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Phase field modeling of quasi-static and dynamic crack propagation: COMSOL implementation and case studies
Shuwei Zhou, Timon Rabczuk, Xiaoying Zhuang
TL;DR
The paper addresses the implementation of phase-field fracture modeling as a coupled multifield problem. It develops quasi-static and dynamic implementations in COMSOL using coupled modules and a staggered scheme, and reports satisfactory accuracy and correct crack patterns in 2D and 3D examples. The study concludes that COMSOL is feasible for implementing and extending phase-field crack models.
Problem
Implementing phase-field fracture as a multifield problem can be laborious in general finite element software, motivating a simpler multifield-oriented implementation.
Method
The authors implement quasi-static and dynamic phase-field models in COMSOL with a scalar crack field, tensile/compressive energy decomposition, coupled modules, and a staggered solution scheme.
Results
2D and 3D benchmark simulations produce correct crack patterns and satisfactory accuracy, supporting the feasibility of the COMSOL implementation.
Takeaways & Limitations
The COMSOL implementation can be extended to phase-field problems involving more coupled physical fields.
Takeaways & Limitations
Selecting the length-scale parameter remains open, and extending its estimation approach to higher dimensions and complex mixed-mode fracture is difficult.
Abstract
from arXiv · showhide
The phase-field model (PFM) represents the crack geometry in a diffusive way without introducing sharp discontinuities. This feature enables PFM to effectively model crack propagation compared with numerical methods based on discrete crack model, especially for complex crack patterns. Due to the involvement of \phased field", phase-field method can be essentially treated a multifield problem even for pure mechanical problem. Therefore, it is supposed that the implementation of PFM based on a software developer that especially supports the solution of multifield problems should be more effective, simpler and more efficient than PFM implemented on a general finite element software. In this work, the authors aim to devise a simple and efficient implementation of phase-field model for the modelling of quasi-static and dynamic fracture in the general purpose commercial software developer, COMSOL Multiphysics. Notably only the tensile stress induced crack is accounted for crack evolution by using the decomposition of elastic strain energy. The width of the diffusive crack is controlled by a length-scale parameter. Equations that govern body motion and phase-field evolution are written into different modules in COMSOL, which are then coupled to a whole system to be solved. A staggered scheme is adopted to solve the coupled system and each module is solved sequentially during one time step. A number of 2D and 3D examples are tested to investigate the performance of the present implementation. Our simulations show good agreement with previous works, indicating the feasibility and validity of the COMSOL implementation of PFM.
1 Introduction
The introduction contrasts discrete crack representations with phase-field modeling and motivates a COMSOL implementation for coupled fracture problems. The paper implements quasi-static and dynamic phase-field models in COMSOL using coupled modules and a staggered scheme.
- Limitations of discrete crack methods: Discrete crack methods represent complex geometries through discontinuities, mesh reconstruction, cohesive interfaces, enrichment functions, or element erosion.Element-erosion methods cannot simulate crack branching correctly.
- Phase-field approach: Phase-field models represent discrete cracks with a scalar field that smoothly transitions material from intact to fully broken states.Crack shape and propagation are determined by the phase-field evolution equations, without requiring additional crack-surface treatments.
- Implementation motivation: Phase-field fracture is essentially a multifield problem even for pure mechanics, making software designed for multifield programming attractive for implementation.The authors contrast this with the laborious implementation of multifield problems in Abaqus.
- COMSOL implementation: The paper exploits COMSOL for a simple and fast phase-field implementation that can be extended by adding modules and coupling terms.The proposed implementation is intended to support additional coupled physical phenomena.
- Scope and validation: Quasi-static and dynamic phase-field models are implemented in COMSOL using a staggered scheme for fracture problems.The paper examines 2D and 3D numerical examples under quasi-static and dynamic loading.
2 Phase-field model for fracture
The phase-field fracture formulation replaces sharp crack surfaces with a scalar diffusive field and derives fracture evolution from a variational energy framework. It incorporates tensile-only degradation, dynamic effects, irreversibility, and a length-scale parameter controlling crack width.
- The body is modeled with displacement, boundary conditions, body forces, and tractions over a domain that may contain an internal discontinuity.
- Crack initiation, propagation, and branching satisfy energy minimization together with irreversibility, so formed cracks cannot return to the uncracked state.A strain-history field is introduced to maintain monotonically increasing phase-field evolution during compression or unloading.
- The scalar phase-field φ(x,t) ∈ [0,1] approximates the crack surface, with φ = 1 denoting a crack and φ = 0 an uncracked body.This produces a diffusive crack topology rather than a sharp discontinuity.
- The length-scale parameter l0 controls the transition region between intact and broken material, with larger l0 producing a wider crack region.
- Only tensile elastic energy drives phase-field evolution, while compressive stress does not contribute to crack propagation.A residual parameter k > 0 prevents positive elastic energy from vanishing and avoids numerical singularity as φ approaches 1.
- The variational formulation combines elastic, fracture, kinetic, and external potential energies to derive the governing equations.The first variation of the total Lagrange functional is set to zero.
3 Implementation method in COMSOL
COMSOL implementation separates displacement, history-strain, and phase-field calculations into coupled modules solved with finite-element spatial and finite-difference temporal discretization. A staggered scheme sequentially updates these fields, with constitutive modifications, initialization procedures, and convergence acceleration supporting quasi-static and dynamic calculations.
- Three COMSOL modules solve displacement, history-strain, and phase-field variables, while a Storage Module calculates and stores internal fields.
- The Solid Mechanics Module uses a linear elastic library, with its elasticity matrix modified during each time step from a fourth-order elasticity tensor.
- The phase-field and History-strain Modules implement the governing phase-field equation and distributed history-strain evolution, respectively.
- Displacement and phase-field variables are discretized using nodal values and shape-function matrices, with strains and phase-field gradients obtained through derivative matrices.
- Anderson acceleration is used because convergence slows when fracture begins, and standard Lagrangian elements discretize the three physical fields.
4 Numerical examples
The numerical examples assess quasi-static and dynamic fracture in 2D and 3D, showing agreement with prior results while revealing effects of length scale, mesh, time step, and critical energy release rate.
- 2D tension: The 2D tensile benchmark reproduces reported horizontal crack patterns and agrees well with Hesch and Weinberg’s load-displacement results.A smaller length scale produces a less diffused crack.
- Numerical sensitivity: Mesh size and displacement increment leave the crack pattern unchanged, but larger mesh size raises peak load and smaller increments produce a steeper post-peak response.The study notes the recommended condition h ≤0.5l0 for precise crack topology.
- 3D tension: The 3D tensile plate gives crack patterns and load-displacement curves similar to the 2D case, while peak load increases as l0 decreases.The simulations use 8-node Lagrangian elements without special refinement along the expected crack path.
- 2D shear: Under shear, the crack follows a curved path, becomes wider for l0 = 1.5 × 10−2 mm, and matches prior crack patterns and load-displacement results.The loads exactly match Hesch and Weinberg’s results particularly for l0 = 7.5 × 10−3 mm.
- Dynamic fracture: In dynamic tension, the crack initiates at 26 µs, reaches an angle of 63◦–67◦, and its speed rises to about 0.6vR before remaining nearly constant.The crack-tip velocity is determined from the φ = 0.75 iso-curve, and results are independent of the two tested time steps.
- Dynamic fracture: Increasing Gc delays crack initiation and reduces maximum crack-tip velocity; smaller Gc produces branching, whereas larger Gc yields a single main crack.The main crack still initiates at the pre-existing crack tip.
- Dynamic fracture: For dynamic crack branching, coarser meshes and larger time steps create wider cracks that are more difficult to reach the boundary, while the staggered scheme delays branching.Dissipated energy increases with time, and velocity remains below 0.5vR.
- Dynamic fracture: Maximum tensile stress concentrates at the crack tip, with similar results for the two meshes and agreement with Liu et al.Elastic strain-energy curves diverge after 60 µs depending on time step and mesh, while dissipated-energy curves remain in good agreement.
5 Conclusions
The paper implements quasi-static and dynamic phase-field fracture modeling in COMSOL using coupled modules and a staggered solution scheme. Benchmark simulations in 2D and 3D produce correct crack patterns and satisfactory accuracy, supporting the feasibility of the approach.
- Implementation: COMSOL implements phase-field crack propagation through coupled modules for the displacement, history-strain, and phase fields, solved with a staggered scheme.The formulation decomposes elastic strain energy into compression and tension, so only tension-induced cracks evolve.
- Implementation: An implicit time-integration scheme with Newton-Raphson iteration computes the individual fields in the coupled system.
- Validation: 2D and 3D benchmark examples cover quasi-static and dynamic crack propagation while examining length scale, critical energy release rate, mesh size, and time-step effects.
- Validation: All simulations produce correct crack patterns and satisfactory accuracy, demonstrating the feasibility of COMSOL phase-field implementation even in 3D.The authors identify future extension to problems involving more fields as a potential use of COMSOL.