Growth-Induced Transitions in Viscoelastic Matter

Abstract

Growth is a fundamental process in living systems. Although the stress-deformation response of growing materials is often described as either purely elastic or purely viscous, many biological tissues, from biofilms to tumors, exhibit both elastic and viscous behavior. Here, we show that this viscoelastic response can crucially control the mechanics of proliferating matter when the growth rate becomes comparable to the rate of stress relaxation. Focusing first on the prototypical case of a growing elastic beam, we find that the dynamics are governed by a single dimensionless parameter, $g\tau$, where $g$ is the growth rate and $\tau$ is the viscoelastic relaxation time. While the limits $g\tau \to 0$ and $g\tau \to \infty$ recover purely viscous and purely elastic behavior, respectively, the intermediate regime is not merely a smooth crossover between them. Instead, qualitatively new dynamics emerge at $g\tau \sim 1$, including rapid transitions between metastable states that occur in neither limiting regime. We then develop a general, growth-compatible theoretical framework in which unconstrained growth is intrinsically stress-free, extending the analysis to other prototypical geometries and enabling simulations of more realistic growing biological materials. Within this framework, sharp mechanical transitions arise when stress generated by exponential growth accumulates faster than it can be dissipated by viscoelastic relaxation.