Dynamic brittle fracture using Lip-field approach in an explicit dynamics context

Technical Summary: Dynamic Brittle Fracture Using Lip-field Approach in an Explicit Dynamics Context

Problem Statement
This paper addresses the numerical simulation of dynamic brittle fracture in materials subjected to high strain-rate loading, such as impacts or explosions. While classical fracture mechanics (e.g., Griffith's model) struggles with crack initiation and branching, modern approaches like Cohesive Zone Models (CZM) and diffuse damage models have been developed. However, local diffuse damage models suffer from pathological mesh sensitivity, requiring regularization to introduce a length scale and ensure mesh-independent results. Existing regularization techniques, such as Phase-Field and Thick Level Set (TLS) methods, often involve non-local energy functionals that can be computationally expensive. The specific challenge addressed here is extending the Lip-field approach—a variational method that introduces length scales via Lipschitz constraints on the damage field—from its previous quasi-static and one-dimensional dynamic applications to two-dimensional dynamic fracture within an explicit time integration framework.

Methodology
The authors propose a variational formulation for dynamic damage mechanics based on the minimization of an incremental potential. The core of the methodology involves:

1. Variational Formulation: The mechanical problem is defined by minimizing an incremental potential $F$ that includes kinetic energy, free energy, and dissipation potential. The state variables are displacement ($u$) and a scalar damage variable ($d$). The constitutive model utilizes generalized standard materials with specific energy degradation ($g(d)$) and softening ($h(d)$) functions.
2. Lip-field Regularization: Unlike Phase-Field models that modify the energy functional with a gradient term, the Lip-field approach keeps the energy functional local (non-regularized). Instead, it imposes Lipschitz constraints on the damage field, requiring that the damage gradient be bounded by a critical value ($1/l$). This restricts the damage field to a specific Lipschitz space $\mathcal{L}$, effectively introducing a length scale $l$ without altering the local energy density.
3. Explicit Staggered Scheme: The paper employs an explicit Newmark time integration scheme (central difference method). A staggered approach is used where displacement and damage fields are solved sequentially. For a given time step, the displacement is updated first, followed by the damage update.
4. Efficient Damage Solver: A key algorithmic contribution is the use of bounds estimation (derived from prior work) to optimize the damage calculation. The process involves:
   • Solving a purely local minimization problem to find a local damage field ($d_{loc}$).
   • Projecting $d_{loc}$ into upper and lower bounds based on the Lipschitz constraint.
   • Identifying regions where these bounds differ.
   • Solving the non-local (Lipschitz-constrained) minimization problem only in the small portion of the domain where the bounds are not equal. In regions where bounds coincide, the local solution is accepted. This significantly reduces computational cost compared to solving the non-local problem over the entire domain.
5. Discretization: The spatial discretization uses linear triangular elements for the displacement field. A separate "lip-mesh" (constructed from element centroids) is used to store and constrain the damage field, ensuring the number of Lipschitz constraints is minimized.

Key Contributions

• Extension to 2D Dynamics: The paper successfully extends the Lip-field approach, previously validated in 1D dynamic fragmentation and quasi-static cases, to two-dimensional dynamic fracture problems.
• Explicit Dynamics Integration: It outlines a specific implementation of the Lip-field approach using an explicit time integration scheme, which is computationally efficient for high-speed impact problems.
• Computational Efficiency via Bounds: The authors demonstrate that utilizing the bounds estimate allows the expensive non-local minimization to be restricted to small sub-domains, making the Lip-field approach computationally competitive for dynamic simulations.
• Variational Distinction: The work clarifies the distinction between Lip-field and Phase-field approaches: Lip-field maintains a local energy functional with non-local constraints, whereas Phase-field uses a non-local energy functional.

Simulation Results
The proposed model was validated through two-dimensional numerical examples:

1. Single Edge Notched Tension (SENT): The model successfully simulated dynamic crack propagation and branching. Results showed mesh convergence in potential and dissipated energy. The crack branching phenomenon was captured without ad-hoc criteria, and the crack tip velocities generally remained below the Rayleigh wave speed, except for transient spikes during branching events. The results were comparable to Phase-field and TLS simulations found in literature.
2. Load Intensity Influence: By varying the applied traction, the model demonstrated the ability to capture multiple crack branches, with branching occurring closer to the notch tip under higher loads.
3. Kalthoff-Winkler Test: The model simulated crack initiation and propagation under dynamic shear loading.
   • At low impact velocities ($33$ m/s), it predicted a Mode-I crack propagation angle of $\approx 62.4^\circ$, consistent with experimental observations of $\approx 70^\circ$.
   • At high impact velocities ($100$ m/s), the model captured successive crack branching and multiple cracks.
   • Limitation: The authors note that while experiments show a transition to ductile failure (shear fracture at $\approx -10^\circ$) at high velocities, the current elastic-damage model does not capture this ductile behavior, as it lacks plasticity.

Significance and Claims
The paper claims that the Lip-field approach offers a robust variational framework for dynamic fracture that preserves the mathematical relevance of the problem while introducing necessary length scales. Its primary significance lies in its computational efficiency achieved through the bounds-based solver, which avoids solving non-local equations globally. The authors assert that the approach effectively handles complex fracture phenomena like crack initiation, branching, and coalescence without ad-hoc criteria.

The paper modestly concludes that while the Lip-field approach is effective for brittle fracture and fragmentation, it is limited to elastic-damage behavior. It acknowledges that simulating ductile failure at high strain rates would require an extension to elastic-plastic damage models, which is outside the scope of the current work. The method is presented as a viable alternative to Phase-field and TLS methods, particularly where computational efficiency in explicit dynamics is critical.