# Cahn-Hilliard equation with capillarity in actual deforming configurations

## TL;DR

This paper formulates the Cahn-Hilliard equation with capillarity effects in deforming configurations, addressing both static and dynamic cases with novel terms arising from actual configuration gradients.

## Contribution

It introduces a formulation of the Cahn-Hilliard equation considering capillarity in actual deforming configurations, including new stress terms and analytical approaches for large strain problems.

## Key findings

- Static analysis via the direct method
- Dynamic problems tackled with Galerkin method
- Emergence of Korteweg-like stress terms

## Abstract

The diffusion driven by the gradient of the chemical potential (by the Fick/Darcy law) in deforming continua at large strains is formulated in the reference configuration with both the Fick/Darcy law and the capillarity gradient term considered at the actual configurations deforming in time. Static situations are analysed by the direct method. Evolution (dynamical) problems are treated by the Galerkin method, the actual capillarity giving rise to various new terms as e.g. the Korteweg-like stress and analytical difficulties related to them. Some other models (namely plasticity at small elastic strains or damage) with gradients at actual configuration allow for similar models and analysis.