The metric description of elasticity in residually stressed soft materials

Living tissue, polymeric sheets and environmentally responsive gel are often described as elastic media. However, when plants grow, plastic sheets deform irreversibly and hydrogels swell differentially the different material elements within an object change their rest lengths often resulting in objects that possess no stress-free configuration making the standard elastic description inappropriate. In this paper we review an elastic framework based on Riemannian geometry devised to describe such objects lacking a stress-free configuration. In this framework the growth or irreversible deformation are associated with the change of a reference Riemannian metric that prescribes local distances within the body, and the elastic problem is one of optimal embedding. We discuss and resolve points of controversy regarding the Riemannian metric formulation. We give examples for dimensionally reduced theories, such as plates and shells theories, which arise naturally and discuss the relation between geometric frustration and residual stress.