Overview
This work focuses on a strength-degradation phase-field approach for cohesive fracture, specifically addressing the antiplane case. The model modifies earlier phase-field frameworks by degrading the material's strength rather than its stiffness. This approach is presented as a regularization of softening plasticity, where localization bands adhere to a cohesive law derived from the strength domain and toughness. The regularization length, when sufficiently small compared to the elasto-cohesive length, primarily serves a numerical function.
A central aspect of this model is its ability to allow strength, stiffness, and toughness to be defined as independent material properties. This enables the integration of limit analysis, perfect plasticity, cohesive fracture, and brittle fracture within a unified variational framework.
Research Context
Phase-field approaches to fracture initially served as a regularization method for the Griffith model of brittle fracture. These approaches are now frequently interpreted as gradient-damage models where a regularization length acts as a material property influencing crack nucleation. A limitation of some earlier phase-field methods is that the shape and magnitude of the strength surface are constrained by the elastic energy and the regularization length.
The model investigated here builds upon an antiplane version introduced by Bourdin, Marigo, Maurini, and Zolesi (arXiv:2506.22558). This prior model was designed to handle crack propagation along unspecified paths and nucleation governed by an arbitrary convex strength surface.
Approach
The research derives closed-form solutions for a simple shear problem using the strength-degradation phase-field model. A numerical scheme combining alternate minimization and conic programming is proposed for implementation. To validate the model, numerical verification was conducted to confirm the equivalent cohesive law and its independence from the regularization. The study also examined the size effect, demonstrating its governance by the elasto-cohesive length.
Two specific simulations were performed: a "surfing" simulation was used to illustrate the structure of the propagating crack. Additionally, a re-entrant V-notch simulation was employed to demonstrate the model's capacity to bridge small-scale yielding, cohesive fracture, and brittle fracture without requiring prior assumptions regarding the fracture mode.
Findings
- The strength-degradation phase-field model allows for independent material data input for strength, stiffness, and toughness.
- The regularization length, when small relative to the elasto-cohesive length, has a purely numerical role in this framework.
- The model unifies limit analysis, perfect plasticity, cohesive fracture, and brittle fracture within a single variational framework.
- Closed-form solutions for a simple shear problem were derived.
- A numerical scheme combining alternate minimization and conic programming was developed.
- Numerical verification confirmed the equivalent cohesive law and its independence from regularization.
- The size effect in the model is governed by the elasto-cohesive length.
- Simulations with a "surfing" crack and a re-entrant V-notch demonstrated the model's ability to span various fracture regimes without a priori hypotheses.
Why This Matters
The ability to independently specify strength, stiffness, and toughness provides a more flexible and comprehensive framework for modeling fracture. By unifying different fracture behaviors—limit analysis, perfect plasticity, cohesive fracture, and brittle fracture—within a single variational framework, the model offers a consistent approach to analyzing material failure.