Functional Error Analysis of Standard and Stress-Driven Reissner’s Mixed Variational Theorems

The partial differential equations ruling the structural mechanics of composites can in general be solved only in an approximate sense. Variational theorems such as Reissner’s mixed variational theorem (RMVT) and stress-driven Reissner’s mixed variational theorem (SDRMVT) have been proposed to solve these equations in weak form and provide an accurate description of the transverse stresses. This work uses the thickness energy functional error analysis recently introduced to reconstruct the Euler–Lagrange equations from the axiomatic laws used to represent the displacements and stresses. As a result of this process, spurious terms appear. In most of the formulated theories for selected orders of expansion in the thickness direction, those terms can be shown to vanish. However, in some situations this does not happen, and a functional error is present. That is, the Euler–Lagrange equations do not correctly reproduce the equilibrium/compatibility. In this work, both RMVT and SDRMVT are investigated by adopting eight different theories. It is shown that under certain conditions RMVT might present functional errors in the first (second) and third equilibrium equations, whereas SDRMVT may introduce a functional error in the first (second) and third compatibility conditions. The present contribution also adopts the elasticity solution of thin cross-ply composites to provide a quantitative evaluation of the functional error and identify where it takes place in the laminate.