A new model for hyperelasticity

The essential criterion for a good mathematical model for hyperelasticity is its ability to match the measured strain energy curves under different deformations over a large range. One group of models for hyperelasticity is to express the strain energy as a function of I ₁, I ₂, I ₃, the invariants of the right Cauchy-Green deformation tensor. Under the assumption of incompressibility, it can be proved that all valid (I ₁, I ₂) pairs fall in a region bounded by the I ₁ – I ₂ locus from deformations under simple extension and equal-biaxial extension (or, equivalently, simple compression). I ₁ – I ₂ locus from planar extension lies inside the region. Since the strain energy curves from simple deformation modes can be measured from experiments, it is possible to approximately obtain the strain energy under other (I ₁, I ₂) values by interpolating data from the three measured curves. The proposed model for hyperelasticity is an interpolation algorithm with all mathematical details. The new model is implemented into a user-defined material subroutine in commercial FEA software. It can not only accurately reproduce the measured data from these simple deformation modes but also predict the stress–strain curve under planar extension in a reasonable good accuracy even without using the measured data from planar extension.