Abstract 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.

中文翻译：

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具有变形截面的高阶几何精确梁的数学理论

摘要 本文研究了具有变形截面的高阶几何精确 Cosserat 型梁的变分公式和数值解，该梁受早期作品中提出的广义运动学的启发。除了来自 Simo-Reissner 梁运动学的其他变形模式之外，概括还包括完全耦合的泊松变形和翘曲变形的影响。手头的运动学使变形图不仅是梁的配置的函数，也是梁配置的切线空间元素（轴向应变矢量、曲率、翘曲幅度及其导数）的函数。虽然这使推导平衡定律和探索梁的变分公式的过程变得复杂，但结果的完整性使其值得。弱和强形式是针对考虑一般边界的动态情况导出的。我们将自己限制在线性小应变弹性本构定律和数值实现的静态情况。该梁的有限元建模具有较高的规律性要求。推导出运动方程的矩阵（离散化）形式。最后，给出了比较各种梁模型的数值模拟。