Balanced quasistatic evolutions of critical points in metric spaces

TL;DR

This paper introduces a novel, constructive approach to modeling quasistatic evolutions of critical points in metric spaces, decoupling energy evolution from state transitions, applicable to nonconvex energies and more general settings.

Contribution

It proposes a new decoupled framework for quasistatic evolutions that is simpler, more general, and computationally feasible, extending previous models to non-degenerate critical points and metric spaces.

Findings

Framework aligns with physical principles of energy minimization.

Method is fully constructive and computationally implementable.

Extends to evolutions in locally compact metric path spaces.

Abstract

Quasistatic evolutions of critical points of time-dependent energies exhibit piecewise smooth behavior, making them useful for modeling continuum mechanics phenomena like elastic-plasticity and fracture. Traditionally, such evolutions have been derived as vanishing viscosity and inertia limits, leading to balanced viscosity solutions. However, for nonconvex energies, these constructions have been realized in Euclidean spaces and assume non-degenerate critical points. In this paper, we take a different approach by decoupling the time scales of the energy evolution and of the transition to equilibria. Namely, starting from an equilibrium configuration, we let the energy evolve, while keeping frozen the system state; then, we update the state by freezing the energy, while letting the system transit via gradient flow or an approximation of it (e.g., minimizing movement or backward differentiation schemes). This approach has several advantages. It aligns with the physical principle that systems transit through energy-minimizing steady states. It is also fully constructive and computationally implementable, with physical and computational costs governed by appropriate action functionals. Additionally, our analysis is simpler and more general than previous formulations in the literature, as it does not require non-degenerate critical points. Finally, this approach extends to evolutions in locally compact metric path spaces, and our axiomatic presentation allows for various realizations.