This paper deals with a numerical analysis of plastic deformation under various conditions, utilizing Radial Basis Function (RBF) approximation. The focus is on the elasto-plastic von Mises problem under plane-strain assumption. Elastic deformation is modelled using the Navier-Cauchy equation. In regions where the von Mises stress surpasses the yield stress, corrections are applied locally through a return mapping algorithm. The non-linear deformation problem in the plastic domain is solved using the Picard iteration. The solutions for the Navier-Cauchy equation are computed using the Radial Basis Function-Generated Finite Differences (RBF-FD) meshless method using only scattered nodes in a strong form. Verification of the method is performed through the analysis of an internally pressurized thick-walled cylinder subjected to varying loading conditions. These conditions induce states of elastic expansion, perfectly-plastic yielding, and plastic yielding with linear hardening. The results are benchmarked against analytical solutions and traditional Finite Element Method (FEM) solutions. The paper also showcases the robustness of this approach by solving case of thick-walled cylinder with cut-outs. The results affirm that the RBF-FD method produces results comparable to those obtained through FEM, while offering substantial benefits in managing complex geometries without the necessity for conventional meshing, along with other benefits of meshless methods.

翻译：本文利用径向基函数近似，对多种条件下的塑性变形进行了数值分析。研究重点在于平面应变假设下的弹塑性von Mises问题。弹性变形采用Navier-Cauchy方程进行建模。在von Mises应力超过屈服应力的区域，通过返回映射算法进行局部修正。塑性域中的非线性变形问题采用Picard迭代求解。Navier-Cauchy方程的解通过径向基函数生成有限差分无网格方法计算，该方法仅使用强形式下的散乱节点。通过对承受不同加载条件的内压厚壁圆筒进行分析，验证了该方法的有效性。这些加载条件会引发弹性膨胀、理想塑性屈服以及线性硬化塑性屈服等状态。计算结果以解析解和传统有限元方法解为基准进行了比较。本文还通过求解带切口的厚壁圆筒案例，展示了该方法的鲁棒性。结果证实，RBF-FD方法能产生与FEM相当的结果，同时在处理复杂几何形状时无需传统网格划分，并具有无网格方法的其他优势。