Although Nature has always been a common source of inspiration in the development of artificial materials, only recently has the ability of man-made materials to produce complex three-dimensional (3D) structures from two-dimensional sheets been explored. Here we present a new approach to the self-shaping of soft matter that mimics fibrous plant tissues by exploiting small-scale variations in the internal stresses to form three-dimensional morphologies. We design single-layer hydrogel sheets with chemically distinct, fibre-like regions that exhibit differential shrinkage and elastic moduli under the application of external stimulus. Using a planar-to-helical three-dimensional shape transformation as an example, we explore the relation between the internal architecture of the sheets and their transition to cylindrical and conical helices with specific structural characteristics. The ability to engineer multiple three-dimensional shape transformations determined by small-scale patterns in a hydrogel sheet represents a promising step in the development of programmable soft matter.

We provide a geometric-mechanical model for calculating equilibrium configurations of chemical systems that self-assemble into chiral ribbon structures. The model is based on incompatible elasticity and uses dimensionless parameters to determine the equilibrium configurations. As such, it provides universal curves for the shape and energy of self-assembled ribbons. We provide quantitative predictions for the twisted-to-helical transition, which was observed experimentally in many systems, and demonstrate it with synthetic ribbons made of responsive gels. In addition, we predict the bi-stability of wide ribbons and also show how geometrical frustration can cause arrest of ribbon widening. Finally, we show that the model's predictions provide explanations for experimental observations in different chemical systems.

We study the mechanical behavior of two-dimensional cellular tissues by formulating the continuum limit of discrete vertex models based on an energy that penalizes departures from a target area A_{0} and a target perimeter P_{0} for the component cells of the tissue. As the dimensionless target shape index s_{0}=(P_{0}/sqrt[A_{0}]) is varied, we find a transition from a soft elastic regime for a compatible target perimeter and area to a stiffer nonlinear elastic regime frustrated by geometric incompatibility. We show that the ground state in the soft regime has a family of degenerate solutions associated with zero modes for the target area and perimeter. The onset of geometric incompatibility at a critical s_{0}^{c} lifts this degeneracy. The resultant energy gap leads to a nonlinear elastic response distinct from that obtained in classical elasticity models. We draw an analogy between cellular tissues and anelastic deformations in solids.

We study the geometry of defects in amorphous materials and their elastic interactions. Defects are defined and characterized by deviations of the material's intrinsic metric from a Euclidian metric. This characterization makes possible the identification of localized defects in amorphous materials, the formulation of a corresponding elastic problem, and its solution in various cases of physical interest. We present a multipole expansion that covers a large family of localized 2D defects. The dipole term, which represents a dislocation, is studied analytically and experimentally. Quadrupoles and higher multipoles correspond to fundamental strain-carrying entities. The interactions between those entities, as well as their interaction with external stress fields, are fundamental to the inelastic behavior of solids. We develop analytical tools to study those interactions. The model, methods, and results presented in this work are all relevant to the study of systems that involve a distribution of localized sources of strain. Examples are plasticity in amorphous materials and mechanical interactions between cells on a flexible substrate.amorphous solid | elasticity | reference metric | plasticity | defects