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Article summary:
1. The article introduces a general phase-field framework for describing the spatial distribution of physical quantities in materials.
2. The framework is extended to include constraints, such as integral and non-integral constraints, by introducing Lagrange multipliers.
3. The kinetic equations with non-integral constraints are derived and the Lagrange multipliers can be solved as functions of certain parameters.
Article analysis:
The article provides a comprehensive overview of the phase-field framework with constraints and its applications to ductile fracture in polycrystals and fatigue. The authors present a clear explanation of the theory behind the framework, including the variational principle, Allen-Cahn equation, and TDGL equation. They also provide detailed descriptions of two types of constraints (integral and non-integral) and how they can be incorporated into the framework using Lagrange multipliers.
The article is generally reliable and trustworthy, as it provides an accurate description of the theory behind the phase-field framework with constraints. However, there are some potential biases that should be noted. For example, while the authors discuss both integral and non-integral constraints, they focus more on non-integral constraints than integral ones; this could lead to readers having an incomplete understanding of how both types of constraints work within this framework. Additionally, while the authors provide a thorough explanation of how to solve for Lagrange multipliers in a general state from non-integral constraints, they do not provide any examples or further discussion on how this solution might be applied in practice; this could leave readers feeling uncertain about how to actually use this information in their own research or projects.
In conclusion, while overall reliable and trustworthy, this article does have some potential biases that should be taken into consideration when reading it.