Principles of Use of Tensile J-Curve Materials in Antagonistic Arrangements

Natural ligaments are soft connective tissues that must simultaneously provide high stretchability to enable dexterous flexibility and high stiffness to protect the musculoskeletal system. These two functions cannot be independently tuned in conventional engineering materials with linear or hyperelasticity. Ligaments achieve this balance through a highly nonlinear tensile response characterized by a J-shaped curve, featuring an extended "toe region" of low force up to intermediate strains followed by an inflection, called the "heel region" which marks the onset of nonlinear stiffening. Here, we present a framework for characterizing the defining features of J-curve behavior. Based on these features, we define measures for protectiveness and mobility to quantitatively describe the effective stiffness and the level of nonlinearity, thereby elucidating how the J-curve enables decoupled fine-tuning of flexibility and damage protection. A simplified mathematical model, supported by experimental validation, reveals the performance advantages of J-curve materials in antagonistic arrangements and highlights their unique design space compared with linear elastic systems. Furthermore, we develop synthetic J-curve materials capable of self-strain sensing via piezoresistive transduction, enabling their integration into practical devices. Collectively, these materials, models, and insights advance the understanding of nonlinear mechanical mechanisms in natural systems and provide a foundation for harnessing J-curve behavior in engineering applications such as bio-inspired robots.