Accretion and Ablation in Deformable Solids using an Eulerian Formulation: A Finite Deformation Numerical Method

TL;DR

This work develops a fully Eulerian finite-element framework for surface growth in deformable solids by coupling an Eulerian growth description with a phase-field representation on a fixed computational domain. A key idea is to use an elastic deformation variable ${\boldsymbol F}_e$ to capture stress while avoiding explicit reference-configurations, with growth governed by surface mass source $M$ and velocity $\boldsymbol v_a$, transported through a phase-field $\phi$ that marks the solid boundary. The method supports non-normal growth and a thermomechanical extension for regelation, enabling stress-driven melting and refreezing via a thermodynamic free-energy formulation rather than the Clausius–Clapeyron relation, and demonstrates these ideas on 2D problems including a nail-like non-normal growth scenario and ice–water phase transitions under load. The fixed-domain approach avoids remeshing and can handle large deformations, complex geometries, and evolving fluid–solid transformations, with potential applications in additive manufacturing, glaciology, and cryomechanics; code is available for broader use and extension.

Abstract

Surface growth, i.e., the addition or removal of mass from the boundary of a solid body, occurs in a wide range of processes, including the growth of biological tissues, solidification and melting, and additive manufacturing. To understand nonlinear phenomena such as failure and morphological instabilities in these systems, accurate numerical models are required to study the interaction between mass addition and stress in complex geometrical and physical settings. Despite recent progress in the formulation of models of surface growth of deformable solids, current numerical approaches require several simplifying assumptions. This work formulates a method that couples an Eulerian surface growth description to a phase-field approach. It further develops a finite element implementation to solve the model numerically using a fixed computational domain with a fixed discretization. This approach bypasses the challenges that arise in a Lagrangian approach, such as having to construct a four-dimensional reference configuration, remeshing, and/or changing the computational domain over the course of the numerical solution. It also enables the modeling of several settings -- such as non-normal growth of biological tissues and stress-induced growth -- which can be challenging for available methods. The numerical approach is demonstrated on a model problem that shows non-normal growth, wherein growth occurs by the motion of the surface in a direction that is not parallel to the normal of the surface, that can occur in hard biological tissues such as nails, horns, etc. Next, a thermomechanical model is formulated and used to investigate the kinetics of freezing and melting in ice under complex stress states, particularly to capture regelation which is a key process in frost heave and basal slip in glaciers.

Figures (7)

• Figure 1: Schematic of a growing body $\Omega_g$ and the ambient exterior $\Omega_c$. The balance equations are derived using the arbitrary subdomain $D$.
• Figure 2: A schematic of the evolution of a growing body in a small interval of time $\Delta t$. The spatial location of the growing boundary depends on the growth velocity and the continuum particle velocity at the boundary.
• Figure 3: A schematic showing the calculations within a single time-step. The thick line depicts the growing boundary with prescribed growth velocity ${\mathbold v}_g \cdot \hat{{\mathbold n}}$. (a) Unconstrained Growth: growth occurs at a boundary where the displacement is not prescribed. (b) Constrained Growth: growth occurs at a boundary where the displacement is prescribed, thereby constraining the relation between the boundary velocity, growth velocity, and mechanical velocity; the figure shows the case that ${\mathbold v}_g$ and ${\mathbold v}$ are coupled through ${\mathbold V}_b = \mathbf{0}$. During growth, we do not consider mechanics, and during mechanical relaxation, we do not consider growth. In $\Omega_g^n$, the region that is not shaded consists of material from the previous time-step $t=t^n$, the region shaded orange is the accreted material with $\rho=\hat{\rho}_a, {\mathbold F}_e = \hat{{\mathbold F}}_e^*$, and the region shaded grey is the ablated material.
• Figure 4: Deformation of a pre-stressed rectangular solid body: (a) the initial shape of the body, (b) the equilibrium shape of the body, and (c) the norm of the Cauchy stress tensor at equilibrium. The white line represents the $\phi=0$ levelset showing the boundary of the body.
• Figure 5: Non-normal growth from a fixed surface. (a) The evolution of the phase function is shown at different times, (b) a zoomed-in view of the computational mesh shown at the end of the second time step, where the thick white line indicates the boundary of the growing body, (c) the result for a mesh that is finer by a factor of 2, at $t=0.2$, suggesting that the method is robust, and (d) the norm of the Cauchy stress tensor at $t=0.2$. The white curve shows the zero levelset of $\phi$ that indicates the shape of the growing body.

Theorems & Definitions (4)

• Remark 2.1: Quasistatic simplification
• Example 4.1: Relaxation of a pre-stressed solid body without growth
• Example 4.2: Non normal growth with a fixed $\theta$
• Example 4.3: Non-normal growth with a varying $\theta$