Weierstrass Institut Berlin 7.November 2006
• Motivation
• Kinematic Relations
• Constitutive Model
• Numerical Treatment
Constitutive Parameters for a Nonlinear Cosserat Theory
• Simple Glide
• Torsion Test
• Imperfection Algorithm
• Compression Test
• Conclusions and Outlook I. Münch , W. Wagner
Karlsruhe University of Technology Institute of Structural Analysis P. Neff
Darmstadt University of Technology Department of Mathematics
Motivation
Applications for Cosserat continua:
• continua with periodical microstructure
• binary media (suspensions)
• plasticity
• …
• stress concentration
• foams
Motivation
homogeneous model without
orientation
homogeneous model with
orientation
Æ regularization Æ size effects
Æ 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:
Æ Cosserat couple modulus penalizes
differences of microrotations to macrorotations:
Experiment
Deformation model uniaxial strain
?
Constitutive Model
phenomenological parameter of inner structure: Lc
Æ acts like torsional spring
Æ influences angular momentum
Nonlinear ansatz for curvature energy:
Æ Lc penalizes curvature Æ curvature increases for
small structures
Æ higher stiffness for smaller structures
Linear Cosserat Model
Free Helmholtz energy:
which turns for the linear Cosserat model into:
Æ 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 displacements in 2-direction
different zones of deformation indicate
simple glide zone
no displacements in 3-direction
+
no gradients in 1- direction
hexagonal or quadrilateral
Simple glide
no displacements in 2- and 3-direction Analytical investigations
in simple glide:
displacements in 1-direction linked in 1-2-plane Æ no
gradients in 1-direction maximal shear
Simple glide
Internal energy expression:
Simple glide
Simple glide
Simple glide
Numerical investigations:
always γ = 0.2
measure 1. P.K. shear stress = reaction force
Simple glide
Simple glide
Simple glide
Simple glide
Conclusions in Simple Glide
Æ Æ
Æ
Æ
Torsion test
deformed mesh boundaries
no displacements in all directions no displacements
in 3-direction
Constant sample values:
all results concern this point
microrotations fixed 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 0,6
lateral 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 concern this point
c c c c c c c
microrotations fixed to zero
Æ
Torsion test
results concern this 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 0,6
lateral 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
Æ
c c c c c c c
microrotations c
fixed 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 0,8
lateral macro-twist (boundary condition is true rotation in the twist) torque
St.V.-Kirchhoff L = 100
L = 10 L = 1 L = 0.1 L = 0.01
Torsion test
results concern this point
Æ Æ
c c c c c
microrotations fixed to zero
Conclusions in Torsion Test
curvature energy
should be responsible for length scale
effects!
Æ Æ
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
lateral 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
c c
c c
c c c c
variation of looks like length scale effect
Æ Æ
Imperfection Algorithm
homogeneous model without
orientation
homogeneous model with
orientation
stochastic rotational imperfections
Boltzmann continua
Cosserat continua
For perfect samples and perfect boundary conditions the full spectrum of possible solutions
can often not be reached numerically Æ disturb perfect
situations
Compression test
d=0.2 stochastic rotational
imperfections microrotations
fixed to zero
no displacements in all directions
displacement only in 3-direction Constant sample values:
deformed mesh boundaries
R
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 10
internal length scale Lc twist [rad]
macro-rot micro-rot
Compression test
Rotations measured at
half height
Macrorotation arround vertical axis for various internal length scale factors
Boltzmann continua
Compression test
Simple model to motivate shortening through twist
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 10
internal length scale Lc energy
strain curv 1,2E+03
1,3E+03 1,4E+03 1,5E+03 1,6E+03
0,0001 0,001 0,01 0,1 1 10
internal length scale Lc energy
total strain
• curvature energy only of second order (maximal 4% of strain energy)
logarithmic scale
• total energy increases for increasing internal length scale factor – but not arbitrary
• pronounced length scale effects for
0 2 4 6 8 10 12 14 16 18 20
0 0,05 0,1 0,15 0,2
displacement d reaction force R
St.Venant-K.
S.V.K. linear Neo-Hooke L = 10 L = 1 L = 0.1 L = 0.03 L = 0.01 L = 0.001 L = 0.0001
Compression test
Æ Size effects not as significant as in torsion test Æ as expected !!!
c c c c c c c
-6.661E-03 min -5.709E-03 -4.758E-03 -3.806E-03 -2.854E-03 -1.902E-03 -9.501E-04 1.731E-06 9.536E-04 1.905E-03 2.857E-03 3.809E-03 4.761E-03 5.713E-03 6.665E-03 max
1 2
3
Compression test
Compression test with various slenderness ratios (different aspect ratio), constant internal length scale factor and constant maximal strain d / h = 10 %
h=1.0: symmetric deformation
-5.078E-02 min -4.354E-02 -3.629E-02 -2.904E-02 -2.179E-02 -1.454E-02 -7.291E-03 -4.232E-05 7.207E-03 1.446E-02 2.170E-02 2.895E-02 3.620E-02 4.345E-02 5.070E-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-02 3.804E-03 max
1 2
3
h=2.0: deformation with twist
h=4: deformation with twist 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 0,10
d / h
detK / detK_0
h=0.4 h=1 h=1.5 h=2 h=4 h=8
h=8
h=4
h=2
h=1.5 h= h=1
0.4
buckling
twist
0 2 4 6 8 10 12 14 16 18 20
0,00 0,02 0,04 0,06 0,08 0,10
d / h
reaction force R
h=0.4 h=1 h=1.5 h=2 h=4 h=8
size effect
Compression test
Reaction force for various heights of structure
buckling
Æ
size effect (caused by twist) is much less significant than buckling
Æsize effect hardly measureable in practical tests
Conclusions and Outlook
Æ
Æ
Æ Æ
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 Berlin 7.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