Modelling biological puncture: a mathematical framework for determining the energetics and scaling

Abstract

Biological puncture systems use a diversity of morphological tools (stingers, teeth, spines etc.) to penetrate target tissues for a variety of functions (prey capture, defence, reproduction). These systems are united by a set of underlying physical rules which dictate their mechanics. While previous studies have illustrated form–function relationships in individual systems, these underlying rules have not been formalized. We present a mathematical model for biological puncture events based on energy balance that allows for the derivation of analytical scaling relations between energy expenditure and shape, size and material response. The model identifies three necessary energy contributions during puncture: fracture creation, elastic deformation of the material and overcoming friction during penetration. The theoretical predictions are verified using finite-element analyses and experimental tests. Comparison between different scaling relationships leads to a ratio of released fracture energy and deformation energy contributions acting as a measure of puncture efficiency for a system that incorporates both tool shape and material response. The model represents a framework for exploring the diversity of biological puncture systems in a rigorous fashion and allows future work to examine how fundamental physical laws influence the evolution of these systems.

Investigation of the rate-mediated form-function relationship in biological puncture

“…2 exhibits overall decreasing trends of , indicating a diminishing puncture performance with increasing values (i.e., decreasing sharpness/slenderness) across a wide range of tested speeds. Underlying this dependence are the changes in the transfer and distribution of the local energy with respect to the variations, as demonstrated theoretically by our previously established puncture energy model 24 . The initial energy investment/loss of kinetic energy due to puncture, , is divided into three energy contributions, i.e.…”

“…1 ). Work on dynamic puncture events (> 1 m/s) has shown that at high speeds strain-stiffening occurs in target materials, particularly when the material is soft and deformable 22 – 24 . Given the shift in material response at high rates of loading, does the relationship between tool shape and puncture performance change at higher puncture speeds?…”

Biomechanics of cutting: sharpness, wear sensitivity and the scaling of cutting forces in leaf-cutter ant mandibles

“…For our experiments with PDMS, G c ≈ 100 J m −2 and t ≈ 200 μm (see Methods), so that F ≈ 20 mN. This simple argument lends itself to a definition of an intuitive, quantitative and functionally relevant index for sharpness, S : the required cutting force is equal to the minimum possible force, and independent of tool geometry, if and if only the dimensionless group S = G c t F −1 is unity; the cutting tool may then be considered ideally sharp (for a conceptually similar suggestion, see [ 101 ]). The fracture forces measured for pristine mandibles of small workers are indeed very close to this theoretical minimum ( figure 2 b ), suggesting that a further reduction in cutting force through changes in mandible morphology may not be possible.…”

“…However, a robust definition of sharpness as such is not a trivial task, as suitably illustrated by the large number of sharpness metrics suggested in the literature (e.g. [13,14,17,42,74,[98][99][100][101]).…”

Experimental observation of near-wall effects during the puncture of soft solids

“…3 Post-puncture regime Analysis of deep indentation and puncture in the post-puncture regime is an active area of investigation. [22][23][24][27][28][29] The current understanding of crack morphology during deep indentation and puncture is that it relates to the indenter tip geometry, materials properties, distribution of materials defects, and large strain stiffening behavior. 21,30 Currently this link is qualitative and prohibits researchers from predicting the crack morphology without making measurements.…”