Simple shear loading

In the previous subsections, we studied the effects of the evolution of the porosity and the aspect ratio on the overall response of a porous material consisting of aligned cylindrical voids with initially circular cross-section subjected to biaxial loading conditions. In this subsection, we study the effective behavior of these materials when subjected to simple shear loading conditions. In this case, the in-plane principal axes of the void evolve during the deformation process. More specifically, the applied load is such that the only non-zero components of the macroscopic strain-rate and spin tensor are D12and Ω12, respectively. The material is subjected to total shear strain 2ε12=γ. For comparison, F EM results are also included, which were discussed in detail in section3.5. Because of the applied load (isochoric loading), the porosity does not evolve during the deformation process. For numerical reasons related to theF EM calculations, the initial porosity has been chosen to befo= 1%.

Fig.5.13presents results for the evolution of the orientation angle ψand the components of the macroscopic stress tensor σ as a function of the applied shear strain γ for various nonlinearities n = 1,2,4. Fig. 5.13a shows the evolution of the orientation angle of a void with initially circular cross-section for a nonlinear exponent n = 4. Because of the loading, the initial orientation of the major axis of the void lies at 45^o. As the deformation progresses, the orientation angle ψ evolves reaching a value of∼32^o at shear strain 100%. Both theSOM and theV ARestimates are in good agreement^† with theF EM predictions. It should also be noted that the evolution of the orientation angleψ depends very slightly on the nonlinearity, This is the reason that we do not include graphs for other values of n. In turn, Fig. 5.13b, Fig. 5.13c and Fig. 5.13d show evolution curves for the

†TheSOM and theV ARestimates coincide and this is the reason that the two curves are not distinguishable in Fig.5.13a.

0 0.2 0.4 0.6 0.8 1

Figure 5.13: 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. The matrix phase exhibits a viscoplastic behavior with exponentsn= 1,2,4, while the composite is subjected to simple shear loading (D12>0 andD11=D22= 0) conditions. SOM andF EM estimates are shown for the evolution of the (a) orientation angleψ(theV ARestimate coincides with the SOM curve) and the components of the macroscopic stress tensorσ for (b)n= 1, (c)n= 2, and (d)n= 4.

macroscopic components of the stress tensorσ normalized by the flow stressσoin the matrix phase as a function of the applied shear strain γ and the nonlinearity n = 1,2,4. The SOM estimates are in very good agreement with the F EM results for all the nonlinearities considered. In all the cases, the shear stressσ12 starts from a finite value, whereas σ11 and σ22 are initially zero. During the deformation process, the two components, σ11 and σ22, evolve similarly, except at sufficiently large shear strain γ, where they start to deviate from each other. On the other hand, the shear stress σ12 remains almost unaffected by the evolution process. For clarity, in Fig. 5.14, we present various deformed states of the unit-cell for n= 4, whereas the undeformed initial mesh is shown in

(a) (b)

(c) (d)

Figure 5.14: Contour plots for simple shear loading at various shear strainsγ.

the background for comparison. In this figure, it is obvious that the principal axes of the void evolve with the increase of the shear strainγ.

5.3.5 Brief summary

In this subsection, we summarize the main results obtained in this section. First of all, the SOM is found to be in very good agreement with the F EM results for the majority of the loadings, nonlinearities and stress triaxialities considered. Furthermore, it is found to improve significantly on the earlier V AR method particularly at high stress triaxialities, where the V AR method fails to predict accurately the evolution of the porosity and consequently the evolution of the effective behavior of the porous material. This is a direct consequence of the fact that the SOM model is constructed such that it recovers the analytical result obtained when a composite-cylinder assemblage is subjected to purely hydrostatic loading conditions.

Next, it is worth noting that the SOM is also capable of predicting with sufficient accuracy the evolution of the aspect ratio (shape of the void) at low triaxialities. On the other hand, it is not in good agreement with the F EM predictions for the evolution of the aspect ratio at high stress triaxialities. In this case, theF EM results confirm the observation made initially by Budiansky et al. (1982), and later by Fleck and Hutchinson (1986) and Lee and Mear (1992b) in the context of dilute porous media, where it was found that at sufficiently high triaxialities and nonlinearities, the void elongates in a direction that is transverse to the maximum macroscopic principal stretching. This counterintuitive result however, was found to have a minor effect on the overall response of the porous medium. This can be easily explained by noting that the evolution of porosity is much more significant

than the corresponding evolution of the aspect ratio at high stress triaxialities. As a consequence, the evolution of the porosity controls the effective behavior of the porous material. In this connection, theSOM is found to predict accurately the evolution of the porosity and consequently the effective behavior of the porous material during the deformation process.

Finally, the SOM estimates are compared withF EM andV AR results for the evolution of the orientation angle of the voids and the macroscopic stress, when the porous material is subjected to simple shear loading conditions. It is worth noting that — to the knowledge of the author — the SOM and theV ARmethods are the only available methods in the literature, apart from theF EM method, to be able to provide estimates fornon-dilute porous media consisting of cylindrical voids with elliptical cross-section, that are subjected to general plane-strain loading conditions.