Fold propagation in tapered-springs: Part I

Many elastic systems can localize during deformation into coexistent 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 generalized 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-sectional geometry. Two special cases are considered in which the geometry varies systematically: ‘constant- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> ’ and ‘constant- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>b</mml:mi> </mml:math> ’ 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.