Whole animal measurements of shear and adhesive forces in adult tree frogs: insights into underlying mechanisms of adhesion obtained from studying the effects of size and scale

Abstract

This allometric study of adhesion in 15 Trinidadian tree frog species investigates how relationships between length, area and mass limit the ability of adult frog species of different sizes to adhere to inclined and overhanging surfaces. Our experiments show that hylid frogs possess an area-based wet adhesive system in which larger species are lighter than expected from isometry and adhere better than expected from their toe pad area. However, in spite of these adaptations, larger species adhere less well than smaller species. In addition to these adhesive forces, tree frogs also generate significant shear forces that scale with mass, suggesting that they are frictional forces. Toe pads detach by peeling and frogs have strategies to prevent peeling from taking place while they are adhering to surfaces, including orienting themselves head-up on slopes. The scaling of tree frog adhesion is also used to distinguish between different models for adhesion, including classic formulae for capillarity and Stefan adhesion. These classic equations grossly overestimate the adhesive forces that tree frogs produce. More promising are peeling models, designed to predict the pull-off forces of adhesive tape. However, more work is required before we can qualitatively and quantitatively describe the adhesive mechanism of tree frogs.

Appendix

Derivation of formula for coefficient of friction (μ) that takes account of adhesion

The standard formula for μ is:

$$ \mu = \frac{T} {N} $$

(8)

where T is the tangential force and N is the normal force at the point where sliding begins. Since adhesion contributes to the normal force, this equation must be rewritten as:

$$ \mu = \frac{T} {{N + A}} $$

(9)

where A is the adhesive force, calculated as the force normal to the surface at the angle at which the frog falls from the surface.

$$ \begin{aligned}{} {\text{From}}\;{\text{Fig}}.\,8{\text{a}},\quad T & = {\text{mg }}\sin \theta _{{\text{S}}} \\ N & = {\text{mg }}\cos \theta _{{\text{S}}} \\ {\text{From}}\;{\text{Fig}}.\,8{\text{b}},\quad {\text{ }}A & = {\text{mg }}\cos (180 - \theta _{{\text{F}}} ) = - {\text{mg }}\cos \theta _{{\text{F}}} . \\ \end{aligned} $$

Fig. 8

Diagrammatic representation of tangential (T), normal (N) and adhesive (A) forces acting on a frog (hatched rectangle) at angles between 0º and 90º (a) and 90º and 180º (b). g force of gravity, m mass of frog, θ _F angle of fall, θ _S angle of slip

Substituting values of T, N and A into Eq. 9 we get:

$$ \mu = \frac{T} {{N + A}} = \frac{{{\text{mg}}\sin \theta _{{\text{S}}} }} {{{\text{mg}}\cos \theta _{{\text{S}}} + ( - {\text{mg}}\cos \theta _{{\text{F}}} )}} = \frac{{\sin \theta _{{\text{S}}} }} {{\cos \theta _{{\text{S}}} - \cos \theta _{{\text{F}}} }}. $$

(10)