A pseudo-dynamic phase-field model for brittle fracture

TL;DR

This work addresses the persistent violation of energy conservation in phase-field models of brittle fracture under unstable crack growth. It introduces a pseudo-dynamic framework that enforces energy balance without solving the full dynamic equations by coupling an overload factor $\eta$ in the phase-field evolution with a global energy-balance constraint controlled by a dissipation parameter $\zeta$. The augmented formulation, together with a nested alternating-minimization solver, enables crack evolution that ranges from fully energy-conserving to maximally dissipative, and it demonstrates close agreement with adjusted load-displacement data from PMMA compact-tension experiments across multiple geometries. The results highlight the practical potential of energy-balanced pseudo-dynamics for realistically capturing unstable fracture processes and guiding toughness estimation, while also pointing to limitations such as the uniformity of $\eta$ and the spectral-decomposition model's handling of complex crack paths.

Abstract

The enforcement of global energy conservation in phase-field fracture simulations has been an open problem for the last 25 years. Specifically, the occurrence of unstable fracture is accompanied by a loss in total potential energy, which suggests a violation of the energy conservation law. This phenomenon can occur even with purely quasi-static, displacement-driven loading conditions, where finite crack growth arises from an infinitesimal increase in load. While such behavior is typically seen in crack nucleation, it may also occur in other situations. Initial efforts to enforce energy conservation involved backtracking schemes based on global minimization, however in recent years it has become clearer that unstable fracture, being an inherently dynamic phenomenon, cannot be adequately resolved within a purely quasi-static framework. Despite this, it remains uncertain whether transitioning to a fully dynamic framework would sufficiently address the issue. In this work, we propose a pseudo-dynamic framework designed to enforce energy balance without relying on global minimization. This approach incorporates dynamic effects heuristically into an otherwise quasi-static model, allowing us to bypass solving the full dynamic linear momentum equation. It offers the flexibility to simulate crack evolution along a spectrum, ranging from full energy conservation at one extreme to maximal energy loss at the other. Using data from recent experiments, we demonstrate that our framework can closely replicate experimental load-displacement curves, achieving results that are unattainable with classical phase-field models.

Figures (13)

• Figure 1: Single-notch specimen with initial crack modeled as row of elements where $\phi = 1$.
• Figure 2: Effect of mesh refinement on the length of simulated cracks, showing (a) the accurracy of crack length prediction as a function of the ratio $\ell/h^e$ and (b) the simulated phase-field profile for a fully developed crack in the case where $\ell/h^e = 0.25$.
• Figure 3: Geometry of different modified compact tension specimens taken from Cavuoto2022, with dimensions given in mm. The specimens in (a) and (b) differ in the actual location of the added hole. For the specimen shown in (c), the holes are additionally notched at their surfaces in order to control the precise location of crack nucleation.
• Figure 4: Influence of mesh refinement at loading point regions on the simulated load-displacement curve.
• Figure 5: Adjustment of experimental results for modified CT specimens. The fitted function $f_0$ for the different specimens is shown in (a), while the adjusted load-displacement curves are shown in (b)--(d), together with the original data from Cavuoto2022.