Have you ever tried to model crack growth or material failure? If so, you may have faced limitations. Traditional methods struggle when crack paths are unknown in advance. They also fail when multiple cracks branch and merge unpredictably.
Classical approaches like FEM or XFEM cannot handle such complexity effectively. This is where the Phase Field model demonstrates its strength. According to our experience, phase field modeling allows cracks to initiate, grow, branch, and merge naturally.
Moreover, Phase field fracture simulation does this without requiring explicit tracking. It seamlessly couples with other physical phenomena. As two approaches for implementing this model, the use of Abaqus and Julia code language has been discussed in this blog.
Phase-Field Models for Fracture: Q&A
Phase-field models regularize sharp cracks over a diffuse region using a continuous scalar variable, avoiding the need to explicitly track complex crack topologies. This contrasts with sharp interface models, which treat cracks as two-dimensional surfaces and require complex remeshing or enrichment techniques to handle crack propagation.
The three main length scales for modeling failure in composite laminates are microscopic, mesoscopic, and macroscopic. The microscopic scale considers the microstructure (fibers, matrix); the mesoscopic scale models each lamina as a homogenized anisotropic material; and the macroscopic scale treats the entire laminate as a single homogenized plate.
A degradation function, often denoted ω(d) or g(φ), models the loss of material stiffness as damage evolves from an intact state to a fully broken state. Its value must be 1 for the intact state (φ=0) and 0 for the broken state (φ=1), and it must be a monotonically decreasing function, with its derivative being zero at the fully broken state.
CZMs can be incorporated by using zero-thickness interface elements along potential crack paths or by modifying the PFM’s degradation function. A key challenge is that the crack surface separation, which is central to a CZM for relating cohesive traction, cannot be calculated directly in a PFM, where the crack is smeared over a region.
A staggered scheme, also known as an alternating minimization (AM) algorithm, solves the coupled displacement and phase-field problems independently and sequentially at the same time. The solution for each field is based on the variables obtained from the previous iteration, a process that is repeated until convergence is achieved.
The heat transfer analogy exploits the mathematical similarity between the steady-state heat transfer equation and the phase-field evolution equation. By defining the phase-field variable as analogous to temperature (T ≡ φ) and the fracture driving force as an internal heat source, PFM can be implemented using built-in heat transfer capabilities (like UMATHT subroutines) without needing custom user elements.
The presence of cracks significantly increases local electrical resistivity, interrupting the flow of electrical current. This is modeled by degrading the material’s electrical conductivity or permeability using a degradation function, h2(φ), which is dependent on the magnitude of the phase-field variable φ.
The history field variable H ensures the irreversibility of damage by storing the maximum value of the fracture driving force (e.g., strain energy density) that a material point has experienced over time. This prevents the model from exhibiting non-physical “self-healing” behavior if the material is unloaded.
Phase-field models capture hydrogen embrittlement by defining the material’s fracture resistance, or toughness (Gc), as a function of the local hydrogen concentration (cH). The model couples the mechanical deformation problem with a hydrogen diffusion problem, where hydrogen accumulation in regions of high hydrostatic stress leads to a local reduction in toughness, promoting crack growth.
The main advantages are the ability to automatically handle complex crack phenomena like nucleation, branching, and coalescence in both 2D and 3D with a straightforward implementation. The primary disadvantages include high computational cost, due to the need for very fine meshes to resolve the damage zone, and ambiguities in precisely locating the crack tip for calculating quantities like velocity.
What is the role of Phase Field Modeling in Materials Science?
Phase field modeling (PFM) addresses critical limitations in traditional fracture mechanics. Classical approaches like FEM struggle when crack paths are unknown beforehand, and methods like XFEM face challenges when multiple cracks interact, branch, or merge.
PFM overcomes these obstacles by treating cracks as evolving diffuse interfaces governed by thermodynamic principles, eliminating the need for complex geometric tracking while naturally capturing arbitrary crack topologies. The central idea is straightforward yet elegant.
Instead of explicitly tracking sharp boundaries, the method defines interfaces as diffuse regions. For instance, consider a crack surface separating intact and broken material. The phase field model spreads this interface over a finite volume.
We achieve this diffusion through an auxiliary field variable, φ. Scientists often call this the phase field or order parameter. This variable smoothly transitions between different material states.
In the phase-field method, a diffuse region (also called the diffuse interface) is the transition zone between two different phases — for example, between solid and liquid, or two crystal grains.
You can think of the diffuse region as a “blurry boundary” that connects two distinct phases in a physically realistic and computationally stable way. It captures interface motion, curvature effects, and interfacial energy, all within a unified framework.
What is the theoretical background behind the phase-field approach?
The phase field takes distinct values for different phases. Specifically, φ = 0 represents intact bulk material. Conversely, φ = 1 indicates a fully broken material or crack. Between these values, the field exhibits a smooth, continuous change near the interface.
The evolution of φ follows a partial differential equation (PDE). This equation derives from irreversible thermodynamics principles. In fracture mechanics, PFM successfully embeds Griffith’s thermodynamic framework.
The method regularizes sharp cracks using an internal length scale, l₀. This parameter controls the width of the diffused crack region. As a result, it ensures mesh objectivity. Therefore, we can interpret damage as a scalar variable.
This paradigm provides a robust numerical platform. It predicts highly complex cracking behaviors effectively. These include crack branching, deflection, coalescence, and nucleation. Importantly, it works in arbitrary geometries and dimensions. Best of all, it requires no ad hoc crack propagation criteria or manual mesh adjustments.
What are the main applications of the phase-field method?
The versatility of phase field modeling allows applications across diverse materials. It also tackles complex physics problems. Often, these involve coupled multi-field phenomena.
Fundamental Fracture Types
Depending on the problem for which the phase-field model is intended, different fracture models can be implemented. In the following, we discuss the types of fractures that can be used.
Brittle Fracture
Researchers widely use PFM for quasi-static and dynamic brittle fracture problems. This applies to both 2D and 3D scenarios. The method excels at capturing sudden crack propagation in ceramic materials and glass.
Ductile Fracture
Scientists have successfully applied the phase field model to ductile damage scenarios. This includes elasto-plastic solids. Typically, these applications involve plastic degradation functions. They also incorporate plasticity effects on dynamic crack propagation.
Fatigue Fracture
Modern PFM approaches now incorporate fatigue degradation functions. Consequently, they predict crack growth in metals under cyclic loading. This covers High-, Low-, and Extremely low-cycle fatigue regimes effectively.
Coupled Multi-Physics Problems
The variational nature of PFM makes it ideal for coupling. Specifically, it connects crack evolution with other physical fields seamlessly.
Hydrogen Assisted Cracking
This application investigates hydrogen transport toward fracture process zones. Subsequently, it models cracking through a coupled framework. This framework integrates mechanical, diffusion, and phase field components.
Stress Corrosion Cracking
Engineers model dissolution-driven stress corrosion cracking using PFM. The method also handles pitting corrosion effectively. Furthermore, it incorporates the film-rupture–dissolution–repassivation (FRDR) process.
Thermo-Mechanical Fracture
Researchers analyze thermal fracture scenarios, such as quenching in ceramic materials. They achieve this by coupling temperature fields with mechanical deformation and damage evolution.
Hydraulic Fracture
The phase field model simulates fluid-driven fractures in porous media. It couples phase field evolution with fluid flow and pressure fields. This approach proves valuable in petroleum and geothermal engineering.
Electro-Mechanical Fracture
Scientists model piezoresistive CNT-based composites using this technique. The approach couples deformation, electrical potential, and the phase field fracture variable. This enables accurate prediction of electrically-coupled failure modes.
Li-Ion Battery Degradation
Engineers model chemo-mechanical induced fracture in battery electrode particles. This application helps predict battery lifespan and performance degradation.
Specific Materials and Structures
The phase field method successfully handles various specialized materials:
- Fiber-reinforced composites (intralaminar and translaminar fracture)
- Functionally Graded Materials (FGMs)
- Rock-like materials (including compressive-shear fractures)
- Shape Memory Alloys (SMAs)
What are the Available Phase-Field Models (Constitutive Choices)?
The first decision in phase-field modeling is selecting the appropriate constitutive model based on the material behavior and failure mechanism you want to capture. The term “constitutive” refers to the material’s constitutive law or constitutive relationship – essentially, the mathematical description of how a material behaves under loading. In phase-field fracture modeling, constitutive models define:
- How damage evolves from intact (φ = 0) to fully broken (φ = 1)
- How material stiffness degrades as damage increases (via degradation functions)
- What drives crack propagation (strain energy, specific energy components)
- When and where cracks initiate and grow
We provide more details on the phase-field constitutive models in the following table.
| Model | Description | Suitability/Purpose |
|---|---|---|
| AT2 Model (Standard PFM) | The standard formulation, typically based on a crack density function \( \omega(\phi) \). | Brittle fracture; damage initiates immediately upon loading. |
| AT1 Model | Uses \( \omega(\phi) = \phi \). | Suitable for materials that exhibit a linear elastic regime before the onset of damage (i.e., it includes a minimum fracture driving force \( H_{min} \)). |
| PF-CZM (Phase-Field Cohesive Zone Model) | Provides an explicit link to material strength. | Quasi-brittle failure and matching analytical/experimental softening curves. |
| Strain Energy Splits | Necessary extensions to prevent damage under compression (e.g., rock-like or concrete materials). | Includes the spectral split (Miehe et al.), volumetric-deviatoric split (Amor et al.), and the Drucker-Prager based split (suited for rock materials). |
How to Solve the Coupled System of Equations for the phase field model? (Solution Schemes)
Once the constitutive model is chosen, the next step is deciding how to solve the coupled system of equations numerically. Phase-field fracture problems involve two coupled partial differential equations (PDEs):
- Mechanical equilibrium equation (for displacement field, u)
- Phase-field evolution equation (for damage field, φ)
These equations are interdependent:
- The displacement field affects the strain energy, which drives damage evolution
- The damage field degrades material stiffness, which affects the mechanical response
The Challenge:
You cannot solve one without knowing the other. They must be solved together, but how?
To address the issue, engineers can solve phase field models using two main strategies in Abaqus. The following table presents these approaches:
| Approach | Description |
|---|---|
| Staggered (Alternate Minimization) | Solves displacement and phase field equations separately in alternating steps. Widely used but computationally intensive. |
| Monolithic | Solves both equations simultaneously. More robust and efficient, especially with quasi-Newton methods. |
How to Implement the Phase Field Fracture Model in Abaqus
Having selected both the model and solution strategy, we need to implement them in software like Abaqus. The phase field fracture model is not a built-in feature in commercial FE software like Abaqus. Therefore, engineers must use various user-defined subroutines.
These subroutines are programmed in Fortran. Generally, implementations fall into two categories.
UEL-Based Implementations
The earliest implementations use a User Element (UEL) subroutine. This approach defines the phase field (φ) as an additional nodal degree of freedom. It sits alongside displacement (u).
Process: The UEL subroutine allows users to define element tangent stiffness matrices manually. It also lets them define nodal force vectors explicitly.
Drawback: This approach uses Abaqus purely as a solver. Consequently, it sacrifices many built-in features. Moreover, it complicates mesh generation and visualization.
Coupled Problems: UELs are necessary for complex multi-physics problems. These involve several coupled fields. For example, deformation (u), hydrogen concentration (C), and the phase field (φ).
Some implementations structure the model using layered finite element systems. For instance, three layers consisting of displacement elements, phase field elements, and a UMAT for post-processing.
Integration Point-Level Implementations
A simpler and more robust approach exploits an analogy. Specifically, it uses the similarity between phase field evolution and heat transfer equations.
Mechanism: The phase field maps onto the temperature degree of freedom (φ → θ). This allows using Abaqus’s built-in coupled temperature-displacement elements. Examples include CPE4T.
Subroutines: Engineers implement this primarily at the integration point level. They use User Material (UMAT) and User Material Heat Transfer (UMATHT/HETVAL) subroutines.
- For Abaqus 2020 or newer: Only a UMAT subroutine is required. It defines the volumetric heat generation source (r) and its derivative. You can implement this in as few as 33 lines of code in its simplest form.
- For Abaqus versions older than 2020: A UMAT combined with a HETVAL subroutine is typically necessary. These define the heat source terms.
- For multiple diffusion equations, A generalized framework often uses UMAT and UMATHT subroutines. For example, phase field and hydrogen transport. This may require a “twin-part method” to define additional temperature-like DOFs.
Do you need Open-Source Abaqus Phase Field Models? Check the following table
A substantial amount of open-source code is available. These resources implement various phase field models. They primarily utilize Abaqus user subroutines (Fortran) or general FE frameworks (Python).
Abaqus Implementations
Most of the following codes are available for download at https://www.empaneda.com/codes/, https://molnar-research.com/tutorials_PH.html, or https://mechmat.web.ox.ac.uk/codes. More references are presented in the following table.
| IMPLEMENTATION TYPE | MODELS & PURPOSE | CODE/SUBROUTINES | SOURCE/CONTRIBUTOR |
|---|---|---|---|
| UMAT/HETVAL | Integration Point Level (Simple, General PFM). Unified implementation for AT1, AT2, PF-CZM models, including various strain energy splits (spectral, volumetric-deviatoric, Drucker-Prager). | PFF-UMAT.f (Abaqus 2020+ UMAT only) and PFF-HETVAL.f (UMAT + HETVAL for older versions). Simple versions: UMATs.f / HETVALs.f. AT1 model: UMAT and UEL (PhaseFieldUMAT.zip, Tutorial_9_AT1.zip) | Y. Navidnehrani, E. Martínez-Pañeda. |
| UEL for Coupled HE/HAC | Coupled deformation-hydrogen diffusion-phase field fracture using 8-node quadratic UEL elements. | PhaseFieldH.for (UEL subroutine) | E. Martínez-Pañeda. |
| UEL for Dynamic/Ductile Fracture | Elasto-plastic phase-field approach for brittle and ductile static and dynamic fracture; widely used robust staggered solution scheme. | Open-source UEL subroutine (layered system used previously) | G. Molnár, A. Gravouil. |
| UEL for SCC/Corrosion | Phase field model for dissolution-driven stress corrosion cracking and pitting corrosion, implemented using a UEL subroutine. | UEL subroutine files (Fortran) | C. Cui, E. Martínez-Pañeda. |
| UEL for Quasi-Newton Solvers | Phase field models (for fracture and fatigue) using the efficient quasi-Newton (BFGS) monolithic solution scheme. | FatigueQN.f (UEL subroutine) | P.K. Kristensen, E. Martínez-Pañeda. |
| UEL for Electromechanical PFM | Coupled deformation-electric-phase field model, specifically for piezoresistive materials (5 DOFs per node in 3D). | UEL piezoresistive phase field for (UEL subroutine) | L. Quinteros, E. Martínez-Pañeda. |
| PF-CZM Implementation | Implementation of the phase-field regularized cohesive zone model (PF-CZM) into ABAQUS. | pfczm_bfgs.for (BFGS monolithic solver for PF-CZM) | J.Y. Wu, Y. Huang. |
| UEL/UMAT for Elasto-Plastic Fracture | 2D and 3D UEL/UMAT for phase-field modeling of fracture in elasto-plastic solids using a staggered algorithm. | UEL and UMAT subroutines | J. Fang. |
| UEL for Brittle/Ductile Fracture (Residual Control) | UEL/UMAT for brittle and ductile fracture using a residual norm-based staggered scheme (2D/3D linear elements). | UEL and UMAT subroutines (Fortran) | K. Seles. |
| Fatigue crack propagation in quasi-brittle materials | Phase field cohesive zone approach for fatigue crack simulation in brittle materials. | HETVAL and UMAT subroutines | Abedulqader Bakthere. |
| Dynamic phase-field simulation | Elasto-plastic solids (dynamic phase-field fracture simulation). | UMAT and UEL | Molnar-research group. |
How to implement the phase-field model in Julia language code?
One of the challenges many people face is that they may not have proper access to commercial finite element software like Abaqus for modeling phase-field problems. To address this issue, the use of free software and programming languages has attracted significant attention in recent years.
One such open-source and free programming language that you can use for simulating phase-field problems is the Julia code language. Today, the Julia code language has gained popularity due to its open-source nature and extensive capabilities compared to other programming languages.
You can access a free and open-source Julia code language source for analyzing phase-field problems, which you can download and use at no cost. Unlike commercial software, the Julia code language does not require any license or purchase, and you can use it completely for free. This Julia code language source can be used for phase-field modeling of fracture and fatigue problems.
Conclusion
Phase field model technology has solidified its position in computational mechanics. It provides a robust, elegant, and versatile paradigm for simulating material failure. Phase field modeling also excels at modeling interface evolution. By regularizing sharp discontinuities into diffuse damage bands, PFM naturally captures complex phenomena.
These include crack nucleation, propagation, coalescence, and branching. The method works in arbitrary 2D and 3D geometries. Therefore, it effectively circumvents the need for ad hoc fracture criteria. As a result, it is highly amenable to multi-field coupling. This inherent variational structure has facilitated successful applications across diverse materials.
Applications extend far beyond simple brittle fracture. Examples include hydrogen-assisted cracking, stress corrosion cracking, and thermo-mechanical fracture. Implementation within commercial software like Abaqus relies on two main strategies. Each offers distinct advantages: 1)User Element Implementation, and 2) Integration Point Level Implementation.
However, phase field fracture modeling is an advanced topic that needs a certain level of professionalism in Abaqus Simulation. As another approach for the implementation of the phase field model, the use of the Julia language code has been suggested.
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