Surface Growth and Accretion
Motivation
Surface growth — the continuous deposition or removal of material at the boundary of an elastic body — is ubiquitous in nature and engineering. In biology, it drives actin treadmilling (by which cells move), bone remodeling, and tumor expansion. In manufacturing, it underlies layer-by-layer additive processes such as 3D printing with cementitious materials. The distinctive challenge of surface growth is kinematic: freshly deposited material arrives in a stress-free state and joins a pre-stressed body. As a result, the reference configuration is not fixed but grows over time, and the standard kinematic framework of continuum mechanics — built around a single, unchanging reference body — must be extended.
My research in this area started from a collaboration with R. Abeyaratne (MIT) and E. Puntel (Udine) and has grown into a multi-paper program addressing existence, stability, and discrete models.
Details
Morphoelastic rods (2016)
Before addressing surface growth directly, early work by Tiero & Tomassetti (2016) studied morphoelastic rods — slender bodies that undergo inelastic changes of natural length due to growth. The paper established a variational framework for quasi-static morphoelastic rods and derived closed-form solutions for simple growth patterns, providing a testbed for later, more ambitious accretion models.
Accretion on a spherical support: the four-dimensional reference space (2016)
The foundational paper of this research program is:
- Tomassetti, Cohen & Abeyaratne (2016) — Steady accretion of an elastic body on a hard spherical surface and the notion of a four-dimensional reference space. The problem studied is a spherical elastic shell growing by steady surface accretion against a rigid mold. Each material particle arrives stress-free and is immediately constrained by the mold; the accumulated residual stress depends on the entire deposition history. To describe this kinematically, the paper introduced the concept of a four-dimensional reference space (three spatial + one temporal), in which the deformation gradient is well-defined even though the reference body has no fixed material identity. This construction provided the mathematical infrastructure for all subsequent work.
Treadmilling stability (2020)
Actin networks in motile cells grow at the leading edge and dissolve at the trailing edge in a steady process called treadmilling. A one-dimensional model for this was the subject of:
- Abeyaratne, Puntel & Tomassetti (2020) — Treadmilling stability of a one-dimensional actin growth model. This paper derived the equations governing steady treadmilling of a 1D elastic rod growing at one end and dissolving at the other, and carried out a linear stability analysis of the treadmilling solution. The key finding is that stability is controlled by a single dimensionless parameter combining the accreted elastic modulus and the growth velocity.
- Abeyaratne, Puntel & Tomassetti (2020, AIMETA proceedings) extended the stability analysis to 2D treadmilling membranes and summarized the dimensional reduction needed to connect membrane theories to the 3D framework.
Pre-stretch, focal adhesions, and soft-matter accretion (2022–2023)
The next set of papers brought two new physical ingredients: pre-stretch in the accreted material, and discrete attachment points (focal adhesions) through which cells exert traction.
- Abeyaratne, Puntel & Tomassetti (2022) — Surface accretion of a pre-stretched half-space. Rather than a 1D model, this paper considered a semi-infinite elastic half-space that grows by accretion of pre-stretched material at its surface. Exact solutions were derived, showing how the pre-stretch controls the residual stress distribution in the bulk and, critically, the stability of the accreted layer.
- Abeyaratne, Puntel & Tomassetti (2022) — Focal adhesion detachment. Focal adhesions are integrin clusters that anchor the actin cytoskeleton to the extracellular matrix. This paper modeled adhesion detachment as a fracture event driven by the elastic energy stored in the actin network during treadmilling. The connection to the earlier stability results is that detachment triggers a sudden redistribution of residual stress, potentially destabilizing the growth front.
- Abeyaratne, Puntel & Tomassetti (2023) — Stability of surface growth. This paper provided a systematic stability analysis for a broad class of surface growth problems, unifying the earlier 1D treadmilling results with the half-space problem and establishing sufficient conditions for Lyapunov stability.
Discrete and layered growth models (2024)
Two 2024 papers moved toward more concrete models relevant to additive manufacturing and biological layer deposition:
- Renzi & Tomassetti (2024) — Discrete model for layered growth. This paper proposed a discrete (layer-by-layer) version of the accretion problem, in which each layer is deposited in a stress-free state and bonded to the growing body. The continuum limit recovers the four-dimensional reference-space framework, validating that construction from a discrete perspective.
- Davoli & Tomassetti (2024) — Existence result for accretive growth. The first rigorous existence theorem for the full quasistatic accretion problem at finite strain. The proof combines monotone operator theory with a careful treatment of the time-varying reference domain.
Growth and shell theory (2025)
- Rubin & Tomassetti (2025) — Eulerian Cosserat shell with growth. This paper introduced an Eulerian (spatial) formulation of the Cosserat shell theory that accommodates surface growth on either the inner or outer surface of the shell. The Eulerian description sidesteps the difficulties with time-varying reference configurations and connects naturally to the four-dimensional framework introduced in 2016.
Summary and open questions
The thread runs from the construction of a kinematic framework (2016) → stability analysis of canonical models (2020) → physical enrichment with pre-stretch and adhesion (2022–2023) → rigorous analysis and discrete approximation (2024) → shell-geometric formulation (2025). Open questions include the extension to large-deformation 3D bodies with non-spherical geometry, the coupling of accretion to growth-induced instabilities (wrinkling, buckling), and the mathematical analysis of the stability criterion beyond linearization.