Surface Growth in Deformable Solids using an Eulerian Formulation

TL;DR

This paper develops an Eulerian formulation for modeling surface growth in deformable solids, allowing for the analysis of accretion and ablation without needing a reference configuration, and provides closed-form solutions using the method of characteristics.

Contribution

It introduces an Eulerian approach with an elastic deformation descriptor and solves simplified growth problems analytically, advancing modeling of surface growth in solids.

Findings

Closed-form solutions for surface growth problems obtained

Eulerian formulation effectively models accretion and ablation

Method of characteristics simplifies complex growth analysis

Abstract

Accretion and ablation, i.e. the addition and removal of mass at the surface, is important in a wide range of physical processes including solidification, growth of biological tissues, environmental processes, and additive manufacturing. The description of accretion requires the addition of new continuum particles to the body, and is therefore challenging for standard continuum formulations for solids that require a reference configuration. Recent work has proposed an Eulerian approach to this problem, enabling the side-stepping of the issue of constructing the reference configuration. However, this raises the complementary challenge of determining the stress response of the solid, which typically requires the deformation gradient that is not immediately available in the Eulerian formulation. To resolve this, the approach introduced the elastic deformation as an additional kinematic descriptor of the added material, and its evolution has been shown to be governed by a transport equation. In this work, the method of characteristics is applied to solve concrete simplified problems motivated by biomechanics and manufacturing. Specifically, (1) for a problem with both ablation and accretion in a fixed domain, and (2) for a problem with a time-varying domain, the closed-form solution is obtained in the Eulerian framework using the method of characteristics without explicit construction of the reference configuration.