Is fracture a bigger problem for smaller animals? Force and fracture scaling for a simple model of cutting, puncture and crushing

Abstract

Many of the materials that are challenging for large animals to cut or puncture are also cut and punctured by much smaller organisms that are limited to much smaller forces. Small organisms can overcome their force limitations by using sharper tools, but one drawback may be an increased susceptibility to fracture. We use simple contact mechanics models to estimate how much smaller the diameter of the tips or edges of tools such as teeth, claws and cutting blades must be in smaller organisms in order for them to puncture or cut the same materials as larger organisms. In order to produce the same maximum stress when maximum force scales as the square of body length, the diameter of the tool region that is in contact with the target material must scale isometrically for punch-like tools (e.g. scorpion stings) on thick targets, and for crushing tools (e.g. molars). For punch-like tools on thin targets, and for cutting blades on thick targets, the tip or edge diameters must be even smaller than expected from isometry in smaller animals. The diameters of a small sample of unworn punch-like tools from a large range of animal sizes are consistent with the model, scaling isometrically or more steeply (positively allometric). In addition, we find that the force required to puncture a thin target using real biological tools scales linearly with tip diameter, as predicted by the model. We argue that, for smaller tools, the minimum energy to fracture the tool will be a greater fraction of the minimum energy required to puncture the target, making fracture more likely. Finally, energy stored in tool bending, relative to the energy to fracture the tool, increases rapidly with the aspect ratio (length/width), and we expect that smaller organisms often have to employ higher aspect ratio tools in order to puncture or cut to the required depth with available force. The extra stored energy in higher aspect ratio tools is likely to increase the probability of fracture. We discuss some of the implications of the suggested scaling rules and possible adaptations to compensate for fracture sensitivity in smaller organisms.