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 yields 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 approxiation 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.

翻译：本文提出了一种用于几何与材料非线性梁瞬态弹性动力学的稳定混合等几何有限元公式，其中采用了具有可伸长方向向量的Cosserat梁公式。可伸长方向向量产生了一个包含恒定面内横截面应变的线性构型空间。引入了高阶（非协调）应变以修正刚度，其额外的自由度通过单元凝聚法消除。此外，当前对初始方向向量场的离散化使得近似应变度量具有客观性，与基函数的阶次无关。对于物理应力合力和应变，我们采用基于B样条基函数的全局分片近似，其高阶连续性使得所需自由度远少于单元近似法。在时间步进方面，我们采用了一种隐式能量-动量一致格式，与标准的梯形法则和中点法则相比，该格式展现出更优异的数值稳定性。文中给出了若干数值算例以验证本方法的有效性。