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The conch shell as a model for tougher composites

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The conch shell is 95% by volume CaCO3 (chalk), yet it's toughness is 103 times greater than that of monolithic CaCO3. In this review paper we look at how this increase in toughness is achieved and what lessons can be learnt for designing new tough composites. Essen- tially, we nd that the CaCO 3 is nely divided into single crystals whose relevant dimensions are below the Grith aw size for the anticipated stresses. Thus upon failure intergranular cracking dominates. Furthermore, failure is encouraged to proceed in a controlled way, which frustrates crack growth and maximises crack surface area. This strategy of maximising damage can only be successful in combination with self-healing properties. Examples are given of synthetic analogues, so-called biomimetic materials.
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Int. J. Materials Engineering Innovation, Vol. 2, No. 2, 2011 149
Copyright © 2011 Inderscience Enterprises Ltd.
The conch shell as a model for tougher composites
D.M. Williamson* and
W.G. Proud
SMF Group, Cavendish Laboratory,
University of Cambridge,
Cambridge CB3 0HE, UK
Email: dmw28@cam.ac.uk
Email: w.proud@imperial.ac.uk
*Corresponding author
Abstract: The conch shell is 95% by volume CaCO
3
(chalk), yet its toughness
is 10
3
times greater than that of monolithic CaCO
3
. In this review paper we
look at how this increase in toughness is achieved and what lessons can be
learnt for designing new tough composites. Essentially, we find that the CaCO
3
is finely divided into single crystals whose relevant dimensions are below the
Griffith flaw size for the anticipated stresses; thus upon failure intergranular
cracking
dominates. Furthermore, failure is encouraged to proceed in a controlled
way, which frustrates crack growth and maximises crack surface area. This
strategy of maximising damage can only be successful in combination with
self-healing properties. Examples are given of synthetic analogues, so-called
biomimetic materials.
Keywords: nacre; Griffith flaw; self-healing composite; biomimetic; light-
weight armour.
Reference to this paper should be made as follows: Williamson, D.M. and
Proud, W.G. (2011) ‘The conch shell as a model for tougher composites’,
Int. J. Materials Engineering Innovation, Vol. 2, No. 2, pp.149–164.
Biographical notes: D.M. Williamson studied Physics at degree and PhD level
in Loughborough University (2002) and University of Cambridge (2006). He is
currently post-doctoral researcher at University of Cambridge, Cavendish
Laboratory, investigating physically based descriptions of material properties,
and in particular, composites.
W.G. Proud obtained his degree and PhD from the University of Newcastle
upon Tyne in 1987 and 1990 in Biosenors. He was post-doctoral researcher
on corrosion protection at the University of Barcelona, Spain (1990–1992).
He conducted and directed research in the area of shock waves, energetic
and ballistics in the Cavendish Laboratory, University of Cambridge during
1994–2009.
He was also research fellow of Clare Hall Cambridge (1997–2000).
He was appointed Reader in Shock Physics, Imperial College London, in the
Institute of Shock Physics in October 2009.
This paper is a revised and expanded version of a paper entitled ‘The Conch
shell as a model for tougher composites’ presented at ‘LWAG 2009 Conference
on Security and Use of Innovative Technologies against Terrorism’, Aveiro,
Portugal, 18–19 May 2009.
150 D.M. Williamson and W.G. Proud
1 Brief introduction to classical fracture
Before we consider what we mean by toughness, it is useful to consider some key results
from the field of fracture mechanics.
There are two seemingly different approaches to fracture mechanics, which can be
shown to be mathematically equivalent; stress- and energy-based criteria for failure
(discussed in following sub-sections).
1.1 The stress-based approach
The first approach to be considered was the stress intensity approach developed by
Inglis (1913), who determined that cracks, as the limiting case of sharp ellipses, and
sharp corners, weaken a plate of material by magnifying remotely applied stresses to the
levels required for bond rupture.
Within a plate, consider an elliptical hole of major axis 2c and minor axis 2b, such
that the major axis lies perpendicular to a remote tensile stress
σ
L
.
In the Inglis analysis the remote tensile stress is magnified by the sharp tips of the
ellipse by an amount
σ
C,0) =
σ
L
(1 + 2c/b), (1)
which in terms of the radius of curvature
ρ
= b
2
/c is
σ
C,0) =
σ
L
(1 + 2(c/
ρ
)
½
). (2)
When b<<c then equation (2) reduces to the approximation
σ
C,0) 2
σ
L
(c/
ρ
)
½
. (3)
From equation (1) we conclude that circular hole will magnify a remote stress in ideally
brittle materials by a factor of three. Elliptical holes magnify the stresses at either end
of their major axes to a greater extent; when the ratio of major to minor axis is equal
to 100, the tensile stress is raised 200 times. When the ratio is 1000, the stress is raised
2000 times. Inglis claimed that: “the ellipse in this latter case would appear as a fine
straight crack, and a very small pull applied to the plate the across the crack would set
up a tension at the ends sufficient to start a tear in the material. The increase in the
length due to the tear exaggerates the stress yet further and the crack continues to spread
in the manner characteristic of cracks”.
Inglis concluded that the dominant cause of stress magnification was not so much the
length of the ellipse, but its sharp tip.
1
As such, equation (3) can be used to estimate the
stress concentration in many scenarios, where c and
ρ
represent a characteristic length
and radius of curvature.
The Inglis description, whilst doubtlessly insightful, is a mathematical one, of single
macroscopic
cracks in an idealised elastic material. Whilst his ideas did strongly influence
ship design,
2
it was Griffith (1920) who built upon these ideas, and suggested that
the reason real materials did not achieve their theoretical strengths was because of the
inherent presence of flaws, each of which magnifies the applied stress in a manner
described by the Inglis analysis, leading to premature failure. Later studies showed that
flaws are indeed responsible, and that their size populations are governed by statistical
functions, e.g. the Weibull distribution (Weibull, 1951). Griffith pointed out that the
effective strength of technical materials might be increased many tens of times if these
flaws could be eliminated.
The conch shell as a model for tougher composites 151
1.2 The energy-based approach
Importantly, Griffith also invoked the theorem of minimum energy to describe crack
propagation. He stated that: “the equilibrium state of an elastic solid body, deformed
by specified surface forces, is such that the potential energy of the whole system is a
minimum. The new criterion of rupture is obtained by adding to this theorem the
statement that the equilibrium position, if equilibrium is possible, must be one in which
rupture of the solid has occurred, if the system can pass from the unbroken to the broken
condition by a process involving a continuous decrease in potential energy”. Or else, in
slightly less precise terms, a stressed body will relieve itself of stored elastic energy by
creating new surface energy, if it may do so without increasing its overall potential
energy.
However in relieving itself of strain-energy, a stressed body below its elastic limit
may not simply spontaneously rupture to create new surface where there was none
before. Griffith pointed out that to do so would actually increase the potential energy,
since work must be done against those attractive forces which act over a small distance,
that Griffith described as the ‘radius of molecular attraction’, to bind the molecules
together. In general the surfaces of a spontaneously formed rupture would not be at a
distance greater than the radius of molecular attraction, and the process would not meet
the criterion of one involving a continuous decrease in potential energy. An important
exception to the above statement is the situation involving the presence of a pre-existing
crack, or flaw, within a body whose width is already greater than the radius of molecular
attraction. In such a scenario only the tips of the crack are subject to the molecular
attractions, and moreover being small, the increase in potential energy associated with
driving them forward when the crack propagates is negligible.
3
In such a scenario the
crack may advance under a state of equilibrium, or else such that the total potential
energy is diminished. Or else, once again in slightly less precise terms, an elastically
loaded body cannot spontaneously rupture to reduce its potential energy, but a pre-
existing flaw or crack may extend under equilibrium conditions, or else act to diminish
the overall potential energy by converting stored strain energy into new surface energy.
If we let the U represent the totality of the stored energy of the system, composed of
the sum of the work done by applied loads in causing global displacement W
L
(which acts
to reduce the potential energy), the strain energy U
E
and the surface energy U
S
, then
U = –W
L
+ U
E
+ U
S
. (4)
If c is the length of the crack, then the mathematical expression for the condition of crack
advance as described above is given by
dU/dc = 0, (5)
which is the famous ‘Griffith energy balance’ equation.
It also follows that if equation (5) is satisfied, the strain energy U
E
is turned into new
surface energy γ:
U
E
= 2c γ. (6)
Equation (6) was experimentally verified by Griffith in his 1920 paper by comparing the
computed fracture surface energy of glass to that of the surface energy computed by more
conventional means, and achieving order of magnitude equality.
152 D.M. Williamson and W.G. Proud
To calculate the fracture surface energy, Griffith imagined the scenario of a flaw in a
plate, exposed to a remote constant stress
σ
L
. He then used the Inglis analysis to deduce
the stain energy density in the vicinity of the crack tip:
U
E
= π c
2
σ
L
2
/E plane stress). (7)
Making use of the fact W
L
= 2U
E
for such a system, and that U
S
= 4c γ, equation (4) may
be written specifically as
U = –π c
2
σ
L
2
/E + 4cγ, (8)
for which the application of condition (5) allows us to deduce that the critical condition
for fracture is
σ
L
= (2Eγ/πc)
½
, (9)
which is the famous ‘Griffith flaw’ equation for a 2D plate.
4
Here for the first time
was insight into why large flaws are more detrimental than small ones, an observed
phenomenon which had hitherto gone unexplained.
Equation (9) also reveals why materials are weaker than their theoretical strengths.
Orowan (1948) pointed out that, on the basis of simple Hookian behaviour up to failure,
that we might expect the strengths of perfect materials to be the order of (2Eγ/a)
½
, where
a is the inter-atomic spacing. The ratio between Orowan and Griffith strengths is (c/a)
½
,
which, if we take c ~ 1 μm and a ~ 1 Å, is equal to 100; which is to say materials with
even micron sized flaws are approximately 100 times weaker than their theoretical
strengths.
The equilibrium condition is an inherently unstable one, since once the applied stress
exceeds the critical level given by equation (9), the crack is free to propagate without
further hindrance. In an extension to the Griffith analysis outlined by Mott (1948), the
excess released energy is manifest as kinetic energy. Mott suggested that, in the case of a
brittle material, the crack speed will tend to a value of the order of the velocity of sound
in the material.
5
If we let G be the rate of loss of strain energy with crack length
G = –U
E
/c, (10)
then it is clear from equation (6) that the critical energy release rate
6
G
C
is given by
G
C
= –2 γ. (11)
Equation (11) is simply telling us that the critical condition for crack initiation and
propagation is that as the crack extends, sufficient energy must be released to make new
surface.
Note that G
C
is a material constant, and represents a useful fracture toughness
parameter; a material with a high G
C
value is more resilient to fracture than a material
with a low G
C
value.
The Griffith description is one of a reversible process. In general, however, the
stresses at the crack-tip will be sufficiently great so as to cause local plastic deformation.
As work must be done when the material flows, this represents a dissipative process
which is not accounted for in equation (11).
The conch shell as a model for tougher composites 153
If the plastic zone is small, i.e. diamond fracture, the effect may be neglected, but for
many materials the plastic zone cannot be ignored, and nor can the dissipated energy. For
such materials both Irwin and Orowan independently extended the Griffith analysis and
wrote the more general equation
G
C
= –2Γ, (12)
where Γ represents the work of fracture, including dissipative processes, and which
reduces to γ for an ideally brittle solid. Thus equation (12) allows us to discriminate
between brittle (Γ γ) and tough materials (Γ >> γ).
In this review we will be look at how shelled molluscs have adopted methods to
maximise this inequality.
2 Why are conch shells interesting?
Conch shells, and the shells of other molluscs, are interesting because they are of the
order of 95% by volume CaCO
3
(chalk), yet they can 10
3
times tougher greater than that
of monolithic CaCO
3
, that is Γ
shell
γ
chalk
× 10
3
. The remaining 5% is organic, mostly
proteinaceous matter.
Thus they would appear to be good models for the design of high performance
composites. And have been heralded as such: Luz and Mano (2009), Munch et al.
(2008a, 2008b), Ballarini and Heuer (2007), Darder et al. (2007), Podsiadlo et al. (2007),
Mayer (2006, 2005), Mayer and Sarikaya (2002); and this list is certainly not exhaustive
– Figure 1 is a graph of data from the ISI database showing the number of nacre-related
articles published each year.
Figure 1 Number of nacre related articles published per year (data from ISI database)
154 D.M. Williamson and W.G. Proud
As we shall see molluscan shells are able to achieve their advantage through having
evolved a complex architecture on the micro/nano-scale.
Some of the features are comparable to the wavelength of light; this gives the ability
to reflect and refract light, and it is this attribute which gives iridescence to nacre, or as it
is more commonly known mother-of-pearl.
3 The nature of nacre
Some molluscan architectures are shown in Figure 2, reproduced from Currey and Taylor
(1974).
These authors pointed out that the crossed lamellar structure is evolutionarily more
advanced than the more primitive nacre, yet nacre has better mechanical performance.
The answer to this apparent contradiction may be that the crossed lamellar structure is
easier to produce, and on balance, this advantage may outweigh the cost of a weaker
shell.
Figure 2 Molluscan shell architectures (reproduced from Currey and Taylor, 1974)
The queen conch (Strombus gigas) utilises the crossed lamellar structure. This is given in
more detail by Ballarini and Heuer (2007), shown in Figure 3. We see that the structure
has a hierarchy of length scales. The coarsest is represented by three layers designated
inner, middle and outer in relation to the animal’s body. Each layer is made up of smaller
first-order lamellar beams, each of which are in turn composed of even smaller second
order lamellar beams. A key observation is that at each of these length scales the
construction of the structure is cross-ply in nature.
At
the smallest length scale are the basic building blocks of the structure: single crystals
of aragonite (the orthorhombic polymorph of CaCO
3
). Which, according to Su et al.
(2004), are of order 60–130 nm thick and 100–380 nm wide, and can be many microns
The conch shell as a model for tougher composites 155
long. Each crystal is enveloped by a proteinaceous layer 5–10 nm thick. It is these
proteins which are thought to dictate CaCO
3
nucleation, polymorph and growth direction,
Thomson et al. (2000), Metzler et al. (2007).
Figure 3 The crossed lamellar structure of the conch is more detail (reproduced from Ballarini
and Heuer, 2007)
It is the width and thickness of the aragonite crystals which are important from a load
bearing perspective. Of critical note is the following, for the global stresses at which
these shells fail, equation (9) tells us that these dimensions are below the Griffith flaw
size.
7
Thus they themselves are unlikely to fracture, and this is borne out be experiment,
which shows predominantly intergranular failure; Currey (1977).
4 How to win at failing
Failure is observed to occur as debonding of the proteinaceous layer from the aragonite
or else as a cohesive failure of the proteinaceous layer, in either case passing around the
periphery of the aragonite crystals. In general, toughness can be increased by maximising
true surface area generated per nominal fracture plane. In the conch case this is achieved
by virtue of the cross-ply construction.
Apparently, the major threat to conches comes from the mouths of turtles and the
claws of crabs, and the bending moments that they can introduce; the basic failure
behaviour of the crossed lamellar structure in bending was documented by Currey and
Kohn (1976), and can be found in more detail in later articles, e.g. Kamat et al. (2004). In
a three point bending scenario cracks are observed to progress in the manner illustrated in
Figure 4. Cracks are initiated at the bottom layer between lamella (b), but are arrested at
the interface with the middle layer. The crack density increases (c) (d). A crack
propagates through an inner layer lamella (e), note the crack plane is orientated 90° to
adjacent lamella of the same layer (f).
156 D.M. Williamson and W.G. Proud
Figure 4 How failure is observed to progress in a three point bending scenario; (a) initial crack in
bottom layer; (b) crack density in inner layer increases; (c) (d) crack through inner
layer lamella; (e) note this is 90° to adjacent lamella; (f) crack can now propagate into
top layer, but will result in crack bridging (see online version for colours)
Ideally, this will result in alternate cracking, Figure 5a, and bridging of the inner layer
lamella, Figure 5b. Second-order, in-plane, crack-bridging can also observed, Figure 6.
Figure 5 (a-left) Alternate cracking of middle-layer lamella (reproduced from Ballarini and
Heuer, 2007); (b-right) a good illustration of the primary crack-bridging phenomena
that results (reproduced from Kamat et al., 2004)
(a) (b)
The conch shell as a model for tougher composites 157
Figure 6 Secondary crack bridging (reproduced from Kamat et al., 2000)
The bridging lamella pin the would-be crack faces together. The pull-out and differential
sliding of neighbouring aragonite crystals associate with further loading is performed
against a background of a viscous matrix which itself will expend energy when sheared.
In fact many authors suggest it is the transfer of strain energy to the viscous matrix which
expends the most energy.
A potential weakness of materials optimised against threats form a specific direction
is built-in anisotropy. Indeed, Currey (1977) showed an order of magnitude difference in
the work of fracture across the grain compared to along it.
5 Measuring up
Jackson and Vincent (1990) list properties normalised by the specific gravity of nacre
and some other common materials. These authors point out an ‘average’ overall
performance, and state ‘clearly, if nacre does conceal an improvement in some
mechanical properties over and above what is to be expected from density or volume
fracture of filler alone then a much more careful comparison is needed to detect it’.
Table 1 Comparison of material properties (data from Jackson and Vincent, 1990).
σ
is
flexural strength, E flexural Young’s modulus, R is work of fracture
Material Specific gravity (
ρ)
σ
/
ρ
(MPa) E/
ρ
(GPa) R/
ρ
(kJ m–2)
Oryx Horn 1.3 170 5 28
Femur bone 2.1 120 7 2
Wood 0.5 200 25 40
Glass 2.4 70 25 0.005
Mild steel 7.8 50 25 100
Aragonite 2.9 30 34 0.0002
Nacre 2.7 110 26 0.4
GFRP (Vf = 0.5) 1.5 730 27 3
158 D.M. Williamson and W.G. Proud
It is worth pointing out, as do Menig et al. (2001), that comparing nacre (and indeed
other bio-materials) is very difficult. Some researchers go to pains to use fresh samples
which have been kept in salt water, others use dusty dried up specimens from zoological
collections. This is important because as Jackson et al. (1988) point out, water does have
a significant effect, acting to plasticise the proteinaceous layer and increase its tenacity.
Similarly, very few researchers appear to discriminate between quasi-static and
dynamic experiments, although Menig et al. (2001) did address just this issue, showing
an approximately 50% increase in compressive strength in the dynamic case.
But it is the 10
3
increase in work of fracture that nacre possesses over aragonite
which captures the attention most researchers. The question arises as to whether modern
composites can achieve the same performance.
At this point is worth noting the attributes which make nacre and other molluscan
shell materials desirable. The first, which is often overlooked, is the ability to self-heal.
Clearly if the strategy of survival depends not upon resilience in the sense of emerging
from an encounter unscathed, but rather on allowing damage to occur in a extensive yet
controlled manner which maximises energy expenditure, it is crucial that self-healing can
occur between encounters to prevent cumulative damage.
As we have seen, having filler particles less than the Griffith flaw size for the
anticipated stress is key to tough material behaviour; this forces the cracks to follow the
microstructural contours of the materials, which can be torturous, maximising the real
surface area generated. Otherwise cracks will have a tendency to follow geometric paths
of highest tension, and the crack surface area will be relatively small in comparison. This
necessitates that the binder adhesion/cohesion should be the weakest link; but only just so
if we want to maximise the overall strength.
The binding matrix itself should be viscoelastic such that it can dissipate energy when
sheared. Ideally, the glass transition condition would be made to coincide with the
anticipated environmental temperatures and strain rates. However, Kamat et al. (2004)
reported a glass transition temperature of +180°C, which would imply brittle behaviour at
ambient temperatures; it may well be worth repeating these experiments to confirm this
value.
A final point, which ties in with the last, is a high degree of perfection is required.
Since imperfections such as voids are the classic Griffith flaw.
6 Laboratory analogues
This section of the paper is only meant as a brief introduction to the topics covered.
6.1 Self-healing composites
A good overview of self-healing in polymer-based concepts is given by Yuan et al.
(2008). In it materials are classified as either intrinsically or extrinsically self-healing; the
former are able to heal on their own, the latter use a pre-embedded healing agent.
Intrinsic self-healing can be either physical or chemically based. Physical self-healing
occurs
when thermoplastics are heated above their glass transition temperatures and slight
pressure
is applied. Under such conditions inter-diffusion of polymer chains can occur across
the intimate crack surfaces, the degree of recovery can be quiet respectable. Importantly
The conch shell as a model for tougher composites 159
for armour applications some polymers can heal through melting caused by frictional
heating, for example during ballistic penetration. Chemically based self-healing is based
upon thermally reversible cross-linking behaviour; at temperatures above ambient the
cross-linking is ‘undone’ and is restored upon cooling, this allows for healing if the crack
faces are in intimate contact, Liu and Chen (2007).
Extrinsic self-healing polymers carry either enclosed tubes or capsules of healing
agent. The concept is that a crack will break through capsules of polymer precursor,
which will spread via capillary action through the crack and react with embedded catalyst
in order to crosslink and heal the sample, Figure 7. In the example given by White et al.
(2001), the authors claim 75% recovery of the original toughness.
Figure 7 Concept of a extrinsically self-healing polymer based on embedded microcapsules of
healing agent (based on a figure from White et al., 2001). From left to right:
microstructure with embedded catalyst and microcapsules of healing agent (polymer
precursor), crack ruptures microcapsules and healing agent coats crack surfaces via
capillary action, catalyst cross-links healing agent (see online version for colours)
6.2 Microstructured composites
A group at Lawrence Berkley National Laboratory has been pursing a technique called
ice-templating to form composite structures, taking bio-materials as their inspiration
(Deville et al., 2006; Deville et al., 2007; Munch 2008a; Munch, 2008b). In their latest
publication (2008b) the fabrication process of (Al
2
0
3
/PMMA) composites is described
thus; ceramic alumina powder suspensions in water are frozen using directional freezing.
This creates a ceramic scaffold which is lamella like in structure. This can be further
crushed and sintered to form a nacre-like ‘brick and mortar’ scaffold, Figure 8. The
thicknesses of the layers are of order unit microns. The ice is then removed by
sublimation and the remaining void infiltrated with PMMA.
Similar
to nacre, the authors claim a greatly enhances fracture toughness over monolithic
alumina, >300. A very important observation is that an unstructured homogeneous
mixture of alumina particles dispersed in PMMA shows negligible increase in toughness;
the advantage is strongly dependant on the architecture, Figure 9.
160 D.M. Williamson and W.G. Proud
Figure 8 Structure of ice-templated materials. The scale bar in each is 100 μm (reproduced from
Munch et al., 2008a, 2008b)
Figure 9 Crack resistance curves for ice-templated materials, nacre and a homogeneous
dispersion of alumina in PMMA, note the latter show no advantage (reproduced from
Munch et al., 2008a, 2008b)
6.3 Defect-free composites
Here we give the example of so-called Macro Defect Free cement (MDF-cement). An
article by Kendal et al. (1983) makes clear the feebleness of cement (compared to its
theoretical strength) is due to the voids created when the water is removed.
The more water is used, the greater the porosity and the weaker the resultant material,
which
goes back to Feret’s Law (1897). Since 1 cm
3
of Portland cement, on full hydration,
converts to 2.2 cm
3
of hydrate, itself 28% porous, it follows for every 1 cm
3
of water
used ~0.5 cm
3
of porous volume is created.
The authors were able to show that the weakness could be reconciled by Griffith
theory as applied to the pores resulting from the above process.
Large pores were removed by the addition of plasticisers, since when using them it
was found that (a) less water was required in the first place, and (b) the resulting dough
had its entrained air more readily squeezed out by application of external pressure.
In doing so the strengths could be dramatically increased, Table 2.
The conch shell as a model for tougher composites 161
Table2 Improved properties of MDF cement compared with ordinary cement (reproduced
from Kendal et al., 1983)
Property Ordinary cement MDF cement
Flex strength/MPA 10 40–150
Young modulus/GPa 20 35–50
Compressive strength/MPa 40 100–300
Fracture energy/J m
–2
20 40–200
In this case the strength limiting flaw size became correlated with the solid particle
loading size. Adding sand grains of increasing size progressively dropped the strength,
which went someway to confirming this hypothesis.
7 Summary and conclusions
Through controlling the architecture of composite materials it is possible to make them
orders of magnitude tougher than their constituents. Molluscan shells are an excellent
example
of this in nature; nacre is often cited as a ‘gold standard’. The following attributes
are deemed necessary:
A high degree of architectural perfection is required; flaws must be smaller than the
filler particles in order to prevent premature failure.
The filler particles should be smaller than the Griffith flaw size for the anticipated
stress regime; this forces cracking to be confined to either debonding the filler
particles or else cohesive failure within the matrix.
The filler particles must be arranged in such a way that as much true surface area is
generated per nominal fracture plane as possible. Crack bridging by filler particles
is desirable as this promotes shear strains in the matrix. In the crossed lamellar
structure this is achieved by hierarchical cross-plying of lamella at multiple length
scales. A limitation of materials which are optimised to such to meet a single threat
is their anisotropic properties.
The nature of the matrix should be viscoelastic, such that induced shear strains
dissipate as much energy as possible. This could be achieved by tailoring the glass
transition condition to coincide with the anticipated strain-rate/temperature regime.
For a synthetic analogue polycarbonate would be a good off-the-shelf candidate
material for ballistic conditions.
If one accepts the philosophy of maximising damage, in a controlled way, so as to
dissipate as much energy as possible, the ability to self-heal becomes very desirable
to prevent cumulative damage occurring.
Finally, it is important to treat these bio-materials as sources of inspiration, rather
than prototypes to be replicated in exquisite detail. After all, if nature had access to
materials like TiB
2
, would seashells look the same as they do now?
162 D.M. Williamson and W.G. Proud
Acknowledgements
Dr. Leslie Payne is foremost acknowledged for suggesting such an interesting title. AWE
are thanked for their continued funding of the authors.
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Notes
1 Note that absolute size does not enter in the description of stresses around ellipses; a small
circular hole has just as detrimental an effect as a large one.
2 Although is questionable as to how much attention was paid; the catastrophic failure of some
of the subsequent US Liberty ships, due in part to stress concentrations caused by square
hatches, could have been avoided if Inglis’ results had been applied.
3 In a 2D scenario this quantity is zero, but in general a crack or flaw is a 3D entity, and
advance will necessitate an increase in the perimeter subject to the molecular attractions.
164 D.M. Williamson and W.G. Proud
4 The corresponding 3D result was deduced by Sack (1946), and represents ~50% increase in
the required critical stress:
σ
L
= (
π
Eγ/2c)
½
.
5 This has since been proven not to be the upper bound it was once thought to be.
6 Note this is a spatial derivative, not a time derivative.
7 Using an aragonite surface energy of 0.58 J m
–2
, modulus of 98.6 GPa (from Jackson and
Vincent, 1990) (see Table 1), and a tensile failure stress of 75 MPa (Currey and Taylor, 1974)
the critical flaw size is calculated to be ~ 5 μm.
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Mother-of-pearl (nacre) is a platelet-reinforced composite, highly filled with calcium carbonate (aragonite). The Young modulus, determined from beams of a span-to-depth ratio of no less than 15 (a necessary precaution), is of the order of 70 GPa (dry) and 60 GPa (wet), much higher than previously recorded values. These values can be derived from `shear-lag' models developed for platey composites, suggesting that nacre is a near-ideal material. The tensile strength of nacre is of the order of 170 MPa (dry) and 140 MPa (wet), values which are best modelled assuming that pull-out of the platelets is the main mode of failure. In three-point bending, depending on the span-to-depth ratio and degree of hydration, the work to fracture across the platelets varies from 350 to 1240 J m-2. In general, the effect of water is to increase the ductility of nacre and increase the toughness almost tenfold by the associated introduction of plastic work. The pull-out model is sufficient to account for the toughness of dry nacre, but accounts for only a third of the toughness of wet nacre. The additional contribution probably comes from debonding within the thin layer of matrix material. Electron microscopy reveals that the ductility of wet nacre is caused by cohesive fracture along platelet lamellae at right angles to the main crack. The matrix appears to be well bonded to the lamellae, enabling the matrix to be stretched across the delamination cracks without breaking, thereby sustaining a force across a wider crack. Such a mechanism also explains why toughness is dependent on the span-to-depth ratio of the test piece. With this last observation as a possible exception, nacre does not employ any really novel mechanisms to achieve its mechanical properties. It is simply `well made'. The importance of nacre to the mollusc depends both on the material and the size of the shell. Catastrophic failure will be very likely in whole, undamaged shells which behave like unnotched beams at large span-to-depth ratios. This tendency is increased by the fact that predators act as `soft' machines and store strain energy which can be fed into the material very quickly once the fracture stress has been reached. It may therefore be advantageous to have a shell made of an intrinsically less tough material which is better at stopping cracks (e.g. crossed lamellar). However, nacre may still be preferred for the short, thick shells of young molluscs, as these have a low span-to-depth ratio and can make better use of ductility mechanisms.
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A theory is formulated to connect the strength of cement paste with its porosity. The theory shows that bending strength is largely dictated by the length of the largest pores, as in the Griffith (1920) model, but there is also an influence of the volume of porosity, which affects toughness through changing elastic modulus and fracture energy. Verification of this theory was achieved by observing the large pores in cement, and then relating bending strength to the measured defect length, modulus and fracture energy. The argument was proved by developing processes to remove the large pores from cement pastes, thereby raising the bending strength to 70 MPa. Further removal of colloidal pores gave a bending strength of 150 MPa and compression strength up to 300 MPa with improved toughness. Re-introduction of controlled pores into these macro-defect-free (MDF) cements allowed Feret's law (1897) to be explained.
Mother of pearl, or nacre, is one of a number of characteristic skeletal structures of molluscs, occurring in cephalopods, gastropods and bivalves. It consists of plates of aragonite, about 0.3 μ m thick, arranged in sheets, with a tenuous protein matrix. Mechanical tests of nacre from all three classes show that it has a tensile strength of between 35 and 110 MN m-2. It is slightly viscoelastic, and shows marked, though not extensive, plastic deformation. The maximum measured strain was 0.018. While undergoing plastic deformation the material shows considerable optical changes. The regions where plastic flow is occurring show 'tension lines', probably equivalent to similar lines in bone. These are probably caused by voids forming in the protein matrix. The work of fracture is very different in different loading directions, being about 1.65 × 103 J m-2 when fractured across the grain, and 1.5 × 102 J m-2 when fractured along it. Nacre shows considerable ability to stop cracks. An attempt is made to explain qualitatively the mechanical behaviour of nacre in terms of its submicroscopic structure. It is concluded that the precise geometric arrangement of the plates is most important, and that this constraint may make nacre less suitable for shells that must be built quickly.
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A shell is a composite of calcium carbonate and protein. Separated by thin layers of protein are sheets of hard but brittle calcium carbonate, which in turn gives the shell stiffness and strength. The shell also has energy-dissipating microcracks that make it much harder to break. Meanwhile, the calcium carbonate that are produced by the organisms can come from calcite, aragonite and vaterite. A shell's microarchitecture is another important variable. For instance, the makeup of the crossed-lamellar shell is a tiny plank of crystalline aragonite encased in protein sheath, which are then bundled into sheets called lamellae. Shells are also characterized by their strength and toughness. For instance, a conch shell is at least 1,000 times tougher than mineral aragonite but only slightly stronger. Meanwhile, a creature makes a shell by means of the synthesizing of the tissue called the mantle, a snug covering that encloses the head and foot of the snail. Shells grow first by the emerging of a thin, transparent sheet of protein called periostracum from a a fold in the mantle. Then the cells of the mantle epithelium secrete proteins into the extrapallial space, together with the calcium and bicarbonate ions that will combine to form calcium carbonate.