Figure 1

(a) Schematic of rheometry experiment. In the limit $L\ll R$ and for adhesive boundary conditions, axial compression induces a volume change $\sim \pi R^2\mathrm{\Delta }L$ [8, 9]. Outside this limit, the sample is free to bulge laterally, away from a cylindrical geometry [6], and any volume change associated with the compression is $\ll \pi R^2\mathrm{\Delta }L$. Torsion measurements were carried out at fixed axial strain $\mathrm{\Delta }L/L$, generating axial stress deviations $\delta \sigma$ from the zero torsion prestress $\sigma$, but these were typically small, i.e., $|\delta \sigma /\sigma |\ll 1$. (b) Shear storage modulus $G^{\prime }$ versus axial compressive prestress $\sigma$ for fibrotic liver tissue, 2 and 6 weeks after disease onset, replotted from Fig. 5(a) of Perepelyuk et al. [6]. The data points in each series correspond to axial strains of 0, 10, 15, 20, and 25%. (c) Compression stiffening data from normal mammalian brain tissue. Purple triangles are replotted from Fig. 5(c) of Pogoda et al. [5]. Axial strain ranges from zero to 40%, in 5 and 10% increments, respectively. (d) Compression stiffening data from the fat tissue samples studied by Mihai et al. [10]. Axial strain again ranges from 0 to 40% in 10% increments. (e) Compression stiffening behavior of 2% agarose gel. Different data series correspond to different sample aspect ratios (all have $R=1$ cm), with all three samples having been cut from the same master gel. Strain goes from 0 to 25% in 5% increments. The horizontal shifts required to collapse the $L=9.5\mathrm{mm}$ and $L=3.8\mathrm{mm}$ curves onto the $L=1.8\mathrm{mm}$ curve are a rough measure of the gravitational stresses present in the thicker samples—see Ref. [11]. (f) Compression stiffening behavior of mango fruit flesh, with sample aspect ratio $L/2R=1/2$.