Elasticity without a reference state: continuum mechanics of active tension nets

A stress-free reference state and a constitutive stress–strain relation are the basic elements of elasticity theory. In living tissues, rapid turnover rules out a fixed reference state, and stress is governed by (motor molecule) activity rather than a constitutive law. We therefore study the mechanics of 2D tissues starting from a prescribed cell-associated active stress, instead of a reference shape. Intuitively, one expects cells to adjust their shapes and positions to achieve force balance by rearranging local sources of active stress. To make this picture precise, we represent the active stress configuration by a Riemannian "tension metric". By computing the coarse-grained stress tensor, we show that the tension metric defines a stress-free reference state, and its transformation law a stress–strain relation. The emergence of the key elements of elasticity theory explains how shape can be controlled by adiabatic changes in active stress during morphogenesis. This continuum theory identifies the large-scale limit of certain cell-level models (active tension networks). More generally, it elucidates the unconventional mechanical phase realized by living tissues, and may apply to foams and granular materials.