I too I 2 : A new class of hyperelastic isotropic incompressible models based solely on the second invariant

In contemporary elasticity theory, the strain–energy function predominantly relies on the first invariant, I1 of the deformation tensor; a practice that has been influenced by models derived from rubber elasticity. However, this approach may not fully capture the complexities of materials exhibiting pronounced shear deformations, such as very soft biological tissues. Here, we explore the implications and potential benefits of constitutive models where the strain–energy function is exclusively a function of the second invariant, I2. By shifting the focus towards I2, we aim to address the limitations of current models in accurately describing shear-dominated behaviors and to provide a more comprehensive understanding of material responses, particularly for materials that do not conform to the assumptions underlying I1-centric theories. Through theoretical musings, data analysis, and automated model discovery, we investigate the feasibility of this approach and its consequences for predicting material behavior under various loading conditions. We show that the so-called “second-invariant materials” conforming to I2-only have interesting properties that are found in biological tissues and are fundamentally different from the traditional “first-invariant materials”.