On the Post-Peak Structural Response due to Softening with Localization

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. With general nonlinear assumptions on the constitutive relations, it turns out that the governing equations can be viewed as a system of parametric equations, which couple the size effect and the nonlinear effect. Compared with the bilinear assumptions in previous literature, we find that the nonlinear assumptions herein capture more details in the post-peak structural response. After doing standard mathematical analysis to the nonlinear equations, 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 of the constitutive curve of the softening part. Mathematical examples are also given to illustrate the proposed procedures.