两个基本屈服准则几何轨迹建立的探索

Exploration on Establishing Geometric Trajectories of Two Fundamental Yield Criteria

摘要: Mises屈服准则和Tresca屈服准则是力学领域两个基本的屈服准则，在不同的应力状态下，两个屈服准则对应的几何轨迹在相关的力学教材中都有介绍，但对于三向应力状态下Tresca屈服准则几何轨迹——正六棱柱面的建立过程，在相关教材中没有具体的描述。本文首先利用平面坐标旋转公式建立平面应力状态下Mises屈服准则的几何轨迹，再通过严格的数学分析建立了三向应力状态下Tresca屈服准则的几何轨迹——内接于Mises圆柱面的正六棱柱面。

Abstract: Mises and Tresca yield criteria are two fundamental yield criteria in the field of mechanics. The geometric trajectories corresponding to the two yielding criteria under different stress states are introduced in related mechanical textbooks. However, the process of establishing the geometric trajectory of the Tresca yield criterion in three-dimensional stress state-the regular hexagonal prism surface is not detailly described in relevant textbooks. In this paper, the geometric trajectory of Mises yield criterion under plane stress state was first established by the formula of plane coordinate rotation. Then, through rigorous mathematical analysis, the geometric trajectory of Tresca yield criterion under triaxial stress state—the regular hexagonal cylindrical surface inscribed on Mises cylinder surface—was established.