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.

在本文中，在相场方法的概念中并利用热力学定律，得出了纳米空隙的固-气界面的热力学一致界面应力。Cahn-Hilliard（CH）方程描述了纳米空隙浓度的演变，该浓度在理想固体的0和气体的1之间变化。考虑到处于变形状态的纳米空隙浓度的梯度项，导致应力张量中的非弹性项，该应力张量仅取决于纳米空隙浓度的梯度并且沿着纳米空隙浓度的梯度。同样，将混合的自由能和梯度能乘以未变形和变形状态下的质量密度之比会导致非弹性静水应力。上述非弹性应力的组合给出了正确的非弹性界面应力。事实证明，对于不支持弹性应力的固定界面，所获得的界面应力会减小为双轴张力，这与急剧的界面逼近极限相吻合。界面应力会改变总应力分布并影响弹性应力场。因此，由于弹性能对空隙动力学的显着影响，它可以间接影响空隙的形核和生长。使用有限元方法求解耦合的CH和弹性方程，并研究了空隙演化的几个示例，包括热诱导的平面空隙的引发和传播，热诱导的圆形空隙生长，双轴压缩下的空隙纳米结构的演化以及空隙的演化具有最初随机分布的空隙浓度的纳米结构。获得的结果表明界面应力对总应力分布有显着影响，因此，热力学驱动力的不同分布会影响纳米结构的演化和变形。界面应力代表对空隙生长的促进作用，其导致更快的空隙生长和更大的空隙浓度。而且，结果表明温度，弹性和样品尺寸对界面应力有显着影响。界面应力代表对空隙生长的促进作用，其导致更快的空隙生长和更大的空隙浓度。而且，结果表明温度，弹性和样品尺寸对界面应力有显着影响。界面应力代表对空隙生长的促进作用，其导致更快的空隙生长和更大的空隙浓度。而且，结果表明温度，弹性和样品尺寸对界面应力有显着影响。