The mathematical theory of a higher-order geometrically-exact beam with a deforming cross-section

• Published in: International journal of solids and structures Vol. 202; pp. 854 - 880
• Main Authors: Chadha, Mayank, Todd, Michael D.
• Format: Journal Article
• Language: English
• Published: New York Elsevier Ltd 01.10.2020; Elsevier BV
• Subjects: Axial strain; Configurations; Coupled Poisson’s and warping effect; Cross-sections; Deformation effects; Equations of motion; Finite element formulation; Finite element method; Geometrically-exact beam; Kinematics; Large deformations; Mathematical analysis; Mathematical models; Matrix methods; Variational formulation; Warping; Coupled Poisson’s and warping effect; Geometrically-exact beam; Finite element formulation; Large deformations; Variational formulation
• ISSN: 0020-7683; 1879-2146
• DOI: 10.1016/j.ijsolstr.2020.06.002

This paper investigates the variational formulation and numerical solution of a higher-order, geometrically exact Cosserat type beam with deforming cross-section, instigated from generalized kinematics presented in earlier works. The generalizations include the effects of a fully-coupled Poisson’s and warping deformations in addition to other deformation modes from Simo-Reissner beam kinematics. The kinematics at hand renders the deformation map to be a function of not only the configuration of the beam but also elements of the tangent space of the beam’s configuration (axial strain vector, curvature, warping amplitude, and their derivatives). While this complicates the process of deriving the balance laws and exploring the variational formulation of the beam, the completeness of the result makes it worthwhile. The weak and strong form are derived for the dynamic case considering a general boundary. We restrict ourselves to a linear small-strain elastic constitutive law and the static case for numerical implementation. The finite element modeling of this beam has higher regularity requirements. The matrix (discretized) form of the equation of motion is derived. Finally, numerical simulations comparing various beam models are presented.