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Accretion and ablation, i.e., the addition and removal of mass at the surface, are 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 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.

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Grain boundaries in polycrystalline materials migrate to reduce the total excess energy. It has recently been found that the factors governing migration rates of boundaries in bicrystals are insufficient to explain boundary migration in polycrystals. We first review our current understanding of the atomistic mechanisms of grain boundary migration based on simulations and high-resolution transmission electron microscopy observations. We then review our current understanding at the continuum scale based on simulations and observations using high-energy diffraction microscopy. We conclude that detailed comparisons of experimental observations with atomistic simulations of migration in polycrystals (rather than bicrystals) are required to better understand the mechanisms of grain boundary migration, that the driving force for grain boundary migration in polycrystals must include factors other than curvature, and that current simulations of grain growth are insufficient for reproducing experimental observations, possibly because of an inadequate representation of the driving force. Expected final online publication date for the Annual Review of Materials Research, Volume 53 is July 2023. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.