On the Accuracy of Standard and Stress-Driven Reissner's Mixed Variational Theorems Assessed via Functional Error Analysis

The partial differential equations ruling structural mechanics 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. Equilibrium and Compatibility Equations are obtained as Euler-Lagrange Equations of the variational statements. Possible source of errors are in principle inaccurate mesh description of the structure's domain, insufficient order of expansion for the displacements/stresses to correctly reproduce the behaviors of the primary fields, and various potential locking phenomena. But the error can also be in the functional itself. 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 correct reproduce the equilibrium/compatibility. In this work both RMVT and SDRMVT are investigated by adopting 8 different theories. It is shown that under certain conditions RMVT might present 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.