Hyperelasticity models extending Hooke’s law from small to moderate strains and experimental verification of their scope of application

Abstract

Lagrangian formulations of Hill’s linear isotropic hyperelastic material models based on the one-parameter (r) Itskov family of strain tensors (including the Hencky, Pelzer, and Mooney strain tensors generating the H, P, and M material models, respectively) were developed. Explicit basis-free expressions for fourth-order elasticity tensors were obtained for these material models. These expressions use the right Cauchy–Green deformation tensor. The developed formulations of isotropic hyperelastic material models were implemented in the commercial finite element MSC.Marc code. Simple tension stress–strain diagrams were obtained for dog-bone shaped flat specimens made of Duothane QA965 polyurethane. The experimental data were used to fit the parameters E and ν (having the meaning of Young’s modulus and Poisson’s ratio for specimens at small strains) for the H, P, and M material models and the parameters C1 and C2 for the Mooney–Rivlin (M–R) isotropic incompressible material model included in the library of standard material models of the MSC.Marc code. For the investigated material, the H and M material models were shown to adequately reproduce stress–strain diagrams up to 50% elongation, and the P and M–R models up to 150% elongation. In addition, cylindrical specimens were made of the same polyurethane and tested for simple torsion (with constrained edges) and generalized torsion (with edges free to move in the longitudinal direction). In both problems, the resultant moments obtained by computer simulations using the P and M–R material models agree well with the experimental data on rod torsion up to buckling; computer simulations using the H and M material models give less satisfactory agreement with the experimental data. Similar qualitative agreement is obtained between the experimental and simulated resultant axial force and axial elongation; however, in this case, the quantitative agreement is less satisfactory than for the resultant moment.