Geometry and Mechanics of Growing, Nonlinearly Elastic Plates and Membranes

Abstract

Until the twentieth century, theories of elastic rods and shells arose from collections of geometric and mechanical assumptions and approximations. These theories often lacked internal consistency and were appropriate for highly proscribed and sometimes unknown geometries and deformation sizes. The pioneering work of Truesdell, Antman, and others converted mechanical intuition into rigorous mathematical statements about the physics and mechanics of rods and shells. The result is the modern, geometrically exact theory of finite deformations of rods and shells.In the latter half of the twentieth century, biomechanics became a major focus of both experimental and theoretical mechanics. The genesis of residual stress by non-elastic growth has significant impact on the shape and mechanical properties of soft tissues. Inspired by the geometry of blood vessels and adopting a formalism found in elasto-plasticity, mechanicians have produced rigorous and applied results on the effect of growth on finite elastic deformations of columns and hollow tubes. Less attention has been paid to shells.A theory of growing elastic plates has been constructed in the context of linear elasticity. It harnessed many results in the theory of Riemann surfaces and has produced solutions that are surprisingly similar to experimental observations. Our intention is to provide a finite-deformation alternative by combining growth with the geometrically exact theory of shells. Such a theory has a clearer and more rigorous foundation, and it is applicable to thicker structures than is the case in the current theory of growing plates.This work presents the basic mathematical tools required to construct this alternative theory of finite elasticity of a shell in the presence of growth. We make clear that classical elasticity can be viewed in terms of three-dimensional Riemannian geometry, and that finite elasticity in the presence of growth must be considered in this way. We present several examples that demonstrate the viability and tractability of this approach.