A hierarchy of higher order and higher grade continua Application to the plasticity and fracture of metallic foams

Transcription

1 A hierarchy of higher order and higher grade continua Application to the plasticity and fracture of metallic foams Samuel Forest Centre des Matériaux/UMR 7633 Mines Paris ParisTech /CNRS BP 87, Evry, France

2 Plan 1 The micromorphic approach to plasticity Continuum thermomechanics Full micromorphic and microstrain theories Internal constraint in the micromorphic approach 2 Application to nickel foams Micromorphic anisotropic compressible plasticity model Simulation of crack propagation 3 Micromechanical approach

3 Plan 1 The micromorphic approach to plasticity Continuum thermomechanics Full micromorphic and microstrain theories Internal constraint in the micromorphic approach 2 Application to nickel foams Micromorphic anisotropic compressible plasticity model Simulation of crack propagation 3 Micromechanical approach

4 Plan 1 The micromorphic approach to plasticity Continuum thermomechanics Full micromorphic and microstrain theories Internal constraint in the micromorphic approach 2 Application to nickel foams Micromorphic anisotropic compressible plasticity model Simulation of crack propagation 3 Micromechanical approach

5 State space observable and controllable variables (temperature, strain...) {T, F } 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} gradient of internal variable (extra entropy flux) {α, α} [Maugin, 1990] [Maugin, 1997] The micromorphic approach to plasticity 5/59

6 The micromorphic approach (1) Start from an initial classical elastoviscoplastic model with internal variables DOF 0 = {u }, STATE0 = {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: DOF = {u, χ φ}, STATE = {F, T, α, Extend the power of internal forces P (i) (v, χ φ ) = p (i) (v, χ φ ) dv p (i) (v, χ φ ) = σ : v + a χ φ + b. χ φ D χ φ, χ φ} a, b generalized stresses, microforces (Gurtin, 1996) Derive additional balance equation and boundary conditions div b a = 0, x Ω, b.n = a c, x Ω The micromorphic approach to plasticity 6/59

7 The micromorphic approach (2) More generally, in the presence of volume generalized forces: div (b b e ) a+a e = 0, x Ω, (b b e ).n = a c, x Ω 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 Consider the constitutive functionals: ψ = ˆψ(F e, T, α, χ φ, χ φ), η = ˆη(F e, T, α, χ φ, χ φ) σ = ˆσ (F e, T, α, χ φ, χ φ) a = â(f e, T, α, χ φ, χ φ), b = ˆb (F e, T, α, χ φ, χ φ) Derive the state laws (Coleman and Noll, 1963) σ = ρ ˆψ F e.f et, η = ˆψ T, X = ρ ˆψ α, a = ˆψ χ φ, b = ˆψ χ φ Residual dissipation D res = W p X α q T. T 0 The micromorphic approach to plasticity 7/59

8 The micromorphic approach (3) Take a simple quadratic potential ψ(f, T, α, χ φ, χ φ) = ψ 1 (F, α, T )+ψ 2 (e = φ χ φ, χ φ, T ) ρψ 2 = 1 2 H χ(φ χ φ) A χ φ. χ φ a = ρ ψ χ φ = H χ(φ χ φ), b = ρ ψ χ φ = A χ φ Simple form of the partial differential equation (homogeneous, isothermal...) a = div b = χ φ A H χ χ φ = φ Helmholtz equation with a minus sign and a source term Coupling modulus H χ and characteristic length of the medium lc 2 = A H χ Stability H χ > 0, A > 0 The micromorphic approach to plasticity 8/59

9 Plan 1 The micromorphic approach to plasticity Continuum thermomechanics Full micromorphic and microstrain theories Internal constraint in the micromorphic approach 2 Application to nickel foams Micromorphic anisotropic compressible plasticity model Simulation of crack propagation 3 Micromechanical approach

10 Micromorphic continuum Micromorphic continuum according to (Eringen and Suhubi, 1964; Mindlin, 1964) Select variable: φ F, χ φ χ p (i) = σ : u + a : χ + B. χ application of the principle of (infinitesimal) material frame indifference, (infinitesimal) change of observer of rate w : u = u + w, χ = χ + w = σ + a must be symmetric. Rewrite the virtual power: two balance equations: p (i) = σ : ε + s : ( u χ ) + S. χ div (σ + s ) + ρf = 0, div S + s = 0 The micromorphic approach to plasticity 10/59

11 Microstrain continuum Microstrain continuum after (Forest and Sievert, 2006) Select φ C = F T.F, χ φ χ C, or φ ε, χ φ χ ε p (i) = σ : ε + a : χ ε + b. χ ε Constitutive coupling between macro and microstrain via the relative strain e := ε χ ε ψ(ε e, T, α, e := ε χ ε, K := χ ε ) Take a quadratic potential Extra balance equation a = H χe, b = A χ ε χ ε l 2 c χ ε = ε, with l 2 c = A H χ example: microfoams (Dillard et al., 2006) The micromorphic approach to plasticity 11/59

12 Cosserat continuum Cosserat continuum φ = R, χ φ χ R p (i) = σ s : ε σ a : (( u ) a Ṙ.R T ) + M : κ The micromorphic approach to plasticity 12/59

13 A hierarchy of higher order continua name number DOF DOF references of DOF (finite case) (infinitesimal case) Cauchy 3 u u [Cauchy, 1823] micromorphic 12 u, χ u, χ s + χ a [Eringen, 1964] [Mindlin, 1964] The micromorphic approach to plasticity 13/59

14 A hierarchy of higher order continua name number DOF DOF references of DOF (finite case) (infinitesimal case) Cauchy 3 u u [Cauchy, 1823] microdilatation 4 u, χ u, χ [Goodman, Cowin, 1972] [Steeb, Diebels, 2003] micromorphic 12 u, χ u, χ s + χ a [Eringen, 1964] [Mindlin, 1964] The micromorphic approach to plasticity 14/59

15 A hierarchy of higher order continua name number DOF DOF references of DOF (finite case) (infinitesimal case) Cauchy 3 u u [Cauchy, 1823] microdilatation 4 u, χ u, χ [Goodman, Cowin, 1972] [Steeb, Diebels, 2003] Cosserat 6 u, R u, Φ [Kafadar, Eringen, 1976] micromorphic 12 u, χ u, χ s + χ a [Eringen, 1964] [Mindlin, 1964] The micromorphic approach to plasticity 15/59

16 A hierarchy of higher order continua name number DOF DOF references of DOF (finite case) (infinitesimal case) Cauchy 3 u u [Cauchy, 1823] microdilatation 4 u, χ u, χ [Goodman, Cowin, 1972] [Steeb, Diebels, 2003] Cosserat 6 u, R u, Φ [Kafadar, Eringen, 1976] microstretch 7 u, χ, R u, χ, Φ [Eringen, 1990] micromorphic 12 u, χ u, χ s + χ a [Eringen, 1964] [Mindlin, 1964] The micromorphic approach to plasticity 16/59

17 A hierarchy of higher order continua name number DOF DOF references of DOF (finite case) (infinitesimal case) Cauchy 3 u u [Cauchy, 1823] microdilatation 4 u, χ u, χ [Goodman, Cowin, 1972] [Steeb, Diebels, 2003] Cosserat 6 u, R u, Φ [Kafadar, Eringen, 1976] microstretch 7 u, χ, R u, χ, Φ [Eringen, 1990] microstrain 9 u, χ C u, χ ε [Forest, Sievert, 2006] micromorphic 12 u, χ u, χ s + χ a [Eringen, 1964] [Mindlin, 1964] The micromorphic approach to plasticity 17/59

18 Plan 1 The micromorphic approach to plasticity Continuum thermomechanics Full micromorphic and microstrain theories Internal constraint in the micromorphic approach 2 Application to nickel foams Micromorphic anisotropic compressible plasticity model Simulation of crack propagation 3 Micromechanical approach

19 Internal constraint and gradient of internal variable approach Impose the internal constraint that χ φ φ = K φ Then, the generalized stress a becomes a Lagrange multiplier. Examples φ F second gradient model (Mindlin, 1965) φ p Aifantis model (Aifantis, 1987; Fleck and Hutchinson, 2001) φ ε p strain gradient plasticity (Forest and Sievert, 2003; Gurtin, 2003) p (i) = σ : ε e + s : ε p + S... ε p, div S = s σ dev The yield condition becomes a PDE (Aifantis, Fleck Hutchinson, Gurtin): σ eq = σ Y + Hp A p What do the boundary conditions become? For the second gradient theory, intricate b.c. involving surface curvature For gradient of plastic strain, S.n = m The micromorphic approach to plasticity 19/59

20 Plan 1 The micromorphic approach to plasticity Continuum thermomechanics Full micromorphic and microstrain theories Internal constraint in the micromorphic approach 2 Application to nickel foams Micromorphic anisotropic compressible plasticity model Simulation of crack propagation 3 Micromechanical approach

21 Plan 1 The micromorphic approach to plasticity Continuum thermomechanics Full micromorphic and microstrain theories Internal constraint in the micromorphic approach 2 Application to nickel foams Micromorphic anisotropic compressible plasticity model Simulation of crack propagation 3 Micromechanical approach

22 Processing of nickel foams from PU template Application to nickel foams 22/59

23 Foam sheet axes and cell shape a b c Application to nickel foams 23/59

24 Anisotropic Compressible Continuum Plasticity Model Yield criterion f (σ ) = σ eq R, σ eq = 2 [H] = 6 4 h a h b h c h d h e h f Normality rule ε p = ṗ σ f = ṗ Consistency condition (cumulative plastic strain p) Nonlinear isotropic hardening ṗ = «1 3 2 Cσ dev : H : σ dev + F (P : σ ) , [P] = f : σ E : ε f : σ E : f + R σ p R = R 0 + Hp + Q(1 exp( bp)) p q r CH : σ dev + F (P : σ ) P Application to nickel foams 24/59

25 Parameter identification C 1 F 6, R 0 (MPa) 0,30 Q (MPa) 0,37 b 3,3 coiling direction H (MPa) 8,17 h a 0,41 h b 1,8 h c 1,75 h d h e 1,45 1 p q 1 37 F i t t a g e p a r a m è t r e s r h f 1 23 transverse direction Application to nickel foams 25/59

26 Metal foam as a micromorphic continuum : The microfoam model Degrees of freedom at material point displacement vector u, after [Mindlin, 1964] [Eringen, 1964] Deformation measures microdeformation tensor χ strain tensor ε = (u + u )/2 relative deformation tensor e = u χ microdeformation gradient K = χ Power density of internal forces / generalized stress tensors p (i) = σ : ε + s : ė + M : K Balance of momentum (σ + s). = 0 Balance of micromorphic momentum Boundary conditions (traction and double traction vectors) M. + s = 0 t = (σ + s ).n, T = M.n Variational formulation for finite element implementation Z Z p (i) dv = t.ṅ + T : χ ds V V Application to nickel foams 26/59

27 The microfoam model : Constitutive equations Strain partition ε = ε e + ε p, e = e e + e p, K = K e p + K Linear elasticity σ = c : ε e, s = a : e e, e M = A :K simplified moduli as proposed by [Shu et al., 1999] M = l 2 c c : K e characteristic length l c when a is very large, and if e p = 0, then u χ gradient theory) (second Application to nickel foams 27/59

28 The microfoam model : Constitutive equations Yield criterion f (σ ) = σ eq R ( 3 σ eq = 2 Cσ dev : H : σ dev + F (P : σ ) 2 + a 1 s dev : s dev + a 2 s dev : s devt + Normality rule +b 1 M :M ) 1 2 p = ṗ f, ε σ ė p = ṗ f s, K p = ṗ f M Microfoam: a 1 = a 2 = a 3 = b 1 = 0, a very large constrained microdeformation (second gradient), linear relationship between gradient of microdeformation and hyperstress tensor Application to nickel foams 28/59

29 Microstrain continuum degrees of freedom (u, χ s ) strain measures : (ε, ε χ s, χ s ) S ijk,k + s ij = 0 Aχ s ij,kk + b(ε ij χ s ij) = 0 σ = C : (ε ε p ) s = b(ε χ s ) S = Aχ s div (σ + s) = 0 div S + s = 0 ε ij = χ s ij l 2 χ s ij Link with implicit gradient enhanced elastoplasticity models [Engelen et al., 2003] Boundary conditions : (u i, χ s ij ) or (σ ij + s ij )n j, S ijk n k Application to nickel foams 29/59

30 Plan 1 The micromorphic approach to plasticity Continuum thermomechanics Full micromorphic and microstrain theories Internal constraint in the micromorphic approach 2 Application to nickel foams Micromorphic anisotropic compressible plasticity model Simulation of crack propagation 3 Micromechanical approach

31 Fracture of nickel foams experiment model stress (MPa) simple fracture criterion p = p crit = 0.08 limited scatter in p crit strain softening law for p > p crit R = R(p > p crit ) tensile test in direction RD Application to nickel foams 31/59

32 Fracture anisotropy of nickel foams prediction of fracture stress for tension along TD experiment RD model RD experiment TD model TD stress (MPa) strain Application to nickel foams 32/59

33 Tension of central crack nickel foam plate crack length : 10 mm Application to nickel foams 33/59

34 Simulation with a classical compressible plasticity model force/ligament (MPa) nodes experiment displacement/gauge length strain component ε 11 Application to nickel foams 34/59

35 Simulation with a classical compressible plasticity model force/ligament (MPa) nodes 509 nodes experiment displacement/gauge length strain component ε 11 Application to nickel foams 35/59

36 Simulation with a classical compressible plasticity model force/ligament (MPa) nodes 1609 nodes 509 nodes experiment displacement/gauge length strain component ε 11 Application to nickel foams 36/59

37 Simulation with the micromorphic foam model force/ligament (MPa) experiment microfoam displacement/gauge length strain component ε 11 Application to nickel foams 37/59

38 Damage zone size Application to nickel foams 38/59

39 Convergence of the load displacement curve force/ligament (MPa) experiment microfoam 6667 nodes microfoam 3927 nodes microfoam 1609 nodes microfoam 509 nodes displacement/gauge length Application to nickel foams 39/59

40 Fracture anisotropy of nickel foams prediction of fracture stress for tension along TD force/ligament (MPa) experiment microfoam displacement/gauge length Application to nickel foams 40/59

41 Plan 1 The micromorphic approach to plasticity Continuum thermomechanics Full micromorphic and microstrain theories Internal constraint in the micromorphic approach 2 Application to nickel foams Micromorphic anisotropic compressible plasticity model Simulation of crack propagation 3 Micromechanical approach

42 Behaviour of Nickel foils elastoplastic model with linear isotropic hardening Micromechanical approach 42/59

43 Computing one real single cell Micromechanical approach 43/59

44 Computing one real single cell Micromechanical approach 44/59

45 Computing one real single cell Micromechanical approach 45/59

46 Computing one real single cell Micromechanical approach 46/59

47 Computing one real single cell Micromechanical approach 47/59

48 Computing one real single cell Micromechanical approach 48/59

49 Computing one real single cell Micromechanical approach 49/59

50 Computing one real single cell Micromechanical approach 50/59

51 Computing one real single cell Micromechanical approach 51/59

52 Computing one real single cell Micromechanical approach 52/59

53 Mesh sensitivity Micromechanical approach 53/59

54 Prediction of elastoplastic properties Young s modulus (GPa) RD TD ND E RD /E TD simulation experiment Micromechanical approach 54/59

55 Eight cell volume element DOF 36 GB 300 hours Micromechanical approach 55/59

56 Eight cell volume element DOF 36 GB 300 hours red: p > 0.03 Micromechanical approach 56/59

57 Eight cell volume element DOF 36 GB 300 hours red: p > 0.03 Micromechanical approach 57/59

58 Eight cell vs. one cell volume element Micromechanical approach 58/59

59 Perspectives Micromorphic approach in the mechanics of generalized continua Plasticity and fracture with the microfoam model Micromechanical definition of the additional degrees of freedom : link between the microdeformation tensor and the deformation of individual cells = see the presentation of Anthony Burtheau Micromechanical approach 59/59

60 Aifantis E.C. (1987). The physics of plastic deformation. International Journal of Plasticity, vol. 3, 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 Dillard T., Forest S., and Ienny P. (2006). Micromorphic continuum modelling of the deformation and fracture behaviour of nickel foams. European Journal of Mechanics A/Solids, vol. 25, pp Engelen R.A.B., Geers M.G.D., and Baaijens F.P.T. (2003). Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behaviour. International Journal of Plasticity, vol. 19, pp Eringen A.C. and Suhubi E.S. (1964). Nonlinear theory of simple microelastic solids. Int. J. Engng Sci., vol. 2, pp , References 59/59

61 Fleck N.A. and Hutchinson J.W. (2001). A reformulation of strain gradient plasticity. Journal of the Mechanics and Physics of Solids, vol. 49, pp Forest S. and Sievert R. (2003). Elastoviscoplastic constitutive frameworks for generalized continua. Acta Mechanica, vol. 160, pp Forest S. and Sievert R. (2006). Nonlinear microstrain theories. International Journal of Solids and Structures, vol. 43, pp Gurtin M.E. (1996). Generalized Ginzburg Landau and Cahn Hilliard equations based on a microforce balance. Physica D, vol. 92, pp Gurtin M.E. (2003). On a framework for small deformation viscoplasticity: free energy, microforces, strain gradients. International Journal of Plasticity, vol. 19, pp References 59/59

62 Mindlin R.D. (1964). Micro structure in linear elasticity. Arch. Rat. Mech. Anal., vol. 16, pp Mindlin R.D. (1965). Second gradient of strain and surface tension in linear elasticity. Int. J. Solids Structures, vol. 1, pp Mindlin R.D. and Eshel N.N. (1968). On first strain gradient theories in linear elasticity. Int. J. Solids Structures, vol. 4, pp Micromechanical approach 59/59