Strain localisation modes in the mechanics of materials

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Strain localisation modes in the mechanics of materials

Samuel Forest

Centre des Mat´eriaux/UMR 7633 Ecole des Mines de Paris/CNRS

BP 87, 91003 Evry, France Samuel.Forest@ensmp.fr

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Outline

1 Bifurcation modes in elastoplastic solids Linear incremental formulation Loss of uniqueness

2 Compatible bifurcation modes Hadamard jump conditions

Orientation of a strain localization band

3 Stability / ellipticity of the boundary value problem Mandel–Rice criterion

Orientation of localization bands in 3D and 2D

4 Summary of strain localization criteria

5 Regularization methods

Mesh dependence of the results Mechanics of generalized continua Application to the case of metal foams

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Plan

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Plan

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Loi de comportement ´ elastoplastique

• Elasticity ε_∼˙=ε_∼˙^e+ε_∼˙^p, σ_∼=E_∼

∼:ε_∼^e

• Yield function g(σ_∼, α), N_∼ = ∂g

∂σ_∼

• Plastic flow ε_∼˙^p= ˙λP_∼, P_∼6=N_∼

• Hardening modulus α˙ = ˙λh, H =−∂g

∂α.h

• Tangent operator (non symmetrical if non associative) σ_∼˙ =L_∼

∼

:ε_∼˙ L∼

∼

= E_∼

∼

si g <0 ou (g = 0 et N_∼ :E_∼

∼

:ε_∼˙≤0) L∼

∼

= D_∼

∼

si g = 0 et N_∼ :E_∼

∼

:ε_∼˙>0 with

D_∼

∼ =E_∼

∼

− (E_∼

∼

:P_∼)N (N_∼ :E_∼

∼

) H+N_∼ :E_∼

∼:P_∼

Bifurcation modes in elastoplastic solids 5/74

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Rate form of the boundary value problem

Ω

∂Ω2

∂Ω1

V

T˙











˙

ε∼= ¹₂(∇v +∇^tv) div σ_∼˙ +˙f = 0, σ_∼˙ =L_∼

∼

:ε_∼˙

˙

σ_∼ .n =T˙ in ∂Ω₁ v =V in ∂Ω₂

• Linear comparison solid: takeL_∼

∼

=D_∼

∼

• Variational formulation (virtual power theorem) Z

Ω

˙

σ_∼:ε_∼˙^?dV = Z

Ω

˙f.v^?dV+ Z

∂Ω1

T˙.v^?dS+ Z

∂Ω2

(σ_∼˙.n).V dS for any fieldv^? such thatv^?=V in ∂Ω₂

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Plan

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Conditions of uniqueness of the solution

Let (σ_∼˙_A,ε_∼˙_A) and (σ_∼˙_B,ε_∼˙_B) be two solutions on Ω at time t Z

Ω

˙

σ_∼_A :ε_∼˙_AdV = Z

Ω

˙f.v_AdV + Z

∂Ω1

T˙.v_AdS+ Z

∂Ω2

(σ_∼˙_A.n).V dS

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Conditions of uniqueness of the solution

Ω

˙

σ_∼_A :ε_∼˙_AdV = Z

Ω

˙f.v_AdV + Z

∂Ω1

T˙.v_AdS+ Z

∂Ω2

(σ_∼˙_A.n).V dS Z

Ω

˙

σ_∼_A :ε_∼˙_BdV = Z

Ω

˙f.v_BdV + Z

∂Ω1

T˙.v_BdS+ Z

∂Ω2

(σ_∼˙_A.n).V dS

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Conditions of uniqueness of the solution

Ω

˙

σ_∼_A :ε_∼˙_AdV = Z

Ω

˙f.v_AdV + Z

∂Ω1

T˙.v_AdS+ Z

∂Ω2

(σ_∼˙_A.n).V dS

− Z

Ω

˙

σ_∼_A:ε_∼˙_BdV = Z

Ω

˙f.v_BdV+ Z

∂Ω1

T˙.v_BdS+

Z

∂Ω2

(σ_∼˙_A.n).V dS

Z

Ω

˙

σ_∼_A :∆ε_∼˙dV = Z

Ω

˙f.∆v dV + Z

∂Ω1

T˙.∆v dS

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Conditions of uniqueness of the solution

Ω

˙

σ_∼_A :∆ε_∼˙dV = Z

Ω

˙f.∆v dV + Z

∂Ω1

T˙.∆v dS

− Z

Ω

˙

σ_∼_B :∆ε_∼˙dV = Z

Ω

˙f.∆v dV + Z

∂Ω1

T˙.∆v dS

Z

Ω

∆σ_∼˙ :∆ε_∼˙dV = 0

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Conditions of uniqueness of the solution

Ω

∆σ_∼˙ :∆ε_∼˙dV = 0

If the solutions (σ_∼_A,ε_∼_A) and (σ_∼_B,ε_∼_B) have coincided until timet, one can assume

˙ σ_∼_A =D_∼

∼ :ε_∼˙_A, σ_∼˙_B =D_∼

∼ :ε_∼˙_B

A necessary condition for the existence of several solutions then is Z

Ω

∆ε_∼˙:D_∼

∼

:∆ε_∼˙dV = 0

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A sufficient condition for uniqueness

A sufficient condition ensuring the uniqueness of the solution (∆ε_∼˙= 0) is the positivity ofsecond order power:

˙ ε∼:D_∼

∼

:ε_∼˙>0, ∀ε_∼˙6= 0

˙ ε∼:D_∼

∼

s :ε_∼˙>0, avec D_∼

∼

s ijkl = 1

2(D_ijkl+D_klij) Hill’s condition for uniqueness: D_∼

∼

s positive definite (Hill, 1958) Uniqueness may be lost (possible bifurcation) when

detD_∼

∼

s = 0

which givesH^u= 1 2(q

N_∼ :E_∼

∼

:N_∼q P_∼ :E_∼

∼

:P_∼−N_∼ :E_∼

∼

:P_∼)

H>H^u: uniqueness ensured associative plasticity: H^u= 0 softening!

uniaxial + Large Strain= Consid`ere criterion...

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Example : von Mises plasticity

• Von Mises plasticity:

g(σ_∼,R) =J₂(σ_∼)−R, J₂(σ_∼) = r3

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, 2µH H+ 3µ

WhenH= 0 (perfect plasticity, begining of a softening regime)

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Example : necking mode

Tensile test, [d_∼₁] =





−1 0 0

0 2 0

0 0 −1





possible perturbations when H = 0...

˙ σ_∼_A =D_∼

∼ :ε_∼˙_A =D_∼

∼ : (ε_∼˙_A+λd_∼₁)

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Example : necking mode





−1 0 0

0 2 0

0 0 −1





˙ σ_∼_A =D_∼

∼

:ε_∼˙_A =D_∼

∼

: (ε_∼˙_A+λd_∼₁)

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Example : necking mode





−1 0 0

0 2 0

0 0 −1





˙ σ_∼_A =D_∼

∼ :ε_∼˙_A =D_∼

∼ : (ε_∼˙_A+λd_∼₁)

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Example : necking mode

0 20 40 60 80 100 120

0 0.04 0.08 0.12 0.16

F/S

δ/L

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Example : necking mode

0 20 40 60 80 100 120

0 0.04 0.08 0.12 0.16

F/S

δ/L

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Example : necking mode

0 20 40 60 80 100 120

0 0.04 0.08 0.12 0.16

F/S

δ/L

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Example : necking mode

0 20 40 60 80 100 120

0 0.04 0.08 0.12 0.16

F/S

δ/L

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Example : necking mode

0 20 40 60 80 100 120

0 0.04 0.08 0.12 0.16

F/S

δ/L

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Example : necking mode

0 20 40 60 80 100 120

0 0.04 0.08 0.12 0.16

F/S

δ/L

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Plan

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Plan

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Compatible discontinuous bifurcation modes : Hadamard jump conditions

Strain localization is the precursor of fracture.

Continuity of displacement is assumed through a surfaceS

[[u]] = [[u˙]] = 0

but discontinuities of some components

of its gradient∇u are considered ¹ 2

S n

Compatible bifurcation modes 26/74

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Compatible discontinuous bifurcation modes : Hadamard jump conditions

Let us work in the frame attached to the surface of discontinuity:

[[u₁]] = 0 =⇒[[u_1,1]] = 0 [[u2]] = 0 =⇒[[u2,1]] = 0 but, in general,∃g such that

[[u1,2]] =g1 6= 0, [[u2,2]] =g2 6= 0 ¹

2 S

n

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Compatible discontinuous bifurcation modes : Hadamard jump conditions

Let us work in the frame attached to the surface of discontinuity:

[[u1]] = 0 =⇒[[u1,1]] = 0 [[u₂]] = 0 =⇒[[u_2,1]] = 0 but, in general,

[[u_1,2]] =g₁ 6= 0, [[u_2,2]] =g₂ 6= 0 [[∇u]] =

0 g₁ 0 g₂

= g₁

g₂

⊗ 0

1

[[∇u]] =g ⊗n

1

2 S

n

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Compatible discontinuous bifurcation modes : Hadamard jump conditions

[[∇u˙]] =g ⊗n [[∇ε_∼˙]] = 1

2(g ⊗n +n ⊗g) in generalg.n 6= 0

Criterion for the occurence of compatible bifurcation modes:

∆ε_∼˙:D_∼

∼ : ∆ε_∼˙= (g⊗n) :D_∼

∼ : (g⊗n) = 0 Theacoustic tensor Q

∼ is introduced:

g.Q

∼.g =g.Q

∼

sym.g = 0, Q

∼ =n.D_∼

∼.n

=⇒ detQ_∼^sym= 0

1

2 S

n

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Compatible discontinuous bifurcation modes : Hadamard jump conditions

One can imagine two parallel surfaces of discontinuity: it is astrain localization bandorshear bandin a wide sense. In- deed, the band does generally not un- dergo pure shear...

• g.n = 0: shear band

• g kn: opening mode

1 2

S1

S2

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Plan

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Example: compatible strain localization mode under plane stress conditions

Consider the mode

[d_∼

1] =





−1 0 0

0 2 0

0 0 −1





but write it with respect to the frame (m,n)

[d_∼

1] =





2 sin²θ−cos²θ 3 cosθsinθ 0 3 cosθsinθ 2 cos²θ−sin²θ 0

0 0 −1





to be identified with¹₂(g ⊗n+n⊗g) =

0 g1/2 g1/2 g2

=⇒ tan²θ= 1

2, ou sin²θ=1 3 θ= 35,26^◦, 90^◦−θ= 54.74^◦

n m

1 2

θ

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Plan

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Plan

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Mandel–Rice criterion

Add the equilibrium contitions at the interface which were not taken into account until now:

[[σ_∼˙]].n = 0, [[D_∼

∼

:ε_∼˙]].n= 0

If the solutions on each side ofS are identical until timet, one has [[D_∼

∼]] = 0 D_∼

∼

: [[_∼ε˙]].n= 0 Let us consider the compatible modes:

D_∼

∼: (g ⊗n).n = 0 Dijklgknlnj=njDjiklnlgk= 0

Q∼.g = 0 This is theacoustic tensor: Q

∼=n.D_∼

∼

.n

1

2 S

n

Mandel–Rice criterion:

detQ_∼(n) = 0

stable propagation of perturbations [Mandel, 1966] [Rice, 1976]

Stability / ellipticity of the boundary value problem 35/74

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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 in the equations of equilibrium (static case), one gets a system of partial differential equations of order 2 in ˙u_i. The system is said to beellipticwhen the differential operator has definite positive eigen–values. It can be shown that the problem then is well–posed(meaning that the solution continuously depends on the data).

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Plan

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Orientation of strain localization bands

The resolution of the equation detn.D_∼

∼

.n= 0 provides us with the plastic modulusH(n) for which a surface of discontinuity of normaln can arise. To determine the first possible band, one must findH^cr andn such that

H^cr = max

kn^k=1 H(n)

In the case of isotropique elasticity and incompressible associative plasticity, one finds

dimension critical modulus orientation

of the problem H^cr n

plane 0 n²₁ = _P^P¹

1−P₂

stress n₂²= 1−n₁²

3D −EP_k² n²_i = ^P_Pⁱ^+νP^k

i−P_j

nk = 0,n²_j = 1−n²_i

plane 0 n²₁ =n₂²= ¹₂

strain

ThePi are the eigenvalues of the plastic flow directionP_∼(_∼ε˙^p= ˙λP_∼).

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Why is the 3D case less restrictive than the 2D case?

Tensile test, von Mises plasticity:Hplane stress^cr = 0, H3D^cr =−^E₄

Answer: the modes found under plane stress are incompatible with respect to the third direction...

n m

1 2

θ

3 2

1 2 3

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Example: the elliptic criterion

• Plasticity criterion

g(σ_∼) =σ?−R, σ_?² = 3

2Cσ_∼^dev :σ_∼^dev+F(trace σ_∼)²

• Normality rule N_∼ = ∂g

∂σ_∼ = 1 σ_?

3

2Cσ_∼^dev+F(traceσ_∼)1_∼

• Tensile test

[P] = signeσ₂

√C +F







F −^C₂ 0 0

0 C +F 0

0 0 F −C

2







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Example: the elliptic criterion

Bifurcation analysis under plane stress conditions:

H^cr = 0, n²₁ = 1 3

1−2F

C

F = 0 von Mises θ= 35.26^◦ with

respect to the horizontal axis

F = C 4 θ= 25^◦ with respect to the horizontal axis

F = C 2 θ= 0^◦ with respect

to the horizontal

Stability / ellipticity of the boundary value problem axis 41/74

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Example: the elliptic criterion

H^cr = 0, n²₁ = 1 3

1−2F

C

to the horizontal

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Example: the elliptic criterion

H^cr = 0, n²₁ = 1 3

1−2F

C

to the horizontal

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Example: compression of an aluminium foam

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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

stress (MPa)

strain experiment

simulation

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Plan

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Crit` eres de localisation

• C’est une question encore ouverte!

• En 2D, le critère de Rice est plausible (à tester en compressible et en non associé!)

• En 3D, le crit`ere d’apparition de modes

compatibles est sans doute le plus plausible

(ellipticit´e forte;

le crit`ere de Rice est trop

optimiste!)

H^cr H Qg 0 det Q 0

H^se gQg 0 det Q^s 0

H^u ε: D :ε 0

det D^s 0 ellipticity

strong ellipticity uniqueness

Summary of strain localization criteria 47/74

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Plan

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Plan

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Spurious / pathological mesh–dependence of finite element results

0 0.0072 0.0144 0.0216 0.0288 0.036

0 40 80 120 160 200

déplacement (mm) charge (N)

maillage 10x20 maillage 20x40 maillage 30x60

Regularization methods 50/74

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Plan

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M´ ecanique des milieux continus g´ en´ eralis´ es

Principe del’action locale: seule compte l’histoire d’un voisinage arbitrairement petit de la particuleX

[Truesdell, Toupin, 1960] [Truesdell, Noll, 1965]

Milieu Continu

action locale

action non locale

th´eorie non locale: formulation int´egrale [Eringen, 1972]

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M´ ecanique des milieux continus g´ en´ eralis´ es

milieu mat´eriellement simple: F

milieunon mat´eriellement simple

milieu de Cauchy (1823) (classique / Boltzmann)

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M´ ecanique des milieux continus g´ en´ eralis´ es

milieu mat´eriellement simple: F

milieunon mat´eriellement simple

milieu de Cauchy (1823) (classique / Boltzmann)

milieu d’ordren

milieu dedegr´en

Cosserat(1909) uR

micromorphe

[Eringen, Mindlin 1964]

uχ

second gradient [Mindlin, 1965]

FF ∇

gradient de variable interne[Maugin, 1990]

uα

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Le milieu micromorphe: cin´ ematique

• 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

? Milieumicrostrain χ

∼≡χ

∼

s

? Th´eorie du second gradient χ

∼≡u ⊗∇

• Mesures de d´eformation

STRAIN :={ε_∼, e_∼:=u ⊗∇−χ

∼, K

∼ :=χ

∼⊗∇}

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Le milieu micromorphe: statique

• M´ethode des puissances virtuelles [Germain, 1973]

P⁽ⁱ⁾ = Z

D

p⁽ⁱ⁾ dV, P^(c)= Z

∂D

p^(c) dS

p⁽ⁱ⁾=σ_∼ :ε_∼˙+s_∼:_∼˙e+S

∼

...K˙

∼

p^(c) =t.u˙ +M_∼ :χ˙

∼

• Equations de champ

? Quantit´e de mouvement (σ_∼+_∼s).∇= 0

? Moment cinétique généralisé S

∼.∇+_∼s= 0

• Conditions aux limites

t = (σ_∼+_∼s).n, M_∼ =S

∼.n

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Le milieu micromorphe: thermodynamique

• Equation locale de l’´energie

ρ˙ = p⁽ⁱ⁾ − q.∇ + r

• Second principe et in´egalit´e de Clausius–Duhem ρη˙ + q

T

.∇ − r T ≥ 0 ρ(T η˙ − ) +˙ p⁽ⁱ⁾ − q

T.(∇T) ≥ 0

• Variables d’état et énergie libre de Helmholtz ε∼=ε_∼ê+ε_∼^p, e_∼=e_∼ê+_∼e^p, K

∼ =K

∼

e+K

∼

p

Z:={T, ε_∼^e, e_∼^e, K

∼

e, α}

Ψ =−Tη

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Le milieu micromorphe: potentiel de dissipation

• Exploitation du second principe `a la Coleman–Noll

? Lois d’´etat η=−∂Ψ

∂T, σ_∼ =ρ∂Ψ

∂ε_∼^e, _∼s=ρ∂Ψ

∂e_∼^e, S

∼=ρ∂Ψ

∂K∼

e, R=ρ∂Ψ

∂α

? Lois d’´evolution dissipation r´esiduelle

D=σ_∼ :ε_∼˙^p+_∼s:_∼˙e^p+S

∼

...K˙

∼

p−Rα˙

potentiel de dissipation

Ω(σ_∼,_∼s,S

∼,R)

˙ ε∼

p=∂Ω

∂σ_∼, _∼˙e^p= ∂Ω

∂s_∼, K˙

∼

p= ∂Ω

∂S∼

, α˙ =−∂Ω

∂R

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Le mod` ele microstrain

Degr´es de libert´e (u,χ

∼

s) Mesures de d´eformation :

(ε_∼,ε_∼−χ

∼

s,χ

∼

s⊗∇)

σ_∼=C_∼

∼

: (ε_∼−ε_∼^p)

∼s=b(ε_∼−χ

∼

s) S∼=Aχ

∼

s ⊗∇

div(σ_∼+_∼s) = 0 divS

∼+s_∼= 0

Sijk,k+sij = 0 Aχ^s_ij,kk+b(ε_ij−χ^s_ij) = 0

ε_ij =χ^s_ij −l²∆χ^s_ij Lien avec “implicit gradient–enhanced elastoplasticity models”

[Engelenet al., 2003]

Conditions aux limites : (u_i, χ^s_ij) or (σij +sij)nj,Sijknk

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Programmation en ´ el´ ements finis (1)

Exemple: milieu micromorphe 2D

[DOF] = [U₁ U₂ X₁₁ X₂₂ X₁₂ X₂₁]^T [STRAIN] = [ε11 ε22 ε33 ε12 e11 e22 e33 e12e21

K₁₁₁ K₁₁₂ K₁₂₁ K₁₂₂ K₂₁₁ K₂₁₂ K₂₂₁ K₂₂₂]^T [STRAIN] = [B] [DOF]

[FLUX] = [σ11 σ22σ33 σ12 s11 s22 s33s12 s21

S₁₁₁ S₁₁₂ S₁₂₁ S₁₂₂ S₂₁₁ S₂₁₂ S₂₂₁ S₂₂₂]^T

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Programmation en ´ el´ ements finis (2)

[B] =







∂x1 0 0 0 0 0

0 ∂x2 0 0 0 0

0 0 0 0 0 0

1 2∂x2

1

2∂x1 0 0 0 0

∂_x₁ 0 −1 0 0 0

0 ∂_x₂ 0 −1 0 0

0 0 0 0 0 0

∂x2 0 0 0 −1 0

0 ∂_x₁ 0 0 0 −1

0 0 ∂x1 0 0 0

0 0 ∂x2 0 0 0

0 0 0 0 ∂_x₁ 0

0 0 0 0 ∂x2 0

0 0 0 ∂x1 0 0

0 0 0 ∂_x₂ 0 0











 S111

S₁₁₂ S₁₂₁ S122

S₂₁₁ S₂₁₂ S221

S₂₂₂







= [A]





 K111

K₁₁₂ K₁₂₁ K122

K₂₁₁ K₂₁₂ K221

K₂₂₂







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Programmation en ´ el´ ements finis (3)

matrice d’élasticité généralisée [Mindlin, 1964]







AA 0 0 A1,4,5 0 A2,5,8 A1,2,3 0

0 A_3,10,14 A_2,11,13 0 A_1,11,15 0 0 A_1,2,3

0 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 0

0 A_1,11,15 A_5,11,14 0 A_4,10,13 0 0 A_1,4,5

A2,5,8 0 0 A5,11,14 0 A8,10,15 A2,11,13 0

A1,2,3 0 0 A1,11,15 0 A2,11,13 A3,10,14 0

0 A_1,2,3 A_2,5,8 0 A_1,4,5 0 0 AA







avec

AA= 2A1+2A2+A3+A4+2A5+A8+A10+2A11+A13+A14+A15

etA_i,j,k =A_i +A_j +A_k

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Plan

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Bande de localisation de largeur finie

Cas des bandes horizontales obtenues `a l’aide d’un crit`ere elliptique σeq =√

C +F|σ₂₂|, ε˙^p₂₂= ˙p√

C +F, p˙ = 2µ√

C +Fε˙22

2µ(C +F) +H







(σ₂₂+s₂₂)_,2 = 0, S_222,2+s₂₂= 0 σ₂₂= 2µ(ε₂₂−ε^p₂₂) = _2µ(C^2µ_+F)+H(Hε₂₂+R₀√

C +F) s₂₂= 2µ(ε₂₂−χ₂₂), S₂₂₂ =Aχ_22,2

Finalement

χ_22,222− 2µH¯

A( ¯H+ 2µ)χ_22,2= 0, H¯ = 2µH 2µ(C +F) +H Longueur d’onde

1 ω =

s

A( ¯H+ 2µ) 2µ|H|¯

(65)

Simulations par ´ el´ ements finis ` a l’aide du milieu micromorphe

-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 elements 20x2 elements 50x2 elements 100x2 elements

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0 20 40 60 80 100

analytical FE analysis

(66)

Comportement et rupture des mousses de nickel

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

(67)

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

stress (MPa)

strain experiment

model

Direction de traction RD

Crit`ere simple(iste) de rupture p =p_crit= 0.08 dispersion limit´ee enp_crit adoucissement explicite pour p>p_crit

R =R(p >pcrit)

(68)

Traction d’une plaque fissur´ ee : 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

force/ligament (MPa)

displacement/gauge length 1609 nodes

experiment

0 0.05

0.1 0.15

0.2 0.25

0.3 0.35

0.4 0.45

0.5 0.55

0.6 0.65

0.7 0.75

0.8 0.85

0.9 0.95

1

strain component ε11

(69)

Traction d’une plaque fissur´ ee : 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

displacement/gauge length 1609 nodes

509 nodes experiment

0 0.05

0.1 0.15

0.2 0.25

0.3 0.35

0.4 0.45

0.5 0.55

0.6 0.65

0.7 0.75

0.8 0.85

0.9 0.95

1

strain component ε₁₁

(70)

Traction d’une plaque fissur´ ee : 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

displacement/gauge length 3927 nodes 1609 nodes 509 nodes experiment

0 0.05

0.1 0.15

0.2 0.25

0.3 0.35

0.4 0.45

0.5 0.55

0.6 0.65

0.7 0.75

0.8 0.85

0.9 0.95

1

strain component ε₁₁

(71)

Traction d’une plaque fissur´ ee : 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

displacement/gauge length experiment

microfoam

0 0.02

0.04 0.06

0.08 0.1

0.12 0.14

0.16 0.18

0.2 0.22

0.24 0.26

0.28 0.3

0.32 0.34

0.36 0.38

0.4

strain component ε11