For consistency with the two-dimensional results presented in chapter 4, we present results on the effective behavior of isotropic porous media subjected to isochoric loading conditions. For complete-ness, we consider two distinct values for the Lode angle θ = 0, π/6, which is related to the third invariant of the macroscopic stress tensor, as defined by (6.30)2. Results are obtained by making
‡The reader is referred in the original work of Lee and Mear (1992c) for details on the linear case.
use of the “second-order” method (SOM), the “variational” method (V AR) and the sequentially laminated microstructures (LAM), discussed in section3.4. In addition, the well-known Voigt bound is also shown whenever applicable. At this point, it is important to note that, for isochoric loadings, the reference stress tensor ˇσ, discussed in subsection (2.6.2), reduces to ˇσ=σ0 (the prime denotes the deviatoric part of a second-order tensor). Moreover, it is emphasized that by making use of the homogeneity of the local potential (2.23) inσ, it follows that in the case of purely deviatoric loadings the effective stress potential can be written as
0.5
Figure 6.4: Estimates and exact results for isotropic porous materials subjected to isochoric axisymmetric loadings (XΣ= 0 andθ = 0). Effective flow stressσecurves normalized by the flow stress of the matrixσo
are shown, (a) as a function of the strain-rate sensitivity parametermfor porosity (f = 25%), and (b) as a function of the porosityf in the case of an ideally-plastic matrix (m= 0).
Ue(σ) =ε˙oσeo(θ)
whereσeois theeffective flow stress of the composite and, in general, depends on the third invariant of the macroscopic stress tensor, which is denoted here with the Lode angleθ, defined by (6.30)2.
More specifically, Fig. 6.4 presents results for the normalized effective flow stress eσo/σo (σo is the flow stress of the matrix phase) as function of the strain-rate sensitivity parameter m and the porosity f for an axisymmetric loading, i.e., for a Lode angleθ = 0. The main observation in the context of this figure is that the SOM is in good agreement with theLAM results, and certainly much better than theV ARestimates for all the porosities and nonlinearities considered. Obviously, the Voigt bound is much stiffer and hence is much less accurate. Fig.6.4a shows corresponding results for eσo/σo as a function of the strain-rate sensitivity parameterm for a porosity f = 25%. In this figure, theSOM improves on theV ARestimates by being in much better agreement with theLAM results, even though it tends to be slightly softer than the rest of the methods. In turn, Fig.6.4b shows corresponding results for eσo/σo as a function of the porosity f for an ideally-plastic matrix (m = 0). Here, we observe that both the SOM and the LAM estimates satisfy the V AR bound
0.5
Figure 6.5: Estimates and exact results for isotropic porous materials subjected to isochoric in-plane shear loadings (XΣ= 0 andθ=π/6). Effective flow stresseσcurves normalized by the flow stress of the matrixσo
are shown, (a) as a function of the strain-rate sensitivity parametermfor porosity (f = 25%), and (b) as a function of the porosityf in the case of an ideally-plastic matrix (m= 0).
in the entire range of the porosities. Note, however, that in this case all the methods — except the Voigt bound — are in quite good agreement for the entire range of the porosities considered.
Nonetheless, it is remarked that all the methods — except the Voigt bound — are expected to be sufficiently accurate in the entire range of nonlinearities and porosities when the material is subjected to isochoric axisymmetric loading conditions.
For completeness, Fig. 6.5 shows corresponding results for the normalized effective flow stress e
σo/σo(σois the flow stress of the matrix phase) as a function of the strain-rate sensitivity parameter m and the porosity f for in-plane shear loading, i.e., for a Lode angle θ =π/6. Note that, while the SOM and the LAM depend on all the three invariants of the macroscopic stress tensor, the V AR method depends only on the first two invariants, i.e., on the mean Σm and the equivalent Σeq macroscopic stress tensor, and thus is independent of the Lode angle θ. By comparing the corresponding results for axisymmetric loading, presented in Fig.6.4with the estimates for in-plane shear in Fig. 6.5, we observe that, for isochoric loadings, both theSOM and the LAM estimates depend only slightly onθand hence the observations made in the previous figure forθ= 0 apply also in this case ofθ=π/6.
Fig. 6.6 presents results for the average equivalent strain-rate in the void D(2)eq normalized by the macroscopic equivalent strain-rate Deq as a function of the strain-rate sensitivity parameter m (part (a)) and the porosity f (part (b)) for a Lode angle θ = 0 (axisymmetric shear loading). In particular, in Fig. 6.6a, we observe that D(2)eq/Deq is not a strong function of the m for the given porosityf = 25%. The correspondingSOM improves on theV ARby being in good agreement with theLAM estimates for the entire range of the nonlinearities. In turn, theV ARestimate is found to be independent of the nonlinearity m and thus underestimate the average strain-rate in the pores.
1
Figure 6.6: Estimates and exact results for transversely, isotropic porous materials subjected to isochoric axisymmetric loadings (XΣ= 0 andθ= 0). The average equivalent strain-rate in the voidD(2)eq normalized by the macroscopic equivalent strain-rateDeqis shown, (a) as a function of the strain-rate sensitivity parameter m for porosity (f = 25%), and (b) as a function of the porosity f in the case of an ideally-plastic matrix (m= 0).
Figure 6.7: Estimates and exact results for transversely, isotropic porous materials subjected to isochoric in-plane shear loadings (XΣ= 0 andθ=π/6). The average equivalent strain-rate in the voidD(2)eq normalized by the macroscopic equivalent strain-rateDeqis shown, (a) as a function of the strain-rate sensitivity parameter m for porosity (f = 25%), and (b) as a function of the porosity f in the case of an ideally-plastic matrix (m= 0).
Looking now at Fig. 6.6b, it is evident that the SOM is in much better agreement with the LAM than the corresponding V AR, particularly for low and moderate porosities (f .0.2). It is further remarked that the maximum difference between theSOM-LAM estimates and theV ARpredictions
is observed form= 0.
Fig. 6.7 presents corresponding results for the normalized average equivalent strain-rate in the voidD(2)eq/Deq as a function of the strain-rate sensitivity parameterm(part (a)) and the porosityf (part (b)) for a Lode angleθ=π/6 (in-plane shear loading). More specifically, in part (a), theSOM still remains in good agreement with the LAM estimates for the entire range of nonlinearities m, in contrast with theV AR estimate, which shows no dependence on the nonlinearity. Note that the V ARestimates do not depend on the Lode angleθand hence they are identical to those presented in Fig.6.6. On the other hand, Fig.6.7b shows corresponding results forD(2)eq/Deq as a function of the porosityf for an ideally-plastic matrix. Here, it is interesting to observe that theSOM estimate is found to increase significantly at sufficiently small porosities, whereas the LAM estimates deliver a lower value for dilute concentrations. In this context, theV ARis found to underestimate significantly the corresponding estimate forD(2)eq/Deqwhen compared with theSOM and theLAM method. Note that the dependance of theSOM and the LAM methods on the third invariant of the macroscopic stress tensorσ, or equivalently on the Lode angleθis more evident in the plots for theD(2)eq/Deq.
In summary, the SOM method is found to improve on the earlier V AR method by being in much better agreement with the LAM estimates for the determination of the effective behavior of isotropic porous materials subjected to isochoric loading conditions. Moreover, the SOM and the LAM estimates for the normalized effective flow stressσeo/σo of the porous material depend on all three invariants of the macroscopic stress tensor σ, although only slightly on the third invariant, denoted here with the Lode angleθ. In contrast, the V ARpredictions depend only on the first two invariants, i.e., on the mean Σm and equivalent Σeq macroscopic stress. In this connection, it is interesting to remark that the correspondingSOM estimates for the average strain-rate in the void exhibit a much stronger dependence on the Lode angle, particularly at dilute concentrations, whereas the corresponding LAM estimates depend slightly on θ. In the next section, we will examine the behavior of isotropic porous materials under general loading conditions.