strain localisation modes in the mechanics of...

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






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


Plan






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




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




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




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




∆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


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...


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)



Tensile test,

[d∼1] =

−1 0 00 2 00 0 −1


σ̇∼A = D∼∼: ε̇∼A = D∼∼

: (ε̇∼A + λd∼1)



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


0

20

40

60

80

100

120

0 0.04 0.08 0.12 0.16

F/S


Plan






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


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



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



[[∇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



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


Plan






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


Plan






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).


Plan






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∼).


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


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


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


Hcr = 0, n21 =1

3

(1− 2F

C

)

F = 0 von Mises


F =C

4


F =C

2



Example: compression of an aluminium foam


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


Plan






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






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






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]

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eu

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3)(c

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indl

in19

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u

�χ �

seco

ndgr

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nt[M

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F ��F �

�

∇

grad

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deva

riab

lein

tern

e[M

augi

n,19

90]

u�α


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∼ ⊗∇}


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


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


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


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


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



[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



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


Plan






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|


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


Comportement et rupture des mousses de nickel

longueur de fissure : 10 mm, taille de cellule : 0.5mm


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)


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



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)


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




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)


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



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)


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




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)


experimentmicrofoam 3927 nodesmicrofoam 1609 nodes

microfoam 509 nodes

influence of mesh size on the overall curves



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



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)


experimentlc = 0.06 mm

lc = 0.1 mm

Influence of the characteristic length on fracture : l2c = a8/µ withµ = 167 MPa


PlanBifurcation modes in elastoplastic solidsincrementaluniquenessHill

Compatible bifurcation modeshadamardbandes

Stability / ellipticity of the boundary value problemRice3D-2D

Summary of strain localization criteriaRegularization methodsmeshmmcgmicromorphnumericelliptic

strain localisation modes in the mechanics of...

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