Applied stress triaxiality

In this subsection, we study the possible development of shear localization in porous media with an ideally-plastic matrix phase consisting of cylindrical voids with initially circular cross-section subjected to plane-strain loading with fixed stress triaxiality XΣ during the deformation process.

Similarly to the previous case, the only non-zero components of the strain-rate and stress tensor are D11, D22, σ11 and σ22, respectively. The mean and equivalent strain-rates, Dm and Deq and the strain-rate triaxiality,XE, are defined by relation (5.21), whereas the corresponding stress quantities

are recalled here to be σeq =

√3

2 |σ11−σ22|, σm= σ11+σ22

2 , XΣ= σm

σeq. (5.23)

The applied load is such that the condition |σ22| >|σ11| is always satisfied during the deformation process. Then making use of definitions (5.23), we could write the stress components in terms of the stress triaxialityXΣand the equivalent stressσeq, such that

σ11= µ

XΣ−ρ 1

√3

¶

σeq, σ22= µ

XΣ+ρ 1

√3

¶

σeq. (5.24)

Here, ρ=±1 such that the condition |σ22|>|σ11|is always true. It is further noted thatσeq is the unknown of the problem, which is computed such that Deq = 1. In turn, the stress triaxiality XΣ

is given. Because of the plane-strain loading the only sufficient condition for instability is H = 0, similarly to the previous case of the fixed strain-rate triaxialityXE.

Fig. 5.17 shows evolution curves for the porosity f, the aspect ratio w, the hardening rate H and the equivalent stressσeq normalized by the flow stress in the matrix phase σo as a function of the stress triaxiality (XΣ=−0.1,0.5) and the macroscopic strainεeq for initial porosity fo = 10%.

Since the stress triaxiality remains fixed during the deformation process, the strain-rate triaxiality is expected to change in time. As a consequence, the corresponding predictions of theSOM, theV AR and theGU R model for the evolution of the porosity are different. In Fig.5.17a, forXΣ= 0.5, the SOM and the V ARmethods predict initially an increase in the porosity, which later decreases to zero (at sufficiently high strain not shown here). In contrast, theGU Rmodel predicts a continuously increasing porosity during the deformation, which has as a consequence the softening of the porous material. In order to have a complete view of the softening or hardening of the material as predicted by the SOM and the V AR methods, it is necessary to study, as well, the evolution of the aspect ratiow, in Fig.5.17b. ForXΣ= 0.5,wis greater than unity for both theSOM and theV AR, which implies that the porous medium hardens in the direction of the maximum principal stress, i.e., in the 2−direction. This hardening is in contrast with the softening induced by the initial increase of the porosity.

However, as we observe in Fig.5.17c, theSOM predicts initially softening, which is translated to a negative hardening rateH, whereas theV ARestimate predicts initially hardening, indicated by the positiveH. In the sequel theSOM estimate for the hardening rateH becomes positive, which implies thatH crosses zero at a critical equivalent strainεeq ∼10% and consequently the material becomes unstable at this point, in contrast to the correspondingV ARestimate, which remains positive during the deformation and therefore does not satisfy the condition for shear localization. On the contrary, theGU Rmodel predicts softening during the deformation process and hence exhibits a behavior that is substantially different than theSOM and theV AR. Finally, Fig.5.17d shows that the equivalent stressσeq as predicted by theSOM is lower than the corresponding estimates obtained by theV AR and theGU Rmodels, which implies that theSOM predicts, at least initially, a much softer response for the porous medium. In large deformations, however, the SOM predicts a rapid decrease in the porosity and hence a significant hardening for the composite.

0 0.2 0.4 0.6 0.8 1

Figure 5.17: Results are shown for the evolution of the relevant microstructural and macroscopic variables for a porous material consisted of cylindrical pores with an initially, circular, in-plane cross-section and initial porosityfo= 10%. The matrix phase exhibits an ideally-plastic behavior, while the composite is subjected to biaxial tension and compression loading conditions with fixed stress triaxiality, XΣ =−0.1,0.5. SOM, V ARand GU R(Gurson, 1977) estimates are shown for the evolution of the (a) porosityf, (b) the aspect ratiow, (c) the hardening rateH and (d) the normalized, macroscopic equivalent stressσeq/σo (σo denotes the flow stress of the matrix phase) of the composite as a function of the macroscopic, equivalent strainεeq. In (c) the symbols◦and¤denote the loss of stability for the porous medium as predicted by theSOM and theV AR, respectively.

Similar to the previous case, the SOM and the V ARmethods exhibit a very different behavior than the correspondingGU Rmodel for an applied stress triaxialityXΣ=−0.1. While, theSOM and theV ARmethods predict a decrease in the porosity, as shown in Fig.5.17a, the correspondingGU R estimate forf remains almost constant during the deformation process. In addition, both theSOM and theV ARestimates exhibit a sharp decrease in the aspect ratio,w <1, which induces a softening

-1 -0.5 0 0.5 1 0

0.1 0.2 0.3 0.4 0.5

LOE, f =10%, m=0

SOM VAR

X

cr

ε

Figure 5.18: Macroscopic onset-of-failure curves as predicted by theSOM and V AR calculations, for an initially transversely, isotropic porous medium with ideally-plastic matrix phase and initial porosityfo= 10%.

The plot shows the critical equivalent strainε^creq as a function of the applied stress triaxialityXΣ.

in the direction of the maximum (absolute) principal stress (|σ22|>|σ11|withσ22<0) as shown in Fig.5.17b. Looking now at the corresponding estimates for the hardening rateH in Fig.5.17c, both theSOM and theV ARexhibit softening initially, while at a finite strain they become unstable (see the points whereH crosses zero). On the other hand, the GU Rmodel predicts slight hardening for the porous medium and as already anticipated it never becomes unstable for this loading. It is worth mentioning at this point that for the case of aligned loading, the GU Rmodel can give a hardening rate equal to zero only in the case of isochoric loadings, i.e.,XΣ= 0 orXE= 0.

Finally, Fig.5.18shows a map of the critical strainsε^cr_eq for loss of stability, as determined by the SOM and theV ARmethods for the entire range of the stress triaxialitiesXΣ. In the context of this figure, it is obvious that according to theSOM and theV ARestimates the material is unstable for a wide range of stress triaxialities. In particular, theSOM and theV AR estimates exhibit several differences especially at positive stress triaxialities where theV ARestimates become unstable just in a small vicinity ofXΣ, close to uniaxial tension loading conditions. In contrast theSOM estimates has a small branch of unstable behavior at stress triaxialities 0.XΣ.0.3. It is remarkable to note that there is a sharp transition (see both theSOM and theV ARlines close toXΣ= 0.5), where the material passes from loadings that induce hardening (0.3.XΣ.0.5) to ones that predict softening (XΣ &0.5). This transition region lies closely to the uniaxial tension loading (i.e.,XΣ= 0.57735), while it is interesting to note that none of the two methods loses stability for this loading. In turn, for negative triaxialities, theSOM is more unstable at lower critical strains, while theV ARbecomes unstable at higher values ofε^cr_eq. In overall, it is observed that, similarly to the case of fixed strain-rate triaxialitiesXE, instabilities may occur when the porous medium is subjected to low stress triaxiality

loadings, where the shear strains are expected to dominate over the corresponding dilatational strains.

5.5 Concluding Remarks

In this chapter, we have studied the evolution of microstructure in porous media consisting of cylin-drical voids with initially circular or elliptical cross-section. The “second-order” estimates have been compared with the Lee and Mear (1999) results, finite element unit-cell calculations and the earlier

“variational” predictions for a wide range of stress triaxialities and nonlinearities. It has been found that the “second-order” model improves significantly on the earlier “variational” method by giving much better agreement with the corresponding Lee and Mear and finite element calculations. The im-provement was dramatic at higher stress triaxialities, where it was already known (Ponte Casta˜neda and Zaidman, 1994; Kailasam and Ponte Casta˜neda, 1998) that the “variational” method is overly stiff and thus underestimates significantly the evolution of the porosity.

More specifically, the “second-order” estimates were found to be in good agreement with the Lee and Mear (1999) results, for the entire range of stress triaxialities considered. However, it should be remarked that the Lee and Mear dilute results were found to underestimate slightly the evolution of the porosity at high stress triaxialities, when compared with the corresponding “second-order”

estimates for very small but finite porosityfo= 10⁻⁶. To validate further this observation, we have included finite element, unit-cell results for a porosityfo= 10⁻⁴. It should be noted that the “second-order” predictions corresponding tofo= 10⁻⁴andfo= 10⁻⁶did not exhibit any difference indicating that both values could be used to approximate sufficiently the dilute limit. In this connection, the finite element results have been found to lie closer to the “second-order” estimates, leading to the conclusion that the Lee and Mear results underestimate slightly the evolution of the porosity at high stress triaxialities and nonlinearities.

To understand the difference in the predictions of the various models it was necessary to recall that Lee and Mear (see also Fleck and Hutchinson, 1986) make the assumption that the void evolves through a sequence of elliptical shapes during the deformation process, which is not the case in the finite element calculations where the void can take non-elliptical shapes. On the other hand, the “second-order” model, which is based on an homogenization procedure, makes use of theaverage fields in the vacuous phase to compute anaverage ellipticalshape for the void. In this regard then, the accuracy of the “second-order” model is related to the estimation of the average fields in the porous material. Obviously, this procedure is different than the one adopted by Lee and Mear who solve the problem locally. Moreover, it is also important to mention that, according to Huang (1991b), in the method used by Lee and Mear to predict the evolution of microstructure in dilute porous media, a very large number of terms needs to be considered in the Rayleigh-Ritz eigen-function expansion to achieve sufficient accuracy at high stress triaxialities and nonlinearities. In this regard, the author believes that these two aforementioned observations could explain the fact that the Lee and Mear technique predicts lower values for the evolution of porosity, while the “second-order” model is able to predict with sufficient accuracy the evolution of porosity and thus be in closer agreement with the finite element results.

In the sequel, both the “second-order” and the Lee and Mear (1999) methods were found to be in good agreement for the prediction of the evolution of the porosity and the orientation angle of a void with initially elliptical cross-section, misaligned with the principal loading directions, when the porous medium is subjected to uniaxial tension or compression loading. In addition, the principal axes of the void evolved with a tendency to align themselves with the principal loading directions, which is intuitively expected, at least for the loadings considered here. In this connection, the “second-order” model was found to predict both qualitatively and quantitatively the evolution of the axes of anisotropy (or equivalently the principal axes of the voids) for tensile and compressive loadings.

Next, the “second-order” estimates were compared with corresponding results obtained by the finite-element and the “variational” method. The improvement of the “second-order” method over the “variational” method was dramatic at higher stress triaxialities. For instance, at a stress triaxiality X_Σ = 5 (biaxial tension) the “second-order” method and the finite element results predict that the porosity can increase up to hundred times the initial porosity at a strain of ∼ 2.5%, whereas the corresponding “variational” estimate delivers an increase in the porosity that is negligible at this strain. On the other hand, for a stress triaxiality XΣ = −5 (biaxial compression), the porosity approaches the zero value at a strain of ∼ 15% as determined by the “variational” method, in contrast to the corresponding “second-order” and finite element estimates that predict zero porosity at a strain value of∼2.5%. A second remark in the context of this set of results is associated with the evolution of the shape of the voids (or the evolution of the in-plane aspect ratio). In this case, it has been found that the evolution of the aspect ratio is expected to affect significantly the effective behavior of the porous material at low stress triaxialities, since for such loadings the void evolves rapidly in a crack shape. For this case of low stress triaxialities, the “second-order” model has been found to provide fairly accurate estimates for the evolution of the aspect ratio when compared with finite element results for all the range of the nonlinearities considered.

On the other hand, at high stress triaxialities and nonlinearities, the “second-order” method has been shown to be in disagreement with the corresponding finite element results for the evolution of the aspect ratio. In this particular case, the finite element results confirm the observation made initially by Budiansky et al. (1982), and later by Fleck and Hutchinson (1986) and Lee and Mear (1992b,1999) in the context of dilute porous media, where it was found that the void elongates in a direction that is transverse to the maximum macroscopic principal stretching at sufficiently high triaxialities and nonlinearities. This counterintuitive result however, was found to have a minor effect on the overall response of the porous medium. The reason for this lies in the fact that at high stress triaxialities the evolution of porosity controls the effective behavior of the porous medium. Consequently, the “second-order” method, which predicts accurately the evolution of porosity at high stress triaxialities, is also capable of predicting with remarkable accuracy the macroscopic strain-rates and thus the effective response of the porous medium.

In the sequel, the “second-order” method has been compared with corresponding finite element results and “variational” estimates in the case of simple shear loading conditions. The main result in the context of this case, is that both the “second-order” and the “variational” methods predict with sufficient accuracy the evolution of the orientation angle of the elliptical void during the deformation

process, when compared with corresponding finite element results. Furthermore, the “second-order”

estimates for the macroscopic stress tensor have been found to be in very good agreement with the corresponding finite element predictions for all the nonlinearities considered. It is worth noting that

— to the best knowledge of the author — the “second-order” and the “variational” methods are the only available methods in the literature, apart from numerical techniques (such as the finite element or FFT methods), to be able to provide estimates fornon-diluteporous media consisting of cylindrical voids with elliptical cross-section, that are subjected to general plane-strain loading conditions.

Finally, the “second-order” model has been used to predict possible development of instabilities in porous media with an ideally-plastic matrix phase. The “variational” method and the Gurson criterion have also been used for comparison. The main observation in the context of this section was that the porosity and the aspect ratio of the void may have opposite effects on the macroscopic behavior of the porous material and as a consequence they lead to possible development of instabilities. However, at large strain-rate or stress triaxialities the evolution of the porosity dominates over the the change of the aspect ratio and hence instability is unlikely to occur. Furthermore, the “second-order” and the “variational” methods were shown to have similar qualitative behavior as far as the prediction of instabilities is concerned. However, the “second-order” method has been shown to predict more accurately the evolution of the porosity and the aspect ratio in the case of finite nonlinearities and thus is expected to deliver more accurate results than the “variational” method in the case of porous media with an ideally-plastic matrix phase, particularly for the prediction of shear localization instabilities.

On the contrary, the Gurson model does not involve any information on the shape of the voids and thus was incapable of predicting unstable behaviors except in the case of purely isochoric loadings (XΣ = 0) where it delivers zero hardening during the deformation process. However, even in this case, the Gurson model is expected to be highly inaccurate, since for isochoric loadings the change in the shape of the void is significant.

Chapter 6

Instantaneous behavior: spherical and ellipsoidal voids

This chapter deals with the instantaneous effective behavior of porous materials consisting of ellip-soidal voids distributed randomly in the specimen. First, the study will be focused on isotropic porous materials made out of spherical voids. The “second-order” estimates (SOM) of Ponte Casta˜neda (2002a), discussed in section2.6, will be compared with corresponding results obtained by the “vari-ational” method (V AR) of Ponte Casta˜neda (1991), discussed in section 2.5, the Lee and Mear (1992c) method for dilute porous media methods, the high-rank sequential laminates (LAM) and the Leblond et al. (1994) (LP S) model – as well as the Flandi and Leblond, (2005) (F L) – for a wide range of nonlinearities and loadings (see chapter3for these last models). In the sequel, we will consider anisotropic porous materials consisting of ellipsoidal voids initially aligned with the loading directions. A thorough study of the effect of the pore shape on the effective behavior of the porous material will be attempted. Finally, this chapter will be concluded with corresponding estimates for the effective behavior of anisotropic porous media whose microstructure is misaligned with the loading directions.