Apropos Julian's reply to Richard Bonser:Julian's final year projects at
Reading were excellent and no doubt will be worth revisiting.

The question of maths in biomimetics is interesting.Clearly for 'those who
can't count'(as Jim Gordon used to say) there's no point in putting them off
when the outcome can be achieved with little or no maths.But if an
experiment on 'strength',say,is to be more than qualitatively
observational,the data have to be interpreted employing maths, cf. the
reporting of 'breaking loads' without regard to area over which the loads
act etc. I recall Julian comparing the 'state-of-the-art' biochemical
knowledge in the food industry with
its antediluvian approach to mechanical properties----all those
indentation/probe tests mixing up different physical properties all at once,
rather at the street market level of pushing one's thumb into an avacado to
see if it is ripe.

I disagree with Julian that maths in biomimetics is 'a snare and delusion'.
It all depends!Without maths, how would we have interpreted the results from
the instrumented microtome those years ago? The problem, however, is at two
levels: (i) getting those people who have little mathematical training up to
speed in order to understand standard engineering stress analysis; and (ii)
teaching them to be aware that direct application of these standard methods
is often inappropriate in biomimetics for all the reasons highlighted by
Julian---in particular non-linearity and anisotropy.

I guess concepts of stress and strain are straightforward, and providing one
defines how something has been determined (eg what is meant, for a
non-linear elastic material,by what people in papers often call 'Young's
modulus') then that's OK.

The problem is that a little knowledge can be dangerous, and that 'in the
land of the blind,the one-eyed man is king'. Measurement of fracture
toughness well illustrates this point: non-engineers will dip into fracture
textbooks and discover something called the 'critical stress intensity
factor' with those strange units of Pa m^1/2. They may believe this is what
they should be measuring, and go off and do so in materials which are
non-linear and anisotropic, and which may also have suffered gross
irreversible deformation (if they don't periodically unload and reload
before fracture they will not know whether non-linearity in
load-displacement is caused by reversible non-linear elasticity or by
irreversibilities remote from the crack tip). But the effort in calculating
a critical K,and more particularly using that same value predictively
elsewhere, is probably nugatory, because critical stress intensity factors
apply
**only** to small strain,reversible linear elastic isotropic behaviour where
you can refit the broken bits back together after fracture.Fracture
mechanics of linear elastic anisotropic materials is tricky, and fracture
mecanics of non-linear elastic anisotropic
solids is plain difficult!

As we all know, it is easier to employ energy to model fracture and this
works whatever form non-linearities take,**providing** the behaviour is
constrained within the reversible region. As soon as the broken bits cannot
be refitted to regain the original shape and size of the starting testpiece,
some of the work consumed is remote flow,some fracture. How to do the
partitioning---particularly during propagation--is the subject of current
research in mainstream ductile fracture mechanics.(This is so whether you
are trying to solve the problem entirely mathematically---Jc and Jr curves
etc or whether you are trying to interpret the graphical meaning of areas
under load-displacement-crack length diagrams).

Perhaps Julian's mathematical 'snare and delusion' relates to getting odd
results when data are interpreted using the wrong analysis. It is
unfortunate that we don't know all the answers yet for the 'correct'
way of dealing with large-deformation, non-linear, elastoplastic,
anisotropic,time-dependent (and God-knows what else) behaviour of many
natural materials. (These problems, which make biomimetics fun and
challenging, are things that mainstream engineering stress analysts avoid if
at all possible!). Nevertheless, it must be offputting to the
non-mathematician to be told that having learnt all about stress, strain,
toughness etc, some of the (difficult) stuff they've learnt is not
applicable, and that we not fully know what they should be employing.

All of the above should not stop us from pushing on, however.Jim Gordon's,
Julian's and George's pioneering work in interpreting the strength,
stiffness and toughness of biological materials in engineering terms has
shown what can be achieved. Just be careful about crack resistance!

Tony


from Prof Tony Atkins ScD FREng
School of Construction Management and Engineering
Engineering Bldg
University of Reading
READING RG6 6AY
Tel +44 118 931 8562
Fax +44 118 931 3327