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We find the strain energy function for isotropic incompressible solidsexhibiting a linear relationship between shear stress and amount of shear, andbetween torque and amount of twist, when subject to large simple shear ortorsion deformations. It is inclusive of the well-known neo-Hookean and theMooney-Rivlin models, but also can accommodate other terms, as certainarbitrary functions of the principal strain invariants. Effectively, the extraterms can be used to account for several non-linear effects observedexperimentally but not captured by the neo-Hookean and Mooney-Rivlin models,such as strain stiffening effects due to limiting chain extensibility.

Strain energy function for isotropic non-linear elastic incompressible solids with linear finite strain response in shear and torsion