Planar folding of shallow tape springs: The rod model with flexible cross-section revisited as a regularized Ericksen bar model

Abstract

This work concerns the parallel that can be made between two models involving propagating instabilities: (1) a reguralized Ericksen bar model and (2) a rod model with flexible cross-section dedicated to the folding of tape springs. This comparison confirms and complements the estimates obtained by Seffen and Pellegrino (1999) and gives some new insights on the formation and growth of folds, including their number and their location. We begin by studying a reguralized Ericksen bar model in statics. The complete bifurcation diagram is analyzed, together with the post-buckling deformed shapes. The influence of the reguralization parameter and of the boundary conditions is also studied. Then we propose a simplified model derived from Guinot et al. (2012) and Picault et al. (2014) to account for the folding of shallow tape springs in opposite sense bending. The equations that govern the problem, involving only two kinematic variables, can be easily written in this case. An analytical expression is found for the fundamental solution that is characteristic of an Ericksen bar model and it is shown that higher order terms that appear in the strain energy account for the transition zones. The obvious parallel with a reguralized Ericksen bar model is made by proceeding to a complete study of the post-bifurcation diagram. Estimates of the length of the transition zones are proposed and compared to those obtained with a FE shell model. A FE shell model is also used to validate the fundamental solution obtained with the proposed simplified rod model with flexible cross-section. These comparisons lead to good agreements, except for the peak moment of the moment-curvature response for the bending test. An enriched model is then proposed that brings significant improvements.