Balanced quasistatic evolutions of critical points in metric spaces

TL;DR

This work introduces a decoupled, constructive framework for balanced quasistatic evolutions of time-dependent energies in metric spaces by evolving the energy while freezing the state and then applying action-driven transitions (gradient flow, minimizing movements, or BDF2). It proves convergence of discrete evolutions to a limit curve in a quotient space that accounts for degenerate critical points, establishes an energy balance with an atomic jump measure, and characterizes energy jumps via transition actions. The theory unifies continuous-time gradient dynamics and discrete-time variational schemes under a common axiomatic and variational structure, and it validates the approach with numerical experiments on a rod-fracture model. The methodology is broadly applicable to metric-path spaces and provides a practical, implementable route to balanced quasistatic evolutions without requiring nondegenerate critical points. The results highlight a transparent link between energy landscape evolution, transition rules, and dissipation captured by an action-driven jump mechanism.

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.

Figures (6)

• Figure 1: The spaces involved in the definition of the quasistatic evolution $\hat{\eta}$. If $E$ does not have degenerate critical points, both $q$ is the identitity and $\hat{\eta}$ takes values in $[0, T] \times \mathbb{X}$.
• Figure 2: We display here a scheme explaining how a quasistatic evolution of critical point is constructed through MMS transition rule.
• Figure 3: The proof strategy for \ref{['lemma:good_curves_fulfill_continuous_triangular_inequality']} and the scheme for shortening competitor curves: We create a shortcut from the first point where a curve enters a neighborhood $B_r(\mathscr{C}_i)$ to the last point where one exits $B_r(\mathscr{C}_i)$, where $\mathscr{C}_i$ is a connected component of critial points. This approach ensures that $B_r(\mathscr{C}_i)$ is traversed at most once. In the formal proof, rather than constructing a new competitor curve by concatination, we employ the triangle inequality to estimate $c_t(X_1, X_4) \leq \mathfrak{c}_t(X_1, p_{\mathrm{in}}^1) + \mathfrak{c}_t(p_{\mathrm{in}}^1, \overline{p}_{\mathrm{in}}^1) + \mathfrak{c}_t(\overline{p}_{\mathrm{in}}^1, \overline{p}_{\mathrm{out}}^1) + \mathfrak{c}_t(\overline{p}_{\mathrm{out}}^1, p_{\mathrm{out}}^1) + \mathfrak{c}_t(p_{\mathrm{out}}^1, X_4)$.
• Figure 4: The simulation for an elastic rod, once for (a) $\delta=\frac{1}{15}$ and once for (b) $\delta=\frac{1}{240}$. From left to right, we plot (i) the initial configuration, (ii) the configuration right before the system transition which leads to the breakage of the rod, (iii) the configuration right after the transition, and (iv) the configuration at the end of the time horizon. In (a.ii), the stress is concentrated on the two end segments, which leads to the rod breaking early. This phenomenon is the result of the approximation error due to the large step size $\delta = \frac{1}{15}$.
• Figure 5: Comparing $E_t(\eta^\delta(t))$ and $\int_{0}^{t} \mathcal{D}^\delta(s) \,\mathrm{d} s$, for $\delta=\frac{1}{15}$ on the left and $\delta=\frac{1}{240}$ on the right.

Theorems & Definitions (123)

• Theorem 1 (simplified)
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6
• Remark 7
• Remark 8
• Definition 1: Discrete quasi-static evolution