A statistical mechanics-based investigation of a solute atmosphere interacting with a crack: Solute redistribution and image effects

Abstract

Although continuum frameworks have been developed in the past for studying how material defects behave in solid solutions, recent studies have demonstrated that the self-stress field of the solutes which enter the chemical potential in the prevalent continuum models must be excluded, and instead the image stresses included, in a correct treatment. Here, working within the framework of Gibbs canonical ensemble of statistical mechanics, and accounting for the image stresses of the solutes arising due to the presence of the boundaries, we examine the interaction between a solute atmosphere and a crack in the presence of an external loading. Relying on a plane strain model, the solute atmosphere is treated as a distribution of misfitting solute rods, infinitely long in the direction normal to the plane of analysis. Since solutes interact through their image stress fields and can move around in the solid, Monte Carlo simulations, fully accounting for both the energy of interaction between the solute rods and their interaction with the external loading, are performed to explore various configurations of the solute rods in the host, and to compute the temperature-dependent ensemble averages of the enthalpy, equilibrium solute distribution, and the energy release rate (ERR) of crack growth. It is shown that solute redistribution induced by a tensile loading of the solid results in an increase in the ERR. Our simulations further interestingly reveal that even in the limit of large temperatures where a nearly uniform solute distribution prevails, the interaction between the collective image stress field of the solutes and the stress field of the loading results in an added contribution to the ERR which, at a fixed average concentration and a fixed loading-induced stress intensity factor, does not fade away with increasing solid dimensions, and further strengthens both with the solute concentration and with the external loading.