A geometrically non-linear theory is developed for shells of generic shape allowing for third-order thickness stretching, higher-order shear deformation and rotary inertia by using eight parameters; geometric imperfections are also taken into account. The geometrically non-linear strain-displacement relationships are derived retaining full non-linear terms in the in-plane and transverse displacements and are presented in curvilinear coordinates, ready to be implemented in computer codes. Higher order terms in the transverse coordinate are retained in the derivation so that the theory is suitable also for thick laminated shells. The theory uses the three-dimensional constitutive equations and does not need the introduction of traction/compression free hypothesis at the shell inner and outer surfaces. The traction/compression free condition is introduced only to obtain a simplified model with six parameters instead of eight. The third-order thickness stretching theory is applied to cross-ply symmetrically laminated circular cylindrical shells complete around the circumference and simply supported at both ends. Geometrically non-linear forced vibrations are studied by using the present theory and results are compared to those obtained by using a refined higher-order shear deformation non-linear shell theory, which neglects thickness stretching, and to results obtained by using first-order and second-order thickness stretching theories. Results obtained by using the reduced third-order thickness stretching model with six parameters are also presented and compared.
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