3D plasticity. Write 3D equations for inelastic behavior. Georges Cailletaud, Ecole des Mines de Paris, Centre des Matériaux

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1 3D plasticity 3D viscoplasticity 3D plasticity Perfectly plastic material Direction of plastic flow with various criteria Prandtl-Reuss, Hencky-Mises, Prager rules Write 3D equations for inelastic behavior

2 3D plasticity and viscoplasticity - Strain partition ε e = Λ 1 : (σ σ I) th = (T T ε I ) α = Λ 1 : (σ ε σ I ) + ε th + ε p vp + ε - Criterion - Flow rule - Hardening rule f ε p =... Ẏ I =...

3 Formulation of viscoplastic constitutive equations The easiest way of writing a viscoplastic model is to define a viscoplastic potential, Φ, depending on stress and hardening variables. A standard model will then be characterized using the yield function f to define Φ, and deriving viscoplastic strain rate and hardening rate from Φ, Φ := Φ(f(σ, Y I )). Viscoplastic strain rate: State variable rate: vp = Φ ε σ α I = Φ Y I Introducing v = Φ f, n = f/ σ, and M I = f/ Y I ε vp = v n α I = v M I

A standard model will then be characterized using the yield function f to define Φ, and deriving viscoplastic strain rate

4 Viscoplastic potential, standard model stress ε. p. α I strain Elasticity f <0 f <0 low... Stress Hardening variables high... viscous effect ε vp = vn α I = vm I

5 Examples of simple viscoplastic models - Norton rule and von Mises criterion f = J(σ ), and : Φ = K n + 1 ( J(σ ) K ) n+1 ε vp = ( J K ) n J σ J = J s : = 3 s σ s σ 2 J : (I 1 3 I I) = 3 2 The elastic domain is reduced to one point. s J - Bingham model: ( J(σ ) σ y ) 2 Φ = 1 2 η

6 NOTE: partial derivative of σ with respect to s Tensor J = I 1 3 I I = J : σ s Index notation: J ijkl = 1 2 (δ ikδ jl + δ il δ jk ) 1 3 δ ijδ kl Other solution: J 2 = 3 2 s ijs ij then 2JdJ = 3s ij dσ ij J σ ij = 3 2 J σ = 3 2 s ij J s J

7 Viscoplastic potential, associated vs standard model Viscoplastic flow ε. p Elasticity f <0 Viscoplastic flow f>0 Elasticity f <0 f>0 α. I Hardening Stress Internal variables Stress Internal variables Standard model Associated model

8 Viscoplastic potential, general vs associated model Viscoplastic flow ε. p Viscoplastic flow ε. p Elasticity f <0 f>0 α. I Hardening Elasticity f <0 f>0 Stress Internal variables Stress Associated model Non associated model

9 From viscoplasticity to plasticity Φ Ind(f) σ ~ Y I a. Viscoplastic potential σ Y I b. Plastic pseudo-potential as a limit case - Viscoplasticity = after the choice of the function defining viscous effect, v is known - Plasticity = λ to be defined from the consistency condition

10 Formulation of the plastic constitutive equations elastic domain : f(σ, Y I )< 0 ( ε = Λ 1 : σ ) elastic unloading : f(σ, Y I )= 0 and f(σ, Y I )< 0 ( ε = Λ 1 : σ ) plastic flow : f(σ, Y I )= 0 and f(σ, Y I )= 0 ( ε = Λ 1 : σ + ε p ) ε p =... Ẏ I =...

11 Plastic pseudo-potential, associated vs standard model Elasticity. α I ε. p Plastic flow :. f=0, f=0 Plastic flow : f Elasticity.. ε = λ n. α I. f=0, f=0 f <0 f>0 forbidden area f <0 f>0 forbidden area Stress Internal variables Stress Internal variables Standard model Associated model

α I. f=0, f=0 f <0 f>0 forbidden area f <0 f>0 forbidden area Stress

12 Plastic pseudo-potential, general vs associated model Plastic flow : f Elasticity.. ε = λ n. α I. f=0, f=0 f g Elasticity ε. p =. λ Ν Plastic flow :. f=0, f=0 Stress f <0 Internal variables f>0 forbidden area Stress f <0 f>0 forbidden area Associated model Non associated model

13 Principle of maximal work The power of the real stress tensor σ associated to the real plastic strain rate ε p is larger than the power computed with any other admissible stress tensor σ id est a tensor respecting the plasticity condition associated to ε p. (Hill, 1951) (σ σ ) : ε p 0 - σ on the loading surface, σ in the domain, ε p = 0 - Normality rule, with σ in the tangent plane, k t : ε p 0 and k t : ε p 0 so that : t : ε p = 0 - Sign of the multiplyier, by setting σ on the interior normal, (σ on the surface), (σ σ ) = kn colinear to n (k > 0), and : k n : λn 0 then : λ 0

(Hill, 1951) (σ σ ) : ε p 0 - σ on the loading surface, σ in the domain, ε p = 0 - Normality rule, with σ in the tangent plane, k t : ε p 0 and k

14 Convexity of the loading surface f<0. p ε ~ n ~. σ ~ σ* ~ n ~ σ ~ a. Illustration of the normality rule b. Convexity of f

15 Perfectly plastic behavior (1) During plastic flow, the current point representing stress state can only follow the elastic domain. The plastic multiplyier cannot be determined using stress rate For f(σ ) = 0 and f(σ p f ) = 0 : ε = λ σ During plastic flow : n : σ = 0 = λ n

The plastic multiplyier cannot be determined using stress rate For

16 Perfectly plastic behavior (2) σ = Λ : ( ε ε p ) and n : σ = 0 n : σ = n : Λ : ( ε ε p ) = n : Λ : ε n : Λ : λ n λ = n : Λ n : Λ : ε : n Case of isotropic elasticity and von Mises criterion Λ ijkl = λ δ ij δ kl + µ(δ ik δ jl + δ il δ jk ) ; n ij = 3 2 s ij J n ij Λ ijkl = 2 µ n kl ; n ij Λ ijkl n kl = 3µ ; n ij Λ ijkl ε kl = 2 µ n kl ε kl λ = 2 3 n : ε For onedimensional loading, with ε = ε 11, this last expression can be written: λ = ε sign(σ) leading to: ε p = ε

jk ) ; n ij = 3 2 s ij J n ij Λ ijkl = 2 µ n kl ; n ij Λ ijkl n kl = 3µ ; n ij Λ ijkl ε kl = 2 µ n kl ε kl λ = 2 3 n

17 Flow directions associated with von Mises criterion f(σ ) = J(σ ) σ y (no hardening) n = f σ = J σ = J s : s σ where : n ij = J s kl s kl σ ij s kl σ ij = δ ik δ jl 1 3 δ ij δ kl Pure tension along direction 1 : s = 2 σ / /2 n ij = 3 2 s ij J where: n = 3 2 ; J = σ ; n = s J / /2 sign(σ)

δ ik δ jl 1 3 δ ij δ kl Pure tension along direction 1 : s = 2 σ 3 1 0 0 0 1/2 0

18 Flow directions associated with Tresca criterion - If σ 1 > σ 2 > σ 3, : f(σ ) = σ 1 σ 3 σ y, p then ε 22 = 0 (shear type deformation) : ε p = λ For pure tension (for instance σ 1 > σ 2 = σ 3 = 0) : f(σ ) = σ 1 σ 2 σ y, where f(σ ) = σ 1 σ 3 σ y ε p = λ µ

0 0 1 - For pure tension (for instance σ 1 > σ 2 = σ 3 = 0) : f(σ ) = σ 1 σ 2

19 Flow directions associated with Drucker Prager criterion Volume increase for any type of load: f(σ ) = J(σ ) (σ y α Tr (σ )) /(1 α) n = 3 2 s J + trace( ε p ) = α 1 α I 3 α 1 α λ

20 Prandtl-Reuss law (1) f(σ, R) = J(σ ) σ y R(p) - Hardening curve for onedimensional monotonic loading: σ = σ y + R(p). - Plastic modulus: H = dr/dε p = dr/dp For pure tension: n 11 = sign(σ), n 22 = n 33 = ( 1/2)n 11 ε p 11 = εp = sign(σ) λ, ε 22 = ε 33 = ( 1/2) ε p ṗ = ε p = λ For general 3D case: ε p : ε p = λ 2 n : n = 3 2 λ then ṗ = ( ) 1/2 2 3 ε p p : ε

- Plastic modulus: H = dr/dε p = dr/dp For pure tension: n 11 = sign(σ), n 22 = n 33 = (

21 Prandtl-Reuss law (2) - Use of the consistency condition: f σ : σ + f R Ṙ = 0 writes: n : σ Hṗ = 0 and: λ = n : σ H with n = 3 2 s J For pure tension: p = λ n ε = n : σ H n n 11 = sign(σ), n : σ = σ sign(σ) and: λ = ṗ = ε p 11 so that : ε p = n 11 σ H n 11 = σ H

22 Hencky-Mises Law Assumption of simple load The applied load in terms of stresses starts from an initial virgin state, and remains proportional to one scalar parameter k σ = k σ M ; σ = k σ M ; s = k s M ; J = k J M with 0 k 1 n = 3 2 s M J M constant n : σ H = 3 2 p = t 0 ṗ dt = σ M J M t 0 : σ M k = J M k λdt = - axial components: ε 11 = (σ 11 σ y )/H - shear components: 2 3 ε 12 = (σ 12 3 σy )/H 1 k e J M H dk

23 Prager rule (1) f(σ, X ) = J(σ X ) σ y with J(σ X ) = ( (3/2)(s X ) : (s X ) ) 0,5 Onedimensional loading : Tensile curve modeled by: σ X σ y = 0 σ = X(ε p ) + σ y Since X is proportional to ε p, its components for onedimensional loading are X 11, X 22 = X 33 = (1/2)X 11 Let us define: For onedimensional loading, assume: X = (2/3)Hε p X = (3/2)X 11 = Hε p 11 then s X = diag ((2/3)σ X 11, (1/3)σ + X 11 /2, id) = diag ((2/3)(σ X), (1/3)(σ X), id)) J(σ X ) = σ X

24 Prager rule (2) Consistency condition: f σ : σ + f X : Ẋ = 0 then : n : σ n : Ẋ = 0 with : n = 3 2 s X J(σ X ) n : σ = n : Ẋ = n : 2 3 H λ n = H λ so that : λ = (n : σ )/H p = λ n ε = n : σ H n - Same formal expression than for isotropic hardening, nevertheless n is different; - Under onedimensional loading, σ = σ 11, with X = (3/2)X 11 : σ X = σ y, σ = Ẋ = H εp

25 Plastic flow under strain control σ = Λ : ( ε ε p ) and : n : σ = Hṗ λ = n : Λ : ε H + n : Λ : n σ = Λ : ( ε ε p ) = ( Λ (Λ : n ) (n : Λ ) H + n : Λ : n ) : ε Isotropic elasticity and von Mises material: λ = 2µ n : ε H + 3µ

26 Non associated plasticity Threshold function Flow Hardening (1) f ε p = (2) f ε p = (3) f ε p = λ f σ λ f σ λ g σ α I = α I = α I = λ f Y I λ h Y I λ h Y I Model (1) is standard, Model (2) is simply associated (function f is not used for determining hardening evolution, but it is still used for flow direction), the shape (3), the most general, characterizes a non-associated model

27 Tangent behavior in non associated plasticity n : σ + f Y I Ẏ I = 0 with n = f/ σ λ = n : σ H f Y I, with : H = Y I α I h Y I Assuming N = g/ σ : λ = n : Λ : ε n : Λ : N ( = Λ σ (Λ ) : N ) (n : Λ ) H + n : Λ : N : ε

28 3D plasticity A classical rule for (visco)plastic flow p = λ n ε Time independent plasticity λ comes from the consistency condition Viscoplasticity λ comes from the viscoplastic potential n comes from the loading function (associated) or not (non associated) model Write 3D equations for inelastic behavior

Lecture 12: Fundamental Concepts in Structural Plasticity

Lecture 12: Fundamental Concepts in Structural Plasticity Plastic properties of the material were already introduced briefly earlier in the present notes. The critical slenderness ratio of column is controlled