# A Phase-Field Description for Pressurized and Non-Isothermal Propagating Fractures

## TL;DR

This paper extends phase-field modeling of pressurized fractures to include non-isothermal conditions, incorporating thermodynamical principles, and demonstrates efficient numerical algorithms with applications in 2D and 3D.

## Contribution

It introduces a thermodynamically consistent phase-field model for pressurized, non-isothermal fractures and develops robust numerical methods for its simulation.

## Key findings

- Effective semi-smooth Newton solution approach
- Successful verification with manufactured solutions
- Parallel implementation reduces computational cost

## Abstract

In this work, we extend a phase-field approach for pressurized fractures to non-isothermal settings. Specifically, the pressure and the temperature are given quantities and the emphasis is on the correct modeling of the interface laws between a porous medium and the fracture. The resulting model is augmented with thermodynamical arguments and then analyzed from a mechanical perspective. The numerical solution is based on a robust semi-smooth Newton approach in which the linear equation systems are solved with a generalized minimal residual method and algebraic multigrid preconditioning. The proposed modeling and algorithmic developments are substantiated with different examples in two- and three dimensions. We notice that for some of these configurations manufactured solutions can be constructed, allowing for a careful verification of our implementation. Furthermore, crack-oriented predictor-corrector adaptivity and a parallel implementation are used to keep the computational cost reasonable. Snapshots of iteration numbers show an excellent performance of the nonlinear and linear solution algorithms. Lastly, for some tests, a computational analysis of the effects of strain-energy splitting is performed, which has not been undertaken to date for similar phase-field settings involving pressure, fluids or non-isothermal effects.