Fundamental Equations in Principal Curvilinear Coordinates

The fully-coupled hygro-thermo-magneto-electro-elastic anisotropic equations are derived for a three-dimensional (3D) solid with double curvature in curvilinear principal coordinates from the Hamilton principle. In particular, the 3D mechanical, electrostatic, magnetostatic, thermal and hygrometric definition equations are derived by computing in principal coordinates the gradient of a scalar quantity and of a vector field. Then, the constitutive equations are obtained from the Landau free energy and the Onsager principle under thermodynamic equilibrium conditions. Then, the balance equations are obtained by applying the Gauss-Green theorem, and they are specialized for a doubly-curved shell solid. A semi-analytical Navier solution is obtained for cross-ply lamination schemes and structures with uniform curvature.