Extended Data Fig. 1: Nonlinear material model.

The nonlinear material model divides the mechanical response of individual fibers into a region where fiber stiffness (ω^″) decreases exponentially with decreasing strain under compression (buckling), a region of constant fiber stiffness for small strains (straightening), and a region of exponentially increasing fiber stiffness for larger strains (stretching)¹². Non-linear materials are modelled according to Eq. (1) using four material parameters: The linear stiffness k₀, the buckling coefficient d₀, the characteristic strain λs, and the stiffening coefficient ds. For small extensional strain in the linear range (0 < λ < λs), fibers exhibit a constant stiffness k. For compressive strain (− 1 < λ < 0), fibers buckle and show an exponential decay of the stiffness with a characteristic buckling coefficient d₀. For larger extensional strain beyond the linear range (λs < λ), the fibers show strain stiffening with an exponential increase of stiffness with a characteristic stiffening coefficient ds. Dashed lines indicate the different regimes. Figure adapted with permission from ref. ¹², Springer America Inc.