AN INTRODUCTION TO DIFFERENTIAL GEOMETRY WITH APPLICATIONS TO ELASTICITY

The applications of three-dimensional (3D) differential geometry to the field of elasticity are described. Covariant derivatives constitute a generalization of the usual partial derivatives of vector fields defined defined by means of their Cartesian components. By studying the equations of nonlinear and linearized elasticity it is found that covariant derivatives appear when a system of partial differential equations with a vector field as the unknown is expressed in terms of curvilinear coordinates. Koiter's equations can be fully justified for all types of shells, since it is clear that Koiter's equations can not be recovered as the outcome of an asymptotic analysis of the three-dimensional equations.