Modeling Challenges and Limitation Principles of Reissner’s Mixed Approaches to Laminates

Displacement-based formulations for composite structures are directly linked to the functional called the Principle of Virtual Displacements and present the minimum number of independent modeled fields, which are the displacements. However, these models usually provide an inadequate approximation of the transverse stress fields and suffer from pathological numerical problems. These issues have been addressed with the introduction of mixed variational formulations where independent stress fields are included in the definition of the corresponding functionals. Particularly effective and widely used are Reissner’s mixed variational theorem and Hellinger–Reissner Principle. Both Reissner’s mixed variational theorem and Hellinger–Reissner principle allow the a priori enforcing of the interlaminar equilibrium because of the axiomatic modeling of the transverse stresses. However, these functionals are different and present their own challenges. Under certain circumstances, Reissner’s Mixed Variational Theorem was shown to produce thickness oscillations of the displacements and transverse stresses. It was not clear if Hellinger-Reissner Principle had similar properties. It is shown that both functionals may present numerical oscillations. Moreover, the reconstructed displacement and stress fields are actually identical between Reissner’s Mixed Variational Theorem and Hellinger-Reissner Principle if the orders of the in-plane stresses of the Hellinger–Reissner Prinicple case are sufficiently large. In other words, a Hellinger–Reissner Principle/Reissner’s Mixed Variational Theorem Limitation Principle is shown to exist in addition to the well-studied limitation principle between Hellinger–Reissner principle and principle of virtual displacements.