On the range of applicability of linearized elasticity

Most contemporary derivations of the linearized theory of elasticity start with the basic equations of the non-linear theory organized into kinematical, balance, and constitutive equations. The linear approximations of these equations, under the assumption that the displacement gradient is small are found, and the higher-order terms are discarded to get the basic equations of the linear theory. Consequently, determination of the range of applicability of solutions to particular linear elasticity problems has generally been based on the size of the displacement gradient. However, all solution procedures involve commingling of the basic equations, and this raises the question of whether or not the gradients of the discarded terms are small. We establish that generally they are not unless the (non-dimensionalized) second gradient of the displacement is also small. An example shows that this additional restriction on the range of applicability can be significant.