Equilibrium equations for large deformations

Lagrangian or referential equilibrium equations for materials undergoing large deformations are of interest in the developing fields of mechanics of soft biomaterials and nanomechanics. The main feature of these equations is the necessity to deal with the First Piola-Kirchhoff, or nominal, stress tensor which is a two-point tensor referring simultaneously to the reference and current configurations. This two-point nature of the First Piola-Kirchhoff tensor is not always appreciated by the researchers and the total covariant derivative necessary for the formulation of the equilibrium equations in curvilinear coordinates is sometimes inaccurately confused with the regular covariant derivative. Surprisingly, the traditional continuum mechanics literature does not discuss this issue properly, except for some brief notions on the two-point nature of the Piola-Kirchhoff tensor. We aim at partially filling this gap by giving a full yet simple derivation of the Lagrangian equilibrium equations in cylindrical and spherical coordinates.