An objective isogeometric mixed finite element formulation for nonlinear elastodynamic beams with incompatible warping strains

TL;DR

This paper introduces a stable, objective isogeometric mixed finite element method for nonlinear elastodynamic beams, utilizing Cosserat theory, incompatible strains, and energy-momentum consistent time integration for improved accuracy and stability.

Contribution

The paper develops a novel mixed finite element formulation for nonlinear beams that ensures objectivity, reduces degrees of freedom, and enhances numerical stability using isogeometric analysis and incompatible strains.

Findings

The method achieves objectivity of strain measures regardless of basis function degree.

It reduces degrees of freedom through element-wise condensation of incompatible strains.

Numerical examples demonstrate superior stability of the implicit energy-momentum scheme.

Abstract

We present a stable mixed isogeometric finite element formulation for geometrically and materially nonlinear beams in transient elastodynamics, where a Cosserat beam formulation with extensible directors is used. The extensible directors yield a linear configuration space incorporating constant in-plane cross-sectional strains. Higher-order (incompatible) strains are introduced to correct stiffness, whose additional degrees-of-freedom are eliminated by an element-wise condensation. Further, the present discretization of the initial director field leads to the objectivity of approximated strain measures, regardless of the degree of basis functions. For physical stress resultants and strains, we employ a global patch-wise approximation using B-spline basis functions, whose higher-order continuity enables to use much less degrees-of-freedom, compared to element-wise approximation. For time-stepping, we employ an implicit energy-momentum consistent scheme, which exhibits superior numerical stability in comparison to standard trapezoidal and mid-point rules. Several numerical examples are presented to verify the present method.