Incorporating the arc-length constraint, the dynamic relaxation strategy has been widely used to trace full equilibrium path in the post-buckling analysis of structures. This combined numerical scheme has been shown to be successful for solving snap-through problems, but its applicability to snap-back problems has been rarely investigated and remains unclear. This paper proposes a direct and more general finite-difference equation to investigate the numerical stability of this combined numerical scheme, which is dominated by the spectral radius of amplification matrix. And a key discovery of this paper is that a first minor of the tangent stiffness matrix is always negative once snap back occurs. Due to this negative minor stiffness, the spectral radius is invariably greater than one, resulting in unconditional instability, which demonstrates the invalidity of dynamic relaxation arc-length method for snap-back problems. These important conclusions are corroborated by the numerical results of three representative examples in one-, two- and three-dimensional spaces.

结合弧长约束，动态松弛策略已被广泛用于跟踪结构的后屈曲分析中的完整平衡路径。这种组合数值方案已被证明可以成功地解决快速通过问题，但其对快速回退问题的适用性很少被研究并且仍不清楚。本文提出了一个直接且更通用的有限差分方程来研究这种组合数值方案的数值稳定性，该数值稳定性由放大矩阵的谱半径决定。本文的一个关键发现是，一旦发生回弹，切线刚度矩阵的第一个次要总是为负。由于这种负次刚度，谱半径总是大于 1，导致无条件的不稳定，这证明了动态松弛弧长方法对于快速返回问题的无效性。这些重要结论得到了一维、二维和三维空间中三个具有代表性的例子的数值结果的证实。