We present a novel formulation for the dynamics of geometrically exact Timoshenko beams and beam structures made of viscoelastic material featuring complex, arbitrarily curved initial geometries. An $\textrm{SO}(3)$-consistent and second-order accurate time integration scheme for accelerations, velocities and rate-dependent viscoelastic strain measures is adopted. To achieve high efficiency and geometrical flexibility, the spatial discretization is carried out with the isogemetric collocation (IGA-C) method, which permits bypassing elements integration keeping all the advantages of the isogeometric analysis (IGA) in terms of high-order space accuracy and geometry representation. Moreover, a primal formulation guarantees the minimal kinematic unknowns. The generalized Maxwell model is deployed directly to the one-dimensional beam strain and stress measures. This allows to express the internal variables in terms of the same kinematic unknowns, as for the case of linear elastic rate-independent materials bypassing the complexities introduced by the viscoelastic material. As a result, existing $\textrm{SO}(3)$-consistent linearizations of the governing equations in the strong form (and associated updating formulas) can straightforwardly be used. Through a series of numerical tests, the attributes and potentialities of the proposed formulation are demonstrated. In particular, we show the capability to accurately simulate beams and beam systems featuring complex initial geometry and topology, opening interesting perspectives in the inverse design of programmable mechanical meta-materials and objects.

翻译：本文提出了一种新颖的几何精确铁木辛柯梁及梁结构动力学公式，该结构由粘弹性材料构成，具有复杂且任意弯曲的初始几何形态。我们采用了基于$\textrm{SO}(3)$群一致性且对加速度、速度及率相关粘弹性应变度量具有二阶精度的时域积分格式。为实现高计算效率与几何灵活性，空间离散化采用等几何配点法进行，该方法在保持等几何分析高阶空间精度与几何表示优势的同时，避免了单元积分。此外，原始变量公式保证了最少的运动学未知量。广义麦克斯韦模型被直接应用于一维梁的应变与应力度量。这使得内部变量能够用与线弹性率无关材料情形相同的运动学未知量表示，从而规避了粘弹性材料引入的复杂性。因此，现有的强形式控制方程基于$\textrm{SO}(3)$群一致性的线性化处理（及其相应的更新公式）可直接应用。通过一系列数值测试，验证了所提公式的特性与潜力。特别地，我们展示了该方法能够精确模拟具有复杂初始几何与拓扑结构的梁及梁系，为可编程机械超材料与物体的逆向设计开辟了新的前景。