We propose a novel approach to the linear viscoelastic problem of shear-deformable geometrically exact beams. The generalized Maxwell model for one-dimensional solids is here efficiently extended to the case of arbitrarily curved beams undergoing finite displacement and rotations. High efficiency is achieved by combining a series of distinguishing features, that are i) the formulation is displacement-based, therefore no additional unknowns, other than incremental displacements and rotations, are needed for the internal variables associated with the rate-dependent material; ii) the governing equations are discretized in space using the isogeometric collocation method, meaning that elements integration is totally bypassed; iii) finite rotations are updated using the incremental rotation vector, leading to two main benefits: minimum number of rotation unknowns (the three components of the incremental rotation vector) and no singularity problems; iv) the same $\rm SO(3)$-consistent linearization of the governing equations and update procedures as for non-rate-dependent linear elastic material can be used; v) a standard second-order accurate time integration scheme is made consistent with the underlying geometric structure of the kinematic problem. Moreover, taking full advantage of the isogeometric analysis features, the formulation permits accurately representing beams and beam structures with highly complex initial shape and topology, paving the way for a large number of potential applications in the field of architectured materials, meta-materials, morphing/programmable objects, topological optimizations, etc. Numerical applications are finally presented in order to demonstrate attributes and potentialities of the proposed formulation.

翻译：针对剪切变形几何精确梁的线性粘弹性问题，我们提出了一种新颖方法。将一维固体的广义Maxwell模型高效推广至经历有限位移和转角的任意曲梁情况。通过结合一系列显著特征实现高效率：i) 该公式基于位移，因此除增量位移和转角外，无需针对率相关材料内部变量引入额外未知量；ii) 控制方程采用等几何配点法进行空间离散，完全避免了单元积分；iii) 利用增量旋转向量更新有限转角，带来两大优势：旋转未知量数量最少（增量旋转向量的三个分量）且无奇异性问题；iv) 可采用与非率相关线弹性材料相同的$\rm SO(3)$一致线性化控制方程与更新流程；v) 标准二阶精度时间积分方案与运动学问题的底层几何结构保持一致。此外，充分利用等几何分析特性，该公式能够精确描述具有高度复杂初始形状和拓扑结构的梁与梁结构，为建筑材料、超材料、变形/可编程物体、拓扑优化等领域的大量潜在应用铺平道路。最后通过数值算例展示所提公式的特性与潜力。