How bending scales: Geometric scaling differentially affects force responses in bending biomechanical structures

The size of an animal's body greatly affects which biomechanical forces (muscle forces, elasticity, viscosity, and inertia) dominate its behavior. By knowing how the relative contributions of these forces scale with the animal's size and geometry, we can make predictions about the neural control required to generate its behaviors and to respond to perturbations. The scaling laws of force-producing and energy-transforming structures utilized in neuromechanics focus on spring-like muscles and elastic structures -- those that generate forces by changing their length. However, many systems generate and transform energy by bending their tissues, that is, changing their curvature. Even though bending structures are common in biomechanical systems, their scaling laws have not been thoroughly studied. In this paper, we investigate how the forces produced by bending structures scale with the size and geometry of an animal and how that scaling differs from standard spring-like muscles and elastic elements. Through analytical and computational investigations, we demonstrate that particular allometries yield different force scaling laws for bending structures. These different scaling laws impact the duration of quasistatic behaviors, may place limits on animal size, and provide material-efficient methods of remodeling in energy-transforming systems. Together, this work demonstrates the differences between bending and straining biomechanical structures and the need for integrative approaches to scaling analysis that account for specific morphologies and deformations, not just a structure's cross-sectional area.