Tensile loadings

Uniaxial tension loading. Fig. 5.7 presents results for the evolution of the normalized porosity f /fo, the aspect ratiow, and the normalized macroscopic axial component of the strain-rateD22/ε˙^∞_eq as a function of the nonlinear exponent n = 1,2,4,10 and the macroscopic axial strain ε22 for uniaxial tension (i.e., XΣ = 1/√

3 or T /S = 0 with S/|S| > 0). In Fig. 5.7a, the predictions of the SOM for the evolution of porosity are in very good agreement with the F EM results for all the nonlinearities considered, except for the case of n = 10 and large deformation ε22 > 60%, where a small difference between the two estimates is observed. On the other hand, the V AR estimates for f /fo are independent of n so that all the V AR predictions coincide with the n = 1 curve. This implies that theV ARmethod underestimates the evolution of the porosityf /fo at high nonlinearities. Furthermore, a main feature of this loading predicted by all the methods shown here, is that the porosity initially grows but finally it approaches an asymptote for sufficiently large strains.

Forn= 10 the initial porosityf = 0.01% increases four times to take a valuef ∼0.04%, after 100%

deformation. In turn, looking at Fig. 5.7b, the corresponding aspect ratio grows substantially for large deformations. In this case, both predictions are in good agreement up to a nonlinearityn= 4, whereas forn= 10 the SOM overestimates the evolution ofwwhen compared with theF EM. On the other hand, theV ARestimates forw are independent ofnand thus they all coinceide with the n= 1 curve.

In turn, the corresponding macroscopic axial strain-rateD22/ε˙^∞_eq is in good agreement for all the nonlinearities, certainly in much better agreement than the correspondingV ARestimate, which tends to underestimate significantlyD22/ε˙^∞_eq, especially at large nonlinearities. Of course all the estimates coincide for the linear case (i.e.,n= 1). Finally, Fig.5.7d shows undeformed and deformed meshes of the unit-cell forn= 10. In this figure, it is clearly observed that for this stress triaxialityXΣ= 1/√

3 the elongation of the pore is much more significant than the total increase of the void surface.

Biaxial tension loading with XΣ= 1. In the previous loading conditions, the stress triaxiality was sufficiently small (XΣ<0.6). In Fig.5.8, we consider a biaxial tension loading such that the stress triaxiality isXΣ= 1 (orT /S= 0.268 withS/|S|>0). Corresponding results for the evolution of the normalized porosityf /fo, the aspect ratiowand the normalized macroscopic equivalent strain-rate Deq/ε˙^∞_eq are shown as a function of the nonlinearity and the total equivalent strainεeq. In particular, in Fig. 5.8a, the porosity f /fo grows rapidly to high values especially at high nonlinearities. The corresponding SOM andF EM are in good agreement, while theV AR estimate is independent of the nonlinearity so that all the predictions coincide with the n = 1 curve. This last observation shows clearly the improvement of theSOM estimate on the earlierV ARestimates. This is a direct consequence of the fact that theSOM is constructed such that it recovers the analytical hydrostatic point predicted by a cylindrical shell subjected to hydrostatic pressure (see subsection2.6.2). On the

0 0.2 0.4 0.6 0.8 1

w Uniaxial Tension - X

Ó=0.5773

Figure 5.7: Results are shown for the evolution of the relevant microstructural and macroscopic variables for a porous material consisting of cylindrical pores with an initially, circular cross-section and porosity fo = 0.01%. The matrix phase exhibits a viscoplastic behavior with exponents n = 1,2,4,10, while the composite is subjected to uniaxial tension (XΣ = 1/√

3 or T /S = 0 with S/|S| > 0) loading conditions.

SOM,F EM andV AR estimates are shown for the evolution of the (a) normalized porosity f /fo, (b) the aspect ratiowand (c) the normalized axial strain-rateD22/ε˙eq ( ˙εeq is the corresponding remote strain-rate in the absence of voids). TheV ARestimate for the evolution of the normalized porosityf /foand the aspect ratio wis found to be independent of the nonlinear exponentn and the corresponding predictions coincide with then= 1 curves. Part (d) shows a typical deformed and undeformedF EM mesh at a given instant in time.

other hand, in Fig. 5.8b, theSOM estimates for the aspect ratio wdeviate from the corresponding F EM predictions, especially at high nonlinearities (i.e., forn= 10). Nonetheless, this difference in the prediction ofwdoes not affect the estimation of the normalized macroscopic strain-rateDeq/ε˙^∞_eq, where theSOM and the F EM are found to be in very good agreement. This last result indicates

0 0.2 0.4 0.6 0.8 1

n=10 Biaxial Tension - X

Ó=1

Figure 5.8: Results are shown for the evolution of the relevant microstructural and macroscopic variables for a porous material consisting of cylindrical pores with an initially, circular cross-section and porosity fo = 0.01%. The matrix phase exhibits a viscoplastic behavior with exponents n = 1,2,4,10, while the composite is subjected to biaxial tension (XΣ= 1 orT /S= 0.268 withS/|S|>0) loading conditions. SOM, F EM andV AR estimates are shown for the evolution of the (a) normalized porosityf /fo, (b) the aspect ratiowand (c) the normalized equivalent strain-rateDeq/ε˙eq ( ˙εeq is the corresponding remote strain-rate in the absence of voids). TheV ARestimate for the evolution of the normalized porosityf /fo and the aspect ratio wis found to be independent of the nonlinear exponentn and the corresponding predictions coincide with then= 1 curves. Part (d) shows a typical deformed and undeformedF EM mesh at a given instant in time.

that, for triaxialityXΣ= 1, the macroscopic behavior of the composite is mainly controlled by the evolution of the porosity and not the one of the aspect ratio. In turn, theV ARestimate, which fails to predict well the evolution of the porosity, fails also to predict the right evolution of the macroscopic

strain-rate Deq/ε˙^∞_eq. Finally, Fig. 5.8d shows undeformed and deformed meshes of the unit-cell for n= 10. In this figure, it is clearly observed that for this stress triaxiality XΣ= 1 the volume of the void increases significantly, while the aspect ratio of the void remains close to unity.

0 0.05 0.1 0.15 0.2

Figure 5.9: Results are shown for the evolution of the relevant microstructural and macroscopic variables for a porous material consisting of cylindrical pores with an initially, circular cross-section and porosity fo = 0.01%. The matrix phase exhibits a viscoplastic behavior with exponents n = 1,2,4,10, while the composite is subjected to biaxial tension (XΣ= 5 orT /S= 0.793 withS/|S|>0) loading conditions. SOM, F EM andV AR estimates are shown for the evolution of the (a) normalized porosityf /fo, (b) the aspect ratiowand (c) the normalized equivalent strain-rateDeq/ε˙eq ( ˙εeq is the corresponding remote strain-rate in the absence of voids). TheV ARestimate for the evolution of the normalized porosityf /fo and the aspect ratio wis found to be independent of the nonlinear exponentn and the corresponding predictions coincide with then= 1 curves. Part (d) shows a typical deformed and undeformedF EM mesh at a given instant in time.

Biaxial tension loading with XΣ= 5. In Fig.5.9, a high stress-triaxiality loading is applied

such thatXΣ = 5 (orT /S = 0.793 withS/|S|>0). Corresponding results for the evolution of the normalized porosityf /fo, the aspect ratiowand the normalized macroscopic equivalent strain-rate Deq/ε˙^∞_eq are shown as a function of the nonlinearity and the total equivalent strainεeq. The agreement between the SOM and the F EM is remarkable for the evolution of the normalized porosity f /fo

and the normalized macroscopic strain-rate Deq/ε˙^∞_eq. This is somewhat expected since, as already discussed earlier, the SOM method recovers the analytical result of a shell subjected to purely hydrostatic loading. It is interesting to note that while for n = 1 the porosity f becomes twenty times higher than the initial porosityfoat a total strain of 20%, for n= 10, the porosity climbs up to hundred times the initial porosity fo in just 2.5% deformation. In contrast, the V ARestimate is independent of the nonlinearity and the corresponding results coincide with the n= 1 curve. As a consequence, theV ARestimates fail completely to predict the evolution off /foat high nonlinearities.

Looking now at Fig.5.9b, it is observed that the SOM estimates for the aspect ratiow are not in good agreement with theF EM results, although it is evident that the aspect ratio remains very close to the unity value (i.e., the void remains almost circular) and hence is not expected to affect the macroscopic behavior of the porous material. Nonetheless, an interesting effect is observed in the context of this figure. For high nonlinearities, n = 4,10, the void elongates in the transverse direction of that defined by the maximum principal loading (see Fig. 5.9d). In other words, while the maximum stress is applied in the 2−direction, the voids elongates in the 1−direction. This effect has been observed very early in the work of Budiansky et al. (1982) and later by Fleck and Hutchinson (1986) and Lee and Mear (1992a) in the context of dilute porous media. Since then many authors tried to model this effect with ad-hoc approximations, sometimes with success (see Flandi and Leblond (2005) for the three dimensional case). However, it is important to mention that, at high stress triaxialities, the evolution of porosity controls the effective behavior of the material, which is illustrated by the remarkable agreement between theSOMand theF EM predictions for the normalized macroscopic strain-rateDeq/ε˙^∞_eq in the context of Fig.5.9c. On the other hand, theV AR overly underestimatesDeq/ε˙^∞_eq, so that it is not possible to distinguish the correspondingV ARcurve forn= 10. Lastly, Fig. 5.9d shows undeformed and deformed meshes of the unit-cell forn= 10. In this figure, it is clearly observed that for this high stress triaxiality XΣ = 5 the volume of the void increases significantly, while the major axis of the void elongates in the transversely (2− direction) to the direction of maximum principal stress (2−direction).