Theory
******

Welcome to the theory section of this documentation, where we delve into the fundamental concepts underlying phase-field models. This section provides a comprehensive overview of the theoretical frameworks essential for understanding phase-field approaches.

Phase-field fracture studies encompass a multidisciplinary approach, integrating principles from elasticity theory, phase-field modeling, and fracture mechanics. We explore four key problem domains that form the foundation of our understanding:

1. **Elasticity Problem** (:ref:`theory_elasticity`): This problem focuses on the analysis of material behavior under mechanical deformation within the framework of elasticity theory. Understanding the elastic properties of materials is crucial for characterizing their response to external forces and predicting the onset of fracture.

2. **Phase-Field Problem** (:ref:`theory_phase_field`): The phase-field method is a powerful computational technique used to model and simulate phase transitions and microstructural evolution in materials. In the context of fracture, the phase-field approach enables the representation of crack initiation, propagation, and interaction in a continuous manner, eliminating the need for explicit crack tracking.

3. **Phase-Field Fracture Problem** (:ref:`theory_phase_field_fracture`): Building on the concepts from elasticity and phase-field problems, the phase-field fracture problem combines these principles to study the initiation and propagation of fractures within heterogeneous materials. By incorporating phase-field techniques into fracture mechanics, this approach provides a versatile framework for simulating complex fracture phenomena observed in real-world materials.

4. **Allen-Cahn** (:ref:`theory_allen_cahn`)
   
Through a systematic exploration of these interconnected problem domains, we aim to provide insights into the underlying mechanisms governing fracture behavior. This documentation is structured to offer in-depth explanations and practical guidance for understanding and applying the theoretical principles discussed in the subsequent sections.


.. toctree::
   :hidden:

   elasticity/main
   phase_field/main
   phase_field_fracture/main
   energy_pff/main
   crack_measurement/main
   Allen_Cahn/main
   errors/main
