The rate-independent constitutive modeling for porous and multi-phase nanocrystalline materials

To determine the time-independent constitutive modeling for porous and multiphase nanocrystalline materials and understand the effects of grain size and porosity on their mechanical behavior, each phase was treated as a mixture of grain interior and grain boundary, and pores were taken as a single phase, then Budiansky's self-consistent method was used to calculate the Young's modulus of porous, possible multi-phase, nanocrystalline materials, the prediction being in good agreement with the results in the literature. Further, the established method is extended to simulate the constitutive relations of porous and possible multi-phase nanocrystalline materials with small plastic deformation in conjunction with the secant-moduli approach and iso-strain assumption. Comparisons between the experimental grain size and porosity dependent mechanical data and the corresponding predictions using the established model show that it appears to be capable of describing the time-independent mechanical behaviors for porous and multi-phase nanocrystalline materials in a small plastic strain range. Further discussion on the modification factor, the advantages and limitations of the model developed were present.