Procedia Engineering

ProcediaEngineering Engineering (2009) 13–16 Procedia 01 1(2009) 000–000 www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia

Mesomechanics 2009

Grain shape effect on the biaxial elastic-inelastic behavior of polycrystals with a self-consistent approach M. Radi and A. Abdul-Latif * Laboratoire de Mécanique, Matériaux et Modélisation (L3M) IUT de Tremblay, 93290 Tremblay-en-France, France, France Received 4 March 2009; revised 20 April 2009; accepted 24 April 2009

Abstract A simplified interaction law is derived describing the non-linear elastic-inelastic behavior of FCC polycrystals. The approach considers that the inclusion (grain) shape is ellipsoidal. A parametric study of the model parameters is carried out, particularly related to the inclusion shape and the new parameter of interaction law. The influence of these parameters on the polycrystal hardening (kinematic and isotropic) is studied using the yield surface evolution concept. © 2009 Elsevier B.V. All rights reserved Keywords: Self-consistent model; ellipsoidal inclusion; elastic-inelastic behavior

1. Introduction The solution of an ellipsoidal inclusion (grain) embedded in an infinite linear elastic homogeneous equivalent medium (matrix) has been theoretically developed for elastic-inelastic behavior of polycrystals by many research programs [1-5]. Hence, the model is based on the generalized non-incremental interaction law [3]. The developed interaction law is simplified considering the overall isotropic elastic behavior using the self-consistent approach. The overall viscous behavior is modeled through a new viscous parameter (γ). A parametric study is conducted for these parameters related notably to the grain shape (α) and the new viscous model parameter (γ). Their effects on the polycrystal hardening are investigated. The initial and subsequent yield surfaces are numerically constructed. 2. Self-consistent model By assuming an ellipsoidal inclusion embedded in infinite homogeneous matrix, a generalized elastic-viscoplastic self-consistent model proposed in [3] is expressed as follows:

* Corresponding author. Tel.: +33-1 4151-1234; fax: +33-1 4151-1249. E-mail address: [email protected]

doi:10.1016/j.proeng.2009.06.005

14

M. Radi, A. Abdul-Latif / Procedia Engineering 1 (2009) 13–16 Radi M., Abdul-Latif A./ Procedia Engineering 01 (2009) 000–000

⎛⎜ s −1 + C ⎞⎟ ℑ ⎝ ⎠

−1

−1 g : (σ& − Σ& ) + ⎛⎜ ℑ' + A ⎞⎟

⎝

−1

⎠

g g : (s − S) = (ε& − E& )

(1)

s ' ℑ and ℑ are respectively fourth rank tensors which have to be computed using A and C with Green function and integral methods. C is the global stiffness tensor. The fourth order tensor A represents the macroscopic tangent

's -1 −1 + A ) becomes negligible modulus. In the case where the elastic response dominates, the viscoplastic term ( ℑ with respect to the elastic part, and the interaction law can be thus written as deduced in [4]: −1

⎛⎜ ℑs −1 + C ⎞⎟ : (σ& g − Σ& ) = (ε& g − E& e ) e ⎝ ⎠

(2)

For a fully viscoplastic behavior dominating at stationary state, Eq. (1) can be therefore expressed approximately as: −1 ⎛⎜ ' −1 + A ⎞⎟ : (s g − S) = (ε& g − E& in ) (3) ℑ in

⎝

⎠

The isotropic elasticity tensor C for infinite homogeneous matrix under macroscopic loading is written as: Cijkl = λδijδ kl + μ(δik δ jl + δilδ jk )

(4)

The Eshelby’s tensor [6] for an ellipsoidal inclusion is used. The symmetrical tensor of elasticity ℑs is for isotropic behavior:

s ℑijkl =

(2) ⎡ ⎤ ~ QIJ 4πabc (1) I I I I ⎢ − (3λ + 8μ ))Iijkl⎥ : Cklop ((λ + μ ) + QIK )δijδkl + ( I I 2 15μ(3λ + 6μ ) ⎢ ⎥

⎣

(5)

⎦

where a, b and c are the half axes of the ellipsoidal inclusion and λI and μI are its Lame’s coefficients. 1 Ι ijkl = (δ ik δ jl + δ il δ jk ) is the fourth order unit tensor. For the viscoplastic part, the fourth order inelastic 2

′ interaction tensor ℑ is defined by:

1 ℑ′ijkl = (Pijkl + Pjikl + Pijlk + Pjilk ) (6) 4 The used fourth order macroscopic tangent modulus tensor proposed in [7] has the following form: in 2 ⎡2 ⎤ Α ijkl (E& ) = σ M (p& eq ) m −3 ⎢ (m − 1)E& inij E& inikl + (p& eq ) 2 K ijkl ⎥ 3 ⎣3 ⎦

(7)

with

K ijkl =

1

1 (δik δ jl + δil δ jk ) − δ ijδ kl 2 3

(8)

σ M is the stress in the matrix, p& eq is overall equivalent strain rate, m is the parameter characterizing the strain rate & in represents the macroscopic inelastic strain rate tensor. The tangent viscoplastic modulus η sensitivity and E ij

dependant on the strain rate state is defined by:

η(p& eq ) =

σ M eq m−1 (p& ) 3

The tangent modulus becomes:

(9)

and

p& eq =

2 & in & in E ij E ij 3

(10)

15

M. Radi, A. Abdul-Latif / Procedia Engineering 1 (2009) 13–16 Radi M., Abdul-Latif A./ Procedia Engineering 01 (2009) 000–000

⎤ eq -2 ⎡ 2 eq 2 & in & in Α ijkl ( E& ) = 2η (p& ) ⎢⎣ 3 (m − 1)E ij E kl + (p& ) K ijkl ⎥⎦

(11)

′ By rewriting the macroscopic tangent modulus tensor A and the inelastic interaction tensor ℑ , their matrix forms

[ ]

[ ′ ] and [ℑ′ ] = 1η [ℑ″ ]. By calculating the inverse of these matrices, we find:

can be defined as : A = η A

−1 1 ⎛ ″−1 ′⎞ +A ⎟ = ⎜ℑ (12) η⎝ ⎠

ProcediaEngineering Engineering (2009) 13–16 Procedia 01 1(2009) 000–000 www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia

Mesomechanics 2009

Grain shape effect on the biaxial elastic-inelastic behavior of polycrystals with a self-consistent approach M. Radi and A. Abdul-Latif * Laboratoire de Mécanique, Matériaux et Modélisation (L3M) IUT de Tremblay, 93290 Tremblay-en-France, France, France Received 4 March 2009; revised 20 April 2009; accepted 24 April 2009

Abstract A simplified interaction law is derived describing the non-linear elastic-inelastic behavior of FCC polycrystals. The approach considers that the inclusion (grain) shape is ellipsoidal. A parametric study of the model parameters is carried out, particularly related to the inclusion shape and the new parameter of interaction law. The influence of these parameters on the polycrystal hardening (kinematic and isotropic) is studied using the yield surface evolution concept. © 2009 Elsevier B.V. All rights reserved Keywords: Self-consistent model; ellipsoidal inclusion; elastic-inelastic behavior

1. Introduction The solution of an ellipsoidal inclusion (grain) embedded in an infinite linear elastic homogeneous equivalent medium (matrix) has been theoretically developed for elastic-inelastic behavior of polycrystals by many research programs [1-5]. Hence, the model is based on the generalized non-incremental interaction law [3]. The developed interaction law is simplified considering the overall isotropic elastic behavior using the self-consistent approach. The overall viscous behavior is modeled through a new viscous parameter (γ). A parametric study is conducted for these parameters related notably to the grain shape (α) and the new viscous model parameter (γ). Their effects on the polycrystal hardening are investigated. The initial and subsequent yield surfaces are numerically constructed. 2. Self-consistent model By assuming an ellipsoidal inclusion embedded in infinite homogeneous matrix, a generalized elastic-viscoplastic self-consistent model proposed in [3] is expressed as follows:

* Corresponding author. Tel.: +33-1 4151-1234; fax: +33-1 4151-1249. E-mail address: [email protected]

doi:10.1016/j.proeng.2009.06.005

14

M. Radi, A. Abdul-Latif / Procedia Engineering 1 (2009) 13–16 Radi M., Abdul-Latif A./ Procedia Engineering 01 (2009) 000–000

⎛⎜ s −1 + C ⎞⎟ ℑ ⎝ ⎠

−1

−1 g : (σ& − Σ& ) + ⎛⎜ ℑ' + A ⎞⎟

⎝

−1

⎠

g g : (s − S) = (ε& − E& )

(1)

s ' ℑ and ℑ are respectively fourth rank tensors which have to be computed using A and C with Green function and integral methods. C is the global stiffness tensor. The fourth order tensor A represents the macroscopic tangent

's -1 −1 + A ) becomes negligible modulus. In the case where the elastic response dominates, the viscoplastic term ( ℑ with respect to the elastic part, and the interaction law can be thus written as deduced in [4]: −1

⎛⎜ ℑs −1 + C ⎞⎟ : (σ& g − Σ& ) = (ε& g − E& e ) e ⎝ ⎠

(2)

For a fully viscoplastic behavior dominating at stationary state, Eq. (1) can be therefore expressed approximately as: −1 ⎛⎜ ' −1 + A ⎞⎟ : (s g − S) = (ε& g − E& in ) (3) ℑ in

⎝

⎠

The isotropic elasticity tensor C for infinite homogeneous matrix under macroscopic loading is written as: Cijkl = λδijδ kl + μ(δik δ jl + δilδ jk )

(4)

The Eshelby’s tensor [6] for an ellipsoidal inclusion is used. The symmetrical tensor of elasticity ℑs is for isotropic behavior:

s ℑijkl =

(2) ⎡ ⎤ ~ QIJ 4πabc (1) I I I I ⎢ − (3λ + 8μ ))Iijkl⎥ : Cklop ((λ + μ ) + QIK )δijδkl + ( I I 2 15μ(3λ + 6μ ) ⎢ ⎥

⎣

(5)

⎦

where a, b and c are the half axes of the ellipsoidal inclusion and λI and μI are its Lame’s coefficients. 1 Ι ijkl = (δ ik δ jl + δ il δ jk ) is the fourth order unit tensor. For the viscoplastic part, the fourth order inelastic 2

′ interaction tensor ℑ is defined by:

1 ℑ′ijkl = (Pijkl + Pjikl + Pijlk + Pjilk ) (6) 4 The used fourth order macroscopic tangent modulus tensor proposed in [7] has the following form: in 2 ⎡2 ⎤ Α ijkl (E& ) = σ M (p& eq ) m −3 ⎢ (m − 1)E& inij E& inikl + (p& eq ) 2 K ijkl ⎥ 3 ⎣3 ⎦

(7)

with

K ijkl =

1

1 (δik δ jl + δil δ jk ) − δ ijδ kl 2 3

(8)

σ M is the stress in the matrix, p& eq is overall equivalent strain rate, m is the parameter characterizing the strain rate & in represents the macroscopic inelastic strain rate tensor. The tangent viscoplastic modulus η sensitivity and E ij

dependant on the strain rate state is defined by:

η(p& eq ) =

σ M eq m−1 (p& ) 3

The tangent modulus becomes:

(9)

and

p& eq =

2 & in & in E ij E ij 3

(10)

15

M. Radi, A. Abdul-Latif / Procedia Engineering 1 (2009) 13–16 Radi M., Abdul-Latif A./ Procedia Engineering 01 (2009) 000–000

⎤ eq -2 ⎡ 2 eq 2 & in & in Α ijkl ( E& ) = 2η (p& ) ⎢⎣ 3 (m − 1)E ij E kl + (p& ) K ijkl ⎥⎦

(11)

′ By rewriting the macroscopic tangent modulus tensor A and the inelastic interaction tensor ℑ , their matrix forms

[ ]

[ ′ ] and [ℑ′ ] = 1η [ℑ″ ]. By calculating the inverse of these matrices, we find:

can be defined as : A = η A

−1 1 ⎛ ″−1 ′⎞ +A ⎟ = ⎜ℑ (12) η⎝ ⎠