FeynKrack: a continuum model for quasi-brittle damage through Feynman–Kac killed diffusion

Continuum damage mechanics (CDM), a popular modelling framework for crack propagation in solids, uses a damage parameter to quantitatively assess ‘material degradation’. While this parameter is sometimes given a physical meaning, its evolution is generally not consistent with such physical interpretations. Curiously, degradation in CDM may be viewed as a change of measures, wherein the damage variable appears as the Radon–Nikodym derivative. By using this worldview and a measure-valued description for the random microcracks underlying quasi-brittle damage, we show that the underlying density evolves via killed diffusion as in the Feynman–Kac theory. Damage growth is then the reduction in this measure, which in turn quantifies the disruption of bonds, i.e. force-transmitting mechanisms between nearby material points. Remarkably, this evolution admits an approximate closed-form solution, bringing forth substantive computational ease and facilitating fast yet accurate simulations of large dimensional problems. With an appropriate killing rate, one accounts for damage irreversibility and thus eliminates ad hoc history-dependent routes. Our proposal Feynman–Kac crack (FeynKrack) propagator is validated and demonstrated for its efficacy through several simulations on quasi-brittle damage. It also offers a promising stochastic route for future explorations of non-equilibrium thermodynamic aspects of damage.