This thesis focuses on the mathematical modelling of soft and active solid matter using continuum mechanics. An elastic body is said to be soft if it can undergo large deformations; it is said to posses an active behaviour when it can rearrange its micro-structure in presence of external stimuli, not necessarily of mechanical nature. Examples of active processes are biological growth or the contraction of dielectric elastomers provoked by an electromagnetic field. The research activities undertaken concerned both analytical and numerical tasks to solve some physical problems in this field. In particular, we focused on: - the constitutive theory of soft materials with initial stresses, - the mathematical modelling of active phenomena in biological matter, - the formation of patterns in soft solids due to a mechanical instability. The thesis is organized as follows. In Chapter 1, we briefly expose some basic notions of non-linear elasticity. We review the fundamental literature on the mathematical modelling of biological growth and muscle contraction, and on an emerging field in mechanics, called morpho-elasticity. In Chapter 2, we investigate the mathematical description of elastic bodies possessing a non-vanishing distribution of initial stress, i.e. the Cauchy stress in the undeformed reference configuration. We provide new mathematical and physical interpretations of the required constitutive restrictions, proving the existence of energy minimizers in the framework of the theory of initially stressed materials. In Chapter 3, we propose new mathematical models of active processes in soft biological matter, particularly focusing on tumour growth and muscular contraction. We show that it is not possible to recover the experimental stress-stretch curve corresponding to a uniaxial deformation of a skeletal muscle using the active strain method, based on a multiplicative decomposition of the deformation gradient. Instead, we propose an alternative model based on a mixture approach, called “mixture active strain”. Moreover, we show that solid tumours behave as growing poroelastic materials, where the growth is modulated by a chemo-mechanical feedback. The results of our model are in very good agreement with both “in-vitro” and “ex-vivo” experimental data. In Chapter 4, we model morpho-elastic phenomena in both living and inert soft matter. First, we investigate the mechanics of tumour capillaries, showing that the incompatible axial growth of the straight vessel can trigger an elastic instability, generating a tortuous shape. Second, we study how residual stresses can induce mechanical instabilities in soft spheres, e.g. in growing tumour masses. Considering several spatial distributions of the residual stress field, we prove that different topological transitions occur in the sphere where the hoop residual stress reaches its maximum compressive value. Third, we show that gravity bulk force can cause an elastic instability in soft elastic bilayers. We show that the non-linear elastic effects saturate the dynamic instability of the bifurcated solutions that characterize fluid-like matter, displaying a rich morphological diagram where both digitations and stable wrinkling can emerge. Finally, the results of this thesis prove how the combination of nonlinearities and nonconvexity in elastic mixed boundary value problems may emerge as complex physical phenomena, whose understanding requires the development of novel mathematical tools (Chapter 5).
