Reissner's Mixed Variational Theorem and variable kinematics in the modelling of laminated composite and FGM doubly-curved shells

The present article proposes a mixed displacements/transverse stresses approach for the free vibration analysis of laminated composite and FGM doubly-curved shells. The theoretical formulation is derived by combining Reissner's Mixed Variational Theorem (RMVT), Carrera's Unified Formulation (CUF) and the Ritz method. With the application of the RMVT the interlaminar equilibrium of the transverse normal and shear stresses is fulfilled a priori by exploiting the use of Lagrange multipliers. The transverse normal and shear stresses become primary variables within the formulation and are modelled with a Layer-Wise (LW) kinematics description. However, on the other hand, displacement variables, which describe the kinematics of the shell structures, are defined using Equivalent Single Layer (ESL), Zig-Zag (ZZ) and LW shell theories. The Mixed Hierarchical Trigonometric Ritz Formulation (MHTRF) is then used as solution technique to compute the natural frequencies of laminated composite and FGM doubly-curved shells. Several study-cases are addressed and the proposed RMVT-based shell models are assessed by comparison with both 3D elasticity and 3D Ritz solutions. The effect of significant parameters such as orthotropic ratio, stacking sequence, aspect ratio, lamination angle, length-to-thickness and radius-to-length ratios as well as volume fraction index on the dimensionless frequency parameters is discussed.