An analytical study is taken to investigate the relationship between material softening and structural softening through the use of a model problem in one dimension. General nonlinear constitutive relations are used. Compared with the bilinear assumptions in previous literature, we find that the nonlinear assumptions herein capture more details in the post-peak structural response. We manage to derive necessary and sufficient conditions for the occurrence of four important post-peak cases, which are often observed in experiments. In particular, our analysis reveals that the mechanism of the snap-through phenomenon is due to the convexity change in the constitutive curve of the softening part.