SLIDE 1 Micromorphic Crystal Plasticity
Samuel Forest, Nicolas Cordero, Ana¨ ıs Gaubert∗, Esteban Busso
Mines ParisTech / CNRS Centre des Mat´ eriaux/UMR 7633 BP 87, 91003 Evry, France Samuel.Forest@mines-paristech.fr
∗ ONERA, 29, Av. de la Division Leclerc, F–92322 Chˆ
atillon, France
SLIDE 2
Plan
1
The “microcurl” model Kinematics and balance equations Constitutive equations Internal stress arising from the model Internal constraint and strain gradient plasticity Linearized formulation
2
Size effect in a two–phase single crystal laminate Boundary value problem Interface conditions Strain gradient plasticity as a limit case
3
Grain size effect in polycrystalline aggregates Hall–Petch effect Strain localization in ultra–fine grains
SLIDE 3
Plan
1
The “microcurl” model Kinematics and balance equations Constitutive equations Internal stress arising from the model Internal constraint and strain gradient plasticity Linearized formulation
2
Size effect in a two–phase single crystal laminate Boundary value problem Interface conditions Strain gradient plasticity as a limit case
3
Grain size effect in polycrystalline aggregates Hall–Petch effect Strain localization in ultra–fine grains
SLIDE 4
Plan
1
The “microcurl” model Kinematics and balance equations Constitutive equations Internal stress arising from the model Internal constraint and strain gradient plasticity Linearized formulation
2
Size effect in a two–phase single crystal laminate Boundary value problem Interface conditions Strain gradient plasticity as a limit case
3
Grain size effect in polycrystalline aggregates Hall–Petch effect Strain localization in ultra–fine grains
SLIDE 5 Enhancing classical continuum mechanics
Within the framework of generalized continuum mechanics, we introduce additional degrees of freedom DOF = {u , ˆ χ
∼
p}
where ˆ χ
∼
p is a generally non compatible plastic microdeformation
tensor. In a first gradient theory, only the first gradients of the DOF intervene in the model GRAD = {F
∼ := 1 ∼ + u ⊗ ∇X,
K
∼ := Curl ˆ
χ
∼
p}
In the context of crystal plasticity, only the curl part of the plastic microdeformation is considered, instead of its full gradient. The following definition of the Curl operator is adopted: K
∼ := Curl ˆ
χ
∼
p :=
∂ ˆ χ
∼
p
∂Xk × e k, Kij := ǫjkl ∂ ˆ χp
ik
∂Xl
The “microcurl” model 5/54
SLIDE 6 Method of virtual power
The method of virtual power is used to derive the balance and boundary conditions, following [Germain, 1973].
- Power density of internal forces; it is a linear form with
respect to the velocity fields and their Eulerian gradients: p(i) = σ
∼ : (˙
u ⊗ ∇x) + s
∼ : ˙
ˆ χ
∼
p + M
∼ : curl ˙
ˆ χ
∼
p,
∀x ∈ V where the conjugate quantities are the Cauchy stress tensor σ
∼, which is symmetric for objectivity reasons, the microstress
tensor, s
∼, and the generalized couple stress tensor M ∼ . The
curl of the microdeformation rate is defined as curl ˙ ˆ χ
∼
p := ǫjkl
∂ ˙ ˆ χp
ik
∂xl e i ⊗ e j = ˙ K
∼ · F ∼
−1
The “microcurl” model 6/54
SLIDE 7 Method of virtual power
The method of virtual power is used to derive the balance and boundary conditions, following [Germain, 1973].
- Power density of contact forces;
p(c) = t · ˙ u + m
∼ : ˙
ˆ χ
∼
p,
∀x ∈ ∂V where t is the usual simple traction vector and m
∼ the double
traction tensor.
The “microcurl” model 7/54
SLIDE 8 Method of virtual power
The method of virtual power is used to derive the balance and boundary conditions, following [Germain, 1973].
- Application of the principle of virtual power, in the absence
- f volume forces and in the static case, for brevity:
−
p(i) dV +
p(c) dS = 0 for all virtual fields ˙ u , ˙ ˆ χ
∼
p, and any subdomain D ⊂ V .
The “microcurl” model 8/54
SLIDE 9 Method of virtual power
The method of virtual power is used to derive the balance and boundary conditions, following [Germain, 1973].
- Application of the principle of virtual power, in the absence
- f volume forces and in the static case, for brevity:
−
p(i) dV +
p(c) dS = 0 for all virtual fields ˙ u , ˙ ˆ χ
∼
p, and any subdomain D ⊂ V .
- By application of Gauss divergence theorem, assuming
sufficient regularity of the fields, this statement expands into:
∂σij ∂xj ˙ ui dV +
∂Mik ∂xl − sij
ˆ χp
ij dV
+
(ti − σijnj) ˙ ui dS+
(mik − ǫjklMijnl) ˙ ˆ χp
ik dS = 0, ∀˙
ui, ∀ ˙ ˆ χp
ij
The “microcurl” model 9/54
SLIDE 10 Method of virtual power
This leads to the two field equations of balance of momentum and generalized balance of moment of momentum: div σ
∼ = 0,
curl M
∼ + s ∼ = 0,
∀x ∈ V and two boundary conditions t = σ
∼ · n ,
m
∼ = M ∼ · ǫ ∼ · n ,
∀x ∈ ∂V the index notation of the latter relation being mij = Mikǫkjlnl.
The “microcurl” model 10/54
SLIDE 11
Plan
1
The “microcurl” model Kinematics and balance equations Constitutive equations Internal stress arising from the model Internal constraint and strain gradient plasticity Linearized formulation
2
Size effect in a two–phase single crystal laminate Boundary value problem Interface conditions Strain gradient plasticity as a limit case
3
Grain size effect in polycrystalline aggregates Hall–Petch effect Strain localization in ultra–fine grains
SLIDE 12 State variables
- The deformation gradient is decomposed into elastic and plastic parts in
the form F
∼ = E ∼ · P ∼
- The elastic strain is defined as
E
∼
e := 1 2(E
∼
T · E
∼ − 1 ∼)
The “microcurl” model 12/54
SLIDE 13 State variables
- The deformation gradient is decomposed into elastic and plastic parts in
the form F
∼ = E ∼ · P ∼
- The elastic strain is defined as
E
∼
e := 1 2(E
∼
T · E
∼ − 1 ∼)
- The microdeformation is linked to the plastic deformation via the
introduction of a relative deformation measure defined as e
∼
p := P
∼
−1 · ˆ
χ
∼
p − 1
∼
It measures the departure of the microdeformation from the plastic deformation, which is associated with a cost in the free energy potential. When e
∼
p ≡ 0, the microdeformation coincides with plastic deformation.
The “microcurl” model 13/54
SLIDE 14 State variables
- The deformation gradient is decomposed into elastic and plastic parts in
the form F
∼ = E ∼ · P ∼
- The elastic strain is defined as
E
∼
e := 1 2(E
∼
T · E
∼ − 1 ∼)
- The microdeformation is linked to the plastic deformation via the
introduction of a relative deformation measure defined as e
∼
p := P
∼
−1 · ˆ
χ
∼
p − 1
∼
It measures the departure of the microdeformation from the plastic deformation, which is associated with a cost in the free energy potential. When e
∼
p ≡ 0, the microdeformation coincides with plastic deformation.
- The state variables are assumed to be the elastic strain, the relative
deformation, the curl of microdeformation and some internal variables, α: STATE := {E
∼
e,
e
∼
p,
K
∼,
α} The specific Helmholtz free energy density, ψ, is a function of these
- variables. In this simple version of the model, the curl of
microdeformation is assumed to contribute entirely to the stored energy. The “microcurl” model 14/54
SLIDE 15 Entropy principle
The dissipation rate density is the difference:
D := p(i) − ρ ˙ ψ ≥ 0
which must be positive according to the second principle of thermodynamics. When the previous strain measures are introduced, the power density of internal forces takes the following form:
p(i) = σ
∼ : ˙
E
∼ · E ∼
−1 + σ
∼ : E ∼ · ˙
P
∼ · P ∼
−1 · E
∼
−1
+ s
∼ : (P ∼ · ˙
e
∼
p + ˙
P
∼ · e ∼
p) + M
∼ : ˙
K
∼ · F ∼
−1
= ρ ρi Π
∼
e : ˙
E
∼
e + ρ
ρi Π
∼
M : ˙
P
∼ · P ∼
−1
+ s
∼ : (P ∼ · ˙
e
∼
p + ˙
P
∼ · e ∼
p) + M
∼ : ˙
K
∼ · F ∼
−1
where Π
∼
e is the second Piola–Kirchhoff stress tensor with respect to the
intermediate configuration and Π
∼
M is the Mandel stress tensor:
Π
∼
e := JeE
∼
−1 · σ
∼ · E ∼
−T,
Π
∼
M := JeE
∼
T · σ
∼ · E ∼
−T = E
∼
T · E
∼ · Π ∼
e
The “microcurl” model 15/54
SLIDE 16 State laws
On the other hand, ρ ˙ ψ = ρ ∂ψ ∂E
∼
e : ˙
E
∼
e + ρ ∂ψ
∂e
∼
p : ˙
e
∼
p + ρ∂ψ
∂K
∼
: ˙ K
∼ + ρ∂ψ
∂α ˙ α We compute JeD = (Π
∼
e − ρi
∂ψ ∂E
∼
e ) : ˙
E
∼
e + (JeP
∼
T · s
∼ − ρi
∂ψ ∂e
∼
p ) : ˙
e
∼
p
+ (JeM
∼ · F ∼
−T − ρi
∂ψ ∂K
∼
) : ˙ K
∼
+ (Π
∼
M + Jes
∼ · ˆ
χ
∼
pT) : ˙
P
∼ · P ∼
−1 − ρi
∂ψ ∂α ˙ α ≥ 0 Assuming that the processes associated with ˙ E
∼
e, ˙
e
∼
p and ˙
K
∼ are
non–dissipative, the state laws are obtained: Π
∼
e = ρi
∂ψ ∂E
∼
e ,
s
∼ = J−1
e P
∼
−T · ρi
∂ψ ∂e
∼
p ,
M
∼ = J−1
e ρi
∂ψ ∂K
∼
· F
∼
T
The “microcurl” model 16/54
SLIDE 17 Evolution laws
The residual dissipation rate is JeD = (Π
∼
M + Jes
∼ · ˆ
χ
∼
pT) : ˙
P
∼ · P ∼
−1 − R ˙
α ≥ 0, with R := ρi ∂ψ ∂α At this stage, a dissipation potential, function of stress measures, Ω(S
∼, R), is introduced in order to formulate the evolution
equations for plastic flow and internal variables: ˙ P
∼ · P ∼
−1 = ∂Ω
∂S
∼
, with S
∼ := Π ∼
M + Jes
∼ · ˆ
χ
∼
pT
˙ α = −∂Ω ∂R where R is the thermodynamic force associated with the internal variable α, and S
∼ is the effective stress conjugate to plastic strain
rate, the driving force for plastic flow.
The “microcurl” model 17/54
SLIDE 18
Plan
1
The “microcurl” model Kinematics and balance equations Constitutive equations Internal stress arising from the model Internal constraint and strain gradient plasticity Linearized formulation
2
Size effect in a two–phase single crystal laminate Boundary value problem Interface conditions Strain gradient plasticity as a limit case
3
Grain size effect in polycrystalline aggregates Hall–Petch effect Strain localization in ultra–fine grains
SLIDE 19 Application to crystal plasticity
In the case of crystal plasticity, a generalized Schmid law is adopted for each slip system s in the form:
f s(S
∼, τ s
c ) = |S
∼ : N ∼
s| − τ s c ≥ 0,
with N
∼
s = l s ⊗ n s
for activation of slip system s with slip direction, l s, and normal to the slip plane, n s. We call N
∼
s the orientation tensor. The critical resolved shear stress
is τ s
c which may be a function of R in the presence of isotropic hardening.
The “microcurl” model 19/54
SLIDE 20 Application to crystal plasticity
In the case of crystal plasticity, a generalized Schmid law is adopted for each slip system s in the form:
f s(S
∼, τ s
c ) = |S
∼ : N ∼
s| − τ s c ≥ 0,
with N
∼
s = l s ⊗ n s
for activation of slip system s with slip direction, l s, and normal to the slip plane, n s. We call N
∼
s the orientation tensor. The critical resolved shear stress
is τ s
c which may be a function of R in the presence of isotropic hardening.
The generalized resolved shear stress can be decomposed into two contributions:
S
∼ : N ∼
s = τ s−xs,
with τ s = Π
∼
M : N
∼
s
and xs = −s
∼·ˆ
χ
∼
pT : N
∼
s
The usual resolved shear stress is τ s whereas xs can be interpreted as an internal stress or back–stress leading to kinematic hardening. [Steinmann, 1996] The back–stress component is induced by the microstress s
∼ or, equivalently, by
the curl of the generalized couple stress tensor, M
∼ , via the balance equation
xs = curl M
∼ · ˆ
χ
∼
pT : N
∼
s
The “microcurl” model 20/54
SLIDE 21 Application to crystal plasticity
In the case of crystal plasticity, a generalized Schmid law is adopted for each slip system s in the form: f s(S
∼, τ s
c ) = |S
∼ : N ∼
s| − τ s c ≥ 0,
with N
∼
s = l s ⊗ n s
for activation of slip system s with slip direction, l s, and normal to the slip plane, n s. The critical resolved shear stress is τ s
c which
may be a function of R in the presence of isotropic hardening. The kinematics of plastic slip follows from the choice of a dissipation potential, Ω(f s), that depends on the stress variables through the yield function itself, f s: ˙ F
∼
p·F
∼
p−1 = N
∂Ω ∂f s ∂f s ∂S
∼
=
N
˙ γs N
∼
s,
with ˙ γs = ∂Ω ∂f s sign(S
∼ : N ∼
s)
The “microcurl” model 21/54
SLIDE 22
Plan
1
The “microcurl” model Kinematics and balance equations Constitutive equations Internal stress arising from the model Internal constraint and strain gradient plasticity Linearized formulation
2
Size effect in a two–phase single crystal laminate Boundary value problem Interface conditions Strain gradient plasticity as a limit case
3
Grain size effect in polycrystalline aggregates Hall–Petch effect Strain localization in ultra–fine grains
SLIDE 23 From micromorphic to strain gradient plasticity
If the following internal constraint is enforced: e
∼
p ≡ 0
⇐ ⇒ ˆ χ
∼
p ≡ P
∼
the curl part of the plastic microdeformation is directly related to the dislocation density densor: K
∼ := Curl ˆ
χ
∼
p ≡ Curl P
∼ = Jα ∼ · F ∼
−T
The microcurl theory reduces to strain gradient plasticity according to [Gurtin, 2002]. As a result the microcurl model incorporates, as wanted, a dependence of material behaviour on the dislocation density tensor.
The “microcurl” model 23/54
SLIDE 24
Plan
1
The “microcurl” model Kinematics and balance equations Constitutive equations Internal stress arising from the model Internal constraint and strain gradient plasticity Linearized formulation
2
Size effect in a two–phase single crystal laminate Boundary value problem Interface conditions Strain gradient plasticity as a limit case
3
Grain size effect in polycrystalline aggregates Hall–Petch effect Strain localization in ultra–fine grains
SLIDE 25 Small deformation microcurl model
When deformations and rotations remain sufficiently small, the previous equations can be linearized as follows: F
∼ = 1 ∼ + H ∼ = 1 ∼ + H ∼
e + H
∼
p,
H
∼
e = ε
∼
e + ωe,
H
∼
p = ε
∼
p + ωp
where ε
∼
e, ω
∼
e (resp. ε
∼
p, ω
∼
p) are practically equal to the symmetric
and skew–symmetric parts of E
∼ − 1 ∼ (resp. P ∼ − 1 ∼).
When microdeformation is small, the relative deformation is linearized as e
∼
p = (1
∼ + H ∼
p)−1 · (1
∼ + χ ∼
p) − 1
∼ ≃ χ ∼
p − H
∼
p,
with χ
∼
p = ˆ
χ
∼
p − 1
∼
The “microcurl” model 25/54
SLIDE 26 Small deformation microcurl model
When deformations and rotations remain sufficiently small, the previous equations can be linearized as follows: F
∼ = 1 ∼ + H ∼ = 1 ∼ + H ∼
e + H
∼
p,
H
∼
e = ε
∼
e + ωe,
H
∼
p = ε
∼
p + ωp
where ε
∼
e, ω
∼
e (resp. ε
∼
p, ω
∼
p) are practically equal to the symmetric
and skew–symmetric parts of E
∼ − 1 ∼ (resp. P ∼ − 1 ∼).
When microdeformation is small, the relative deformation is linearized as e
∼
p = (1
∼ + H ∼
p)−1 · (1
∼ + χ ∼
p) − 1
∼ ≃ χ ∼
p − H
∼
p,
with χ
∼
p = ˆ
χ
∼
p − 1
∼
When linearized, the state laws become: σ
∼ = ρ ∂ψ
∂ε
∼
e ,
s
∼ = ρ ∂ψ
∂e
∼
p ,
M
∼ = ρ∂ψ
∂K
∼
The evolution equations read then: ˙ ε
∼
p =
∂Ω ∂(σ
∼ + s ∼),
˙ α = −∂Ω ∂R
The “microcurl” model 26/54
SLIDE 27 Small deformation microcurl model
We adopt the most simple case of a quadratic free energy potential: ρψ(ε
∼
e, e
∼
p, K
∼) = 1
2ε
∼
e : C
≈ : ε ∼
e + 1
2Hχe
∼
p : e
∼
p + 1
2AK
∼ : K ∼
The usual four–rank tensor of elastic moduli is denoted by C
≈. The
higher order moduli have been limited to only two additional parameters: Hχ (unit MPa) and A (unit MPa.mm2). It follows that: σ
∼ = C ≈ : ε ∼
e,
s
∼ = Hχe ∼
p,
M
∼ = AK ∼
Large values of Hχ ensure that e
∼
p remains small so that χ
∼
p remains
close to H
∼
p and K
∼ is close to the dislocation density tensor:
curl χ
∼
p ≡ curl H
∼
p = α
∼
The “microcurl” model 27/54
SLIDE 28 Small deformation microcurl model
We adopt the most simple case of a quadratic free energy potential: ρψ(ε
∼
e, e
∼
p, K
∼) = 1
2ε
∼
e : C
≈ : ε ∼
e + 1
2Hχe
∼
p : e
∼
p + 1
2AK
∼ : K ∼
The usual four–rank tensor of elastic moduli is denoted by C
≈. The
higher order moduli have been limited to only two additional parameters: Hχ (unit MPa) and A (unit MPa.mm2). It follows that: σ
∼ = C ≈ : ε ∼
e,
s
∼ = Hχe ∼
p,
M
∼ = AK ∼
The yield condition for each slip system becomes: f s = |τ s − xs| − τ s
c
with xs = −s
∼ : P ∼
s = (curl M
∼ ) : P ∼
s = A(curl curl χ
∼
p) : N
∼
s
The “microcurl” model 28/54
SLIDE 29
Plan
1
The “microcurl” model Kinematics and balance equations Constitutive equations Internal stress arising from the model Internal constraint and strain gradient plasticity Linearized formulation
2
Size effect in a two–phase single crystal laminate Boundary value problem Interface conditions Strain gradient plasticity as a limit case
3
Grain size effect in polycrystalline aggregates Hall–Petch effect Strain localization in ultra–fine grains
SLIDE 30
Plan
1
The “microcurl” model Kinematics and balance equations Constitutive equations Internal stress arising from the model Internal constraint and strain gradient plasticity Linearized formulation
2
Size effect in a two–phase single crystal laminate Boundary value problem Interface conditions Strain gradient plasticity as a limit case
3
Grain size effect in polycrystalline aggregates Hall–Petch effect Strain localization in ultra–fine grains
SLIDE 31 Simple shear of a two–phase laminate
γ h 2 h 2 γ l 1 2 O n s
(s) (h+) (h−)
The microstructure is composed of a hard elastic phase (h) and a soft elasto–plastic phase (s) where one slip system with slip direction normal to the interface between (h) and (s) is considered. A mean simple glide ¯ γ is applied in the crystal slip direction of the phase (s). We consider displacement and microdeformation fields
u1 = ¯ γx2, u2(x1), u3 = 0, χp
12(x1),
χp
21(x1)
within the context of small deformation theory.
Size effect in a two–phase single crystal laminate 31/54
SLIDE 32 Simple shear of a two–phase laminate
u1 = ¯ γx2, u2(x1), u3 = 0, χp
12(x1),
χp
21(x1)
ˆ H
∼
˜ = 2 4 ¯ γ u2,1 3 5 ˆ H
∼
p˜
= 2 4 γ 3 5 ˆ H
∼
e˜
= 2 4 ¯ γ − γ u2,1 3 5 h χ
∼
pi
= 2 4 χp
12(x1)
χp
21(x1)
3 5 h curl χ
∼
pi
= 2 4 −χp
12,1
3 5 Size effect in a two–phase single crystal laminate 32/54
SLIDE 33 Simple shear of a two–phase laminate
u1 = ¯ γx2, u2(x1), u3 = 0, χp
12(x1),
χp
21(x1)
ˆ H
∼
˜ = 2 4 ¯ γ u2,1 3 5 ˆ H
∼
p˜
= 2 4 γ 3 5 ˆ H
∼
e˜
= 2 4 ¯ γ − γ u2,1 3 5 h χ
∼
pi
= 2 4 χp
12(x1)
χp
21(x1)
3 5 h curl χ
∼
pi
= 2 4 −χp
12,1
3 5 The resulting stress tensors are: ˆ σ
∼
˜ = µ 2 4 ¯ γ − γ + u2,1 ¯ γ − γ + u2,1 3 5 ˆ s
∼
˜ = −Hχ 2 4 γ − χp
12
−χp
21
3 5 ˆ M
∼
˜ = 2 4 −Aχp
12,1
3 5 ˆ curl M
∼
˜ = 2 4 −Aχp
12,11
3 5 These forms of matrices are valid for both phases, except that γ ≡ 0 in the hard elastic phase. Each phase possesses its own material parameters, Hχ and A, the shear modulus, µ, being assumed for simplicity to be identical in both phases. Size effect in a two–phase single crystal laminate 33/54
SLIDE 34 Simple shear of a two–phase laminate
The balance equation, s
∼ = −curl M ∼ , gives χp
21 = 0 and the plastic slip:
γ = χp
12 − A
Hχ χp
12,11
In the soft phase, the plasticity criterion stipulates that σ12 + s12 = τc + Hγcum where H is a linear hardening modulus considered in this phase. We obtain the second order differential equation for the microdeformation variable in the soft phase, χps
12,
1 ωs2 χps
12,11 − χps 12 = τc − σ12
H , with ωs = s Hs
χH
As ` Hs
χ + H
´ where 1/ωs is the characteristic length of the soft phase for this boundary value problem. Size effect in a two–phase single crystal laminate 34/54
SLIDE 35
Plan
1
The “microcurl” model Kinematics and balance equations Constitutive equations Internal stress arising from the model Internal constraint and strain gradient plasticity Linearized formulation
2
Size effect in a two–phase single crystal laminate Boundary value problem Interface conditions Strain gradient plasticity as a limit case
3
Grain size effect in polycrystalline aggregates Hall–Petch effect Strain localization in ultra–fine grains
SLIDE 36 Simple shear of a two–phase laminate
The force stress balance equation requires σ12 to be uniform. It follows that the non–homogeneous part of the differential equation is constant and then the hyperbolic profile of χps
12 takes the form:
χps
12 = C s cosh (ωsx) + D
where C s and D are constants to be determined. Symmetry conditions (χps
12(−s/2) = χps 12(s/2)) have been taken into account.
In the elastic phase, where the plastic slip vanishes, an hyperbolic profile of the microdeformation variable, χph
12, is also obtained:
χph
12 = C h cosh
„ ωh „ x ± s + h 2 «« , with ωh = s Hh
χ
Ah where, again, C h is a constant to be determined and symmetry conditions have been taken into account. It is remarkable that the plastic microvariable, χph
12,
does not vanish in the elastic phase, close to the interfaces, although no plastic deformation takes place. Size effect in a two–phase single crystal laminate 36/54
SLIDE 37 Simple shear of a two–phase laminate
The coefficients C s, D and C h can be identified using the interface and periodicity conditions:
12 at x = ±s/2:
C s cosh “ ωs s 2 ” + D = C h cosh „ ωh h 2 « (1)
- Continuity of the double traction, m12 = −M13 at x = ±s/2:
AsωsC s sinh “ ωs s 2 ” = −AhωhC h sinh „ ωh h 2 « (2) Size effect in a two–phase single crystal laminate 37/54
SLIDE 38 Simple shear of a two–phase laminate
- Periodicity of displacement component u2. We have the constant stress
component σ12 = µ(¯ γ − γ + u2,1) whose value is obtained from the plasticity criterion in the soft phase: σ12 = τc + Hγcum − Asχps
12,11
us
2,1 = σ12
µ − ¯ γ + γ = τc µ − ¯ γ + Asωs2C s H cosh (ωsx) + H + µ µ D in the soft phase and uh
2,1 = σ12
µ − ¯ γ = τc µ − ¯ γ + H µ D in the hard phase. The average on the whole structure, Z (s+h)/2
−(s+h)/2
u2,1 dx = 0 must vanish for periodicity reasons and gives „τc µ − ¯ γ « (s + h) + 2AsωsC s H sinh “ ωs s 2 ” + H (s + h) + µs µ D = 0 (3) Size effect in a two–phase single crystal laminate 38/54
SLIDE 39 Plastic microdeformation and plastic slip
µ = 30000 MPa s = 0.7µm h = 0.3µm As = Ah = 100 MPa.µm2 Hs
χ = Hh χ = 100000 MPa
τc = 20 MPa lc = q
A µ ≃ 60 nm
0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018
0.2 0.4
x
γ(x)
Size effect in a two–phase single crystal laminate 39/54
SLIDE 40 Plastic microdeformation and plastic slip
µ = 30000 MPa s = 0.7µm h = 0.3µm As = Ah = 100 MPa.µm2 Hs
χ = Hh χ = 100000 MPa
τc = 20 MPa lc = q
A µ ≃ 60 nm
0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018
0.2 0.4
x
γ(x) γ(x) FE
Size effect in a two–phase single crystal laminate 39/54
SLIDE 41 Plastic microdeformation and plastic slip
µ = 30000 MPa s = 0.7µm h = 0.3µm As = Ah = 100 MPa.µm2 Hs
χ = Hh χ = 100000 MPa
τc = 20 MPa lc = q
A µ ≃ 60 nm
0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018
0.2 0.4
x
χp
12(x)
γ(x)
Size effect in a two–phase single crystal laminate 39/54
SLIDE 42 Plastic microdeformation and plastic slip
µ = 30000 MPa s = 0.7µm h = 0.3µm As = Ah = 100 MPa.µm2 Hs
χ = Hh χ = 100000 MPa
τc = 20 MPa lc = q
A µ ≃ 60 nm
0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018
0.2 0.4
x
χp
12(x)
γ(x) χp
12(x) FE
γ(x) FE
Size effect in a two–phase single crystal laminate 39/54
SLIDE 43 Overall cyclic response
10 20 30 40 50
0.005 0.01 <σ12> <2ε12> Microcurl Classic
Size effect in a two–phase single crystal laminate 40/54
SLIDE 44 Size effects
1e+02 1e+03 1e+04 1e+05 1e-06 1e-05 1e-04 1e-03 1e-02 1e-01 1e+00
Σ12|0.2 (MPa) l (mm)
curl H p (Hχ→ ∞) microcurl (Hχ=109 MPa) microcurl (Hχ=108 MPa) microcurl (Hχ=107 MPa) microcurl (Hχ=106 MPa)
Size effect in a two–phase single crystal laminate 41/54
SLIDE 45 Size effects
Flow stress, Σ Microstructural length scale, l
∆Σ
Effect of H
c
l Effect of H Scaling law l
n n
Effect of A
χ χ
Hχ and A can be calibrated to account for a scaling law 1/ln, 0 < n < 2 [Cordero et al., 2010]
Size effect in a two–phase single crystal laminate 42/54
SLIDE 46
Plan
1
The “microcurl” model Kinematics and balance equations Constitutive equations Internal stress arising from the model Internal constraint and strain gradient plasticity Linearized formulation
2
Size effect in a two–phase single crystal laminate Boundary value problem Interface conditions Strain gradient plasticity as a limit case
3
Grain size effect in polycrystalline aggregates Hall–Petch effect Strain localization in ultra–fine grains
SLIDE 47 Strain gradient plasticity as a limit case
- Size effect according to SGP
lim
Hχ→∞ Σ12 = τc + 12Asγ
f 3
s l2
physical meaning of a 1/l2 scaling law?
Size effect in a two–phase single crystal laminate 44/54
SLIDE 48
Plan
1
The “microcurl” model Kinematics and balance equations Constitutive equations Internal stress arising from the model Internal constraint and strain gradient plasticity Linearized formulation
2
Size effect in a two–phase single crystal laminate Boundary value problem Interface conditions Strain gradient plasticity as a limit case
3
Grain size effect in polycrystalline aggregates Hall–Petch effect Strain localization in ultra–fine grains
SLIDE 49 Boundary value problem for micromorphic crystals
- Shear test
- Periodic grains with random
- rientations
- Periodic mesh
- Periodic boundary
conditions: u (x) = E
∼.x + v (x), v
periodic χ
∼
p(x) periodic
- 2 slip systems / grain
- The grain size d is ranging
from tens of nanometers to hundreds of microns.
l2
d
2
n n1 l1
E = 70000 MPa ν = 0.3 A = 0.01MPa.mm2 Hχ = 106 MPa Intrinsic length scale: lc =
Hχ = 0.1µm
Grain size effect in polycrystalline aggregates 46/54
SLIDE 50 Boundary value problem for micromorphic crystals
d d
φ θ φ x y Y X l l n n1
1 2 2
Grain size effect in polycrystalline aggregates 47/54
SLIDE 51 Boundary value problem for micromorphic crystals
50 100 150
0.005 0.01
Σ12 [MPa] E12
Grain size effect in polycrystalline aggregates 48/54
SLIDE 52
Plan
1
The “microcurl” model Kinematics and balance equations Constitutive equations Internal stress arising from the model Internal constraint and strain gradient plasticity Linearized formulation
2
Size effect in a two–phase single crystal laminate Boundary value problem Interface conditions Strain gradient plasticity as a limit case
3
Grain size effect in polycrystalline aggregates Hall–Petch effect Strain localization in ultra–fine grains
SLIDE 53 Grain size effect
20 40 60 80 100 120 140 160 0.01 0.02 0.03 0.04 0.05
Σ12 [MPa] E12 d = 0.4µm
2.0µm 4.0µm 10.0µm 200.0µm
Grain size effect in polycrystalline aggregates 50/54
SLIDE 54 Grain size effect
10 100 1000 1e-03 1e-02 1e-01 1e+00 1e+01 1e+02 1e+03
Σ12|1% [MPa]
d [µm]
m = -0.78 (set (e)) m = -0.67 (f) m = -0.50 (g) m
Grain size effect in polycrystalline aggregates 51/54
SLIDE 55
Plan
1
The “microcurl” model Kinematics and balance equations Constitutive equations Internal stress arising from the model Internal constraint and strain gradient plasticity Linearized formulation
2
Size effect in a two–phase single crystal laminate Boundary value problem Interface conditions Strain gradient plasticity as a limit case
3
Grain size effect in polycrystalline aggregates Hall–Petch effect Strain localization in ultra–fine grains
SLIDE 56 Plastic strain field
d = 200µm d = 20µm d = 10µm d = 4µm d = 2µm d = 1µm
0.004 0.008 0.012 0.015 0.019 0.023 0.027 0.031 0.035 0.039
Grain size effect in polycrystalline aggregates 53/54
SLIDE 57 Dislocation density tensor field
d = 200µm d = 20µm d = 10µm d = 4µm d = 2µm d = 1µm
0.5 1.0 1.5 1.9 2.4 2.9 3.4 3.9 4.4 4.9 [mm ]
Grain size effect in polycrystalline aggregates 54/54
SLIDE 58
Cordero N.M., Gaubert A., Forest S., Busso E., Gallerneau F., and Kruch S. (2010). Size effects in generalised continuum crystal plasticity for two–phase laminates. Journal of the Mechanics and Physics of Solids, vol. 58, pp 1963–1994. Forest S. (2009). The micromorphic approach for gradient elasticity, viscoplasticity and damage. ASCE Journal of Engineering Mechanics, vol. 135, pp 117–131. Forest S. and Sievert R. (2003). Elastoviscoplastic constitutive frameworks for generalized continua. Acta Mechanica, vol. 160, pp 71–111. Forest S. and Sievert R. (2006). Nonlinear microstrain theories.
Grain size effect in polycrystalline aggregates 54/54
SLIDE 59
International Journal of Solids and Structures, vol. 43, pp 7224–7245. Germain P. (1973). The method of virtual power in continuum mechanics. Part 2 : Microstructure. SIAM J. Appl. Math., vol. 25, pp 556–575. Gurtin M.E. (2002). A gradient theory of single–crystal viscoplasticity that accounts for geometrically necessary dislocations. Journal of the Mechanics and Physics of Solids, vol. 50, pp 5–32. Gurtin M.E. and Anand L. (2009). Thermodynamics applied to gradient theories involving the accumulated plastic strain: The theories of Aifantis and Fleck & Hutchinson and their generalization. Journal of the Mechanics and Physics of Solids, vol. 57, pp 405–421.
Grain size effect in polycrystalline aggregates 54/54
SLIDE 60
Sedl´ aˇ cek R. and Forest S. (2000). Non-local plasticity at microscale : A dislocation-based model and a Cosserat model. physica status solidi (b), vol. 221, pp 583–596. Steinmann P. (1996). Views on multiplicative elastoplasticity and the continuum theory of dislocations. International Journal of Engineering Science, vol. 34, pp 1717–1735.
Grain size effect in polycrystalline aggregates 54/54
