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

TL;DR

This work examines tensile J-curve materials as a route to decouple mobility (nonlinearity) from protectiveness (stiffness) in antagonistic architectures, inspired by natural ligaments. It introduces a structure–property–performance framework, a toe-heel–linear J-curve characterization with a mobility metric, and a simplified model for how pre-stretch tunes system stiffness. The authors fabricate bio-inspired twisted and coiled artificial ligaments (TCALs) from PU, nylon, and SIS, demonstrate tunable nonlinear tensile behavior across hierarchical designs, and validate antagonistic and rotary configurations that exploit J-curve nonlinearity. They also integrate self-sensing by CNT coatings to enable piezoresistive readouts, broadening the potential for practical bio-inspired robotic applications. Collectively, the study highlights a versatile design space where J-curve materials outperform linear springs in adaptability and function, providing a foundation for robust, tunable, and self-sensing soft robotic components.

Abstract

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.

Figures (6)

• Figure 1: Structure and properties of biological J-curve materials. a) Macroscale musculoskeletal structure of the human arm; b) The microstructures of muscle, tendon, ligament, and skin; c) The nanostructure of fibers in biological tissues; d) The representative tensile J-curve of biological tissues and illustrations for its mechanism; e) The property of ligaments with joint bending schematic; f) The property of tendons with an illustration of their role in the leg joint of a running turkey Roberts1997MuscularWork; g) The protectiveness of the natural ligament under damaging force; and h) The mobility of the natural ligament quantified via the resistance energy.
• Figure 2: Synthetic J-curve materials inspired by natural ligaments. a) Fabrication steps of making twisted and coiled artificial ligaments (TCALs); b) Computer-aided design (CAD) models and optical images of different types of TCALs fabricated with polyurethane (PU) fibers (scale bar is 1 mm). TC: Twisted and coiled, PL: Plied, SC: Supercoiled, and HC: Hypercoiled; and c) The hierarchical architecture ranking of artificial ligaments.
• Figure 3: Mechanical behavior of artificial ligaments. a) Tensile behavior of PU precursor fibers and PU fabricated TCALs; b-c) Experimental tensile loading and unloading of PU-fabricated TCALs. The former is focused on hierarchical TCAL variants, while the latter is only evaluating the coiling force effect on three-plied supercoiled (3PL-SC) artificial ligaments; d) Hysteresis and stress-dependent features of TCAL during four training cyclic tests at two incremental stress levels; and e-f) Tensile behavior of nylon and polystyrene-block-polyisoprene-block-polystyrene (SIS) fibers, as well as 3PL-SC artificial ligaments made by those two materials. Extreme coiling forces are applied to demonstrate tunability.
• Figure 4: Features of J-curve. a) An iteration method that determines the toe region zone and the effective modulus of the whole J-curve; b) J-curve shape evaluation method with the toe-region strain and effective modulus using the coiling force effect on TCAL as an example; c) The definition of mobility to evaluate the nonlinearity of the J-curve; d) The mobility corresponding to several stress levels with experimental results from the linear elastic spring and PU-made TCALs; e) The toe-region strain and mobility of existing synthetic J-curve materials and multi-material TCALs. Non-hatched bars are showing the toe-region strain, while the hatched bars are presenting the mobility; and f) The Ashby chart compares linear elastic materials and J-curve materials using experimental data from all TCAL samples. The yellow region illustrates the range of protectiveness that can be tuned in J-curve materials while maintaining constant mobility, whereas the green region highlights the mobility tunability achievable at a fixed level of protectiveness.
• Figure 5: Principles of use of J-curve materials in antagonistic arrangements. a) Theoretical force-displacement predictions of linear antagonistic structure with & without pre-stretch; b) Schematics of the linear antagonistic structure with & without pre-stretch; c) Results of linear antagonistic test with theoretical and experimental data. Theoretical predictions are plotted in dashed curves (the zero pre-stretch curve is from tensile experiments), and experimental data are plotted in dots. A line of equivalent antagonistic linear spring systems is also given for comparison. The color bar shows the pre-stretch level, and different dot shapes indicate the total tension force $F_{tension}$ of TCALs. The red dashed curve represents the damaging force at different pre-stretch levels; d) The exchange of the protectiveness and mobility values for the linear antagonistic structure with different $F_{tension}$ and pre-stretch strains. Orange solid dots are the change of protectiveness, while purple hollow dots are the alteration of mobility; and e) Schematics of the rotary antagonistic structure with a pivot joint. A normal external force is applied at the end of the lower arm to rotate the lower arm with angle $\theta$. The right side shows the predicted result of the rotary antagonistic structure.