Surface Growth in Deformable Solids using an Eulerian Formulation

Growth occurs in a wide range of systems ranging from biological tissue to additive manufacturing. This work considers surface growth, in which mass is added to the boundary of a continuum body from the ambient medium or from within the body. In contrast to bulk growth in the interior, the description of surface growth requires the addition of new continuum particles to the body. This is challenging for standard continuum formulations for solids that are meant for situations with a fixed amount of material. Recent approaches to handle this have used, for instance, higher-dimensional time-evolving reference configurations. In this work, an Eulerian approach to this problem is formulated, 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 introduces additional kinematic descriptors, namely the relaxed zero-stress deformation and the elastic deformation; in contrast to the deformation gradient, these have the important advantage that they are not required to satisfy kinematic compatibility. The zero-stress deformation and the elastic deformation are used to eliminate the deformation gradient from the formulation, with the evolution of the elastic deformation shown to be governed by a transport equation. The resulting model has only the density, velocity, and elastic deformation as variables in the Eulerian setting. The proposed method is applied to simplified examples that demonstrate non-normal growth and growth with boundary tractions. The introduction in this formulation of the relaxed deformation and the elastic deformation provides a description of surface growthwhereby the addedmaterial can bring in its own kinematic information. Loosely, the added material “brings in its own reference configuration” through the specification of the relaxed deformation and the elastic deformation of the added material. This kinematic description enables, e.g., modeling of non-normal growth using a standard normal growth velocity and a simple approach to prescribing boundary conditions.