In order to complete the study of the effective behavior of transversely isotropic porous materials, it is necessary to include plots for the entire range of the stress triaxialities. For this reason, we specialize the more general expression for the gauge function (Leblond et al., 1994), defined in subsection2.2.4 for ellipsoidal particulate microstructures, to the case of transversely isotropic porous media, such that
Φen(Σ;f) = Γn(Σ;f)−1. (4.34)
Here, Γn is the gauge factor defined by (2.151) for the “variational” method and (2.184) for the
“second-order” method, whereasΣ is a normalized stress tensor, defined in (2.30), that is homoge-neous of degree zero in σ. Then, the equation Φen = 0 describes the corresponding gauge surface defined in (2.29).
In turn, the normalized, macroscopic strain-rate is given by
E= D
˙
εo(Γn(σ;f)/σo)n = ∂Γn(σ;f)
∂σ , (4.35)
and the macroscopic stress and strain-rate triaxialities are expressed as XΣ= Σm
Σeq
= σm
σeq and XE = Em
Eeq
= Dm
Deq
, (4.36)
where the normalized, in-plane mean stress and strain-rate are defined as Σm = (Σ11+ Σ22)/2, Em= (E11+E22)/2, and Σeq andEeqdenote the von Mises equivalent parts of the normalized stress and strain-rate, respectively.
In the sequel, the “second-order” estimates (SOM), discussed in section2.6, are compared with corresponding results generated by the sequential laminates (LAM) described in section 3.4. In addition, these results are compared with the earlier “second-order” (SOM S) estimates using the simpler prescription ˇσ = σ0 for the reference stress tensor (see subsection 2.6.2), as well as with the “variational” estimates (V AR) and the Gurson criterion (GU R) (for the case of ideally-plastic materials). Note that Leblond et al. (1994) only considered axisymmetric loadings for the cylindrical microstructures, and therefore it was not possible to compare with their estimates for the case of plane-strain loading. However, relevant comparisons will be carried out in chapter 6 for isotropic porous media.
Fig. 4.7 shows the various estimates for the gauge surface and the corresponding macroscopic triaxialities, for moderate values of the porosity (f = 10%) and nonlinearity (m= 0.2). The main observation in the context of this figure is that the SOM estimates proposed in this work are in very good agreement with the exact LAM results, for the entire range of the stress triaxialities. In contrast, the agreement exhibited by the SOM S estimates is very good for low triaxialities, but deteriorates for sufficiently large triaxialities.
0.2 0.6 1 1.4 1.8
0 0.2 0.4 0.6 0.8 1
f =10%, m=0.2
LAM SOM
VAR
Shell
SOMS
Σeq
Σm
0.1 1 10 100
0.01 0.1 1 10
f =10%, m=0.2
LAM
X
ESOM VAR
SOMS
XΣ
(a) (b)
Figure 4.7: (a) Gauge surfaces and (b) strain-rate - stress triaxiality plots for a transversely, isotropic porous material with porosity f = 10% and strain-rate sensitivity parameter m = 0.2. The SOM is compared with corresponding estimates obtained by the variational method (V AR), the high-rank sequential laminates (LAM), and the earlier second-order estimate (SOM S) ( ˇσ=σ0).
In the hydrostatic limit, theSOM estimates coincide, by construction, with the exact result for CCAs, denoted by¤in Fig.4.7a. In turn, theLAM results, which correspond to high but finite rank laminates, also tend to this exact result — further support to the fact that theLAM results should agree exactly with the hydrostatic behavior ofCCAs in the limit of infinite rank has been provided by Idiart (2007). On the other hand, both the SOM S curve, and the V AR estimates are found to deviate from the analytical solution (2.195) in the hydrostatic limit. Note that Fig. 4.7b shows that, although the V ARpredictions for the strain-rate triaxiality XE tend to infinity as the stress triaxialityXΣbecomes infinite, they deviate significantly from theSOM and theLAM estimates for the entire range of the stress triaxialities. Correspondingly, the SOM S curve in this figure reaches
an asymptotic value at high stress triaxialities (i.e.,Deq 6= 0 as|XΣ| → ∞), which is consistent with
Figure 4.8: SOM gauge surfaces for a transversely, isotropic porous material, as a function of (a) the porosityf for a strain-rate sensitivity parameterm= 0.2, and (b) the strain-rate sensitivity parameter m for a porosityf= 10%. Corresponding results by the “second-order” (SOM) and the “variational” method (V AR), as well as the high-rank sequential laminates (LAM) are included for comparison. In the case of ideally-plastic matrix (m= 0), the Gurson criterion (GU R) is also shown.
At this point, it is worth noting that there exist evidence (Pastor and Ponte Casta˜neda, 2002) suggesting that the yield surface of porous materials with an ideally-plastic matrix phase may exhibit a corner on the hydrostatic axis (but see also Bilger et al. (2005)). A possible explanation for this vertex-like behavior could be the development of shear bands in the matrix phase. However, it is unrealistic to expect formation of shear bands for a porous material whose matrix phase is described by an exponentm >0. For this reason, we have assumed our new gauge surfaces to be smooth on the hydrostatic axis, even in the ideally-plastic limit.
Fig. 4.8 shows effective gauge surfaces for different values of the porosity f and the strain-rate sensitivity parameterm. The gauge surfaces delivered by theSOM are in very good agreement with the LAM estimates for the entire range of Σeq−Σm. In particular, gauge surfaces are shown in Fig. 4.8a as functions of the porosity f for a given value of m = 0.2, where it is clearly observed that theV ARmethod significantly overestimates the resistance of the porous medium at high stress triaxialities and low porosities, when compared with the SOM and LAM estimates. However, it is worth noting that the estimate delivered by the V AR method in the hydrostatic limit improves at higher porosities (see corresponding curve forf = 20% in Fig. 4.8a). This observation is a mere consequence of the fact that the V AR estimate (2.197) approaches the exact hydrostatic solution, given by relation (2.196), at high porosities.
Fig. 4.8b shows gauge surfaces as a function of the strain-rate sensitivity parameter m for a given value of porosityf = 10%. Due to numerical difficulties, LAM results are only provided for m≥0.2. Similarly to Fig. 4.8a, the SOM and theLAM estimates are in very good agreement for the entire range of macroscopic triaxialities, while theV ARestimate, albeit a rigorous upper bound,
remains too stiff as the nonlinearity of the matrix phase increases (i.e.,mdecreases). In addition, for the special case of ideally-plastic materials, theGU Restimate deviates significantly from the SOM estimate, despite the fact that it recovers the exact hydrostatic solution. As already anticipated, it violates the variational bound, V AR, for low stress triaxialities, in which case it tends to the Voigt bound. In any event, all the methods indicate a softening of the composite as the porosity or the nonlinearity increases, which is consistent with the contraction of the effective gauge surfaces.
4.4.1 Macroscopic strain-rates
0.010 0.1 1 10
0.5 1 1.5
m=0.2
Eeq
SOM VAR LAM
f =10%
f =1%
f =20%
XE 0.010 0.1 1 10
0.1 0.2 0.3 0.4 0.5 0.6
m=0.2
SOM VAR LAM
f =10%
f =1%
f =20%
XE
Em
(a) (b)
Figure 4.9: Macroscopic strain-rates obtained by theSOMas a function of the porosityfand the strain-rate triaxialityXE. (a) shows the equivalent partEeq and (b) shows the hydrostatic partEmof the macroscopic strain-rate for a strain-rate sensitivity parameterm= 0.2 and several porositiesf = 1,10,20%. V AR and LAM estimates are shown for comparison.
In order to complete the study of the macroscopic properties of the porous medium, Fig. 4.9 shows the two “modes” of the normalized, macroscopic strain-rate E, the equivalent (Eeq) and the hydrostatic (Em) mode, as functions of the strain-rate triaxialityXE and the porosityf, for a fixed value of the nonlinearity m = 0.2. The estimates obtained by the SOM for the modes Eeq and Em are found to be quite good, when compared with the LAM estimates, for the whole range of triaxialities and porosities, considered here. On the other hand, the V AR bound underestimates Eeq at low triaxialities and Em at high triaxialities, when compared with the LAM and the SOM estimates. In particular, at high triaxialities, it predicts an asymptotic value for Em that is 40%
lower than theSOM estimate (see Fig.4.9forf = 1%) and thus, the exact result forEmobtained by direct derivation of the exact relation (2.195) with respect toσm.
Finally Fig. 4.10showsEeq and Em, as functions of the strain-rate triaxialityXE and the non-linearity m, for a fixed value of the porosity f = 10%. Note that for the case of m = 0 (i.e., ideal-plasticity) theGU Restimates are also shown. TheSOM estimates are in very good agreement with theLAM results for the entire range of the strain-rate triaxiality and nonlinearities considered.
On the other hand, as already anticipated, the V AR method underestimates the two modes Eeq
and Em at low and high strain-rate triaxialities, respectively. In addition, it is further noted that,
0.010 0.1 1 10
Figure 4.10: Macroscopic strain-rates obtained by the SOM as a function of the strain-rate sensitivity parameter m and the strain-rate triaxiality XE. (a) shows the equivalent part Eeq and (b) shows the hydrostatic partEmof the macroscopic strain-rate for a porosityf= 10% and several exponentsm0,0.2,0.5.
V AR and LAM estimates are shown for comparison. In the case of ideally-plastic matrix (m = 0), the Gurson criterion (GU R) is also shown.
although GU R recovers by construction the analytical estimate for Em and Eeq = 0 in the limit of hydrostatic loading for the case of ideally-plastic materials (provided that the case of a corner in the yield surface is excluded), it is found to underestimate both of these modes of the strain-rate for moderate to low triaxialities, when compared with theSOM estimates. It should be emphasized that the accurate prediction ofEm is critical, as it controls the dilatation rate of the voids, which is even more sensitive at high triaxialities. The dilatation of the voids may lead to a significant increase of the porosity measure and eventually to the failure of the material. Therein lies the significance of theGU Rmodel which was the first to be able to account for this effect.
In summary, the new SOM estimates are found to be in very good agreement with the LAM results for the case of transversely isotropic porous media consisting of cylindrical voids. In particular, the new prescription for the reference stress tensor, discussed in the context of section2.6, improves significantly on earlier choices by being able to recover the exact hydrostatic shell result. On the other hand, theV ARmethod overestimates significantly the effective strength of the porous material and as a consequence underestimates the hydrostatic part of the strain-rate at high triaxialities.