Navigating with Stability: Local Minima, Patterns, and Evolution in a Gradient Damage Fracture Model

Abstract

We investigate the computation of stable fracture paths in brittle thin films using one-dimensional damage models with an elastic foundation. The underlying variational formulation is non-convex, making the evolution path sensitive to algorithmic choices. In this paper, we inquire into the effectiveness of quasi-Newton algorithms as an alternative to conventional Newton-Raphson solvers. These algorithms improve convergence by constructing a positive definite approximation of the Hessian, trading improved convergence for the risk of missing bifurcation points and stability thresholds. In the absence of irreversibility constraints, we construct an equilibrium map that represents all stable and unstable equilibrium states as a function of the external load, using well-known branch-following bifurcation techniques. Our main finding is that quasi-Newton algorithms fail to select stable evolution paths without exact second variation information. To overcome this, we introduce a spectral stability criterion based on the full Hessian, which enables the identification of optimal perturbations and improves path-following accuracy. We then extend our analysis to the irreversible case, where admissible perturbations are constrained to a cone. We develop a nonlinear constrained eigenvalue solver to compute the minimal eigenmode within this restricted space and show that it plays a key role in distinguishing physical instabilities from numerical artefacts. Our results provide practical guidance for robust computation of fracture paths in irreversible, non-convex settings.

References in corpus (14)

• SciPy 1.0--Fundamental Algorithms for Scientific Computing in Python
• Assessment and optimization of the fast inertial relaxation engine (FIRE) for energy minimization in atomistic simulations and its implementation in LAMMPS
• Ferroelastic Dynamics and Strain Compatibility
• An accelerated staggered scheme for phase-field modeling of brittle fracture
• On the critical nature of plastic flow: one and two dimensional models
• Landau theory of crystal plasticity
• Discrete models of dislocations and their motion in cubic crystals
• Fluctuations in crystalline plasticity
• Homogeneous nucleation of dislocations as bifurcations in a periodized discrete elasticity model
• Origin of stabilization of macrotwin boundaries in martensites
• De-localizing brittle fracture
• Homogeneous nucleation of dislocations as a pattern formation phenomenon
• Elastic instability behind brittle fracture
• Fragmentation of shells: An analogy with the crack formation in tree bark