Strain gradient and micromorphic plasticity

Transcription

1 Strain gradient and micromorphic plasticity Samuel Forest Mines ParisTech / CNRS Centre des Matériaux/UMR 7633 BP 87, Evry, France Samuel.Forest@mines-paristech.fr

2 Plan 1 Introduction Mechanics of generalized continua 2 Strain gradient plasticity theory Method of virtual power Constitutive equations Shearing of a laminate microstructure 3 Micromorphic plasticity theory Balance and constitutive equations Rate independent plasticity Shearing of a laminate microstructure 4 Discussion

3 Cartesian basis Notations A = A i e i, A = A ij e i e j, A = A = A = A ijk e i e j e k tensor products a b = a i b j e i e j, A B = A ij B kl e i e j e k e l contractions A B = A ik B jl e i e j e k e l nabla operators A B = A i B i, A : B = A ij B ij, A. B = A ijk B ijk x =,i e i, X =,I E I u = u i,j e i e j, σ. = σ ij,j e i 3/53

4 Plan 1 Introduction Mechanics of generalized continua 2 Strain gradient plasticity theory Method of virtual power Constitutive equations Shearing of a laminate microstructure 3 Micromorphic plasticity theory Balance and constitutive equations Rate independent plasticity Shearing of a laminate microstructure 4 Discussion

5 Plan 1 Introduction Mechanics of generalized continua 2 Strain gradient plasticity theory Method of virtual power Constitutive equations Shearing of a laminate microstructure 3 Micromorphic plasticity theory Balance and constitutive equations Rate independent plasticity Shearing of a laminate microstructure 4 Discussion

6 Mechanics of generalized continua Principle of local action: the stress state at a point X depends on variables defined at this point only [Truesdell, Toupin, 1960] [Truesdell, Noll, 1965] local action Continuous Medium nonlocal action nonlocal theory: integral formulation [Eringen, 1972] Introduction 6/53

7 Mechanics of generalized continua Simple material: A material is simple at the particle X if and only if its response to deformations homogeneous in a neighborhood of X determines uniquely its response to every deformation at X. [Truesdell, Toupin, 1960] [Truesdell, Noll, 1965] simple material F Cauchy medium (1823) (classical / Boltzmann) local action Continuous non simple material Medium nonlocal action nonlocal theory: integral formulation [Eringen, 1972] Introduction 7/53

8 Mechanics of generalized continua Simple material: A material is simple at the particle X if and only if its response to deformations homogeneous in a neighborhood of X determines uniquely its response to every deformation at X. [Truesdell, Toupin, 1960] [Truesdell, Noll, 1965] Continuous Medium local action nonlocal action simple material F non simple material Cauchy continuum (1823) (classical / Boltzmann) medium of order n medium of grade n nonlocal theory: integral formulation [Eringen, 1972] Cosserat (1909) u,r micromorphic [Eringen, Mindlin 1964] u,χ second gradient [Mindlin, 1965] F,F gradient of internal variable [Maugin, 1990] u,α Introduction 8/53

9 Plan 1 Introduction Mechanics of generalized continua 2 Strain gradient plasticity theory Method of virtual power Constitutive equations Shearing of a laminate microstructure 3 Micromorphic plasticity theory Balance and constitutive equations Rate independent plasticity Shearing of a laminate microstructure 4 Discussion

10 Plan 1 Introduction Mechanics of generalized continua 2 Strain gradient plasticity theory Method of virtual power Constitutive equations Shearing of a laminate microstructure 3 Micromorphic plasticity theory Balance and constitutive equations Rate independent plasticity Shearing of a laminate microstructure 4 Discussion

11 Power of internal forces Model variables according to a first gradient theory MODEL = { v, v, ṗ, ṗ } velocity v and cumulative plastic strain p are assumed to be independent degrees of freedom Virtual power of internal forces of a subdomain D B of the body P (i) (v, ṗ ) = p (i) (v, ṗ ) dv D simple stress tensor σ, generalized stresses a (unit MPa), b (unit MPa.mm), microforces according to [Gurtin, 2002] The virtual power density of internal forces is a linear form on the fields of virtual modeling variables p (i) = σ : (v ) + a ṗ + b ṗ The virtual power density of internal forces is invariant with respect to superimposed rigid body motion σ is symmetric [Germain, 1973a] Strain gradient plasticity theory 11/53

12 Power of contact forces Application of Gauss theorem to the power of internal forces p (i) dv = v σ n ds + ṗ b n ds D D D v σ dv ṗ (b a) dv D The form of the previous boundary integral dictates the form of the power of contact forces acting on the boundary D of the subdomain D B P (c) (v, ṗ ) = p (c) (v, ṗ ) ds D D p (c) (v, ṗ ) = t v + a c ṗ simple traction t (unit MPa), double traction a c (unit MPa.mm) Strain gradient plasticity theory 12/53

13 Power of forces acting at a distance P (e) (v, ṗ ) = D p (e) (v, ṗ ) dv p (e) (v, ṗ ) = f v + c : (v ) + a e ṗ + b e ṗ simple body forces f (unit N.mm 3 ), double body forces c and a e (unit N.mm 2 ), triple body force b e (unit N.mm 1 ) P (e) (v, ṗ ) = ( v c n + ṗ b e n ) ds D D ( v (c f ) dv + ṗ (b e a e ) ) dv Strain gradient plasticity theory 13/53

14 Principle of virtual power In the static case, v, ṗ, D B, [Germain, 1973b] P (i) (v, ṗ ) = P (c) (v, ṗ ) + P (e) (v, ṗ ) Strain gradient plasticity theory 14/53

15 Principle of virtual power In the static case, v, ṗ, D B, P (i) (v, ṗ ) = P (c) (v, ṗ ) + P (e) (v, ṗ ) which leads to v (t (σ c ) n ) + ṗ (a c (b b e ) n ) ds D + v ((σ c) + f ) + ṗ ((b b e ) a + a e ) dv = 0 D cf. [Germain, 1973b, Forest and Sievert, 2003, Kirchner and Steinmann, 2005, Lazar and Maugin, 2007, Hirschberger et al., 2007] Strain gradient plasticity theory 15/53

16 Balance and boundary conditions The application of the principle of virtual power leads to the balance of momentum equation (static case) (σ c ) + f = 0, x B balance of generalized moment of momentum equation (static case) (b b e ) a + a e = 0, x B boundary conditions (σ c ) n = t, x B (b b e ) n = a c, x B Strain gradient plasticity theory 16/53

17 Plan 1 Introduction Mechanics of generalized continua 2 Strain gradient plasticity theory Method of virtual power Constitutive equations Shearing of a laminate microstructure 3 Micromorphic plasticity theory Balance and constitutive equations Rate independent plasticity Shearing of a laminate microstructure 4 Discussion

18 Continuum thermodynamics Enhance the local balance of energy and the entropy inequality ρ ɛ = p (i) div q + ρr, ρ( ψ + ηṫ ) + p(i) q T. T 0 Decomposition of total strain v = u, ε = 1 2 (u + u ) ε = ε e p + ε Consider the constitutive functionals: ψ = ˆψ(ε e, T, p, α, p), η = ˆη(ε e, T, p, α, p) σ = ˆσ (ε e, T, p, α, p) a = â(ε e, T, p, α, p), b = ˆb (ε e, T, p, α, p) Strain gradient plasticity theory 18/53

19 State laws Clausius Duhem inequality (isothermal) (σ ρ ψ ε e ) : ε p +(a ρ ψ p ψ )ṗ+(b ρ p ) ṗ+σ : ε p ρ ψ α α 0 Derive the state laws [Coleman and Noll, 1963] σ = ρ ˆψ ε e, Residual dissipation ˆψ X = ρ α, R = ˆψ p, b = ˆψ p D res = σ : ε p + (a R)ṗ X α 0 Strain gradient plasticity theory 19/53

20 Flow rule and evolution law Introduce an equivalent stress measure σ eq such that σ : ε p = σ eq ṗ which defines the cumulative plastic strain rate ṗ D res = (σ eq + a R)ṗ X α 0 Introduce the viscoplastic potential Ω(σ eq + a R, X ) such that ε p Ω = (σ eq + a R), Ω α = X Ω convex with respect to the first, concave with respect to the second variable for positive dissipation Rate independent case; introduce the yield function f (σ, a, R) = σ eq R 0 R + a where R 0 is the initial yield stress p = ṗ f, σ ε eq = σ σ : f σ Strain gradient plasticity theory 20/53

21 Specific constitutive equations Free energy function ρψ(ε e, p, p) = 1 2 ε e : Λ : ε e Hp p A p State laws σ = Λ : ε e, R = Hp, b = A p Balance of generalized momentum (homogeneous material) a = div b = div (A p) = A : ( p) Yield function f (σ, R, a) = J 2 (σ ) R 0 Hp + a Consistency condition; ṗ is solution of a p.d.e. (H + f σ : Λ : f σ )ṗ + A : ( ṗ) = f σ : Λ : ε Isotropic case + von Mises : Aifantis model [Aifantis, 1987] A = c 2 1, σ eq = R 0 + Hp c 2 p Strain gradient plasticity theory 21/53

22 Plan 1 Introduction Mechanics of generalized continua 2 Strain gradient plasticity theory Method of virtual power Constitutive equations Shearing of a laminate microstructure 3 Micromorphic plasticity theory Balance and constitutive equations Rate independent plasticity Shearing of a laminate microstructure 4 Discussion

23 Confined plasticity start animate end periodic simple shear test: classical solution [Ashby, 1970] conventional continuum plasticity predicts homogenenous plastic deformation in the layers Strain gradient plasticity theory 23/53

24 Confined plasticity start animate end periodic simple shear test: classical solution [Ashby, 1970] conventional continuum plasticity predicts homogenenous plastic deformation in the layers Strain gradient plasticity theory 23/53

25 Confined plasticity start animate end periodic simple shear test: classical solution [Ashby, 1970] conventional continuum plasticity predicts homogenenous plastic deformation in the layers Strain gradient plasticity theory 23/53

26 Confined plasticity start animate end periodic simple shear test: classical solution [Ashby, 1970] conventional continuum plasticity predicts homogenenous plastic deformation in the layers Strain gradient plasticity theory 23/53

27 Confined plasticity start animate end periodic simple shear test: classical solution [Ashby, 1970] conventional continuum plasticity predicts homogenenous plastic deformation in the layers Strain gradient plasticity theory 23/53

28 Confined plasticity start animate end periodic simple shear test: classical solution [Ashby, 1970] conventional continuum plasticity predicts homogenenous plastic deformation in the layers Strain gradient plasticity theory 23/53

29 Confined plasticity start animate end periodic simple shear test: classical solution [Ashby, 1970] conventional continuum plasticity predicts homogenenous plastic deformation in the layers Strain gradient plasticity theory 23/53

30 Confined plasticity start animate end periodic simple shear test: classical solution [Ashby, 1970] conventional continuum plasticity predicts homogenenous plastic deformation in the layers Strain gradient plasticity theory 23/53

31 Confined plasticity start animate end periodic simple shear test: classical solution [Ashby, 1970] conventional continuum plasticity predicts homogenenous plastic deformation in the layers Strain gradient plasticity theory 23/53

32 Confined plasticity start animate end periodic simple shear test: classical solution [Ashby, 1970] conventional continuum plasticity predicts homogenenous plastic deformation in the layers Strain gradient plasticity theory 23/53

33 Confined plasticity start animate end periodic simple shear test: classical solution [Ashby, 1970] conventional continuum plasticity predicts homogenenous plastic deformation in the layers Strain gradient plasticity theory 23/53

34 Confined plasticity start animate end periodic simple shear test: classical solution [Ashby, 1970] conventional continuum plasticity predicts homogenenous plastic deformation in the layers Strain gradient plasticity theory 23/53

35 Confined plasticity start animate end periodic simple shear test: classical solution [Ashby, 1970] conventional continuum plasticity predicts homogenenous plastic deformation in the layers Strain gradient plasticity theory 23/53

36 Confined plasticity start animate end periodic simple shear test: classical solution [Ashby, 1970] conventional continuum plasticity predicts homogenenous plastic deformation in the layers Strain gradient plasticity theory 23/53

37 Confined plasticity start animate end periodic simple shear test: classical solution [Ashby, 1970] conventional continuum plasticity predicts homogenenous plastic deformation in the layers Strain gradient plasticity theory 23/53

38 Confined plasticity start animate end periodic simple shear test: classical solution [Ashby, 1970] conventional continuum plasticity predicts homogenenous plastic deformation in the layers Strain gradient plasticity theory 23/53

39 Confined plasticity start animate end periodic simple shear test: classical solution [Ashby, 1970] conventional continuum plasticity predicts homogenenous plastic deformation in the layers Strain gradient plasticity theory 23/53

40 Confined plasticity start animate end periodic simple shear test: classical solution [Ashby, 1970] conventional continuum plasticity predicts homogenenous plastic deformation in the layers Strain gradient plasticity theory 23/53

41 Confined plasticity start animate end periodic simple shear test: classical solution [Ashby, 1970] conventional continuum plasticity predicts homogenenous plastic deformation in the layers Strain gradient plasticity theory 23/53

42 Confined plasticity start animate end periodic simple shear test: classical solution [Ashby, 1970] conventional continuum plasticity predicts homogenenous plastic deformation in the layers Strain gradient plasticity theory 23/53

43 Confined plasticity start animate end periodic simple shear test: classical solution [Ashby, 1970] conventional continuum plasticity predicts homogenenous plastic deformation in the layers Strain gradient plasticity theory 23/53

44 Confined plasticity start animate end periodic shear test [Ashby, 1970] Additional interface conditions associated with strain gradient plasticity induce size dependent non homogenous deformation Strain gradient plasticity theory 24/53

45 Confined plasticity start animate end periodic shear test [Ashby, 1970] Additional interface conditions associated with strain gradient plasticity induce size dependent non homogenous deformation Strain gradient plasticity theory 24/53

46 Confined plasticity start animate end periodic shear test [Ashby, 1970] Additional interface conditions associated with strain gradient plasticity induce size dependent non homogenous deformation Strain gradient plasticity theory 24/53

47 Confined plasticity start animate end periodic shear test [Ashby, 1970] Additional interface conditions associated with strain gradient plasticity induce size dependent non homogenous deformation Strain gradient plasticity theory 24/53

48 Confined plasticity start animate end periodic shear test [Ashby, 1970] Additional interface conditions associated with strain gradient plasticity induce size dependent non homogenous deformation Strain gradient plasticity theory 24/53

49 Confined plasticity start animate end periodic shear test [Ashby, 1970] Additional interface conditions associated with strain gradient plasticity induce size dependent non homogenous deformation Strain gradient plasticity theory 24/53

50 Confined plasticity start animate end periodic shear test [Ashby, 1970] Additional interface conditions associated with strain gradient plasticity induce size dependent non homogenous deformation Strain gradient plasticity theory 24/53

51 Confined plasticity start animate end periodic shear test [Ashby, 1970] Additional interface conditions associated with strain gradient plasticity induce size dependent non homogenous deformation Strain gradient plasticity theory 24/53

52 Confined plasticity start animate end periodic shear test [Ashby, 1970] Additional interface conditions associated with strain gradient plasticity induce size dependent non homogenous deformation Strain gradient plasticity theory 24/53

53 Confined plasticity start animate end periodic shear test [Ashby, 1970] Additional interface conditions associated with strain gradient plasticity induce size dependent non homogenous deformation Strain gradient plasticity theory 24/53

54 Confined plasticity start animate end periodic shear test [Ashby, 1970] Additional interface conditions associated with strain gradient plasticity induce size dependent non homogenous deformation Strain gradient plasticity theory 24/53

55 Confined plasticity start animate end periodic shear test [Ashby, 1970] Additional interface conditions associated with strain gradient plasticity induce size dependent non homogenous deformation Strain gradient plasticity theory 24/53

56 Confined plasticity start animate end periodic shear test [Ashby, 1970] Additional interface conditions associated with strain gradient plasticity induce size dependent non homogenous deformation Strain gradient plasticity theory 24/53

57 Confined plasticity start animate end periodic shear test [Ashby, 1970] Additional interface conditions associated with strain gradient plasticity induce size dependent non homogenous deformation Strain gradient plasticity theory 24/53

58 Confined plasticity start animate end periodic shear test [Ashby, 1970] Additional interface conditions associated with strain gradient plasticity induce size dependent non homogenous deformation Strain gradient plasticity theory 24/53

59 Confined plasticity start animate end periodic shear test [Ashby, 1970] Additional interface conditions associated with strain gradient plasticity induce size dependent non homogenous deformation Strain gradient plasticity theory 24/53

60 Confined plasticity start animate end periodic shear test [Ashby, 1970] Additional interface conditions associated with strain gradient plasticity induce size dependent non homogenous deformation Strain gradient plasticity theory 24/53

61 Confined plasticity start animate end periodic shear test [Ashby, 1970] Additional interface conditions associated with strain gradient plasticity induce size dependent non homogenous deformation Strain gradient plasticity theory 24/53

62 Confined plasticity start animate end periodic shear test [Ashby, 1970] Additional interface conditions associated with strain gradient plasticity induce size dependent non homogenous deformation Strain gradient plasticity theory 24/53

63 Confined plasticity start animate end periodic shear test [Ashby, 1970] Additional interface conditions associated with strain gradient plasticity induce size dependent non homogenous deformation Strain gradient plasticity theory 24/53

64 Confined plasticity start animate end periodic shear test [Ashby, 1970] Additional interface conditions associated with strain gradient plasticity induce size dependent non homogenous deformation Strain gradient plasticity theory 24/53

65 Laminate microstructure under shear Unit cell of a periodic two phase laminate 2 O s Aifantis material in the white (soft) phase, purely elastic gray (hard) phase Form of the solution for imposed mean shear γ u 1 = γ x 2, u 2 (x 1 ) = u(x 1 ), u 3 = 0 1 h l = s + h unknown periodic functions u(x 1 ), p(x 1 ) Deformation gradient and strain 1 0 γ 0 [ ] 0 2 ( γ + u,1) 0 [ u ] = u,1 0 0, ε = 1 2 ( γ + u,1) Strain gradient plasticity theory 25/53

66 Laminate microstructure under shear Unit cell of a periodic two phase laminate 2 O s Aifantis material in the white (soft) phase, purely elastic gray (hard) phase Form of the solution for imposed mean shear γ u 1 = γ x 2, u 2 (x 1 ) = u(x 1 ), u 3 = 0 1 h l = s + h unknown periodic functions u(x 1 ), p(x 1 ) Deformation gradient and strain 1 0 γ 0 [ ] 0 2 ( γ + u,1) 0 [ u ] = u,1 0 0, ε = 1 2 ( γ + u,1) Strain gradient plasticity theory 25/53

67 Resolution of the b.v.p. Let us consider homogeneous isotropic elasticity and no hardening in the plastic phase for simplicity Equilibrium: homogeneous shear stress σ 12 throughout the laminate Displacement in the hard phase σ 12 = µ( γ + u h,1) = u h,1 = C, u h = Cx 1 + D Plastic strain in the soft phase s ε p = 3 2ṗ J 2 (σ ), from the yield condition we get 3 ε p = 2 ṗ(e 1 e 2 + e 2 e 1 ) σ 12 = µ( γ + u h,1) = u h,1 = C, u h = Cx 1 + D so that the plastic strain is parabolic p = α(x 2 1 s2 4 ) Continuity of plastic strain at the interface p(±s/2) = 0 Strain gradient plasticity theory 26/53

68 Resolution of the b.v.p. Displacement in the soft phase σ 12 = µ( γ + u ș 1 3p) = u ș 1 = C + 3p u s = (C α 3 s2 3 4 )x 1 + α 3 x 1 3 Strain gradient plasticity theory 27/53

69 Interface conditions Displacement continuity at x 1 = ±s/2 u s ( s 2 ) = uh ( s 2 ) = 3α s3 12 = D Displacement periodicity at x 1 = s/2 and x 1 = s/2 + h u s ( s 2 ) = uh ( s 2 + h) = 3α s3 12 = Cl + D Continuity of the stress vector at x 1 = ±s/2 R 0 2cα = µ 3( γ + C) The wanted constants are deduced from the previous equations C = R 0 3µ γ, D = C l 12cl 2, α = 12 D 3µ + 3 s 3 3s 3 Strain gradient plasticity theory 28/53

70 Plastic strain profile in the channel p/ γ u2/l γ b1/lr x/l µ (MPa) R 0 (MPa) c (MPa.mm 2 ) f l (µm) γ Characteristic length: l c = c/µ = 0.4 µm, leading to strong size effects in the micron range and below Strain gradient plasticity theory 29/53

71 Plastic strain profile in the channel p/ γ u2/l γ b1/lr x/l µ (MPa) R 0 (MPa) c (MPa.mm 2 ) f l (µm) γ The higher order stress b 1 = 2cα experiences a jump at the interface s = ±s/2: b 1 ( s+ 2 ) b 1( s 2 ) = 0 cαs, [[b 1]]( s 2 ) = cαs Strain gradient plasticity theory 30/53

72 Overall size effect Macroscopic stress strain relation σ 12 µ = 1 µfs 2 + 4c ( 3 3 fs2 R 0 + 4c γ bilinear response depending explicitly on channel size s Macroscopic stress vs mean plastic strain; p = 1 l s/2 s/2 p(x 1 ) dx 1 = 3 p = f γ C(1 f ) f σ 12 µ σ 12 = R c f 3 l 2 p microstructure induced linear hardening depending on unit cell size l Limit cases thick channels: size independent threshold σ 12 = R 0 / 3 thin films: scaling law σ 12 / p 1/l 2 Strain gradient plasticity theory 31/53 )

73 Plan 1 Introduction Mechanics of generalized continua 2 Strain gradient plasticity theory Method of virtual power Constitutive equations Shearing of a laminate microstructure 3 Micromorphic plasticity theory Balance and constitutive equations Rate independent plasticity Shearing of a laminate microstructure 4 Discussion

74 Plan 1 Introduction Mechanics of generalized continua 2 Strain gradient plasticity theory Method of virtual power Constitutive equations Shearing of a laminate microstructure 3 Micromorphic plasticity theory Balance and constitutive equations Rate independent plasticity Shearing of a laminate microstructure 4 Discussion

75 General scalar microstrain gradient plasticity Classical and generalized plasticity DOF = {u, p χ } STATE = {ε e, p, α, p χ, p χ } scalar plastic microstrain variable p χ Enhanced power of internal forces and extra balance equation p (i) = σ : ε + a ṗ χ + b. ṗ χ, p (c) = t. u + a c ṗ χ div b a = 0, x Ω, b.n = a c, x Ω Micromorphic plasticity theory 34/53

76 General scalar microstrain gradient plasticity State laws σ = ρ ψ ε e, ε = ε e + ε p R = ρ ψ p, X = ρ ψ α a = ρ ψ p χ + a v, b = ρ ψ p χ Evolution laws D res = σ : ε p + (a v R)ṗ X α 0 [Forest, 2009] p = ε λ f σ, f ṗ = λ R, f α = λ X Micromorphic plasticity theory 35/53

77 Explicit constitutive equations Quadratic free energy potential ρψ(ε e, p, p χ, p χ) = 1 2 ε e : Λ : ε e Hp Hχ(p pχ) pχ.a. pχ [Forest, 2009, Dimitrijevic and Hackl, 2011] Constitutive equations σ = Λ : ε e, a = H χ (p p χ ), b = A. p χ, R = (H+H χ )p H χ p χ Substitution of constitutive into extra balance equations p χ 1 H χ div (A. p χ ) = p Homogeneous and isotropic materials p χ A H χ p χ = p, b.c. b n = A p χ.n = a c A = A1 same p.d.e. as in the implicit gradient enhanced elastoplasticity with a c = 0 [Engelen et al., 2003] Micromorphic plasticity theory 36/53

78 Link to Aifantis strain gradient plasticity Yield function Hardening law Under plastic loading f (σ, R) = σ eq R 0 R R = ψ p = (H + H χ)p H χ p χ σ eq = R 0 + Hp χ A(1 + H ) p χ H χ compare with Aifantis model [Aifantis, 1987] σ eq = R 0 + R(p) c 2 p The equivalence is obtained for H χ = (internal constraint): p χ p, A = c 2 Micromorphic plasticity theory 37/53

79 Plan 1 Introduction Mechanics of generalized continua 2 Strain gradient plasticity theory Method of virtual power Constitutive equations Shearing of a laminate microstructure 3 Micromorphic plasticity theory Balance and constitutive equations Rate independent plasticity Shearing of a laminate microstructure 4 Discussion

80 Consistency condition Consistency condition Plastic multiplier ḟ = f : σ σ + + f R Ṙ = σeq σ : Λ : ( ε ε p ) R p ṗ R ṗ χ = 0 p χ N : Λ : ε R ṗ χ p χ ṗ = N : Λ : N + R, with N = σeq σ p where ε and ṗ χ are controllable variable. Even though the yield condition can be written as a partial differential equation, there is no need for a variational formulation of the consistency condition contrary to [Mühlhaus and Aifantis, 1991, Liebe et al., 2001]. There is no need for a plastic front tracking technique. The plastic microstrain p χ and the generalized traction b.n are continuous across the elastic/plastic domain. Micromorphic plasticity theory 39/53

81 Thermal effects For temperature dependent parameters a = div b = div (A p χ ) = A p χ + A T T. p χ Consistency condition p χ A H χ p χ 1 H χ A T T. p χ = p ṗ = N : Λ : ( ε ε th ) R p χ ṗ χ R T Ṫ N : Λ : N + R p Micromorphic plasticity theory 40/53

82 Plan 1 Introduction Mechanics of generalized continua 2 Strain gradient plasticity theory Method of virtual power Constitutive equations Shearing of a laminate microstructure 3 Micromorphic plasticity theory Balance and constitutive equations Rate independent plasticity Shearing of a laminate microstructure 4 Discussion

83 Laminate microstructure under shear Unit cell of a periodic two phase laminate 2 O s 1 h l = s + h Micromorphic material in the white (soft) phase, purely elastic micromorphic gray (hard) phase Form of the solution for impose mean shear γ u 1 = γ x 2, u 2 (x 1 ) = u(x 1 ), u 3 = 0 unknown periodic functions u(x 1 ), p(x 1 ), p χ (x 1 ) Deformation gradient and strain 1 0 γ 0 [ ] 0 2 ( γ + u,1) 0 [ u ] = u,1 0 0, ε = 1 2 ( γ + u,1) Micromorphic plasticity theory 42/53

84 Resolution of the b.v.p. Let us consider homogeneous isotropic elasticity, homogeneous H χ and no hardening in the plastic phase for simplicity The shear stress is uniform throughout the laminate and takes the value 3σ12 = R 0 + R = R 0 + H χ (p p χ ) = R 0 Ap χ,11 Derivation of the previous equations with respect to x 1 shows that p χ,111 = 0 which leads to the parabolic profile of the micro plastic deformation in the soft phase p χ (x) = αx 2 + β, x s 2 Note that 3σ12 = R 0 2Aα The parabolic plastic strain profile follows p = αx 2 + β 2A H χ Micromorphic plasticity theory 43/53

85 Resolution of the b.v.p. A new feature of the model is that the microplastic strain p χ does not vanish in general in the hard phase, whereas p does: p h χ = α h cosh ω h (x l 2 ), p χ Ah H χ p χ = 0 the p h χ profile is of hyperbolic nature s 2 x s 2 + h, with ω2 h = H χ A h Micromorphic plasticity theory 44/53

86 Interface conditions Continuity of micro plastic deformation at x = s/2: α s2 4 + β = α h cosh ω h h 2 Continuity of the generalized stress component b 1 : Aαs = A h α h ω h sinh ω h h 2 Micromorphic plasticity theory 45/53

87 Interface conditions The displacement in the plastic and elastic phases can be expressed as u s = α x «3 3β R γ + 2Aα( + ) x 3 3µ 3µ H χ u h = «1 (R 0 2Aα) γ x + C 3µ They are used to exploit two additional interface conditions Continuity of the displacement at x = s/2: u s (s/2) = u h (s/2) α s3 8 3µ + 3(β 2Aα ) s H χ 2 = C Periodicity of the displacement component u s ( s/2) = u h (s/2 + h) ( σ 12 µ γ)l + 3(β + 2Aα H χ ) s 2 α s3 8 3 = C Micromorphic plasticity theory 46/53

88 Plastic strain profiles in the channel p χ/ γ p/ γ u 2/l γ b 1/lR x/l µ (MPa) R 0 (MPa) H χ (MPa) A (MPa.mm 2 ) f l (µm) γ Micromorphic plasticity theory 47/53

89 Overall size effect The scaling law results from the expression of the overall stress σ 12 as a function of the mean plastic strain over the unit cell: p = 1 l s 2 s 2 (αx 2 + β 2Aα ) dx = βf (1 1L ( s 2 H 2 χ 12 2A )) H χ with L 2 = s2 4 + A s h cotanh(ω h A h ω h 2 ) = β. The uniform stress α component can now be expressed as a function of the volume fraction f of the soft phase and of the unit cell size l: 3σ12 = R 0 + 2A f p f 2 l A + A fl h cotanh (ω h H χ A h ω h 2 ) displaying a size dependent overall linear hardening Micromorphic plasticity theory 48/53

90 Scaling laws Two limit cases naturally arise Internal constraint H χ for which the strain gradient plasticity model is retrieved Unit cell size l 0 leads to saturation stress 1 f 3σ12 R 0 H χ p f 1e+10 1e+08 1e+06 Aifantis model micromorphic model σ12 R0/ 3 (MPa) e-06 1e-06 1e l (mm) Micromorphic plasticity theory 49/53

91 Plan 1 Introduction Mechanics of generalized continua 2 Strain gradient plasticity theory Method of virtual power Constitutive equations Shearing of a laminate microstructure 3 Micromorphic plasticity theory Balance and constitutive equations Rate independent plasticity Shearing of a laminate microstructure 4 Discussion

92 Main comments The choice of quadratic potentials with respect to strain gradients leads to the existence of a size dependent overall linear hardening modulus in a laminate microstructure The corresponding scaling law according to Aifantis strain gradient plasticity (one new material parameter) is 1/l 2 In contrast the micromorphic model (two new material parameters) leads to a saturation at nano scales Physical metallurgy hints rather at scaling laws of the Orowan (1/l) or Hall-Petch (1/ l) types Discussion 51/53

93 Needed improvements Improve constitutive equations; for instance ρψ( p) = p A p [Conti and Ortiz, 2005, Okumura et al., 2007] Add higher order dissipative parts [Forest, 2009, Anand et al., 2012] Enhance interface conditions [Gurtin and Needleman, 2005, Acharya, 2007, Gurtin and Anand, 2008] Discussion 52/53

94 Extension to crystal plasticity In crystal plasticity, the relevant variable is not the gradient of the cumulative plastic strain but rather the dislocation density tensor Γ = curl H p, [Cermelli and Gurtin, 2001, Svendsen, 2002] but no effects in laminates... H = u = H e + H p A strain gradient and a micromorphic theory can be designed based on the introduction of the dislocation density tensor in the free energy function [Aslan et al., 2011] Similar effects arise in laminate microstructures but the overall linear hardening is of kinematic nature x = curl curl H p : m n [Cordero et al., 2010, Cottura et al., 2012] Hall Petch related size effects can be predicted in polycrystals based on full field simulations [Forest et al., 2000, Bayley et al., 2007, Neff et al., 2009, Bargmann et al., 2010, Cordero et al., 2012, Wulfinghoff and Boehlke, 2012] Constitutive equations are still unrealistic... Discussion 53/53

95 Acharya A. (2007). Jump condition for GND evolution as a constraint on slip transmission at grain boundaries. Philosophical Magazine, vol. 87, pp Aifantis E.C. (1987). The physics of plastic deformation. International Journal of Plasticity, vol. 3, pp Anand L., Aslan O., and Chester S.A. (2012). A large deformation gradient theory for elastic plastic materials: Strain softening and regularization of shear bands. International Journal of Plasticity, vol , pp Aslan O., Cordero N. M., Gaubert A., and Forest S. (2011). Micromorphic approach to single crystal plasticity and damage. International Journal of Engineering Science, vol. 49, pp Bargmann S., Ekh M., Runesson K., and Svendsen B. (2010). Discussion 53/53

96 Modeling of polycrystals with gradient crystal plasticity: A comparison of strategies. Philosophical Magazine, vol. 90, pp Bayley C.J., Brekelmans W.A.M., and Geers M.G.D. (2007). A three dimensional dislocation field crystal plasticity approach applied to miniaturized structures. Philosophical Magazine, vol. 87, pp Cermelli P. and Gurtin M.E. (2001). On the characterization of geometrically necessary dislocations in finite plasticity. Journal of the Mechanics and Physics of Solids, vol. 49, pp Coleman B.D. and Noll W. (1963). The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Rational Mech. and Anal., vol. 13, pp Conti S. and Ortiz M. (2005). Discussion 53/53

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102 Numerical approximation of incremental infinitesimal gradient plasticity. International Journal for Numerical Methods in Engineering, vol. 77, pp Okumura D., Higashi Y., Sumida K., and Ohno N. (2007). A homogenization theory of strain gradient single crystal plasticity and its finite element discretization. International Journal of Plasticity, vol. 23, pp Svendsen B. (2002). Continuum thermodynamic models for crystal plasticity including the effects of geometrically-necessary dislocations. J. Mech. Phys. Solids, vol. 50, pp Wulfinghoff S. and Boehlke T. (2012). Equivalent plastic strain gradient enhancement of single crystal plasticity: theory and numerics. Proceedings of the Royal Society A, vol. 468, pp Discussion 53/53