Covariance in Linearized Elasticity

In this paper we covariantly obtain the governing equations of linearized elasticity. Our motivation is to see if one can make a connection between (global) balance of energy in nonlinear elasticity and its counterpart in linear elasticity. We start by proving a Green-Naghdi-Rivilin theorem for linearized elasticity. We do this by first linearizing energy balance about a given reference motion and then by postulating its invariance under isometries of the Euclidean ambient space. We also investigate the possibility of covariantly deriving a linearized elasticity theory, without any reference to the local governing equations, e.g. local balance of linear momentum. In particular, we study the consequences of linearizing covariant energy balance and covariance of linearized energy balance. We show that in both cases, covariance gives all the field equations of linearized elasticity.