Results characterizing the behavior of dilute porous materials consisting of isolated cylindrical voids with circular cross-section subjected to plane-strain conditions are discussed in this section. In par-ticular, theSOM and theV ARmethods are compared with corresponding results obtained by the works of Fleck and Hutchinson (1986) (F H) and Lee and Mear (1992b) (LM), which have been discussed briefly in chapter3.
Remark 4.2.1. However, before proceeding to the discussion of the results it is important to recall some observations made in the context of section 3.2, where the dilute methods of Fleck and Hutchinson (1986) and Lee and Mear (1992b) have been discussed. More specifically, these methods are based on a dilute expansion of the effective stress potential, as well as on the construction of a stream function to approximate the actual velocity field around the pore surface by making use of the Rayleigh-Ritz method. A first point that should be made is related to the fact that the range of validity of this dilute expansion diminishes to zero for purely hydrostatic loading and high nonlinearities close to the ideally plastic limit. This result is consistent with the observations made by Huang (1991b), who showed that a large number of terms in the representation of the stream function needs to be considered for convergence at high nonlinearities and stress triaxialities. In contrast, use of a small number of terms could lead to underestimation of the dilatation rate at high stress triaxialities and nonlinearities.
Remark 4.2.2. A second remark that should be made here is that Duva and Hutchinson (1984) and Duva (1986) found that the dilute methods, such as the one of Lee and Mear (1992b) under consid-eration here, become less accurate for small but finite concentrations of voids (fo ∼ 10−2) at high nonlinearities and stress triaxialities. The reason for this has been attributed by the aforementioned authors to the fact that these dilute techniques are not able to take into account the interactions between voids, which may become large at high nonlinearities even when the concentration of voids is small. For this reason, the SOM and V AR estimates are obtained numerically for a very low porosity offo= 10−6.
To begin with, the initial dilatation rate of an isolated void of circular cross-section is studied for an exponent n= 5 (orm= 0.2) as a function of a special measure of the stress triaxiality, denoted as σm/τ =√
3XΣ(see (3.17)). Fig.4.1shows the normalized dilatation rate ˙V2/( ˙γV2) as predicted
0 1 2 3 4 5 6
Figure 4.1: Dilatational rate for a dilute, transversely, isotropic porous medium, whose matrix phase is described by an exponentn = 5, predicted by theSOM. Corresponding homogenization estimates by the variational method (V AR) and numerical results by Fleck and Hutchinson (1986) (F H) and Lee and Mear (1992) (LM) are shown for comparison.
by the SOM, the V AR, the F H and the LM methods for dilute concentration of voids. A first observation that can be made in the context of Fig. 4.1 is that the V AR method underestimates significantly the dilatation rate as the stress triaxiality increases, while the SOM remains in very good agreement with the numerical F H and LM results. The improvement of theSOM over the V ARestimate is due to the fact that the SOM recovers — by construction — the analytical shell result for purely hydrostatic loadings (see subsection2.6.2). On the other hand, it has already been shown that, in the dilute limit and for purely hydrostatic loadings, theV ARestimate is inconsistent with the analytical shell result (see equation (3.45)).
0 0.5 1 1.5 2 2.5 3
Figure 4.2: Homogenization (SOM andV AR) and numerical estimates (LM) for (a)Pcir, which is related to the hydrostatic part of the strain-rate and (b)Qcir, which is related to the deviatoric part of the strain-rate, are shown for a porous medium consisting of dilute concentration of voids with circular cross-section and matrix phase described by exponentsn= 1,3,5,10 as a function of the stress triaxialityXΣ.
A more thorough study of the macroscopic strain-rate Din a dilute porous medium is discussed
in Fig. 4.2, where various values for the nonlinearity exponentn= 1,3,5,10 are considered. In this figure,Pcir andQcir denote measures of the hydrostatic and the deviatoric part of the strain-rateD, respectively (for further details see relations (3.28) and (3.29)), which are shown as a function of the stress triaxialityXΣ. The main observation in the context of this figure is that both theSOM and theLM estimates are strong functions of the stress triaxiality and the nonlinearity. In addition, the SOM estimates are found to be in good agreement with theLM results, although theLM predicts lower values for bothPcir andQcir than the SOM method. These differences could be explained by the comments made in remark 4.2.1. On the other hand, it is emphasized that the SOM model is based on a rigorous variational principle but does not constitute an exact solution. In this regard, it could lead to overestimation ofPcir andQcir. Note however that for purely hydrostatic loading the SOM is exact and thus, it is expected to predict accurately the effective behavior of porous media at high stress triaxialities.
On the other hand, the V ARunderestimates both the hydrostaticPcir and deviatoricQcir part of the macroscopic strain-rate when compared with theSOM and theLM results. In particular, the V ARestimate forPciris independent of the nonlinearityn, such that all the estimates coincide with then= 1 curve. This peculiar result can be explained by noting that theV ARestimate forPciris not linear in the porosityf in the dilute limit, as discussed in (3.45). On the other hand, the equivalent part of D (i.e.,Qcir) predicted by the V ARis linear in f as f →0 (Idiart et al., 2006). However, even in this case, the V AR method still underestimates Qcir when compared with corresponding estimates by theSOM and theLM methods, especially at high triaxialities and nonlinearities.
2 4 6 8 10
Figure 4.3: Homogenization (SOM) and numerical estimates (LM) for (a)P/Pcir, which is related to the hydrostatic part of the strain-rate and (b)Q/Qcir, which is related to the deviatoric part of the strain-rate, are shown for a porous medium consisting of dilute concentration of voids with elliptical cross-section and matrix phase described by exponentsn= 1,3,5. The results are plotted as a function of the in-plane aspect ratio 1/wof the elliptical voids for a uniaxial tension loading.
For completeness, Fig. 4.3shows results for the normalized P/Pcir andQ/Qcir for a fixed stress triaxiality XΣ= 1/√
3 or equivalently uniaxial tension loading, as a function of the in-plane aspect ratio 1/wfor nonlinearities of n= 1,3,5. As already mentioned previously, the quantitiesP andQ are associated with the hydrostatic and deviatoric part of the macroscopic strain-rateD, respectively, whereas Pcir and Qcir correspond to estimates for voids with circular cross-section (see definitions
2 4 6 8 10
Biaxial Tension, XΣ=3
P/Pcir Biaxial Tension, XΣ=3
SOM LM
(fo=10- 6) (dilute)
(a) (b)
Figure 4.4: Homogenization (SOM) and numerical estimates (LM) for (a)P/Pcir, which is related to the hydrostatic part of the strain-rate and (b)Q/Qcir, which is related to the deviatoric part of the strain-rate, are shown for a porous medium consisting of dilute concentration of voids with elliptical cross-section and matrix phase described by exponentsn= 1,3,5. The results are plotted as a function of the in-plane aspect ratio 1/wof the elliptical voids for a biaxial tension loading withXΣ= 3.
(3.28) and (3.29), respectively). The main observation in the context of this figure is that the SOM estimates for both P/Pcir and Q/Qcir are in very good agreement with the corresponding LM predictions for all aspect ratios shown here†. In particular, we observe in Fig. 4.3a that the normalized ratio P/Pcir increases almost linearly with the aspect ratio 1/w as predicted by both theSOM and theLM methods. On the other hand, both estimates exhibit no dependence on the nonlinear exponentn, since for all thenshown here, the curves coincide.
In turn, Fig. 4.3b shows corresponding SOM and LM estimates for the normalized equivalent part ofD, denoted withQ/Qcir, as a function of the aspect ratio 1/wfor a uniaxial tension loading.
Similar to the previous case, the increase of Q/Qcir is almost linear with respect to the in-plane aspect ratio 1/w. In contrast to theP/Pcir curves, the Q/Qcir estimates depend on the nonlinear exponent n, as predicted by both SOM and LM methods. Note that Q/Qcir increases at higher nonlinearities for the special case of uniaxial loading. As we will see in the following figure this trend is not preserved for higher triaxial loadings.
Fig. 4.4 shows results for the normalized dilatation P/Pcir and deviatoric Q/Qcir part of the macroscopic strain-rate D, with a fixed stress triaxiality XΣ = 3 (biaxial tension loading), as a function of the in-plane aspect ratio 1/w for nonlinearities of n = 1,3,5. The main observation in the context of this figure is that the SOM and the LM predictions for Q/Qcir are in better agreement than forP/Pcir. More specifically, we observe in Fig.4.4a, that as the nonlinear exponent nincreases, the rate of change ofP/Pcir decreases. This trend is obtained by both theSOM and the LM method, with the first one providing a lower rate of increase forP/Pcir than the LM method, especially for higher values ofn. It is important to emphasize though that this does not mean that theSOM underestimates P, since from Fig.4.2a it has already been deduced that, forn = 5, the SOM estimate forPcir lies higher than the correspondingLMprediction. However, in Fig.4.4a, the only relevant conclusion that we can draw is that the qualitative behavior ofP, as predicted by the
†In this case,V ARestimates are not included, since it has already been found in Fig.4.2that the they underestimate significantlyPcirandQcirand thus the comparison is not meaningful.
SOM is consistent with theLMresult for any aspect ratio 1/w. On the other hand, it is important to mention here that theSOM makes use of a prescription for the reference stress tensor, as described in subsection2.6.2, which is related with the evaluation of the hydrostatic behavior of a porous medium consisting of cylindrical voids with arbitrary aspect ratiow. Indeed this prescription is approximate and in general it could lead to “conservative” estimates for the estimation of the dilatation rate at high stress triaxialities.
On the other hand, in Fig. 4.4b, the corresponding SOM estimate for Q/Qcir is in very good agreement with theLM prediction. It is interesting to observe that the curves forn= 3 lie higher than the ones for n = 5 and n = 1. This interesting nonlinear behavior is captured by the SOM model, which is found to agree, at least qualitatively, with theLM predictions for the entire range of nonlinearities and aspect ratios at this high triaxial loading (XΣ = 3). Note that in Fig. 4.3 associated with a uniaxial tension loading, such a trend was not observed.
In summary, the above results provide a first indication of the improvement of the SOM on the earlier V AR method. Furthermore, the SOM has been found to compare well with the LM results for all nonlinearities, triaxialities and aspect ratios considered here. However, it is worth noting that the accuracy of the LM results (as well as the F H results) is expected to deteriorate at high nonlinearities and stress triaxialities for the reasons explained in remark4.2.1. This could result in an underestimation of the dilatational rate, as already observed by Huang (1991b). On the other hand, the SOM is constructed such that it recovers the exact hydrostatic solution of a CCA (composite cylinder assemblage), and as a consequence of this it is expected to be accurate at high stress triaxialities. In the sequel, we study the effective response of porous media consisting of cylindrical voids with circular (or elliptical) cross-section for finite porosities.