Transient morphing and optimal shape design of synthetic and natural active structures

Living organisms often display shape morphing capabilities allowing them to efficiently perform tasks that are fundamental for survival. Understanding the way biological activity is exploited to perform shape changes has a deep impact both on natural sciences and technology, often through a process of reverse engineering. In this thesis, we examine four instances of shape morphing both in synthetic and natural, active structures. In the first Chapter, we analyze the transient shaping of a linear poroelastic plate and investigate how mechanical parameters, strains, and stresses influence the swelling dynamics. We obtain an approximate analytical solution for the case of stress-free evolutions and investigate the effect of stresses in the case of an axisymmetric plate. We show that compressive stresses promote faster swelling with respect to the stress-free case, and vice-versa. In the the second Chapter, we address the question of devising efficient morphing strategies for the attainment of specific shape changes in active structures. We set up an optimal control problem which selects, among the activation patterns producing a prescribed shape change, the one minimizing an objective functional, designed to quantify the complexity of the activation. We provide analytical insights for the case of affine shape changes and, with the aid of numerics, we explore the outcome of different objective functionals in a more general context. Chapter 3 is devoted to the study of active reconfigurations in axons, slender cylindrical structures of neurons, which are responsible for the transmission of electro-chemical signals. Axons are able to actively regulate their thickness trough a contractile coating, named cortex, surrounding the cytoplasm (axoplasm). Here, we develop a continuum model describing the interplay between the cortex contractility and the axoplasm elastic response inherited by a network of microtubules. The validity of our modelling assumptions are supported by an excellent match between numerical simulations and experiments. Finally, in the last Chapter, we develop a teleological model to interpret leaves morphogenesis by accounting for the simultaneous growth of both the venation pattern and the blade. Inspired by previous works in the relevant literature, we develop a continuum model by which leaves growth is driven by a gradient flow maximizing the net power absorbed by the leaf. The numerical solution of the ensuing equations provides preliminary results showing some qualitative agreement with features of existing leaves.