Numerical implementation

. (2.297)

All the quantities involved in (2.297) are evaluated in the hydrostatic limit XΣ → ±∞, i.e., for σ= sgn(XΣ) (2σew/3)I. In particular,d||=σewS·M(kc ^H)I, whereM(kc ^H) is independent ofµand is given by

M(kc ^H) =M(k^H) + f

1−f Q(kb ^H)⁻¹, (2.298)

in the context of the “second-order” method. Similarly,dvar= eσ_w^varS·cMI, withcMgiven by (2.143) or equivalently by (2.298) withk^H = 1. Furthermore, for the computation of the anisotropy ratiok^H in (2.297), it is necessary to solve the nonlinear equation (2.267), in the hydrostatic limit.

In the context of these results for the special case of porous materials with ideally-plastic matrix phase, expressions (2.206) (or (2.297)) have been assumed to continue to hold for the determination of αeq, which implies that the resulting effective yield surfaceΦesomremains smooth for the entire range of the stress triaxialities. Further support for this last assumption arises from the fact that — to the best knowledge of the authors — there is no definitive numerical or experimental evidence implying the existence of a vertex in the hydrostatic limit for general isotropic or ellipsoidal microstructures (but see also Bilger et al., 2005), in contrast with transversely isotropic microstructures, where the existence of a corner may be observed in yield surfaces obtained by limit analysis procedures (Pastor and Ponte Casta˜neda, 2002). Note that, for the case of a vertex-like yield surface, the strain-rateD would not be uniquely determined at the hydrostatic point.

2.8.3 Numerical implementation

In this subsection, we describe the numerical implementation of the equations developed in the context of porous media with ideally-plastic matrix phase, which are similar for the “variational”

and the “second-order” method. For the “variational” method a complete numerical description of the equations has been provided by Kailasam (1998) and Aravas and Ponte Casta˜neda (2004) and for completeness will be repeated briefly here. On the other hand, the numerical implementation of the “second-order” equations for porous media with ideally-plastic matrix phase is presented in more detail.

First, it is essential to define the boundary conditions in the problem. For convenience, we will consider here that velocity boundary conditions are given such that

v=Lx, ∀x ∈ ∂Ω, (2.299)

whereL is the macroscopic velocity gradient. The symmetric and skew-symmetric part ofL denote the macroscopic strain-rate D = 1/2

h L+L^T

i

and Ω = 1/2 h

L−L^T i

. Note that in the ideally-plastic limit the problem is rate-independent and thus the magnitude ofv(orL) does not affect the

final result. Thus, we can define the total displacementuin terms of the velocityvvia

u=vt, 0< t < tf, (2.300) wherev is constant in timet andtf is the total time.

In the following, the problem is solved incrementally using implicit and explicit (backward and forward Euler) schemes. First, we present the system of equations for the “variational” method, and then for the system for the “second-order” method. In the sequel, we update the values for the microstructural variables, which is the same for both methods considered here.

“Variational” method

At the beginning of the incrementt=tj, the known quantities are

D, sα|_j. (2.301)

At the end of the incrementt=tj+1, the following quantities need to be computed

σj+1, ˙Λj+1. (2.302)

In the general case, the total number of unknowns are 7. More specifically, the unknowns are computed by the following equations:

Φevar(σj+1;sα|_j) = 0, (#1) (2.303) and

D= ˙Λ∂Φevar

∂σ

¯¯

¯¯

¯j+1

(#6). (2.304)

The above set of equations is solved by using a standard Newton-Raphson technique, while the symbol

# has been used to denote the number of equations.

“Second-order” method

At the beginning of the incrementt=tj, the known quantities are

D, s_α|_j. (2.305)

At the end of the incrementt=tj+1, the following quantities need to be computed

σj+1, ˙Λ_j+1, kj+1, αm|_j+1, k^H_j+1, (2.306) where k is the anisotropy ratio of the matrix phase in the LCC and αm is a factor involved in the computation of the reference stress tensor ˇσgiven by relation (2.204). It is recalled here that for the computation of αm, the hydrostatic limit (|XΣ| → ∞) needs to be considered also in the equation (2.267) for the anisotropy ratio denoted ask^Hin this limiting case. Note that the number of unknowns in the more general case is 10.

More specifically, the unknowns are computed by the following equations:

Φe_som(σ_j+1;k_j+1, α_m|_j+1, k^H_j+1, s_α|_j) = 0, (#1) (2.307) D= ˙Λ∂Φesom

∂σ

¯¯

¯¯

¯j+1

, (#6) (2.308)

(1−kj+1) ˆσ_||¯

¯j+1−ˇσeq|_j+1= 0, (#1) (2.309)

e

σ_w^som(αm|_j+1, k^H) =eσw, (#1) (2.310)

(1−k^H_j+1) ˆσ_||^H

¯¯

¯j+1−ˇσ_eq^H¯

¯j+1= 0, (#1) (2.311)

where the superscriptH has been used to denote that the relevant quantities need to be computed in the hydrostatic limit|XΣ| → ∞, whereas eσw is given by (2.196). For clarity, it is worth mentioning that ˇσ = ˇσ(σ;αm, sα) and ˆσeq(σ; ˇσ, sα). Thus, all the above equations are coupled in the general case and need to be solved simultaneously.

Update of the microstructural variables

Once, the macroscopic stressσand the plastic multiplier ˙Λ are known, we can update the microstruc-tural variables sα|_j+1. Note that this step is the same for the “variational” and the “second-order”

method. Then, we use an explicit scheme to update the microstructural variables by noting that the time increment is ∆t=tj+1−tj, such that

fj+1=fj+ (1−fj)Dii∆t, (2.312) w1|_j+1= w1|_j+

³

n⁽³⁾_j · D⁽²⁾n⁽³⁾_j −n⁽¹⁾_j · D⁽²⁾n⁽¹⁾_j

´

∆t (2.313)

w2|_j+1= w2|_j+³

n⁽³⁾_j · D⁽²⁾n⁽³⁾_j −n⁽²⁾_j · D⁽²⁾n⁽²⁾_j ´

∆t (2.314)

n⁽ⁱ⁾_j+1=n⁽ⁱ⁾_j +ωn⁽ⁱ⁾_j ∆t, i= 1,2,3. (2.315) Note that in the last relation for the update of the orientation vectorsn⁽ⁱ⁾, the above explicit scheme leads to non-unit vectors due to the incremental approximation introduced in the problem. This can be resolved by integrating exactly the orientation vectors n⁽ⁱ⁾ (Aravas and Ponte Casta˜neda, 2004), which leads to

n⁽ⁱ⁾_j+1= exp(ω)n⁽ⁱ⁾_j , i= 1,2,3. (2.316) The exponential of the skew-symmetric tensorωis an orthogonal tensor that can be determined from the following formula, attributed to Gibbs (Cheng and Gupta, 1989)

exp(ω) =I+sinx

x ω+1−cosx

x² ω², with x= r1

2ω·ω. (2.317) It is important to emphasize that the above described formulation is adequate for the purposes of this study, however, due to the explicit scheme used here, a small time increment is required. Instead, for a faster solution of the system of equations use of an implicit scheme should be made (see Aravas and Ponte Casta˜neda, 2004), which would allow for a larger time increment.

Finally, in many cases of interest, the applied boundary conditions are such that the stress triax-ialityXΣ is constant during the deformation process. The above mentioned system of equations can

be easily modified by imposing the constraint of constant stress triaxiality. This would imply that one of the equations associated with the flow rule, i.e., D = ˙Λ∂Φ/∂e σ needs to be replaced by the one ofXΣ=const.