Griffith's postulate: Grand Canonical Monte Carlo approach for fracture mechanics of solids

A Grand Canonical Monte Carlo Approach (GCMC) is proposed for the fracture analysis of solids discretized as mass points and bond interactions. In contrast to classical load-driven fracture processes, the GCMC approach introduces an auxiliary field, the bond rupture potential μ, to which the system is subjected; in addition to changes in volume V and temperature T. In this μVT-ensemble, bond isotherms that link the average number of bonds to the bond rupture potential (Nk-μ) are obtained that carry critical information for fracture analysis. Specifically, the slope of the bond isotherm reflects bond fluctuations, permitting identification of (1) a fluctuation-based damage variable, and (2) the competition in energy fluctuations between the redistribution of strain energy induced by bond rupture, and the dissipation of the groundstate energy. Based on these fluctuations, it is shown that the GCMC-approach allows the identification of a critical bond energy release rate of material samples, when strain energy fluctuations equal groundstate energy fluctuations - much akin to Griffith's 1921 stationarity postulate to “predicting the breaking loads of elastic solids”. This is illustrated by means of thermodynamic integration of bond isotherms to determine force-displacement curves, for both notched and unnotched homogeneous samples discretized by regular 2-D lattices with bonds exhibiting harmonic potentials.