Mixed plate theories based on the Generalized Unified Formulation. Part I: Governing equations

Abstract

Abstract The generalized unified formulation (GUF) is a formal technique which was introduced in the framework of displacement-based theories. GUF is extended for the first time to the case in which a mixed variational statement (Reissner’s mixed variational theorem) is used. Each of the displacement variables and out-of-plane stresses is independently treated and different orders of expansions for the different unknowns can be chosen. Since infinite combinations can be freely chosen for the displacements u x , u y , u z and for the stresses σ xz , σ yz , σ zz , the generalized unified formulation allows the user to write, with a single formal theory, ∞ 6 theories which can be successfully implemented in a single FEM code. In addition, this formulation allows the user to treat each unknown independently and, therefore, different numerical approaches can be used in the FEM codes based on this generalized unified formulation. All the theories are originated from 13 independent fundamental nuclei (kernels of the generalized unified formulation) which are formally invariant and the layerwise mixed theories (analyzed in Part II), mixed higher order shear deformation theories (analyzed in Part III) and advanced mixed zig-zag models (analyzed in Part IV) can be studied without extra implementations or theoretical developments. Numerical performances and convergence properties of a very large amount of new mixed theories are discussed (Part V) with particular focus on the effects of the orders of expansion in the thickness direction of the displacements and modeled stresses. Multilayered composite plates will be analyzed. Different mixed variational statements could be used and the formulation could be easily adopted for multifield problems such as thermoelastic applications and multilayered plates embedding piezo-layers.