We report a discrete differential geometry-based numerical framework to simulate the rate-independent, elasto-plastic behavior of one dimensional rod-like structures. Our numerical tool first discretizes the rod centerline into a number of nodes and edges in a manner similar to the well-established Discrete Elastic Rods (DER) method – a fast geometrically exact simulation for elastic filaments. The cross section of each vertex is next meshed into multiple fiber-like elements, such that the nonlinear constitutive relation of each element can be individually described by an increment-based associated flow rule and updated through return mapping algorithm. Equations of motion numerically obtained from energy variation at each degree of freedom are integrated through an implicit Euler time marching scheme, for its unconditional convergence and numerical stability. For quantitative comparison, we derive the analytical solutions for several simple cases, and a good match between the analytical solution and the numerical results in the geometrically linear regime indicates the accuracy of our discrete model. Our simulation can seamlessly handle the nonlinearity from both geometric and material sides, which is often not amenable to an analytical approach.

我们报告了离散的基于微分几何的数值框架，以模拟一维杆状结构的速率独立，弹塑性行为。我们的数值工具首先将杆中心线离散化为多个节点和边缘，其方式类似于公认的离散弹性杆（DER）方法-弹性长丝的快速几何精确模拟。接下来，将每个顶点的横截面划分为多个纤维状元素，以便可以通过基于增量的关联流规则分别描述每个元素的非线性本构关系，并通过返回映射算法进行更新。通过隐式的Euler时间行进方案对从每个自由度的能量变化数值获得的运动方程进行积分，由于其无条件收敛和数值稳定性。为了进行定量比较，我们导出了几种简单情况的解析解，解析解与几何线性方案中的数值结果之间的良好匹配表明了离散模型的准确性。我们的仿真可以从几何和材料两个方面无缝处理非线性，这通常不适用于分析方法。