Gran Material Notes

Cosserat Continuum Mechanics, I. Vardoulakis 2009 1

3rd National Meeting on “Generalized Continuum Theories and Applications”

Thessaloniki February 13 and 14, 2009

LECTURE NOTES ON COSSERAT CONTINUUM

MECHANICS WITH APPLICATION TO THE MECHANICS

OF GRANULAR MEDIA

Ioannis Vardoulakis, N.T.U.A.

MEDIGRA

(EU Programme “Ideas”, ERC-2008-AdG 228051)

January 2009


© COSSERAT CONTINUUM MECHANICS, 2009. Lecture Notes by Ioannis Vardoulakis, Dr-Ing., Professor of Mechanics at N.T.U. Athens, Greece, P.O. box 144, Paiania Gr-19002, http://geolab.mechan.ntua.gr/, [email protected]


Table of Contents LECTURE NOTES ON COSSERAT CONTINUUM MECHANICS WITH APPLICATION TO THE MECHANICS OF GRANULAR MEDIA ............................................................................. 1 Preface ............................................................................................................................................. 5 1 Introduction ............................................................................................................................ 6 2 Mathematical preliminaries .................................................................................................... 7

2.1 Line integrals ................................................................................................................. 7 2.2 The transport law of a von Mises “motor”................................................................... 12 2.3 Rigid body motion ....................................................................................................... 14 2.4 General rigid-body motion........................................................................................... 16

3 Cosserat continuum kinematics ............................................................................................ 20 3.1 Description of Cosserat kinematics in curvilinear coordinates .................................... 20 3.2 Integrability conditions and compatibility equations ................................................... 22 3.3 Compatibility equations in Cartesian coordinates........................................................ 24 3.4 Strain, spin, curvature and torsion in Cartesian coordinates ........................................ 28 3.5 2D Cosserat kinematics in Cartesian coordinates ........................................................ 30 3.6 Exercise: Deformation of Cosserat continuum in special curvilinear coordinates ...... 36

3.6.1 Polar cylindrical coordinates ................................................................................... 36 3.6.2 Polar spherical coordinates ...................................................................................... 38

4 Cosserat continuum statics ................................................................................................... 40 4.1 The virtual work equation ............................................................................................ 40 4.2 Equilibrium equations in curvilinear coordinates ........................................................ 45 4.3 Equilibrium equations in Cartesian coordinates .......................................................... 47 4.4 The Mohr circle of non-symmetric stress in 2D .......................................................... 48 4.5 Exercise: Differential equilibrium equations in special curvilinear coordinates ......... 55

4.5.1 Polar cylindrical coordinates ................................................................................... 55 4.5.2 Polar spherical coordinates ...................................................................................... 55

5 Cosserat continuum dynamics .............................................................................................. 56 5.1 Balance of mass ........................................................................................................... 56 5.2 Balance of linear momentum ....................................................................................... 57 5.3 Balance of angular momentum .................................................................................... 58 5.4 The micro-morhic continuum interpretation................................................................ 60 5.5 Exercise: Balance of angular momentum in curvilinear coordinates........................... 64 5.6 Exercise: Dynamic equations in plane polar coordinates............................................. 65 5.7 Stress power in micro-morhic media ........................................................................... 66

6 Cosserat continuum energetics ............................................................................................. 67 6.1 Energy balance equation .............................................................................................. 67 6.2 Entropy balance............................................................................................................ 72 6.3 Linear, isotropic Cosserat elasticity theory.................................................................. 72 6.4 A 2D linear, isotropic Cosserat- elasticity theory ........................................................ 75 6.5 Examples of simple Cosserat elasticity b.v. problems ................................................. 76

6.5.1. Pure bending of a Cosserat-elastic beam............................................................. 76 6.5.2. Annular shear of a cylindrical hole in Cosserat-elastic solid .............................. 79 6.5.3. Sphere under uniform radial torsion.................................................................... 87

6.6 Cosserat thermo-elasto-plasticity ................................................................................. 90 7 Micromechanics of solid granular materials......................................................................... 94

7.1 Stress and couple stress in granular media................................................................... 94 7.1.1 Definitions ............................................................................................................... 95 7.1.2 The virtual work equation for a discrete assembly of particles in contact............... 97 7.1.3 Compatibility in the discrete setting ........................................................................ 98


7.1.4 Remark on incompatible deformations in granular media..................................... 102 7.1.5 Example: Buckling of rigid-plastic, frictional hinged mechanism ........................ 104 7.1.6 Equilibrium conditions for compatible virtual kinematics .................................... 105 7.1.7 A micromechanical definition of average stress and couple stress........................ 107 7.1.8 Example: Computation of the Love stress in a regular hinged lattice ................... 110

7.2 Mass and moment of inertia considerations............................................................... 113 7.3 Grain scale energy dissipation considerations ........................................................... 114 7.4 The 2-grain circuit of homothetically rotating grains ................................................ 116 7.5 Statistical averaging ................................................................................................... 122

8 References .......................................................................................................................... 129 9 Appendix: The meaning of the Lode angle in Boltzmann Continuum Mechanics............. 134


Preface

These Lectures were prepared for and presented at the 3rd National Meeting on

“Generalized Continuum Theories and Applications” in Thessaloniki, February 13 and

14, 2009. For the presentation of the material at hand we use the vectorial and the indicial

tensorial notation, and fixed in space Cartesian- or curvilinear coordinate description. For

easy reading of this material a basic course in Continuum Mechanics is considered as a

prerequisite1. For easy reference, some concepts and definitions that derive from basic

Analysis and Mechanics are summarized in sect. 2. In the first part of these Lecture Notes

(sects. 1 3 to 6) we compile the basic results that pertain to the mechanics of infinitesimal

deformations of Cosserat Continua. In the second part (sect. 7) and in the light of

micromechanical considerations the general Cosserat Continuum Theory is applied to the

Mechanics of Solid Granular Media. Some of the material presented is published here for

the first time and the author would appreciate any comments and critique. The support of

the EU project MEDIGRA (ERC-2008-AdG 228051) and the help of my co-worker Dr.

Stefanos-Aldo Papanicolopulos are acknowledged.

1 http://geolab.mechan.ntua.gr/teaching/lectnotes.html#postcm


1 Introduction There is an ongoing discussion concerning the “origins” of “generalized” continuum

theories such as the Cosserat Theory. On p. 2 of these Notes the reader can find the

learned opinion on the subject put on paper in 1966 by late Maurice Antoine Biot. As

precursors of the Cosserat theory are mentioned in the literature the theory of Lord

Kelvin, concerning the light-aether and the works of W. Voight on the physics of

crystalic matter. A nice historical note on the subject can be found in the introduction of

the CISM Lecture on “Polar Continua” by Rastko Stojanović2.

Classical continuum mechanics is based on the axiom that the stress tensor is symmetric.

According to Schaefer [43] it is Hamel [26] who has named this statement the Boltzmann

axiom, since it is Boltzmann who has pointed first already in the year 1899 to the fact that

the assumption about the symmetry of the stress tensor has an axiomatic character. Thus

the continuum mechanics of media with non-symmetric stress tensor may be termed also

as non-Boltzmann continuum mechanics. Such a theory is the theory of the Cosserat

Continuum, that derives from the seminal work of the brothers Eugène and François

Cosserat [14].

A three dimensional Boltzmann continuum is a continuous manifold of material points

that possess 3 degrees of freedom (dofs) of displacement. As already pointed the

Boltzmann continuum is juxtaposed to the Cosserat continuum that is in turn a manifold

of oriented particles, called “trièdres rigides” or rigid crosses, with 6 dofs, namely 3 dofs

of displacement and 3 dofs of rotation. This property is the reason why Cosserat continua

are also called polar continua. For example the Timoshenko beam is a typical example of

an one-dimensional Cosserat continuum [23], [47], [54] [54]. The Bernoulli-Euler beam is

seen as a special case of a one-dimensional Cosserat continuum for which the rotation of

the material point (i.e. of its cross-section) is related to its displacement through the well

known orthogonality condition of J. Bernoulli. Such a continuum is called a restricted

Cosserat continuum [34].

2 Stojanović, R., Recent developments in the theory of polar continua, CISM, No. 27, Springer, 1970.


2 Mathematical preliminaries

2.1 Line integrals

The following summary on the basic properties of line integration is taken from the

reference book on Tensor Analysis by Duschek & Hochrainer [18]. Let

( ) , 1, 2,3iA x i = (2.1)

be a continuous function of the Cartesian coordinates of point ( )iP x . Let also ( )Γ be a piecewise smooth Jordan curve. The expression,

( )( )

i k iI A x dxΓ

= ∫ (2.2)

is called a line integral. If the curve is closed then we write,

( )( )

i k iI A x dxΓ

= ∫ (2.3)

The above introduced line integral can be transformed to an ordinary (Riemann) Integral

if we select a parametric description of the considered curve, say

1 2( ) : ( ) [ , ]i ix u u u uχΓ = ∈ (2.4)

and evaluate the definition eq. (2.2)

( )( )2

1

ui

i ku

dI A u duduχχ= ∫ (2.5)

The selection of the parameter u is irrelevant, since the transformation

( ) du t du dtdtυυ= ⇒ = (2.6)

leads to

1 2( ) : ( ) [ , ]i ix t t t tχΓ = ∈ (2.7)

and

( )( )( ) ( )( )2 2

1 1

t ti i

i k kt t

d ddI A t dt A t dtdu dt dtχ χυχ υ χ= =∫ ∫ (2.8)

where the integration limits are derived from the equations

( )( )

1 1

2 2

0

0

t u

t u

υ

υ

− =

− = (2.9)


The representation of the line integral given above by eq. (2.5) includes the case of

closed-curve line integral, if we assume that

2 1( ) ( )i iu uχ χ= (2.10)

It is obvious that this definition of the line integral includes the case where the line ( )Γ

consists of a number of consecutive smooth segments,

1 2

1 2

1 12 1

( ) ( ) ( ) ( )

( ) : ( ) [ , ] , 1

( ) ( ) , 1 1i i

i i

x u u u u

u u

ν

α α α α αα

α α α α

χ α ν

χ χ α ν+ +

Γ = Γ ∪ Γ ∪ Γ

Γ = ∈ =

= = −

……

… (2.11)

since

( ) ( ) ( ) ( )1 2( ) ( ) ( ) ( )

i k i i k i k i k iI A x dx I A x dx A x dx A x dxνΓ Γ Γ Γ

= = = + +∫ ∫ ∫ ∫ (2.12)

Note that a special case of such a curve is a polygonal line, consisting of straight line

segments (Figure 2-1).

Figure 2-1: A polygonal curve consisting of consecutive straight-line segments

From the definition of the line integral follows also that if ( )kA x is a scalar, then iI is a

vector. To prove this let us consider the coordinate transformation

r rm m r rm mx a x dx a dx= ⇒ = (2.13)

and assume that

( ) ( ) ( )r rm m mA x A a x A x= = (2.14)

then


( ) ( ) ( )( ) ( ) ( )

i k i k im m im k m im mI A x dx A x a dx a A x dx a IΓ Γ Γ

= = = =∫ ∫ ∫ (2.15)

i.e. it transforms like a vector.

Similarly we prove that the line integral of a vector

( )( )

ij j k iI A x dxΓ

= ∫ (2.16)

is a 2nd order tensor, and so on.

A special case of the line integral, eq. (2.1) arises, if we select

2

1

2 11 ( ) ( )u

i i i iu

A I dx u uχ χ= ⇒ = = −∫ (2.17)

In that case the vector iI is the vector that connects the starting point (1) with the

endpoint (2) on the considered curve and coincides thus with the oriented secant of that

curve through these points (Figure 2-2).

Figure 2-2: Oriented secant on a curve, passing through points (1) and (2)

The integral representation of the secant (12)→

, eq. (2.17), is not to be confused with the integral,

2

1

,u

ii i i

u

ds x x du xduχ′ ′ ′= =∫ (2.18)

that computes the arc length between points (1) and (2).

Let us consider the tensor ijI , eq. (2.16). With indices contraction we may define the scalar,


( )

ii i iI I A dxΓ

= = ∫ (2.19)

In continuum mechanics applications this is the most commonly appearing line integral.

Thus some authors will call eq. (2.19) a line integral of the vector iA . For example, if a

force if applies on a particle that moves along a curve ( )Γ , then the work done by this

force during the passage of the particle from point (1) to point (2) along this path is given

by the “line integral”,

( )

i iW f dxΓ

= ∫ (2.20)

The sign of a line integral depends on the orientation of the curve ( )Γ , since it changes

sign if we change the orientation,

2 1

1 2

u u

i iu u

Adx Adx= −∫ ∫ (2.21)

We consider now two curves

(1)

1 1 2(2)

2 1 2

( ) : ( ) [ , ]

( ) : ( ) [ , ]i i

i i

x u u u u

x v u v v

χ

χ

Γ = ∈

Γ = ∈ (2.22)

that are having the same start- and end-points (Figure 2-3),

(1) (2)

1 1(1) (2)

2 2

( ) ( )

( ) ( )i i

i i

u v

u v

χ χ

χ χ

=

= (2.23)

Figure 2-3: Two curve segments with common end-points

In general the two line integrals

1 2

(1) (2)

( ) ( )

,i i i iI Adx I AdxΓ Γ

= =∫ ∫ (2.24)

will have different values.


If we change the orientation of one of the two curves, say of 2( )Γ we may compute the

line integral along the closed loop (1,2,1) :

1 2 1 2

1 2( ) ( ) ( ) ( ) ( )

, ( )i i i i iAdx Adx Adx Adx AdxΓ −Γ Γ Γ Γ

+ = − = Γ = Γ ∪ −Γ∫ ∫ ∫ ∫ ∫ (2.25)

If the value of the line integral does not depend on the integration path, then for the two

considered curves that possess the same end-points we have that,

1 2( ) ( )

i iAdx AdxΓ Γ

=∫ ∫ (2.26)

and that the closed path integral, eq. (2.25), vanishes,

( )

0iAdxΓ

=∫ (2.27)

The converse is also true, since any partition of the closed loop ( )Γ , would lead from eq.

(2.27) to (2.26).

We return now to the line integral, eq. (2.19), and ask the question when the integral

( )

i iI A dxΓ

= ∫ (2.28)

is path independent.

It is evident that this true, if the vector field iA derives as the gradient of a scalar (called

the potential), i.e. if

i iA U= ∂ (2.29)

Indeed for

( )( ) ( ) ( )i iU x U u uχ= = ϒ (2.30)

we have

2 2 2

1 1 1

2 1( )

u u Ui

i iiu u U

dU dI Udx du du dU U Ux du du

χ

Γ

∂ ϒ= ∂ = = = = −

∂∫ ∫ ∫ ∫ (2.31)

This means that the value of the line integral I depends only on the values of the

underlying potential function at the end points of the connecting curve. It is thus path

independent.


In textbooks of Tensor Analysis we find finally the following fundamental,

Theorem Necessary and sufficient condition for a vector field to be the gradient of potential scalar

function is that its rotor vanishes,

0i i ijk j kA U Aε= ∂ ⇔ ∂ = (2.32)

This means that sufficient and necessary for the path independence of the line integral

(2.28) is the condition

rot 0 0ijk j kA Aε= ⇔ ∂ = (2.33)

2.2 The transport law of a von Mises “motor”3

Figure 2-4: Equipollent reductions of a system of forces

Let us consider a system of forces 1,F … that are acting on a rigid body. These forces

can be reduced by their resultant force F that is acting at a point P . We denote this

setting by ( )F P . This force ( )F P can be replaced by a force, that is passing through

another point 1O , denoted as 1( )F O , and a moment, denoted as 1( )M O The force 1( )F O

arises through parallel translation of the force F and the moment of the pair of forces

1( ), ( )F P F O− . The moment 1( )M O is a free vector, computed as

3 Schaeffer, H. (1968). The basic affine connection in a Cosserat Continuum, In: Mechanics of Generalized Continua, Springer, p. 57-62.


1 1( ) ( )M O O P F P→

= × (2.34)

The point 1O is called a reduction point. We notice that for all reduction points ( )P ε′∈

along the straight line ( )ε , that is parallel to F and passes through point P , the moment

( ) 0M P′ = (2.35)

We may select another point 2O with,

2 2 2 1 1 2 1 1 1

1 1 2 1 1 1 1 2 1 1 2 1

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

M O O P F P O O O P F P O O F O M O

M O F O O O M O F O O O M O F O OO OO

→ → → →

→ → → →

⎛ ⎞= × = + × = × +⎜ ⎟⎝ ⎠

⎛ ⎞= − × = + × = + × −⎜ ⎟⎝ ⎠

(2.36)

or

( )2 12 1( ) ( ) O OM O M O F= + × ℜ − ℜ (2.37)

In summary, we have above two reductions that are called equivalent or equipollent,

when the following transport law is holding,

( )2 1

2 1

2 1

( ) ( )

( ) ( ) O O

F O F O

M O M O F

=

= + × ℜ − ℜ (2.38)

The compound of the two vectors

FM

⎛ ⎞= ⎜ ⎟

⎝ ⎠∆ (2.39)

is called a v. Mises motor4, if these vectors obey the transport law, eq. (2.38). The motor

eq. (2.39) in particular is a dynamic motor5.

The same is the case with the velocity field of a rigid body. Let w be the angular velocity

and v the displacement velocity. From two points A and B we have the same transport

law,

4 Mises, R.v. (1924). Motorrechnung, ein neues Hilfsmittel der Mechanik, and Anwendungen der Motorrechnung, ZAMM, 4, 155-181, 193-213. 5 ∆ύναµις, Greek for dynamic action.


( )( ) ( )

( ) ( ) B A

w B w A

v B v A w

=

= + × ℜ − ℜ (2.40)

The corresponding motor is the vector pair,

wv

⎛ ⎞= ⎜ ⎟

⎝ ⎠Η (2.41)

The motor eq. (2.41) in particular is a kinematic motor6.

We remark that in the motor space 6V we can define the power,

: i ii i

wFA F v M w F v M wvM

⎛ ⎞ ⎛ ⎞= = = ⋅ + ⋅ = +⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠∆ Η (2.42)

It can be shown that power A = ∆ Η is independent of the choice of the reduction point!

2.3 Rigid body motion

We consider Cartesian coordinates. The affine mapping 7 of the points of a body Β

( ) ( )i i ij jx c t R t ξ= + (2.43)

that has the property to keep the distance constant between two arbitrary points of that

body is called a rigid body motion.

Let two such points ( )iA ξ και ( )iB ψ , and let

( ) ( )

( ) ( )i i ij j

i i ij j

x c t R t

y c t R t

ξ

ψ

= +

= + (2.44)

From the above definition we get that,

( )( ) ( )( )i i i i i i i ix y x y ξ ψ ξ ψ− − = − − (2.45)

where

( )i i ij j jx y R ξ ψ− = − (2.46)

thus

6 ‘Ελίκωσις, Greek for spiral motion. 7 Lat. affinitas=neighboring


( )( ) ( )( )ij ik j j k k jk j j k k

ij ik jk

R R

R R

ξ ψ ξ ψ δ ξ ψ ξ ψ

δ

− − = − − ⇒

= (2.47)

We recall that a square matrix [ ]R is called orthogonal, if

[ ]det 1R = ± (2.48)

For an orthogonal matrix we have the following relations

[ ] [ ] [ ][ ] [ ]T TR R R R I= = (2.49)

where [ ]I is the unit matrix, and

[ ] [ ] 1TR R −= (2.50)

Or in components

,ik il kl ki li klR R R Rδ δ= = (2.51)

From the above we conclude that if the affine mapping eq. (2.43) is describing a rigid

body motion, then ijR must be orthogonal, since from eq. (2.47) we get the condition, eq.

(2.48). The case where det[ ] 1ijR = + corresponds to a real motion, whereas the case

det[ ] 1ijR = − is not, since the corresponding mapping is a reflection! In case when

ij ijR δ= (2.52)

From eq. (2.43) we get that

i i ix c ξ= + (2.53)

In that case the motion is a translation of the rigid body (Figure 2-5) and the displacement

vector of all material points of the rigid body is a unique function of time only,

( )i i i iu x c tξ= − = (2.54)

Since for

0 : 0 (0) 0i i i it u x cξ= = ⇒ ≡ ⇒ = (2.55)

Thus from eq. (2.55) follows that the vector ( )ic t describes the motion of that material

point, which at 0t = was posiotined at the origin of the selected coordinate system.


Figure 2-5: Translation of a rigid body

2.4 General rigid-body motion

A pont of a body is called a fixed point if after the application of the motion, eq. (2.43), is

mapped onto. Based on the definition of the affine mapping and the definition of the

fixed point of a motion, we can prove the follwing theorems8:

Theorem 1: If a motion, eq. (2.43), has four fixed points, that are not on the same plane, then the

motion is an identy mapping of all points onto themselves

( ) 0 , ( )i ij ijc t R t δ= = (2.56)

Theorem 2: If a motion, eq. (2.43), has a fixed plane and the motion is not the identy mapping, eq.

(2.56), then this motion is not a real motion; it is a pseudomotion that corresponds to a

reflection of all points of the considered body with respect to the given fixed plane.

Therome 3: If a motion, eq. (2.43), possesses a fixed straight line, then this motion is a rotation with

respect to that line.

8 cf. Grottemeyer, K.P., Analytische Geometrie, Göschen, Bd. 65-65a, 1964.


For the determination of the fixed points of a motion we consider the following

equations,

0i i ij j ij j ic R T cξ ξ ξ= + ⇒ + = (2.57)

where

ij ij ijT R δ= − (2.58)

To that we have te follwing matix

[ ] [ ] [ ]T R I= − (2.59)

The classification of the solutions of the problem, eqs. (2.57), is done on the basis of the

rank of matrix [ ]T as follows:

1) [ ] 3ijrng T = : In that det[ ] 0ijT ≠ , and there exists only one fixed-point,

1k ki iT cξ −= − (2.60)

If we assume that the mapping ijR corresponds to a real motion, i.e. if det[ ] 1ijR = + , then

( )( ) TTij jk ji jk jk ik ki ik ikR T R R R Rδ δ δ= − = − = − (2.61)

thus

det[ ( )] ( 1)det[ ] det[ ]

det[ ] 0 [ ] 3ji jk jk jk jk ik ik

ij ij ij

R R R R

R rng T

δ δ δ

δ

− = + − = − − ⇒

− = ⇒ < (2.62)

This is in contradiction we the initial assumption. This case is impossible.

2) [ ] 2ijrng T = : Inthat case the homogeneous system of equations,

0ij jT ξ = (2.63)

has a solution of the form

i iξ λρ= (2.64)

Thus

0 ( ) 0

( ) 0 ( ) 0ij j j ji j ji ji

j ji ji ik j jk jk

T T R

R R R

ρ ρ ρ δ

ρ δ ρ δ

= ⇒ = − = ⇒

− = ⇒ − = (2.65)

or


11 12 13

1 2 3 21 22 23

31 32 33

11 0 0 0

1

R R RR R RR R R

ρ ρ ρ−⎡ ⎤

⎢ ⎥− =⎢ ⎥⎢ ⎥−⎣ ⎦

(2.66)

This means that the vector iρ is normal to the column-vectors of the matrix

[ ] [ ]T R I= − , and two such vectors are be hypothesis linearly independent.

From eq. (2.57) we get,

( ) 0 0i ij ij j i i i iR c cρ δ λρ ρ ρ− + = ⇒ = (2.67)

Thus vector iρ must be normal to vector ic . In the considered case eq. (2.67) is a

necessary condition for eq. (2.57) two have a solution. Thus a sufficient condition for a

solution to exist in the considered case is that the vector ic is a linear combination of

two vector-columns of the matrix [ ] [ ]T R I= − . In that case the solution we seek has the

form,

i i iξ µ λρ= + (2.68)

where iµ is any particular solution of eq. (2.57).

We consider now the vector

( )i k k i k k im c c m cρ ρ ρ ρ ρ ρ= − ⇔ = × × (2.69)

Thus,

0i i i k k i i k k im c cρ ρ ρ ρ ρ ρ ρ= − = (2.70)

and

1 ,

i i i

i i

i i i i

c mc

α βρρα β

ρ ρ ρ ρ

= +

= = (2.71)

With the help of this analysis we can replace the motion, eq. (2.43),

i ij j ix R cξ= + (2.72)

by the following consequtive motions:

i ij j ix R mξ α= + (2.73)


i i ix x βρ= + (2.74)

Based on Theorem 2, above, the first component of the motion that is given by eq. (2.73),

is a rotation with respect an axis that has the orientation of vector iρ . This is so

because eq.(2.69) holds by construction. The second component of the motion, eq. (2.74),

is a translation in the direction of iρ . A motion like that is called helicoidal. Figure 2-6

shows the geometric characteristics of a helix.

Figure 2-6: Right, circular helix

3) [ ] 1ijrng T = : This case is impossible when όταν det[ ] 1ijR = + , since in this case the

existence of a fixed plane is only compatible with a reflection.

4) [ ] 1ijrng T = : In this case ij ijR δ= and the motion is a translation.

Theorem of Chasles9: The general motion of a rigid body is a helicoidal motion, that combines a rotation with

respect to an axis and a parallel translation 10. This theorem is originally attributed by

some authors 11 to Giulio Mozzi12, and is considered as the basis of the mechanical theory

of screws 13.

9 Chasles, M. (1830). Note sur les propriétés générales du système de deux corps semblables entre eux, Bulletin de Sciences Mathématiques, Astronomiques Physiques et Chimiques, Baron de Ferussac, Paris, 321-326. 10 P. Chadwick, Continuum Mechanics, Chapt.2, Dover, 1976. 11 Ceccarelli, M. (2000). Screw axis defined by Giulio Mozzi in 1763 and early studies on helicoidal motion, Mechanism and Machine Theory, 35, 761-770. 12 Mozzi, G. Discorso matematico sopra il rotamento momentaneo dei corpi, Stamperia di Donato Campo, Napoli, 1763, 13 R.S. Ball, Treatise on the Theory of Screws, Hodges Dublin 1876, Cambridge University Press 1900, Cambridge Mathematical Library 1998.


3 Cosserat continuum kinematics

3.1 Description of Cosserat kinematics in curvilinear coordinates

Figure 3-1: Cartesian and curvilinear coordinates of point in the plane

We restrict our analysis here to infinitesimal deformations. The following demonstrations

follow closely the lines of the seminal paper of late Professor Wilhelm Günther [23].

The position vector of the material point is (Figure 3-1)

iiOP R x e

→

= = (3.1)

where ix are the underlying Cartesian coordinates of the position vector with,

( ) ; 0i

i i kkx χχ ∂

= Θ ≠∂Θ

(3.2)

This transformation allows us to write the position vector as a function of the curvilinear

coordinates ( 1,2,3)i iΘ = of the point P

( )iR = ℜ Θ (3.3)

We introduce the local affine basis

, ,, ( )i i ii ig ∂ℜ ∂= = ℜ ⋅ ≡

∂Θ ∂Θ (3.4)

and we assume that the vectors, 1 2 3, ,g g g , in the given order build a right handed system.

The infinitesimal displacement vector at point P is denoted as (Figure 3-2),


( )i kiu u g= Θ (3.5)

and the infinitesimal rotation of the rigid cross attached to the material point P , is given

by another vector,

( )i kigψ ψ= Θ (3.6)

Figure 3-2: Dofs of a material polar point of a 2D Cosserat continuum projected on to the local affine basis ( 1 2 3, ,g g g ).

We introduce the following vector deformation measures,

,i i iu gγ ψ= + × (3.7)

,i iκ ψ= (3.8)

Note that the i-th component of the vector product of two vectors is computed

( ) k li iklx y e x y× = (3.9)

where klme is the corresponding Levi-Civita 3rd-order fully antisymmetric tensor,

: ( , , ) (1, 2,3)

: ( , , ) (2,1,3) ,0

⎧ =⎪⎪= − = =⎨⎪⎪⎩

klm ij

g if k l m cycl

e g if k l m cycl g gelse

(3.10)

where ijg is the co-variant metric tensor.

In sects. 3.4 and 3.5 . we will demonstrate that the above introduced measures, eqs. (3.7)

and (3.8), really describe deformation in a Cosserat continuum.

We observe that the gradient of a vector is expressed by means of its covariant derivative,


,i

j iju u g= (3.11)

The 2nd term on r.h.s. of eq. (3.7) becomes,

k k l k li i k ikl ilkg g g e g e gψ ψ ψ ψ× = × = = − (3.12)

or,

li ilg gψ ψ× = (3.13)

where

kil ilkeψ ψ= − (3.14)

With

ki i kgγ γ ⋅= (3.15)

eqs. (3.7), (3.11) and (3.13) yield the following expression for the deformation tensor,

k k ki iiuγ ψ⋅⋅ ⋅= + (3.16)

where

k lki il gψ ψ⋅ = (3.17)

Similarly from

ki i kgκ κ ⋅= (3.18)

and

,k

i ki gψ ψ= (3.19)

we get the following expression for the Cosserat rotation gradient

k ki iκ ψ⋅ = (3.20)

Thus from the 6 placements iu and iψ ( 1, 2,3)i = we have generated 18 deformations k

iγ ⋅ and kiκ ⋅ .

3.2 Integrability conditions and compatibility equations

Let now ( )Γ be a curve in space that is passing through points 0P and P . Starting from a

point 0P we can compute the value of one of the kinematic fields, say the particle


rotation, by means of a line integral that is evaluated along the considered curve ( )Γ .

Thus from

0

0 ,( ) ( )P

kk

P

P P dψ ψ ψ= + Θ∫ (3.21)

and eq. (3.8) we get

0

0( )P

kk

P

P dψ ψ κ= + Θ∫ (3.22)

For uniqueness purposes we require that the value of the Cosserat rotation at point P , as

computed from eq. (3.22), is independent of the particular choice of the curve ( )Γ that

joins the points 0P and P ; assuming that at point 0P the value of ψ is known. According

to the fundamental theorem of Tensor Analysis [18] the sufficient and necessary

condition for this integrability requirement is that

rot 0kκ = (3.23)

or

(1)

, 0i ijkk je κℑ = = (3.24)

The first-order system (1)

kℑ is called the 1st incompatibility form.

With

;k k l l ki i k i kl il il i klg g g g gκ κ κ κ κ κ⋅ ⋅ ⋅= = = = (3.25)

eq. (3.24) yields

( ) ( ) ( ), , , ,,0 ijk ijk l ijk l q ijk l q l

k j k l k j l k q j k j k qj lje e g e g g e gκ κ κ κ κ κ⋅ ⋅ ⋅= = = + = + Γ (3.26)

or

(1)

0kl kpq lq pI e κ ⋅= = (3.27)

Similarly from,

( )0 0

0 , 0( ) ( )P P

k kk k k

P P

u P u P u d u g dγ ψ= + Θ = + − × Θ∫ ∫ (3.28)

and due to eq. (3.4) through partial integration we get


( ) ( )( )0

0 0 0( )P

kP k P k

P

u P u dψ γ κ= + × ℜ − ℜ + + ℜ − ℜ × Θ∫ (3.29)

The integrability of eq. (3.29) results to the following condition

( )( )

( )( ),

, , ,

0 ijkk P k j

ijkk j j k P k j

e

e

γ κ

γ κ κ

= + ℜ − ℜ ×

= + ℜ × + ℜ − ℜ × (3.30)

Due to eqs. (3.24) and (3.4), eq. (3.30) yields the following condition,

( )(2)

, 0k ijkk j j ke gγ κℑ = + × = (3.31)

The first-order system (2)

kℑ is called the 2nd incompatibility form.

In order to derive the analytic form of the 2nd compatibility condition we start from eq.

(3.28) and set it as,

( )

0 0

0 0( )

,

P Pm p lm k k

k klp m kP P

m m m lm pk k m k k klp

u P u e g g d u E d

E E g E e g

γ ψ

γ ψ

⋅

⋅

= + + Θ = + Θ

= = +

∫ ∫ (3.32)

After some extended algebraic manipulations eq. (3.32) yields [23],

(2)

i ijk i i kr r r kkr jI e γ κ δ κ⋅ ⋅ ⋅= − + (3.33)

where

pprrk j k jgγ γ= (3.34)

3.3 Compatibility equations in Cartesian coordinates

As an application of the above derivations we consider a Cartesian description of the

motion of the Cosserat continuum. In that setting the infinitesimal displacement vector

and Cosserat rotation of the polar material point P are given as,

;i i i iu u e eψ ψ= = (3.35)

where ie is the orthonormal Cartesian basis.


In Figure 3-3 we see in a 2D setting that the polar material point is symbolized as a

material cross that is moved from position ( )iP x to position ( )( )i i kP x u x′ + and at the

same time is rotated in positive sense by a small angle ( )3 ixψ ψ= .

Figure 3-3: Displacement and rotation of the polar material point in a 2D setting.

In Cartesian coordinates the relative deformation tensor, as defined above in eq. (3.16),

becomes

;ik i k ik ii

ux

γ ψ ∂= ∂ + ∂ =

∂ (3.36)

with

ij ijk kψ ε ψ= − (3.37)

where ijkε is the Cartesian permutation tensor,

1 : ( , , ) (1, 2,3)1 : ( , , ) (2,1,3)

0ijk

if i j k cyclif i j k cycl

elseε

=⎧⎪= − =⎨⎪⎩

(3.38)

Note that an antisymmetric system of 2nd order is always determined by a system of 1st

order,

3 2

3 1

2 1

0[ ] 0

0ij ij ijk k

ψ ψψ ψ ψ ψ ε ψ

ψ ψ

− −⎡ ⎤⎢ ⎥= − ↔ = −⎢ ⎥⎢ ⎥⎣ ⎦

(3.39)

and with that ,

12l mnl mnψ ε ψ= − (3.40)


In Cartesian coordinates eq. (3.20) yields that the gradient of the Cosserat rotation is,

; 1

ik i k

k k k k

κ ψψ ψν ν ν

= ∂= =

(3.41)

where the angle ψ is infinitesimal and iν is the axial unit vector. In particular we call the

components ( )( )i iκ torsions and the rest components curvatures. Similarly from eqs.(3.36)

and (3.37) we get,

ik i k ikl luγ ε ψ= ∂ − (3.42)

For completeness we write down first in Cartesian form the line integrals for the

computation of the relative displacement and the relative rotation between two distant

points 2P and 1P , eqs. (3.21) and (3.28),

2 2

2 1

1 1

( , )2 1( ) ( )

P PP P

i i i k i k ki kP P

P P dx dxψ ψ ψ ψ κ∆ = − = ∂ =∫ ∫ (3.43)

2 2 2

2 1

1 1 1

( , )2 1( ) ( )

P P PP P

i i i k i k ki k ikl l kP P P

u u P u P u dx dx dxγ ε ψ∆ = − = ∂ = −∫ ∫ ∫ (3.44)

Eq. (3.43) can be written formally as

( ) ( ) ( )( )

,i i k k i k kiI A dx A κΓ

= =∫ (3.45)

The path independence if this line integral is guaranteed, if

( ) 0pjk j i kAε ∂ = (3.46)

or if

(1)

0pi pjk j kiI ε κ= ∂ = (3.47)

Similarly, eq. (3.44) formally reads,

2

1

( ) ( ) ( );P

i i k k i k ki ikl lP

J B dx B γ ε ψ= = −∫ (3.48)

leading to

( )( ) 0 0pjk j i k pjk j ki ikl lBε ε γ ε ψ∂ = ⇒ ∂ − = (3.49)

or


( )0

0

0

pjk j ki pjk ilk j l

pjk j ki pi jl pl ji j l

pjk j ki pi ll ip

ε γ ε ε ψ

ε γ δ δ δ δ κ

ε γ δ κ κ

∂ + ∂ =

∂ + − =

∂ + − =

(3.50)

Thus, in Cartesian coordinates the compatibility conditions, eqs. (3.27) and (3.33),

become

(1)

0kl kpq p qlI ε κ= ∂ = (3.51)

(2)

0pi pjk j ki pi ll ipI ε γ δ κ κ∂ + − = (3.52)

Explicitly these compatibility equations read as follows,

(1)

11 1 1 123 2 31 132 3 21 2 31 3 21

(1)

12 1 2 123 2 32 132 3 22 2 32 3 22

0

0

pq p q

pq p q

I

I

ε κ ε κ ε κ κ κ

ε κ ε κ ε κ κ κ

= = ∂ = ∂ + ∂ = ∂ − ∂

= = ∂ = ∂ + ∂ = ∂ − ∂

… (3.53)

and

(2)

11 1 1 11 11 2 31 3 21 22 33

(2)

12 1 2 21 21 2 32 132 3 22 21

0

0

jk j k kk

jk j k kk

I

I

ε γ κ δ κ γ γ κ κ

ε γ κ δ κ γ ε γ κ

= = ∂ − + = ∂ − ∂ + +

= = ∂ − + = ∂ − ∂ −

… (3.54)

If ijκ are the components of the gradient of a vector field kψ , eq. (3.41), then the

compatibility eqs. (3.53) reduce to the differentiability conditions for the named vector

field,

(1)

11 2 3 1 3 2 1(1)

12 2 3 2 3 2 2

0

0

I

I

ψ ψ

ψ ψ

= = ∂ ∂ − ∂ ∂

= = ∂ ∂ − ∂ ∂…

(3.55)

Similarly, if the ijγ are given by eqs. (3.42), then the compatibility eqs. (3.54) reduce to

the differentiability conditions for the vector field iu ,

( ) ( )

(2)

11 2 3 1 31 3 2 1 21 2 2 3 3

2 3 1 3 2 1 2 2 3 3 2 2 3 3 2 3 1 3 2 1

0 l l l lI u uu u u u

ε ψ ε ψ ψ ψψ ψ ψ ψ

= = ∂ ∂ − − ∂ ∂ − + ∂ + ∂

= ∂ ∂ − ∂ ∂ − ∂ − ∂ + ∂ + ∂ = ∂ ∂ − ∂ ∂…

(3.56)


3.4 Strain, spin, curvature and torsion in Cartesian coordinates

Let ix be the coordinates of a point of a rigid body before the motion and ix′ the

coordinates of the same point after the motion. It can be shown that the motion [5],

( )( )( )sin 1 cosi i ikj k i j ij jx x n n n xθ ε θ δ′ = + + − − (3.57)

where

1 , 0i in n = ≤ ≤θ π (3.58)

is a rigid-body rotation at a finite angle θ around a fixed axis with unit director in .

We consider two neighboring points ( )iP x and ( )iQ y in the undeformed configuration

of a Cosserat continuum, such that i i iy x dx= + . The material line element that connects

these two points is given by the vector,

i iPQ dx e→

= (3.59)

Using eq. (3.57) the positions of points P and Q in the deformed configuration are

computed as,

( )( )( )

( )( )( )( )( )

sin 1 cos

sin 1 cos

i i i ikj k i j ij j

i i ikj k j

i i i i j i j ikj k i j ij j j

i i i j i j ikj k j j

x x u x

x u x

y x dx u u dx x dx

x dx u u dx x dx

ψ ε ν ψ ν ν δ

ψε ν

ψ ε ν ψ ν ν δ

ψε ν

′ = + + + − −

≈ + +

′ = + + + ∂ + + − − +

≈ + + + ∂ + +

(3.60)

Thus

( ) ( )( )

i i i

i i i j i j ikj k j j i i ikj k j

ij j i j ikj k j

dx y x

x dx u u dx x dx x u x

u dx dx

ψε ν ψε ν

δ ε ψ

′ ′ ′= −

= + + + ∂ + + − + +

= + ∂ +

(3.61)

or

( )i i i j i ikj k jdx dx dx u dxε ψ′∆ = − = ∂ + (3.62)

From eqs. (3.62), (3.37) and (3.36) we get


( ) ( )( ) ( )

i j i ikj k j j i ijk k j

j i ij j j i ji j

ji j

dx u dx u dx

u dx u dx

dx

ε ψ ε ψ

ψ ψ

γ

∆ = ∂ + = ∂ −

= ∂ − = ∂ +

=

(3.63)

In matrix notation eq. (3.63) reads,

1 11 21 31 1

2 12 22 32 2

3 13 23 33 3

dx dxdx dxdx dx

γ γ γγ γ γγ γ γ

∆⎧ ⎫ ⎡ ⎤ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎢ ⎥∆ =⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥∆⎩ ⎭ ⎣ ⎦ ⎩ ⎭

(3.64)

or

[ ] Tdx dxγ∆ = (3.65)

The length of the line element before and after the deformation is

2

2 2( )2

i i

i i ij i j

ds dx dx

ds dx dx ds dx dxγ

=

′ ′ ′= = + (3.66)

where ( )ijγ is the symmetric part of the relative deformation,

( )

( )

( )

( )12

1212

ij ij ji

i j ij j i ji

i j j i

u u

u u

γ γ γ

ψ ψ

= +

= ∂ + + ∂ +

= ∂ + ∂

(3.67)

This means that the infinitesimal strain in the Cosserat continuum is given as the

symmetric part of the relative deformation tensor and coincides with common

infinitesimal strain tensor in the Boltzmann continuum

( )ij ijγ ε= (3.68)

where

( )12ij i j j iu uε = ∂ + ∂ (3.69)

Let us now consider the antisymmetric part of the relative deformation tensor,


( ) ( )

( ) ( ) ( )

( ) ( )

[ ]

[ ]

1 12 2

1 1 12 2 21 12 2

ij ij ji i j ij j i ji

i j j i ij ji i j j i ij

i j j i ij i j j i ijk k

u u

u u u u

u u u u

γ γ γ ψ ψ

ψ ψ ψ

ψ ε ψ

= − = ∂ + − ∂ −

= ∂ − ∂ + − = ∂ − ∂ +

= ∂ − ∂ + = ∂ − ∂ −

(3.70)

The antisymmetric part of the transposed displacement gradient is denoted as

( )12ij j i i ju uω = ∂ − ∂ (3.71)

thus

[ ]ij ij ijγ ψ ω= − (3.72)

We may define the axial vector that corresponds to ijω ,

ij ijk kω ε ω= − (3.73)

with

; 1k k k kω ωµ µ µ= = (3.74)

With that, eq. (3.73) becomes,

( )

[ ]ij ij ij ijk k ijk k

ijk k k

γ ψ ω ε ψ ε ω

ε ψ ω

= − = − +

= − − (3.75)

Summarizing the above results, from eqs. (3.67) to (3.75) we get,

( )( )

ij ij ij ij

ij ijk k k

γ ε ψ ω

ε ε ψ ω

= + −

= − − (3.76)

3.5 2D Cosserat kinematics in Cartesian coordinates

As an application we assume a 2D setting. In this case we have the following placements

[44],

1 1 2 2

3 3

u u e u eeψ ψ

= +=

(3.77)

The relative deformation tensor becomes


11 1 1

12 1 2 12 1 2 123 3 1 2

21 2 1 21 2 1 213 3 2 1

22 2 2

uu u uu u uu

γγ ψ ε ψ ψγ ψ ε ψ ψγ

= ∂= ∂ + = ∂ − = ∂ −= ∂ + = ∂ − = ∂ += ∂

(3.78)

The curvature tensor becomes

11 12 13 1 3 1

21 22 23 2 3 2

11 22 33

0 ;0 ;

0

κ κ κ ψ ψκ κ κ ψ ψκ κ κ

= = = ∂ = ∂= = = ∂ = ∂= = =

(3.79)

The compatibility conditions, eq. (3.27) and eq. (3.33), yield

(1)

33 3 3 321 2 13 312 1 23

2 13 1 23 0pq p qI ε κ ε κ ε κ

κ κ

= ∂ = ∂ + ∂

= −∂ + ∂ = (3.80)

(2)

31 3 1 13 312 1 21 321 2 11 13

1 21 2 11 13

(2)

32 3 2 23 312 1 22 321 2 12 23

1 22 2 12 23

0

0

jk j k

jk j k

I

I

ε γ κ ε γ ε γ κ

γ γ κ

ε γ κ ε γ ε γ κ

γ γ κ

= ∂ − = ∂ + ∂ −

= ∂ − ∂ − =

= ∂ − = ∂ + ∂ −

= ∂ − ∂ − =

(3.81)

Introducing the infinitesimal strain tensor and the infinitesimal background spin tensor,

( )11 1 1

12 21 1 2 2 1

22 2 2

12

u

u u

u

ε

ε ε

ε

= ∂

= = ∂ + ∂

= ∂

(3.82)

( )

( )

11

12 2 1 1 2 123 3

21 1 2 2 1 213 3

22

012120

u u

u u

ω

ω ε ω ω

ω ε ω ω

ω

=

= ∂ − ∂ = − = −

= ∂ − ∂ = − = +

=

(3.83)

we get,


( )

( )

11 11

12 1 2 2 1 12

21 2 1 1 2 21

22 2 2

1212

u u

u u

u

γ ε

γ ψ ε ω ψ

γ ψ ε ω ψ

γ

=

= ∂ ± ∂ − = + −

= ∂ ± ∂ + = − −

= ∂

(3.84)

Thus

( )ij ijγ ε= (3.85)

and

( )

( )( ) ( )

[12] 12 21

1 2 2 1 1 2 2 1

12

1 12 2

u u u u

γ γ γ

ψ ψ ψ

ω ψ

= −

= ∂ − − ∂ + = ∂ − ∂ −

= −

(3.86)

In order to visualize this decomposition, we consider a line element PQ→

that is originally

parallel to the 1x -axis, eq. (3.59), with

1

2

10

dxdx

dx⎧ ⎫ ⎧ ⎫

=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭

(3.87)

With,

[ ] 11 21

12 22

T γ γγ

γ γ⎡ ⎤

= ⎢ ⎥⎣ ⎦

(3.88)

we get from eq. (3.63),

1 11 21 11

2 12 22 12

10

dx dxdx

dx dxγ γ γγ γ γ

∆⎧ ⎫ ⎡ ⎤ ⎧ ⎫⎧ ⎫= =⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎢ ⎥∆ ⎩ ⎭⎩ ⎭ ⎣ ⎦ ⎩ ⎭

(3.89)

We consider a line element PR→

that is originally parallel to the 2x -axis,

1

2

01

dxdy

dx⎧ ⎫ ⎧ ⎫

=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭

(3.90)

Then

1 11 21 21

2 12 22 22

10

dx dydy

dx dyγ γ γγ γ γ

∆⎧ ⎫ ⎡ ⎤ ⎧ ⎫⎧ ⎫= =⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎢ ⎥∆ ⎩ ⎭⎩ ⎭ ⎣ ⎦ ⎩ ⎭

(3.91)


In Figure 3-4 we show the geometrical visualization of the deformation of the solid

orthogonal element ,PQ PR→ →

, that is computed from eqs. (3.89) and (3.91).

Figure 3-4: The deformation of a solid orthogonal element

From this figure it becomes clear that the diagonal terms of the relative deformation

matrix describe normal strains,

( )( ) ( ) ( ) ( )

( )

2 2 211 12 11 11

1111 1 1 11

( ) 1 1 2 1

1( ) ( )( )

PQ dx dx dx dx

dx dxPQ PQ uPQ dx

γ γ γ γ

γγ ε

′ = + + ≈ + ≈ +

+ −′ −≈ = = ∂ =

(3.92)

Similarly we get that

22 2 2 22( ) ( )

( )PR PR u

PRγ ε

′ −≈ = ∂ = (3.93)

Angular strains are given by,

( ) ( )12 21

12 21 12 2111 22

( ) 2 22 1 1

dx dyQ PRdx dy

γ γπ γ γ ε ε γγ γ

′ ′− ≈ + ≈ + = = =+ +

≺ (3.94)

We return now to eq. (3.64) and we consider a deformation that is generated by [ ]ijγ

[11] [21] [21]1 1 1

[12] [22] [12]2 2 2

00

dx dx dxdx dx dx

γ γ γγ γ γ

∆ ⎡ ⎤ ⎡ ⎤⎧ ⎫ ⎧ ⎫ ⎧ ⎫= =⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥∆⎩ ⎭ ⎩ ⎭ ⎩ ⎭⎣ ⎦ ⎣ ⎦

(3.95)


or

( )( )( )

( )( )

( )

213 3 31 1

123 3 32 2

3 3 1

3 3 2

1

2

00

00

00

dx dxdx dx

dxdx

dxdx

ε ψ ωε ψ ω

ψ ωψ ω

ω ψω ψ

− −⎡ ⎤∆⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎢ ⎥− −∆⎩ ⎭ ⎩ ⎭⎣ ⎦

−⎡ ⎤ ⎧ ⎫= ⎨ ⎬⎢ ⎥− − ⎩ ⎭⎣ ⎦

− −⎡ ⎤ ⎧ ⎫= ⎨ ⎬⎢ ⎥− ⎩ ⎭⎣ ⎦

(3.96)

We visualize this motion using again the concept of the rigid orthogonal element (Figure

3-5): For the line element PQ we get,

( )

( )1

2

0 1 00 0

dxdx

ω ψω ψ ω ψ

− −⎡ ⎤∆⎧ ⎫ ⎧ ⎫ ⎧ ⎫= =⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎢ ⎥−∆ −⎩ ⎭ ⎩ ⎭⎩ ⎭ ⎣ ⎦

(3.97)

and for the line element PR we get,

( )

( )( )1

2

0 00 1 0

dxdx

ω ψ ω ψω ψ

− −⎡ ⎤∆ ⎧− − ⎫⎧ ⎫ ⎧ ⎫= =⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎢ ⎥−∆ ⎩ ⎭⎩ ⎭ ⎩ ⎭⎣ ⎦

(3.98)

Figure 3-5: The relative rotation of a solid orthogonal element

As is shown in Figure 3-6, this motion is the relative rotation of the polar material points

in the neighborhood of point P that is due to the displacement field with respect to the

rotation that is due to their spin. In Figure 3-7, for the visualization of the curvature of the


Cosserat deformation we consider the relative rotation of the rigid crosses attached at

points Q and R , with respect to the rigid cross attached at point P ,

( ) ( )( ) ( )

13 1

23 2

Q P x

R P x

ψ ψ κ

ψ ψ κ

≈ + ∆

≈ + ∆ (3.99)

The curvature tensor is seen as a measure of the bend of the neighbourhood of point P .

Figure 3-6: Visualization of the relative spin

Figure 3-7: Visualization of the curvature of the deformation


3.6 Exercise: Deformation of Cosserat continuum in special curvilinear coordinates

3.6.1 Polar cylindrical coordinates

Figure 3-8: Cartesian and polar cylindrical coordinates

The polar cylindrical coordinates of a point ( , , )P r zθ , are related to its Cartesian

coordinates by the following set of equations (Figure 3-8),

1 1 2

2 1 2

3 3

cos cos (0 2 )sin sin

x x ry x rz x

θ θ π

θ

= = Θ Θ = ≤ ≤

= = Θ Θ =

= = Θ

(3.100)

for (0, )r ∈ ∞ and [0,2 )θ π∈ .

Prove that the deformation tensors in cylindrical polar cylindrical coordinates are as

follows,

[ ] 1 1 1

r z

r r z

r z

uu ur r r

u uu u uu

r r r r ruu u

z z z

θ

θ θ

θ

θ θ θ

∂∂ ∂⎡ ⎤⎢ ⎥∂ ∂ ∂⎢ ⎥

∂∂ ∂⎢ ⎥∇ = − +⎢ ⎥∂ ∂ ∂⎢ ⎥∂∂ ∂⎢ ⎥⎢ ⎥∂ ∂ ∂⎣ ⎦

(3.101)


[ ]

1 1 12 2

1 1 1 1 12 2

1 1 12 2

rr r rz

r z

zr z zz

r r r z

r r z

r z z z

sym

u uu u u ur r r r z r

u u u uu u ur r r r r z r

uu u u uz r z r z

θ

θ θθ θ

θ

θ θ

θ θ θ θ

θ

ε ε εγ ε ε ε

ε ε ε

θ

θ θ θ

θ

⎡ ⎤⎢ ⎥= =⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ∂∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞+ − +⎢ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠⎢⎢ ∂ ∂ ∂∂ ∂⎛ ⎞ ⎛ ⎞= + − + +⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

∂∂ ∂ ∂ ∂⎛ ⎞⎛ ⎞+ +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣

⎤⎥⎥⎥

⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎦

(3.102)

[ ]

1 1 102 2

1 1 1 102 2

1 1 1 02 2

r z rz

r zz r

r z zr

asym

u uu u ur r r r z

u u uu ur r r r z

uu u uz r r z

θ θθ

θ θ θ

θθ

γ

ψ ψθ

ψ ψθ θ

ψ ψθ

=

⎡ ⎤∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞− + − − +⎜ ⎟⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠⎢ ⎥⎢ ⎥∂ ∂∂ ∂⎛ ⎞ ⎛ ⎞= − − + + − −⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎢ ⎥⎢ ⎥∂∂ ∂ ∂⎛ ⎞⎛ ⎞− − − − − +⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

(3.103)

[ ] 1 1 1

r z

r r z

r z

r r r

r r r r r

z z z

θ

θ θ

θ

ψψ ψ

ψ ψψ ψ ψκ

θ θ θψψ ψ

∂∂ ∂⎡ ⎤⎢ ⎥∂ ∂ ∂⎢ ⎥

∂∂ ∂⎢ ⎥= − +⎢ ⎥∂ ∂ ∂⎢ ⎥∂∂ ∂⎢ ⎥⎢ ⎥∂ ∂ ∂⎣ ⎦

(3.104)


3.6.2 Polar spherical coordinates

Figure 3-9: Cartesian and polar spherical coordinates

The polar spherical coordinates of a point be ( , , )P r θ φ are related ti its Cartesian

coordinates by the following set of equations (Figure 3-9)

1 1 2 3

2 1 2 3

3 1 2

sin cos sin cossin sin sin sincos cos

x x ry x rz x r

θ φ

θ φ

θ

= = Θ Θ Θ =

= = Θ Θ Θ =

= = Θ Θ =

(3.105)

for (0, )r ∈ ∞ , [0, )θ π∈ and [0,2 )ϕ π∈ .

Prove that the deformation tensors in polar spherical coordinates are as follows,

[ ]

1 1 1 12 2 sin

1 1 1 1 1 1 cot2 2 sin

1 1 1 1 1 cot2 sin 2 sin

r r r

r r

r

sym

u uu uu u ur r r r r r r

uu u u uu uu

r r r r r r r r

u u uuuu

r r r r r r

φ φθ θ

φθ θ θ θφ

φ φ φθ

γ

θ θ φ

θθ θ θ φ θ

θθ φ θ φ θ

=

∂⎛ ⎞∂∂ ∂ ∂⎛ ⎞+ − + −⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

∂⎛ ⎞∂ ∂ ∂∂⎛ ⎞+ − + + −⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

∂ ∂⎛ ⎞ ∂∂+ − + −⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

1 cotsin

ru uur r r

φ θφ θ

θ φ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥∂⎛ ⎞ ⎛ ⎞⎢ ⎥+ +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥∂⎝ ⎠ ⎝ ⎠⎣ ⎦ (3.106)


[ ]1 1 1 102 2 sin

1 1 1 1 1 cot02 2 sin

1 1 1 1 1 cot2 sin 2 sin

r r

rr

r

asym

u uu uu ur r r r r r

uu u uuu

r r r r r r

u u u uuu

r r r r r r

φ φθ θφ θ

φθ θ θφ φ

φ φ φ θθ φ

γ

ψ ψθ θ φ

θψ ψθ θ θ φ

θψθ φ θ θ φ

=

∂⎛ ⎞∂ ∂ ∂⎛ ⎞− + + − + −⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠∂⎛ ⎞∂ ∂∂⎛ ⎞− − + − − + +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

∂ ∂⎛ ⎞ ⎛ ∂∂− − + + − − +⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝

0rψ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

⎞⎢ ⎥−⎜ ⎟⎢ ⎥⎢ ⎥⎠⎣ ⎦ (3.107)

1 1 1[ ]

1 1 1 1 1sin sin tan sin tan

r

r r

r r

r r r

r r r r r

r r r r r r r

φθ

φθ θ

φ φθθ θ

ψψψ

ψψ ψψ ψκ

θ θ θψ ψψψ ψ

ψ ψθ φ θ φ θ θ φ θ

∂⎡ ⎤∂∂⎢ ⎥∂ ∂ ∂⎢ ⎥

∂⎢ ⎥∂∂= − +⎢ ⎥

∂ ∂ ∂⎢ ⎥⎢ ⎥∂∂∂

− − + +⎢ ⎥∂ ∂ ∂⎢ ⎥⎣ ⎦

(3.108)


4 Cosserat continuum statics

4.1 The virtual work equation

We consider a Cosserat continuum Β, that occupies a domain with volume V that has the

boundary V∂ . Body Β is assumed to be in a state of stress in static equilibrium. In order

to formulate the equilibrium conditions we consider fields ( )i kuδ Θ and ( )i kδψ Θ , that are

defined uniquely at all points of the given body. These fields will be called virtual

particle displacement and virtual particle rotation fields, respectively and it will be

assumed that they are sufficiently differentiable. We define the virtual relative

deformation tensor,

ik ikk iuδγ δ δψ= + (4.1)

where

lik ikleδψ δψ= − (4.2)

and the virtual curvature tensor

ik k iδκ δψ= (4.3)

We define the tensor fields ( )ij mσ Θ and ( )ij mµ Θ , through the so called virtual work of

the internal forces, that is in turn defined per unit volume of the considered continuum,

(int) ik ikik ikwδ σ δγ µ δκ= + (4.4)

We assume that (int)wδ is an invariant scalar quantity. In this case the quantities σ and µ

are called the stress- and couple-stress tensors, respectively. From the point of view of

continuum thermodynamics the stress- and couple stress tensors are intensive quantities,

that are dual in energy to the deformation gradient an spin, that are in turn the

corresponding mechanical extensive quantities of the considered continuum14.

We remark at this point that the above definition of the virtual work of internal forces is

consistent with the Cosserat continuum energetics, that are discussed below in sect. 6.

14 An intensive property (also called a bulk property) is a physical property of a system that does not depend on the system size or the amount of material in the system. By contrast, an extensive property of a system does depend on the system size or the amount of material in the system


In view of eq. (3.76) we decompose additively the virtual relative deformation into

symmetric and antisymmetric part

( ) [ ]ij ij ijδγ δγ δγ= + (4.5)

where

( ) ( )

( )

( )

[ ]

1 12 21 12 2

k l l kij i j i j kl ij ji ij

klpij ijp kl ij ji ij ije e

δγ δ δ δ δ δγ δγ δγ δε

δγ δγ δγ δγ δψ δω

⋅ ⋅ ⋅ ⋅= + = + =

= = − = − (4.6)

or

( )( )( )

ij ij ij ij

k kij ijk

k kij ijk

e

e

δγ δε δψ δω

δε δψ δω

δε δω δψ

= + −

= − −

= + −

(4.7)

where

( )12ij j i i ju uδε δ δ= + (4.8)

( )12

kij ijk i j j ie u uδω δω δ δ= − = − (4.9)

and

kij ijkeδψ δψ= − (4.10)

Thus

( )[ ]k k

ij ijkeδγ δω δψ= − (4.11)

Similarly we decompose additively the stress tensor into symmetric and antisymmetric

part

( ) [ ]ij ij ijσ σ σ= + (4.12)

where

( ) ( )

( ) ( )

( )

[ ]

1 12 21 1 12 2 2

ij i j i j kl ij jik l l k

ij ijp kl i j i j kl ij jiklp k l l ke e

σ δ δ δ δ σ σ σ

σ σ δ δ δ δ σ σ σ

⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅

= + = +

= = − = − (4.13)

With this decomposition the virtual work of the internal forces, eq. (4.4), becomes,


( )( )

(int) ( ) [ ][ ]

( ) [ ]

( ) [ ]

ij ij ijij ij ij

ij ij ijij ij ij ij

ij ij k k ijij ijk ij

w

e

δ σ δε σ δγ µ δκ

σ δε σ δψ δω µ δκ

σ δε σ δω δψ µ δκ

= + +

= + − +

= + − +

(4.14)

Eq. (4.14) is summarized in the following,

Theorem In a Cosserat continuum the virtual work of the internal forces is such that: a) the

symmetric part of the stress tensor ( )ijσ is dual in energy to the strain ( )ij i juε = (i.e. to the

symmetric part of the displacement gradient), b) the antisymmetric part of the stress

tensor [ , ]i jσ is dual in energy to the relative spin15 ( )k kijke δψ δω− − , and c) the couple

stress tensor ijµ is dual in energy to the distortion tensor ij j iδκ δψ= .

We remark that the term in eq. (4.14) that pertains to the virtual work of the

antisymmetric part of the stress tensor can be written as follows,

( )( )( )

( )

[ ] [ ][ ]

[ ]

*2

ij ijij ij ij

ij m mijm

m mm

e

t

σ δγ σ δψ δω

σ δψ δω

δω δψ

= −

= − −

= −

(4.15)

where it∗ is the axial vector, that corresponds to the non-symmetric part of the stress

tensor.

[ ] [ ]1 12 2

jk jk jk ijki ijk ijk it e e e tσ σ σ∗ ∗= = ⇔ = (4.16)

With this remark eq. (4.14) becomes

( )(int) ( ) *2ij i i ijij i ijw tδ σ δε δω δψ µ δκ= + − + (4.17)

The total work of the internal forces is defined as the integral of the density function

(int)wδ over the volumeV ,

(int) (int)

( )V

W w dVδ δ= ∫ (4.18)

15 For this reason we call [ ]ijσ the relative stress tensor (see sect. 5.7 ).


For the formulation of the principle of virtual work (p.v.w.), we assume that on the

considered Cosserat continuum three types of external forces are acting16: a) Volume

forces if dV , b) surface tractions it dS and c) surface couples km dS . In these expressions

dV is the volume element and dS is the surface element. These external actions are

related to the internal forces and it is the virtual work equation that defines this relation.

As we will see in sect. 4.2, the equations that couple locally the internal and the external

forces are the equilibrium equations.

Figure 4-1: Local coordinates in a point at the bounding surface

The bounding material surface V∂ of a material volume V is seen as a two dimensional

smooth point manifold17 with each point of that manifold possessing two vectorial

degrees of freedom, the one of particle displacement and that of particle rotation. At any

point P V∈∂ we define a local coordinate system, say ( )1 2 3, ,α α α , where the

coordinates 1α and 2α , describe the position of the considered point in the bounding

surface and 3α is the coordinate positive along the outward normal to it At the arbitrary

point ( )1 2, ,0P α α on the surface we can define the corresponding covariant basis,

16 In general one may assume the existence of body couples as well; cf. sect. 5.4 . 17 Bounding surfaces with sharp corners and edges are not considered here.


( )1 2 3, ,α α α as shown in Figure 4-1. From that basis we may construct the corresponding

contavariant basis ( )1 2 3, ,α α α 18..

A set of admissible boundary conditions at point ( )1 2, ,0P α α are of purely Diriclet type,

with data projected on the local contravariant basis

[ ] [ ]1 2 3

1 2 3

: : :D NP P

u u uP V S p S q

ψ ψ ψ⎧ ⎫⎧ ⎫ ⎧ × × × ⎫⎡ ⎤ ⎡ ⎤⎪ ⎪ ⎪ ⎪∈∂ = ∪ = ∅ =⎨ ⎨ ⎬ ⎨ ⎬⎬⎢ ⎥ ⎢ ⎥× × ×⎪ ⎪ ⎣ ⎦⎣ ⎦⎪ ⎪⎩ ⎭⎩ ⎭⎩ ⎭

(4.19)

Accordingly Neumann type boundary conditions are expressed on the covariant basis,

[ ] [ ]1 2 3

1 2 3: : :D NP P

t t tP V S p S q

m m m

⎧ ⎫⎧ ⎫⎧ × × × ⎫ ⎡ ⎤⎡ ⎤⎪ ⎪ ⎪⎪∈∂ = ∅ = ∪ =⎨ ⎨ ⎬ ⎨ ⎬⎬⎢ ⎥⎢ ⎥× × × ⎪ ⎪⎣ ⎦ ⎣ ⎦⎪ ⎪⎩ ⎭ ⎩ ⎭⎩ ⎭ (4.20)

Mixed-type boundary conditions are also allowed, however if some information, say ijp

is given no information concerning ijq can be given et vice versa; e.g.

[ ] [ ]1 2

31 2

3

: : :D NP P

u t tP V S p S q

m mψ

⎧ ⎫⎧ ⎫⎧ × × ⎫ ⎡ ⎤⎡ ⎤ ×⎪ ⎪ ⎪ ⎪ ⎪ ⎪∈∂ = ∪ =⎨ ⎨ ⎬ ⎨ ⎬ ⎬⎢ ⎥⎢ ⎥× × ×⎪ ⎪ ⎪ ⎪⎣ ⎦ ⎣ ⎦⎪ ⎪⎩ ⎭ ⎩ ⎭⎩ ⎭ (4.21)

In the example given above by eq. (4.21) in the neighbourhood of point ( )1 2, ,0P α α and

along the αα -surface lines ( 1, 2)α = (i.e. in the tangential plane) tractions and couples

are prescribed, whereas in normal to the surface direction the displacement and the spin

are restricted.

On the basis of the above definitions we define a functional that is called the virtual work

of external forces,

( )( )

( ) ( )N

ext i i ii i i

V S

W f u dV t u m dSδ δ δ δψ= + +∫ ∫ (4.22)

We assume that on DS , the virtual kinematics vanish,

0 0i iuδ δψ= ∧ = (4.23)

18 For a concise presentation of the geometry of a surface in the Euclidean space we refer to: McConnell, A.J., Applications of Tensor Analysis, Ch. XIV, Dover, 1957, and to: Klingbeil, E., Tensorrechnung für Ingenieure, Kap. 4, BI, 1966.


We assume that these data are continuously extended into V and on the disjoint parts of

the boundary. Thus the functional can be extended over the whole boundary,

( )( )

( ) ( )

: 0 0

ext i i ii i i

V V

D i i

W f u dV t u m dS

on S u

δ δ δ δψ

δψ δ∂

= + +

= ∧ =

∫ ∫ (4.24)

On the basis of the above definitions the p.v.w. in a Cosserat continuum is defined as

follows: The system , ; , , ij ij i i if t mσ µ is called an equilibrium set, if for any choice of

the virtual fields of displacement and rotation that satisfy eq. (4.23), the virtual work

equation holds,

( ) (int)extW Wδ δ= (4.25)

From eq. (4.25) and the definitions for the virtual work of internal- and external forces,

eqs. (4.18), (4.4) and (4.24), we obtain the following integral equation, that is the

expression of the virtual work equation in a Cosserat continuum:

( )( ) ( ) ( ) ( )

i i i ik iki i i ik ik

V V V V

f u dV t u dS m dS dVδ δ δψ σ δγ µ δκ∂ ∂

+ + = +∫ ∫ ∫ ∫ (4.26)

4.2 Equilibrium equations in curvilinear coordinates

We remark first that the density of the virtual work of the internal forces can be written as

follows,

( )

( ) ( )(int) ik ik

ikk i k i

ik ik ik ik ik lk k k k ikli ii

w u

u u e

δ σ δ δψ µ δψ

σ δ µ δψ σ δ µ δψ σ δψ

= + +

= + − + − (4.27)

With the notation

i ik ikk kq uσ δ µ δψ= + (4.28)

we observe that

|iidivq q= (4.29)

and with the use of Gauss’ theorem we get,


( )( )

ik ikk k i

V V vik ik

k k iV

u dV divq dV q n dS

u n dS

σ δ µ δψ

σ δ µ δψ

∂

∂

+ = = ⋅

= +

∫ ∫ ∫

∫ (4.30)

With eq. (4.30) the virtual work equation (4.26) becomes

( )

( )

( ) ( ) ( )

( ) ( )

k k kk k k

V V V

ik ikk k i

Vik ik lk im

k iml ki iV V

f u dV t u dS m dS

u n dS

u dV e g dV

δ δ δψ

σ δ µ δψ

σ δ µ σ δψ

∂ ∂

∂

+ + =

+ −

− − +

∫ ∫ ∫

∫

∫ ∫

(4.31)

or

( ) ( )

( ) ( )( ) ( )

( ) ( )

ik k ik lk imk iml ki i

V V

ik k ik ki k i k

V V

f u dV e g dV

n t u dS n m dS

σ δ µ σ δψ

σ δ µ δψ∂ ∂

+ + + =

− + −

∫ ∫

∫ ∫ (4.32)

The test functions ( )kkuδ Θ and ( )k

kδψ Θ can be chosen arbitrarily. In particular they

may be chosen in such a way that from eq. (4.32) the following equations follow,

( )

( )( )

( )

0 '

0 '

ik kki

V

ik lk imiml ki

V

f u dV V V

e g dV V V

σ δ

µ σ δψ

′

′

+ = ∀ ⊆

+ = ∀ ⊆

∫

∫ (4.33)

( )

( )( )

( )

0 '

0 '

ik ki k

V

ik ki k

V

n t u dS V V

n m dS V V

σ δ

µ δψ

′∂

′∂

− = ∀∂ ⊆ ∂

− = ∀∂ ⊆ ∂

∫

∫ (4.34)

These equations result finally to the following set of stress equilibrium equations,

0 ( )ik k ii f P Vσ + = ∀ Θ ∈ (4.35)

( )ik k iin t P Vσ = ∀ Θ ∈∂ (4.36)

and moment stress equilibrium equations,


0 ( )ik lk im iimli e g P Vµ σ+ = ∀ Θ ∈ (4.37)

( )ik k iin m P Vµ = ∀ Θ ∈∂ (4.38)

We remark here that in [23] we find the following equivalent form of eq. (4.37),

0 0ik lk im i imkp iml kp impi p ig e g g eµ σ µ σ⋅+ = ⇒ + = (4.39)

4.3 Equilibrium equations in Cartesian coordinates

We apply eqs. (4.35) to (4.38) for a Cartesian description, thus yielding

ik i in tσ = (4.40)

0i ik kfσ∂ + = (4.41)

and

ik i kn mµ = (4.42)

0i ik imk imµ ε σ∂ + = (4.43)

We observe that the equilibrium equations (4.40) and (4.41) are identical to the ones

holding for the Boltzmann continuum and that the equilibrium equations (4.40) and

(4.42) introduce the stress- and couple stress tensors as lineal densities for the internal

forces in the sense of Cauchy. Due to the moment equilibrium equation (4.43), however,

the stress tensor in a Cosserat continuum is in general non-symmetric.

Figure 4-2:Stress and couple stress in the sense of Cauchy in 2D.

As an example we apply the above equilibrium equations for a 2D setting, thus yielding

[44] (Figure 4-2),


1 11 1 21 2

2 12 1 22 2

3 13 1 23 2

t n nt n nm n n

σ σσ σµ µ

= += += +

(4.44)

and (Figure 4-3)

11 211

1 2

12 222

1 2

13 2312 21

1 2

0

0

0

fx x

fx x

x x

σ σ

σ σ

µ µ σ σ

∂ ∂+ + =

∂ ∂∂ ∂

+ + =∂ ∂

∂ ∂+ + − =

∂ ∂

(4.45)

Figure 4-3: Stress and moment stress equilibrium in 2D

4.4 The Mohr circle of non-symmetric stress in 2D19

We consider a 2D state of stress in the 1 2( , )O x x -plane and we identify 1x x= , 2x y= . As

shown in Figure 4-4 we may introduce another coordinate system ( , )O ξ η that results

from the original coordinate system ( , )O x y by a rotation of the coordinate axes by an

angle ϕ and we want to compute the components of the stress vector in planes parallel to

the rotted system.

19. For a 3D non-symmetric state of stress the concept of the Mohr “circle” is meaningfull only for very special cases [41]. However, as we will see in this section, it is always meaningful for a 2D setting.


Figure 4-4: Plane state of stress

As can be seen from Figure 4-4 the normal and tangential vectors on the plane that is

normal to the ξ − axis are,

1 2

1 2

cos ; sinsin ; cos

n nm m

ϕ ϕϕ ϕ

= == − =

(4.46)

and with that

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1 1 1cos 2 sin 22 2 2

1 1 1sin 2 cos 22 2 21 1 1sin 2 cos 22 2 2

1 1 1cos 2 sin 22 2 2

xx yy xx yy yx xy

xx yy xy yx xy yx

xx yy xy yx xy yx

xx yy xx yy yx xy

ξξ

ξη

ηξ

ηη

σ σ σ σ σ ϕ σ σ ϕ

σ σ σ ϕ σ σ σ σ ϕ

σ σ σ ϕ σ σ σ σ ϕ


= + + − + +

= − − + − + +

= − − − − + +

= + − − − +

(4.47)

We observe that, if the stress tensor is symmetric ( xy yx ξη ηξσ σ σ σ= ⇔ = ), then the

transformation rule, eq. (4.47), collapses to the one known from Boltzmann Continuum

Mechanics.


Figure 4-5: Normal and shear stresses on an arbitrary plane

In order to check the applicability of concept of the Mohr circle of stresses for non-

symmetric states of stress, we consider here the geometric locus of the normal- and shear

stress vector, acting on a plane with unit outward normal in with the angle ϕ as curve

parameter (Figure 4-5),

( ) ( ) ( )

( ) ( ) ( )

1 1 1cos 2 sin 22 2 21 1 1sin 2 cos 22 2 2

n xx yy xx yy yx xy

n xx yy xy yx xy yx


τ σ σ ϕ σ σ σ σ ϕ

= + + − + +

= − − + − + + (4.48)

Let the mean normal stress is an in-plane invariant and is denoted as

( ) ( )1 12 2M xx yy ξξ ηησ σ σ σ σ= + = + (4.49)

The shear stress difference,

( ) ( )31 12 2a xy yxt ξη ηξτ σ σ σ σ∗= = − = − (4.50)

is also an invariant and measure of the stress-tensor asymmetry20.

With this notation we get from eq. (4.48),

20 cf. eq. (4.17).


( )

( )

( )

( )

1 cos 2 sin 221 sin 2 cos 22

n n M xx yy xy

n n a xx yy xy

s

t

σ σ σ σ ϕ σ ϕ

τ τ σ σ ϕ σ ϕ

= − = − +

= − = − − + (4.51)

Thus

2

2 2 2( )2

xx yyM n n xys t

σ στ σ

−⎛ ⎞= + = +⎜ ⎟

⎝ ⎠ (4.52)

( )( )tan 2n

n

ts

ϕ ξ= − (4.53)

where

( )2tan 2 xy

xx yy

σξ

σ σ=

− (4.54)

Figure 4-6: Mohr circle for a non-symmetric stress tensor in 2D

As can be seen from Figure 4-6, Mτ is the radius of the Mohr circle. The center

( , )M aσ τΜ of the Mohr circle is shifted normal to the nOσ -axis by the amount of the

asymmetry of the stress tensor, given by aτ .This observation allows us to transfer some

geometrical results from Mohr-circle analysis, holding for symmetric stress, to the 2D


Mohr-circle holding for non-symmetric stress. For example the pole of normals Π is

defined as in the Boltzmann continuum. From Figure 4-6 we can also compute the

angular displacement of the center Μ of the Mohr circle, expressed by the stress

obliquity of the antisymmetric part of the stress tensor,

( )tan tan ( ) xy yxaa

M xx yy

σ στφσ σ σ

−′= Ο ΟΜ = =

+≺ (4.55)

The opening angle of the tangent lines, drwn from the origin to the shifted Mohr circle is

2 sφ , where

( ) ( ) ( )( )

( )( ) ( )2 2

2 2 2

tan tan ( ) tan ( )

tan

s

Ms

M a M

C CC C

C Cφ

τφσ τ τ

Μ Μ′= ΜΟ = ΜΟ = = ⇒

Ο ΟΜ − Μ

=+ −

≺ ≺

(4.56)

With,

( )

( )

2 2( )

2

4tan xx yy xyM

MM xx yy

σ σ στφσ σ σ

− += =

+ (4.57)

eq. (4.56) becomes,

2 2

tantan1 tan tan

Ms

a M

φφφ φ

=+ −

(4.58)

We observe that for symmetric states of stress we retrieve a well known result,

2

tan0 tan1 tan

Ma s

M

φφ φφ

= ⇒ =−

(4.59)

or

( )

( )

2 2( )

2

4sin tan xx yy xy

s M

xx yy

σ σ σφ φ

σ σ

− += =

+ (4.60)

With eqs. (4.55) and (4.57) eq. (4.58)becomes,

( ) ( )2 2

1tan2

xx yy xy yxs

xx yy xy yx

σ σ σ σφ

σ σ σ σ

− + +=

− (4.61)


The slopes of these tangents with respect to to the nσ - axis are,

( )

2

2 2 2

2

2 2 2

tan tantan1 tan tan

1

aM

M a M Ms as a

s a aM

M a M M

ττσ τ τ σφ φφ φ

φ φ ττσ τ τ σ

±+ −±

± = =

+ −

∓∓

(4.62)

With the notation,

; 0 ;M M ap q rσ τ τ= = ≥ = (4.63)

and

tan ; tana sφ λ φ µ= = (4.64)

from eqs. (4.55) and (4.56) we get,

( )2 2

22 2 2

tan

tan1

aa

M

Ms

M a M

r p

q p r

τφ λσ

τ µφµσ τ τ

= ⇒ =

= ⇒ = +++ −

(4.65)

With that from eq. (4.62) we get,

( ) tan tantan1 tan tan 1

s as a

s a

φ φ µ λφ φφ φ µλ± ±

± = =∓ ∓

(4.66)

If 0aτ ≥ ( 0 , 0aφ λ> > ), then the critical Mohr-Coulomb envelope has the slope angle,

s aφ φ+ ; for 0aτ < ( 0 , 0aφ λ< < ), the critical value is s aφ φ− . Thus we set

1

mµ λ

µ λ+

=−

(4.67)

If we assume that m is a constitutive friction parameter, we get a constraint equation

between the friction parametrs

01m m

mµλ µµ

−= ⇒ < <

+ (4.68)

This means that

01

r m pmr p r m pm r m p

µ µµ

− +−= ⇒ = ≥ ⇒ ≤

+ + (4.69)

From eqs. (4.65) and (4.69) we get


( )( )2

1 ;1

q p r m p m r r m pm

= + − ≤+

(4.70)

In ( , , )p q r -space of stresses this is a compound conical surface, as shown in Figure 4-7.,

that degenerates into straight lines on the varios coordinate planes,

2

10 : sin1 sfor r q p p

mφ= = =

+ (4.71)

and (Figure 4-8),

20 : sin sfor q r m p pφ= = ≈ (4.72)

Figure 4-7: Limit condition in the ( , , )p q r -stress space of the Mohr-Colomb type for a 2D frictional Cosseart material ( 0.5m = ).

Figure 4-8: Relation between friction coefficients m and sin sφ


4.5 Exercise: Differential equilibrium equations in special curvilinear coordinates

4.5.1 Polar cylindrical coordinates Prove that the equilibrium equations for a Cosserat continuum in terms of physical

components in polar, cylindrical coordinates are the following:

( )

( )

1 1 0

1 1 0

1 1 0

rrr zrrr r

r zr r

zrz zzrz z

fr r r z

fr r r z

fr r r z

θθθ

θ θθ θθ θ θ

θ

σσ σσ σθ

σ σ σσ σ

θσσ σσ

θ

∂∂ ∂+ + − + + =

∂ ∂ ∂∂ ∂ ∂

+ + + + + =∂ ∂ ∂

∂∂ ∂+ + + + =

∂ ∂ ∂

(4.73)

and

( )

( )

1 1 0

1 1 0

1 1 0

rrr zrrr z z r

r zr r zr rz

zrz zzrz r r z

r r z r

r r z r

r r z r

θθθ θ θ

θ θθ θθ θ θ

θθ θ

µµ µ µ µ σ σθ

µ µ µµ µ σ σ

θµµ µ µ σ σθ

∂∂ ∂+ + + − + − + Φ =

∂ ∂ ∂∂ ∂ ∂

+ + + + + − + Φ =∂ ∂ ∂

∂∂ ∂+ + + + − + Φ =

∂ ∂ ∂

(4.74)

4.5.2 Polar spherical coordinates Prove that the equilibrium equations for a Cosserat continuum in terms of physical

components in polar, cylindrical coordinates are the following:

( )

( ) ( )

( ) ( )

1 1 cot 1 2 0sin

1 1 1 cot2 0sin

1 1 1 cot 0sin

rrrr fr rr rr r r r r

r fr rr r r r r

r fr rr r r r r

σσσ θφθ σ σ σ σθ φφ θθθ θ φσσ σ θφθθ θθ σ σ σ σθ θ θθ φφ θθ θ φ

σ σ σ θφ θφ φφ σ σ σ σφ φ θφ φθ φθ θ φ

∂∂∂+ + + + − − + =

∂ ∂ ∂∂∂ ∂

+ + + + + − + =∂ ∂ ∂

∂ ∂ ∂+ + + + + + + =

∂ ∂ ∂

(4.75)

( )

( ) ( )

( ) ( )

1 1 cot 1 2 0sin

1 1 1 cot2 0sin

1 1 1 cot2 0sin

rrrrr rr rr r r r r

rr r r rr r r r r

rr r r rr r r r r

µµµ θφθ µ µ µ µ σ σθ φφ θθ φθ θφθ θ φµµ µσ θφθθ θθ µ µ µ µ σ σθ θ θθ φφ φ φ θθ θ φ

µ µ µ θφ θφ φφ µ µ µ µ σ σφ φ θφ φθ θ θ φθ θ φ

∂∂∂+ + + + − − + − + Φ =

∂ ∂ ∂∂∂

+ + + + + − + − + Φ =∂ ∂ ∂

∂ ∂ ∂+ + + + + + + − + Φ =

∂ ∂ ∂

(4.76)


5 Cosserat continuum dynamics The equations that describe mass balance and balance of linear momentum in a Cosserat

continuum are the same as the ones holding for a Boltzmann continuum. The difference

between the two types of continua arises while considering the action of the extra dofs of

the Cosserat continuum, i.e. in the formulation of the momentum balance- and energy

balance equations. For completeness we derive here also the equations that describe

balance of mass and balance of linear momentum.

5.1 Balance of mass

The material point of a Cosserat continuum is equipped with a linear particle velocity

i

i DuvDt

= (5.1)

We remark that within a small deformation theory we have that

i i i i i

k k ikk k

Du u u u uv vDt t x x t

δ⎛ ⎞∂ ∂ ∂ ∂

= + ⇒ − =⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ (5.2)

or

( )1 1

1

k i i k i j i jk t k i t i

j i j i lt i t k

v F u v F F u F

v u F u O u

− −⋅ ⋅ ⋅ ⋅

−⋅

= ∂ ⇒ = ∂

⇒ = ∂ ≈ ∂ + ∂ (5.3)

The mass of the particle is,

dm dVρ= (5.4)

where density ( ),i tρ Θ is the mass density at the considered point. The total mass of a

body B at a given time t is,

( )

( )V

M t dVρ= ∫ (5.5)

Mass balance is expressed by the requirement,

0dMdt

= (5.6)

We recall Reynolds transport theorem,


( )( ) ( )

( ) ( , )k it iV V

dSS t s t dV s sv dVdt

⎛ ⎞= Θ ⇒ = ∂ +⎜ ⎟

⎝ ⎠∫ ∫ (5.7)

Thus, from eqs. (5.5) and (5.6) follows that mass balance is expressed as

( )( )( )

0 0it i

V

v dV V Vρ ρ′

′∂ + = = ∀ ⊂∫ (5.8)

If we assume that mass balance holds for any subdivision of the considered body, then

from eq. (5.8) we get the following local form for the mass balance equation,

( ) 0 ( )i i

iv P V

tρ ρ∂

+ = ∀ Θ ∈∂

(5.9)

Note that if eq. (5.9) holds, then Reynold’s transport theorem applied for the global

quantity

( ) ( )

( )B V

S t sdm sdVρ= =∫ ∫ (5.10)

yields,

( )V

dS sdVdt

ρ= ∫ (5.11)

where s is the material time derivative of the specific quantity ( , )ks tΘ :

kk

Ds ss v sDt t

∂= = +

∂ (5.12)

5.2 Balance of linear momentum

The total force that is acting on a body B at a given time t is,

( ) ( )

( )i i i

V V

F t f dV t dS∂

= +∫ ∫ (5.13)

The total linear momentum of the consider body is,

( )

( )i i

V

t v dVρΙ = ∫ (5.14)

Balance of linear momentum is expressed as,

i

id FdtΙ

= (5.15)

From Reynold’s transport theorem, eq. (5.12), we get that,


( )( ) ( )

ii i k i

t kV V

d d v dV v v v dVdt dt

ρ ρ ⋅

Ι= = ∂ +∫ ∫ (5.16)

Thus from eqs. (5.13) to (5.16) we get,

( )( ) ( ) ( )

i k i i it k

V V V

v v v dV f dV t dSρ ⋅∂

∂ + = +∫ ∫ ∫ (5.17)

We assume that linear momentum balance holds for any subdivision of the considered

body. If we apply eq. (5.17) in particular for the elementary tetrahedron under suitable

mathematical restrictions [24] the volume integrals tend to zero and the remaining surface

integral yields Cauchy’s theorem21,

( )ki i mkn t P Vσ = ∀ Θ ∈∂ (5.18)

From eqs. (5.17) and (5.18) and Gauss’ theorem we get

( )

( ) ( ) ( )

( ) ( )

i i k i kit kk

V V V

i kik

V V

v v v dV f dV n dS

f dV dV V V

ρ σ

σ

⋅′ ′ ′∂

′ ′

∂ + = +

′= + ∀ ⊂

∫ ∫ ∫

∫ ∫ (5.19)

We observe that the material time derivative of the velocity coincides with the particle

acceleration,

i

i i i kt k

Dva v v vDt ⋅= = ∂ + (5.20)

From eq. (5.19) and (5.20) we get the dynamic equations,

i

ki ik

DvfDt

σ ρ+ = (5.21)

We observe that if we assume that the particle acceleration is vanishing, then eqs. (5.21)

reduce to the static equilibrium eqs. (4.35).

5.3 Balance of angular momentum

The total moment of the forces and couples acting on a body B at a given time t is,

( ) ( ) ( )V V V

M tdS fdV mdS∂ ∂

= ℜ× + ℜ× +∫ ∫ ∫ (5.22)

21 cf. eq. (4.36).


where ℜ is the position vector.

On the other hand the total angular momentum is

( )( ) ( )V V

L v dV dVρ ρ= ℜ× + ϑ∫ ∫ (5.23)

where ρϑ is the angular momentum of the spinning polar material point.

Balance of angular momentum is expressed as,

dL Mdt

= (5.24)

Assuming that mass balance is holding we have that,

( )

( )

( ) ( )

V

V V

dL d v dVdt dt

Dv DdV dVDt Dt

ρ

ρ ρ

= ℜ× + ϑ

ϑ⎛ ⎞= ℜ× +⎜ ⎟⎝ ⎠

∫

∫ ∫ (5.25)

We consider the 1st term on the r.h.s. of eq. (5.22), and evaluate it for convenience in a

Cartesian description,

( )

( ) ( )

( )

( ) ( ) ( )

( ) ( )

( )

ijk j k ijk j mk miV V V

ijk m j mk ijk mj mk j m mkV V

ijk jk ijk j m mkV

t dS x t dS x n dS

x dV x dV

x dV

ε ε σ

ε σ ε δ σ σ

ε σ ε σ

∂ ∂ ∂

ℜ× = =

= ∂ = + ∂

= + ∂

∫ ∫ ∫

∫ ∫

∫

(5.26)

Let,

1,2i i i ijk jkt t e t ε σ∗ ∗ ∗= = (5.27)

and with that eq.(5.22) becomes

( ) ( )( )

( ) ( ) ( ) ( )

( ) ( )

2

2

i i ijk j m mk ijk j k ki kV V V V

i k ki ijk j m mk kV V

M t dV x dV x f dV n dS

t dV x f dV

ε σ ε µ

µ ε σ

∗

∂

∗

= + ∂ + +

= + ∂ + ∂ +

∫ ∫ ∫ ∫

∫ ∫ (5.28)

By combining eqs. (5.24), (5.25) and (5.28) we obtain

( )( ) ( ) ( )

2i ki k ki ijk j m mk k

V V V

D DvdV t dV x f dVDt Dt

ρ µ ε σ ρ∗ϑ ⎛ ⎞⎛ ⎞= + ∂ + ∂ + −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

∫ ∫ ∫ (5.29)


If we assume that balance of linear momentum holds, then the last term on the r.h.s. of

eq.(5.29) is vanishing, thus yielding

( )( ) ( )

2ii k ki

V V

D dV t dVDt

ρ µ∗ϑ= + ∂∫ ∫ (5.30)

The local form of eq. (5.30) is

ik ki ikl kl

DDt

µ ε σ ρ ϑ∂ + = (5.31)

5.4 The micro-morhic continuum interpretation

Figure 5-1: The microstructure of an (REV) with sub-particles sharing a rigid-body motion

For the determination of the angular momentum of the spinning polar material point we

resort to the micro-morphic continuum interpretation22. In this case we assign to the

material (polar) particle (or macro-particle) of the continuum the average properties of a

Representative Elementary Volume (REV) of an assembly of sub-particles, as shown in

Figure 5-1. The (REV) may consist of ν sub-particles (or micro-particles)”. The spatial

position of the polar macro-particle is identified with the position of the center of mass

22 The terminus “micro-morphic” is introduced by Eringen: A volume element of a micro-morphic medium consists of micro-elements which undergo micro-motions and micro-deformations. Micro-polar media are a subclass, in which the micro-elements behave like rigid bodies; Eringen, A.C. (1965). Theory of micropolar continua. Proc. 9th Midwestern Mechanics Conference, Madison.


( )iS x of the sub-particles in the (REV). The velocity iv of the center of mass ( )iS x is

defined as the velocity of the particle itself,

i iv x= (5.32)

The sub-particle, at position ( )i iP x y+ , has the mass pm Vρ= , where pρ is the sub-

particle mass density, V , and a velocity

( ) ( )i i iv P v S v= + (5.33)

The total mass of the macro-particle is the sum of the masses of its constituents,

m mνν

= ∑ (5.34)

The linear- and angular momentum of the macro-particle are computed as follows,

i ii mv= (5.35)

( )k ijk i j ijk i jd m x v m y vν

ν

ε ε= + ∑ (5.36)

The volume of the material (REV) is V and the total volume of the sub-particles inside

the (REV) is

pV Vνν

= ∑ (5.37)

The volume fraction

1pVV V

V V

ννφ

−= = −

∑ (5.38)

is the porosity of the (REV). The density of the macro-particle is

(1 ) p

m m V

V V V

ν νν ν ν

ν

ρ φ ρ= = = −∑ ∑ ∑

∑ (5.39)23

Similarly we introduce the linear momentum of the macro-particle,

ii i

mvs vV

ρ= = (5.40)

and its angular momentum

23 If the particles consist of different substances then we should replace in eq. (5.39) instead of pρ with an average particle density pρ< > .


( )1kk ijk i j ijk i j

dD x v m y vV V ν

ν

ρε ε= = + ∑ (5.41)

The relative velocity iv of the sub-particle at point ( )i iP x y+ with respect to the center

of mass ( )iS x is assumed to be function of its position inside the (REV) and of time. We

expand this function in a Taylor series in the vicinity of the center of mass ( )iS x of the

REV,

( ) ( )i ij j ijk j kv v t y v t y y≈ + + (5.42)

We can develop a special theory, if we consider only the linear term in the series

expansion, eq. (5.42),

( )i ij jv v t y≈ (5.43)

This assumption is interpreted as a statement for local homogeneity of the micro-

deformation; i.e. of the deformation inside the (REV), and our demonstration will follow

the steps of [6]. In this case from eqs. (5.41), and (5.43) we get

( ) ( )k ijk i j ijk il jlD x v t J v tρε ε= + (5.44)

where jlJ is the inertia tensor of the (REV) with respect to its center of mass,

( ) ( )1 1il i l p i lJ my y y y V

V Vν νν ν

ρ= =∑ ∑ (5.45)

For simplicity we assume that on the (REV) only volume external forces are acting. In

this case the moment per unit volume of external forces acting on the (REV) is,

k ijk i j ijk jix f fµ ε ε= + (5.46)

where,

( )

( )

1

1

i i

ij i j

f f VV

f f VyV

νν

νν

=

=

∑

∑ (5.47)

If we integrate eq. (5.46) over the volume of the continuum body B we get the expression

for the total moment of body forces acting on B,


( . .)

( ) ( ) ( )

b fk k ijk i j ijk ji

V V V

M dV x f dV f dVµ ε ε= = +∫ ∫ ∫ (5.48)

In view of eq. (5.22) we recognize the 1st term on the r.h.s. of eq. (5.48), as the moment

of body forces. The 2nd term is the contribution of body-couples, that were systematically

ignored in the previous derivations since there was no real motivation to introduce such

body-couples until this point in the demonstration. Thus we introduce here body couples,

k ijk jifεΦ = (5.49)

and eq. (5.48) becomes

( . .)

( ) ( )

b fk ijk i j k

V V

M x f dV dVε= + Φ∫ ∫ (5.50)

With this background we may re-write the linear- and angular momentum equations for

the considered special micro-morphic continuum; these are,

( ) ( )

i iV V

d v dV f dVdt

ρ = +∫ ∫ (5.51)

( ) ( ) ( )

k ijk i j kV V V

d D dV x f dV dVdt

ε= + Φ +∫ ∫ ∫ (5.52)

where the dots stand for the actions of surface tractions and surface couples.

Eq. (5.52) with (5.44) becomes,

( )( ) ( ) ( )

ijk i j ijk il jl ijk i j kV V V

d x v J v dV x f dV dVdt

ρε ε ε+ = + Φ +∫ ∫ ∫ (5.53)

The balance equations for the Cosserat continuum can be derived from eq. (5.53) if we

set,

ij ij ijk kv wψ ε= = − (5.54)

where

kk t k

DwDtψ ψ= ≈ ∂ (5.55)

This assumption means that the considered macro-element is a swarm of ν sub-particles

that they all share a rigid body motion: The center of mass of these sub-particles is


translated by the velocity iv and at the same time all the sub-particles rotate around an

instantaneous axis with director kn and have an angular velocity w , such that,

k kw n w= (5.56)

Thus, the spin and the velocity of a sub-particle inside the (REV) is given by the v. Mises

motor24 that characterizes the rigid-body motion of the sub-particles,

( )( ) ( )

( ) ( )i

i ijk k jP s

w P w S wv w yv P v S w ε

=⎛ ⎞ ⎛ ⎞⎜ ⎟ = ⎜ ⎟⎜ ⎟ −= + × ℜ − ℜ ⎝ ⎠⎝ ⎠

(5.57)

In this case we have that

( )ijk il jl ijk il jlm m km m kJ v J w J wε ε ε ρ= − = = ϑ (5.58)

With these results we return to the momentum balance eq. (5.30), that is written now as

follows,

( )( ) ( )

mim i k ki i

V V

DwJ dV t dVDt

ρ µ∗= + ∂ + Φ∫ ∫ (5.59)

Its local form reads,

mp pk kpq pq k km

DwJDt

µ ε σ ρ∂ + + Φ = (5.60)

In general curvilinear coordinates the dynamic eq. (5.60) takes the following form,

m

p pqkpq k kmk p

Dwe JDt

µ σ ρ⋅ + + Φ = (5.61)

5.5 Exercise: Balance of angular momentum in curvilinear coordinates

Prove that in curvilinear coordinates eq. (5.31) becomes25,

k

ji i jkjkj

DeDt

µ σ ρ⋅ ⋅⋅⋅⋅⋅

ϑ+ = (5.62)

Proof:

From,

24 see sect. 2.2 25 cf. eq. (5.58)


( )( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )

i i j k i j k i ijk jk

V V V V

i j lk i j k ji ijk l jk j

V V V V

ji i jk i i j lk kjk jkj l

V

M e x t dS e x f dV m dS dV

e x n dS e x f dV n dS dV

e e x f dV

σ µ

µ σ σ

⋅⋅ ⋅⋅⋅ ⋅

∂ ∂

⋅⋅ ⋅⋅⋅ ⋅

∂ ∂

⋅ ⋅⋅ ⋅⋅ ⋅⋅ ⋅⋅⋅ ⋅⋅

= + + + Φ

= + + + Φ

= + + Φ + +

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫

∫

(5.63)26

( ) ( )

i i j k ijk

V V

L e x v dV dVρ ρ⋅⋅⋅= + ϑ∫ ∫ (5.64)

and eq. (5.24) we get,

( )

0i k

ji i jk i i j lk kjk jkj l

V

D Dve e x f dVDt Dt

µ σ Φ ρ σ ρ⋅ ⋅⋅ ⋅⋅ ⋅⋅ ⋅⋅⋅ ⋅⋅

⎛ ⎞⎛ ⎞ϑ+ + − + + + =⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠∫ (5.65)

q.e.d.

5.6 Exercise: Dynamic equations in plane polar coordinates

In the absence of body forces and body couples the dynamic equations in physical

components and in plane polar coordinates are (Figure 5-2),

( )

( )

1 1

1 1

rrr rrr

rr r

Dvr r r Dt

Dvr r r Dt

θθθ

θ θθ θθ θ

σσ σ σ ρθ

σ σσ σ ρ

θ

∂∂+ + − =

∂ ∂∂ ∂

+ + + =∂ ∂

(5.66)

1 1zrz zrz r r

Dr r r Dt

θθ θ

µµ µ σ σ ρθ

∂∂ ϑ+ + + − =

∂ ∂ (5.67)

Figure 5-2: Dynamic equilibrium in a Cosserat medium

26 For completeness in eq. (5.63) we considered the action of body couples idVΦ as well.


5.7 Stress power in micro-morhic media

At this point we would like to make a reference to the more general formulation of the

stress power that applies for a micro-morphic medium [32]. In this context we define the

following kinematic variables: The macro-strain rate

( )12ij i j j iv vε = ∂ + ∂ (5.68)

the micro-deformation ijv , the relative strain-rate

ij i k ijv vΓ = ∂ − (5.69)

and the micro-strain rate gradient,

ijk i jkvΚ = ∂ (5.70)

Based on these definitions we define the stress power as

(int)ij ij ij ij ijk ijkw τ ε α µ= + Γ + Κ (5.71)

The tensor ijτ is called the macro-stress, the tensor ijα is the relative stress and the tensor ijkµ is the double stress [54].

We specialize now the micro-deformation so as to correspond to a rigid-body rotation (cf.

eq. (5.54))

ij ij ijk kv wψ ε= = − (5.72)

and with that,

ijk i jk jkl i l jkl ilv vε εΚ = ∂ = − ∂ = − Κ (5.73)

In that case from eq. (5.71) we get,

( )

( ) ( ) ( )

(int)

( ) [ ]

ij ij ij ij ijk ijk ij ij ij i j ij jkl ijk il

ij ij ij ij ij ij jkl ijk il ij ij ij ij ij ij jkl ijk il

w vτ ε α µ τ ε α ψ ε µ

τ ε α ε ω ψ ε µ τ α ε α ω ψ ε µ

= + Γ + Κ = + ∂ − − Κ

= + + − − Κ = + + − − Κ(5.74)

If we compare eqs. (4.14) and (5.74) we obtain the following identification among the

stress fields defined in the micro-morphic and the Cosserat continuum, respectively,

( ) ( )

[ ] [ ]

ij ij ij

ij ij

il jkl ijk

σ τ α

σ α

µ ε µ

= +

=

= −

(5.75)


6 Cosserat continuum energetics

6.1 Energy balance equation

The total energy ( )E t of a continuum body B is split into two parts: One part that

depends on the position of the observer, that is called the kinetic energy ( )K t of the

considered body and another part that does not depend on the observer, called the internal

energy ( )U t ,

( ) ( ) ( )E t K t U t= + (6.1)

The kinetic energy of a Cosserat continuum consists of the contribution that is due to the

translationary motion of the particles,

12

kkdmv v (6.2)

and of the contribution that is due to their spin,

12

kkdm wϑ (6.3)

Thus the total kinetic energy is computed as,

( ) ( )

( )

1 1( )2 2

1 12 2

k kk k

V V

k kk k

V

K t v v dV w dV

v v w dV

ρ ρ

ρ

= + ϑ

⎛ ⎞= + ϑ⎜ ⎟⎝ ⎠

∫ ∫

∫ (6.4)

The internal energy is assumed to be given by means of a specific internal energy density

function, ( ),ie tΘ ,

( ) ( )

( )B V

U t edm edVρ= =∫ ∫ (6.5)

The 1st Law of Thermodynamics requires that the change of the total energy of a material

body B is due to two factors: a) the power ( )extW of all external forces acting on B in the

current configuration, and b) the non-mechanical energy Q , which is supplied per unit

time to B from the exterior domain; i.e.:

( )extdE W Qdt

= + (6.6)


By eliminating /dE dt from eqs. (6.1) and (6.6) we arrive to the fundamental energy

balance equation27

( )extdU dK W Qdt dt

+ = + (6.7)

The work of external forces is computed as follows,

( ) ( )( )

( ) ( )

ext k k k kk k k k

V V

W t v m w dS f v w dV∂

= + + + Φ∫ ∫ (6.8)

The influx of energy in the form of heat flow is defined through a heat-flow vector

( ),i kq tΘ , that is taken positive if heat flows into the considered body,

( )

kk

V

Q q n dS∂

= − ∫ (6.9)

We introduce into eq. (6.8) the stress- and couple-stress tensors, according to eqs. (4.36)

and (4.38),

( ) ( )( )

( ) ( )

ext ik ik k kk k i k k

V V

W v w n dS f v w dVσ µ∂

= + + + Φ∫ ∫ (6.10)

and we apply Gauss’ theorem,

( ) ( ) ( )( )

( ) ( ) ( )

ext ik k ik k ik ikk ki i k i k i

V V V

W f v dV v dV v w dVσ µ σ µ= + + + Φ + +∫ ∫ ∫ (6.11)

Similarly from eq. (6.9) we get

( )

kk

V

Q q dV⋅= − ∫ (6.12)

The l.h.s. of eq. (6.6) is computed by means of eq. (6.1) to (6.5) and Reynold’s transport

theorem

( ) ( )( ) ( ) ( )

1 12 2

k kk k

V V V

dE D D dev v dV w dV dVdt Dt Dt dt

ρ ρ ρ= + ϑ +∫ ∫ ∫ (6.13)

We remark that the following relations hold,

27 According to Truesdell & Toupin [50] [50] the first formulation of the Energy balance Law is due to

Duhem (1892).


( ) ( )1 12 2

k kk k l l

k kl kl kD D Dv Dvv v v g v g v vDt Dt Dt Dt

= = = (6.14)

k

k kk

DwD wDt Dtϑ

= ϑ (6.15)

The latter will be demonstrated below. With that we get,

( )1 12 2

k kk k k

k k kDwD D Dw w w

Dt Dt Dt Dt⎛ ⎞ϑ ϑ

ϑ = + ϑ =⎜ ⎟⎝ ⎠

(6.16)

By combining eqs. (6.6), (6.11) to (6.16) we get

( ) ( )( )

( ) ( ) ( )

( ) ( )

( ) ( )

k k

k kV V V

ik k ik kk ki i

V V

ik ik kk i k i k

V V

Dv D Dev dV w dV dVDt Dt Dt

f v dV w dV

v w dV q dV

ρ ρ ρ

σ µ Φ

σ µ ⋅

ϑ+ + =

= + + +

+ + − ⇒

∫ ∫ ∫

∫ ∫

∫ ∫

(6.17)

or

( )

( )

( ) ( )

( ) ( )

V

k kik k ik k

k ki iV V

ik ik kk i k i k

V V

De dVDt

Dv Df v dV w dVDt Dt

v w dV q dV

ρ

σ ρ µ Φ ρ

σ µ ⋅

⎛ ⎞ ⎛ ⎞ϑ= + − + + −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

+ + −

∫

∫ ∫

∫ ∫

(6.18)

If we assume that linear- and angular momentum balance equations, eqs. (5.21) and

(5.62), are holding, then we get from eq. (6.18)

( )( ) ( )

lk im ik ik kiml k k i k i k

V V

De dV e g w v w q dVDt

ρ σ σ µ ⋅= − + + −∫ ∫ (6.19)

With

lk lim iml k imlw e g w e w= − = − (6.20)

we get

( )( )( ) ( )

im im kimm i m i k

V V

De dV v w w q dVDt

ρ σ µ ⋅= + + −∫ ∫ (6.21)

In accordance to eq. (4.1) we define the rate of deformation tensor,


ik ikk iv wΓ = + (6.22)

and the rate of curvature tensor,

ik i kK w= (6.23)

With this notation in accordance to eq. (4.4) we define the stress power in a Cosserat

continuum as

im imim miP Kσ µ= Γ + (6.24)

With this definition, from eq. (6.21) we obtain the following local form of the energy

balance equation,

kk

De P qDt

ρ ⋅= − (6.25)

Remark In order to have the above derivation complete we must prove the validity of eq. (6.15);

cf. [6]. We use Cartesian coordinates and eqs. (5.45) and (5.58) as starting points,

*k km mJ wϑ = (6.26)

where ijJ ∗ is the (symmetric) inertia tensor referred the unit of mass

( )* *1 1 ;il il i l ij ilJ J my y J Jm ν

ν

ρρ

= = =∑ (6.27)

In view of the l.h.s. of eq. (6.15) ,eq. (6.26) and the symmetry of the moment of inertia

tensor we get,

k km m km mk m km k k m km k

km mk m mk k

D DJ Dw DJ Dww w J w w w J wDt Dt Dt Dt Dt

DJ Dww w J wDt Dt

∗ ∗∗ ∗

∗∗

⎛ ⎞ϑ= + = +⎜ ⎟

⎝ ⎠

= +

(6.28)

In a Cosserat continuum the material points move like rigid bodies. This means that for a

fixed coordinate system, the inertia tensor is,

( ) ( ) ( ) (0)ij in jm nmJ t Q t Q t J∗ ∗= (6.29)


The proper orthogonal tensor ( )ijQ t describes the (finite) rotation of the of the material

point between its configuration at time 0t = and time 0t > . We recall that tensor ( )ijQ t

fulfills the orthogonality conditions,

,ik il kl ki li klQ Q Q Qδ δ= = (6.30)

Thus,

* (0) ( ) ( ) (0)ij in jm nm in jm nmD J Q Q J Q t Q t JDt

∗ ∗= + (6.31)

If we take the current configuration as reference configuration, then eq. (6.31) yields

*ij in jm nm in jm nm in nj jm im

D J Q J Q J Q J Q JDt

δ δ∗ ∗ ∗ ∗= + = + (6.32)

or

*ij in nj im mj

D J J JDt

∗ ∗= Ω − Ω (6.33)

where the tensor

kl klQΩ = (6.34)

is antisymmetric and has the angular velocity vector kw as an axial vector,

3 2

3 1

2 1

0[ ] 0

0

w ww ww w

−⎡ ⎤⎢ ⎥Ω = −⎢ ⎥⎢ ⎥−⎣ ⎦

(6.35)

or

ij ijk kwεΩ = − (6.36)

Thus,

*ij ink k nj mjk k im

D J w J w JDt

ε ε∗ ∗= − + (6.37)

and

( )( )2 0

iji j i imk k mj mjk k im j

mik k i mj j mjk k j mi i m mik i k

DJw w w w J w J w

Dtw w J w w w J w r w w

ε ε

ε ε ε

∗∗ ∗

∗ ∗

= − +

= + = = (6.38)


6.2 Entropy balance

Let Η be the total entropy of the considered material body B in the current configuration

( )V

t sdVρΗ = ∫ (6.39)

In eq. (6.39) ( , )ks s x t= , is the specific entropy. Let also ( , ) 0kT T x t= > be the absolute

temperature. We define further the following quantities:

a) The Helmholtz free energy is defined as that portion of the internal energy, which is

available for doing mechanical work at constant temperature

f e sT= − (6.40)

b) The local dissipation is

locDf DTD P sDt Dt

ρ ⎛ ⎞= − +⎜ ⎟⎝ ⎠

(6.41)

With the above definitions the energy balance eq. (6.25) becomes,

klock

DsT q DDt

ρ ⋅⎛ ⎞ = − +⎜ ⎟⎝ ⎠

(6.42)

This equation is called the balance law for local entropy production. The entropy balance

law, eq.(6.42), is further worked out by introducing appropriate constitutive assumptions.

6.3 Linear, isotropic Cosserat elasticity theory

For an elastic Cosserat material that is stressed under isothermal conditions, the energy

balance equation (6.25) provides the means to compute the rate of the internal energy

density function. In Cartesian notation we have,

ij ij ij ijDe KDt

ρ σ µ= Γ + (6.43)

Within the frame of a small-deformation theory we assume that the density remains

practically constant,

(0)

(0) (0)(0) (0) (1 )

1 kkkk

dVdV

ρ ρρ ρ ε ρρ ε

= ⇒ = ≈ − ≈+

(6.44)


where (0)ρ is the density of the material in the initial, unstrained configuration. Thus for

small deformations we get,

ijij t ij

ijij t ij

DDtD

KDt

γγ

κκ

Γ = ≈ ∂

= ≈ ∂ (6.45)

and

(0)ij ij ij ij

De DeDt Dt

ρ ρ σ µ≈ ≈ Γ + Κ (6.46)

We assume that the elastic energy density,

( ) (0)elw eρ= (6.47)

is a function of the 18 kinematic variables, ijγ and ijκ ,

( )( ) ,elij ijw F γ κ= (6.48)

Then from eqs. (6.48) and (6.46) we get that

( ) ( )

;el el

ij ijij ij

w wσ µγ κ

∂ ∂= =

∂ ∂ (6.49)

Within the frame of linear elasticity we assume that ( )( ) ,elij ijw F γ κ= is homogeneous of

degree 2 with respect to its arguments.

From

( ) ( ) [ ] [ ]ij ij ij ij ij ijσ γ σ γ σ γ= + (6.50)

we get that the elastic strain energy is split into three terms, as

( ) ( ) ( )( ) ( ) ( ) ( )1 ( ) 2 [ ] 3

el el el elij ij ijw w w wγ γ κ= + + (6.51)

A simple elasticity model arises if we assume that the 1st term on the r.h.s. of eq. (6.46)

reflects Hooke’s law for linear isotropic elastic materials [43],

( )1 ( )

12

elij ijw σ ε= ⇒ (6.52)

( )1

1 2;2 1 2

elmm nn mn mnw G Gνλε ε ε ε λ

ν= + =

− (6.53)


thus yielding,

( )( )1

( ) 21 2

elel

ij ij kk ijij ij

ww G νσ ε ε δε ε ν

∂∂ ⎛ ⎞= = = +⎜ ⎟∂ ∂ −⎝ ⎠ (6.54)

We notice also that both, [ ]ijσ and [ ]ijγ , are antisymmetric tensors of 2nd order. Thus both

are possessing axial vectors, say

*[ ] [ ];ij ijk k ij ijk ktσ ε γ ε γ ∗= = (6.55)

where according to eq. (4.11)

k k kγ ω ψ∗ = − (6.56)

Isotropy calls for proportionality between the axial vector of the antisymmetric stress and

the antisymmetric part of the deformation, thus yielding

1 1 [ ] 1 [ ]2 ( 0) 2i i ij ijt G Gη γ η σ η γ∗ ∗= > ⇒ = (6.57)

and with that

( )2 [ ] [ ] 1 [ ] [ ] 1

1 62

elij ij ij ij k kw G Gσ γ η γ γ η γ γ∗ ∗= = = (6.58)

The couple stress tensor and the gradient of the Cosserat rotation kψ are decomposed

additively also into symmetric and antisymmetric parts,

( ) [ ] ( ) [ ];ij ij ij ij ij ijµ µ µ κ κ κ= + = + (6.59)

Then the isotropic linear-elastic law for the couple stress reads,

( )

( )23

( ) ( ) 2 2( )

( )23

[ ] 3 [ ] 3[ ]

, 0

, 0

el

ij ij ij kkij

el

ij ijij

w G

w G

µ κ η δ κ ηκ

µ η κ ηκ

∂= = + >

∂

∂= = >

∂

(6.60)

where is a material constant with the dimension of length, called also material or

internal length. Thus

( )( ) 23 ( ) ( ) 2 ( ) ( ) 3 [ ] [ ]

12

elmn mn mm nn ij ijw G κ κ η κ κ η κ κ= + + (6.61)

Note that the general anisotropic Cosserat elasticity was formulated in a paper by Kessel

[28].


6.4 A 2D linear, isotropic Cosserat- elasticity theory

Here we summarize some results from the paper of M. Schäfer [44], that pertain to a

proposition for a simple two-dimensional, linear elasticity theory for isotropic materials

of the Cosserat type. This is a theory of plane stress states.

First we assume that the symmetric part of the stress tensor is related to the symmetric

part of the deformation tensor (i.e. to the symmetric part of the displacement gradient,

that is identified with the infinitesimal strain tensor), trough the usual equations of plane-

stress, isotropic Hooke elasticity,

( )

( )

11 11 22

22 22 11

12 (12)

1

1

12

E

E

G

ε σ νσ

ε σ νσ

ε σ

= −

= −

=

(6.62)

where

( ), ,12ij i j j iu uε = + (6.63)

The antisymmetric parts of the relative deformation and of the stress are also linked by a

linear relation,

[12] [12] 11 , 0

2 cc

G GG

γ σ η= = > (6.64)

where according to eqs. (3.86) and (4.13)

( )

( )

[12] 12 21

[12] 12 21

1212

γ γ γ ω ψ

σ σ σ

= − = −

= − (6.65)

Due to the isotropy requirement we assume that the couple stress are linked to the

curvatures by means of only one additional material constant,

13 13 23 23; , 0D D Dµ κ µ κ= = > (6.66)

where

13 1 3 1

23 2 3 2

κ ψ ψκ ψ ψ

= ∂ = ∂= ∂ = ∂

(6.67)


6.5 Examples of simple Cosserat elasticity b.v. problems

6.5.1. Pure bending of a Cosserat-elastic beam We consider a beam, with rectangular cross section, as shown in Figure 6-1. The only

stresses that are considered are the axial stress xxσ and the couple stress xyµ . Motivated

by the classical beam theory we assume that [44],

;xx xycz cσ µ= = (6.68)

where c and c are positive constants.

Figure 6-1: Pure bending of a Cosserat beam

In the considered case the only significant equilibrium equations are,

0 ; 0xyxx

x xµσ ∂∂

= =∂ ∂

(6.69)

The stress fields, eqs. (6.68), are equilibrium fields. The elasticity equations and the

ansatz (6.68) yield,

1xx xx

c zE E

ε σ= = (6.70)

1xy xy

cD D

κ µ= = (6.71)

The only surviving compatibility condition is

(2)

21 0xxxyI

zε κ∂

= − =∂

(6.72)

which in turn yields a restriction for the introduced constants,


0c c Dc cE D E

− = ⇒ = (6.73)28

We normalize the material constant D by the Young’s modulus, by setting.

2112

D E= (6.74)

where is a micro-mechanical length that is in most cases considered to be small, if

compared with the typical geometric dimension of a structure. As we will see below, the

factor 1/12 is put for convenience in the computation.

With this remark from eqs. (6.71) and (6.74) we get

2112xy xyEµ κ= (6.75)

and

xycE

κ = (6.76)

As in classical beam bending theory, from Figure 6-2 we read

1

1

x x xxu dx dxz R z R

cE R

ε∂= ⇒ = ⇒

= (6.77)

where R is the radius of curvature of the beam.

On other hand from eqs. (6.77) and (6.76) we get

1yydx Rd

x Rψ

ψ∂

= ⇒ =∂

(6.78)

28 In the paper by M. Schäfer [44] we find the derivation of stress functions that satisfy equilibrium and compatibility conditions.


Figure 6-2: Pure bending

The total bending moment that is taken by the rectangular cross-section of the beam is

computed as

/ 2 / 2 / 2 / 22

/ 2 / 2 / 2 / 2

23 32 1

12 12 12

h h h h

xx xyh h h h

M zbdz bdz cb z dz cb dz

bh bh bhc cbh c cIh

σ µ− − − −

= + = +

⎛ ⎞⎛ ⎞ ⎛ ⎞= + = + = +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠

∫ ∫ ∫ ∫ (6.79)

where I is the surface moment of inertia of the rectangular cross-section of the beam,

3

12bhI = (6.80)

Then from eq. (6.79) we get

EIMR

′= (6.81)

where

2

1I Ih

⎛ ⎞⎛ ⎞′ = +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ (6.82)

This is a typical result of Cosserat elasticity theory, meaning that a structure made of

Cosserat elastic material is stiffer then the corresponding classical elastic structure.


6.5.2. Annular shear of a cylindrical hole in Cosserat-elastic solid

Figure 6-3: Cylindrical hole in plane strain annular shear

We consider a cylindrical hole of radius R under the action of internal shear as shown in

Figure 5-2. Axial symmetry yields to the following equilibrium equations,

( )1 0rr r

ddr r

θθ θ

σσ σ+ + = (6.83)

1 0rzrz r r

ddr r θ θµ µ σ σ+ + − = (6.84)

Remark Note that for Boltzmann continua the considered problem is isostatic; i.e. for the

determination of the stress field one does need to specify the constitutive equation.

Indeed in that case the only valid equilibrium equation is ,

2 0rr

ddr r

θθ

σσ+ = (6.85)

The boundary conditions for the classical problem are,

: rr R θσ τ= = (6.86)

: rr θσ→ ∞ < ∞ (6.87)

These b.c. admit the solution


2

rrRθσ τ

−⎛ ⎞= ⎜ ⎟⎝ ⎠

(6.88)

Figure 6-4: Stress state at the the cavity lips in case of a Boltzamann continuum

Figure 6-5: Principal stress trajrectories in case of Boltzamann continuum, indicating the isoststaticity of the considered problem.

As mentioned, the probem of a cylindrical cavity under annular shear is isostatic (Figure

6-4, Figure 6-5). In this case principal stresses exist and their trajectories are logarithmic

spirals,

( ) ( )

( ) ( )

10 0

20 0

( ) : Re cot , cos , sin4

3( ) : Re cot , cos , sin4

r xp x r y r

r xp x r y r

πσ θ θ θ θ θ

πσ θ θ θ θ θ

⎛ ⎞⎛ ⎞= = + = +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

⎛ ⎞⎛ ⎞= − = − + = − +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

(6.89)


In the case, however, of a Cosseart continuum, none of the above holds. The problem is

not isostatic, the solution depends on the constitutive assumptions met fror the stresses

and the couple stresses and a boundary layer is forming close to the cavity wall, where

Cosserat effects are dominant.

Indeed in case of a Cosserat continuum, from sect. 3.6 we get that the following

expressions for the deformation measures,

10 02

1[ ] 0 02

0 0 0

rr r rz

r z

zr z zz

u ur r

u usymr r

θ θ

θθ θ

θ θθ θ

θ

ε ε εγ ε ε ε

ε ε ε

⎡ ⎤∂⎛ ⎞−⎜ ⎟⎢ ⎥∂⎝ ⎠⎢ ⎥⎡ ⎤⎢ ⎥∂⎛ ⎞⎢ ⎥= = −⎢ ⎥⎜ ⎟⎢ ⎥ ∂⎝ ⎠⎢ ⎥⎢ ⎥⎣ ⎦ ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(6.90)

[ ]

10 02

1 0 02

0 0 0

z

z

u ur r

u uasymr r

θ θ

θ θ

ψ

γ ψ

⎡ ⎤∂⎛ ⎞+ −⎜ ⎟⎢ ⎥∂⎝ ⎠⎢ ⎥⎢ ⎥∂⎛ ⎞= − + +⎢ ⎥⎜ ⎟∂⎝ ⎠⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(6.91)

[ ]

0 0

0 0 00 0 0

z

rψ

κ

∂⎡ ⎤⎢ ⎥∂⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(6.92)

These expressions are combined here with the constitutive equations of linear isotropic

Cosserat elasticity, that are derived in sect. 6.3and provide the following set of

generalized stress-strain relationships,

( ) 2r rdu uG Gdr r

θ θθ θσ ε ⎛ ⎞= = −⎜ ⎟

⎝ ⎠ (6.93)

[ ] 1 [ ] 12 2r r zdu uG Gdr r

θ θθ θσ η γ η ψ⎛ ⎞⎛ ⎞= = + −⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠ (6.94)

And couple-stress-curvarure relationhips


2 2( ) ( )

2 2[ ] 3 [ ] 3

12

12

zrz rz

zrz rz

dG Gdr

dG Gdr

ψµ κ

ψµ η κ η

= =

= = (6.95)

Thus

( ) ( )( ) [ ] 1 1 11 1 2r r r zdu u

Gdr r

θ θθ θ θσ σ σ η η η ψ⎛ ⎞= + = + − − −⎜ ⎟

⎝ ⎠ (6.96)

and

( )2( ) [ ] 3

1 12

zrz rz rz

dGdrψµ µ µ η= + = + (6.97)

Introducing the above set of constitutive equations into the equilibrium eqs. (6.83) and

(6.84) we get a set of two coupled differential equations for the particle displacement in

tangential direction uθ and for the particle rotation, zψ ,

2

2 21 2 zd u du u d

ar dr drdr r

θ θ θ ψ+ − = (6.98)

and

2

2 22

1 2z zz

du ud db b

r dr dr rdrθ θψ ψ

ψ ⎛ ⎞+ − = − +⎜ ⎟⎝ ⎠

(6.99)

where

1 1

1 3; 4

1 1a bη η

η η= =

+ + (6.100)

For 1 0 ( 0)a bη = = = the stress tensor is symmetric and eqs. (6.98) and (6.99) become

decoupled,

2

2 21 0d u du ur drdr r

θ θ θ+ − = (6.101)

2

21 0z zd dr drdr

ψ ψ+ = (6.102)

The solution of eq. (6.101) and eq. (6.102) is

3 41u C r Crθ = + (6.103)


1 2 lnz C C rψ = + (6.104)

The boundeness condition at infinity for the particle rotation angle zψ and for the

circumferential displacement uθ is fulfilled, if 2 0C = and 3 0C = . The solution for

1 0C ≠ is physically meaningless, thus we accept the solution

4 ; 0zCurθ ψ= = (6.105)

The integration constant 4C is determined from the boundary condition for the shear

stress,

: rr R θσ τ= = (6.106)

Thus

;2

Ru u u Rr Gθ

τ= − = (6.107)

In the uncoupled case ( 1 0η = ), the valid solution for the displacement, eq.(6.107),

together with the constitutive equation for the symmetric part of the stress, eq. (6.93),

yield the classical solution, eq. (6.88).

In the general case ( 1 0η > ), eqs. (6.99) and eq. (6.98) yield

2

2 22

1 0z zz

d dr drdr

ψ ψ η ψ⎛ ⎞

+ − =⎜ ⎟⎜ ⎟⎝ ⎠

(6.108)

where 0> and

2 11

3 1

18 0 ( 0)1 1

ηη ηη η

= > >+ +

(6.109)

Let

rρ η= (6.110)

the general solution of eq. (6.108) is given in terms of 0th-order modified Bessel functions

( ) ( )1 0 2 0z C I C Kψ ρ ρ= + (6.111)

and from that


( ) ( )1 1 2 1zd C I C K

dzψ η ηρ ρ= − (6.112)

The extra boundary conditions are given in terms of the particle rotation and/or of the

couple stress. In order to introduce these extra boundary conditions within the Cosserat

continuum description we resort to the concept of ortho-fiber [21] An ortho-fiber is a

rigid bar of length ′ aligned normally to the surface of the considered Cosserat

continuum body and it is pointing outwards. On the end of this fiber we assume either

displacements or tractions are applied thus giving to the surface actions the meaning of v.

Mises motors (Figure 6-6). Accordingly we assume here that at the cavity wall the shear

stress and the couple stress are prescribed and at infinity the particle displacement and

rotation must vanish,

: r

rz

r R θσ τµ τ

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟′−⎝ ⎠⎝ ⎠

(6.113)

0

:0

zr Ruθ

ψ∗ ⎛ ⎞ ⎛ ⎞= → ∞ =⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠ (6.114)

The sign of the surface couple in eq. (6.113) as follows: As shown in Figure 6-6 the shear

traction of magnitude τ is assumed to be applied on an ortho-fiber of length ′ , thus

yielding and equivalent set of surface actions, a surface traction and a surface couple. If

the surface traction is positive then the surface couple must negative,

Figure 6-6: Shear traction applied on an ortho-fiber at distance ′ , resulting into a surface traction and a surface couple when transported to the surface.

For large argument ρ we have the following asymptotic expression for the solution, eq.

(6.111)


( ) ( )1 2

expexp

22z C Cρ πψ ρ

ρπρ= + − (6.115)

From eqs. (6.114), (6.113) and (6.115) we get

( ) ( )

*1 2

1 1 1 2

e e 0 ( 1)1R R

c cI c K c

ρ ρ ρρ ρ

∗ ∗−+ = >>

+ = − (6.116)

where

( )3

( 1, 2) ;1 21

ii R

C Rc i

G

ρ ητη η

= − = =′+

(6.117)

Using Kramer’s rule, the solution of the system of linear eqs. (6.116) takes the following

form,

2

1 21 1

2 211 1

0 ( )( ) ( )

1 1 ( )( )( ) ( )

R R

RR R

ecK I e

cKK I e

ρ

ρ

ρ

ρρ ρ

ρρρ ρ

∗

∗

∗

−∗

−

∗

−

= → → ∞−

= → → ∞−

(6.118)

This means that the valid solution here is the logarithmic one 1( 0)C =

( )2 0z C Kψ ρ= (6.119)

and with that

( ) ( )22 3 11rz C G Kηµ η ρ= − + (6.120)

The b.c. at the cavity wall ,eq. (6.113) for the couple stress, yields,

( ) ( )2

3 1

1 4 11 R

uCR Kη η ρ

′⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟+ ⎝ ⎠ ⎝ ⎠ (6.121)

and with that (Figure 6-7)

( )

( )( )

0

3 1

1 4 01z

R

KuR K

ρψ

η η ρ′⎛ ⎞⎛ ⎞= >⎜ ⎟⎜ ⎟+ ⎝ ⎠⎝ ⎠

(6.122)


Figure 6-7: Boundary particle displacement and rotation

For small values of the internal length 0 Rη< << we have the following asymptotic

solution for the particle rotation.

( )4 RRz

u eR

ρ ρρψρ

− −′⎛ ⎞⎛ ⎞≈ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

(6.123)

This means that the particle rotations are confined in boundary layer and they decay

faster then exponentially with the radial distance from the cavity wall. On the other hand

we observe that the particle rotation depends linearly on the ratio of the “roughness”

length scale ′ to the material length scale , and on the ratio of first-order imposed

hoop displacement to the radius of the cavity

Eq. (6.98) becomes

( )2

0 22 21 0 ; 2d u du u dC K C aCr dr drdr r

θ θ θ ρ+ − − = = (6.124)

or

( )3 4 2 11 2u C r C aC Krθ ρ

η= + − (6.125)

As already explained above the only meaningful solution is the one for 3 0C = , thus

( )( )

14

1 1

1 21 R

KCu Cr Kθ

ρη η ρ

= −+

(6.126)

where


( ) ( )

12 1

3 1

4 11 R

uC CK R

ηηη η ρ

′= =

+ (6.127)

The b.c. at the cavity wall ,eq. (6.113) for the shear stress, yields,

( )( ) ( )02

4 01 1

1 121

RR

R R

KC u CR K

Kρ

ρη ρ ρ

⎛ ⎞= − + + −⎜ ⎟⎜ ⎟+⎝ ⎠

(6.128)

and with that

(0)u u uθ θ θ≈ − (6.129)

In this expression with (0)uθ we denote the classical solution eq. (6.107)

(0) ;2

Ru u u Rr Gθ

τ= − = (6.130)

and uθ is the perturbation that stems from the Cosserat terms,

( )

( ) ( )

2

1

12

3 1

21

1 21

RR R R

R

Cu e

CK G

ρ ρθ

ρ ρ ρη ρ η ρ ρ

η τη η η ρ

− −⎛ ⎞⎛ ⎞≈ + −⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟+ ⎝ ⎠⎝ ⎠

′=+

(6.131)

The perturbation for the displacement contains both hyperbolically and exponentially

decaying terms, it is proportional to the classical solution, eq. (6.107), and it scales

linearly with the interfacial length scale , ′ ,

(0) ( , )RRu u u u C fr Rθ θ θ ρ ρ

′⎛ ⎞⎛ ⎞= − ≈ − + ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ (6.132)

6.5.3. Sphere under uniform radial torsion We Consider a spherical body of radius R made of linear-elastic, isotropic Cosserat-type

material that is subjected on its surface to uniform radial-torsional laoding of intensity

( )rr R mµ = ( 6Figure 6-8). We want to analyze its state of stress and the deformation that

this object suffers.


6Figure 6-8: Sphere under uniform surface torsion

In the considered setting the deformations, distortions and torsions are given in pola

spherical coordinates as follows:

[ ]0 0 00 00 0

r

r

asym γ ψψ

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥−⎣ ⎦

(6.133)

0 0

[ ] 0 0

0 0

r

r

r

ddr

r

r

ψ

ψκ

ψ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(6.134)

0 0

[ ] 0 00 0

rr

θθ

φφ

µµ µ

µ

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

(6.135)

The only significant stress and couple-stress components are (Figure 6-9):

1 1;r rG Gθφ φθσ η ψ σ η ψ= = − (6.136)

( )

( )

22 2

22 2

1 2

1 2

r rrr

r r

dGdr rdGdr rθθ φφ

ψ ψµ η η

ψ ψµ µ η η

⎛ ⎞= + +⎜ ⎟⎝ ⎠

⎛ ⎞= = + +⎜ ⎟⎝ ⎠

(6.137)


Figure 6-9: Stress state in the element of a sphere under uniform torsion

The governing equilibrium equation is:

( )1 2 0rrrr

ddr r φφ θθ φθ θφµ µ µ µ σ σ+ − − + − = (6.138)

The solution that is acceptable (as being regular at the origin) is (Figure 6-10):

2

cosh sinh( ) ; ,rrC ρ ρ ρψ ρ ρ

ρ−

= Ψ Ψ = = (6.139)

Figure 6-10: Torsional boundary layer solution

In this case the torsion is also confined in a boundary layer. For example the solution for

the isotropic part of the torsion reads,


( ) ( )2

2

3

1 1 1 3 2 ( )3 3sinh ( )

r rT rr

dG Cdr r

O

θθ φφψ ψµ µ µ µ η ρ

ρ ρρ

⎛ ⎞= + + = + + = Μ⎜ ⎟⎝ ⎠

Μ = = + (6.140)

At this point we should make a remark concerning the so called the restricted Cosserat

Continuum theory. In this theory we assume that the relative particle spin is zero,

meaning that the particle rotation coincides with the antisymmetric part of the

displacement gradient [34]. This theory, although widely used, it has led to difficulties,

since the isotropic part of the couple-stress remains indeterminate [33]. This observation

has lead to some controversies that we believe that it has been resolved recently by Froiio

et al. [21], who have shown, by using the concept of ortho-fiber, that such a restricted

Cosserat continuum is incapable to absorb boundary conditions that refer to the torsional

dof. In other words the above example of the sphere under uniform torsion illustrates

vlearly the ability of the Cosserat continuum theory to a provide a unique solution for the

torsion and the mean torsion!

6.6 Cosserat thermo-elasto-plasticity

As for a Boltzmann Continuum so for the Cosserat Continuum within the realm of small-

deformations elastoplasticity thery the total rate-of-deformation measures are

decomposed additively into elastic and plastic part; say

;e p e pij ij ij ij ij ijΓ = Γ + Γ Κ = Κ + Κ (6.141)

For isotropic, linear thermo-elastoplatic materials the small strain-theory is based on the

following constitutive equations for the elastic rates-of-deformation,

( ) ( )( ) [ ]

1

1 1 12 2 1

e ekl ijij ik jl il jk ij kl ij

DTG Dt

νδ δ δ δ δ δ σ σ α δν η

∇ ∇⎛ ⎞⎛ ⎞Γ = + − + +⎜ ⎟⎜ ⎟+⎝ ⎠⎝ ⎠ (6.142)

( ) ( ) [ ]23

1 1 12

ekl ijij ki lj kj li ij klK

Gδ δ δ δ δ δ µ µ

η

∇ ∇⎛ ⎞= + − +⎜ ⎟

⎝ ⎠ (6.143)

where ( )eα is the coefficient of elastic thermal expansion and ( , )kT x t is the absolute

temperature field. With a superimposed ∇ we denote the Zaremba-Jaumann derivative of

a tensor,


( )1;2

ijJij ij ik kj ik kj ij j i i jDSdS S S S v v

dt Dtω ω ω

∇= = + − = ∂ − ∂ (6.144)

The elastic rate-of-deformation tensors are set dual in free energy to the stress and couple

stress respectively29,

e eij ij ij ij

Df DTsDt Dt

ρ σ µ⎛ ⎞+ = Γ + Κ⎜ ⎟⎝ ⎠

(6.145)

The local dissipation is

locDf DTD P sDt Dt

ρ ⎛ ⎞= − +⎜ ⎟⎝ ⎠

(6.146)

and with

ij ij ij ijP σ µ= Γ + Κ (6.147)

we conclude that the local dissipation coincides here with the power of stress and of the

couple stress on the plastic part of the rate of the corresponding rate of deformation

tensors,

p p ploc ij ij ij ijD P σ µ= = Γ + Κ (6.148)

We recall here the balance law for local entropy production, eq. (6.42),

pk k

DsT q PDt

ρ ⎛ ⎞ = −∂ +⎜ ⎟⎝ ⎠

(6.149)

We assume that the rate of entropy production consists of two terms as follows,

( / ) ( )th e iDs s sDt

= + (6.150)

The first contribution is due to thermo-elastic effects and the second is due to in-elastic

effects, such as grain breakage and grain attrition.

For the thermo-elastic part we assume the following form:

( / )( )

th e ekl kl

DTs jc ADt

Τ = + Γ (6.151)

29 see esct. 6.2


In eq. (6.151) jc is the specific heat of the material and ijA is the stress temperature

tensor, which gives the stress resulting from a given temperature distribution, when the

strain vanishes30. In the isotropic case we have

( )2(1 ),1 2

eij ijA A A Gνδ α

ν+

= =−

(6.152)

where ( )eα is the thermal expansion coefficient of skeleton.

From the above constitutive assumptions we get

( )( )

( )( )

e i pkk k k

p e ik k kk

Ds DTT jc A s q PDt Dt

DTjc q P A sDt

ρ ρ ρ

ρ ρ ρ

⎛ ⎞ ⎛ ⎞= + Γ + = −∂ + ⇒⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= −∂ + − Γ − (6.153)

In this expression of the energy balance law, we observe that only a part of the local

dissipation is converted to heat. One part of the entropy production goes to thermo-elastic

expansion and another is dissipated in other forms than heat, such attrition and breakage

of grains. If we neglect these effects, we end up with the simplest form of the energy

balance law,

pk k

DTjc q PDt

ρ = −∂ + (6.154)

We consider further Fourier's constitutive law of isotropic heat conduction

k F kq k T= − ∂ (6.155)

where Fk is the thermal conductivity of the material.

From classical thermodynamics we retain as an expression of the 2nd Law the axiom that

heat must flow opposite to the temperature gradient (i.e. from ‘hot’ regions to ‘cold’

regions). This means that the entropy production by heat conduction must be non-

decreasing

2

1 0con k kq Tηρ

= − ∂ ≥Τ

(6.156)

30 As an example think of an elastic bar heated and clamped at both ends.


With Fourier’s law the entropy production by heat conduction becomes a quadratic form

in the temperature gradient, and Ineq. (6.156) reduces simply to a constitutive inequality

for the thermal conductivity,

2 0 0Fcon k k F

k T T kηρ

= ∂ ∂ ≥ ⇒ >Τ

(6.157)

With these assumptions and restrictions eq. (6.154) results in the following heat-

conduction/generation equation,

( )2 1 p pij ij ij ij

DT TDt jc

κ σ µρ

= ∇ + Γ + Κ (6.158)

where

0Fkc

κρ

= > (6.159)

is Kelvin’s coefficient of thermal diffusion, with dimensions [ 2 1L T − ].

Thus for the evaluation of the heat eq. (6.158) we need to specify the plastic part of the

rate-of-deformation measures.


7 Micromechanics of solid granular materials Granular materials are large assemblies of particles interacting with each other through

pair-wise frictional contacts. In this section the emphasis will be laid on “solid” granular

matter, where particle contacts are semi-permanent31, i.e. intergranular contacts exist for

finite time intervals. We assume that deterministic solid granular assembly configurations

exist for substantial time intervals, making thus time averaging unnecessary to describe

the dynamics of the assembly. However, spatial averaging is applied here as a tool for

bridging the gap between discrete and continuum realizations.

7.1 Stress and couple stress in granular media

The application of Cosserat continuum mechanics to the description of the mechanical

behavior of granular media is traced by many authors to the pioneering work of Oshima

[39] and later by Kanatani [27] and Mühlhaus & Vardoulakis [35]. In recent years a

discussion could be followed where the pros and cons of the applicability of the Cosserat

continuum model to granular materials were vividly debated [2], [1], [29], [30], [3], [4],

[19], [10], [15]. The central issue in these papers was the discussion concerning the

validity of stress asymmetry hypothesis. Froiio et al. [20] followed closely this debate

and tried to provide a platform where the various viewpoints could find their position.

This deductive mathematical approach is encouraging us to try to learn more about the

basic physics that may govern the mechanics of granular media and try to describe them

within the frame of Cosserat Continuum Mechanics.

It is obvious that the application of the Cosserat continuum concept will be meaningful if

the phenomena that we want to describe run at grain scale. Indeed the analysis presented

here is meant for the mathematical description of localization phenomena such as shear-

bands occurring in the interior of a granular body [35] or interfacial bands, occurring at

shear interfaces between hard material and granular material [7], [57].

31 As opposed to granular gases and granular fluids, granular solids possess granular contacts that survive finite time intervals (not impacting contacts).


7.1.1 Definitions

Figure 7-1: (REV) containing set B of particles in contact among each other and other exterior particles

Following Bardet & Vardoulakis [2] we consider an (REV) that consists of N sub-

particles (“grains”), some of which are subjected to external forces or couples, applied

from the exterior of the considered (REV), Figure 7-1. These particles inside the (REV)

are grouped in the set 1, ,B p Nα α= = … . The forces and couples acting on the

particles of B are reduced at M points that form the set of “contact points,

1, ,cC c c M= = … . The subset I C⊂ contains the contact points between two particles

of B , whereas the subset E C⊂ contains the points where external actions are applied,

11, , , , ,

,I IM M MI c c E c c

C I M I M+

= =

= ∪ ∅ = ∩

… … (7.1)

Sets Iα , Eα and Cα denote the contact points on particle pα corresponding internal

actions, external actions and all actions. Sets Cα , Iα , Eα , I , E and C are related as

follows:

,p B

p B

p B

C C C I E

I I

E E

α

α

α

α α α α

α

α

∈

∈

∈

= = ∪

=

=

∪

∪

∪

(7.2)


The intersections of Iα and Eα are either empty or reduced to a single (contact) point,

( )

. . ( )i

E E

I I c I orα β

α β

α β

α β

∩ = ∅ ≠

∩ = ∈ ∅ ≠ (7.3)

The particle assembly inside the (REV) is in equilibrium, when each sub-particle or grain

is in equilibrium. Let us assume that the action of an internal or an external grain onto the

considered grain pα is reduced to the dynamic v. Mises motor of a force acting on

“contact” point c , ( ) ci if c fα α≡ and the moment c

imα :

( ) ( 1, 2,3)c

ii c

i

fc i

m

αα

α

⎛ ⎞∆ = =⎜ ⎟

⎝ ⎠ (7.4)

The resultant of forces acting on particle pα is the force

( 1, 2,3)ci i

c Cf f i

α

α α

∈

= =∑ (7.5)

We transport all forces and moments acting on grain pα to a “center” point α 32 of the

considered particle, thus obtaining the equivalent total dynamic v. Mises motor,

( ) ( 1, 2,3)ii

i

fi

m

αα

αα⎛ ⎞

∆ = =⎜ ⎟⎝ ⎠

(7.6)

where

( )( )

( ) ( )

( )

ci i i

c C

ac c ci i ijk j j k

c C

f f c f

m m x x fα

α

α α α

α α α

α

α ε

∈

∈

= =

= + −

∑

∑ (7.7)33

Note that force and moment equilibrium for grain pα is expressed by34

0

( )0i

α α⎛ ⎞

∆ = ⎜ ⎟⎝ ⎠

(7.8)

32 At this point of the derivation this point α should be seen as an arbitrarily chosen point inside the grain 33 Note that in eq. (7.7) summation of repeated lower indices is meant! 34 As an exercise the reader should replace this equilibrium condition by Newton’s dynamic equation between the dynamic and the kinematic motor of the considered grain.


7.1.2 The virtual work equation for a discrete assembly of particles in contact The virtual kinematics of a grain pα is given by the virtual rotation i

αδω ′ of it along an

axis passing through a point α′ and the virtual displacement iuαδ ′ of that point35. If we

select the center α α′≡ , then from eqs. (7.7) and (7.8) we get

( )( )( ) 0ac c ci i i ijk j j k i

c Cf u m x x f

α

α α α α αδ ε δω∈

⎛ ⎞+ + − =⎜ ⎟

⎝ ⎠∑ (7.9)

or

( )( )( )( ) 0c ac c ci i i ijk j j k i

c Cf u m x x f

α

α α α α αδ ε δω∈

+ + − =∑ (7.10)

This equation holds for all grains in B , thus,

( )( )( )( ) 0c ac c ci i i ijk j j k i

B c Cf u m x x f

α

α α α α α

α

δ ε δω∈ ∈

+ + − =∑ ∑ (7.11)

We note that the double sum over Cα and B can be split into two separate sums over I and E , respectively. In addition we observe that for any two grains pα and pβ in contact at point c we have from Newton’s 3rd law that,

( , )

( , )

c c ci i i

c c ci i i

f f f

m m m

α β α β

α β α β

= − =

= − = (7.12)

Thus from eq. (7.11) we get

(int) ( )extW Wδ δ= (7.13)

where

( ) ( )( )(int) ( , )c c c ci i i i i i

c IW f u u mα β α β α βδ δ δ δω δω

∈

= − + −∑ (7.14)

( )( )ext e e e ei i i i

e EW f u mδ δ δω

∈

= +∑ (7.15)

35 A silent assumption made here is that in granular media particle displacement and particle rotation are independent degrees of freedom.


Figure 7-2: Relative displacement of two grains in contact

The relative displacement and relative rotation of two homothetically rotating grains α

and β in contact is (Figure 7-2)

( ) ( )( )c c c c c

i i i i i ijk j k k j k k

ci i i

u u u u u x x x xβ α β α β β α α

β α

δ δ δ δ δ ε δω δω

δω δω δω

∆ = − = − + − − −

∆ = − (7.16)

or

( ) ( ) ( )c c c

i i i ijk j k k ijk i k k

ci i i

u u u x x x xβ α α α β β

β α

δ δ δ ε δω ε δω

δω δω δω

∆ = − + − + ∆ −

= + ∆ (7.17)

7.1.3 Compatibility in the discrete setting Let us consider an open line of grains36, as is shown in Figure 7-3. If we apply eq.(7.16)

consecutively, then we get the following expressions for the difference in rotation and

displacement between two grains in “remote” contact,

3 1 2 3c ci i i iδω δω δω δω− = ∆ + ∆ (7.18)

and

( ) ( ) ( )( )3 1 (2) (3)

1 (2) 1 2 (3) (2) 3 3 (3)

c ci i i i

c c c cijk j k k j k k j k k

u u u u

x x x x x x

δ δ δ δ

ε δω δω δω

− = ∆ + ∆

+ − + − + − (7.19)

36 usually termed also a “granular column”.


Figure 7-3: Open line of homothetically rotating grains

In general for a column of N -grains we shave,

1

( ,1) 1 ( 1)

1

NN N c

i i i iα

α

δω δω δω δω−

+

=

∆ = − = ∆∑ (7.20)

and

( )( )1

( ,1) 1 ( 1) ( 1) ( 1)

1

NN N c c c

i i i i ikj k k ju u u u x xα α α α

α

δ δ δ δ ε δω−

+ + −

=

∆ = − = ∆ − −∑ (7.21)

where

(1) 1 ( 1),c c N Nk k k kx x x x+≡ ≡ (7.22)

Following a remark by Satake [41], eqs. (7.20) and (7.21), should be seen as the discrete

manifestation of the integrability, holding for a Cosserat continuum, eqs. (3.43) and

(3.44):

2

2 1

1

( , )2 1

1( ,1) 1 ( 1)

1

( ) ( )P

P Pi i i ki k

P

NN N c

i i i i

P P dx

α

α

ψ ψ ψ κ

δω δω δω δω−

+

=

∆ = − =

↔

∆ = − = ∆

∫

∑

(7.23)

and


( )

2 2

2 1

1 1

( , )2 1

1 1( ,1) 1 ( 1) ( 1) ( 1)

1 1

( ) ( )P P

P Pi i i ki k ikj j k

P P

N NN N c c c

i i i i ikj j k k

u u P u P dx dx

u u u u x xα α α α

α α

γ ε ψ

δ δ δ δ ε δω− −

+ + −

= =

= − = −

↔

∆ = − = ∆ − −

∫ ∫

∑ ∑

(7.24)

Satake’s analogy, displayed above, allows us to identify: a) the Cosserat continuum

rotation as that kinematical property of the continuum that is meant to reproduce the

particle rotation, b) the relative deformation of the Cosserat continuum as measure for

relative or inter-particle displacement37, and c) the curvature of the Cosserat continuum

as the measure for the relative inter-particle rotation.

The discrete and the continuous realization of the relative displacement and relative

rotation between two points are given by eqs. (7.23) and (7.24). If these relative motions

are path independent then we are dealing with a “compatible” deformation. In particular

the relative motions in a compatible deformation should vanish, if evaluated in a closed

loop. This is not always the case, in granular media. To demonstrate this we consider the

paradigm of the planar, 3-rain circuit of Figure 7-4.

Figure 7-4: Three-particle assembly of two homothetically and one antithetically rotating particle, forming two rolling contacts and one sliding contact.

37 cf. eq. (7.17) and sect. 7.3


For simplicity we assume that the grains are circular rods of equal radius gR and that

grains (1) and (2) are spinning homothetically, grain (3) is spinning antithetically, all with

the same strength δω . We see immediately that in this constellation provides two pure

rolling contacts ( )rc at 1c and 3c , and a pure sliding contact ( )sc at 2c . We note that the

relative displacement between to neighboring grains is null across pure rolling contacts.

For this circuit we compute,

(2,1) 2 13 3 3(3,2) 3 23 3 3(1,3) 1 33 3 3

( , )3

0

2

( ) 2

0cycl

α β

δω δω δω δω δω

δω δω δω δω δω δω

δω δω δω δω δω δω

δω

∆ = − = − =

∆ = − = − − = −

∆ = − = − − =

⇒ ∆ =∑

(7.25)

and

( ) ( )( ) ( )

(2,1) (2) 1 (2) 1 1 (2) 12 2 2 3 3 213 3 1 1

(3,2) (3) 2 (3) (2) 2 (3) (2)2 2 2 3 3 213 3 1 1

(1,3) (1) 3 1 (2 2 2 3 3

2 2

0 ( )( 2 )

c c ck k k g g g g

c c c c ck k k g g g

c ck k k

u u x x R x x R R R

u u x x x x R R R

u u x x

δ δ ε δω δω ε δω δω δω δω

δ δ ε δω ε δω δω δω

δ δ ε δω

∆ = ∆ − − = − − − = − + =

∆ = ∆ − − = − − = + + − = −

∆ = ∆ − −( ) ( )1) 3 1 (1)213 3 1 1

( , )2

0 ( )( )

0

cg g

gcycl

x x R R

u Rα β

ε δω δω δω

δ δω

= − − = + − − =

⇒ ∆ = ≠∑ (7.26)

( ) ( )

( ) ( )

( ) ( )

(2,1) (2) 1 (2) 1 1 (2) 11 1 1 3 3 123 3 2 2

(3,2) (3) 2 (3) (2) 2 (3) (2)1 1 1 3 3 123 3 2 2

(1,3) (1) 3 1 (1) 3 1 (1)1 1 1 3 3 123 3 2 2

0 0

30430 ( )(

c c ck k k

c c c c ck k k g

c c ck k k

u u x x x x

u u x x x x R

u u x x x x

δ δ ε δω ε δω

δ δ ε δω ε δω δω

δ δ ε δω ε δω δω

∆ = ∆ − − = − − =

∆ = ∆ − − = − − = −

∆ = ∆ − − = − − = − − −

( , )2

3)4 4

3 02

g

gcycl

R

u Rα β

δω

δ δω

= −

⇒ ∆ = − ≠∑ (7.27)

This means that incorporation of antithetically rotating particles into our consideration,

would mean to extend the Cosserat model to incompatible deformations as is the case for

example in Günther’s interpretation of Kröner’s theory of dislocations [23]. In the

following we will restrict our analysis to compatible deformations.


7.1.4 Remark on incompatible deformations in granular media The question arises if incompatible deformations are important in the study of the

mechanics of “dense” granular matter. The answer to this question seems today to be

affirmative. To this end we recall an early statement by Oda & Kazama [38] who

remarked that: “..that a shear band grows through buckling of columns together with

rolling at contacts; it can be said that the thickness of a shear band is determined by the

number of particles involved in a single column...”. Indeed from the micro-mechanical

point of view an important structure that appears to dominate localized deformation in 2D

simulations is the formation and collapse (buckling) of grain columns, as this was

demonstrated experimentally by Oda and was given a simple theoretical description by

Satake [42]. These load-carrying columns belong to the so-called “competent grain

fraction” introduced by Dietrich [16] and Vardoulakis [52] (cf. also [40], [45]) and their

current length reflects more or less the current shear band thickness. Recently Tordessilas

[48] picked up on this matter and pointed that “… One such unjamming mechanism is the

buckling of force chains and associate growth of surrounding voids….This mechanism is

characteristically non-affine …”. The term “non-affine” in connection to an open line of

grains is in our understanding not outside Günther’s original idea of incompatible

deformations and Satake’s integrability and dislocation concepts. This can be seen in

Figure 7-5 taken from [48], where we clearly observe that the line of grains that caries

non-affine deformation information includes antithetically rotating grains. Indeed, as we

have sketched in Figure 7-6, these columns could contain cross-links between

antithetically rotating lines, such as the grain pairs (3,3 )′ and (5,6 )′ in that figure. These

cross-links contain strong rolling contacts, of reduced resistance, if compared to sliding

contacts and serve as buckling hinges in case they belong to the strong-force network. To

our understanding, the essential feature here is the incompatibility of grain rotation across

a grain contact, that is leading to the possibility of an internal instability in the form of an

internal (frictional) plastic hinge. Thus shear-banding in the sense of Oda should include

the formation of plastic hinges among grains along strong force chains as an internal

instability.


Figure 7-5: Picture taken from Tordesillas [48]

Figure 7-6: Two lines of compatible rotations (1,2,3,4,5) and (3 ,4 ,5 ,6 )′ ′ ′ ′ . If we “integrate” along the lines (1,2,3,4,5) and (1,2,3,3 ,4 ,5 ,6 ,5)′ ′ ′ ′ we may end up with different results, reflecting the “dislocation” that is trapped inside the closed loop (3,3 ,4 ,5 ,6 ,5,4,3)′ ′ ′ ′ .Antithetically rotating grains-pairs, such as (3,3 )′ and (5,6 )′ , are connected with plastic “hinges” (colored in red) and are leading to “buckling” of strong force columns (colored here in yellow and light blue).


7.1.5 Example: Buckling of rigid-plastic, frictional hinged mechanism38

Figure 7-7: Buckling of frictional hinged mechanism consisting of two rigid rods

In order to explain grain-column buckling it is not necessary to make any assumption

concerning the elasticity of grains. It suffices only to consider rolling friction. In order to

demonstrate this statement let us consider the mechanism of Figure 7-7, consisting of two

rigid bars of length L connected with a frictional hinge. We may assume that the shear

forces acting as reactions at the end-points of the mechanism are limited by Coulomb’s

law,

s nf f µ≤ (7.28)

The limit moment that the plastic frictional hinge can sustain is

l nm f e= (7.29)

where e is the “rolling friction” coefficient, with dimensions of length. The mechanism

is stable as soon as

s lf L m< (7.30)

Thus, stability is achieved as soon as

n nf L f e e Lµ µ< ⇒ > (7.31)

The “shear-band thickness is then estimated as

2 2B cred Lµ

≈ = (7.32)

38 This model problem is a very crude first attempt to expalain the effect of plastic hinges forming in grain columns, that should lead to instabilities like shear banding and micro-structural softening.


7.1.6 Equilibrium conditions for compatible virtual kinematics The virtual displacement and rotation of the grains are assumed to be continuous

functions of position of the center-points of the grains and expand them in series of

position vector of the center of the particle. We truncate these (test) functions to the

second order for the displacement,

i i ij j ijk j ku a b x c x xα α α αδ = + + (7.33)

and to the first order for the rotations

i i ij jxα αδω α β= + (7.34)

where , , , ,i ij ijk i ija b c α β are arbitrary coefficients.

The above (continuity) assumption, that allows us to write down eqs. (7.33) and (7.34),

restricts the variations of the kinematic variables to compatible sets39. From eqs.(7.16) to

(7.34) we get

( ) ( ) ( )

( ) ( )( )ci ij j j ijk j k j k ijk j k k

c cijk jl k k l k k l

u b x x c x x x x x x

x x x x x x

β α β β α α β α

β β α α

δ ε α

ε β

∆ = − + − − −

+ − − − (7.35)

( )ci ij j jx xβ αδω β∆ = − (7.36)

( )

( ) ( )

e e ei i ijk j k k

e e e e e e e ei ij j ijk j k ijk j k k ijk jl l k k

u u x x

a b x c x x x x x x x

α α α

α α α α α α

δ δ ε δθ

ε α ε β

= + −

= + + + − + − (7.37)

where eixα is the position of the center of particle pα , where contact e takes place.

From eqs. (7.35) to (7.37) we get the following expressions for the virtual work of

internal and external actions in the considered (REV),

( )

( )

( )

( ) ( )( ) ( )( )

(int) ( , )

( , )

( , )

( , ) ( , )

cij i j j

c I

cijk i j k j k

c I

cj i ijk k k

c I

c c c cji ijk i l k k l k k j l l

c I

W b f x x

c f x x x x

f x x

f x x x x x x m x x

α β β α

α β β β α α

α β β α

α β β β α α α β β α

δ

α ε

β ε

∈

∈

∈

∈

= −

+ −

− −

+ − − − + −

∑

∑

∑

∑

(7.38)

39 cf. sect. 3.3 .


( )( )( )( )

( )ext e e ei i ij i j

e E e E

e e eijk i j k

e E

e e e ej j ijk i k k

e E

e e e e eji j ijk i k k l

c I

W a f b f x

c f x x

m f x x

m f x x x

α

α α

α

α α

δ

α ε

β ε

∈ ∈

∈

∈

∈

= +

+

+ + −

+ + −

∑ ∑

∑

∑

∑

(7.39)

The virtual work equation (7.13), with eqs. (7.38) and (7.39), applies for arbitrary choice

of the coefficients , , , ,i ij ijk i ija b c α β . Thus by independent variation of these coefficients

we get the following set of algebraic equations:

0ei

e Ef

∈

=∑ (7.40)

( ) ( , )c e ej j i j i

c I e Ex x f x fβ α α β α

∈ ∈

− =∑ ∑ (7.41)

( )( , )c e e ei j k j k i j k

c I e E

f x x x x f x xα β β β α α α α

∈ ∈

− =∑ ∑ (7.42)

( ) ( , ) ( , )c eijk j j k i

c I e E

x x f mβ α α β αε∈ ∈

− = −∑ ∑ (7.43)

( ) ( )( ) ( )( )( , ) ( , ) ( , )c c c c e eikl l j k k j k k i j j i j

c I e Ef x x x x x x m x x m xα β β β α α α β β α α αε

∈ ∈

− − − + − =∑ ∑ (7.44)

where

( )( , )e e e e ei i ijk j j km m x x fα αε= + − (7.45)

is the moment acting that results by transporting the external contact force and couple

from point e on particle pα to its centerα .

Eq. (7.40) is expressing the equilibrium of external forces that are applied to the whole

assembly of particles in the considered (REV). On the other hand from eq. (7.41) we get

( ) ( , )c e eijk j j k ijk j k

c I e Ex x f x fβ α α β αε ε

∈ ∈

− =∑ ∑ (7.46)

and with that eq. (7.43) transforms into

( )( , ) 0e e ei ijk j k

e E

m x fα αε∈

+ =∑ (7.47)

or the moment equilibrium equation for all external actions on the considered (REV),


( )( )

( )

0

0

e e e e e ei ijk j j k ijk j k

e E

e e ei ijk j k

e E

m x x f x f

m x f

α αε ε

ε∈

∈

+ − + = ⇒

+ =

∑

∑ (7.48)

If we combine eqs.(7.42) and (7.44) we obtain,

( ) ( ) ( )( , ) ( , )c e e e e c ci j j i inm n m j ikl l j j k

c I e E c I

m x x m x f x f x x xα β β α α α β β αε ε∈ ∈ ∈

− = + − −∑ ∑ ∑ (7.49)

We summarize below the set of equations that we derived by applying the virtual work

equation on an (REV) of particles that are in a state of static equilibrium under the action

of external forces and couples,

0ei

e Ef

∈

=∑ (7.50)

( ) 0e e ei ijk j k

e E

m x fε∈

+ =∑ (7.51)

( ) ( , )c e ej j i j i

c I e Ex x f x fβ α α β α

∈ ∈

− =∑ ∑ (7.52)

( ) ( , )c e e ej k j k i i j k

c I e E

x x x x f f x xβ β α α α β α α

∈ ∈

− =∑ ∑ (7.53)

( ) ( ) ( )( , ) ( , )c e e e e c ci j j i inm n m j ikl l j j k

c I e E c I

m x x m x f x f x x xα β β α α α β β αε ε∈ ∈ ∈

− = + − −∑ ∑ ∑ (7.54)

7.1.7 A micromechanical definition of average stress and couple stress We consider now a strategy for a transition from the discrete medium to the continuum.

This is by far not a unique procedure, thus having always the character of a working

hypothesis. The mathematical limitations of such strategies are discussed in detail by

Froiio et al. [20].

The analysis starts from the stress equilibrium equations that apply for the continuum,

e.g. eqs. (4.41). We assume the small volume V that is occupied by the (REV) and

observe that in that case (as was done already in the discrete medium analysis) the effect

of volume forces can be neglected. Thus we assume the existence of a stress field that

satisfies the following equations

0ijk REV

i

x Vxσ∂

= ∀ ∈∂

(7.55)

ij i j k REVn t x Vσ = ∀ ∈∂ (7.56)


For the computation of a mean value of the stress within the (REV) we follow a standard

procedure40: We multiply eq. (7.55) with kx , integrate over V and apply Gauss’ theorem,

( ) ( )0

0REV REV REV

REV REV

i ij k i ij k ij i kV V V

ij k i ij kiV V

x dV x dV x dV

x n dS dV

σ σ σ

σ σ δ∂

= ∂ = ∂ − ∂ ⇒

− =

∫ ∫ ∫

∫ ∫ (7.57)

or

REV REV

kj j kV V

dV t x dSσ∂

=∫ ∫ (7.58)

First we observe that the quantity

1

REV

ij ijREV V

dVV

σ σ= ∫ (7.59)

is describing the volume-averaged stress.

Secondly we juxtapose eqs. (7.58) and (7.52)

( ) ( , )

REV REV

e e ck j kj k j k k j

e E c IV V

x t dS dV x f x x fα β α α βσ∈ ∈∂

= ↔ = −∑ ∑∫ ∫ (7.60)

This identification suggest a formula for the computation of the mean stress inside the

(REV) using micromechanical information41,

( , )1 ( ) cij i i j

c IREV

x x fV

β α α βσ∈

≈ −∑ (7.61)

Eq. (7.61)is a celebrated formula that is attributed to Love42 and has been used by many

authors since.

Similarly we assume the existence of a couple stress field that satisfies the following

equations

ik i kn mµ = (7.62)

0i ij imj imµ ε σ∂ + = (7.63)

From these equations we derive,

40 L.D. Landau and E.M. Lifshitz, Theory of Elasticity, Vol.7, sect. 2, p.7, Pergamon Press, 1959. 41 e.g. information stemming from a DEM simulation of a granular medium. 42 A.E.H. Love, A Treatise of the Mathematical Theory of Elasticity, Cambridge University Press, 1927.


( ) ( )R

0REV REV REV EV

i ij imj im k i ij k ij i k imj im kV V V V

x dV x dV x dV x dVµ ε σ µ µ ε σ∂ + = ∂ − ∂ + =∫ ∫ ∫ ∫ (7.64)

or

R

R

0REV REV EV

REV REV EV

kj ij i k imj im kV V V

kj j k imj im kV V V

dV n x dV x dV

dV m x dS x dV

µ µ ε σ

µ ε σ∂

= + = ⇒

= +

∫ ∫ ∫

∫ ∫ ∫ (7.65)

We remark that if surface couples and couple stresses are zero then eq. (7.65) reduces to a

condition that implies symmetry of stress tensor. In general however, with

( ) ( ) ( )

( )

0REV REV REV

REV REV

i ij k l i ij k l ij i k lV V V

kj l lj k j k lV V

x x dV x x dV x x dV

x x dV t x x dS

σ σ σ

σ σ∂

= ∂ = ∂ − ∂ ⇒

+ =

∫ ∫ ∫

∫ ∫ (7.66)

the last integral on the r.h.s. of eq. (7.65) becomes,

R R R

R R

[ ] [ ]

EV EV EV

EV EV REV

imj im k imj im k imj mi kV V V

imj mi k imj ki m imj i k mV V V

x dV x dV x dV

x dV x dV t x x dS

ε σ ε σ ε σ

ε σ ε σ ε∂

= = −

= − = −

∫ ∫ ∫

∫ ∫ ∫ (7.67)

and with that

RREV REV EV REV

kj j k imj ki m imj i k mV V V V

dV m x dS x dV t x x dSµ ε σ ε∂ ∂

= + −∫ ∫ ∫ ∫ (7.68)

or

( ) ( )REV REV

kj jmi m ki j jmi m i kV V

x dV m x t x dSµ ε σ ε∂

+ = +∫ ∫ (7.69)

First we observe that the quantity

1

REV

ij ijREV V

dVV

µ µ= ∫ (7.70)

is the mean couple stress computed over the considered volume.

Secondly we juxtapose eqs. (7.69) and (7.54), that is written as follows,

( )( ) ( )( , ) ( , )c c c e e e ek k j jmi m i j mij m i k

c I e Ex x m x f m x f xβ α α β α β αε ε

∈ ∈

− + = +∑ ∑ (7.71)

This comparison suggests the introduction of the following definitions fort the

computation of the mean couple stress inside the (REV):


1) The mean couple stress referred to particle centroid, introduced by Oda [37]:

( ) ( , )1 ckj k k j

c IREV

x x mV

β α α βµ∈

= −∑ (7.72)

2) The “transported” couple stress at interparticle contact points, introduced by Bardet &

Vardoulakis [2] and Tordessilas & Walsh [49]:

( )( )( ) ( , ) ( , )c c c ckj k k j jmi m i

c I

x x m x fβ α α β α βµ ε∈

= − +∑ (7.73)

The existence of these two definitions explains in part also the controversy that exists in

relation to the statement that couple stresses in granular media are: a) only due to contact

couples [37], an assumption that would support definition (7.72), or b) that they are also

generated in part also by the contact forces, as in definition (7.73). Both definitions are

meaningful and valid. We will demonstrate, however in sect. 7.3 , that indeed the

transported to the contact point couple stress ( )cijµ is the one that does work on the rolling

contact.

7.1.8 Example: Computation of the Love stress in a regular hinged lattice For the illustration of Love’s formula for the computation of the stress in a continuum

that is supposedly carrying the average stress of an underlying discrete medium, we

consider here the example of a regular truss under the action of a force F as shown in

Figure 7-8.

Figure 7-8: Regular lattice under shear


Table 7-1: Evaluation of the regular lattice of Figure 7-8

We focus on the central node a and we define an (REV) that includes all connecting rods

(1) to (6) . Love’s formula applied in this example will yield a symmetric stress tensor,

because the hinged rods cannot transmit an moments, thus

( )6

1

1 c c c cij i j j i

cREV

S SV

σ=

= +∑ (7.74)

In this expression ci are the projections of the lengths of the rods on the coordinate axes

and iS are the corresponding components of the forces of the respective rods. The

Volume of the (REV) is computed as follows: The central region considered is span by a

hexagon whose total surface is

23 32

S = (7.75)

Each rod has a volume

rodV A= (7.76)

where

2 / 4A Dπ= (7.77)

is the cross-sectional area of each rod (assuming that they are equal). Thus the volume

occupied by this structure is

23 32REVV SD D= = (7.78)

Note that the “solid” fraction has a volume


2362s rodV V D= = (7.79)

and the “porosity” of the structure is

11 ( )3

REV s

REV

V V D DV

φ − ⎛ ⎞= = − <<⎜ ⎟⎝ ⎠

(7.80)

In Table 7-1 we summarize the results concerning the forces carried by the central rods,

that are computed using an elastic analysis of the considered truss.

The components of the resulting stress tensor, evaluated from eq. (7.74) in the considered

Cartesian coordinate system are

[ ]0 / 4 0/ 4 0 00 0 0

τσ τ

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

(7.81)

where

2 ;3 3REV

F D FV A

πτ σ σ⎛ ⎞= = =⎜ ⎟⎝ ⎠

(7.82)

Exercises: 3) Verify the accuracy of Love’s formula and the result given by eq. (7.82), by solving

numerically the problem of a triangular disc of thickness D , loaded by a concentrated

force43 as in Figure 7-8. The disc should be made of poro-elastic material with

porosity φ as given above by eq. (7.80).

4) Solve the same problem as above in sect. , by assuming that 7.1.8 by assuming that all

rods are clamped and not hinged. Develop the formulae for the stress and couple

stress at collocation point ( )a and verify the result numerical as in Exercise 3).

43 The problem of wedge loaded with a force at is tip is an ill-posed problem within linear Boltzmann elasticity, thus the linear Cosserat elastic solution is needed here; cf. Bogy, D.B. and Sternberg. (1968). The effect of couple-stress on the corner singularity due to an asymmetric shear loading. Int. J. Solids Structures, 4, 159-174.


7.2 Mass and moment of inertia considerations

Within the realm of Cosserat continuum mechanics the smallest elementary unit in

granular medium is the grain, which moves in space in good approximation as a rigid

body. For example, a spherical grain with radius gR has a volume

343s gV Rπ= (7.83)

The grain is made of mineral with bulk density gρ and has the mass,

g g sm Vρ= (7.84)

The statistical (REV) has a volume V and includes grains that do not fill all the available

space, but leave some void space, vV Vφ= free of solid mass,

1

ss v s

VV V V V V Vφφ

= + = + ⇒ =−

(7.85)

where φ is the porosity of the granular medium. Thus the mass of (distributed) solid

material per unit volume is

( )1gg

mV

ρ φ ρ= = − (7.86)

On the other hand the moment of inertia of a homogeneous sphere is,

5 28 215 5s g g g gJ R m Rπρ= = (7.87)

The moment of inertia distributed over the volume V is44

22;5

sg

JJ J J RV

ρ ∗ ∗= = = (7.88)

This means that even on that naïve level of simplification a physical property with

dimension of length enters the mathematical description of the material behavior of

granular media through the micro-polar inertia of that medium.

44 cf. eq. (6.27).


Let ( )gN D D< be the cumulative distribution curve (or “sieve” curve) of the grain

diameters per unit mass that appear in a statistical (REV) that is given as function of the

grain diameter 2g gD R= . Then the representative material moment of inertia is

* * 21;10 gJ J J Dρ< >= = (7.89)

where

max

min

2 2 ,D

gD

dND N D dD NdD

′ ′= =∫ (7.90)

7.3 Grain scale energy dissipation considerations

Figure 7-9: Picture taken from Cole & Peters [12]

As stated by Cole & Peters [12] (Figure 7-9)“…the relationship between the contact

motions and resisting forces define the micro-scale properties of the medium. The

constitutive response of the material at the macro-scale is an emergent property that is the

result of the collective response of the aggregate, and depends on the micro-scale

properties, the stochastic nature of the particle arrangement and boundary conditions ….

A similar response is modeled for rolling resistance which is assumed to be proportional

to the magnitude of the normal force. Not shown is the torsional mode in which the


relative rotation vector is aligned with the normal axis. The torsional mode is of greatest

interest in bonded materials....”.

Figure 7-10: Two grain circuit with sliding contact and rolling contact respectively.

As shown in Figure 7-10, the contact of two homothetically rotating grains will involve

strong contact sliding and weak contact rolling, whereas the contact of two antithetically

rotating grains will involve strong contact rolling and weak contact sliding. In this

context we like to quote directly from Tordesillas & Walsh [49]: “…Johnson45 classified

the various sources of rolling resistance to be: (a) those arising from microslip and

friction at the contact interface, (b) those due to the inelastic properties of the contacting

bodies, (c) those due to the roughness of the rolling surfaces. The latter two may be safely

ignored in quasistatic loading of densely packed assemblies: inelastic deformation may

be ignored for small particle deformations while surface irregularities only influence

rolling resistance in the following two ways. Firstly, they intensify the contact pressure at

certain points in the contact area, causing local plastic deformation, but this may be

ignored for the same reasons for neglecting inelastic deformation. Secondly, as particles

move past each other, energy is lost from impacts between irregularities on the opposing

contacting surfaces, but this may be considered negligible in systems undergoing quasi-

static deformation…”.

A basic hidden assumption met in earlier studies was that almost all energy dissipation is

localized at sliding contacts [31]. In general, however, energy dissipation due to rolling

45 K.L. Johnson, Contact Mechanics, Cambridge Univ. Press, Cambridge, 1985.


cannot be excluded due to micro slip and friction at the contact interface. Thus, in recent

numerical simulations, energy dissipation is admitted to rolling contacts as well. This

assumption is in line with the Cosserat plasticity theory of granular materials [11], [49].

7.4 The 2-grain circuit of homothetically rotating grains

Figure 7-11: Two-grain circuit: kinematic embedment

Let two homothetically rotating grains of equal radius gR with a strong sliding contact,

as seen in Figure 7-11. The branch vector that connects the centers of the two grains is

1 2( ) 2 ; 1i i i g i k kK K l R n n n= = , = (7.91)

The velocities of the centers of the grains (1) and (2) are denoted by (1)iv and (2)

iv , and the

grains are rotating homothetically with angular velocities (1)kω and (2)

kω , respectively. At

the midpoint c of the center line 1 2( )K K the velocities of the grains are

( )

( )( )

(1 ) (1) (1) (1) (1)

(2 ) (2) (2) (2) (2)

ci i g i ilk l ki

ci i g i ilk l ki

v v R n v

v v R n v

ω ε ω

ω ε ω

= + × = +

= + × − = − (7.92)

Thus the relative velocity and relative rotation of grain (2) with respect to grain (1) at

the contact point c are [11], [49]46:

46 cf eq. (7.16)


( )( )2(2,1) (2 ) (1 ) (2) (1) (1)c ci i i i i ijk j j kv v v v v ε ω ω= − = − − + (7.93)

(2,1) (2) (1)i i iω ω ω= − (7.94)

We assume that the particle velocity is embedded now into a continuous field, such that

(1) (2), 2i i i i j i jv v v v v= ≈ + ∂ (7.95)

Similarly for the particle spin we assume that

(1) (2), 2l l l l m m lw w wω ω= ≈ + ∂ (7.96)

With this notation eq. (7.93) yields

( )( )( ) ( )

( )( )

2(2,1) (2) (1) (1)

2,1 2 2 2

2

2

i i i ijk j j k

i j i j ikj j m m j k

k i ikj j ikj m m j k

k i ki ikj m m j k

v v v

v v w w

v w w

v w w

ε ω ω

ε

ε ε

ε

= − − +

= ∂ + + ∂

= ∂ + + ∂

= ∂ + + ∂

(7.97)

With

;ki k i ki ik k iv w K wΓ = ∂ + = ∂ (7.98)

we get

( )(2,1) 2i ki ikl lm m kv Kε= Γ + (7.99)

Similarly we get that

(2,1) 2i m m iwω = ∂ (7.100)

Since the two grains are in contact, we assume that they interact with contact forces and

contact couples. The force and the couple acted upon grain (1) by grain (2) are denoted

as (2,1)if and (2,1)

im , respectively; their reactions are the contact force (1,2)if and the

contact couple (1,2)im that are acted by grain (1) on grain (2) . These force- and couple

pairs satisfy Newton’s 3rd law,

(1,2) (2,1) (1,2) (2,1);i i i if f m m= − = − (7.101)

The interface at the contact of the two grains is identified as an intergranular surface.

This is a continuum material band of vanishing thickness, whose boundaries share the


motion of the two adjacent faces of the contact 47 (1,2)c and (2,1)c . On the faces of this

infinitesimal slip the reactions of the intergranular forces are acting. On the face of the

intergranular surface that contains point (1,2)c , with the outer unit normal in− the force

(1,2)if is acting and on the face that contains point (2,1)c with the outer unit normal in+ the

force (2,1)if is acting.

The rate of work per unit volume, done by these forces at the considered contact due to

sliding is (Figure 7-12)

( ) ( )( ) (1,2) (1 ) (2,1) (2 ) (1,2) (1 ) (2 ) (1,2) (2,1)1 1 1ns c c c ci i i i i i i i iP f v f v f v v f v

V V V= + = − = (7.102)

Figure 7-12: Two-grain circuit: strong sliding contact

Similarly the rate of work of contact couples at the considered contact due to (weak)

rolling is (Figure 7-13),

( ) ( )( ) (1,2) (1) (2,1) (2) (1,2) (1) (2)1 1nri i i i i i iP m m m

V Vω ω ω ω= + = − (7.103)

47 In the terminology of Tribology this interface is called the “third body”. As stated by Godet [22], “…Interfaces, or third bodies can be defined in a material sense, as a zone which exhibits a marked change in composition from that of the rubbing specimens or in a kinematic sense, as the thickness across which the difference in velocity between solids is accommodated…”.


Figure 7-13: Two-grain circuit: weak rolling sliding contact

We assume that the force (1,2)if and the couple (1,2)

im are generated by a stress field and a

couple stress field, which are defined in turn at the center of the considered grain.

Thus

( ) ( )(1,2) (2,1)

,i iki k ki k

f fn nS S

σ σ= − = + (7.104)

where 2S a= is the area of a the face of a cube with unit volume , 3V a= , and with that

( )

( )

( ) (1 ) (2 )

(2 ) (1 ) (2,1)

( )

1 1 ;

ns c cki k i ki k i

c cki k i i ki k i

SP n v n vV

Vn v v n v aa a S

σ σ

σ σ

⎛ ⎞= − +⎜ ⎟⎝ ⎠

= − = = (7.105)

or, due to eq. (7.99)

( )

( ) ( ) ( )( )

( ) 1 2

2

nski k ni inl lm m n

gki k n ni inl lm m n

P n Ka

Rn n K n n

a

σ ε

σ ε

= Γ +

= Γ + (7.106)

Let

2 gR

aλ = (7.107)

With the notation

( )ni ki kt nσ= (7.108)


( )

( ) ;

ni ki k

ni im m m m i i m m

n

K n n w Dw D n

Γ = Γ

Κ = = ∂ = = ∂ (7.109)

the expression eq. (7.106) for the work per unit volume done by the grain contact forces

becomes,

( )( ) ( ) ( ) ( )ns n n ni i ikl k lP tλ ε= Γ + Κ (7.110)

We observe that the compound

(1)( )

(2) ( ) ( )

n

n n

vv

v

⎛ ⎞ ⎛ ⎞Κ⎜ ⎟= = ⎜ ⎟⎜ ⎟⎜ ⎟ Γ + × Κ⎜ ⎟ ⎝ ⎠⎝ ⎠

(7.111)

is a v. Mises kinematic motor. Moreover the component

(2)

( ) ( )n nv = Γ + ×Κ (7.112)

is dual in energy to the stress vector

(1)

( )ntσ = (7.113)

Since according to eqs. (7.110), (7.112) and (7.113),

(1) (2)

( )nsP vλσ= ⋅ (7.114)

We postulate the couple-stress field ( )cijµ , that we call hereafter the contact couple stress,

such that,

( )cij ij ijk l lkµ µ ε σ= − (7.115)

and we define the moment vectors,

( ) ( ) ( );cn c ni ij k i ki kn nµ µ µ µ= = (7.116)

Thus

( )( ) ( )

( ) ( )

cn cj ij i ij ijk l lk i

ij i ijk lk l i ij i ijk i lk l

n nj jik i k

n n

n n n n

t

µ µ µ ε σ

µ ε σ µ ε σ

µ ε

= = −

= − = −

= +

(7.117)

We observe that the compound


(1)( )

(2) ( ) ( )

n

n n

ttt

tt µ

⎛ ⎞ ⎛ ⎞⎜ ⎟= = ⎜ ⎟⎜ ⎟⎜ ⎟ + ×⎜ ⎟ ⎝ ⎠⎝ ⎠

(7.118)

is a v. Misses dynamic motor. Indeed the selection of

( ) ( ) ( )cn n nj j jik i ktµ µ ε= + (7.119)

is meaningful, because this corresponds the transport of the traction and the couple

defined at the center of the grain to the contact point, according to the transport law of

statics,

( ) ( ) ( )

1 1 1

( ) ( ) ( ) ( )1 1 1

( ) ( ) ; ( ) ( )

( ) ( ) ; ( ) ( )

n n nki k

n n n nki k

t c t K t K K n

c K t K K n

σ

µ µ µ µ

= =

= + × = (7.120)

.With these remarks we introduce the inner product

( ) ( )( )( ) ( )( )

(1) (2)(2) (1)( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

n

n n n n n n

n n n n n n n n

P t v t v t v

t t

t t t

λ

λ µ

λ µ

= = ⋅ + ⋅

= ⋅ Γ + ×Κ + + × ⋅Κ

= ⋅Γ + ⋅ ×Κ + ⋅Κ + × ⋅Κ

(7.121)

or

( )( ) ( ) ( ) ( ) ( )n n n n nP tλ µ= ⋅Γ + ⋅Κ (7.122)

We showed already that the first on the r.h.s. of eq. (7.122) reflects the work done by the

forces due to (strong) sliding, eq. (7.114). The 2nd term corresponds on the r.h.s. of eq.

(7.122) corresponds to the work done by the couples at the considered contact due to

(weak) rolling,

( )

( )

( )

( ) (1,2) (1) (2,1) (2)

( ) (1) ( ) (2)

( ) (2) (1) ( ) (2,1) ( )

1

( )

21 1

nri i i i

c cki k i ki k i

gc c cki k i i ki k i ki k m m i

P m mV

S n nV

Rn n n w

a a a

ω ω

µ ω µ ω

µ ω ω µ ω µ

= +

⎛ ⎞= − +⎜ ⎟⎝ ⎠

= − = = ∂

(7.123)

or due to eqs. (7.100), (7.116) and (7.109)

( ) ( ) ( )nr cn nP λµ= ⋅Κ (7.124)


where the componenets of the couple stress vector at the contact ( )cnµ are given by eq.

(7.117). Thus

( )(2) (1)

( ) ( ) ( ) ( )nr n n nP t t vλ µ λ= + × ⋅Κ = ⋅ (7.125)

and with that

( ) ( ) ( )n ns nrP P P= + (7.126)

7.5 Statistical averaging

As a starting point we consider Cauchy’s fundamental theorem, that relates the

components of stress vector on a plane with unit outward normal in as linear functions of

the stress tensor,

j ij it nσ= (7.127)

Let now in be the unit normal that characterizes an intergranular contact plane, as this

was discussed in the previous section. We select all such contact-plane normal vectors

and transfer them parallelly to the center of the unit sphere in 3R (Figure 7-14). Let E′

be the corresponding point on the surface of the unit sphere. At this point we attach the

stress vector, that derives from eq. (7.127).

Figure 7-14: Mapping of the unit contact plane vectors on the unit sphere

Following this reasoning one could ask for example the question as of what is the mean

value of the normal component of the stress vector if one considers all probable normal-

contact directions in the considered statistical (REV). Thus we define first the

scalarquantity


( )ni i ki k it t n n nσ= = (7.128)

and try to compute its mean value

( )np t=< > (7.129)

The simplest possible model derives from the assumption that the probability distribution

of the unit contact-plane normal vectors is uniform. This assumption is rather crude as

gar as granular media is concerned, and for realistic modeling considerations should be

replaced by suitable anisotropic probability distributions [49]. In case of isotropy,

averaging over all contact normals is done on the unit sphere as follows [27], [35], [11]:

2 2

( ) ( ) ( )

0 0 0 0

1 1sin sin4 4

n n nt t d d t d dπ π π π

θ θ φ θ θ φπ π

< > = =∫ ∫ ∫ ∫ (7.130)

where 1r = , θ and φ are the polar, spherical coordinates of point E′ . We observe that

the corresponding Cartesian coordinates of the position vector OE n→

′ = on the unit sphere

are (Figure 7-15):

1 2 3sin cos ; sin sin ; cosn n nθ φ θ φ θ= = = (7.131)

Figure 7-15: Position vector on the unit sphere: spherical and Cartesian coordinates.

It can be shown that following identities hold [27]:


( )

( )

2

0 0

ln

0

1 1sin4 3

0

1 13 5 15

0

1 13 5 7 105

i

i j i j ij

i j k

i j k l ij kl ik jl il jk ijkl

i j k l m

i j k l m n in jklm jn klmi kn lmij mijk mn ijkl ijklmn

n

n n n n d d

n n n

n n n n

n n n n n

n n n n n n

π π

θ θ φ δπ

δ δ δ δ δ δ δ

δ δ δ δ δ δ δ δ δ δ δ

< >=

< > = =

< > =

< >= + + =⋅

< > =

< > = + + + + =⋅ ⋅

∫ ∫

(7.132)

Thus

( ) 1 13 3

nji j i ji j i ij ij kkt n n n nσ σ σ δ σ< > =< > = < > = = (7.133)

We recover here the well known statement that the trace of the stress tensor48 is a

measure for the mean normal traction,

( ) 13

nkkp t σ=< >= (7.134)

Before we proceed with the statistical interpretation of further stress- and couple stress

invariants we return to the expression for the work of contact forces and contact couples

done at intergranular contact, eq. (7.122) [49],

( )( )( )

( )

( ) ( ) ( ) ( ) ( )n n n cn n

ki k li l ij ijk i lk i jm m

ki li k l ij jm i m g ijk lk jm l i m

P t

n n n K n

n n K n n R K n n n

λ µ

λ σ µ ε σ

λ σ µ ε σ

= ⋅Γ + ⋅Κ

= Γ + −

= Γ + −

(7.135)

Thus

( )( )

( )

( )

3

3

nki li k l ij jm i m g ijk lk jm l i m

ki li kl ij jm im

ki ki ij ji

P n n K n n R K n n n

K

K

λ σ µ ε σ

λ σ δ µ δ

λ σ µ

< >= Γ < > + < > − < >

= Γ +

= Γ +

(7.136)

By comparing this result with eq. (6.24) we conclude that for

48 i.e. the mean normal stress


2 23

3g

g

R Va Ra S

λ = = ⇒ = = (7.137)

and with that

( )nki ki ij jiP Kσ µ< >= Γ + (7.138)

the particular choice of micromechanical variables at the level of intergranular contact

has allowed us to recover the stress power of the Cosserat continuum as the average value

of the work done by contact forces and contact couples at the third body of strong sliding

contact.

We may now return to the stress analysis. From eqs. (7.127) and (7.128) we get the

expression for the shear stress vector, that is tangential to unit sphere,

( ) ( )t ni i i ki k kl k l it t t n n n n nσ σ= − = − (7.139)

We remark that by introducing the decomposition of the stress tensor in spherical and

deviatoric part,

1 , 03ij kk ij ij kks sσ σ δ= + = (7.140)

we get

( ) 13

nij i j kkt s n n σ= + (7.141)

( )( )tj ik i jk j kt s n n nδ= − (7.142)

Thus from eq. (7.141) we retrieve eq. (7.133),

( ) 1 1 1 1 13 3 3 3 3

nij i j kk ij ij kk ii kk kkt s n n s sσ δ σ σ σ< >= + = + = + = (7.143)

From eq. (7.142) we get

( ) ( )

( )

( ) ( )t tj j ik i jk j k nm n jm j m

ik i nm n mk k m ik nk i n ik nm i n k m

t t s n n n s n n n

s n s n n n s s n n s s n n n n

δ δ

δ

= − −

= − = − (7.144)

and it’s average value

( ) ( ) 4 13 5 3 5

t ti i ki ki kp pkt t s s s s< >= −

⋅ ⋅ (7.145)

Based on this computation we define here the shearing stress intensity as,


( ) ( )5 2 12 3 6

t ti i ki ki kp pkt t s s s sΤ = < > = − (7.146)

Remark We notice that if the stress tensor is symmetric, then the average of the square of the

shear stress magnitude is related to the shearing stress intensity [56],

( ) ( ) 2

( ) ( )

1 2 1 2 1;5 5 2 5 252

t ti i ki ki ki ki ki ki

t ti i

t t s s s s s s

t t

< >= = = Τ Τ =

⇒ Τ = < >

(7.147)

or

25meanτ = Τ (7.148)

We note that the so-called shearing stress intensity Τ in case of a Boltzamnn continuum,

differs but little from the maximum shear stress49,

( )

( )max

max

min0.87 max

ττ

⎧⎪Τ = ⎨⎪⎩

(7.149)

We may now repeat the above procedure for the contact couple-stress tensor, ( )cijµ ,

defined above through eq. (7.115)

( )cij ij g ijk lk lR nµ µ ε σ= − (7.150)

The corresponding couple stress vector is defined through Cauchy’s fundamental theorem

( ) ( )c cj ij i ij i g ijk lk l in n R n nµ µ µ ε σ= = − (7.151)

The normal component

( ) ( ) ( )cn c cj j ij i j ij i j g ijk lk l i jn n n n n R n n nµ µ µ µ ε σ= = = − (7.152)

The 1st statistical moment of the contact couple-stress tensor ( )cijµ is a measure for the

mean torsion at the contact,

49 see sect. 9


( ) 13

cnij i j g ijk lk l i j kkn n R n n nµ µ ε σ µ< >= < > − < >= (7.153)

or

( ) 13

cnT kkm µ µ=< >= (7.154)

With

( ) ( ) ( ) ( ) ( )ct c cn c ci i i ki k kl k l in n n n nµ µ µ µ µ= − = − (7.155)

we may also compute the 2nd statistical moment of the contact couple-stress tensor, ( )cijµ ,

( )( ) ( ) 24 1 1 43 5 3 5 3 5

ct cti i ki ki qi iq g mr mr lm mlm m m m R s s s sµ µ< >= − + −

⋅ ⋅ ⋅ (7.156)

where ijm is the couple-stress deviator,

13ij ij kk ijmµ µ δ= + (7.157)

Similarly we define the intensity of deviator couples, as

( ) ( ) 2 25 2 1 12 3 6 6

ct cti i ki ki qi iq gm m m m Rµ µΜ = < > = − + Τ (7.158)

The above introduced stress- and couple-stress invariants can be used in the formulation

of constitutive equations for granular media. In case that someone wishes for example to

generalize plasticity models that incorporate in their formulation the effect of the 3rd

invariant, then one could consider the computation of 3rd order “moments” [56] of the

deviators of the stress tensor ijσ and the contact couple stress tensor ( )cijµ . For example

we may introduce the standard deviations

2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 13 3

1 13 3

t t t t t t t ti j p p ij i j q q ij

ct ct ct ct ct ct ct cti j p p ij i j q q ij

t t t t t t t tσ δ δ

µ µ µ µ µ δ µ µ µ µ δ

⎛ ⎞⎛ ⎞= − −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

⎛ ⎞⎛ ⎞= − −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

(7.159)

Excersise: Compute the above expressions for the statistically meaningful 3rd invariants

and see if in case of Boltzmann continuum give simple relations that can be expressed in

terms of the Lode angle (see sect. 9).



8 References [1] Bagi, K. (2003). Discussion on ‘‘The asymmetry of stress in granular media’’. Int. J.

Solids Struct. 40, 1329–1331

[2] Bardet, J.P., Vardoulakis, I. (2001). The asymmetry of stress in granular media. Int. J.

Solids Struct. 38, 353–367.

[3] Bardet, J.P., Vardoulakis, I. (2003a). Reply to discussion by Dr. Katalin Bagi. Int. J.

Solids Struct. 40, 1035.

[4] Bardet, J.P., Vardoulakis, I. (2003b). Reply to Dr. Kuhn’s discussion. Int. J. Solids

Struct. 40, 1809.

[5] Beatty, M. F. (1963). Vector representation of rigid body rotation. American Journal

of Physics, 31, (2), 134-135.

[6] Becker, E. und Bürger, W., Kontinuumsmechanik, Teubner Stuttgart, 1975.

[7] Bogdanova-Bontcheva, N. and Lippmann, H. (1975). Rotationssymmetrisches ebenes

Fließen eines granularen Modellmaterials, Acta Mech., 21, 93–113.

[8] Buggisch, H. (173). Herleitung der mechanischen Bilanzgleichungen des Cossert-

Kontinuums aus der Energiegleichung und ihrem Verhalten beim Übergang zum

rotierenden System. ZAMM, 53, T68-T69.

[9] Chambon, R., Caillerie, D. and El Hassan N. (1996). Etude de la localisation

unidimensionelle à l’aide d’un modèle de second gradient. C.R. Acad. Sci. Paris,

Série II b, 231-238.

[10] Chang, C.S., Kuhn, M.R. (2005). On virtual work and stress in granular media. Int.

J. Solids Struct. 42, 3773–3793.

[11] Chang, C.S. and Ma, L. (1991). A micromechanically-based micropolar theory for

deformation of granular solids. Int. J. Solids Struct., 28, 67-86.

[12] Cole, D.M. and Peters, J.F. (2007). A physically based approach to granular media

mechanics: Grain-scale experiments, initial results and implications to numerical

modeling., Granular Matter 9, 309-321, DOI: 10.1007/s10035-007-0046-2.

[13] Corfdir, A., Lerat, P. and Vardoulakis, I. (2004). A cylinder shear apparatus. ASTM

GTJ, 27(5),1-9.

[14] Cosserat; E. & F., Théorie de Corps déformables, Paris, 1909.


[15] D’ Addetta, G.A., Ramm, E., Diebels, S. And Ehlers W. (2004). A particle center

based homogenization strategy for granular assemblies. Engineering Computations,

21, 360-383.

[16] Dietrich, Th.. Der Psammische Stoff als mechanisches Modell des Sandes.

Dissertation Universität Karlsruhe 1976.

[17] Dietsche, A. and Willam, K. (1997). Boundary effects in elastoplastic Cosserat

Continua. Int. J. Solids Structures, 34, 877-893.

[18] Duschek, A. Und Hochrainer, A., Tensorrechnung in Analytischer Darstellung, Bd.

II. Tensoranalysis, Springer, 1970.

[19] Ehlers, W., Ramm, E., Diebels, S., D’Addetta, G.A. (2003). From particle

ensembles to Cosserat continua: homogenization of contact forces.

[20] Froiio, F., Tomassetti, G. and Vardoulakis, I. (2006). Mechanics of granular

materials: the discrete and the continuum descriptions juxtaposed, Int. J. Solids

Structures, 43, 7684–7720.

[21] Froiio, F., Zervos, A. and Vardoulakis, I. (2008). Linear 2nd-grade elasticity

revisited, J. of Elasticity, submitted.

[22] Godet, M. (1990). Third-bodies in tribology. Wear, 136, 29-45.

[23] Günther, W. (1958). Zur Sataik und Kinematik des Cosseratschen Kontinuums.

Abh. Braunschweigische Wiss. Ges., 10, 195-213.

[24] Gurtin, M.E., Martins, L.C. (1976). Cauchy’s theorem in classical physics. Arch.

Rat. Mech. An. 60, 305-324.

[25] Gurtin, M.E., Williams, W.O. (1967). An axiomatic foundation of continuum

thermodynamics. Arch. Rat. Mech. Anal. 26, 83–117, Gurtin, M.E., Martins, L.C.,

1976. Cauchy’s theorem in classical physics. Arch. Rat. Mech. Anal. 60, 305–324.

[26] Hamel, G., Elementare Mechanik, Leipzig u. Berlin, 1912.

[27] Kanatani, K. (1979). A micropolar continuum theory for flow of granular materials.

International Journal of Engineering Science, 17, 419-432.

[28] Kessel, S. (1964). Lineare Elastizitätstheorie des anisotropen Cosserat-Kontinuums.

Abh. Braunschweig. Wiss. Ges., 16, 1-22.

[29] Kruyt, N.P. (2003). Static and kinematics of discrete Cosserat-type granular

materials. Int. J. Solids Struct. 40, 511–534.


[30] Kuhn, M. (2003). Discussion on ‘‘The asymmetry of stress in granular media’’. Int.

J. Solids Struct. 40, 1805–1807.

[31] Mehrabadi, M.M. and Cowin, S.C. (1978). Initial planar deformation of dilatant

granular materials. J. Mech. Phys. Solids, 269-284.

[32] Mindlin, R. D. (1964). Microstructure in linear elasticity. Arch. Rational Mech.

Anal. 16, 51-78.

[33] Mindlin, R.D. and Eshel, N.N. (1968). On first strain-gradient theory in linear

elasticity. Int. J. Solids Struct., 4, 109-124

[34] Mindlin, R. D. and Tiersten, H. F. (1962) Effects of Couple stresses in linear

elasticity. Arch. Rational Mech. Anal., 11, 415-448.

[35] Mühlhaus, H.-B. and Vardoulakis, I. (1987). The thickness of shear bands in

granular materials. Géotechnique, 37, 271-283

[36] Noll, W. (1959). The foundation of classical mechanics in the light of recent

advances in continuum mechanics. In: The axiomatic method, with special reference

to Geometry and Physics, North-Holland Publishing Co., Amsterdam pp. 266–281.

[37] Oda, M. and Iwashita, K. (2000). Study on couple stress and shear band

development in granular media based on numerical simulation analyses. Int. J.

Engineering Sci., 38, 1713-1740.

[38] Oda, M. and Kazama, H. (1998). Microstructure of shear bands and its relation to

the mechanisms of dilatancy and failure of dense granular soils. Géotechnique, 48,

465-481.

[39] Oshima, N. (1955). Dynamics of granular media. Memoirs of the Unifying Study of

the Basic Problems in Engineering Sciencs by means of Geometry, Vol. 1, Division

D, 563-572.

[40] Radjai, F., Wolf, D.E., Jean, M. and Moreau J.-J. (1998). Bimodal character of

stress transmission in granular packings. Physics Review Letters, 80, 61-64.

[41] Satake, M. (1968). Some considerations on the mechanics of granular materials. In:

Mechanics of Generalized Continua (E. Kröner, Ed.), Springer, 156-159.

[42] Satake, M. (1998). Finite Difference Approach to the Shear Band Formation from

Viewpoint of Particle Column Buckling. 13th South Asian Geotechnical Conference,

16-20 November, 1998,Taipei, Taiwan, ROC.


[43] Schaefer, H. (1967). Das Cosserat-Kontinuum, ZAMM, 47 (8), 485-498.

[44] Schäfer, M. (1962). Versuch einer Elastizitätstheorie des zweidimensionalen

Cosserat-Kontinuums. Miszellaneneen der Angewandten Mechanik, Akademie

Verlag, Berlin, 277-292.

[45] Staron, L., Vilotte and Radjai, F. (2001). Friction and mobili-zation of contacts in

granular numerical avalanches. Powders and Grains 2001, Kishino (ed.), c 2001

Swets & Zeitlinger.

[46] Tejhman , J. and Wei Wu. (1995). Experimental and numerical study of sand-stell

inter-faces. Int. J. Num. Anal. Meth. Geomech., 19, 513-536.

[47] Timoshenko, S. P. (1921) On the correction for shear of the differential equation

for transverse vibrations of prismatic beams. Philosophical Magazine, Sec. 6., 41,

744-746.

[48] Tordesillas, A. (2007). Force chain buckling, unjamming transitions and shear

banding in dense granular assemblies. Phil. Magazine, 87, 4987-5016.

[49] Tordesillas, A. and Walsh, D.C.S. (2002). Incorporating rolling resistance and

contact anisotropy in micromechanical models of granular media. Powder

Technology, 124, 106-111.

[50] Truesdell, C and Toupin, R.. The Classical Field Theories. Vol. III/1, Sect. 240ff,

Springer, 1960.

[51] Unterreiner, F. and Vardoulakis, I. (1994). Interfacial localisation in granular

media. Computer Methods in Geomechanics (Ed. Siriwardane & Zaamn), Balkema,

1711-1715.

[52] Vardoulakis I. (1989). Shear banding and liquefaction in granu-lar materials on the

basis of a Cosserat continuum theory. In-genieur Archiv, 59, 106-113.

[53] Vardoulakis, Ι. and Georgopoulos, Ι.-O. (2005) .The “Stress - dilatancy” hypothesis

revisited: shear - banding related instabilities. Soils & Foundations, 45 (2), 61-76.

[54] Vardoulakis, I. and Giannakopoulos, A. (2006). An Example of double forces

taken from Structural Analysis, Int. J. Solids Structures, 43, 4047-4062.

[55] Vardoulakis, I., Shah, K.R. and Papanastasiou, P. (1992). Modelling of tool-rock

interfaces using gradient dependent flow-theory of plasticity. Int. J. Rock Mech. &

Min. Sci. and Geomechanics. Abstr., 29, 573-582.


[56] Vardoulakis, I. and Sulem, J., Bifurcation Analysis in Geomechanics, Blackie

Academic & Professional, 1995.

[57] Zervos, A., Vardoulakis, I., Jean, M. and Lerat, P. (2000). Numerical investigation

of granular kinematics. Mechanics of Cohesive-Frictional Materials , 5, 305–324.


9 Appendix: The meaning of the Lode angle in Boltzmann Continuum Mechanics

In Boltzmann Continuum Mechanics we introduce the so-called stress invariant angle of

similarity or Lode angle

30 03/ 2

2

3 3cos3 , 0 / 32

ss s

s

JJ

α α π= ≤ ≤ (8.1)

where

212s ij jiJ s s= (8.2)

and

313s ij jk kiJ s s s= (8.3)

are the second and third deviatoric stress invariants .

The Lode angle is the angle in the deviatoric plane that defines the position of the stress

deviator (Figure 9-1).

Figure 9-1: Lode angle in the deviatoric plane

The Lode angle in turn defines in turn the deviation of maximum shear stress from its

mean value (Figure 3-1),


1,max 2 3

2,max 3 1

3,max 1 2

/ 2 5 sin( )2

/ 2 5 sin( / 3 )2

/ 2 5 sin( / 3 )2

smean mean

smean mean

smean mean

τ σ σα

τ τ

τ σ σπ α

τ τ

τ σ σπ α

τ τ

−= =

−= = −

−= = +

(8.4)

where

22 25 5mean sJ Tτ = = (8.5)

Figure 9-2: Mean and max deviator in the deviatoric plane

We observe that the least deviation between maximum and mean deviatoric stress holds

for the cases of triaxial extension and compression

max

min

5 5 3 15sin( / 3) 1.372 2 2 8mean

τ πτ

= = = ≈ (8.6)

The maximum deviation takes place at 0 / 6sα π=

max

max

5 5sin( / 3 / 6) 1.582 2mean

τ π πτ

= + = ≈ (8.7)

In this case we have also that the intermediate principal stress is equal to the mean of the

other two,


0 3 1 2 2 3 11/ 6 ( ) ( )2sα π σ σ σ σ σ σ= ⇒ = + ≤ ≤ (8.8)

The case with 0 / 6sα π= characterizes approximately the so-called plane-strain states.

Documents

Gran Material Notes