Phase field approach for void dynamics with interface stresses at the nanoscale

In this paper, a thermodynamically consistent interface stress is derived for the solid-gas interface of nanovoids within the concept of the phase field approach and using the laws of thermodynamics. The Cahn-Hilliard (CH) equation describes the evolution of nanovoid concentration which varies between 0 for perfect solid to 1 for gas. Considering the gradient term of the nanovoid concentration in the deformed state results in an inelastic term in the stress tensor which depends only on and is along the gradient of the nanovoid concentration. Also, multiplying the free energy of mixing and the gradient energy by the ratio of mass densities in the undeformed and deformed states leads to an inelastic hydrostatic stress. The combination of the above inelastic stresses gives the correct inelastic interface stress. It is proved that for a stationary interface which does not support elastic stresses, the obtained interface stress reduces to the biaxial tension which coincides with a sharp-interface approach limit. The interface stress changes the total stress distribution and affects the elastic stress field. Thus, due to the significant effect of the elastic energy on void dynamics, it can indirectly affect the void nucleation and growth. The coupled CH and elasticity equations are solved using the finite element method and several examples of void evolution are studied consisting of thermal induced initiation and propagation of a planar void, thermal induced circular void growth, evolution of void nanostructure under biaxial compression and evolution of void nanostructure with an initially, randomly distributed void concentration. The obtained results show the significant effect of the interface stress on the total stress distribution and consequently, a different distribution of thermodynamic driving force which can affect the nanostructure evolution and deformation. The interface stress represents a promotive effect on the void growth which results in a faster void growth and a larger void concentration. Also, the results reveal a significant effect of temperature, elastic properties and sample size on the interface stress. (C) 2020 Elsevier Ltd. All rights reserved.