Unlike the jamming transition, which occurs when the number of constraints on the motions of a system is equal to the number of degrees of freedom, some materials can undergo a rigidity transition while being underconstrained and maintaining a fixed connectivity. This is known as second order rigidity as it is due to non-linear effects. Several biological materials are known to experience a second order rigidity transition; sub-isostatic fiber networks are typically soft to allow for normal cell motions, but dramatically stiffen at a specific critical strain to protect tissues from rupturing, while confluent epithelial tissues can change their fluidity by adjusting the preferred shapes in individual cells. In addition, these biological systems are ale to tune their internal degrees of freedom in order to cross the second order rigidity transition in specific ways to accomplish their given functions. We wanted to mimic this by designing an underconstrained system that sits right at this critical point while also optimizing a given objective function.

We did this by using the internal stresses in a central force network as a new set of degrees of freedom to parameterize the full space of rigid states of the network, which we call the critical manifold. We then used gradient descent algorithms to search this space for configurations with special structure, such as having equal edge lengths; or with enhanced responses to external deformations, by maximizing either the bulk or shear modulus