  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.
