Table 1.

Force-indentation relationships for Hertzian and hyperelastic strain energy density functions.

Name	Force (F) – indentation (δ) relationship	Parameters	See also
Hertz	$F=\frac{4E{R}^{\frac{1}{2}}{\delta }^{\frac{3}{2}}}{3\left(1-{\nu }^2\right)}a=R^{1/2}{\delta }^{1/2}$	E	(Hertz, 1882)
NeoHookean	$F=C_{10}\pi \left(\frac{a^5-15R{a}^4+75R^2{a}^3}{5R{a}^2-50R^{2}a+125R^3}\right)$	C 10	(Lin et al., 2009)
Mooney-Rivlin	$F=C_{10}\pi \left(\frac{a^5-15R{a}^4+75R^2{a}^3}{5R{a}^2-50R^{2}a+125R^3}\right)+C_{01}\pi \left(\frac{a^5-15R{a}^4+75R^2{a}^3}{-a^3+15R{a}^2-75R^{2}a+125R^3}\right)$	C10, C01	(Lin et al., 2009; Mooney, 1940)
Arruda-Boyce	$F={\mu }_0\sqrt{R\delta }\delta \left(A_0+{\sum }_{i=1}^3{A}_i+\exp \left[-{\lambda }_m/B_i\right]\right)$ Where Ai and Bi vary by δ/R, according to values in Supplementary Table 1.	μ0, λm	(Arruda and Boyce, 1993; Zhang et al., 2014)
Fung	$F=B\pi \left(\frac{a^5-15R{a}^4+75R^2{a}^3}{5R{a}^2-50R^{2}a+125R^3}\right)\exp \left[b\left(\frac{a^3-15R{a}^2}{25R^{2}a-125R^3}\right)\right]$	B, b	(Fung, 1967; Fung et al., 1979; Lin et al., 2009)
Ogden	$F=\frac{B\pi a^2}{\alpha }\left[{\left(1-0.2\frac{a}{R}\right)}^{-\frac{\alpha }{2}-1}-{\left(1-0.2\frac{a}{R}\right)}^{a-1}\right]$	B, α	(Lin et al., 2009; Ogden, 1972)