Micromorphic Crystal Plasticity Samuel Forest, Nicolas Cordero, Ana - PowerPoint PPT Presentation

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

Plan The “microcurl” model 1 Kinematics and balance equations Constitutive equations Internal stress arising from the model Internal constraint and strain gradient plasticity Linearized formulation Size effect in a two–phase single crystal laminate 2 Boundary value problem Interface conditions Strain gradient plasticity as a limit case Grain size effect in polycrystalline aggregates 3 Hall–Petch effect Strain localization in ultra–fine grains

Plan The “microcurl” model 1 Kinematics and balance equations Constitutive equations Internal stress arising from the model Internal constraint and strain gradient plasticity Linearized formulation Size effect in a two–phase single crystal laminate 2 Boundary value problem Interface conditions Strain gradient plasticity as a limit case Grain size effect in polycrystalline aggregates 3 Hall–Petch effect Strain localization in ultra–fine grains

Plan The “microcurl” model 1 Kinematics and balance equations Constitutive equations Internal stress arising from the model Internal constraint and strain gradient plasticity Linearized formulation Size effect in a two–phase single crystal laminate 2 Boundary value problem Interface conditions Strain gradient plasticity as a limit case Grain size effect in polycrystalline aggregates 3 Hall–Petch effect Strain localization in ultra–fine grains

Enhancing classical continuum mechanics Within the framework of generalized continuum mechanics, we introduce additional degrees of freedom p } DOF = { u , ˆ χ ∼ p is a generally non compatible plastic microdeformation where ˆ χ ∼ tensor. In a first gradient theory, only the first gradients of the DOF intervene in the model p } GRAD = { F ∼ := 1 ∼ + u ⊗ ∇ X , K ∼ := Curl ˆ χ ∼ 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: p χ p ∂ ˆ ∂ ˆ χ p := ik K ∼ := Curl ˆ × e k , K ij := ǫ jkl ∼ χ ∂ X k ∂ X l ∼ The “microcurl” model 5/54

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 + M p ( i ) = σ p , ∼ : ˙ ∼ : curl ˙ ∼ : ( ˙ u ⊗ ∇ x ) + s ˆ ˆ ∀ x ∈ V χ χ ∼ ∼ where the conjugate quantities are the Cauchy stress tensor ∼ , which is symmetric for objectivity reasons, the microstress σ tensor, s ∼ , and the generalized couple stress tensor M ∼ . The curl of the microdeformation rate is defined as ∂ ˙ χ p ˆ p := ǫ jkl − 1 curl ˙ ik e i ⊗ e j = ˙ ˆ K ∼ · F χ ∂ x l ∼ ∼ The “microcurl” model 6/54

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 · ˙ p , ∼ : ˙ u + m ∀ x ∈ ∂ V ˆ χ ∼ where t is the usual simple traction vector and m ∼ the double traction tensor. The “microcurl” model 7/54

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 of volume forces and in the static case, for brevity: � � p ( i ) dV + p ( c ) dS = 0 − ∂ D D p , and any subdomain D ⊂ V . u , ˙ for all virtual fields ˙ ˆ χ ∼ The “microcurl” model 8/54

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 of volume forces and in the static case, for brevity: � � p ( i ) dV + p ( c ) dS = 0 − ∂ D D p , and any subdomain D ⊂ V . u , ˙ for all virtual fields ˙ ˆ χ ∼ • By application of Gauss divergence theorem, assuming sufficient regularity of the fields, this statement expands into: � ∂σ ij � � ∂ M ik � χ p ˙ u i dV + ˙ ǫ kjl − s ij ˆ ij dV ∂ x j ∂ x l V V � � ( m ik − ǫ jkl M ij n l ) ˙ χ p u i , ∀ ˙ χ p + ( t i − σ ij n j ) ˙ u i dS + ˆ ik dS = 0 , ∀ ˙ ˆ ij ∂ V ∂ V The “microcurl” model 9/54

Method of virtual power This leads to the two field equations of balance of momentum and generalized balance of moment of momentum: ∼ = 0 , ∼ + s ∼ = 0 , ∀ x ∈ V div σ curl M and two boundary conditions t = σ ∼ · n , ∼ = M ∼ · ǫ ∼ · n , ∀ x ∈ ∂ V m the index notation of the latter relation being m ij = M ik ǫ kjl n l . The “microcurl” model 10/54

Plan The “microcurl” model 1 Kinematics and balance equations Constitutive equations Internal stress arising from the model Internal constraint and strain gradient plasticity Linearized formulation Size effect in a two–phase single crystal laminate 2 Boundary value problem Interface conditions Strain gradient plasticity as a limit case Grain size effect in polycrystalline aggregates 3 Hall–Petch effect Strain localization in ultra–fine grains

State variables • The deformation gradient is decomposed into elastic and plastic parts in the form ∼ = E ∼ · P F ∼ e := 1 T · E • The elastic strain is defined as 2 ( E ∼ − 1 ∼ ) E ∼ ∼ The “microcurl” model 12/54

State variables • The deformation gradient is decomposed into elastic and plastic parts in the form ∼ = E ∼ · P F ∼ e := 1 T · E • The elastic strain is defined as 2 ( E ∼ − 1 ∼ ) E ∼ ∼ • The microdeformation is linked to the plastic deformation via the introduction of a relative deformation measure defined as p := P − 1 · ˆ p − 1 e χ ∼ ∼ ∼ ∼ It measures the departure of the microdeformation from the plastic deformation, which is associated with a cost in the free energy potential. p ≡ 0, the microdeformation coincides with plastic deformation. When e ∼ The “microcurl” model 13/54

State variables • The deformation gradient is decomposed into elastic and plastic parts in the form ∼ = E ∼ · P F ∼ e := 1 T · E • The elastic strain is defined as 2 ( E ∼ − 1 ∼ ) E ∼ ∼ • The microdeformation is linked to the plastic deformation via the introduction of a relative deformation measure defined as p := P − 1 · ˆ p − 1 e χ ∼ ∼ ∼ ∼ It measures the departure of the microdeformation from the plastic deformation, which is associated with a cost in the free energy potential. p ≡ 0, the microdeformation coincides with plastic deformation. When e ∼ • The state variables are assumed to be the elastic strain, the relative deformation, the curl of microdeformation and some internal variables, α : e , p , STATE := { E e 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

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: − 1 + σ − 1 · E p ( i ) ∼ : ˙ ∼ · ˙ − 1 = E ∼ · E ∼ : E P ∼ · P σ ∼ ∼ ∼ p + ˙ p ) + M ∼ : ˙ − 1 + ∼ : ( P ∼ · ˙ ∼ · e ∼ · F s e P K ∼ ∼ ∼ ρ e + ρ e : ˙ M : ˙ − 1 = Π E Π P ∼ · P ρ i ρ i ∼ ∼ ∼ ∼ p + ˙ ∼ : ˙ − 1 p ) + M + ∼ : ( P s ∼ · ˙ e P ∼ · e K ∼ · F ∼ ∼ ∼ e is the second Piola–Kirchhoff stress tensor with respect to the where Π ∼ M is the Mandel stress tensor: intermediate configuration and Π ∼ e := J e E − 1 · σ M := J e E T · σ − T = E T · E − T , e ∼ · E ∼ · E ∼ · Π Π Π ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ The “microcurl” model 15/54

State laws On the other hand, ψ = ρ ∂ψ e + ρ ∂ψ p + ρ∂ψ ∼ + ρ∂ψ ρ ˙ e : ˙ : ˙ E p : ˙ e K ∂α ˙ α ∂ E ∂ e ∂ K ∼ ∼ ∼ ∼ ∼ We compute ∂ψ e + ( J e P ∂ψ e − ρ i T · s e ) : ˙ p = ( Π ∼ − ρ i p ) : ˙ J e D E e ∂ E ∂ e ∼ ∼ ∼ ∼ ∼ ∼ ∂ψ − T − ρ i ) : ˙ + ( J e M ∼ · F K ∂ K ∼ ∼ ∼ ∂ψ M + J e s − 1 − ρ i pT ) : ˙ + ( Π ∼ · ˆ ∼ · P ∂α ˙ α ≥ 0 P χ ∼ ∼ ∼ e , ˙ p and ˙ Assuming that the processes associated with ˙ ∼ are E e K ∼ ∼ non–dissipative, the state laws are obtained: ∂ψ ∂ψ ∂ψ e = ρ i − T · ρ i ∼ = J − 1 ∼ = J − 1 T Π e , s e P p , M e ρ i · F ∂ E ∂ e ∂ K ∼ ∼ ∼ ∼ ∼ ∼ The “microcurl” model 16/54