Compressive loadings

Uniaxial compression loading. Fig.5.10shows results for the evolution of the normalized porosity f /fo, the aspect ratiowand the normalized macroscopic axial strain-rateD22/ε˙^∞_eqas a function of the macroscopic axial strain|ε22|and the nonlinearityn= 1,2,4 for uniaxial compression (XΣ=−1/√

3 orT /S= 0 withS/|S|<0). More specifically, in Fig.5.10a, theSOMpredictions for the evolution of the normalized porosityf /fo are found to be in very good agreement with the corresponding results obtained by theF EM, whereas the corresponding V ARestimates are independent ofnso that all the V ARresults coincide with the n= 1 curve. At this point, it is noted that it was not possible

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Figure 5.10: 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 exponentsn= 1,2,4, while the composite is subjected to uniaxial compression (XΣ=−1/√

3 or T /S= 0 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 axial strain-rate|D22|/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 w is found to be independent of the nonlinear exponentnand the corresponding predictions coincide with then= 1 curves. Part (d) shows a typical deformed and undeformedF EM mesh at a given instant in time.

to have good numerical accuracy with theF EM method for nonlinearities greater thann= 4. The reason for this is linked to the fact that as the porosity becomes smaller the void loses its concave shape and develops contact zones (see Fig.5.10d). In this case, we do not proceed further with the F EM calculations. In fact, this is the reason that the n = 4 curve predicted by the F EM stops before the porosity becomes zero.

Looking now at Fig. 5.10b, theSOM slightly overestimates the decrease of the aspect ratio w when compared with the F EM. However, both methods predict a very sharp change in the aspect ratio, which finally tends to zero as the porosity becomes zero. Similar to the evolution of porosity, theV ARmethod underestimates the evolution ofw, since it delivers results that are independent of nand thus coincide with then= 1 curve.

Next, theSOMestimates for the evolution of the normalized macroscopic axial strain-rateD22/ε˙^∞_eq are certainly in better agreement with the F EM, than the V AR results, while they exhibit a very abrupt change in the slope as the porosity approaches zero. This happens because the matrix phase is incompressible and therefore it cannot sustain any compressive strains as the porosity becomes zero.

Next, Fig.5.10d shows undeformed and deformed meshes of the unit-cell forn= 4. In this figure, it is clearly observed that total surface of the void shrinks during the deformation process, while at a certain strain the void develops contact points losing its elliptical shape. After that point, theF EM calculations are terminated since they would require to redefine the boundary conditions in order to preserve material impenetrability.

Biaxial compression loading with XΣ=−1. Fig.5.11presents results for the evolution of the normalized porosityf /fo, the aspect ratiowand the normalized macroscopic equivalent strain-rate Deq/ε˙^∞_eq as a function of the total equivalent macroscopic strain εeq and the nonlinearityn= 1,2,4 for biaxial compression loadings (XΣ =−1 or T /S = 0.268 with S/|S|<0). For the same reasons explained in the context of Fig.5.10, no results are presented for a nonlinear exponentn= 10. More specifically, in Fig.5.11a, theSOMestimate is in quite good agreement with theF EM results for the evolution of the normalized porosityf /fo. In addition, we observe that the porosity approaches the zero value faster than the corresponding prediction obtained in the context of the uniaxial compression loading. The correspondingV ARestimates are independent ofn and thus coincide with then= 1 curve. Consequently, they underestimate significantly the decrease off /foat high nonlinearities.

On the other hand, the SOM estimates are not in very good quantitative agreement with the F EM results for the evolution of the aspect ratiow, as shown in Fig.5.11b. However, it should be noted that, in theF EM, the aspect ratiowis measured by computing the geometrical ratio between the major and the minor axis of the void, based on the assumption that the void remains elliptical in shape. Now, when the porosity approaches the zero value the void does not have an elliptical cross-section and the comparison is not meaningful after this point. Even so, the corresponding estimates of the SOM for the normalized macroscopic strain-rate Deq/ε^∞_eq, in Fig. 5.11c, are in much better agreement with theF EM, certainly better than theV ARestimates. This observation indicates that the evolution of porosityf /fodominates over the evolution of the aspect ratiowand thus, theSOM is able to capture adequately the effective response of the porous medium for this moderate stress triaxiality. Finally, Fig. 5.11d shows undeformed and deformed meshes of the unit-cell for n = 4.

In this figure, it is clearly observed that total surface of the void shrinks significantly during the deformation process, while the void develops contact points losing its elliptical shape, and similarly to the previous case of uniaxial compression, theF EM calculations are terminated at that point.

Biaxial compression loading with XΣ=−5. Finally, the set of results for aligned loadings is completed with Fig.5.12, where the porous material is subjected to high-triaxiality biaxial

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Figure 5.11: 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 exponentsn= 1,2,4, while the composite is subjected to biaxial compression (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.

sion (XΣ= 5 or T /S = 0.793 withS/|S| <0). Similarly to the rest of the results for compression loadings, we consider here nonlinear exponents n = 1,2,4, while the corresponding predictions for the evolution of the normalized porosity f /fo, the aspect ratio w and the normalized macroscopic

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Figure 5.12: 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 exponentsn= 1,2,4, while the composite is subjected to biaxial compression (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.

equivalent strain-rateDeq/ε˙^∞_eq are plotted as a function of the equivalent macroscopic strainεeq. The main observation in the context of Fig.5.12a is that theSOM estimates for the porosity f /fo are in very good agreement with the F EM results for all nonlinearities considered, whereas the V AR significantly underestimatef /foby being independent ofnand thus coinciding with then= 1 curve.

In this connection, it is interesting to note that while forn= 1 the porosity becomes zero at strain

∼15%, the corresponding prediction forn= 10 indicates that the porosity goes to zero at very low strains of the order ∼2.5%. Moreover, whilef /fo initially decreases rapidly, as it approaches the zero value, the rate of decrease off /fodiminishes leading to an asymptotic behavior for the evolution of the porosity.

On the other hand, theSOM estimate fails to predict qualitatively the change in the shape of the void, in Fig.5.12b. Here, the void elongates parallel to the direction of the maximum (compressive) principal stress, as shown in Fig.5.12d. This phenomenon is equivalent to that observed in Fig.5.9 for high triaxiality tensile loadings, and is intuitively unexpected. However, it is remarkable to note that the evolution of the aspect ratio w has a minor effect on the prediction of the normalized equivalent macroscopic strain-rateDeq/ε˙^∞_eq, as shown in Fig.5.12c. In this figure, theSOM is found to be in very good agreement with the F EM results, which clearly implies that for high-triaxiality loadings, the evolution of porosity mainly controls the effective response of the porous material. In contrast, theV ARmethod underestimates significantly the evolution of theDeq/ε˙^∞_eq, and hence, fails to predict accurately the effective behavior of the porous medium. The set of results for this loading are completed with Fig. 5.12d, which shows undeformed and deformed meshes of the unit-cell for n= 4. As already mentioned before, we can observe the significant growth of the porosity, and the elongation of the void towards the direction of the maximum compressive stress.