Accretion and Ablation in Deformable Solids with an Eulerian Description: Examples using the Method of Characteristics

Dear colleagues,

We invite you to see the preprint of our new paper "Accretion and Ablation in Deformable Solids with an Eulerian Description: Examples using the Method of Characteristics" which will appear in Mathematics and Mechanics of Solids. Recent work has proposed an Eulerian approach to the surface growth 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. 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. Here, we applied the method of characteristics to solve concrete simplified problems motivated by surface growth in biomechanics and manufacturing  (https://journals.sagepub.com/doi/10.1177/10812865211054573)

Accretion and Ablation in Deformable Solids with an Eulerian Description: Examples using the Method of Characteristics

S. Kiana Naghibzadeh, Noel Walkington, Kaushik Dayal

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.