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Strain localisation modes in the mechanics ofmaterials
Samuel Forest
Centre des Matériaux/UMR 7633Ecole des Mines de Paris/CNRS
BP 87, 91003 Evry, [email protected]
Outline
1 Bifurcation modes in elastoplastic solidsLinear incremental formulationLoss of uniqueness
2 Compatible bifurcation modesHadamard jump conditionsOrientation of a strain localization band
3 Stability / ellipticity of the boundary value problemMandel–Rice criterionOrientation of localization bands in 3D and 2D
4 Summary of strain localization criteria
5 Regularization methodsMesh dependence of the resultsMechanics of generalized continuaApplication to the case of metal foams
Plan
1 Bifurcation modes in elastoplastic solidsLinear incremental formulationLoss of uniqueness
2 Compatible bifurcation modesHadamard jump conditionsOrientation of a strain localization band
3 Stability / ellipticity of the boundary value problemMandel–Rice criterionOrientation of localization bands in 3D and 2D
4 Summary of strain localization criteria
5 Regularization methodsMesh dependence of the resultsMechanics of generalized continuaApplication to the case of metal foams
Plan
1 Bifurcation modes in elastoplastic solidsLinear incremental formulationLoss of uniqueness
2 Compatible bifurcation modesHadamard jump conditionsOrientation of a strain localization band
3 Stability / ellipticity of the boundary value problemMandel–Rice criterionOrientation of localization bands in 3D and 2D
4 Summary of strain localization criteria
5 Regularization methodsMesh dependence of the resultsMechanics of generalized continuaApplication to the case of metal foams
Loi de comportement élastoplastique
• Elasticity ε̇∼ = ε̇∼e + ε̇∼
p, σ∼ = E∼∼: ε∼
e
• Yield function g(σ∼ , α), N∼ =∂g
∂σ∼
• Plastic flow ε̇∼p = λ̇ P∼ , P∼ 6= N∼
• Hardening modulus α̇ = λ̇h, H = −∂g∂α
.h
• Tangent operator (non symmetrical if non associative) σ̇∼ = L∼∼ : ε̇∼
L∼∼= E∼∼
si g < 0 ou (g = 0 et N∼ : E∼∼: ε̇∼ ≤ 0)
L∼∼= D∼∼
si g = 0 et N∼ : E∼∼: ε̇∼ > 0
with
D∼∼= E∼∼
−(E∼∼
: P∼)⊗
(N∼ : E∼∼)
H + N∼ : E∼∼: P∼
Bifurcation modes in elastoplastic solids 5/74
Rate form of the boundary value problem
Ω
∂Ω2
∂Ω1
V
Ṫ
ε̇∼ =12(∇v +∇
tv )
div σ̇∼ + ḟ = 0, σ̇∼ = L∼∼: ε̇∼
σ̇∼ . n = Ṫ in ∂Ω1
v = V in ∂Ω2
• Linear comparison solid: take L∼∼ = D∼∼• Variational formulation (virtual power theorem)∫
Ωσ̇∼ : ε̇∼
? dV =
∫Ω
ḟ .v ? dV +
∫∂Ω1
Ṫ .v ? dS+
∫∂Ω2
(σ̇∼ .n ).V dS
for any field v ? such that v ? = V in ∂Ω2
Bifurcation modes in elastoplastic solids 6/74
Plan
1 Bifurcation modes in elastoplastic solidsLinear incremental formulationLoss of uniqueness
2 Compatible bifurcation modesHadamard jump conditionsOrientation of a strain localization band
3 Stability / ellipticity of the boundary value problemMandel–Rice criterionOrientation of localization bands in 3D and 2D
4 Summary of strain localization criteria
5 Regularization methodsMesh dependence of the resultsMechanics of generalized continuaApplication to the case of metal foams
Conditions of uniqueness of the solution
Let (σ̇∼A, ε̇∼A) and (σ̇∼B , ε̇∼B) be two solutions on Ω at time t∫Ω
σ̇∼A : ε̇∼A dV =
∫Ω
ḟ .v A dV +
∫∂Ω1
Ṫ .v A dS +
∫∂Ω2
(σ̇∼A.n ).V dS
Bifurcation modes in elastoplastic solids 8/74
Conditions of uniqueness of the solution
Let (σ̇∼A, ε̇∼A) and (σ̇∼B , ε̇∼B) be two solutions on Ω at time t∫Ω
σ̇∼A : ε̇∼A dV =
∫Ω
ḟ .v A dV +
∫∂Ω1
Ṫ .v A dS +
∫∂Ω2
(σ̇∼A.n ).V dS∫Ω
σ̇∼A : ε̇∼B dV =
∫Ω
ḟ .v B dV +
∫∂Ω1
Ṫ .v B dS +
∫∂Ω2
(σ̇∼A.n ).V dS
Bifurcation modes in elastoplastic solids 9/74
Conditions of uniqueness of the solution
Let (σ̇∼A, ε̇∼A) and (σ̇∼B , ε̇∼B) be two solutions on Ω at time t∫Ω
σ̇∼A : ε̇∼A dV =
∫Ω
ḟ .v A dV +
∫∂Ω1
Ṫ .v A dS +
∫∂Ω2
(σ̇∼A.n ).V dS
−∫
Ωσ̇∼A : ε̇∼B dV =
∫Ω
ḟ .v B dV+
∫∂Ω1
Ṫ .v B dS+
∫∂Ω2
(σ̇∼A.n ).V dS
∫Ω
σ̇∼A : ∆ε̇∼ dV =
∫Ω
ḟ .∆v dV +
∫∂Ω1
Ṫ .∆v dS
Bifurcation modes in elastoplastic solids 10/74
Conditions of uniqueness of the solution
Let (σ̇∼A, ε̇∼A) and (σ̇∼B , ε̇∼B) be two solutions on Ω at time t∫Ω
σ̇∼A : ∆ε̇∼ dV =
∫Ω
ḟ .∆v dV +
∫∂Ω1
Ṫ .∆v dS
−∫
Ωσ̇∼B : ∆ε̇∼ dV =
∫Ω
ḟ .∆v dV +
∫∂Ω1
Ṫ .∆v dS
∫Ω
∆σ̇∼ : ∆ε̇∼ dV = 0
Bifurcation modes in elastoplastic solids 11/74
Conditions of uniqueness of the solution
Let (σ̇∼A, ε̇∼A) and (σ̇∼B , ε̇∼B) be two solutions on Ω at time t∫Ω
∆σ̇∼ : ∆ε̇∼ dV = 0
If the solutions (σ∼A, ε∼A) and (σ∼B , ε∼B) have coincided until time t,one can assume
σ̇∼A = D∼∼: ε̇∼A, σ̇∼B = D∼∼
: ε̇∼B
A necessary condition for the existence of several solutions then is∫Ω
∆ε̇∼ : D∼∼: ∆ε̇∼ dV = 0
Bifurcation modes in elastoplastic solids 12/74
A sufficient condition for uniquenessA sufficient condition ensuring the uniqueness of the solution(∆ε̇∼ = 0) is the positivity of second order power :
ε̇∼ : D∼∼: ε̇∼ > 0, ∀ε̇∼ 6= 0
ε̇∼ : D∼∼s : ε̇∼ > 0, avec D∼∼
sijkl =
1
2(Dijkl + Dklij)
Hill’s condition for uniqueness: D∼∼s positive definite (Hill, 1958)
Uniqueness may be lost (possible bifurcation) when
detD∼∼s = 0
which gives Hu =1
2(√
N∼ : E∼∼: N∼
√P∼ : E∼∼
: P∼ −N∼ : E∼∼ : P∼)H > Hu: uniqueness ensured associative plasticity: Hu = 0softening!
uniaxial + Large Strain= Considère criterion...
Bifurcation modes in elastoplastic solids 13/74
Example : von Mises plasticity
• Von Mises plasticity:
g(σ∼ ,R) = J2(σ∼)− R, J2(σ∼) =√
3
2σ∼
dev : σ∼dev
• Associative plasticity: N∼ = P∼ =3
2
σ∼dev
J2• The tangent operator can be computed easily!
N∼ : E∼∼: N∼ = N∼ : (2µN∼ ) = 3µ
D∼∼= E∼∼
− 3µ2
H/3 + µ
σ∼dev ⊗ σ∼
dev
J22• Principal or eigen–tensors of D∼∼ = D∼∼
s : one tries
d∼1 =σ∼
dev
J2, D∼∼
: d∼1 =2µH
H + 3µd∼1
• Principal or eigen–values: k, 2µ, 2µHH + 3µ
When H = 0 (perfect plasticity, begining of a softening regime)Bifurcation modes in elastoplastic solids 14/74
Example : necking mode
Tensile test,
[d∼1] =
−1 0 00 2 00 0 −1
possible perturbations when H =0...
σ̇∼A = D∼∼: ε̇∼A = D∼∼
: (ε̇∼A + λd∼1)
Bifurcation modes in elastoplastic solids 15/74
Example : necking mode
Tensile test,
[d∼1] =
−1 0 00 2 00 0 −1
possible perturbations when H =0...
σ̇∼A = D∼∼: ε̇∼A = D∼∼
: (ε̇∼A + λd∼1)
Bifurcation modes in elastoplastic solids 16/74
Example : necking mode
Tensile test,
[d∼1] =
−1 0 00 2 00 0 −1
possible perturbations when H =0...
σ̇∼A = D∼∼: ε̇∼A = D∼∼
: (ε̇∼A + λd∼1)
Bifurcation modes in elastoplastic solids 17/74
Example : necking mode
0
20
40
60
80
100
120
0 0.04 0.08 0.12 0.16
F/S
δ/LBifurcation modes in elastoplastic solids 18/74
Example : necking mode
0
20
40
60
80
100
120
0 0.04 0.08 0.12 0.16
F/S
δ/LBifurcation modes in elastoplastic solids 19/74
Example : necking mode
0
20
40
60
80
100
120
0 0.04 0.08 0.12 0.16
F/S
δ/LBifurcation modes in elastoplastic solids 20/74
Example : necking mode
0
20
40
60
80
100
120
0 0.04 0.08 0.12 0.16
F/S
δ/LBifurcation modes in elastoplastic solids 21/74
Example : necking mode
0
20
40
60
80
100
120
0 0.04 0.08 0.12 0.16
F/S
δ/LBifurcation modes in elastoplastic solids 22/74
Example : necking mode
0
20
40
60
80
100
120
0 0.04 0.08 0.12 0.16
F/S
δ/LBifurcation modes in elastoplastic solids 23/74
Plan
1 Bifurcation modes in elastoplastic solidsLinear incremental formulationLoss of uniqueness
2 Compatible bifurcation modesHadamard jump conditionsOrientation of a strain localization band
3 Stability / ellipticity of the boundary value problemMandel–Rice criterionOrientation of localization bands in 3D and 2D
4 Summary of strain localization criteria
5 Regularization methodsMesh dependence of the resultsMechanics of generalized continuaApplication to the case of metal foams
Plan
1 Bifurcation modes in elastoplastic solidsLinear incremental formulationLoss of uniqueness
2 Compatible bifurcation modesHadamard jump conditionsOrientation of a strain localization band
3 Stability / ellipticity of the boundary value problemMandel–Rice criterionOrientation of localization bands in 3D and 2D
4 Summary of strain localization criteria
5 Regularization methodsMesh dependence of the resultsMechanics of generalized continuaApplication to the case of metal foams
Compatible discontinuous bifurcation modes :Hadamard jump conditions
Strain localization is the precursor offracture.Continuity of displacement is assumedthrough a surface S
[[u ]] = [[u̇ ]] = 0
but discontinuities of some componentsof its gradient ∇u are considered
1
2S
n
Compatible bifurcation modes 26/74
Compatible discontinuous bifurcation modes :Hadamard jump conditions
Let us work in the frame attached to thesurface of discontinuity:
[[u1]] = 0 =⇒ [[u1,1]] = 0
[[u2]] = 0 =⇒ [[u2,1]] = 0
but, in general, ∃g such that
[[u1,2]] = g1 6= 0, [[u2,2]] = g2 6= 0
1
2S
n
Compatible bifurcation modes 27/74
Compatible discontinuous bifurcation modes :Hadamard jump conditions
Let us work in the frame attached to thesurface of discontinuity:
[[u1]] = 0 =⇒ [[u1,1]] = 0
[[u2]] = 0 =⇒ [[u2,1]] = 0
but, in general,
[[u1,2]] = g1 6= 0, [[u2,2]] = g2 6= 0
[[∇u ]] =[
0 g10 g2
]=
[g1g2
]⊗
[01
][[∇u ]] = g ⊗ n
1
2S
n
Compatible bifurcation modes 28/74
Compatible discontinuous bifurcation modes :Hadamard jump conditions
[[∇u̇ ]] = g ⊗ n
[[∇ε̇∼]] =1
2(g ⊗ n + n ⊗ g )
in general g .n 6= 0Criterion for the occurence of compatiblebifurcation modes:
∆ε̇∼ : D∼∼: ∆ε̇∼ = (g⊗n ) : D∼∼ : (g⊗n ) = 0
The acoustic tensor Q∼
is introduced:
g .Q∼.g = g .Q
∼sym.g = 0, Q
∼= n .D∼∼
.n
=⇒ detQ∼
sym = 0
1
2S
n
Compatible bifurcation modes 29/74
Compatible discontinuous bifurcation modes :Hadamard jump conditions
One can imagine two parallel surfaces ofdiscontinuity: it is a strain localizationband or shear band in a wide sense. In-deed, the band does generally not un-dergo pure shear...
• g .n = 0: shear band• g ‖ n : opening mode
1
2S 1S 2
Compatible bifurcation modes 30/74
Plan
1 Bifurcation modes in elastoplastic solidsLinear incremental formulationLoss of uniqueness
2 Compatible bifurcation modesHadamard jump conditionsOrientation of a strain localization band
3 Stability / ellipticity of the boundary value problemMandel–Rice criterionOrientation of localization bands in 3D and 2D
4 Summary of strain localization criteria
5 Regularization methodsMesh dependence of the resultsMechanics of generalized continuaApplication to the case of metal foams
Example: compatible strain localization mode underplane stress conditions
Consider the mode
[d∼1] =
24 −1 0 00 2 00 0 −1
35but write it with respect to the frame (m , n )
[d∼1] =
24 2 sin2 θ − cos2 θ 3 cos θ sin θ 03 cos θ sin θ 2 cos2 θ − sin2 θ 00 0 −1
35to be identified with 1
2(g ⊗ n + n ⊗ g ) =
»0 g1/2
g1/2 g2
–
=⇒ tan2 θ = 12, ou sin2 θ =
1
3
θ = 35, 26◦, 90◦ − θ = 54.74◦
mn
1
2
θ
Compatible bifurcation modes 32/74
Plan
1 Bifurcation modes in elastoplastic solidsLinear incremental formulationLoss of uniqueness
2 Compatible bifurcation modesHadamard jump conditionsOrientation of a strain localization band
3 Stability / ellipticity of the boundary value problemMandel–Rice criterionOrientation of localization bands in 3D and 2D
4 Summary of strain localization criteria
5 Regularization methodsMesh dependence of the resultsMechanics of generalized continuaApplication to the case of metal foams
Plan
1 Bifurcation modes in elastoplastic solidsLinear incremental formulationLoss of uniqueness
2 Compatible bifurcation modesHadamard jump conditionsOrientation of a strain localization band
3 Stability / ellipticity of the boundary value problemMandel–Rice criterionOrientation of localization bands in 3D and 2D
4 Summary of strain localization criteria
5 Regularization methodsMesh dependence of the resultsMechanics of generalized continuaApplication to the case of metal foams
Mandel–Rice criterion
Add the equilibrium contitions at the interface whichwere not taken into account until now:
[[σ̇∼]].n = 0, [[D∼∼: ε̇∼]].n = 0
If the solutions on each side of S are identical untiltime t, one has [[D∼∼
]] = 0
D∼∼: [[ε̇∼]].n = 0
Let us consider the compatible modes:
D∼∼: (g ⊗ n ).n = 0
Dijklgknlnj = njDjiklnlgk = 0
Q∼
.g = 0
This is the acoustic tensor: Q∼
= n .D∼∼.n
1
2S
n
Mandel–Rice criterion:
detQ∼
(n ) = 0
stable propagation of perturbations [Mandel, 1966] [Rice, 1976]
Stability / ellipticity of the boundary value problem 35/74
Vocabulary
• Plastic waves [Mandel, 1962]the previous situation corresponds to stationary plastic waves
• Stability: analysis if the growth of perturbations• Loss of ellipticity of the boundary value problem:
when the incremental constitutive equations are inserted inthe equations of equilibrium (static case), one gets a systemof partial differential equations of order 2 in u̇i . The system issaid to be elliptic when the differential operator has definitepositive eigen–values. It can be shown that the problem thenis well–posed (meaning that the solution continuouslydepends on the data).
Stability / ellipticity of the boundary value problem 36/74
Plan
1 Bifurcation modes in elastoplastic solidsLinear incremental formulationLoss of uniqueness
2 Compatible bifurcation modesHadamard jump conditionsOrientation of a strain localization band
3 Stability / ellipticity of the boundary value problemMandel–Rice criterionOrientation of localization bands in 3D and 2D
4 Summary of strain localization criteria
5 Regularization methodsMesh dependence of the resultsMechanics of generalized continuaApplication to the case of metal foams
Orientation of strain localization bandsThe resolution of the equation det n .D∼∼
.n = 0 provides us with the plastic
modulus H(n ) for which a surface of discontinuity of normal n can arise. Todetermine the first possible band, one must find Hcr and n such that
Hcr = max‖n ‖=1
H(n )
In the case of isotropique elasticity and incompressible associative plasticity,
one finds
dimension critical modulus orientationof the problem Hcr n
plane 0 n21 =P1
P1−P2stress n22 = 1− n213D −EP2k n2i =
Pi+νPkPi−Pj
nk = 0, n2j = 1− n2i
plane 0 n21 = n22 =
12
strain
The Pi are the eigenvalues of the plastic flow direction P∼ (ε̇∼p = λ̇P∼).
Stability / ellipticity of the boundary value problem 38/74
Why is the 3D case less restrictive than the 2Dcase?
Tensile test, von Mises plasticity: Hcrplane stress = 0, Hcr3D = − E4
Answer: the modes found under plane stress are incompatible with respect to
the third direction...
mn
1
2
θ
3
2
1
2 3
Stability / ellipticity of the boundary value problem 39/74
Example: the elliptic criterion
• Plasticity criterion
g(σ∼) = σ? − R, σ2? =
3
2Cσ∼
dev : σ∼dev + F (trace σ∼)
2
• Normality rule
N∼ =∂g
∂σ∼=
1
σ?
(3
2Cσ∼
dev + F (trace σ∼)1∼
)• Tensile test
[P] =signe σ2√
C + F
F −C2 0 0
0 C + F 0
0 0 F − C2
Stability / ellipticity of the boundary value problem 40/74
Example: the elliptic criterionBifurcation analysis under plane stress conditions:
Hcr = 0, n21 =1
3
(1− 2F
C
)
F = 0 von Mises
θ = 35.26◦ withrespect to thehorizontal axis
F =C
4
θ = 25◦ withrespect to thehorizontal axis
F =C
2
θ = 0◦ with respectto the horizontal
axisStability / ellipticity of the boundary value problem 41/74
Example: the elliptic criterionBifurcation analysis under plane stress conditions:
Hcr = 0, n21 =1
3
(1− 2F
C
)
F = 0 von Mises
θ = 35.26◦ withrespect to thehorizontal axis
F =C
4
θ = 25◦ withrespect to thehorizontal axis
F =C
2
θ = 0◦ with respectto the horizontal
axisStability / ellipticity of the boundary value problem 42/74
Example: the elliptic criterionBifurcation analysis under plane stress conditions:
Hcr = 0, n21 =1
3
(1− 2F
C
)
F = 0 von Mises
θ = 35.26◦ withrespect to thehorizontal axis
F =C
4
θ = 25◦ withrespect to thehorizontal axis
F =C
2
θ = 0◦ with respectto the horizontal
axisStability / ellipticity of the boundary value problem 43/74
Example: compression of an aluminium foam
Stability / ellipticity of the boundary value problem 44/74
Example: compression of an aluminium foam
0
1
2
3
4
5
6
7
8
0 0.1 0.2 0.3 0.4 0.5 0.6
stre
ss (
MP
a)
strain
experimentsimulation
Stability / ellipticity of the boundary value problem 45/74
Plan
1 Bifurcation modes in elastoplastic solidsLinear incremental formulationLoss of uniqueness
2 Compatible bifurcation modesHadamard jump conditionsOrientation of a strain localization band
3 Stability / ellipticity of the boundary value problemMandel–Rice criterionOrientation of localization bands in 3D and 2D
4 Summary of strain localization criteria
5 Regularization methodsMesh dependence of the resultsMechanics of generalized continuaApplication to the case of metal foams
Strain localisation criteria
• An open question!• In 2D, Rice’s
criterion isplausible (test itin compressibleand nonassociativeplasticity!)
• In 3D, thecriterion foroccurrence ofcompatible modesis more plausible(strong ellipticity;Rice’s criterion istoo optimistic!)
HH
cr
Q �� g
�
0de
tQ ��
0
Hse
g
� Q �� g
�
0de
tQ �
s
�
0
Hu
ε �:D � �
:ε ��
0
detD � �
s
�
0
ellip
tici
ty
stro
ngel
lipti
city
uniq
uene
ss
Summary of strain localization criteria 47/74
Plan
1 Bifurcation modes in elastoplastic solidsLinear incremental formulationLoss of uniqueness
2 Compatible bifurcation modesHadamard jump conditionsOrientation of a strain localization band
3 Stability / ellipticity of the boundary value problemMandel–Rice criterionOrientation of localization bands in 3D and 2D
4 Summary of strain localization criteria
5 Regularization methodsMesh dependence of the resultsMechanics of generalized continuaApplication to the case of metal foams
Plan
1 Bifurcation modes in elastoplastic solidsLinear incremental formulationLoss of uniqueness
2 Compatible bifurcation modesHadamard jump conditionsOrientation of a strain localization band
3 Stability / ellipticity of the boundary value problemMandel–Rice criterionOrientation of localization bands in 3D and 2D
4 Summary of strain localization criteria
5 Regularization methodsMesh dependence of the resultsMechanics of generalized continuaApplication to the case of metal foams
Spurious / pathological mesh–dependence of finiteelement results
0 0.0072 0.0144 0.0216 0.0288 0.0360
40
80
120
160
200
déplacement (mm)
charge (N)
maillage 10x20
maillage 20x40
maillage 30x60
Regularization methods 50/74
Plan
1 Bifurcation modes in elastoplastic solidsLinear incremental formulationLoss of uniqueness
2 Compatible bifurcation modesHadamard jump conditionsOrientation of a strain localization band
3 Stability / ellipticity of the boundary value problemMandel–Rice criterionOrientation of localization bands in 3D and 2D
4 Summary of strain localization criteria
5 Regularization methodsMesh dependence of the resultsMechanics of generalized continuaApplication to the case of metal foams
Mécanique des milieux continus généralisés
Principe de l’action locale: seule compte l’histoire d’un voisinagearbitrairement petit de la particule X
[Truesdell, Toupin, 1960] [Truesdell, Noll, 1965]
Mili
eu
Con
tinu
acti
on
loca
le
acti
on
non
loca
leth
éori
eno
nlo
cale
:fo
rmul
atio
nin
tégr
ale
[Eri
ngen
,197
2]
Regularization methods 52/74
Mécanique des milieux continus généralisés
Principe de l’action locale: seule compte l’histoire d’un voisinagearbitrairement petit de la particule X
[Truesdell, Toupin, 1960] [Truesdell, Noll, 1965]
Mili
eu
Con
tinu
acti
on
loca
le
acti
on
non
loca
leth
éori
eno
nlo
cale
:fo
rmul
atio
nin
tégr
ale
[Eri
ngen
,197
2]
mili
eum
atér
ielle
men
tsi
mpl
e:F �
mili
euno
nm
atér
ielle
men
tsi
mpl
e
mili
eude
Cau
chy
(182
3)(c
lass
ique
/Bol
tzm
ann)
Regularization methods 53/74
Mécanique des milieux continus généralisés
Principe de l’action locale: seule compte l’histoire d’un voisinagearbitrairement petit de la particule X
[Truesdell, Toupin, 1960] [Truesdell, Noll, 1965]
Mili
eu
Con
tinu
acti
on
loca
le
acti
on
non
loca
leth
éori
eno
nlo
cale
:fo
rmul
atio
nin
tégr
ale
[Eri
ngen
,197
2]
mili
eum
atér
ielle
men
tsi
mpl
e:F �
mili
euno
nm
atér
ielle
men
tsi
mpl
e
mili
eude
Cau
chy
(182
3)(c
lass
ique
/Bol
tzm
ann)
mili
eud’
ordr
en
mili
eude
degr
én
Cos
sera
t(1
909)
u
�R �
mic
rom
orph
e[E
ring
en,M
indl
in19
64]
u
�χ �
seco
ndgr
adie
nt[M
indl
in,1
965]
F ��F �
�
∇
grad
ient
deva
riab
lein
tern
e[M
augi
n,19
90]
u�α
Regularization methods 54/74
Le milieu micromorphe: cinématique
• Degrés de liberté et raffinement de la théorie
DOF := {u , χ∼}
χ∼
= χ∼
s + χ∼
a
MODEL := {u ⊗∇, χ∼⊗∇}
Cas particuliers:
? Milieu de Cosserat χ∼≡ χ
∼
a = −�∼.Φ
? Milieu microstrain χ∼≡ χ
∼
s
? Théorie du second gradient χ∼≡ u ⊗∇
• Mesures de déformation
STRAIN := {ε∼, e∼ := u ⊗∇− χ∼ , K∼ := χ∼ ⊗∇}
Regularization methods 55/74
Le milieu micromorphe: statique
• Méthode des puissances virtuelles [Germain, 1973]
P(i) =∫D
p(i) dV , P(c) =∫
∂Dp(c) dS
p(i) = σ∼ : ε̇∼ + s∼ : ė∼ + S∼...K̇∼
p(c) = t .u̇ + M∼ : χ̇∼
• Equations de champ? Quantité de mouvement (σ∼ + s∼).∇ = 0? Moment cinétique généralisé S
∼.∇ + s∼= 0
• Conditions aux limites
t = (σ∼ + s∼).n , M∼ = S∼.n
Regularization methods 56/74
Le milieu micromorphe: thermodynamique
• Equation locale de l’énergie
ρ�̇ = p(i) − q .∇ + r
• Second principe et inégalité de Clausius–Duhem
ρη̇ +(q
T
).∇ − r
T≥ 0
ρ (T η̇ − �̇) + p(i) −q
T.(∇T ) ≥ 0
• Variables d’état et énergie libre de Helmholtz
ε∼ = ε∼e + ε∼
p, e∼ = e∼e + e∼
p, K∼
= K∼
e + K∼
p
Z := {T , ε∼e , e∼
e , K∼
e , α}
Ψ = �− Tη
Regularization methods 57/74
Le milieu micromorphe: potentiel de dissipation
• Exploitation du second principe à la Coleman–Noll? Lois d’état
η = −∂Ψ∂T
, σ∼ = ρ∂Ψ
∂ε∼e, s∼= ρ
∂Ψ
∂e∼e, S
∼= ρ
∂Ψ
∂K∼
e , R = ρ∂Ψ
∂α
? Lois d’évolutiondissipation résiduelle
D = σ∼ : ε̇∼p + s∼ : ė∼
p + S∼
...K̇∼
p− Rα̇
potentiel de dissipation
Ω(σ∼ , s∼,S∼,R)
ε̇∼p =
∂Ω
∂σ∼, ė∼
p =∂Ω
∂s∼, K̇
∼
p=
∂Ω
∂S∼
, α̇ = −∂Ω∂R
Regularization methods 58/74
Le modèle microstrain
Degrés de liberté (u ,χ∼
s)Mesures de déformation :
(ε∼, ε∼− χ∼s ,χ
∼s ⊗∇)
σ∼ = C∼∼: (ε∼− ε∼
p)
s∼ = b(ε∼− χ∼s)
S∼
= Aχ∼
s ⊗∇
div (σ∼ + s∼) = 0
div S∼
+ s∼ = 0
Sijk,k + sij = 0
Aχsij ,kk +b(εij−χsij) = 0
εij = χsij − l2∆χsij
Lien avec “implicitgradient–enhanced
elastoplasticity models”[Engelen et al., 2003]
Conditions aux limites :(ui , χ
sij) or
(σij + sij)nj ,Sijknk
Regularization methods 59/74
Programmation en éléments finis (1)
Exemple: milieu micromorphe 2D
[DOF] = [U1 U2 X11 X22 X12 X21]T
[STRAIN] = [ε11 ε22 ε33 ε12 e11 e22 e33 e12 e21
K111 K112 K121 K122 K211 K212 K221 K222]T
[STRAIN] = [B] [DOF]
[FLUX] = [σ11 σ22 σ33 σ12 s11 s22 s33 s12 s21
S111 S112 S121 S122 S211 S212 S221 S222]T
Regularization methods 60/74
Programmation en éléments finis (2)
[B] =
∂x1 0 0 0 0 00 ∂x2 0 0 0 00 0 0 0 0 0
12∂x2
12∂x1 0 0 0 0
∂x1 0 −1 0 0 00 ∂x2 0 −1 0 00 0 0 0 0 0
∂x2 0 0 0 −1 00 ∂x1 0 0 0 −10 0 ∂x1 0 0 00 0 ∂x2 0 0 00 0 0 0 ∂x1 00 0 0 0 ∂x2 00 0 0 ∂x1 0 00 0 0 ∂x2 0 0
S111S112S121S122S211S212S221S222
= [A]
K111K112K121K122K211K212K221K222
Regularization methods 61/74
Programmation en éléments finis (3)
matrice d’élasticité généralisée [Mindlin, 1964]
AA 0 0 A1,4,5 0 A2,5,8 A1,2,3 00 A3,10,14 A2,11,13 0 A1,11,15 0 0 A1,2,30 A2,11,13 A8,10,15 0 A5,11,14 0 0 A2,5,8
A1,4,5 0 0 A4,10,13 0 A5,11,14 A1,11,15 00 A1,11,15 A5,11,14 0 A4,10,13 0 0 A1,4,5
A2,5,8 0 0 A5,11,14 0 A8,10,15 A2,11,13 0A1,2,3 0 0 A1,11,15 0 A2,11,13 A3,10,14 0
0 A1,2,3 A2,5,8 0 A1,4,5 0 0 AA
avecAA = 2A1+2A2+A3+A4+2A5+A8+A10+2A11+A13+A14+A15et Ai ,j ,k = Ai + Aj + Ak
Regularization methods 62/74
Plan
1 Bifurcation modes in elastoplastic solidsLinear incremental formulationLoss of uniqueness
2 Compatible bifurcation modesHadamard jump conditionsOrientation of a strain localization band
3 Stability / ellipticity of the boundary value problemMandel–Rice criterionOrientation of localization bands in 3D and 2D
4 Summary of strain localization criteria
5 Regularization methodsMesh dependence of the resultsMechanics of generalized continuaApplication to the case of metal foams
Bande de localisation de largeur finie
Cas des bandes horizontales obtenues à l’aide d’un critère elliptique
σeq =√
C + F |σ22|, ε̇p22 = ṗ√
C + F , ṗ =2µ√
C + F ε̇222µ(C + F ) + H
(σ22 + s22),2 = 0, S222,2 + s22 = 0
σ22 = 2µ(ε22 − εp22) =2µ
2µ(C+F )+H (Hε22 + R0√
C + F )
s22 = 2µ(ε22 − χ22), S222 = Aχ22,2Finalement
χ22,222 −2µH̄
A(H̄ + 2µ)χ22,2 = 0, H̄ =
2µH
2µ(C + F ) + H
Longueur d’onde
1
ω=
√A(H̄ + 2µ)
2µ|H̄|
Regularization methods 64/74
Simulations par éléments finis à l’aide du milieumicromorphe
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0 20 40 60 80 100
p
y
10x2 elements20x2 elements50x2 elements
100x2 elements
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0 20 40 60 80 100
analyticalFE analysis
Regularization methods 65/74
Comportement et rupture des mousses de nickel
longueur de fissure : 10 mm, taille de cellule : 0.5mm
Regularization methods 66/74
Comportement et rupture des mousses de nickel
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
stre
ss (
MP
a)
strain
experimentmodel
Direction de traction RD
Critère simple(iste) de rupture
p = pcrit = 0.08
dispersion limitée en pcrit
adoucissement explicite pourp > pcrit
R = R(p > pcrit)
Regularization methods 67/74
Traction d’une plaque fissurée : cas classique
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.01 0.02 0.03 0.04 0.05 0.06
forc
e/lig
amen
t (M
Pa)
displacement/gauge length
1609 nodesexperiment
00.05
0.10.15
0.20.25
0.30.35
0.40.45
0.50.55
0.60.65
0.70.75
0.80.85
0.90.95
1
strain component ε11
Regularization methods 68/74
Traction d’une plaque fissurée : cas classique
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.01 0.02 0.03 0.04 0.05 0.06
forc
e/lig
amen
t (M
Pa)
displacement/gauge length
1609 nodes509 nodesexperiment
00.05
0.10.15
0.20.25
0.30.35
0.40.45
0.50.55
0.60.65
0.70.75
0.80.85
0.90.95
1
strain component ε11
Regularization methods 69/74
Traction d’une plaque fissurée : cas classique
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.01 0.02 0.03 0.04 0.05 0.06
forc
e/lig
amen
t (M
Pa)
displacement/gauge length
3927 nodes1609 nodes509 nodesexperiment
00.05
0.10.15
0.20.25
0.30.35
0.40.45
0.50.55
0.60.65
0.70.75
0.80.85
0.90.95
1
strain component ε11
Regularization methods 70/74
Traction d’une plaque fissurée : cas micromorphe
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.01 0.02 0.03 0.04 0.05 0.06
forc
e/lig
amen
t (M
Pa)
displacement/gauge length
experimentmicrofoam
00.02
0.040.06
0.080.1
0.120.14
0.160.18
0.20.22
0.240.26
0.280.3
0.320.34
0.360.38
0.4
strain component ε11
Regularization methods 71/74
Traction d’une plaque fissurée : cas micromorphe
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.01 0.02 0.03 0.04 0.05 0.06
forc
e/lig
amen
t (M
Pa)
displacement/gauge length
experimentmicrofoam 3927 nodesmicrofoam 1609 nodes
microfoam 509 nodes
influence of mesh size on the overall curves
Regularization methods 72/74
Traction d’une plaque fissurée : cas micromorphe
00.02
0.040.06
0.080.1
0.120.14
0.160.18
0.20.22
0.240.26
0.280.3
0.320.34
0.360.38
0.4
Regularization methods 73/74
Traction d’une plaque fissurée : cas micromorphe
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.01 0.02 0.03 0.04 0.05 0.06
forc
e/lig
amen
t (M
Pa)
displacement/gauge length
experimentlc = 0.06 mm
lc = 0.1 mm
Influence of the characteristic length on fracture : l2c = a8/µ withµ = 167 MPa
Regularization methods 74/74
PlanBifurcation modes in elastoplastic solidsincrementaluniquenessHill
Compatible bifurcation modeshadamardbandes
Stability / ellipticity of the boundary value problemRice3D-2D
Summary of strain localization criteriaRegularization methodsmeshmmcgmicromorphnumericelliptic