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Inverse non-linear stress–strain relations between the logarithmic Hencky strain tensor and the Cauchy stress tensor are derived for incompressible isotropic materials. The constitutive model is based on a new type of complementary energy potential that considered to be the function of the second and third invariants of the deviatoric Cauchy stress tensor in a power-law form. A proper scaling of the third deviatoric stress invariant allows separate fitting to uniaxial tension and equibiaxial tension (or uniaxial compression) data. The predictive capabilities of a four- and a six-parameter model are presented through a parameter fitting procedure using Treloar's experimental data for rubber. The performance of the new constitutive model is also demonstrated by solving the inflation problem of a spherical shell. • Complementary energy-based constitutive model in power-law form • The formulation applies the invariants of the deviatoric Cauchy stress tensor • Explicit strain–stress relations between the Hencky strain and the Cauchy stress • The model allows separate fits to uniaxial and equibiaxial measurement data • Numerical solutions and comparisons for the inflation of a spherical rubber shell. [ABSTRACT FROM AUTHOR]