Anisotropic swelling due to hydration constrains anisotropic elasticity in biomaterial fibers

Abstract

Naturally occurring protein fibers often undergo anisotropic swelling when hydrated. Within a tendon, a hydrated collagen fibril's radius expands by 40% but its length only increases by 5%. The same effect, with a similar relative magnitude, is observed for single hair shafts. Fiber hydration is known to affect elastic properties. Here we show that anisotropic swelling constrains the anisotropic linear elastic properties of fibers. First we show, using data from disparate previously reported studies, that anisotropic swelling can be described as an approximately linear function of water content. Then, under the observation that the elastic energy of swelling can be minimized by the anisotropic shape, we relate swelling anisotropy to elastic anisotropy -- assuming radial (transverse) symmetry within a cylindrical geometry. We find an upper bound for the commonly measured axial Poisson ratio $\nu_{zx}<1/2$. This is significantly below recently estimated values for collagen fibrils extracted from tissue-level measurements, but is consistent with both single hair shaft and single collagen fibril mechanical and hydration studies. Using $\nu_{zx}$, we can then constrain the product $\gamma \equiv (1-\nu_{xy}) E_z/E_x$ -- where $\nu_{xy}$ is the seldom measured transverse Poisson ratio and $E_z/E_x$ is the ratio of axial to radial Young's moduli.