Energetic rigidity. II. Applications in examples of biological and underconstrained materials.

This is the second paper devoted to energetic rigidity, in which we apply our formalism to examples in two dimensions: Underconstrained random regular spring networks, vertex models, and jammed packings of soft particles. Spring networks and vertex models are both highly underconstrained, and first-order constraint counting does not predict their rigidity, but second-order rigidity does. In contrast, spherical jammed packings are overconstrained and thus first-order rigid, meaning that constraint counting is equivalent to energetic rigidity as long as prestresses in the system are sufficiently small. Aspherical jammed packings on the other hand have been shown to be jammed at hypostaticity, which we use to argue for a modified constraint counting for systems that are energetically rigid at quartic order.

Energetic rigidity. I. A unifying theory of mechanical stability.

Rigidity regulates the integrity and function of many physical and biological systems. This is the first of two papers on the origin of rigidity, wherein we propose that "energetic rigidity," in which all nontrivial deformations raise the energy of a structure, is a more useful notion of rigidity in practice than two more commonly used rigidity tests: Maxwell-Calladine constraint counting (first-order rigidity) and second-order rigidity. We find that constraint counting robustly predicts energetic rigidity only when the system has no states of self-stress. When the system has states of self-stress, we show that second-order rigidity can imply energetic rigidity in systems that are not considered rigid based on constraint counting, and is even more reliable than shear modulus. We also show that there may be systems for which neither first- nor second-order rigidity imply energetic rigidity. The formalism of energetic rigidity unifies our understanding of mechanical stability and also suggests new avenues for material design.

Statistics of noisy growth with mechanical feedback in elastic tissues.

Tissue growth is a fundamental aspect of development and is intrinsically noisy. Stochasticity has important implications for morphogenesis, precise control of organ size, and regulation of tissue composition and heterogeneity. However, the basic statistical properties of growing tissues, particularly when growth induces mechanical stresses that can in turn affect growth rates, have received little attention. Here, we study the noisy growth of elastic sheets subject to mechanical feedback. Considering both isotropic and anisotropic growth, we find that the density-density correlation function shows power law scaling. We also consider the dynamics of marked, neutral clones of cells. We find that the areas (but not the shapes) of two clones are always statistically independent, even when they are adjacent. For anisotropic growth, we show that clone size variance scales like the average area squared and that the mode amplitudes characterizing clone shape show a slow [Formula: see text] decay, where n is the mode index. This is in stark contrast to the isotropic case, where relative variations in clone size and shape vanish at long times. The high variability in clone statistics observed in anisotropic growth is due to the presence of two soft modes-growth modes that generate no stress. Our results lay the groundwork for more in-depth explorations of the properties of noisy tissue growth in specific biological contexts.