Gauge surfaces for isotropic porous media

r2

3Σeqcos(θ) (6.33)

and

y=Σ1−Σ2

√2 = r2

3Σeqsin(θ). (6.34)

On the other hand, the macroscopic strain-rate and strain-rate triaxiality measures recalled here to be

E= D

˙

ε_o(Γ_n(σ;f)/σ_o)ⁿ = ∂Γn(σ;f)

∂σ , XE= Em

Eeq

, (6.35)

where the normalized, mean strain-rate is defined as,Em=Eii/3, whereasEeqdenotes the von Mises equivalent part of the normalized strain-rate, defined in terms of the deviatoric strain-rate tensorE⁰ asEeq =

q

2

3E⁰·E⁰.

In summary, the previous definitions allow us to study in an efficient and complete manner the effective response of isotropic porous media subjected to general loading conditions.

6.4.2 Gauge surfaces for isotropic porous media

Before proceeding to the discussion of the results, it is useful to introduce first the material and loading parameters used in the plots that follow. The present study is focused on high nonlinearities such as m= 0.1 and m= 0 (i.e., ideally-plastic materials) and small to moderate porosities,f = 1,5,10%.

Results for porosity levels below 1% — relevant to ductile fracture — have not been included in the present section due to difficulties encountered in the numerical computation of theLAM values. In contrast, it is emphasized that SOM estimates have been obtained for small and dilute porosities and have already been reported in the section6.2. It is noted, however, that the conclusions drawn below are expected to remain valid at those porosity levels.

As already discussed in the previous section, theSOM and theLAM models depend on all the three invariants of the macroscopic stress tensor (i.e., on Σm, Σeq and θ) introduced in relations (6.30) and (6.32). For completeness, gauge curves are shown for three representative values of the Lode angle,θ= 0, π/4, π/2. The value θ= 0 is associated with an axisymmetric shear loading. In

turn, the valueθ=π/2 denotes a simple shear loading condition, whereasθ=π/4 corresponds to a combination of axisymmetric and simple shear loading. It is emphasized that the rest of the models (V AR,LP S andGU R) depend only on the first two invariants, i.e., on Σm and Σeq.

Fig. 6.8 shows effective gauge curves in the Σm−Σeq plane (or meridional plane) for a fixed nonlinearity m = 0.1. In Fig. 6.8a the various models are compared for axisymmetric loadings (θ= 0) and a typical porosityf = 5%. The main result in the context of this figure is that theSOM estimates are in very good agreement with the LAM predictions for the entire range of the stress triaxialities (XΣ ∈ (−∞,∞)), while both models recover the exact effective response of a hollow shell subjected to pure hydrostatic loading, described by relation (2.195) and (2.196). The fact that theLAM results should agree exactly with the hydrostatic behavior of CSAs in the limit of infinite rank has been shown rigorously in (Idiart, 2007). Furthermore, the SOM improves “significantly”

on the earlierV ARestimate, which in spite of being in good agreement with theLAM results at low triaxialities, it is found to be too stiff at high triaxialities. Indeed, for the given porosity f = 5%, the V AR method predicts that the effective response of the porous material is almost 50% stiffer than the exact shell result, when subjected to pure hydrostatic loading. It can be easily verified by comparing relations (2.196) and (2.197), that the V AR estimate deviates significantly from the analytical shell result for small porosities. On the other hand, the LP S model, even though exact in the hydrostatic loading, it deviates from theLAM results for moderate to high triaxialities (i.e., approximately 1≤XΣ≤10 ). In contrast, theLP S model coincides — by construction — with the V ARbound for isochoric loadings (XΣ= 0).

Figs. 6.8b, 6.8c, 6.8d, show gauge curves for three different Lode angles, θ = 0, π/4, π/2, and porosities,f = 1, 5,10%. With these graphs, we verify the very good correlation between theSOM and theLAM estimates for the whole range of stress triaxialities, porosities and Lode angles shown here. Furthermore, it is interesting to note that forθ= 0 andθ=π/4, the predicted gauge curves of both theSOM and theLAM are found to be slightly “asymmetric” about the Σeq-axis, in contrast with the case of θ = π/2, where the SOM and the LAM curves are completely symmetric. This

“asymmetry” is a direct consequence of the coupling between the three invariants (i.e., Σm, Σeq, θ) in the expression for the gauge function introduced in relation (6.32). While theSOM and theLAM curves exhibit this particularly interesting behavior, the V AR and the LP S model show a lack of this asymmetric effect about the Σeq-axis. This is a direct consequence of the fact that theV ARand theLP S models involve no dependence on the third invariant of the macroscopic stress tensor, and thus they cannot capture this effect, which, as will be seen later, can become non negligible at high stress triaxialities.

More specifically, in the case of axisymmetric loading (θ = 0) (see Fig.6.8b), theSOM and the LAM estimates are found to be slightly stiffer in the negative pressure regime (Σm<0), while the converse is observed whenθ =π/4 (see Fig.6.8c), i.e., the porous material is stiffer in the positive pressure regime (Σm>0). On the other hand, for θ=π/2 (see Fig. 6.8d), the corresponding gauge curves are completely symmetric about the Σeq-axis. This can be explained by noting that in the case ofθ =π/2, the term det(Σ⁰/Σeq) = 0, which has the implication that the gauge function becomes an even function of the mean macroscopic stress, and therefore is independent of the sign of Σm. In

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Figure 6.8: Gauge surfaces (on the Σm−Σeq plane) for isotropic, porous materials as predicted by the SOM, the high-rank sequential laminatesLAM, theV ARand the recently updated Leblond-Perrin-Suquet (1994) (LP S) model by Flandi and Leblond (2005) as a function of the porosity and the Lode angle θ(θ is directly related to the third invariant of the macroscopic stress tensorΣ). The strain-rate sensitivity of the matrix phase ism= 0.1. (a) shows gauge surfaces as predicted by the aforementioned models for a porosity f= 5% and Lode angleθ= 0 (axisymmetric loading). The rest of the graphs show gauge surfaces for several porositiesf= 1,5,10% and (b)θ= 0, (c)θ=π/4 (combination of in-plane shear and axisymmetric loading), and (d)θ=π/2 (in-plane shear–pressure loading) as predicted by theSOM and the exactLAM results.

contrast, for θ= 0 and θ=π/4, the term det(Σ⁰/Σeq) is not zero and hence the gauge function is not an even function of Σm. In addition, it is important to remark that the observed asymmetry of theSOM and theLAM gauge curves about the Σeq-axis is more pronounced at moderate porosities (f = 10%) than small ones (f = 1%), as can be seen in Figs.6.8b, 6.8c,6.8d.

The special case of porous materials with an ideally-plastic matrix phase is studied next. Specif-ically, gauge (or yield) surfaces (m = 0) are shown in Fig. 6.9 for a fixed porosity f = 5% and θ = 0, π/2. Here, the GU R model is also included, whereasLAM estimates are not available for this case due to numerical difficulties. The main observation in the context of this figure is that even though all but the V AR estimate recover the analytical hydrostatic point, theSOM exhibits a softer behavior than the LP S and the GU R models at moderate and high stress triaxialities. In addition, similar to Fig. 6.8, an interesting effect of the presence of the third invariant is the asym-metry of the gauge curve predicted by theSOM about the Σeq-axis, in the case of θ= 0, as shown in Fig.6.9a. In this case, theSOM estimates are found to be slightly stiffer in the negative pressure

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Figure 6.9: Gauge surfaces (on the Σm−Σeq plane) for isotropic, porous materials as predicted by theSOM, theV AR, the Gurson model (GU R) and the recently updated Leblond-Perrin-Suquet (1994) (LP S) model by Flandi and Leblond (2005) as a function of the porosity and the Lode angleθ(θis directly related to the third invariant of the macroscopic stress tensor Σ). The matrix phase exhibits an ideally-plastic behavior m= 0. The graphs show gauge surfaces for porosityf= 5% and (a)θ= 0, (b)θ=π/2 as predicted by the aforementioned models.

regime (Σm < 0) than in the positive pressure regime (Σm > 0). On the contrary, for θ = π/2, the corresponding gauge curve is symmetric about the Σeq-axis, since in this case det(Σ⁰/Σeq) = 0.

Furthermore, it is emphasized that theGU Rmodel violates, as already anticipated, theV ARbound at low triaxialities. To amend this drawback of theGU Rmodel at low triaxialities, theLP S model was constructed such that it recovers theV AR bound for isochoric loadings (i.e.,XΣ= 0), while it lies very close to theGU Rmodel for moderate and high triaxialities.

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Figure 6.10: Gauge surfaces (on the Σm−Σeqplane) for isotropic, porous materials as predicted by theSOM and the high-rank sequential laminatesLAM as a function of the nonlinear exponent and the Lode angleθ (θis directly related to the third invariant of the macroscopic stress tensorΣ). The porosity takes the typical value f = 5%. In particular, this figure shows gauge surfaces as predicted by the aforementioned models for nonlinear exponents n= 2,5,∞% and Lode angle (a)θ = 0 and (b)θ =π/2 (in-plane shear–pressure loading).

For completeness, Fig. 6.10shows SOM and LAM gauge surfaces for nonlinear exponents n= 2,5,∞ and a typical porosityf = 5%. Part (a) of this figure corresponds to Lode angle θ= 0 and

part (b) toθ=π/2. The main observation in the context of this figure is that bothSOM andLAM gauge surfaces are functions of the nonlinear exponent, as already expected, while the dependence on n becomes somewhat stronger at higher stress triaxialities. In addition, it is observed that the effect of the third invariant of the macroscopic stress tensor on the effective response of the porous medium, indicated by the slight asymmetry of the gauge curves about the Σeq−axis, diminishes at low nonlinearities, i.e., n = 2. This is expected since in the purely linear case (n= 1) there is no effect of the Lode angle.

In summary, the previous analysis made in the context of Fig. 6.8, Fig. 6.9and Fig. 6.10shows that the effective response of the porous material as predicted by the SOM and theLAM models exhibits somewhat softer behavior when compared with the estimates obtained by theLP S and the GU R models. In addition, theSOM and theLAM estimates depend on the third invariant of the macroscopic stress tensor Σ, in contrast with the rest of the models (V AR, LP S and GU R) that depend only on the first two invariants ofΣ. Similar observations have been made by Shtern et al.

(2002a,b) in the context of unit-cell, porous media, who introduced the effect of the third invariant of the macroscopic stress tensor in a somewhat ad-hoc manner.

For a better understanding of the effect of the third invariant on the effective response of the porous material, we need to study an alternative cross-section of the gauge surface, as shown in Fig. 6.11 for a fixed porosity f = 10% and strain-rate sensitivity parameter m = 0.1. This cross-section lies on the deviatoric or Π−plane which is defined (see subsection 6.4.1) by considering constant mean stresses Σm= 0 and Σm= 0.99Σ_m^H (with Σ_m^H denoting the mean stress delivered by the analytical shell result for a given porosity and nonlinearity, given in Appendix V of Chapter2). The origin of these two graphs corresponds to zero deviatoric macroscopic stressΣ⁰, while Σ₁⁰, Σ₂⁰ and Σ₃⁰are the three principal values ofΣ⁰.

Fig.6.11a thus presents the deviatoric cross-sections of the gauge surface obtained by the SOM, the LAM and the LP S models for Σm = 0. It is recalled that for isochoric loadings, the LP S model is identical to the V ARbound. In this figure, all the methods give very similar predictions for all θ. However, in order to understand further these results, it is useful to recall that overall isotropy of the porous material together with the fact that the corresponding effective gauge function in relation (6.32) is insensitive to the sign of Σ (see subsection6.4.1) yields the continuous and the dotted symmetry lines on the Π-plane (see Fig. 6.11a). This implies that the whole gauge curve may be constructed by considering stress states in any one of the twelve (π/6) segments defined by the continuous and dotted lines, which can be easily verified for all the models shown in Fig. 6.11a.

However, because of the fact that the SOM and the LAM estimates depend slightly on θ in this case of isochoric loadings, the corresponding gauge curves do not form a perfect circular arc at the interval 0< θ < π/6. On the other hand, theLP S (and theV AR) model, which is independent of θ, does form a circle with radiusr=p

2/3 Σeq.

Considering now the second set of gauge curves in Fig. 6.11b, it is observed that the shape of the curves delivered by theSOM and the LAM models no longer conform to the above-mentioned π/6-symmetry. This is because in this case the origin corresponds to zero deviatoric stresses, while the total stress at this point isΣ = ΣmI withI denoting the identity tensor. For this reason, the