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

TL;DR

This paper presents an analytical investigation into how material softening influences the post-peak response of structures, emphasizing the role of nonlinear constitutive relations and identifying conditions for different post-peak behaviors.

Contribution

It introduces a nonlinear modeling approach that captures detailed post-peak responses and derives conditions for various post-peak phenomena, including snap-through.

Findings

Nonlinear assumptions better capture post-peak behavior than bilinear models.

Necessary and sufficient conditions for four key post-peak cases are derived.

Snap-through is caused by a change in the convexity of the softening curve.

Abstract

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.