Motivated by applications of soft-contact problems such as guidewires used in medical and engineering applications, we consider a compressed rod deforming between two parallel elastic walls. Free elastica buckling modes other than the first are known to be unstable. We find the soft constraining walls to have the effect of sequentially stabilizing higher modes in multiple contacts by a series of bifurcations, in each of which the degree of instability (the index) is decreased by one. Further symmetry-breaking bifurcations in the stabilization process generate solutions with different contact patterns that allow for a classification in terms of binary symbol sequences. In the hard-contact limit, all these bifurcations collapse into highly degenerate ‘contact bifurcations’. For any given wall separation at most a finite number of modes can be stabilized and eventually, under large enough compression, the rod jumps into the inverted straight state. We chart the sequence of events, under increasing compression, leading from the initial straight state in compression to the final straight state in tension, in effect the process of pushing a rod through a cavity. Our results also give new insight into universal features of symmetry-breaking in higher mode elastic deformations. We present this study also as a showcase for a practical approach to stability analysis based on numerical bifurcation theory and without the intimidating mathematical technicalities often accompanying stability analysis in the literature. The method delivers the stability index and can be straightforwardly applied to other elastic stability problems.

受软接触问题的应用（例如医疗和工程应用中使用的导丝）的启发，我们考虑了两个平行弹性壁之间的压缩杆变形。已知除第一种之外的自由弹性屈曲模式是不稳定的。我们发现软约束壁具有通过一系列分叉在多个接触中顺序稳定更高模式的效果，在每个分叉中，不稳定性（指数）降低一个。稳定过程中进一步破坏对称性的分岔产生具有不同接触模式的解决方案，允许根据二进制符号序列进行分类。在硬接触极限下，所有这些分叉都会坍缩成高度退化的“接触分叉”。对于任何给定的壁分离，最多可以稳定有限数量的模式，最终，在足够大的压缩下，杆跳入倒直状态。我们绘制了事件序列，在增加的压缩下，从压缩的初始直线状态到拉伸的最终直线状态，实际上是推动杆穿过空腔的过程。我们的结果还为高模弹性变形中对称破坏的普遍特征提供了新的见解。我们还将这项研究作为展示基于数值分岔理论的稳定性分析实用方法的展示，而没有文献中经常伴随稳定性分析的令人生畏的数学技术细节。