Update JOSS paper with latest changes and improvements

joss/paper.bib

@@ -53,6 +53,19 @@ @article{Miehe2010
 author    = {Miehe, C. and Hofacker, M. and Welschinger, F.},
 }
 
+@article{FrancfortMarigo1998,
+title = {Revisiting brittle fracture as an energy minimization problem},
+journal = {Journal of the Mechanics and Physics of Solids},
+volume = {46},
+number = {8},
+pages = {1319-1342},
+year = {1998},
+issn = {0022-5096},
+doi = {10.1016/S0022-5096(98)00034-9},
+url = {https://www.sciencedirect.com/science/article/pii/S0022509698000349},
+author = {Francfort, G.A. and Marigo, J.-J.},
+}
+
 @article{Amor2009,
 title = {Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments},
 journal = {Journal of the Mechanics and Physics of Solids},
@@ -69,7 +82,6 @@ @article{Amor2009
 
 
 @Article{Ambati2015,
-author    = {Ambati, M. and Gerasimov, T. and De Lorenzis, L.},
 title={A review on phase-field models of brittle fracture and a new fast hybrid formulation},
 journal={Computational Mechanics},
 year={2015},
@@ -80,7 +92,8 @@ @Article{Ambati2015
 pages={383-405},
 issn={1432-0924},
 doi={10.1007/s00466-014-1109-y},
-url={https://doi.org/10.1007/s00466-014-1109-y}
+url={https://doi.org/10.1007/s00466-014-1109-y},
+author = {Ambati, M. and Gerasimov, T. and De Lorenzis, L.},
 }
 
 @article{Carrara2020,

joss/paper.md

@@ -3,6 +3,7 @@ title: "PhaseFieldX: An Open-Source Framework for Advanced Phase-Field Simulatio
 tags:
   - Fenicsx
   - Phase field
+  - Fatigue
   - Fracture
   - Finite element
 authors:
@@ -20,27 +21,76 @@ bibliography: paper.bib
 
 # Summary
 
-The **PhaseFieldX** project is designed to simulate and analyze material behavior using phase-field models, which provide a continuous approximation of interfaces, phase boundaries, and discontinuities such as cracks. Leveraging the robust capabilities of *FEniCSx* [@BarattaEtal2023; @ScroggsEtal2022; @BasixJoss; @AlnaesEtal2014], a renowned finite element framework for solving partial differential equations, this project facilitates efficient and precise numerical simulations. It supports a wide range of applications, including phase-field fracture, solidification, and other complex material phenomena, making it an invaluable resource for researchers and engineers in materials science.
+The **PhaseFieldX** project is designed to simulate and analyze material behavior using phase-field models, which provide a continuous approximation for interfaces, phase boundaries, and discontinuities such as cracks. Leveraging the robust capabilities of *FEniCSx* [@BarattaEtal2023; @ScroggsEtal2022; @BasixJoss; @AlnaesEtal2014], a renowned finite element framework for solving partial differential equations, this project enables efficient and accurate numerical simulations related to phase-field models. It supports a broad range of applications, including phase-field fracture, solidification, and other complex material phenomena, making it an invaluable resource for researchers and engineers in materials science.
 
-![Logo](./images/logo_name.png){height="80pt"}
+![PhaseFieldX logo.\label{fig:logo}](./images/logo_name.png){height="100pt"}
+
+- **Phase-Field Fracture**:
+
+The phase-field fracture model was extended by [@Miehe2010] to incorporate a regularized crack field within the variational framework of Griffith’s fracture theory, as developed by [@FrancfortMarigo1998]. This model provides a robust and versatile approach for simulating fracture and material failure, representing cracks implicitly through a continuous phase-field variable rather than requiring explicit crack tracking. This implicit representation simplifies numerical implementation and naturally enables the simulation of complex crack patterns, including branching, merging, and interactions among multiple cracks.
+
+A diffuse variable, the “phase-field” $\phi$, distinguishes between intact ($\phi = 0$) and fractured ($\phi = 1$) regions, with a smooth transition between them. This setup allows cracks to evolve based on energy minimization principles, eliminating the need for explicit crack-path tracking. The functional is given by \autoref{eq:phase_field_fracture_functional},
+\begin{equation}\label{eq:phase_field_fracture_functional}
+\mathcal{E}(\boldsymbol u, \phi) = \int_\Omega g(\phi)  \psi(\epsilon(\boldsymbol u)) \mathrm{d}\Omega + G_c \int_\Omega \left( \frac{1}{2l}\phi^2 + \frac{l}{2} |\nabla \phi|^2 \right) \mathrm{d}\Omega - E_{\text{ext}}[\boldsymbol u].
+\end{equation}
+where $\phi$ is the phase-field variable, $\boldsymbol u$ is the displacement vector field, $g(\phi) = (1 - \phi)^2$ is the degradation function that reduces material stiffness as damage progresses, $\epsilon(\boldsymbol u)= \frac{1}{2} \left( \nabla \boldsymbol u + \nabla^T \boldsymbol u \right)$ is the small strain tensor, $\psi(\epsilon(\boldsymbol u)) = \frac{1}{2}\lambda tr^2(\epsilon(\boldsymbol u)) + \mu tr(\epsilon(\boldsymbol u)^2)$ is the strain energy, $G_c$ is the critical energy release rate, $l$ is the length scale parameter that controls the width of the diffuse crack region and $E_{\text{ext}}[\boldsymbol u] = \int_\Omega f \cdot \boldsymbol u \, \mathrm{d}\Omega + \int_{\partial \Omega} t \cdot \boldsymbol u \, \mathrm{d}S$ represents the external terms, with $f$ being the prescribed volume force in $\Omega$ and $t$ the surface traction force on $\partial \Omega$.
+
+The energy can be split into two components $\psi(\epsilon(\boldsymbol u)) = \psi_a(\epsilon(\boldsymbol u)) + \psi_b(\epsilon(\boldsymbol u))$, where $\psi_a$ represents the energy associated with tensile stresses capable of generating fractures, and the degradation function is applied only to this component. This yields the form $g(\phi) \psi_a(\epsilon(\boldsymbol u)) + \psi_b(\epsilon(\boldsymbol u))$, preventing crack formation due to compressive forces. Various methods exist for performing this energy split, such as the spectral decomposition [@Miehe2010] and the volumetric-deviatoric decomposition [@Amor2009]. Models that incorporate this energy split are known as anisotropic models, while models without energy splitting are referred to as isotropic models.
+
+To enforce crack irreversibility, [@Miehe2010] introduces a history field, defined as the maximum reference energy obtained throughout the material’s history, which drives the phase field. As a result, this variable will always increase, ensuring that cracks cannot heal.
+
+- **Phase-Field Fatigue**:
+
+Following [@Carrara2020], fatigue effects can be modeled by introducing a fatigue degradation function, $f(\bar{\alpha}(t))$, which reduces fracture toughness based on a cumulative history variable, $\bar{\alpha}(t)$, where $t$ represents a pseudotime in the simulation setup. This new history variable, can be any scalar quantity that effectively captures the material’s fatigue history. The functional, in the absence of external terms, is given by \autoref{eq:phase_field_fatigue_functional}.
+\begin{equation}\label{eq:phase_field_fatigue_functional}
+\mathcal{E}(\boldsymbol u, \phi) = \int_\Omega g(\phi) \psi(\epsilon(\boldsymbol u)) \mathrm{d}\Omega + f(\bar{\alpha}(t)) G_c \int_\Omega \left( \frac{1}{2l}\phi^2 + \frac{l}{2} |\nabla \phi|^2 \right)  \mathrm{d}\Omega.
+\end{equation}
+Note that multiple types of fatigue degradation functions have been developed [@Carrara2020], along with various methods for accumulating the cumulative history variable.
+
+- **Crack surface density functional**:
+
+One of the key aspects of phase-field fracture simulations is that the crack surface density functional provides a continuous approximation of cracks, which converges to a discrete crack representation as the length scale parameter approaches zero. This corresponds to the second part of the phase-field fracture functional, as given in \autoref{eq:phase_field_fracture_functional}. Therefore, before studying phase-field fracture models, it is recommended to first focus on this part of the functional in isolation.
+
+The second part of the functional, which controls the regularization of the phase-field, is expressed as:
+\begin{equation}\label{eq:crack_representation_functional}
+W[\phi] = \int_\Omega \left( \frac{1}{2l} \phi^2 + \frac{l}{2} |\nabla \phi|^2 \right) \mathrm{d}\Omega
+\end{equation}
+
+By considering this, an analytical solution can be derived for a one-dimensional problem, where a cracked bar of length $2a$ is represented by the phase-field, as shown in \autoref{fig:bar_one_dimension_solution}. Considering the conditions of $\phi(0)=1$  and  $\phi'(\pm a)=0$ the solution is given by \autoref{eq:one_dimensional_solution},
+\begin{equation}\label{eq:one_dimensional_solution}
+\phi(x) = e^{-|x|/l} + \frac{2}{e^{\frac{2a}{l}}+1} \sinh \left( \frac{|x|}{l} \right)
+\end{equation}
+where $l$ is the length scale parameter and $a$ is half the length of the bar. In the limiting case where $l \to 0$, it follows that $a/l \to \infty$, and the phase-field is represented solely by the $e^{-|x|/l}$ term. The total energy of the bar is given by $W=\tanh(a/l)$, and in the limit case as $l \to 0$, $W$ approaches $1$.
+
+Note that as the length scale parameter tends to zero, the continuous phase-field representation converges to a discrete crack representation. \autoref{fig:bar_one_dimension_solution} illustrates the crack approximation for two different length scale values.
+
+![Left: Cracked bar represented by the phase-field variable. Right: Phase-field representation of the cracked bar with $l=0.1$ and $l=0.5$.\label{fig:bar_one_dimension_solution}](./images/crack_surface_1d_solution.png){width="360pt"}
+
+- **Other Phase-Field Models**:
+
+Although the main capabilities of **PhaseFieldX** focus on fracture, fatigue, crack surface density functional, and elasticity problems, this framework also allows for the study of other phase-field models. Currently, the *Allen-Cahn* equation is included, enabling simulations of phase transformations and other diffuse-interface phenomena.
 
 # Statement of Need
 
-The **PhaseFieldX** project aims to advance phase-field modeling through open-source contributions. By leveraging the powerful *FEniCSx* framework, our goal is to enhance and broaden the application of phase-field simulations across various domains of materials science and engineering. We strive to make these advanced simulation techniques more accessible, enabling researchers and engineers to conduct more accurate and comprehensive scientific investigations. Through collaborative efforts, our mission is to deepen understanding, foster innovation, and contribute to the broader scientific community’s pursuit of knowledge in complex material behaviors.
+**PhaseFieldX** stands out in the landscape of phase-field modeling software as an open-source package, with no commercial solutions currently available that directly incorporate phase-field fracture simulations; the primary alternative is the use of *ABAQUS* subroutines, which necessitate custom coding by the user. Most other software options are in-house codes that can be difficult to share, replicate, or adapt. While some implementations are present in open-source software like *FEniCS* or *Comsol*, these are often limited to specific problems and specific implementations of the phase-field model. In contrast, **PhaseFieldX** integrates the latest advancements in phase-field research, including fatigue simulations, into a user-friendly, ready-to-use package for scientific research. This accessibility encourages broader adoption of phase-field methods, simplifying the process for researchers to engage in cutting-edge investigations without the complexities of bespoke modeling tools.
+
+**PhaseFieldX** simplifies the setup and control of diverse model configurations, enabling users to customize key parameters like degradation functions, energy-splitting schemes, and solver strategies for phase-field fracture simulations. This modular approach allows researchers to explore a wide range of settings, encouraging innovation and experimentation in phase-field studies. **PhaseFieldX** caters to users seeking reliable simulation tools as well as researchers eager to contribute to the package’s development, fostering a collaborative ecosystem that advances the understanding of complex material behaviors and drives innovation within the scientific community.
 
 # Applications
 
-**PhaseFieldX** offers a wide range of applications in materials science and engineering:
+**PhaseFieldX** offers a comprehensive range of applications in materials science and engineering. For phase-field fracture simulations, it supports various degradation functions and energy split methods, such as the volumetric-deviatoric split [@Amor2009] and spectral decomposition [@Miehe2010]. The framework accommodates multiple formulations, including isotropic, anisotropic, and hybrid approaches [@Ambati2015]. For example, two well-known simulations from [@Miehe2010] can be conducted through **PhaseFieldX**, specifically using an anisotropic formulation with spectral energy decomposition.
+
+The first example, a single-edge notched tension test, consists of a square plate with a centrally located notch extending from the left edge, as illustrated in \autoref{fig:example_sentt}. The lower edge of the plate is fully constrained, while the upper edge is free to slide vertically. A vertical displacement is applied to the top until the crack propagates. The final phase-field values displayed in \autoref{fig:example_sentt}.
 
-- **Phase-Field Fracture**: Enables phase-field fracture simulations, considering various degradation functions and energy split methods such as volumetric-deviatoric [@Amor2009] or spectral decomposition [@Miehe2010]. It supports different formulations including isotropic, anisotropic, and hybrid [@Ambati2015].
+![Single-edge notched tension test: setup with boundary conditions and final phase-field distribution at the end of the simulation.\label{fig:example_sentt}](./images/single_edge_notched_tension_test.png){width="280pt"}
 
-- **Elasticity**: Provides tools for analyzing elasticity problems, foundational to phase-field fracture modeling.
+The second example, known as the three-point bending test, involves a rectangular plate with a centrally located notch and is supported at both ends, as shown in \autoref{fig:example_three_point}. The lower left corner is fixed in all directions, while the lower right corner is constrained vertically.
 
-- **Phase-Field Fatigue**: Introduces a phase-field model to study fatigue [@Carrara2020].
+![Three point bending test: setup with boundary conditions and final phase-field distribution at the end of the simulation.\label{fig:example_three_point}](./images/three_point_bending_test.png){width="360pt"}
 
-- **Other Phase-Field Models**: Facilitates the study of other phase-field models like the Allen-Cahn equation, expanding simulation capabilities to include various phase transformation phenomena.
+**PhaseFieldX** also enables fatigue analysis by applying cyclic loading, allowing the simulation of crack propagation as the cycle count increases. Additionally, the platform facilitates studies of other phase-field models, such as the Allen-Cahn equation, broadening its application to various phase transformation phenomena.
 
-These capabilities make **PhaseFieldX** a versatile and powerful tool for researchers and engineers aiming to investigate complex material behaviors through advanced phase-field simulations. A complete list of examples is available, demonstrating the breadth of possible applications and scenarios.
+These capabilities make **PhaseFieldX** a versatile and powerful tool for researchers and engineers investigating complex material behaviors through advanced phase-field simulations. A comprehensive list of examples is available, demonstrating the breadth of possible applications and scenarios.
 
 # Acknowledgements