlin7.November 2006
• Motivation
• Kinematic Relations
• Constitutive Model
• Numerical Treatment
Constitutive Parameters for a NonlinearCosserat Theory
• Simple Glide
• Torsion Test
• Imperfection Algorithm
• Compression Test
• Conclusions and Outlook
I. Münch , W. WagnerKarlsruhe University of Technology
Institute of Structural Analysis
P. NeffDarmstadt University of Technology
Department of Mathematics
Motivation
Applications for Cosserat continua:
• continua with periodicalmicrostructure
• binary media (suspensions)
• plasticity
• …
• stress concentration
• foams
Motivation
homogeneousmodel without
orientation
homogeneousmodel withorientation
regularization
size effects
foam-like behaviour
Boltzmann continua
Cosserat continua
effects of (strong) inner
structure
additional kinematic
Kinematic Relations
Relations for Cosserat theory (e.g. Ehlers, Bluhm [1]):
first Cosserat strain:
second Cosserat strain: (curvature)
macrorotation
additional kinematic
Kinematic Relations
1) linearization:
Euler-Rodrigues formula:
Relations and linearizations for Cosserat theory (e.g. Ehlers, Bluhm [2]):
first Cosserat strain:
2) linearization:
second Cosserat strain:
3) linearization:
(curvature)pull back, similar to
Constitutive Model
Quadratic ansatz in first Cosserat strain:
Cosserat couple modulus penalizesdifferences of microrotations to macrorotations:
Experiment
Deformation modeluniaxial strain
?
Constitutive Model
phenomenologicalparameter of
inner structure: Lc
acts liketorsional spring
influences angularmomentum
Nonlinear ansatz for curvature energy:
Lc penalizes curvaturecurvature increases forsmall structureshigher stiffness for smallerstructures
Linear Cosserat Model
Free Helmholtz energy:
which turns for the linear Cosserat model into:
linear isotropic theory decouples for
Numerical Treatment
• variational formulation and nonlinear 3-d finite element model
• Lagrangean description
• consistent linearization for stiffness matrix and Newtons strategy
• 8 / 27 node brick elements with trilinear / triquadratic shape functions for both fields
• system of algebraic equations:
• additive update of displacements and infinitesimal microrotations for linear theory
• multiplicative update of microrotations for nonlinear theory (Sansour,Wagner [2])
Simple glide
Motivation: from planar shear to simple glide
planar: no displacementsin 2-direction
different zones of deformation indicate
simple glide zone
no displacementsin 3-direction
+no gradients in 1-
direction
hexagonal or quadrilateral
Simple glide
no displacements in 2- and 3-direction
Analytical investigationsin simple glide:
displacements in 1-direction linked in 1-2-plane no gradients in 1-direction
maximal shear
Torsion test
deformedmesh
boundaries
no displacementsin all directions
no displacementsin 3-direction
Constant sample values:
all resultsconcernthis point
microrotationsfixed to zero
upper border rotates
torque
From now on: Nonlinear Cosserat theory
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
0 0,1 0,2 0,3 0,4 0,5 0,6lateral macro-twist (boundary condition is true rotation in the twist)
torque
St.V.-Kirchhoff
L = 10
L = 1
L = 0.1
L = 0.01
L = 0.001
L = 0.0001
L = 0
Torsion test
results concernthis point
c
c
c
c
c
c
cmicrorotationsfixed to zero
Torsion test
results concernthis point
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
0 0,1 0,2 0,3 0,4 0,5 0,6lateral macro-twist (boundary condition is true rotation in the twist)
torque
St.V.-Kirchhoffμ = 10 μ
μ = 1 μ
μ = 0.1 μ
μ = 0.01 μ
μ = 0.003 μ
μ = 0.001 μ
μ = 0.0001 μ
μ = 0.00001 μ
c
c
c
c
c
c
c
cmicrorotationsfixed to zero
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8lateral macro-twist (boundary condition is true rotation in the twist)
torque
St.V.-Kirchhoff
L = 100L = 10
L = 1
L = 0.1L = 0.01
Torsion test
results concernthis point
c
c
c
c
c
microrotationsfixed to zero
Conclusions in Torsion Test
curvature energyshould be responsiblefor length scaleeffects!
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
0 0,1 0,2 0,3 0,4 0,5 0,6lateral macro-twist (boundary condition is true rotation in the twist)
torque
St.V.-Kirchhoff
μ = 10 μ
μ = 1 μ
μ = 0.1 μ
μ = 0.01 μ
μ = 0.003 μ
μ = 0.001 μ
μ = 0.0001 μ
μ = 0.00001 μ
cc
cc
cc
cc
variation of looks like length scale effect
Imperfection Algorithm
homogeneousmodel without
orientation
homogeneousmodel withorientation
stochasticrotational
imperfections
Boltzmann continua
Cosserat continua
For perfect samples and perfectboundary conditions the full
spectrum of possible solutionscan often not be reached
numerically disturb perfectsituations
Compression test
d=0.2
stochastic rotationalimperfections
microrotationsfixed to zero
no displacementsin all directions
displacementonly in
3-direction
Constant sample values:
deformedmesh
boundariesR
twist function
of LC
-4,0E-01
-2,0E-01
0,0E+00
2,0E-01
4,0E-01
0,0001 0,001 0,01 0,1 1 10internal length scale L c
twist [rad]
macro-rotmicro-rot
Compression test
Rotationsmeasured athalf height
Macrorotation arround vertical axis for various internal length scale factors
Boltzmann continua
Compression test
1,0E-03
1,0E-01
1,0E+01
1,0E+03
1,0E+05
0,0001 0,001 0,01 0,1 1 10internal length scale L c
energy
straincurv
1,2E+03
1,3E+03
1,4E+03
1,5E+03
1,6E+03
0,0001 0,001 0,01 0,1 1 10internal length scale L c
energy
totalstrain
• curvature energy only of secondorder (maximal 4% of strain energy)
logarithmicscale
• total energy increases for increasinginternal length scale factor – butnot arbitrary
• pronounced length scale effects for
0
2
4
6
8
10
12
14
16
18
20
0 0,05 0,1 0,15 0,2displacement d
reaction force R
St.Venant-K.S.V.K. linearNeo-HookeL = 10L = 1L = 0.1L = 0.03L = 0.01L = 0.001L = 0.0001
Compression test
Size effects not as significant as in torsion test as expected !!!
ccccccc
-6.661E-03 min-5.709E-03-4.758E-03-3.806E-03-2.854E-03-1.902E-03-9.501E-041.731E-069.536E-041.905E-032.857E-033.809E-034.761E-035.713E-036.665E-03 max
1 2
3
Compression test
Compression test with various slendernessratios (different aspect ratio), constant internallength scale factor and constant maximal straind / h = 10 %
h=1.0: symmetricdeformation
-5.078E-02 min-4.354E-02-3.629E-02-2.904E-02-2.179E-02-1.454E-02-7.291E-03-4.232E-057.207E-031.446E-022.170E-022.895E-023.620E-024.345E-025.070E-02 max
1 2
3
-4.181E-01 min-3.880E-01-3.578E-01-3.277E-01-2.976E-01-2.674E-01-2.373E-01-2.072E-01-1.770E-01-1.469E-01-1.167E-01-8.661E-02-5.647E-02-2.633E-023.804E-03 max1 2
3
h=2.0: deformationwith twist
h=4: deformation withtwist and buckling
deformed mesh and coloured displacement in 1-direction
Compression test
Determinant of global stiffness matrix; various heights of structure
1,0E-02
1,0E-01
1,0E+00
1,0E+01
1,0E+02
1,0E+03
1,0E+04
1,0E+05
1,0E+06
1,0E+07
1,0E+08
0,00 0,02 0,04 0,06 0,08 0,10d / h
detK
/ de
tK_0
h=0.4h=1h=1.5h=2h=4h=8
h=8 h=
4
h=2
h=1.
5
h=1
h=0.
4
buckling
twist
02468
101214161820
0,00 0,02 0,04 0,06 0,08 0,10d / h
reac
tion
forc
e R
h=0.4h=1h=1.5h=2h=4h=8
sizeeffect
Compression test
Reaction force for various heights of structure
buckling
size effect (caused by twist) is much less significant than bucklingsize effect hardly measureable in practical tests
Conclusions and Outlook
Outlook: Extension to micromorphic theory (simulation of foams)
Open question: Right choice of boundary condition for microrotations (is there a physical interpretation of consistent coupling?)
Weierstrass Institut Berlin7.November 2006
END
References:[1] W. Ehlers, J. Bluhm: Porous Media – Theory, Experiments and Numerical Applications,
Springer 2002[2] C. Sansour, W. Wagner: Multiplicative updating of the rotation tensor in the finite
element analysis – a path independent approach, Comp. Mech. 31, Springer 2003