Accretion and Ablation in Deformable Solids using an Eulerian Formulation: A Finite Deformation Numerical Method

TL;DR

This paper introduces a novel Eulerian phase-field finite element method for modeling surface growth and ablation in deformable solids, enabling accurate simulations of complex biological and physical processes without remeshing.

Contribution

It develops a coupled Eulerian-phase field approach with finite elements that overcomes limitations of Lagrangian methods in surface growth modeling.

Findings

Successfully models non-normal biological tissue growth.

Simulates freezing and melting kinetics under complex stress.

Captures regelation in ice, relevant for frost heave.

Abstract

Surface growth, i.e., the addition or removal of mass from the boundary of a solid body, occurs in a wide range of processes, including the growth of biological tissues, solidification and melting, and additive manufacturing. To understand nonlinear phenomena such as failure and morphological instabilities in these systems, accurate numerical models are required to study the interaction between mass addition and stress in complex geometrical and physical settings. Despite recent progress in the formulation of models of surface growth of deformable solids, current numerical approaches require several simplifying assumptions. This work formulates a method that couples an Eulerian surface growth description to a phase-field approach. It further develops a finite element implementation to solve the model numerically using a fixed computational domain with a fixed discretization. This approach bypasses the challenges that arise in a Lagrangian approach, such as having to construct a four-dimensional reference configuration, remeshing, and/or changing the computational domain over the course of the numerical solution. It also enables the modeling of several settings -- such as non-normal growth of biological tissues and stress-induced growth -- which can be challenging for available methods. The numerical approach is demonstrated on a model problem that shows non-normal growth, wherein growth occurs by the motion of the surface in a direction that is not parallel to the normal of the surface, that can occur in hard biological tissues such as nails, horns, etc. Next, a thermomechanical model is formulated and used to investigate the kinetics of freezing and melting in ice under complex stress states, particularly to capture regelation which is a key process in frost heave and basal slip in glaciers.