Elastic Localization With Particular Reference to Tape-Springs

Many elastic systems localize under applied displacement, precipitating into regions of lower and higher strain; further displacement is accommodated by growth of the high strain region at a constant load. Such systems can be studied as propagating instabilities, focusing on the work required to propagate the high strain region, or as two-phase energy minimization problems. It is shown that the Maxwell “equal-areas” construction, and the related common tangent construction, provide the solution to either approach. A new, graphical, proof of the Maxwell equal-areas construction using total strain energy diagrams is presented. Tape-springs are investigated as a case study, with localization presenting as the formation of elastic folds—developable regions with high curvature. One notable property of tape-spring folds is that the fold radius is approximately equal to the initial transverse radius. This result was first proven by Rimrott, and later improved by Calladine and Seffen. A further improvement is obtained here by application of the common tangent construction, and all solutions are shown to be approximations to the Maxwell equal-areas construction in the limit of zero thickness.