The polynomial hyperelastic material model[ 1] rubber elasticity . In this model, thestrain energy density function is of the form of a polynomial in the two invariantsI 1 , I 2 {\displaystyle I_{1},I_{2}}
The strain energy density function for the polynomial model is[ 1]
W = ∑ i , j = 0 n C i j ( I 1 − 3 ) i ( I 2 − 3 ) j {\displaystyle W=\sum _{i,j=0}^{n}C_{ij}(I_{1}-3)^{i}(I_{2}-3)^{j}} whereC i j {\displaystyle C_{ij}} C 00 = 0 {\displaystyle C_{00}=0}
For compressible materials, a dependence of volume is added
W = ∑ i , j = 0 n C i j ( I ¯ 1 − 3 ) i ( I ¯ 2 − 3 ) j + ∑ k = 1 m 1 D k ( J − 1 ) 2 k {\displaystyle W=\sum _{i,j=0}^{n}C_{ij}({\bar {I}}_{1}-3)^{i}({\bar {I}}_{2}-3)^{j}+\sum _{k=1}^{m}{\frac {1}{D_{k}}}(J-1)^{2k}} where
I ¯ 1 = J − 2 / 3 I 1 ; I 1 = λ 1 2 + λ 2 2 + λ 3 2 ; J = det ( F ) I ¯ 2 = J − 4 / 3 I 2 ; I 2 = λ 1 2 λ 2 2 + λ 2 2 λ 3 2 + λ 3 2 λ 1 2 {\displaystyle {\begin{aligned}{\bar {I}}_{1}&=J^{-2/3}~I_{1}~;~~I_{1}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}~;~~J=\det({\boldsymbol {F}})\\{\bar {I}}_{2}&=J^{-4/3}~I_{2}~;~~I_{2}=\lambda _{1}^{2}\lambda _{2}^{2}+\lambda _{2}^{2}\lambda _{3}^{2}+\lambda _{3}^{2}\lambda _{1}^{2}\end{aligned}}} In the limit whereC 01 = C 11 = 0 {\displaystyle C_{01}=C_{11}=0} Neo-Hookean solid model. For acompressible Mooney–Rivlin material n = 1 , C 01 = C 2 , C 11 = 0 , C 10 = C 1 , m = 1 {\displaystyle n=1,C_{01}=C_{2},C_{11}=0,C_{10}=C_{1},m=1}
W = C 01 ( I ¯ 2 − 3 ) + C 10 ( I ¯ 1 − 3 ) + 1 D 1 ( J − 1 ) 2 {\displaystyle W=C_{01}~({\bar {I}}_{2}-3)+C_{10}~({\bar {I}}_{1}-3)+{\frac {1}{D_{1}}}~(J-1)^{2}} ^a b Rivlin, R. S. and Saunders, D. W., 1951, Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber. Phi. Trans. Royal Soc. London Series A, 243(865), pp. 251-288. 
