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

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.

Figures (30)

• Figure 1: A schematic illustration of the beam kinematics. $\mathcal{B}_0$ and $\mathcal{B}_t$ denote the (open) domains of the initial (undeformed) and current configurations, respectively. ${\boldsymbol{e}}_1$, ${\boldsymbol{e}}_2$, and ${\boldsymbol{e}}_3$ represent the global Cartesian base vectors. This figure is redrawn with modifications from choi2023selectively.
• Figure 2: A schematic illustration of the reference domain of a beam having a rectangular cross-section with dimension $h_1\times{h_2}$. $\mathcal{B}$ denotes the (open) domain of the reference configuration. This figure is redrawn with modifications from choi2023selectively.
• Figure 3: Rigid rotation of a stress-free rod (objectivity test 1): Change of the total strain energy under a rigid rotation. The dashed vertical lines indicate full turns. Note that those markers are only plotted in every four rotational increments for better visibility.
• Figure 4: Rigid rotation of a bent rod (objectivity test 2): Change of the total strain energy under a superposed rigid body rotation in the second phase. (a) Here, $n_\mathrm{load}$ represents the number of load steps in the second phase. (b) We use $n_\mathrm{load}=9$ for all cases.
• Figure 5: Straight beam under twisting moment: Initial geometry and boundary conditions. ${\bar{\theta}}_\mathrm{A}$ and ${\bar{\theta}}_\mathrm{B}$ denote the prescribed rotation angles of the cross-sections at the points A and B with respect to the center axis, respectively.

Theorems & Definitions (6)

• Remark 2.1
• Remark 3.1
• Remark 4.1
• Remark 4.2
• Remark 5.1
• Remark A.1