Decomposed theorem of a transversely isotropic elastic plate for extensional deformation

•Elliott–Lodge solution and Lur’e method are utilized.•A decomposed theorem of a transversely isotropic plate for extensional deformation are derived and studied.•The exact equations of the plate are obtained under homogeneous boundary conditions.•The decomposed theorem of the extensional plate is proven strictly for the first time.

Without ad hoc assumptions, a decomposed theorem of a transversely isotropic plate for extensional deformation are derived and studied based on transversely isotropic elastic theory. Firstly, from the Elliott–Lodge solution and Lur’e method, the displacement and stress components are obtained in terms of mid-plane displacements and transverse normal strain. Secondly, the exact equations of the plate are obtained under homogeneous boundary conditions. The general stress state of the plate consists of three parts: the generalized plane-stress state, the shear state, and the Papkovich–Fadle state. At last, the decomposed form of a transversely isotropic elasticity plate for extensional deformation is obtained, and the decomposed theorem is proven strictly for the first time.