Table Of ContentdedicaTion
For Philip, Richard and Ruth
Misce stultitiam consiliis brevem:
Dulce est desipere in loco
Horace 65–8 BC
ODES (book 4, poem 12 (a poem on the pleasures of Spring), lines 27-8)
Mix a little foolishness with your serious moments
Silliness in its place is charming
(i.e don’t be po-faced all the time)
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Preface
This book is about the deliberate or accidental cutting of all sorts of materials from ‘soft,
compliant and weak’ to ‘hard, stiff and strong’. Examples are drawn from the engineering,
physical and biological sciences, history and archaeology, palaeontology, medicine, veterinary
medicine and dentistry, and food technology. It is a very broad canvas over which I attempt
to demonstrate that separation of one part from another and the factors controlling separa-
tion – the mechanics of cutting – are basically the same, whatever the field and whatever the
material. There is much to learn by looking at how other people do similar tasks. I am an
engineering scientist, interested in why things break, with limited knowledge in biological and
other fields, but I have been fortunate for many years to collaborate with members of the
Centre for Biomimetics at the University of Reading. Biomechanics is the study of the physical
properties of all biological materials (including foodstuffs), and their employment in under-
standing biological design and function. The subject should not be confused with the comple-
mentary field of biomedical engineering whose employment of physical properties of human
body parts relates to specific replacements of limbs and organs. Biomimetics (‘mimicking
nature’) is inspired by biology and applies biomechanical knowledge to the manufacture of
new engineering materials and devices. It may be said, perhaps, that biomimetics was invented
at Reading by my predecessor, Jim Gordon, and carried on by George Jeronimidis, Julian
Vincent (now at the University of Bath) and Richard Bonser. Professor Gordon also built up
contacts with archaeologists, classicists and historians, and that has continued in Engineering
at Reading with our group of archaeometallurgists (Henry Blyth, Eddie Cheshire, David Sim
and Alan Williams). Having been fortunate to study under Charles Gurney in Cardiff and
David Tabor in Philip Bowden’s Laboratory for the Physics & Chemistry of Solids in the
Cavendish Laboratory at Cambridge, I am receptive to this sort of wide thinking. The pio-
neering work on metal cutting at the University of Michigan, Ann Arbor, also rubbed off on
me during my tenure in Mechanical Engineering there, as did discussions with Wallace Hirst
and Gerry Hamilton at Reading on contact mechanics and wear, for which I am grateful.
One difficulty in writing an interdisciplinary text is that what may be elementary and well
known to a worker in one field is completely strange to others. I have had to have a dictionary
ix
x Preface
next to me when reading non-engineering papers, and have pestered very many colleagues
across the world for clarification of lots of topics. I owe a great debt to all those who have
advised me on the disparate topics covered in this book and corrected many of my misconcep-
tions. Geoff Rowe once said that I had a habit of ‘setting crossword puzzles for tired minds’.
At one stage it seemed that the list of acknowledgements would be longer than the text. Even
so, any mistakes in this book are entirely down to me. Even within engineering, some branches
have a different language from another: the subject of soil mechanics (geomechanics), for
example, developed separately from solid mechanics. Again, I have had to face the problem
that different fields use different symbols and different nomenclature for identical things.
Some ideas about cutting require mathematics for explanation, and biological scientists
and others do not always have the mathematical background of physical scientists. The math-
ematics in this book is not advanced but, to help those to whom modelling and mathematical
derivations may appear mysterious, I have tried to be as basic as possible. Detailed workings
may be found in original papers. Equally, specialists in the biological and other fields may no
doubt consider that I have over-egged the pudding in many places regarding their subjects.
The author’s own inadequacies will be evident to all.
Sir Charles Inglis, in the Proceedings of the Institution of Mechanical Engineers in 1947,
said that:
‘… Mathematics [required by engineers] though it must be sound and incisive as far as
it goes, need not be of that artistic and exalted quality which calls for the mentality of
a real mathematician. It can be termed mathematics of the tin-opening variety, and in
contrast to real mathematicians, engineers are more interested in the contents of the tin
than in the elegance of the tin-opener employed …’.
In this book we hope to discuss how the tin-opener itself works with such simple mathematics.
The reader will become aware that there is, perhaps, less maths in some of the later sections of
the book that deal with biology, palaeontology, medicine, food, etc. With notable exceptions,
there are fewer workers performing instrumented experiments to provide data to assess the role
of cutting and its interaction with biological microstructure. Sometimes the level of taxonomic
sampling is not yet wide enough to determine the role played by cutting in biological design and
function. Many exciting experiments are waiting to be performed and analysed.
Unfortunately, there is no space for interesting topics from literature, mythology, art and
so on that involve cutting of various sorts: driving of stakes through the chests of Dracula’s
victims; King Arthur’s magical sword Excalibur; St George slaying the dragon; Beowulf’s cut-
ting off the head of Grendel’s mother; the sword of Damocles; Odysseus killing the sleeping
giant Polyphemus by driving a stake through his eye; Hercules’ second labour was to kill the
Hydra, a monster with nine heads where, when one was chopped off, two grew back in its
place; the oldest of the classical Greek Fates Atropus cut the thread of Lachesis’s life with her
shears (Lachesis was the Fate who spun life’s thread and determined its length); the courtly
love of Lancelot and Guinevere (King Arthur’s wife) by Chretien de Troyes from the twelfth
century involved crawling over a bridge made of swords. A knight with his ‘sword and buck-
ler’ is found in poetry. A buckler is a shield with a boss (from the French bocle, a boss).
The word swashbuckler to describe a swaggering bully comes from swash, the noise made
by swords clashing or a sword beating on a shield. In Kipling’s The Glory of the Garden, we
read that ‘… better men than we go out and start their working lives at grubbing weeds from
gravel paths with broken dinner knives …’.
Museums around the world have collections of cutting instruments or illustrate tools and
cutting in various guises: the knife grinder in the Octagonal Room (Tribuna) at the Uffizi
Gallery in Florence; among the lozenge-shaped panels from the old campanile now in the
Cathedral museum in Florence, the panel entitled Logic by Gino Micheli da Castello shows
Preface xi
shears, and an image of David and Goliath appears in relief in the door of the Duomo.
A billhook representing winter cutting of wood may be found in the pediment over Minerva’s
temple at the Roman baths in Bath. Among paintings that show cutting is David Teniers the
Younger’s Interior, Old Woman Peeling Apples at the Fitzwilliam Museum in Cambridge. A
strange pair of scissors that will not function is to be found in Salvador Dali’s painting entitled
Guillaume Tell: the blades of the scissors in his left hand cannot close up because the thumb
and forefinger are already together when the blades are still open. Caillabotte’s Les Raboteurs
de Parquet shows workmen finishing a wooden floor by scraping. A lot of arms, armour and
decapitations (Medusa; John the Baptist) have been painted and sculpted over the centuries. In
an interactive exhibition of modern art in the 1960s Yoko Ono invited people to take part and
cut bits off her dress. Man Ray’s Cadeau at the Pompidou Centre in Paris has fourteen carpet
tacks stuck in line along the axis of the lower surface of a ‘Gendarme’ flat iron.
It is curious that teeth in a number of sculptures of heads are not always represented accu-
rately. In the Louvre, for example, there is a stone lion dating from 350 BC from a cemetery at
Glyphada (near modern Athens): the teeth are ‘human’ dentition, not animal. The same is true
of a number of ceremonial lions guarding doors in China. There is also an oil painting entitled
Surprised! by Henri Rousseau at the National Gallery in London showing a tiger in the jungle.
Are the teeth correct?
The cutting of metals is commercially important and there are many admirable books on the
subject. There are also books devoted to cutting of wood and plastics (polymers). They con-
tain much practical information and detail of industrial processes that it has not been possible
to include here. Apart from numerous empirical formulae for cutting forces, cutting energy
and so on, most models of cutting for different materials in such monographs follow what is
to be found in the metal-cutting monographs, by and large, namely that the work required
for cutting comprises two components: (i) plastic or other irreversible work in forming the
offcut or chip; and (ii) work done against friction. Cutting is different from other deforma-
tion problems in elasticity and plasticity since after cutting, a single starting body has been
separated into a number of entirely separate bodies that are no longer ‘attached’ to the parent
body. The work for separation is absent in traditional models of metal cutting because it was
believed that it is insignificant. That view is challenged in this book, based on tracing the his-
tory of the assumption in original papers (the use of the chemical surface free energy rather
than the fracture toughness). Central to the current theme is the idea of separation of parts
and that cutting is a branch of elastoplastic fracture mechanics. The cutting of floppy and brit-
tle materials principally concerns fracture: the specific work of separation is not negligible, and
calculations for forces and power consumed employ the fracture toughness of the material,
not the surface free energy. When significant work of separation is incorporated in analyses for
the cutting of ductile materials, as well as the customary plasticity and friction, a number of
experimental observations for which the traditional treatment has no explanation, now make
sense. Because the work of separation takes place in thin boundary layers contiguous with the
cut surfaces, the separation work and remote work are essentially uncoupled, which is why
flow fields in the cutting of ductile materials may be estimated without reference to fracture
work. However, when cutting forces and power are to be determined, consideration of separa-
tion work is necessary.
The classification of materials into ductile and brittle is based on the behaviour of laboratory-
size testpieces. Yet brittle glass can be machined at micrometre depths of cut and ductile steel
behaves in a brittle fashion in large sizes. Such behaviour reinforces the concept that cutting is a
branch of elastoplastic fracture mechanics because it is a manifestation of the cube-square scal-
ing inherent in fracture mechanics, where there is competition between energies dependent on
volume and dependent on area. In cutting a given material, ductile chips are produced at very
small depths, but as the thickness of the slice is increased the behaviour becomes less ductile,
xii Preface
so that eventually splits form. This is common experience in wood. Ductility (called tenacity in
old papers), as well as a material’s strength or hardness, had always been known to influence
the ease or difficulty of cutting, but its role was never properly quantified. The use of frac-
ture toughness, as well as yield strength, simply makes that connexion. Both these mechanical
properties may be altered independently of each other by different thermomechanical or other
treatments so that different samples of the same material may have the same hardness but quite
different toughnesses. It is to be expected that they will cut differently.
Energy methods are employed throughout this book. They permit solutions of problems
that might otherwise be difficult: for example, the explanation why cutting with a knife is
always easier when sideways motion is coupled with simple pressing-down, and hence the
importance of the ‘slice–push ratio’ in cutting. Materials are considered as continua having
reproducible mechanical properties, but microstructure is discussed where appropriate. It
comes down to the ‘magnification’ at which a material is being looked. Concrete and salami
to the naked eye are coarse heterogeneous microstructures; other solids require inspection
under the microscope to see the constituents. All biological materials are hierarchical com-
posites of one sort or another and, in trying to understand behaviour, it is important to know
the level at which particular properties are controlled.
The subject and sources of information are scattered over many disciplines and published
in a bewildering number of journals. Whole books have been written on just parts of the
subject. While I hope I have recorded important papers, I expect that I have missed a number,
and can only apologize to the authors. There are far too many references to quote even a
small selection. My choices of papers from a given author/school of work are, to an extent,
arbitrary, but it is possible to trace other publications from the papers that are referred to.
In addition to historically interesting references, some of the ‘working’ references are quite
old. I make no apology for that. Early researchers thought carefully about what they did and
experimental work was carefully and painstakingly done, often under difficult circumstances:
insensitive load and displacement measurement devices; no image recognition schemes for
flow fields; no data acquisition and manipulation software; algebra rather than finite element
methods (FEM), and so on. It seems to be a disappointing trend that some young research-
ers are not familiar with the old literature, and know only about things that can be down-
loaded from a search engine. In consequence, they sometimes reinvent the wheel, and often
have the view that if they have used FEM or similar techniques, then it must be all the better
for it. What is possible with modern elastoplastic computational models is, of course, truly
remarkable, but FEM is not a substitute for thinking and experimentation. Furthermore,
FEM requires physical property inputs and they come from experiments. Calibration of FEM
models has to be done with care: while computational simulations can explain the results of
experiments too complicated to be modelled by simple algebra, the real success comes when
FEM is able to predict events ‘blind’.
Acknowledgements
Many people have helped and advised me on this book. I must mention David Wyeth, Eddie
Cheshire, and all former research and project students; Richard Bonser, Richard Chaplin,
George Jeronimidis, Tony Pretlove and other colleagues at Reading; John Frew is especially
thanked for experimental assistance over many years. Peter Lillford, Peter Lucas, Gordon
Sanson and Julian Vincent have patiently fielded e-mails from me seeking clarification on
the biological side. Gordon Sanson has read drafts, made the most helpful comments and
prevented my making many biological howlers. What is written down though is entirely my
responsibility and I hope that there are few errors.
Other people to whom I owe thanks are:
Julian Allwood, Hilary Arnold-Baker, Daniel Balint, Dick Bassett, Roger Bentley, Henry
Blyth, Malcolm Bolton, Roy Brigden, Brian Briscoe, Andy Brunner, Tim Burns, Byron Byrne,
Peter Chamberlain, Maria Charalambides, Chen Zhong, Tom Childs, Brian Cotterell, Matt
Davies, John Dempsey, Coen Dijkman, Peter Dunn, Caroline Ellick, Bill Endres, Roland
Ennos, David Felbeck, Paul Fenne, Tony Gee, Giacomo Goli, Roger Hamby, Bryan Harris,
Linda Holland, Ian Horsfall, Ian Hutchings, Norman Jones, Dirk Keeley, Kevin Kendall,
Tony Kinloch, Raja Kountanya, Hans Kruuk, Brian Lawn, Ming Li, Ken Ludema, Yiu-Wing
Mai, Remy Marchal, Adrian Marshall, Paulo Martins, Shelagh McKay, Roy Moore, Sue
Mott, Barbara Murray, John Nairn, Kazimierz Orlowski, Andrew Palmer, Lucy Peltz, Gill
Pittman, Tracey Popowics, Charles Preston, Tony Pretlove, Richard Rahdon, Jenny Read,
Steve Reid, Pedro Reis, Peter Roberts, Liz Robertson, Benoit Roman, Pedro Rosa, Marco
Rossi, David Sim, Gerhard Sinn, Stefi Stanzl-Tschegg, Roger Stewart, David Stirling, Hew
Strachan, Frank Tallett, Bernard Thibaut, Michael Thouless, Chris Tufnel, John Videler,
Julian Vincent, Stephen Walley, Celia Watson, Shelley Wiederhorn, Tomasz Wierzbicki, Alan
Williams, Gordon Williams, John Williams, Xianzhong Xu and John Yeo.
xiii
ChaPter 1
Controlled and Uncontrolled Separation of Parts
Cutting, Scraping and Spreading
The design of structures and components in nature and in engineering usually aims to avoid
fracture – at least during life – but there are circumstances where separation of parts is
required. These range all the way from the beast of prey tearing apart its victim with teeth
and claws, to the manufacture of a precision surface in metal using special cutting tools.
Some processes of separation rely on pulling, bending or twisting an object at regions remote
from where the object breaks; others load right at the zone of fracture and this includes cut-
ting. Processes of ‘separation’ that are not cutting include pulling corks out of bottles. Some
processes that are thought to be cutting are really not: it is a common misconception that ice
breakers cut ice fields by splitting; rather, they ride up on the edge of the ice sheet and break
pieces off by bending fracture.
Separation of materials is all around us: in the kitchen (e.g. carving meat, coring apples,
grating cheese, peeling vegetables), when eating (on the dinner plate, in the mouth), in carpen-
try and building (e.g. sawing, planing and drilling wood; cutting bricks and paving stones), in
the office (e.g. paper guillotining and shredding, pencil sharpening), in manufacturing (e.g. all
metal-cutting operations), in agriculture (e.g. ploughing, harvesting of crops, sheep shearing),
in medicine and dentistry (surgery; the drilling of teeth), in nature (hunters, raptors, their
prey and defences; teeth and chewing), in shaving, in opening packaging and in war (arms
and armour: a spear through ancient armour, depleted uranium missiles through modern tank
armour). While, usually, the cutting tool remains undeformed, in the latter field both cutter
and target are deformed. ‘Cutting’ is interpreted very broadly in this book, but even so we
do not consider flame cutting, liquid jets, abrasive water jet cutting, laser cutting, plasma arc
cutting, electrodischarge machining and electrochemical machining.
Different materials respond differently when cut with a knife, well illustrated by the wide
variety of foodstuffs that includes mashed potatoes, boiled potatoes, uncooked potatoes;
cooked and uncooked vegetables; stringy vegetables like celery; squidgy food like blancmange
or tofu; boiled sweets, fudge and sugar; easy-to-chew high-quality meat, or poor-quality
meat with lots of gristle; soft puddings like icecream, or hard puddings like toffee; some are
mixtures of hard and soft (crème brûlée); chocolates may have a hard case with a soft inside.
Properties may change with time and storage: some fruit has to be stored after picking before
it becomes ripe enough to eat. Fresh food and stale food behave differently: when freshly
harvested, foods such as carrot or celery are hard and stiff owing to the turgor pressure that
pressurizes the composite structure from within (from the Latin for ‘to swell’); turgor pressure
is a plant’s internal stressing to keep it erect, among other things. Turgor pressure decreases
with time after harvesting, making fruits and vegetables flaccid (from the Latin for flabby)
when they become rubbery and bendy. Loss of turgor pressure is why flowers wilt. The con-
dition of food affects how they are dealt with on the plate, their ‘mouth feel’ and how we bite
and masticate food.
A wide variety of different types of implement is found, ranging from butcher’s knives to
cheese graters. Kitchen shops offer strange and ingenious gadgets with pointy bits for doing
special cutting jobs in the preparation of food. One of the most exotic, perhaps, is a foie-gras
Copyright © 2009 Elsevier Ltd. All rights reserved.
2 The Science and Engineering of Cutting
cutter. The Swiss have different slicers for potatoes (to prepare roesti) and for apples (for
muesli). A mandolin is a device like a wood-plane over the blade of which foodstuffs are sliced,
grated or shred depending on the blade. A hachoir is a rocking device for cutting up herbs
(from the French hache for axe; hence hatchet). In antique shops may be found old devices
such as sugar cleavers (in Victorian times, sugar used to come in big lumps), mechanical apple
peelers, nutmeg grinders and so on.
Experiments in the kitchen can be very instructive about cutting, and the reader is encouraged
to do so and get a feel for stiff/compliant, strong/weak, tough, etc., materials. For example,
scrape a carrot with a knife and notice the difference depending on the angle of the blade. What
controls the depth of cut in a potato peeler? Why is peeling with a knife more wasteful? Can
you skin an orange with a potato peeler? Indeed, can you shave with a potato peeler? Are
there differences depending on whether the fruit is hard and stiff, or soft and squidgy? What
determines the ease of scraping up a portion of butter on to a knife from a block, or a scoop
of icecream? What are those serrations that appear on the back of the butter after scrap-
ing? What determines the ‘spreadability’ of butter on toast? What is the best way to take the
top off a boiled egg? What are those cracks that appear having scraped the back of a spoon
across the surface of a table jelly? These are not flippant questions or suggestions: the answers
are central to understanding of the mechanics of cutting.
When we eat with the aid of a knife, fork, spoon, chopsticks or fingers, we often separate
(fracture) food into smaller pieces to fit the mouth, where further deformation and fracture takes
place before swallowing. Why is it easier to cut when we ‘slice’ across the food as well as simply
‘press down’? Food on the plate will have been previously prepared from larger pieces and/or
cooked to make eatable and digestible. Cooking alters the properties of food and distinguishes
humans from other creatures. To tell whether potatoes or other vegetables have been cooked
for the requisite time, we stab the vegetable in the saucepan with a knife and see how easy it is
to pierce, or see whether it can be suspended from the knife. The altered properties revealed by
the knife must connect with perception in the mouth and what, for example, al dente means.
Similarly, to get food from plate into mouth, we often pierce, indent or perforate the food with
the prongs of a fork, the mechanics of which are similar to nailing a piece of wood.
Cutting may concern big pieces being separated into two or more still-big pieces (sawing
logs of timber into planks, slitting metal sheets off rolling mills, cutting wedges from ‘rounds’
of cheese, cutting fruit into segments). In other examples, thin slices or chips are removed
from the surface of a larger piece (peeling potatoes, whittling wood, lathe cutting, carving).
Sometimes the piece cut off is important (wood veneer, microtomed sections for histological
examination); at other times the piece left behind is important (true of most manufacturing
processes where the offcut ‘swarf’ is scrap, trimming the edges of bound books); and some-
times both are important (the division of paper sheets into smaller sizes or the slicing and dic-
ing of semiconductor wafers). Sometimes the quality of the resulting surfaces is of particular
concern (limits and fits in engineering assembly) but sometimes it does not matter (chopping
firewood). Sometimes the same mechanism of cutting may be both undesirable in one situa-
tion, yet beneficial in another (erosion versus sandblasting)
Different types of cutting include:
l cutting layers or slices from the surface or edge of a body
l cutting a groove in the surface of the workpiece
l dividing a workpiece into sections by cutting through the thickness
l making profiles (e.g. round shapes on a lathe)
l making some sort of hole down into, or though, the thickness by penetration and perforation.
