On the deformation of slender filaments with planar crimp: theory, numerical solution and applications to tendon collagen and textile materials

Tensile deformation of a slender filament crimped into the form of a plane wave is analysed in terms of the theory of extensible planar elasticas. The load-extension relation is shown to be expressible in terms of the following dimensionless quantities: crimp level; slenderness (defined by the ratio of thickness to contour length of one wave) and shape function (curvature normalized to its peak value) only. A particular family of shapes is introduced where the shape function is specified by means of a single parameter q. All shapes of practical interest lie between the limits of a wave of circular arcs (q-> — ∞) and a planar zigzag(q - + ∞). A numerical solution is described in which the effects of varying crimp level, slenderness and shape are all examined and normalized forms of load and extension are found which bring together the load-extension curves for wide ranges of crimp and slenderness. This leads to a procedure for fitting experimental load-extension curves to those calculated, by simple shifting of double logarithmic plots. The method is illustrated by applying it to published data for tendon (which consists largely of aligned collagen fibrils with planar crimp). Reasonable agreement is obtained and the significance of this is discussed. Suggestions are made of how the method may be applied to the problems of planar crimp that frequently arise in textile materials.