The micromorphic approach to plasticity and phase transformation - PowerPoint PPT Presentation

The micromorphic approach to plasticity and phase transformation Samuel Forest Mines ParisTech / CNRS Centre des Mat´ eriaux/UMR 7633 BP 87, 91003 Evry, France Samuel.Forest@mines-paristech.fr

Objectives The objectives of this presentation are two-fold: • propose a systematic procedure to extend standard elastoviscoplasticity models to include: ⋆ size effects in the hardening behaviour of materials (grain size effects...) ⋆ regularization properties in the softening behaviour (strain localization...) • unify the “zoology” of generalized continuum models: ⋆ “Classical” generalized continua: Cosserat, second gradient, micromorphic media (Mindlin, 1964; Eringen and Suhubi, 1964; Mindlin and Eshel, 1968) ⋆ strain gradient plasticity, “implicit gradient approach”... (Aifantis, 1987; Fleck and Hutchinson, 2001; Gurtin, 2003; Engelen et al., 2003) • establish links between generalized continuum mechanics and phase field approaches 2/31

Plan The micromorphic approach to plasticity 1 Continuum thermomechanics Full micromorphic and microstrain theories Microstrain gradient plasticity 2 Gradient of plastic microstrain Consistency condition Anisothermal strain gradient plasticity Internal constraint in the micromorphic approach 3 Microdiffusion and phase field approach 4

Plan The micromorphic approach to plasticity 1 Continuum thermomechanics Full micromorphic and microstrain theories Microstrain gradient plasticity 2 Gradient of plastic microstrain Consistency condition Anisothermal strain gradient plasticity Internal constraint in the micromorphic approach 3 Microdiffusion and phase field approach 4

Plan The micromorphic approach to plasticity 1 Continuum thermomechanics Full micromorphic and microstrain theories Microstrain gradient plasticity 2 Gradient of plastic microstrain Consistency condition Anisothermal strain gradient plasticity Internal constraint in the micromorphic approach 3 Microdiffusion and phase field approach 4

State space • observable and controllable variables (temperature, strain...) { T , ∼ } ε • internal degrees of freedom (controllable variables that account for some aspects of the microstructre) { α, ∇ α } they have associated stresses and α or its associated force can be prescribed at the boundary • internal variables are the remembrance of internal degrees of freedom; they cannot be controlled { α } The micromorphic approach to plasticity 6/31

The micromorphic approach (1) • Start from an initial classical elastoviscoplastic model with internal variables DOF 0 = { u } , STATE 0 = { F ∼ , T , α } • Select one variable φ ∈ STATE 0 and introduce the associated micromorphic variable χ φ as an additional degree of freedom and, possibly, state variable: χ φ } , χ φ, ∇ χ φ } DOF = { u , STATE = { F ∼ , T , α, • Extend the power of internal forces � P ( i ) ( v ⋆ , χ ˙ p ( i ) ( v ⋆ , χ ˙ φ ⋆ ) = − φ ⋆ ) dV D p ( i ) ( v ⋆ , χ ˙ ∼ : ∇ v ⋆ + a χ ˙ φ ⋆ + b . ∇ χ ˙ φ ⋆ ) = σ φ ⋆ a , b generalized stresses, microforces (Gurtin, 1996) • Derive additional balance equation and boundary conditions b . n = a c , ∀ x ∈ ∂ Ω div b − a = 0 , ∀ x ∈ Ω , The micromorphic approach to plasticity 7/31

The micromorphic approach (2) • More generally, in the presence of volume generalized forces: div ( b − b e ) − a + a e = 0 , ( b − b e ) . n = a c , ∀ x ∈ ∂ Ω ∀ x ∈ Ω , • Enhance the local balance of energy and the entropy inequality T ) + p ( i ) − q ǫ = p ( i ) − div q + ρ r , − ρ ( ˙ ψ + η ˙ ρ ˙ T . ∇ T ≥ 0 • Consider the constitutive functionals: ˆ e , T , α, χ φ, ∇ χ φ ) , η = ˆ e , T , α, χ φ, ∇ χ φ ) ψ = ψ ( F η ( F ∼ ∼ e , T , α, χ φ, ∇ χ φ ) = ˆ ∼ ( F σ σ ∼ ∼ b = ˆ e , T , α, χ φ, ∇ χ φ ) , e , T , α, χ φ, ∇ χ φ ) a = a ( F ˆ b ( F ∼ ∼ • Derive the state laws (Coleman and Noll, 1963) ∼ = ρ ∂ ˆ eT , η = − ∂ ˆ ∂ T , X = ρ∂ ˆ a = ∂ ˆ ∂ ˆ ψ ψ ψ ψ ψ e . F ∂α, ∂ χ φ, b = σ ∂ F ∂ ∇ χ φ ∼ ∼ α − q D res = W p − X ˙ • Residual dissipation T . ∇ T ≥ 0 The micromorphic approach to plasticity 8/31

The micromorphic approach (3) • Take a simple quadratic potential ∼ , T , α, χ φ, ∇ χ φ ) = ψ 1 ( F ∼ , α, T )+ ψ 2 ( e = φ − χ φ, ∇ χ φ, T ) ψ ( F ρψ 2 = 1 2 H χ ( φ − χ φ ) 2 + 1 2 A ∇ χ φ. ∇ χ φ a = ρ ∂ψ ∂ψ ∂ χ φ = − H χ ( φ − χ φ ) , ∂ ∇ χ φ = A ∇ χ φ b = ρ • Simple form of the partial differential equation (homogeneous, isothermal...) ⇒ χ φ − A ∆ χ φ = φ a = div b = H χ Helmholtz equation with a minus sign and a source term • Coupling modulus H χ and characteristic length of the medium c = A l 2 H χ Stability H χ > 0 , A > 0 The micromorphic approach to plasticity 9/31

Plan The micromorphic approach to plasticity 1 Continuum thermomechanics Full micromorphic and microstrain theories Microstrain gradient plasticity 2 Gradient of plastic microstrain Consistency condition Anisothermal strain gradient plasticity Internal constraint in the micromorphic approach 3 Microdiffusion and phase field approach 4

Micromorphic continuum Micromorphic continuum according to (Eringen and Suhubi, 1964; Mindlin, 1964) • Select variable: χ φ ≡ χ φ ≡ F ∼ , ∼ . p ( i ) = σ . ∼ : ∇ ˙ u + a ∼ : ˙ ∼ + B . ∇ ˙ χ χ ∼ ∼ • application of the principle of (infinitesimal) material frame indifference, (infinitesimal) change of observer of rate w ∼ : ∇ ˙ u = ⇒ ∇ ˙ u + w ∼ , ∼ = ⇒ χ ∼ + w χ ∼ = ⇒ σ ∼ + a ∼ must be symmetric. Rewrite the virtual power: . p ( i ) = σ . ∼ : ˙ ∼ + s ∼ : ( ∇ ˙ u − ˙ ∼ ) + S . ∇ ˙ ε χ χ ∼ ∼ • two balance equations: div ( σ ∼ + s ∼ ) + ρ f = 0 , div S ∼ + s = 0 The micromorphic approach to plasticity 11/31

Microstrain continuum Microstrain continuum after (Forest and Sievert, 2006) • Select T . F χ φ ≡ χ C χ φ ≡ χ ε φ ≡ C ∼ = F ∼ , ∼ , or φ ≡ ε ∼ , ∼ ∼ . p ( i ) = σ ∼ : χ ˙ . χ ε ∼ : ˙ ∼ + a ∼ + b . ∇ ε ε ∼ ∼ • Constitutive coupling between macro and microstrain via the relative strain ∼ − χ ε ∼ := ε e ∼ e , ∼ − χ ε χ ε ψ ( ε T , α, ∼ := ε e ∼ , K ∼ := ∇ ∼ ) ∼ • Take a quadratic potential χ ε ∼ = H χ e a ∼ , b ∼ = A ∇ ∼ • Extra–balance equation c = A χ ε ∼ − l 2 χ ε l 2 c ∆ ∼ = ε ∼ , with H χ example: microfoams (Dillard et al., 2006) The micromorphic approach to plasticity 12/31

Cosserat continuum Cosserat continuum χ φ ≡ χ R φ = R ∼ , ∼ p ( i ) = σ s : ˙ a : (( ∇ ˙ u ) a − ˙ T ) + M ∼ − σ R ∼ . R ∼ : ˙ ε κ ∼ ∼ ∼ ∼ The micromorphic approach to plasticity 13/31

Plan The micromorphic approach to plasticity 1 Continuum thermomechanics Full micromorphic and microstrain theories Microstrain gradient plasticity 2 Gradient of plastic microstrain Consistency condition Anisothermal strain gradient plasticity Internal constraint in the micromorphic approach 3 Microdiffusion and phase field approach 4

Plan The micromorphic approach to plasticity 1 Continuum thermomechanics Full micromorphic and microstrain theories Microstrain gradient plasticity 2 Gradient of plastic microstrain Consistency condition Anisothermal strain gradient plasticity Internal constraint in the micromorphic approach 3 Microdiffusion and phase field approach 4

General scalar microstrain gradient plasticity • Classical and generalized plasticity e , DOF 0 = { u } STATE 0 = { ε p , α } ∼ χ φ ≡ χ p φ ≡ p , χ p } e , χ p , χ p } DOF = { u , STATE = { ε p , α, ∇ ∼ • Extra balance equation p ( i ) = σ ∼ + a χ ˙ p + b . ∇ χ ˙ p ( c ) = t . ˙ u + a c χ ˙ ∼ : ˙ p , p ε b . n = a c , div b − a = 0 , ∀ x ∈ Ω , ∀ x ∈ ∂ Ω • State laws e + ε p ∼ = ε ε ∼ ∼ ∼ = ρ ∂ψ R = ρ∂ψ X = ρ∂ψ a = ρ ∂ψ ∂ψ e , ∂ p , ∂α, ∂ χ p , b = ρ σ ∂ ∇ χ p ∂ ε ∼ p − R ˙ D res = σ • Evolution laws ∼ : ˙ p − X ˙ α ≥ 0 ε ∼ λ ∂ f λ ∂ f λ ∂ f p = ˙ p = − ˙ α = − ˙ ˙ , ˙ ∂ R , ˙ ε ∂ σ ∂ X ∼ ∼ Microstrain gradient plasticity 16/31

Simplified scalar microstrain gradient plasticity • Quadratic free energy potential e , p , χ p , ∇ χ p ) = 1 e +1 2 Hp 2 +1 2 H χ ( p − χ p ) 2 +1 e : Λ 2 ∇ χ p . A ∼ . ∇ χ p ρψ ( ε ≈ : ε 2 ε ∼ ∼ ∼ • Constitutive equations e , a = − H χ ( p − χ p ) , b = A ∼ . ∇ χ p , R = ( H + H χ ) p − H χχ p ∼ = Λ ≈ : ε σ ∼ • Substitution of constitutive equation into extra balance equation χ p − 1 ∼ . ∇ χ p ) = p div ( A H χ • Homogeneous and isotropic materials A ∼ = A 1 ∼ χ p − A ∆ χ p = p , ∇ χ p . n = a c b . c . H χ same partial differential equation as in the implicit gradient–enhanced elastoplasticity with a c = 0 (Engelen et al., 2003) Microstrain gradient plasticity 17/31