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An analytical study is taken to investigate the relationship between materialsoftening and structural softening through the use of a model problem in onedimension. With general nonlinear assumptions on the constitutive relations, itturns out that the governing equations can be viewed as a system of parametricequations, which couple the size effect and the nonlinear effect. Compared withthe bilinear assumptions in previous literature, we find that the nonlinearassumptions herein capture more details in the post-peak structural response.After doing standard mathematical analysis to the nonlinear equations, wemanage to derive necessary and sufficient conditions for the occurrence of fourimportant post-peak cases, which are often observed in experiments. Inparticular, our analysis reveals that the mechanism of the snap-throughphenomenon is due to the convexity change of the constitutive curve of thesoftening part. Mathematical examples are also given to illustrate the proposedprocedures.

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