Biomechanics of Hierarchical Elastic Systems
Elastic energy plays important roles in biology across scales, from the molecular to organismal level, and across the tree of life. The ubiquity of elastic systems in biology is partly due to the variety of useful functions they permit such as the simplification of motor control in running cockroaches and the efficient recycling of kinetic energy in hopping kangaroos. Elastic energy is also responsible for ultrafast movements; the fastest movements in animals are not powered directly by muscle, but instead by elastic energy stored in a spring. By demonstrating that the power required to generate ultrafast movements exceeds the limits of muscle, many studies conclude that energy storage is necessary; but, what these studies do not explain is how the properties of a biological structure affect its capacity for energy storage. In this dissertation, I test the general principles of energy storage by investigating elastic systems at three hierarchical levels of organization: a single structure, multiple connected structures, and a spring system connected to muscle. By using a multi-level approach, my aim is to demonstrate, at each of the mentioned levels, how properties of the spring system affect where or how much energy is stored in the system as well as how these conclusions can be combined to inform our understanding of the biomechanics of hierarchical elastic systems.
When considering spring systems at the level of a single structure, morphology is one major structural aspect that affects mechanics. Continuous changes in morphology are capable of dividing a structure into regions that are responsible for the two contradicting functions that are essential for spring function: energy storage (via deformation) and structural support (via resistance to deformation). Using high quality micro computed tomography scans, I quantify the morphology of the mantis shrimp (Stomatopoda) merus, a single structure of the raptorial appendage hypothesized to store the elastic energy that drives ultrafast strikes. Comparing the morphology among the species, I find that the merus in smashers, species that depend heavily on elastic energy storage, have relatively thicker ventral regions and more eccentric cross-sections than spearers, species that strike relatively slower. I also conclude that differential thickening of a region can provide structural support for resisting spring compression as well as facilitate structural deformation by inducing bending. This multi-level morphological analysis offers a foundation for understanding the evolution and mechanics of monolithic systems in biology.
When two or more structures are connected, their relative physical properties determine whether the structures store energy, provide structural support, or some combination of both. Although the majority of elastic energy is stored via large deformations of the merus in smashers, some spearer species show relatively little meral deformation, and it is unclear whether elastic energy is stored in these systems. To determine whether the apodeme (arthropod tendon) provides energy storage in species that exhibit low meral deformation, I measure the physical properties of the lateral extensor apodeme and the merus to which it is connected. Comparisons of these properties show that in the spearer species I tested, the merus has a relatively higher spring constant than the apodeme, which results in the merus providing structural support and the apodeme storing the majority of elastic energy. Comparing the material properties of the apodemes with those of other structures reveals that apodemes and other biological spring systems share similar material characteristics. This study demonstrates that in order to understand the biomechanics of spring systems comprised of connected structures, it is necessary to compare their relative mechanical properties.
Finally, because muscles are responsible for loading spring systems with potential energy, muscle dynamics can affect elastic energy storage in a spring system. Although spring systems can circumvent the limits imposed by muscle via power amplification, they are not entirely independent from muscle dynamics. For example, if an organism has relatively low time to prepare and stretch the spring prior to the onset of movement, the limits of muscle power can dominate energy storage. To test the effects of muscle dynamics on spring loading, I implement a mathematical model that connects a Hookean spring model to a Hill-type muscle model, representing the muscle-tendon complex of the hindlimbs of American bullfrogs, in which the muscle dynamics are well understood and the duration of spring loading is low. I find that the measured spring constants of the tendons nearly maximize energy storage within the duration of in vivo spring loading. Additionally, the measured spring constants are lower than those predicted to produce maximal energy storage when infinite time is available for spring loading. Together, these results suggest that the spring constants of the tendons of American bullfrogs are tuned to maximize elastic energy for small durations of spring loading. This study highlights the importance of assessing muscle dynamics and their effect on energy storage when assessing the functional significance of spring constants.
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