Infinitesimal non-linear Cosserat theory based on the grain size length scale Hamidr´ eza Ram´ ezani1∗ and Jena Jeong2 ´ UMR CNRS 6619- Research Center on Divided Materials, Ecole Polytechnique de l’Universit´ e d’Orl´ eans, 8 rue L´ eonrad de Vinci, 45072 Orl´ eans cedex 2, France.
1∗ CRMD,
´ Sp´ eciale des Travaux Publics, du Bˆ atiment et de Modeling-Ecole l’industrie (ESTP), 28 avenue du Pr´ esident Wilson, 94234 Cachan cedex, France.
2 ESTP/IRC/LM-Lean
New Trends in Fatigue and Fracture 10th Meeting NT2F10 2010-Metz France Universit´ e de Paul Verlaine (Universit´ e de Metz) and ´ Ecole Nationale d’Ing´ enieurs de Metz (ENIM) (30-31 August and 1st September 2010) Hamidr´ eza Ram´ ezani1∗
and Jena Jeong2 Investigation ( 1∗ CRMD, on UMR non-linear CNRS Cosserat 6619theoryResearch August-September Center on 2010 Divided1Mater / 19
Table of Contents 1
2
3
Heterogeneous materials SEM observations of heterogeneous materials: some examples Available approaches treating the heterogeneous materials Classification of the available methodologies A brief review on the Generalized continuum mechanics Cosserat or so-called micro-polar theory State field variables definitions of Cosserat theory Linear Cosserat theory Non-linear or geometrically exact Cosserat theory
4
5
6 7
Size effect and characteristic length scale concept Total energy density for linear isotropic materials Cosserat material moduli: λ, µ, α, β and γ Numerical experiments using the grain size length scale Material choice: Amorphous and heterogeneous materials Summary, conclusion and outlooks Acknowledgements
Hamidr´ eza Ram´ ezani1∗
and Jena Jeong2 Investigation ( 1∗ CRMD, on UMR non-linear CNRS Cosserat 6619theoryResearch August-September Center on 2010 Divided2Mater / 19
Heterogeneous materials
SEM observations of heterogeneous materials: some examples
Heterogeneous materials
a)
c)
b)
d)
Figure: a) Nikel foam, b) Ceramic, c) Limestone, d) Human bone. Hamidr´ eza Ram´ ezani1∗
and Jena Jeong2 Investigation ( 1∗ CRMD, on UMR non-linear CNRS Cosserat 6619theoryResearch August-September Center on 2010 Divided3Mater / 19
Available approaches treating the heterogeneous materials
Classification of the available methodologies
Available approaches treating the heterogeneous materials Classification of the available methodologies
Homogenization technique using classical continuum mechanics: 1
Self-consistent field (Known as Auto coherent method in France)
2
Micro-mechanics
3
Mori-Tanaka method
Advantages: mesh-independent features, mean value calculation, morphological problem, ... Drawbacks: lack of information at interfaces, ...
Homogenization technique using generalized continuum mechanics: 1
Micro-morphic theory (12 DOFs node and 19 material constants)
2
Micro-stretch theory (7 DOFs per node and 10 material constants)
3
Micro-polar theory (6 DOFs per node and 6 material constants)
4
Micro-dilatation theory (4 DOFs per node and 6 material constants)
Advantages: Micro-structural description by means of the characteristic length scale definition and less stress localization. Drawbacks: Difficulties in extracting these material constants by experiments
Hamidr´ eza Ram´ ezani1∗
and Jena Jeong2 Investigation ( 1∗ CRMD, on UMR non-linear CNRS Cosserat 6619theoryResearch August-September Center on 2010 Divided4Mater / 19
Available approaches treating the heterogeneous materials
A brief review on the Generalized continuum mechanics
Generalized continuum mechanics family Comparison among the well-known generalized continuum mechanics
Name Cauchy-Boltzmann Micro-dilatation Micro-polar/Cosserat Micro-stretch Micro-strain Micro-morphic Theory name Classical-Boltzmann Micro-dilatation Micro-polar/Cosserat Micro-stretch Micro-strain Micro-morphic Hamidr´ eza Ram´ ezani1∗
DOFs 3 4 6 7 9 12
MCs 2 6 6 10 ? 19
State field variables u ∈ R3 u ∈ R3 , ϕ ∈ R u ∈ R3 , φ ∈ R3 u ∈ R3 , φ ∈ R3 , ϕ ∈ R u ∈ R3 , χ ∈ sym(3) ⊂ M3×3 u ∈ R3 , χ ∈ M3×3
proposed by: Cauchy Cowin-Nazaro Eringen Kaafadar-Eringen Forest-Sievert Eringen
Date: 1823 1985 1964 1971 2006 1965
2D/3D-FEM X/ X X/ × X/ X X/ × ? /× X/ ×
and Jena Jeong2 Investigation ( 1∗ CRMD, on UMR non-linear CNRS Cosserat 6619theoryResearch August-September Center on 2010 Divided5Mater / 19
Cosserat or so-called micro-polar theory
State field variables definitions of Cosserat theory
Cosserat theory: Basics State field variables: displacement u : Ω ⊂ R3 7→ R3 and micro-rotation φ : Ω ⊂ R3 7→ R3
Cosserat state field variables Displacement vector u : Ω ⊂ R3 7→ R3 Micro-rotation vector φ : Ω ⊂ R3 7→ R3 : Linear Cosserat or infinitesimal Cosserat theory: First Cosserat stretch tensor, ε¯ ∈ M3×3 : ε¯T := ∇u − A¯ where T
A¯ := −eijk φk eˆi ⊗ eˆj
∇u := (∇ ⊗ u) = ui,j eˆi ⊗ eˆj
for
and
i, j, k = 1, 2, 3
(1)
Curvature tensor or wryness tensor k¯ ∈ M3×3 : k¯ := ∇φ
Hamidr´ eza Ram´ ezani1∗
where
∇φ := (∇⊗φ)T = φi,j eˆi ⊗ eˆj
for
i, j, k = 1, 2, 3 (2)
and Jena Jeong2 Investigation ( 1∗ CRMD, on UMR non-linear CNRS Cosserat 6619theoryResearch August-September Center on 2010 Divided6Mater / 19
Cosserat or so-called micro-polar theory
State field variables definitions of Cosserat theory
Cosserat theory: Basics Kinematical relations: Linear and non-linear or geometrically exact Cosserat theory
Geometrically exact or non-linear Cosserat theory: ¯ ∈ M3×3 : First Cosserat stretch tensor, E 1 − cos(kφk) ¯2 ¯ ¯ T := ∇u − sin(kφk) A¯ − sin(kφk) A∇u + A E kφk kφk kφk2 1 − cos(kφk) ¯2 + A ∇u kφk2
(3)
¯ ∈ M3×3 : Curvature tensor or wryness tensor K ¯ T := 2 sin(kφk) − kφk ∇φ − 1 − cos(kφk) A∇φ ¯ K kφk kφk2 kφk − sin(kφk) ¯2 + A ∇φ kφk3 Hamidr´ eza Ram´ ezani1∗
(4)
and Jena Jeong2 Investigation ( 1∗ CRMD, on UMR non-linear CNRS Cosserat 6619theoryResearch August-September Center on 2010 Divided7Mater / 19
Cosserat or so-called micro-polar theory
State field variables definitions of Cosserat theory
Total energy density method for anisotropic materials Total energy density: strain energy Wmp , centro-symmetry Wcs and curvature energy density Wcurv
Total energy density can be divided into three counterparts for the linear anisotropic materials including the zero-centro-symmetry features as below: ¯ K) ¯ = Wmp (E, ¯ K) ¯ + Wcs (E, ¯ K) ¯ + Wcurv (E, ¯ K) ¯ W (E,
(5a)
0 ¯ K) ¯ = Wmp ¯ + 1E ¯:D:E ¯ Wmp (E, +S:E (5b) 2 ¯:C:K ¯ ¯:C:K ¯ + 1K ¯ :C:E ¯ = W0 + E ¯ K) ¯ = W0 + 1E Wcs (E, cs cs 2 2 (5c) ¯:C:K ¯ =K ¯ :C:E ¯ where E 0 ¯ K) ¯ = Wcurv ¯ + 1K ¯ :E:K ¯ Wcurv (E, +B:K 2
Hamidr´ eza Ram´ ezani1∗
(5d)
and Jena Jeong2 Investigation ( 1∗ CRMD, on UMR non-linear CNRS Cosserat 6619theoryResearch August-September Center on 2010 Divided8Mater / 19
Cosserat or so-called micro-polar theory
State field variables definitions of Cosserat theory
Constitutive equations for elastic anisotropic materials including centro-symmetry Total energy density: strain energy Wmp and curvature energy density Wcurv
In the absence of initial energy densities, stress and couple stress and centro-symmetry for the anisotropic constitutive equations: ¯ K) ¯ ∂W (E, ¯ =D:E ¯ ∂E
or
¯ K) ¯ ∂W (E, ¯ m= =E:K ¯ ∂K
or
σ ¯=
¯kl σ ¯ij = Dijkl E
for
i,j,k,l=1,2,3 (6a)
¯ ij = Eijkl K ¯ kl K
for
i,j,k,l=1,2,3 (6b)
Linear isotropic constitutive laws:
Hamidr´ eza Ram´ ezani1∗
¯ + 2µc skew E ¯ + λ tr[E] ¯ 11 σ ¯ = 2µ sym E
(7a)
¯T + γ K ¯ + α tr[K] ¯ 11 m=βK
(7b)
and Jena Jeong2 Investigation ( 1∗ CRMD, on UMR non-linear CNRS Cosserat 6619theoryResearch August-September Center on 2010 Divided9Mater / 19
Cosserat or so-called micro-polar theory
State field variables definitions of Cosserat theory
What is the physical interpretation of the Cosserat theory? Stress tensor σ ∈ M3×3 , stress moment tensor m ∈ M3×3 , micro-rotation φ ∈ R3 and macro-rotation vector ω ∈ R3 ,
009
x3 m33
σ31 σ13 m13
σ11 m11
Cosserat Point Point or Micro‐Polar Point
σ33 m32 m31
m23 m12
i
σ32 σ23
Micro‐rotation
m22
σ22 σ12 σ21 m21
x2 ωi
Macro‐rotation
x1
Hamidr´ eza Ram´ ezani1∗
and Jena Jeong2 Investigation ( 1∗ CRMD, on UMR non-linear CNRS Cosserat 6619theoryResearch August-September Center on2010 Divided10Mater / 19
Size effect and characteristic length scale concept
Total energy density for linear isotropic materials
Size effect and characteristic length scale concept Total energy density: strain energy Wmp and curvature energy density Wcurv
Isotropic centor-symmetric strain energy density: ¯ ¯ = µ k sym Ek ¯ 2 + µc k skew Ek ¯ 2 + λ tr2 [E] Wmp (E) 2 ¯ 2 + µc k skew Ek ¯ 2 + 3λ + 2µ tr2 [E] ¯ = µ k dev sym Ek 6 1 ¯ + (µ + µc ) kEk ¯ 2 + (µ − µc ) tr[E ¯ 2] λ tr2 [E] = 2
(8)
Isotropic centor-symmetric curvature energy density: ¯ 2 + γ − β k skew Kk ¯ 2 + α tr2 [K] ¯ ¯ = γ + β k sym Kk Wcurv (K) 2 2 2 γ+β ¯ 2 + γ − β k skew Kk ¯ 2 + 3α + (β + γ) tr2 [K] ¯ (9) = k dev sym Kk 2 2 6 1 ¯ + γ kKk ¯ 2 + β tr[K ¯ 2] = α tr2 [K] 2 Hamidr´ eza Ram´ ezani1∗
and Jena Jeong2 Investigation ( 1∗ CRMD, on UMR non-linear CNRS Cosserat 6619theoryResearch August-September Center on2010 Divided11Mater / 19
Size effect and characteristic length scale concept
Total energy density for linear isotropic materials
Isotropic linear micro-polar elasticity Curvature energy density cases and its form in terms of characteristic length scale Lc
Toupin has firstly introduced the concept of the characteristic length [Toupin et al. 1962]: r α Lc := ⇒ α = µL2c µ
(10)
We can obtain the following cases for the curvature energy density [Neff et al. 2008,Jeong et al. 2009 and Jeong and Ram´ ezani 2010]: 2 µLc ¯ 2 + tr2 [K] ¯ 1 α=µL2 , β = 0 and γ=µL2 ⇒ W kKk curv = 2 c c µL2c µL2c 1 ¯ 2 1 2 ¯ ¯2 2 α=µL2 , β=γ= c 2 ⇒ Wcurv = 2 2 kKk + 2 tr[K ] + tr [K] µL2c µL2c 1 ¯ 2 µL2c 1 1 2 ¯ ¯2 3 α=− 3 , β=γ = 2 ⇒ Wcurv = 2 2 kKk + 2 tr[K ] − 3 tr [K] µL2c ¯ 2 + tr[K ¯ 2 ] + tr2 [K] ¯ 4 α=β=γ = µL2 ⇒ W kKk curv = c
Hamidr´ eza Ram´ ezani1∗
2
and Jena Jeong2 Investigation ( 1∗ CRMD, on UMR non-linear CNRS Cosserat 6619theoryResearch August-September Center on2010 Divided12Mater / 19
Size effect and characteristic length scale concept
Cosserat material moduli: λ, µ, α, β and γ
Cosserat material moduli: λ, µ, α, β and γ Application of the characteristic length scale Lc and µc =µ
Granular materials and grain size
A 3D pack of spherical grains. Granular material under loading.
Ellipsoid grains assumption including the characteristic length scales
Hamidr´ eza Ram´ ezani1∗
and Jena Jeong2 Investigation ( 1∗ CRMD, on UMR non-linear CNRS Cosserat 6619theoryResearch August-September Center on2010 Divided13Mater / 19
Size effect and characteristic length scale concept
Cosserat material moduli: λ, µ, α, β and γ
Cosserat material moduli: λ, µ, α, β and γ Application of the characteristic length scale Lc and µc =µ
Ellipsoid grains assumption including the characteristic length scales
Assuming that Lc1 , Lc2 and Lc3 deal with three diameters of an ellipsoid grain: Lc1 = 2a,
Lc1 = 2b and Lc1 = 2c
(11)
Assumptions pertaining to Lc and µc : This give rise case4, i.e. α=β=γ = µL2c and we need only λ, µ and LG . Lc1 = Lc2 = Lc3 = LG Hamidr´ eza Ram´ ezani1∗
and µc = µ
(12)
and Jena Jeong2 Investigation ( 1∗ CRMD, on UMR non-linear CNRS Cosserat 6619theoryResearch August-September Center on2010 Divided14Mater / 19
Numerical experiments using the grain size length scale
Material choice: Amorphous and heterogeneous materials
Numerical experiments of the non-linear Cosserat theory Material choice: glass and cement as the amorphous and heterogeneous materials
Float glass and cement mortar SEM images: 20 µm
SEM image of the float glass, i.e. LG =10−6 [mm] dealing with the Cauchy’s medium.
Hamidr´ eza Ram´ ezani1∗
SEM image of the cement mortar at 48h of hydration, i.e. LG =1 [mm] dealing with the Cosserat medium.
and Jena Jeong2 Investigation ( 1∗ CRMD, on UMR non-linear CNRS Cosserat 6619theoryResearch August-September Center on2010 Divided15Mater / 19
Numerical experiments using the grain size length scale
Material choice: Amorphous and heterogeneous materials
Numerical experiments: Independent and dependent material properties and 3D-FEM geometry
3D-FEM geometry and mesh density 50
X 0
X 60 40
-50
0
Z 20
40
20 0
-50
50
Y 0
Y
0
50 -50 0 Z
50 100
Modified Brazilian Disk (MBD) geometry, OD=150 [mm], ID=30 [mm].
One quarter symmetry for our 3D-FEM model.
Mechanical properties E [GPa] ν [-] Lc =LG [mm] µc [GPa]
Hamidr´ eza Ram´ ezani1∗
Float glass 58 0.218 10−6 E µc = µ = 2(1+ν)
Mesh density of our 3D-FEM including 4.6 Mi DOFs.
Cement mortar 20 0.28 1 E µc = µ = 2(1+ν)
and Jena Jeong2 Investigation ( 1∗ CRMD, on UMR non-linear CNRS Cosserat 6619theoryResearch August-September Center on2010 Divided16Mater / 19
Numerical experiments using the grain size length scale
Material choice: Amorphous and heterogeneous materials
Numerical experiments of the MBD specimen via the geometrically exact Cosserat theory including Augmented Lagrangian-Eulerian (ALE) method Micro-rotation distribution, φ : ΩM BD ⊂ R3 7→ R3 ,
Float glass and cement mortar micro-rotation distribution:
Float glass, i.e. LG =10−6 [mm].
Hamidr´ eza Ram´ ezani1∗
Cement mortar at 48h of hydration, i.e. LG =1 [mm].
and Jena Jeong2 Investigation ( 1∗ CRMD, on UMR non-linear CNRS Cosserat 6619theoryResearch August-September Center on2010 Divided17Mater / 19
Summary, conclusion and outlooks
Conclusions and outlooks 1
2
3
4
5
6
The material moduli, i.e. α, β and γ are considered in function of the microstructure state notably, the grain size via LG , The characteristic length scale Lc is considered as an average grain diameter (Lc := LG ), According to the physical and mathematical restrictions on the Cosserat-based media as generalized continuum media, six material moduli have been reduced to only three, i.e. λ, µ and LG , The Cosserat theory is capable of treating the amorphous materials (classical theory) as well as the granular materials with various diameters using our assumptions herein, The numerical implementations open new outlooks in treating the micro-structure, particularly, for the heterogeneous materials. The micro and macro damage issue would be simultaneously considered to obtain a multi-scale damage models, using the Cosserat, micro-stretch and micro-morphic theories.
Hamidr´ eza Ram´ ezani1∗
and Jena Jeong2 Investigation ( 1∗ CRMD, on UMR non-linear CNRS Cosserat 6619theoryResearch August-September Center on2010 Divided18Mater / 19
Acknowledgements
Acknowledgements
Pr. Pierre Mounanga from University of Nantes for SEM images of cement mortar, Pr. Patrizio Neff from University of Duisburg-Essen for his scientific support and his fruitful comments, Pr. Samuel Forest, Ecole des Mines de Paris fo his valuable remarks about the generalized continuum mechanics during our meeting which was held at Ecole des Mines de Paris in Paris last summer.
Hamidr´ eza Ram´ ezani1∗
and Jena Jeong2 Investigation ( 1∗ CRMD, on UMR non-linear CNRS Cosserat 6619theoryResearch August-September Center on2010 Divided19Mater / 19