Fold propagation in tapered-springs: Part I

Many elastic systems can localise during deformation into co-existent modes, or "phases", of lower and higher strain; further deformation is accommodated by growth of the high-strain phase at a constant load, which is sometimes declared a "propagating instability". The strains in each phase can be obtained from applying the Maxwell Equal-Areas Construction to the generalised stress-strain relationship, which must be satisfied on the boundary between the phases. If the elastic domain has a spatially varying stress-strain relationship, this can reveal interesting behaviour: in particular, we investigate systems with spatially varying stress-strain relationships using "folded" tape-springs with varying cross-section geometry. Two special cases are considered in which the geometry varies systematically: "constant-α" and "constant-b" springs. These form folds which are spiral in shape, in contrast to the constant curvature fold in a standard tape-spring. These concepts may have beneficial application in designing new deployable structures.