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

TL;DR

FeynKrack introduces a probabilistic continuum damage model using killed diffusion and Feynman-Kac theory, enabling efficient, physically consistent simulations of quasi-brittle damage without ad-hoc history dependence.

Contribution

It presents a novel measure-valued damage evolution model based on killed diffusion, providing a closed-form solution for quasi-brittle damage simulation.

Findings

Efficient closed-form damage evolution solution

Accurate simulations validated against benchmarks

Elimination of ad-hoc history dependence in damage modeling

Abstract

Continuum damage mechanics (CDM) is a popular framework for modelling crack propagation in solids. The CDM uses a damage parameter to quantitatively assess what one loosely calls `material degradation'. While this parameter is sometimes given a physical meaning, the mathematical equations for its evolution are generally not consistent with such physical interpretations. Curiously, degradation in the CDM may be viewed as a change of measures, wherein the damage variable appears as the Radon-Nikodym derivative. We adopt this point of view and use a probabilistic measure-valued description for the random microcracks underlying quasi-brittle damage. We show that the evolution of the underlying density may be described via killed diffusion as in the Feynman-Kac theory. Damage growth is then interpreted as the reduction in this measure over a region, which in turn quantifies the disruption of bonds through a loss of force-transmitting mechanisms between nearby material points. Remarkably, the evolution of damage admits an approximate closed-form solution. This brings forth substantive computational ease, facilitating fast yet accurate simulations of large dimensional problems. By selecting an appropriate killing rate, one accounts for the irreversibility of damage and thus eliminates the need for ad-hoc history-dependent routes typically employed, say, in phase field modelling of damage. Our proposal FeynKrack (a short form for Feynman-Kac crack 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.