[Paper Review] A smoothed particle hydrodynamics approach for phase field modeling of brittle fracture

Mohammad Naqib Rahimi, Georgios Moutsanidis|arXiv (Cornell University)|Mar 13, 2022

Fluid Dynamics Simulations and Interactions68 references27 citations

TL;DR

This paper presents a novel smoothed particle hydrodynamics (SPH) framework for modeling brittle fracture using a hyperbolic phase field approach, enabling stable, explicit time integration and accurate simulation of complex crack initiation, propagation, branching, and coalescence without explicit crack tracking. The method achieves mesh-free, robust fracture dynamics with excellent agreement to FEM and experimental results in challenging benchmark problems.

ABSTRACT

Fracture is a very challenging and complicated problem with various applications in engineering and physics. Although it has been extensively studied within the context of mesh-based numerical techniques, such as the finite element method (FEM), the research activity within the Smoothed Particle Hydrodynamics (SPH) community remains scarce. SPH is a particle-based numerical method used to discretize equations of continuum media. Its meshfree nature makes it ideal to simulate fracture scenarios that involve extreme deformations. However, to model fracture, SPH researchers have mostly relied on ad-hoc empirical local damage models, cohesive zone approaches, or pseudo-spring models, which come with a set of drawbacks and limitations. On the other hand, phase field models of brittle fracture have recently gained popularity in academic circles and provide significant improvements compared to previous approaches. These improvements include the derivation from fundamental fracture theories, the introduction of non-locality, and the ability to model multiple crack initiation, propagation, branching, and coalescence, in situations where no prior knowledge of the crack paths is available. Nevertheless, phase field for fracture has not been studied within SPH. In this proof-of-concept paper we develop and implement a phase field model of brittle fracture within the context of SPH. Comprehensive mathematical and implementation details are provided, and several challenging numerical examples are computed and illustrate the proposed method's ability to accurately and efficiently simulate complex fracture scenarios.

Motivation & Objective

• To address the lack of robust, physics-based fracture modeling in SPH, which has traditionally relied on empirical or heuristic damage models.
• To overcome limitations of existing SPH fracture methods—such as mesh dependency, spurious damage, and high computational cost—by introducing a variational, non-local phase field approach.
• To develop a compatible, stable, and efficient coupling between SPH solid mechanics and a hyperbolic phase field model for brittle fracture.
• To demonstrate the method’s capability on complex fracture problems with arbitrary crack paths, including branching and coalescence, under large deformations and dynamic loading.

Proposed method

• The SPH formulation is based on a total Lagrangian framework to eliminate tensile instability and ensure stability under large deformations.
• A hyperbolic phase field equation governs crack evolution, enabling explicit time integration without the stiffness or time step restrictions of elliptic or parabolic models.
• The phase field parameter represents material integrity and diffuses the crack discontinuity over a regularized length scale, ensuring mesh independence and convergence.
• The coupled system of solid mechanics and phase field evolution is solved using explicit time integration, with consistent SPH approximations for spatial derivatives.
• Contact forces between bodies are modeled via a penalty-based interface force formulation with adaptive contact detection and stabilization.
• The method uses a regularized Heaviside function and a non-local damage description to avoid spurious crack patterns and ensure physical consistency.

Experimental results

Research questions

• RQ1 Can a hyperbolic phase field model be successfully coupled with SPH for brittle fracture simulation while maintaining stability and efficiency?
• RQ2 How does the proposed SPH-phase field framework perform in simulating complex fracture patterns such as crack branching and coalescence without prior knowledge of crack paths?
• RQ3 To what extent does the method reproduce experimental and FEM reference results in benchmark problems like the notched beam under three-point bending?
• RQ4 Does the use of a total Lagrangian SPH formulation effectively suppress tensile instability in dynamic fracture simulations?
• RQ5 Can the explicit time integration of the hyperbolic phase field PDE be effectively applied in SPH without introducing numerical instabilities or excessive time step constraints?

Key findings

• The proposed SPH-phase field framework successfully simulates crack initiation and propagation in a notched beam under three-point bending, with a critical load at 0.3 mm crack opening that matches well with FEM and experimental results.
• The method accurately captures crack branching and coalescence in a notched circular plate under impact, showing that fracture initiates earlier on the tension side (back of the notch), consistent with physical expectations.
• In the impact simulation, the plate undergoes large deformation, fragmentation, and energy dissipation, with displacements of key points (A and B) showing realistic pre- and post-failure behavior.
• The hyperbolic phase field model enables stable, explicit time integration without solving large linear systems, significantly improving computational efficiency compared to elliptic or parabolic phase field models.
• The framework demonstrates robustness in complex, multi-body contact scenarios involving dynamic fracture and large deformations, with no observed spurious damage or penetration.
• All qualitative features of the fracture process—including crack path evolution, energy release, and structural response—align closely with underlying physics and reference solutions, validating the method’s physical consistency and predictive capability.