constitutive parameters for a nonlinear cosserat theory

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

Simple glide

Internal energy expression:

Simple glide

Simple glide

Numerical investigations:

always γ = 0.2

measure 1. P.K. shearstress = reaction force

Simple glide

Conclusions in Simple Glide

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


0

0,5

1

1,5

2

2,5

3

3,5

4

4,5


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


c

c

c

c

c


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


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


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

Simple model to motivateshortening through twist

Compression test

1,0E-03

1,0E-01

1,0E+01

1,0E+03

1,0E+05


energy

straincurv

1,2E+03

1,3E+03

1,4E+03

1,5E+03

1,6E+03


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

constitutive parameters for a nonlinear cosserat theory

Documents