Appendix 6 – Shape Optimisation in Biological Materials
The shape optimisation of a Tiger and Bear claw have been studied in the following paper by C Mattheuk. There is discussion upon the logarithmic spiral as a structurally efficient and materially optimised structure.
Table of Contents
The subject of efficiency in design is well known to engineers, particularly with the most up-to-date computer packages on the market at present which combine finite element analysis with solid modelling to automatically refine parts in order to optimise material utilisation and simplify manufacture.
The introduction to the following paper provides a brief background to the concept that biological structures are adapted to the mechanical loadings to which they are exposed. It is stated that “it probably cannot be proved generally that biological load carriers are optimised” but that this assumption “is widely accepted”.
The logarithmic spiral first appears in literature with regards to describing a grown design in 1917 in “On Growth and Form” by D’Arcy Thompson (see 2.4 Papers of Further Interest). The principle author of the following paper, Mattheuk, has conducted investigations into a “constant stress hypothesis” and he has shown that this “characterises at least all of the biological structures that are made of elastic material as well as those capable of adaptive growth”.
If a structure is in a constant stress state then there are no stress concentrations and no part of the structure is more likely to fail than any other. If a structure is capable of adaptive growth then this means that it can respond to changes in loading by adding or reducing material as required.
2“The Claw of the Tiger: An Assessment of its Mechanical Shape Optimisation”, (1991) C Mattheck and S Reuss 
Written in 1991, this paper examines claws of the tiger and black bear (Ursa americanus) for adaptive growth, predicting that they will exhibit a high degree of shape optimization and that the inner curve of the claw will be the shape of a logarithmic spiral (see Figure 1) .
The researchers proceed along the path of examining the stress fields of different shapes approximating a claw in profile, introducing a load at a point and direction giving the highest bending moment. Figure 3 shows the different shapes that were analyzed for stress using von Mises criterion for equivalent stress. As a reminder, von Mises stress relates the two principle stresses and is normally used for calculations of the failure stress of metallic structures, where:
Mises = sqrt(21 + 22 – 12)
but due to the self-optimisation for biological structures, the stress fields are “nearly always uni-axial”. In other words, the iso-stress lines are parallel with the contours of the shape and so the von Mises stress is the same as the largest tensile or compressive stress present at the point being considered.
The logarithmic spirals are all of the form:
r = a where r = radius
a = initial radius
= constant angle between r and
the tangent at the spiral
Figure 3 shows the four logarithmic spirals used and Figure 4 shows the corresponding stress results. Note that the von Mises stress for a hook that uses circular profiles, such as are conventionally used in engineering (see Figure 2 previously), is shown for a comparison with the logarithmic profiles (the uppermost of the curves in the figure).
Considering Figure 5 illustrating the differences in stress distribution between a circular hook and a shape-optimized hook, it can be seen that the hook of circular profiles (a) features a stress concentration at short of the tip. The logarithmic spiral (b) however features no such stress concentration. These two cases were created using an engineering shape optimizing program called CAO (Computer Aided Optimization) which is able to simulate biological growth. The first design represents the initial state which is then shaped optimized using adaptive growth under loading. The result is (b).
The image of this profile was then superimposed over that of a tiger’s claw (Figure 6). The paper continues to infer from this result that the claws of all carnivores have a logarithmic spiral as profile shapes since it is a highly shape optimised shape.
“The logarithmic spiral is a shape optimized design which leads to a nearly constant stress state at the surface of the claw.”
“Different logarithmic spirals lead only to slightly different stress profiles; however, the existence of an optimum combination of concave and convex contour has to be expected.”
“Any of the logarithmic spirals considered here is superior to a circular arch design which is widely used in engineering.”
The paper concludes with the statement ;
“The FEM model used is only two-dimensional and it only considered the horny part of the claw. However, because of the fibre orientation observed the authors believe that the horny part does most of the work. None the less, a 3-D model would probably give an even more constant stress state.”
“On Growth and Form” D’Arcy Thomson 1917 Cambridge University Press
“Why they grow, how they grow – the mechanics of trees” Mattheuk, C (1990), Arboricultural J 14, 1-17
“Wound healing in a plane (Platanus acerifolia), a proof of its mechanical stimulation” Mattheck C and Korseska G. (1989). Arboricultural J. 13, 211-218
“Engineering components grow like trees” Report No.: KfK 4648 of the Karlsruhe Nuclear Research Center, 21, 143-168