Thickness Energy Functional Error Analysis for axiomatic models of laminates based on Reissner’s Mixed Variational Theorem

The mathematical modeling of composite structures requires the solution of a set of partial differential equations that rule how strains, displacements, stresses, loads, material properties, and boundary conditions are related. This is usually carried out in weak form, where the solid-mechanics problem is formulated by the means of functionals, such as the Principle of Virtual Displacements (PVD). However, several difficulties related to poor quality of the calculated transverse stresses and numerical pathologies that included various types of locking phenomena motivated the researchers to explore other functionals and work with the so-called mixed variational theorems. In parallel, the practical need to adequately represent the displacement and stress fields, led to the proposal of a vast variety of axiomatic models where the various quantities (often the displacements) where expanded in the thickness direction with chosen functions. However, in some cases oscillations of the stresses were observed even in analytical computational frameworks without a finite element representation. The present contribution will introduce the Thickness Energy Functional Error Analysis (TEFEA), which provides an a-priori theoretical indication of functional errors potentially co-responsible for the poor quality of the numerical evaluations of the stresses without the need of carrying out a-posteriori implementation in a computer code and performing simulations. TEFEA can be considered a bridge between the generation of functionals and the creation of practical models for engineering use. In the present work TEFEA is applied to Reissner’s Mixed Variational Theorem and is used to interpret the transverse stress results for various low-order theories. It is shown that under certain conditions functional error in the equilibrium equations is present. TEFEA can be used to effectively construct the Best Theory Diagrams by a-priori discarding ineffective axiomatic models.