03 Papamichos E Cosserat Theory

8/10/2019 03 Papamichos E Cosserat Theory

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21st ALERT Graduate School Mathematical modeling in Geomechanics, Aussois, France, 07-09 October 2010

Aristotle University of Thessaloniki 1

03 Continua with microstructure:Cosserat Theory

Euripides Papamichos

Department of Civil EngineeringAristotle University of Thessaloniki, Greece

[email protected]


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Aristotle University of Thessaloniki

Classical continuum mechanics do not incorporate any intrinsicmaterial length scale

Classical continua consist of points having three translational dofs Displacement in three directions ux, uy, uz

Material response to the displacement of its points

Symmetric stress tensor ij

Transmission of loads is uniquely determined by a force vector, neglectingcouples

2

Introduction


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Such continua may be insufficient for description of certain physical

phenomena Real materials often have a number of important length scales, which

should be included in a realistic model

Grains

Particles

Fibers

Cellular structures

Building blocks, etc.

Non-classical behavior due to microstructure arises if the material is

subjected to non-homogeneous straining and is mostly observed inregions of high strain gradients, e.g. at

Notches

Holes

Cracks

3


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Typical components of Double stress ijk Gradient deformations

ijk

4


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The Cosserat (or micropolar) continuum theory is one of

the most prominent extended continuum theories Seminal work of brothers Eugene and Francois Cosserat (1909)

Re-discovered in the 1950s (Mindlin, Gunther, etc.)

Milestone: Freudenstadt-IUTAM-Symposium on the Mechanics

of Generalized Continua (Krner 1968)

Introduction of micro-inertia for dynamic effects (Eringen 1970)

Historical account of the theory : Introduction of the CISMLecture on Polar Continua by R Stojanovi (1970)

Has been used extensively since the 1980s

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Kinematics

Continuum of oriented rigid particles, called tridres rigides (or rigidcrosses), with 6 d.o.f.

3 displacements ui 3 rotationsi

c (different from the rotational part iof displacement gradient)

Statics

9 Force stressesij(force per unit area) associated with thedisplacements

9 Couple stresses ij(torques per unit area) associated with the rotations

Constitutive relations (in simplest isotropic Cosserat elasticity)

The 2 Lame constants , of classical elasticity 1 additional elastic constant R =INTERNAL LENGTH scale parameter

Relates to the microstructure

E,g. experimentally by the method of size effects for a particular problem

6

Cosserat continuum description


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Continuous medium consisting of a collection of particles

that behave like rigid bodies E.g. Assembly of rods and bricks

8

Direct shear apparatus 12Lab 3S, Grenoble


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Rotation of

individualparticles differsfrom that of theirneighborhood

9

E Charalambidou (2007)


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Deformation field

10

Large rotations in areas of non-uniform

deformation, e.g. in shear bands

Rotation field

E Charalambidou (2007)


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Brick structure


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x1

x2

13

11

12

2123

22

= +

11 13

Statics of a 2-d Cosserat stress element

4 (force) stresses ij 2 couple stresses ij


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21st

ALERT Graduate School Mathematical modeling in Geomechanics, Aussois, France, 07-09 October 2010


Statics of a 3-d Cosserat stress element

9 (force) stresses ij 9 couple stresses ij

Bending

Torsional

Sign convention

13

11

21 21

31 31

32

1213

22

23

32

3333

12

11

22

23

13

x1

x2

x3


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Derivation of

equilibrium equations(a) Force equilibrium

(b) Moment equilibrium

,

,

0 in

0

0

0 in

0

ji j

yxxx

xy yy

ji j ijk jk

yzxz

xy yx

V

x y

x y

e V

x y

=

+ =

+ =

+ =

+ + =


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1 of the essential features of Cosserat or micropolarcontinua

Modified equation for the balance of angular momentum

All theories in which the stress tensor is not symmetriccan be regarded as polar-continua

Typically predict a size-effect, meaning that smallersamples of the same material behave relatively stiffer than

larger samples

An experimental fact completely neglected in the classical

approach It implies that the additional parameter in the Cosserat model

defines a length-scale present in the material

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Non-symmetry of stress tensor


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Experiments show that Brittle fracture and onset of static yielding inthe presence of stress concentration occur at higher loads thanexpected on the basis of stress concentration factors calculated from

the theory of elasticity (Mindlin 1963)

Kirsch solution => Stress concentration = 3

Couple stress solution

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Example

Circular hole in a field of simple tension

( )

( )

( )

( )

( )

2

0

2

1

3

8 13

1 24

3

3 0.44 12.4 2.6

1 0.44 1

m

m a R

m a R

FF K a Rp F a a

R R K a R

p

p

=

=

+= =+

+ +

=

+ =

+


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Stress concentration factor with increasing holesize (Mindlin 1993)

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ExampleCircular hole in a field of simple tension


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Characteristic differences between classical

and Cosserat continuum elasticity (Lakes 2010)

Classical continuum

In bending and torsion of circular

cylindrical bars the rigidity isproportional to the fourth power ofthe diameter

Wave speed of plane shear and

dilatational waves in anunbounded medium isindependent of frequency

No length scale; hence stressconcentration factors for holes or

inclusions under a uniform stressfield depend only on the shape ofthe inhomogeneity and not its size

Cosserat continuum

Size-effect in bending and torsion of

circular cylinders/square bars Slender cylinders appear stiffer than

expected classically

Experimentally one uses size effects todetermine the internal length

Speed of shear wave depends onfrequency A new kind of wave associated with the

micro-rotation is predicted.

Stress concentration factor for a

circular hole is smaller than theclassical value (Mindlin 1963) Small holes exhibit less stress

concentration than larger ones

This gives rise to enhanced toughness

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Micropolar model has been around for several decadesbut yet there is no compelling evidence for the use of

such a model

Material moduli which characterize the model haveNOT been measured directly

Problems with regard to the prescription of boundaryconditions for the microrotations

.

19

Some problems

I.A. Kunin (1982, 1983)


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Dynamic problems

Differences between classical linear elasticity and experiment appearparticularly in dynamic problems involving elastic vibrations of highfrequencies and short wavelength

The reason lies in the microstructure of the material

Linear dynamic Cosserat model predicts dispersion

Wave speed depends on the wave frequency

Impossible in classical linear elasticity, but an experimental fact

New rotational waves that have not been observed yet experimentally

Micropolar fluid flow For modeling turbulence and augmenting the Navier-Stokes equations

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6 Equilibrium equations

6 Traction boundary conditions

6 Displacement b.c.

25

Variational equilibrium equation

( )

( ) ( )

, ,

c

ji j i ji j ijk jk i

V V

c

ji j i i ji j i i

S S

u dV e dV

n t u dS n m dS

+ + =

= +

,

,

0 in

0 in

ji j

ji j ijk jk

V

e V

=

+ =

in

in

ji j i

ji j i

n t S

n m S

=

=

in

in

i i u

c c

i i u

u U S

S

=

=


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st



Example of b.v. problem

Pure bending of a Cosserat-elastic beam

Deformation of a Timoshenko beamThe normal rotates by an

amount not equal to dw/ dx

Timoshenko beam =1-d Cosserat beam

Vardoulakis 2009


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Pure bending of a Cosserat-elastic beam

x

z

xxxy

y

z

K

b

h

xx z =

Non trivial stresses are the axial stress xxand the couplestress xyand their corresponding strain xxand curvature xy

The geometry of deformation gives

= radius of curvature

Compatibility conditions 0xx xyz

=

1xy

=


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Elasticity relations

Satisfies equilibrium

equations

Total bending moment

28

2

2 1.61.6

xx xx

xy xy

EzE

GRGR

= =

= =

0

0

xx

xy

x

x

=

=

2 2 2 22

2

2 2 2 2

23 2 2

1.6

1.6 0.8 9.61

12 1 1

h h h h

xx xy

h h h h

E GRM zbdz bdz b z dz b dz

E bh GR E R EI Rbh I bh

h

= + = + =

= + = + = +

+ +


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I= equivalent moment ofinertia of a Cosserat

beam I Ifor R/h 0

The smaller the structurethe larger the effect

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Cosserat beam is stiffer

29.6

, 11

EI RM I I

h

= = +

+


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03 Papamichos E Cosserat Theory