Non-hookean mechanics of a random spring

Random elastic structures, from polymer molecules to crumpled sheets, are common across numerous fields and applications. Understanding their mechanics is of broad interest, especially for creating programmable responses that are robust to manufacturing imperfections. In this work, we explore the mechanical properties of a random variant a fundamental elastic structure: a random spring. Our model consists of discrete chains of links connected by joints, where the local tangent follows a random walk sequence. Each joint has a linear torsional hinge, creating a unique force response under traction. Through numerical analyses, we observe a universal transition from a linear to a quadratic force-strain relationship at a critical strain that varies with system size. Remarkably, in the thermodynamic limit, the linear regime disappears entirely. Experimental studies using a rigid random profile and origami structures support our findings. We provide a theoretical framework that describes both the discrete random chain and its continuum analog, the noisy elastica, predicting this characteristic non-linear behavior. Finally, we extend our model to demonstrate how the initial curvature statistics can be harnessed to program the non-Hookean response of these random springs, enabling tunable mechanical performance.