Quantitative non-linearity analysis of single plant fiber tensile curves using Legendre harmonic decomposition

Abstract

The non-linear tensile behavior of single plant fibers carries microstructural information that modulus and strength fail to capture. This work presents a quantitative shape-descriptor framework for single fiber stress–strain curves based on a Legendre harmonic decomposition. Stress and strain are normalized to the unit interval, mapping every curve onto a common shape space independent of stiffness, strength and failure strain. The normalized curve is decomposed onto the shifted Legendre polynomial basis truncated at third order, whose orthogonality ensures that the resulting coefficients are independent descriptors of curve shape. Three descriptors are derived: the non-linearity magnitude R, the behavior type φ (the angle locating each curve in a two-dimensional shape plane), and the strain at inflection εi. The framework is demonstrated on carbon, flax, hemp and cotton fibers. Carbon shows significantly lower R than the plant fibers, consistent with its near-linear response. φ resolves three statistically distinct groups, separating carbon from plant fibers and flax from hemp and cotton. εi isolates cotton, consistent with its higher microfibril angle requiring greater strain before fibril realignment. The framework offers a quantitative alternative to qualitative shape classifications, supporting systematic comparison of plant fibers as composite reinforcements.