Meshless Approximation of Initially Curved Shells: Linear Model

Abstract

This work develops a kinematically linear shell model departing from a consistent nonlinear theory. Starting with a flat reference configuration for the shell midsurface, an initial (curved) geometry is imposed as a stress-free deformation, after which, the actual motion of the shell takes place. This strategy leads to the use of only orthogonal frames, precluding the use of objects as Christoffel symbols, the second fundamental form or 3-D degenerated solids. The resort to a flat reference configuration allows the use of 2-D Moving Least Squares approximations. The resulting model inherits the same features of the original formulation in the sense that (i) cross-sectional stresses and strains are defined in a totally consistent way, rendering a complete stress-resultant theory and (ii) first order shear deformations are accounted for, as Reissner-Mindlin kinematics is assumed, based on an inextensible director (no thickness changes). A variational statement of the shell model is presented, where the domain displacements and kinematic boundary reactions are independently approximated, hence falling in the category of the hybrid displacement formulations. The discretization of this variational form is made using the Multiple Fixed Least-Squares (MFLS) approximation on the domain and simple Lagrange polynomials on the boundary. The presented model is assessed through several numerical examples and results are compared to those in the literature. The smoothness and convergence of the meshless approximation is discussed. The present scheme's use of only orthogonal frames, along with the consistent definition of stress resultants and consequent plane stress definition led to a neat, consistent formulation for the analysis of initially curved shells. The consistent linear approximation, combined with MFLS approximation, lead to fast computations with high continuity, thus, smooth results for the displacement, strain and stress fields.