Isogeometric rotation-free analysis of planar extensible-elastica for static and dynamic applications

Finite deformations of planar slender beams for which shear strain can be neglected are described by the extensible-elastica model, where the strain-displacement relation is geometrically exact and the Biot stress–strain relation is linear. However, if the formulation is expressed in terms of displacements without rotation, the kinematics are described by a partial differential equation involving a fourth-order spatial operator, which cannot be approximated by the classical C0-continuous FE method in the standard Galerkin framework. In this work, we propose the spatial approximation of such high-order PDE by means of NURBS-based isogeometric analysis (IGA) which allows the use of globally high-order continuous basis functions. The employed IGA approach possesses three advantages: first, it facilitates the encapsulation of the exact geometric representation of the beams in the spatial approximation with fewer discrete points, especially useful for curved structures; second, it allows the discretization of high-order spatial operators; and third, it provides an efficient numerical solution of the discrete problem by using a limited number of degrees of freedom since the employed standard Galerkin formulation does not require rotational degrees of freedom. Yet this approach has not been directly compared to appropriate analytical solutions. To this end, we compare and validate numerical results from FE with the closed-form solutions for a set of static beam problems, including a newly derived solution for an initially curved beam, based on the extensible-elastica theory, by estimating the convergence orders of the errors. We also highlight the advantages of this formulation with the numerical solution of three dynamic problems: the swinging of a pinned beam, the propagation of solitons (nonlinear waves) in post-buckled beams, and snap-through buckling of a pinned beam that is axially buckled before transverse loading.