In the previous section, a complete comparison has been made between the “second-order” method (SOM), the “variational“ bound (V AR) and the high-rank, sequentially laminated results (LAM) in the case of transversely isotropic porous materials. In this section, we extend the predictions of the homogenization methods in the context of anisotropic microstructures. This extension is important, since the goal of this work is to propose a model that is able to approximate the effective
behavior of porous materials that are subjected to large deformations. A priori large deformations are synonymous with evolution of microstructure which in turn leads to an overall anisotropic response of the material. For a better understanding of the results that follow, it is useful to refer to Fig.2.3 for a complete definition of the geometry of the voids.
-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3
f=5%, m=0, w=1
PS axis HP axis
Σ22
Σ11
SOM VAR FEM
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2
-1.5 -1 -0.5 0 0.5 1 1.5 2
f=5%, m=0, w=10
PS axis HP axis
Σ22
Σ11
SOM VAR FEM
(a) (b)
Figure 4.11: Isotropic and anisotropic gauge surfaces as predicted by the SOM, the V AR and F EM calculations for porous materials with cylindrical voids with in-plane, aspect ratio (a)w= 1 and (b) w=10.
The matrix phase exhibits an ideally-plastic behavior (m= 0) and the porosityf= 5%.
In this regard, Fig. 4.11 shows comparisons between the SOM, the V AR and F EM results provided by Mariani and Corigliano (2001) in the case of porous media consisting of cylindrical voids with in-plane, aspect ratio w= 1 and w= 10 and an ideally-plastic matrix phase. In the first case, the response of the composite is transversely, isotropic about the 3−axis, whereas, in the second case, the material exhibits anisotropic behavior in the plane 1−2. When w= 1, in Fig.4.11a, the SOM estimates are in very good agreement with theF EM results, as already anticipated from the analysis of the previous section. The V AR estimate although quite accurate at low triaxialities, it overestimates the effective behavior of the composite at high ones. On the other hand, for w= 10 in Fig.4.11b, theSOM, although quite accurate for a large section of the stress space, it becomes too soft compared to the F EM results for Σ22>Σ11 >0 and Σ22<Σ11<0. The V ARestimate, in turn, remains stiffer than both the SOM and the F EM results in the entire stress space. It is noted that the case ofw= 10 corresponds to a material that is highly anisotropic in the plane and, in principle, the SOM shows some of the qualitative effects of this complicated behavior. A main effect of this anisotropy is the non-standard shape of the gauge curve as determined by the SOM and theF EM results, in contrast with theV ARestimate which remains a pure ellipsoid.
Next, Fig.4.12and Fig. 4.13show a set of results obtained by theSOM for three values of the porosity f = 1,5,10% and the aspect ratio w = 1,0.3,0.1 for a porous material with an ideally-plastic matrix phase. They are plotted in two different ways in order to highlight several effects of the anisotropy induced when the voids do not have a circular cross-section. First, in Fig. 4.12, the
-3 -2 -1 0 1 2 3
Figure 4.12: Anisotropic gauge surfaces obtained by theSOM for porous materials made up of cylindrical voids with in-plane, aspect ratiosw= 0.1,0.3,1, and porosities (a)f = 1%, (b)f = 5% and (c)f = 10%.
The matrix phase exhibits an ideally-plastic behavior (m= 0).
porosity is kept fixed and we let the aspect ratio change. For all the porosities, we observe that reduction of the aspect ratio, mainly, induces a rather significant softening of the material at high triaxiality loadings (look at the HP (hydrostatic) axis). While forf = 1%, the gauge curve shrinks with the reduction of the aspect ratio without changing its shape significantly, forf = 10% in turn, the reduction ofwinduces a crucial distortion of the gauge curve. More specifically, forf = 10%, a porous material with aspect ratiow= 0.1 is stiffer than the one withw= 1 in the 1−direction, which of course, is intuitively expected. In contrast, the material softens in the 2−direction as the aspect ratio decreases. Of course, the exactly opposite effect is observed when the aspect ratio increases, i.e.,wÀ1 (see for example Fig.4.11b).
On the other hand, in Fig. 4.13, the aspect ratio is kept fixed, while we let the porosity change.
-3 -2 -1 0 1 2 3
Figure 4.13: Anisotropic gauge surfaces obtained by theSOM for porous materials made up of cylindrical voids with porositiesf = 1,5,10% and in-plane, aspect ratios, (a)w= 1, (b)w= 0.3 and (c)w= 0.1. The matrix phase exhibits an ideally-plastic behavior (m= 0).
As expected, in all the cases shown here, the reduction of porosity has a hardening effect on the effective behavior of the porous medium, especially at high triaxialities. It is interesting to observe, though, that in the case of w= 0.1 the increase of the porosity causes a very small softening effect in the direction that the voids are elongated (see Fig.4.13c in the direction of Σ11). This is a direct consequence of the fact that while in the direction of the major axis of the elliptical void the material hardens, perpendicular to this direction the material softens (see Fig.4.13c in the direction of Σ22).
Hence, in the first case, even though the increase of the porosity induces overall softening for the composite, the elongation of the void acts against this by causing hardening in the direction of the major axis of the void. The result of this “competition” between the aspect ratio and the porosity brings about a very slight softening in the direction of the major axis of the void. On the other hand,
in the direction perpendicular to the major axis of the void the material is softer and together with the increase of the porosity, the material becomes significantly softer in this case.
-4 -3 -2 -1 0 1 2 3 4
Figure 4.14: Anisotropic gauge surfaces obtained by theSOM, theV ARand theGU Rmodels for porous materials made up of cylindrical voids with porositiesf = 1,5,10% and in-plane, aspect ratios, w= 1 and w= 0.1. The matrix phase exhibits an ideally-plastic behavior (m= 0).
Finally, Fig. 4.14 compares the SOM estimates with the “variational” results (V AR) and the Gurson criterion (GU R) for three values of the porosityf = 1,5,10% and two values for the aspect ratiow= 1,0.1. Of course, theGU Rcriterion contains no information about the shape of the void and hence only one curve is shown in each plot. A general comment for all the plots is that forw= 1 (transversely isotropic porous media) the SOM and the GU R models recover by construction the exact shell result defined by relation (2.195). In contrast, theV ARmethod is much stiffer than both theSOM and the GU Rmodel at high stress triaxialities. As long as low stress triaxiality loadings are considered and stillw= 1, theSOM and theV ARare in good agreement while theGU Rmodel
violates slightly theV ARbound.
More specifically, a main observation in the context of Fig. 4.14a, is that for f = 1% the V AR is significantly stiffer than the SOM for both values of the aspect ratio, especially at high stress triaxialities. In turn, the deficiency of theGU Rcriterion to capture any information for the shape of the voids becomes obvious in this plot. While theGU Rmodel is sufficiently accurate for circular voids and high stress triaxialities, when the aspect ratio becomes small, e.g.,w= 0.1 in this case, theGU R predictions become overly stiff. Similar observations can also be made in the context of Fig. 4.14b and Fig.4.14c. It is emphasized, though, that in the case off = 5% andf = 10% theGU Restimates violate considerably the V ARbound as a consequence of not being able to include information on the shape of the voids. Consequently, theGU Rmodel is expected to be highly inaccurate in the case of a porous material with an initially low (or high) aspect ratio.
In summary, the SOM is able to capture several features of the effective behavior of porous materials consisting of cylindrical voids with elliptical cross-section. Moreover, it is found that the aspect ratio of the voids affects significantly the response of the porous medium in all directions. In particular, the elongation of the voids can lead to geometric hardening or softening of the porous medium, which is in “competition” with the hardening or softening induced by the reduction or increase of the porosity. We will see in the following chapter how these two microstructural variables can affect the response of the porous medium when subjected to finite deformations.
4.5.1 Macroscopic strain-rates
For completeness, we present results for the macroscopic strain-rates as predicted by theSOM, the V ARand theGU Rmodels for porous materials consisting of cylindrical voids with (circular) elliptical cross-section. Similar to the previous gauge curves, we show results for two values of the aspect ratio w= 1,0.1 and a representative value of the porosityf = 5%. Note thatGU Rcurves are shown only forw= 1.
Fig. 4.15 shows curves for the deviatoric Ed = E11−E22 and the hydrostatic Em part of the normalized macroscopic strain-rate defined in relation (4.35) as a function of the stress triaxiality XΣ. More specifically Fig.4.15a shows estimates for the deviatoric part Ed. The main observation in the context of this figure is the asymmetry of the w = 0.1 curve about the Ed- and XΣ−axes, in contrast with the w = 1 curve, which is completely symmetric about these axes. Note that for w= 1, all the models deliver a zero deviatoric strain-rate atXΣ→ ±∞. Even so theGU Restimates are significantly lower than the corresponding SOM and V AR estimates in the case of w = 1. In turn, whenw= 0.1, the deviatoric strain-rate, predicted by both theSOM and theV ARmethods, is found to increase for low stress triaxialities, whereas it does not become zero in the purely hydrostatic limit (XΣ → ±∞). In fact, it reaches an asymptotic value for high triaxial loadings. Furthermore, it is worth to mention that, forw = 0.1, the deviatoric strain-rate becomes negative at sufficiently high positive stress triaxialities (i.e.,XΣ&3) and positive for sufficiently negative stress triaxialities (i.e.,XΣ.3).
On the other hand, Fig.4.15b shows results for the macroscopic mean strain-rateEmas a function
-10 -5 0 5 10 -2
-1 0 1 2
f=5% ,m=0
w=0.1 SOM
VAR GUR
w=1
X
ΣE
d-10 -5 0 5 10
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
f=5% ,m=0 w=0.1 SOM
VAR GUR
w=1
E
mX
Σ(a) (b)
Figure 4.15: Macroscopic strain-rates obtained by theSOM as a function of the porosityf and the stress triaxialityXΣ for two values of the aspect ratio w = 1,0.1. (a) shows the deviatoric partE11−E22 and (b) shows the hydrostatic partEm of the macroscopic strain-rate for an ideally-plastic matrix phasem= 0.
V ARandGU Restimates are shown for comparison.
of the stress triaxialityXΣ. First, we observe that theV ARmethod significantly underestimatesEm
for bothw= 1 andw= 0.1 when compared with theSOM method. Note that forw= 1, theSOM and theGU R models recover by construction the exact mean strain-rate obtained by a cylindrical hollow shell when subjected to purely hydrostatic loading. In addition, it is remarkable to observe such an increase ofEmin the case ofw= 0.1. This last result highlights the importance of proposing a model that is capable of handling porous media with more general elliptical microstructures.