Homogenization-based constitutive models for viscoplastic porous media with evolving microstructure

This work is concerned with the application of the second-order nonlinear homogenization procedure of Ponte Castañeda (2002) to generate estimates for the effective behavior of viscoplastic porous materials. The main concept behind this procedure is the construction of suitable variational principles utilizing the idea of a linear comparison composite to generate corresponding estimates for the nonlinear porous media. Thus, the main objective of this work is to propose a general constitutive model that accounts for the evolution of the microstructure and hence the induced anisotropy resulting when the porous material is subjected to finite deformations. The model is constructed in such a way that it reproduces exactly the behavior of a composite-sphere assemblage in the limit of hydrostatic loadings, and therefore coincides with the hydrostatic limit of Gurson\u27s (1977) criterion in the special case of ideal plasticity and isotropic microstructures. As a consequence, the new model improves on earlier homogenization estimates, which have been found to be quite accurate for low triaxialities but overly stiff for sufficiently high triaxialities and nonlinearities. Additionally, the estimates delivered by the model exhibit a dependence on the third invariant of the macroscopic stress tensor, which has a significant effect on the effective response of the material at moderate and high stress triaxialities. Finally, the above-mentioned results are generalized to more complex anisotropic microstructures (arbitrary pore shapes and orientation) and general, three-dimensional loadings, leading to overall an isotropic response for the porous material. The model is then extended to account for the evolution of microstructure when the material is subjected to finite deformations. To validate the proposed model, finite element axisymmetric unit-cell calculations are performed and the agreement is found to be very good for the entire range of stress triaxialities and nonlinearities considered