Table 1.

Hyperelastic strain energy functions and corresponding uniaxial stress-strain equations

Name	Strain energy potential (W) Uniaxial stress (α)—stretch (λ) equation Initial shear modulus (G0)
Mooney-Rivlin, Neo-Hookean (Mooney 1940; Treloar 1975)	W = C1 (I1 – 3) + C2 (I2 – 3); C2 = 0 for Neo-Hookean model
	σ = 2C1 (λ – λ−2) + 2C2 (1 – λ−3)
	G0 = 2 (C1 + C2); Fitting parameters: C1, C2
Reduced polynomial (Mooney 1940)	$W={{\displaystyle {\sum }_{i=1}^N{C}_i(I_1-3)}}^i$
	$\sigma =2(\lambda -{\lambda }^{-2}){\displaystyle {\sum }_{i=1}^{N}i{C}_i{({\lambda }^2+2{\lambda }^{-1}-3)}^{i-1}}$
	G0 = 2C1; Fitting parameters: Ci
Ogden (1972)	$W={\displaystyle {\sum }_{i=1}^N\frac{2C_i}{{\alpha }_l^2}}({\lambda }_x^{{\alpha }_i}+{\lambda }_y^{{\alpha }_i}+{\lambda }_z^{{\alpha }_i}-3)$
	$\sigma ={\displaystyle {\sum }_{i=1}^N\frac{2C_i}{{\alpha }_i}}({\lambda }^{{\alpha }_i-1}-{\lambda }^{-{\alpha }_i/2-1})$
	$G_0={\displaystyle {\sum }_{i=1}^N{C}_i}$; Fitting parameters: Ci, αi
Fung (Fung 1967; Fung et al. 1979)	$W=\frac{C}{2b}\{\text{exp}[b(I_i-3)]-1\}$
	σ = C (λ − λ−2) exp [b (I1 − 3)]
	G0 = C; Fitting parameters: C, b
Van der Waals (Kilian) (Kilian 1985)	$W=C\left\{-(I_{1m}-3)\left[\text{ln}\left(1-\sqrt{\frac{I_1-3}{I_{1m}-3})}\right)+\sqrt{\frac{I_1-3}{I_{1m}-3}}\right]-\frac{2}{3}b{\left(\frac{I_1-3}{2}\right)}^{3/2}\right\}$
	$\sigma =C(\lambda -{\lambda }^{-2})\left[{\left(1-\sqrt{\frac{{\lambda }^2+2{\lambda }^{-1}-3}{{\lambda }_m^2+2{\lambda }_m^{-1}-3}}\right)}^{-1}-b\sqrt{\frac{{\lambda }^2+2{\lambda }^{-1}-3}{2}}\right]$
	G0 = C; Fitting parameters: C, b
	λm is the limiting tensile stretch; I1m is the corresponding first invariant
Gaylord—Douglas (1987,1990), Tschoegl—Gurer (Gurer and Tschoegl 1985; Tschoegl and Gurer 1985)	$W=(C_1/2)(I_1-3)+(2C_2/b^2)\left({\lambda }_x^b+{\lambda }_y^b+{\lambda }_z^b-3\right)$
	σ = C1 (λ − λ−2) + (2C2/b) [λb − λ−b/2]
	G0 = C1 + C2; Fitting parameters: C, C2
	Gaylord–Douglas: b = 1 Tschoegl–Gurer: b = 0.34

Assuming material incompressibility, in uniaxial loading in the x-direction, I₁ first strain invariant = ${\lambda }_x^2+{\lambda }_y^2+{\lambda }_z^2$; I₂ second invariant = ${\lambda }_x^{-2}+{\lambda }_y^{-2}+{\lambda }_z^{-2}$, σ = λ (∂W/∂λ), λ_x = λ, λ_y = λ_z = λ^−1/2 and strain ε = λ − 1