Geometry, Growth, and Instability in morphing Soft and Active Systems

Morphing, intended as the ability of a system to change its shape, is a phenomenon that appears in a variety of contexts, from biological tissues and soft materials to swelling gels, active matter, and engineered structures. A common question associated with morphing is how such systems reach, maintain and dynamically transform their shapes when non-mechanical stimuli interact with their inherent mechanical elastic response. In other words, the challenge associated to morphing is to understand how forces, geometry and internal fields couple to regulate shapes and sizes. Addressing these mechanisms not only deepens our understanding of natural systems but also guides the design of controllable, programmable artificial morphing materials. The course aims to present a coherent framework for modelling morphing in soft, biological and active systems across scales, combining continuum and discrete descriptions. It covers the mathematical foundations of modelling approaches and their mechanical implications via the application to a range of case studies. In the three-dimensional setting, the course introduces the general continuum framework where the stimuli–feedback mechanism is treated via the multiplicative decomposition of the deformation gradient tensor. Growth laws consistent with thermodynamics are derived for both bulk and surface growth and the role of curvature, as a measure of incompatibility and as a mechanism for generating residual stresses and size regulation, is highlighted in examples as the evolution of tumour spheroids and 3D printing. The course discusses swelling, presented as a chemo-mechanical coupling between elasticity and transport, as a morphing mechanism responsible for large shape changes, transient instabilities and pattern formation in gels and biological tissues. Furthermore, the course introduces how to include, phenomenologically, an active description into an effective constitutive relationship to model active metamaterials. The course presents dimensional-reduction approaches, based either on energy expansions or on reduced equilibrium equations, as effective tools to derive how the three-dimensional descriptions transform into geometric nonlinearity, bending–stretching interactions and incompatibility for slender structures. These theories explain morphing phenomena such as vesicle formation, snapping and wrinkling in rods and shells. To address morphing strategies in biological contexts, the course links the continuum description to the discrete behaviour of cells. Vertex and tension-based models illustrate how cellular mechanics give rise to effective tissue-level laws, including active responses. Mechanosensing, active stress generation and active nematic concepts provide a framework for collective flows, defect dynamics and turbulence in epithelial layers and bacterial colonies, offering a consistent picture of how local activity and geometry shape morphogenesis. The course is aimed at doctoral students, postdoctoral researchers and other early-career scientists interested in the mathematical and physical foundation of morphing strategies in three-dimensional, slender and cellular systems.