Summer School "Elasticity and Growth"

Building on the field theory of Elasticity with Disarrangements (Deseri & Owen, 2003–2019), its extensions to instability-driven reconfiguration (Palumbo et al., 2018, 2021), and recent applications to remodeling in cell membranes (Carotenuto et al., 2020–2025), this course aims to develop a preliminary framework for the study of remodeling and growth in simple systems within the setting of Structured Deformations.
Geometric incompatibility is interpreted here as the macroscopic trace of submacroscopic reconfigurations, thereby offering a possible multiscale basis for the phenomena under consideration. From this perspective, growth and remodeling are studied in paradigmatic and simple cases and related to recent incompatibility-driven approaches (e.g., Zurlo et al., 2017–2025).

The lectures will start with the introduction of the basic concepts of mixture theory, from the biphasic in vitro case to go to the in vivo case. Pros and cons of every model will be discussed. In this framework, the lectures will describe how to model the effects of stress on growth (contact inhibition of growth) and the reason of emergence of border instability, due for instance to differential motility. Coming to the stress constitutive model, lectures will start with the easiest fluid-like models to elasto-plastic model, explaining the emergence of yield and residual stresses.

The study of growth and remodeling offers a unifying framework for understanding phenomena across scales, bridging fields as diverse as biology, geology, and engineering. Central to these processes is the generation of stress in the absence of conventional loading. This course explores the mechanics of growth, with a primary focus on strain incompatibility as the kinematic signature of non-Euclidean elastic bodies.
Departing from standard "growth tensor" literature, we propose a paradigm shift: a perspective where micro-scale displacements drive the acquisition of incompatibility. We introduce the micro-displacement tensor, a continuum measure of internal "micro-slips" that produce disarrangements between the material parts of a body. By analyzing surface deposition and volumetric accretion, we demonstrate that residual stress is an inherent, functional feature of growing systems, rather than a mere structural byproduct.
We further examine biological growth through this mechanistic lens, showing how pre-stress is harnessed to regulate tissue size and maintain homeostasis. These processes are analyzed at both the discrete and continuous levels. Through fully resolved case studies, this short course provides the mathematical tools and conceptual frameworks necessary to understand intrinsic stress generation in biology and to design a next generation of non-Euclidean solids with tailored internal states of pre-stress.

Biological soft tissues display a rich mechanical behaviour that reflects their complex internal organization and their ability to undergo large, reversible deformations. Capturing this response is essential for understanding both normal physiological function and the development of diseases. However, classical viscoelastic models based on spring–dashpot assemblies often prove inadequate, particularly for materials exhibiting broad, power-law relaxation. This seminar presents an approach for the efficient computation of the viscoelastic response of biological tissues based on fractional constitutive modelling. By introducing fractional-order elements, complex time-dependent behaviour can be described in a compact and
flexible manner, reducing the number of parameters while improving accuracy.
As a representative example, we focus on epithelial monolayers, where recent experimental observations reveal rich time-dependent behaviour. To enhance computational efficiency and predictive capability, the proposed models are implemented using advanced numerical techniques. In addition, practical tools for parameter estimation and data fitting are developed and presented through short tutorials, enabling robust analysis of experimental rheological data.

Biological tissues exhibit both passive and active responses. Their mechanical behavior reflects not only elasticity, but also viscous and viscoelastic effects, while at the same time being strongly influenced by fluid transport, nutrient supply, reactive constituents, remodeling, and growth. From a material point of view, this rich and multifaceted nature calls for modeling frameworks capable of coupling mechanical and fluid-related aspects in a unified way. In this context, poroelasticity provides a natural setting to describe the interplay between deformation of the solid skeleton, redistribution of interstitial fluids, and evolving microstructural organization, all of which contribute to tissue function and dysfunction.
This lecture presents two case studies illustrating the role of poroelasticity in tissue mechanics. The first concerns the lamina cribrosa, where an anisotropic poroelastic model helps elucidate the coupled mechanical and fluid response of the tissue under variations in intraocular pressure, with possible implications for glaucoma. The second addresses porous soft composites with magnetically activated internal constituents, viewed as simplified prototypes of adaptive tissue-inspired systems in which internal activity can modulate stiffness and permeability. The lecture will finally discuss how these ideas may be extended toward growth and remodeling, building on established poroelastic formulations that incorporate mass transport, volumetric changes, and evolving internal structure.