| Summary: | In contemporary elasticity theory, the strain-energy function predominantly relies on
the first invariant, I<sub>1</sub> 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,
I<sub>2</sub>. By shifting the focus towards I<sub>2</sub>, 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 I<sub>1</sub>-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 I<sub>2</sub>-only have interesting properties that are found in biological tissues and are fundamentally different from the traditional
“first-invariant materials”.
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