The implicit corotational method and its use in the derivation of nonlinear structural models for beams and plates

Abstract

What we call the implicit corotational method is proposed as a tool to obtain geometrically exact nonlinear models for structural elements, such as beams or shells, undergoing finite rotations and small strains, starting from the basic solutions for the three-dimensional Cauchy continuum used in the corresponding linear modelings. The idea is to use a local corotational description to decompose the deformation gradient in a stretch part followed by a finite rigid rotation. Referring to this corotational frame we can derive, from the linear stress tensor and the deformation gradient provided by linear elasticity, an accurate approximation for the nonlinear stress and strain tensors which implicitly assure frame invariance. The stress and strain fields recovered in this way as functions of generalized stress and strain resultants are then used within a mixed variational formulation allowing us to recover an objective nonlinear modeling directly suitable for FEM implementations through a black-box process which maintains the full details of the linear solutions, such as shear warping and other subtle effects. The method is applied to the construction of three-dimensional beam and plate nonlinear models starting from the Saint-Venant rod and Kirchhoff and Mindlin-Reissner plate linear theories, respectively. © 2012 by Mathematical Sciences Publishers.