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Generalized Continuum Modelling of Crystal Plasticity

Samuel Forest

MINES ParisTech Centre des Matériaux CNRS UMR 7633

BP 87 91003 Evry Cedex, France

1 Cosserat crystal plasticity

The classical theory of continuum crystal plasticity is first recalled and then generalized to incorporate the effect of lattice curvature on material hardening. The specific notations used in this chapter are summarized in section 1.6.

1.1 Dislocation densities and classical continuum crystal plastic- ity

Elements from a statistical theory of dislocations The yielding and hardening behaviour of crystals mainly depends on the growth of the dislo- cation population and on the development of dislocation structures inside the volume element V of continuum mechanics. A precise account of the evolution of dislocation distribution in V still lies beyond current comput- ing capacity, although promising results in that field are available Fivel et al. (1998). The incomplete information about the dislocation state per- mits probability predictions and suggests the use of statistical mechanics. Kröner proposed that the information be given in terms of n-point disloca- tion correlation tensor functions Kröner (1969, 1971). If the vectors ξ(x ) and b (x ) describe the line vector and Burgers vector of a dislocation located at x , the first correlation function reads :

α∼ =< b ⊗ ξ > (1) where the brackets denote ensemble averaging. The next correlation func- tion then is :

α∼∼ (x ,x ′) =< (b ⊗ ξ)(x )⊗ (b ⊗ ξ)(x ′) >= α∼∼(x − x

′) (2)

Generalized Continua and Dislocation Theory © CISM, Udine 2012 C. Sansour et al. (eds.),

182 S. Forest

if statistical uniformity is assumed. Second and fourth rank tensors α∼ and α∼∼

are indeed related to classical plastic state indicators used in classical

crystal plasticity. For a large enough volume element V , it is resorted to the ergodic hypothesis so that ensemble averaging is replaced by volume averaging over V . In that case, α∼ turns out to be identical to the so-called dislocation density tensor, or Nye’s tensor, which is the basic variable of the continuum theory of dislocations Nye (1953). On the other hand, one invariant of the tensor α∼∼

can be shown to be

αijij(0 ) = L/V (3)

where L is the length of dislocation lines inside V . The dislocation density � well-known in the field of metallurgy.

Classical thermomechanics of single crystals Classical crystal plas- ticity theory is the paradigm of the model for anisotropic finite deforma- tion plasticity. It has been founded in the original references Teodosiu (1970); Mandel (1971, 1973); Rice (1971); Teodosiu and Sidoroff (1976). More recent accounts of this continuum approach show the success of this approach based on multiplicative decomposition and the use of scalar dis- location densities � Maugin (1992); Teodosiu (1997); Gumbsch and Pippan (2005); Bertram (2005). It is useful to recall the basis of this theory before generalizing it to include the effect of the dislocation density tensor.

The main field equations of continuum thermomechanics are: • energy principle (local form)

ρ�̇ = σ∼ : D∼ + ρr − div q

where D∼ is the strain rate tensor, σ∼ the Cauchy stress and q the heat flux vector.

• entropy principle (local form)

ρη̇ + div (q T

) − ρr

T ≥ 0

where η is the specific entropy function. • Clausius-Duhem inequality

−ρ(ė− T η̇) + σ∼ : D∼ − q

T .grad T ≥ 0

The Helmholtz free energy density ψ = �− Ts is introduced:

−ρ(ψ̇ + ηṪ ) + σ∼ : D∼ − q

T .grad T ≥ 0

Generalized Continuum Modelling of Crystal Plasticity 183

• intrinsic and thermal dissipation

Dth = − q

T .grad T

Di = −ρ(ψ̇ + sṪ ) + σ∼ : D∼ The gradient of the transformation is called the deformation gradient F∼ . It is decomposed into elastic and plastic parts as shown in figure 1(a):

F∼ = F∼ e.F∼

p (4)

The uniqueness of this decomposition is ensured in the case of the single crystal material element if the orientation of the intermediate configuration is such that the lattice vectors do not rotate from the initial to the in- termediate configuration. This unique intermediate configuration is called isoclinic by Mandel (1973). The elastic strain and a stress tensor are defined with respect to the isoclinic configuration:

E∼ e =

1

2 (F∼

e.F∼ eT − 1∼), Π∼ e =

ρi ρ F∼

e−1.σ∼ .F∼ e−T (5)

The power of internal forces makes the link between the different stress tensors:

1

ρ σ∼ : (Ḟ∼ .F∼

−1) = 1

ρi (Π∼

e : Ė∼ e + (F∼

eT .F∼ e.Π∼

e) : (Ḟ∼ p .F∼

p−1)) (6)

The Clausius–Duhem inequality takes now the form

ρ( Π∼

e

ρi − ∂Ψ

∂E∼ e ) : Ė∼

e − ρ(s+ ∂Ψ ∂T

)Ṫ − ρ∂Ψ ∂α

: α̇+Dres ≥ 0 (7)

The state variables are (E∼ e, T, α) with additional internal variables α. The

state functions are the internal energy density e(E∼ e, η, α) and the Helmholtz

free energy density ψ(E∼ e, T, α). The exploitation of the second principle à

la Coleman–Noll provides the state laws

Π∼ e = ρi

∂Ψ

∂E∼ e , X = ρi

∂Ψ

∂α , s = −∂Ψ

∂T (8)

The residual dissipation then reads

Dres = M∼ : (Ḟ∼ p .F∼

p−1)−Xα̇ ≥ 0 (9)

This leads to the definition of the so–called Mandel stress tensor:

M∼ := F∼ eT .F∼

e.Π∼ e = C∼

e.Π∼ e (10)

184 S. Forest

We assume the existence of a convex dissipation potential Ω(M∼ , X) from which the flow rule and evolution equations for internal variables are derived:

Ḟ∼ p .F∼

p−1 = ∂Ω

∂M∼ , α̇ = − ∂Ω

∂X (11)

We present now a template model for single crystal behaviour introducing the specific internal variables: “dislocation density” �s, internal structure αs. Take the free energy function as

ρiψ(E∼ e, �s, αs) =

1

2 E∼

e : c∼∼ : E∼

e + r0

N∑ s=1

�s + 1

2 q

N∑ r,s=1

hrs�r�s + 1

2

N∑ s=1

αs2

(12) The state laws then read

Π∼ e = ρi

∂ψ

∂E∼ e = c∼∼

: E∼ e

rs = ρi ∂ψ

∂�s = r0 + q

N∑ r=1

hsr�s

xs = ρi ∂ψ

∂αs = cαs

The driving force for plastic slip is the resolved shear stress on slip system s

τ s = (M∼ .n s).m s = M∼ : (m

s⊗n s) (13) The multimechanism crystal plasticity yield criterion for single crystal is Schmid’s law:

fs = |τs − xs| − rs (14) The yield threshold is rs. An internal stress (back–stress) xs is introduced for each system s. Take the dissipation potential

Ω(M∼ , r s, xs) =

K

n+ 1

N∑ s=1

〈f s

K 〉n+1 (15)

from which flow and hardening rules are derived

ε̇∼ p =

∂Ω

∂σ∼ =

N∑ s=1

γ̇sm s sym ⊗ n s, �̇s = − ∂Ω

∂rs = v̇s, α̇s = − ∂Ω

∂xs = γ̇s (16)

with

v̇s = 〈f s

K 〉n, γ̇s = v̇s sign(τ s − rs) (17)

Nonlinear hardening rules are more realistic for actual materials in the form:

Generalized Continuum Modelling of Crystal Plasticity 185

• nonlinear isotropic hardening

�s = 1− exp(−bvs), rs = r0 + q ∑ r=1

hsr(1− exp(−bvs)) (18)

• nonlinear kinematic hardening

α̇s = γ̇s − dv̇sαs (19)

For monotonic loading, this equation integrates in

xs = c

d (±1− exp(−dvs)) (20)

The isotropic hardening involves an interaction matrix between slip systems. This corresponds to the notion of latent hardening Besson et al. (2001); Fivel and Forest (2004). The general form of this matrix for FCC crystals is⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

B4 B2 B5 D4 D1 D6 A2 A6 A3 C5 C3 C1 B4 h1 h2 h2 h4 h5 h5 h5 h6 h3 h5 h3 h6 B2 h1 h2 h5 h3 h6 h4 h5 h5 h5 h6 h3 B5 h1 h5 h6 h3 h5 h3 h6 h4 h5 h5 D4 h1 h2 h2 h6 h5 h3 h6 h3 h5 D1 h1 h2 h3 h5 h6 h5 h5 h4 D6 h1 h5 h4 h5 h3 h6 h5 A2 h1 h2 h2 h6 h5 h3 A6 h1 h2 h3 h5 h6 A3 h1 h5 h4 h5 C5 h1 h2 h2 C3 h1 h2 C1 h1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ A simplified form of the matrix is h1 = 1, h2 = h3 = h4 = h5 = h6 = h. If additionally h = 1, we obtain Taylor hardening.

Improved evolution equations for continuum crystal plasticity are ob- tained by comparing continuum and discrete dislocation dynamics methods Šǐska et al. (2009).

1.2 A Cosserat theory for elastoviscoplastic single crystals at finite deformation

Mandel Mandel (1973) introduced the notion of oriented microelements characterized by some hidden directors into the theory of elastoviscoplas- ticity. The epoch-making expression “trièdre directeur” is directly taken

186 S. Forest

from the Cosserat brothers’ well-known work Cosserat and Cosserat (1909). The relative rotation of neighbouring microelements may induce local cou- ple stresses. To a first approximation Mandel neglects th