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 -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 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.

中文翻译：

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具有任意弯曲初始几何形状的几何精确粘弹性梁和梁系统动力学的等距分析公式


我们提出了一种新颖的几何精确铁木辛科梁和由粘弹性材料制成的梁结构的动力学公式，具有复杂、任意弯曲的初始几何形状。采用一致的二阶精确时间积分方案来测量加速度、速度和速率相关的粘弹性应变测量。为了实现高效率和几何灵活性，空间离散化采用等几何配置（IGA-C）方法进行，该方法允许绕过单元集成，保留等几何分析（IGA）在高阶空间精度和精度方面的所有优势。几何表示。此外，原始公式保证了最小的运动学未知数。广义麦克斯韦模型直接应用于一维梁应变和应力测量。这允许用相同的运动学未知数来表达内部变量，就像线弹性速率无关材料的情况一样，绕过了粘弹性材料引入的复杂性。因此，可以直接使用强形式控制方程的现有 SO(3) 一致线性化（以及相关的更新公式）。通过一系列的数值测试，证明了所提出的配方的属性和潜力。特别是，我们展示了精确模拟具有复杂初始几何和拓扑的梁和梁系统的能力，为可编程机械超材料和物体的逆向设计开辟了有趣的视角。