Investigating mathematical methods for high-throughput prediction of the critical buckling load of non-uniform wool fibers

Abstract

The sensation of prickle from textile garments is directly related to the force that a fiber protruding from the fabric surface can exert on the skin without buckling – its critical buckling load (CBL). Finite element modeling (FEM) has previously been used in the literature to predict CBLs for a set of 25 fibers with different along-fiber morphology. With a view to high-throughput analysis of fibers, we investigated two analytical methods that were potentially faster and less computationally intensive than FEM and applied them to calculate CBLs for the same set of fibers. In addition, we tested a numerical integration and gradient search (NIGS) method that we developed by adapting a previously published, non-FEM, numerical approach. The analytical methods that we tested were either inadequately formulated or prone to instability. Our NIGS method was more reliable that the analytical methods (but slower to compute), and its results appeared more accurate than the published FEM results, based on an inconsistency metric that we developed. The published FEM results and the NIGS predictions agreed within 5% for 60% of the fibers, and within 10% for 72% of the fibers (with differences ranging from 0.4% to 19.1%) and generally showed qualitative agreement on the response of CBL to fiber shape, with some notable exceptions. The response of CBL to dimensional variation was complex. This, and the inconsistency between methods, highlights the need for caution when analyzing complicated biological structures, such as wool, and the value of verifying the reliability of any predictions from any approach.

A modular modeling approach for investigating wool critical buckling from biologically variable along-fiber microstructure

“…In our previous paper 19 we calculated the following dimensionless inconsistency score, γ, to use as an indicator of how well the solutions satisfied both Equation (14) and the boundary conditions …”

“…We presented a numerical integration and gradient search method approach that expressed Equation (1) as a system of four first-order differential equations, and used the boundary conditions of clamped/pinned fiber to determine the values of w and its first, second and third derivatives at x=0, and hence the initial conditions for the four-dimensional system. 19 …”

“…This ODE system was solved numerically and β, the only unknown, was found using a Levenberg-Marquard gradient search. 19…”

“…For example, the effect on CBL of smooth sinusoidal variations in diameter along the length of a fiber has been investigated using finite element modeling (FEM). 17,18 Recently we presented a numerical approach 19 that improved upon the accuracy of the calculations of an earlier FEM, 17 which we found sometimes to give inaccurate predictions. 19 Indeed, Burkhart 20 cautioned against automatic acceptance of FEM predictions.…”

Wanted, dead and alive: Why a multidisciplinary approach is needed to unlock hair treatment potential

“…However, clear understanding of protein interconnectedness is in its early stages and is crucial because network alteration by environmental stresses will be determined by the basal organization laid down during hair development. Connecting structural features to single fibre properties is a challenge that will rely heavily on mathematical models and robust data on variation at different molecular scales …”