Lagrangian and Eulerian formulations of second-grade elasticity via convected coordinates

The transformation between the Lagrangian and Eulerian descriptions of the equilibrium equations for second-grade elastic materials is reconsidered in the setting of a convected-coordinate formulation of the relevant kinematics. The third-order contortion tensor, equivalent to the strain gradient and representing the change of the Levi-Civita connection induced by deformation, is adopted as the basic descriptor of the refined kinematics associated with the second-gradient theory.