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Growth-Induced Transitions in Viscoelastic Matter

This paper demonstrates that in growing viscoelastic materials, such as biological tissues, a dimensionless parameter comparing growth rate to stress relaxation time governs the emergence of qualitatively new mechanical dynamics and sharp transitions between metastable states when these rates are comparable.

Original authors: Valentin Slepukhin, Oskar Hallatschek

Published 2026-08-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Valentin Slepukhin, Oskar Hallatschek

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Living things are never static; they are constantly expanding, dividing, and pushing against their surroundings. From the way a tumor swells inside the body to how a colony of bacteria spreads across a petri dish, growth is a fundamental force that shapes the biological world. When this growth happens in a confined space, it creates pressure. For decades, scientists have tried to predict how living matter responds to this pressure by treating it as either a solid that springs back when pushed, like a rubber band, or a fluid that flows and drips, like honey. However, real biological tissues are rarely so simple. They possess a dual nature, behaving partly like a solid and partly like a fluid, a property known as viscoelasticity. The question that has long puzzled researchers is how this dual nature changes when the material is actively growing. Does the speed of growth simply add to the stress, or does it fundamentally alter the rules of how the material moves and deforms?

A team of researchers has now answered this by showing that the relationship between how fast a material grows and how quickly it can relax its internal stress creates a surprising new kind of behavior. They discovered that when the rate of growth matches the speed at which the material can release its built-up tension, the system does not simply settle into a middle ground between solid and fluid. Instead, it undergoes a sharp, sudden shift. In this specific zone, the material can get stuck in a temporary, unstable state before violently snapping into a new shape. This is not a gradual change but a distinct mechanical transition that occurs only when the growth rate and the relaxation time are perfectly balanced.

To understand this, imagine a beam of growing tissue that is pinned at both ends. As the beam grows longer, it has nowhere to go but to bend. If the material were a perfect solid, it would bend into a smooth, teardrop shape and stay there. If it were a perfect fluid, it would slowly coil up over time. But when the material is viscoelastic and the growth rate is just right, the beam behaves erratically. It bends, pauses, and then suddenly snaps into a tighter coil, releasing a burst of stored energy before settling again. The researchers found that this snapping behavior creates a ladder-like pattern of movement, where the beam jumps from one configuration to another in rapid succession. This happens because the stress generated by the rapid growth accumulates faster than the material's internal structure can relax it, leading to a sudden release of tension that forces the beam to reconfigure itself.

The team developed a new mathematical framework to describe this phenomenon, one that treats the growth of the material and the stress it creates as two sides of the same coin. They showed that the entire behavior of the system can be predicted by a single number: the product of the growth rate and the relaxation time. When this number is very small, the material acts like a fluid. When it is very large, it acts like a solid. But in the middle, where the number is close to one, the material exhibits these unique, unstable jumps. This finding challenges the old assumption that the behavior of growing tissues is just a smooth blend of solid and fluid properties. Instead, it reveals a critical threshold where the mechanics of life change qualitatively.

The researchers tested this idea not just on simple beams, but also in more complex scenarios that mimic real biological environments. They looked at how growing matter moves through porous materials, like soil or the space between cells in a tissue. They found that the combination of growth and viscoelasticity makes the material more permeable, allowing fluids to pass through more easily than if the material were just growing or just relaxing. They also examined how layers of growing tissue wrinkle, a common feature in everything from fruit skins to bacterial films. In these cases, the wrinkles reached their largest size only when the growth rate and relaxation time were balanced, again showing that the middle ground is not a compromise but a zone of maximum effect.

These results suggest that the way living things grow and change shape is governed by a delicate timing mechanism. If a tissue grows too slowly compared to its ability to relax, it flows. If it grows too fast, it locks up like a solid. But when the timing is just right, the system becomes dynamic and unstable, capable of rapid, dramatic changes. This has profound implications for understanding how biological structures form and how diseases like cancer expand. The study indicates that to truly understand the mechanics of life, scientists must look beyond the simple categories of solid and fluid and consider the specific timing of growth and relaxation. The researchers used computer simulations to explore these dynamics, confirming that these sharp transitions are a real and robust feature of growing viscoelastic matter, waiting to be observed in the complex, living world around us.

Technical Summary: Growth-Induced Transitions in Viscoelastic Matter

Problem Statement
Living systems, ranging from biofilms to tumors, undergo growth that generates mechanical stresses when constrained. While biological tissues are inherently viscoelastic, existing models often treat them as either purely elastic or purely viscous. A critical gap exists in understanding how the interplay between the rate of growth ( $g$ ) and the rate of stress relaxation ( $\tau^{-1}$ ) governs the mechanics of proliferating matter. Specifically, it is unclear whether the intermediate regime, where the growth rate is comparable to the inverse relaxation time ( $g\tau \sim 1$ ), represents a smooth crossover between elastic and viscous limits or if it hosts qualitatively distinct dynamical behaviors.

Methodology
The authors develop a theoretical framework to address this problem through two primary approaches:

Prototypical Geometry (Growing Beam): The study begins with a slender, growing viscoelastic beam with fixed endpoints. The dynamics are governed by a torque balance equation where the torque is determined by a history-dependent integral of curvature, decaying exponentially with a relaxation time $\tau$ . The growth is modeled as an increase in arc length at rate $g$ , constrained by fixed endpoints, inducing compressive loads.
General Theoretical Framework: To generalize beyond the beam geometry, the authors construct a continuum model for growing viscoelastic matter. This model decomposes the deformation gradient tensor ( $F$ ) into growth ( $F_g$ ) and elastic ( $F_e$ ) components ( $F = F_g F_e$ ). Crucially, the stress tensor is defined to depend only on the elastic part of the deformation, ensuring that uniform, unconstrained growth remains stress-free. Viscoelasticity is incorporated via the Lodge equation, leading to an upper-convected Maxwell model for growing matter. This results in a modified continuity equation ( $\nabla \cdot v = g$ ) and a stress evolution equation where the velocity gradient is shifted by the growth rate ( $\nabla v \to \nabla v - \frac{g}{D}I$ ).

Key Contributions and Results
The central finding is that the dynamics of growing viscoelastic matter are governed by a single dimensionless parameter, $g\tau$ . The behavior of the system is not a simple interpolation between limits but exhibits distinct regimes:

Limiting Cases:
- Viscous Limit ( $g\tau \ll 1$ ): Stress relaxes faster than it accumulates. The system behaves like a viscous fluid; for the beam, this results in slow, continuous coiling.
- Elastic Limit ( $g\tau \gg 1$ ): Growth is fast relative to relaxation. The system behaves like an elastic solid; the beam adopts a static "teardrop" shape that scales with time without coiling.
Intermediate Regime ( $g\tau \sim 1$ ):
- Metastability and Snapping: Contrary to the expectation of a smooth crossover, the intermediate regime exhibits qualitatively new dynamics. The system possesses metastable states and undergoes rapid, discontinuous transitions ("snaps") between configurations. In the beam model, these snaps manifest as sudden jumps in the tip angle (by $\pi/2$ to $\pi$ ), significantly accelerating the coiling rate compared to the viscous limit.
- Energy Landscape: The effective energy functional in this regime exhibits a sawtooth pattern, rising as the beam bends and dropping sharply during snaps, corresponding to transitions between solution branches.
General Geometries:
- Phase Transition: In a 2D channel with frictionless walls, the system undergoes a sharp mechanical transition at $g\tau = 1$ . Below this threshold, stress reaches a stationary value; above it, stress grows exponentially without bound.
- Porous Media: For growth in a porous medium, viscoelasticity combined with growth increases pore permeability by a factor of $1 + 2.5g\tau$ (for small $g\tau$ ). Neither purely growing viscous fluids nor non-growing viscoelastic fluids show this effect; it arises solely from their combination.
- Wrinkling Instabilities: The amplitude of wrinkles in a growing viscoelastic layer is non-monotonic with respect to $g\tau$ , reaching a maximum in the intermediate regime that exceeds the amplitudes found in both purely viscous and purely elastic limits.

Significance and Claims
The paper claims that the interplay between growth and viscoelasticity can produce discontinuous mechanical transitions and metastable dynamics that are absent in purely elastic or purely viscous models. The authors argue that the parameter $g\tau$ serves as a critical control parameter, determining whether a material behaves as a fluid or a solid.

The significance of these findings lies in the necessity of incorporating viscoelastic effects into models of biological morphogenesis. The authors note that in biological systems like biofilms, relaxation times can span seconds to minutes, and growth rates can result in $g\tau$ values approaching unity. Consequently, many biological tissues may operate in this intermediate regime where "smooth" elastic or viscous approximations fail to capture phenomena such as rapid shape transitions, enhanced permeability, or amplified wrinkling. The work provides a self-consistent theoretical foundation for analyzing these complex, growth-driven mechanical transitions.

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