268
P. Franciosi and M. Spagnuolo
f (ε(r)):
σ (r) = σ o (r) + l ε o (r) (r) : (ε(r) − ε o (r)) = l ε o (r) (r) :
ε(r) − ε
∗
o (r)
, (15.11)
with ε
∗
o (r) = ε o (r) − l
−1
ε o (r) (r) : σ o (r) and l ε o (r) (r) = ∂ f (ε(r))/∂ε| ε=ε o the tangent
moduli at previous iteration “o”, to step-wise determining the thermo-elastic (affine)
effective composite of comparison (see Masson et al. 2000 and given references
there). Using the Levin results (1967), the thermal properties entered quite easily
into the picture of elastic behavior modeling, hence the affine, thermo-elastic-like,
model success in practice. Also, with progress in second moment theory, it was shown
that the inner phase strain and stress fields were better represented by their second
moment than by simple averages (first order moments) owing to their nonuniformity
from the phase nonlinear behavior (Suquet 1995, 1997; Ponte-Castaneda and Suquet
1998; Ponte-Castaneda and Willis 1999; Buryachenko 2001: Brenner et al. 2001;
Lahellec and Suquet 2004). These two directions have allowed substantial modeling
improvements, although still complicated to deal with unless in elementary cases.
This so-called affine procedure could have been applied to the follow-up of the
here examined evolving microstructures, at least in its classical first order form,
although even the modified, second order, form seems quite at hand too,
4 . This
framework could likely be selected to develop a pretty well rigorous modeling of
the here examined structure types, what we did not need to do so far, for it remains
quite complex to fully implement. Simpler alternatives are often preferred, like the
use of secant moduli as linear comparison materials or even the true elastic behavior
of the phases in the so-called NTFA
5 modeling (Michel and Suquet 2004). Such an
attempt has been reported in (Franciosi and Berbenni 2007, 2008) for polycrystal
plasticity, with a particular use of the RT-IRT. Available analytical forms for the
mGOs representative of microstructures in concern (as those exemplified in this
lecture) can avoid the computations to become too huge machinaries. The same
route can be applied to other pairs of conjugate variables generalizing stress and
strain tensors.
15.5.2 From Elasticity-Type to Coupled Piezo-Type
Properties
These coupled problems (Li and Dunn 1998) which assemble the sub-combinations
from the thermo-electro-magneto-elastic (TMEE) type share into several more or
less difficult problems according to which properties are carried by the matrix phase:
namely, either the matrix has uncoupled properties (they can be isotropic then) or it
4 It consists in substituting the phase φ mean strains
ε φ
with the square root of their mean second
moments
ε φ ⊗ ε φ 1/2 of relatively easy calculations from the derivatives of the effective elastic
energy by the moduli of the phase (Brenner et al. 2001) compared with other difficulties.
5 Nonuniform Transformation Field Analysis.
P. Franciosi and M. Spagnuolo
f (ε(r)):
σ (r) = σ o (r) + l ε o (r) (r) : (ε(r) − ε o (r)) = l ε o (r) (r) :
ε(r) − ε
∗
o (r)
, (15.11)
with ε
∗
o (r) = ε o (r) − l
−1
ε o (r) (r) : σ o (r) and l ε o (r) (r) = ∂ f (ε(r))/∂ε| ε=ε o the tangent
moduli at previous iteration “o”, to step-wise determining the thermo-elastic (affine)
effective composite of comparison (see Masson et al. 2000 and given references
there). Using the Levin results (1967), the thermal properties entered quite easily
into the picture of elastic behavior modeling, hence the affine, thermo-elastic-like,
model success in practice. Also, with progress in second moment theory, it was shown
that the inner phase strain and stress fields were better represented by their second
moment than by simple averages (first order moments) owing to their nonuniformity
from the phase nonlinear behavior (Suquet 1995, 1997; Ponte-Castaneda and Suquet
1998; Ponte-Castaneda and Willis 1999; Buryachenko 2001: Brenner et al. 2001;
Lahellec and Suquet 2004). These two directions have allowed substantial modeling
improvements, although still complicated to deal with unless in elementary cases.
This so-called affine procedure could have been applied to the follow-up of the
here examined evolving microstructures, at least in its classical first order form,
although even the modified, second order, form seems quite at hand too,
4 . This
framework could likely be selected to develop a pretty well rigorous modeling of
the here examined structure types, what we did not need to do so far, for it remains
quite complex to fully implement. Simpler alternatives are often preferred, like the
use of secant moduli as linear comparison materials or even the true elastic behavior
of the phases in the so-called NTFA
5 modeling (Michel and Suquet 2004). Such an
attempt has been reported in (Franciosi and Berbenni 2007, 2008) for polycrystal
plasticity, with a particular use of the RT-IRT. Available analytical forms for the
mGOs representative of microstructures in concern (as those exemplified in this
lecture) can avoid the computations to become too huge machinaries. The same
route can be applied to other pairs of conjugate variables generalizing stress and
strain tensors.
15.5.2 From Elasticity-Type to Coupled Piezo-Type
Properties
These coupled problems (Li and Dunn 1998) which assemble the sub-combinations
from the thermo-electro-magneto-elastic (TMEE) type share into several more or
less difficult problems according to which properties are carried by the matrix phase:
namely, either the matrix has uncoupled properties (they can be isotropic then) or it
4 It consists in substituting the phase φ mean strains
ε φ
with the square root of their mean second
moments
ε φ ⊗ ε φ 1/2 of relatively easy calculations from the derivatives of the effective elastic
energy by the moduli of the phase (Brenner et al. 2001) compared with other difficulties.
5 Nonuniform Transformation Field Analysis.