DOI: 10.1126/science.1220854
, 773 (2013);
339 Science et al.Marc André Meyers Connections
Structural Biological Materials: Critical Mechanics-Materials
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w w w .s c i e n c e m a g .o r g D o w n l o a d e d f r o m
Structural Biological
Materials:Critical
Mechanics-Materials Connections
Marc AndréMeyers,1,2*Joanna McKittrick,1Po-Yu Chen3
Spider silk is extraordinarily strong,mollusk shells and bone are tough,and porcupine quills and feathers resist buckling.How are the notable properties achieved?The building blocks of the materials listed above are primarily minerals and biopolymers,mostly in combination;the first weak in tension and the cond weak in compression.The intricate and ingenious hierarchical structures are responsible for the outstanding performance of each material.Toughness is conferred by the prence of controlled interfacial features(friction,hydrogen bonds,chain straightening and stretching);buckling resistance can be achieved by filling a slender column with a lightweight foam.Here,we prent and interpret lected examples of the and other biological materials.Structural bio-inspired materials design makes u of the biological structures by inrting synthetic materials and process that augment the structures’capability while retaining their esntial features.In this Review,we explain this idea through some unusual concepts.
M aterials science is a vibrant field of in-tellectual endeavor and rearch.This
field applies physics and chemistry, melding them in the process,to the interrela-tionship between structure,properties,and perform-ance of complex materials with technological applications.Thus,materials science extends the rigorous scientific disciplines into complex ma-terials that have structures providing properties and synergies beyond tho of pure and simple solid
s.Initially geared at synthetic materials,ma-terials science has recently extended its reach into biology,especially into the extracellular matrix, who mechanical properties are of utmost im-portance in living organisms.Some of the mi-nal work and important contributions in this field are either prented or reviewed in(1–5).There are a number of interrelated features that define biological materials and distinguish them from their synthetic counterparts[inspired by Arzt(6)]: (i)Self-asmbly.In contrast to many synthetic process to produce materials,the structures are asmbled from the bottom up,rather than from the top down.(ii)Multi-functionality.Many com-ponents rve more than one purpo.For exam-ple,feathers provide flight capability,camouflage, and insulation,whereas bones provide structural framework,promote the growth of red blood cells, and provide protection to the internal organs.(iii) Hierarchy.Different,organized scale levels(nano-to ultrascale)confer distinct and translatable prop-erties from one level to the next.We are starting to develop a systematic and quantitative understanding
of this hierarchy by distinguishing the character-
istic levels,developing constitutive descriptions
of each level,and linking them through appro-
priate and physically bad equations,enabling a
full predictive understanding.(iv)Hydration.The
properties are highly dependent on the level of
water in the structure.There are some exceptions,
such as enamel,but this rule applies to most
biological materials and is of importance to me-
chanical properties such as strength(which is
decread by hydration)and toughness(which is
incread).(v)Mild synthesis conditions.The
majority of biological materials are fabricated at
ambient temperature and pressure as well as in an
aqueous environment,a notable difference from
synthetic materials fabrication.(vi)Evolution and
environmental constraints.The limited availabil-
ity of uful elements dictates the morphology
and resultant properties.The structures are not
necessarily optimized for all properties but are
the result of an evolutionary process leading to
satisfactory and robust solutions.(vii)Self-healing
capability.Whereas synthetic materials undergo
damage and failure in an irreversible manner,
biological materials often have the capability,
due to the vascularity and cells embedded in the
structure,to rever the effects of damage by
healing.
The ven characteristics listed above are
prent in a vast number of structures.Nevertheless,
the structures of biological materials can be
divided into two broad class:(i)non-mineralized
(“soft”)structures,which are compod of fibrous
constituents(collagen,keratin,elastin,chitin,
lignin,and other biopolymers)that display widely
varying mechanical properties and anisotropies
depending on the function,and(ii)mineralized
(“hard”)structures,consisting of hierarchically
asmbled composites of minerals(mainly,but
not solely,hydroxyapatite,calcium carbonate,
and amorphous silica)and organic fibrous com-
ponents(primarily collagen and chitin).
The mechanical behavior of biological con-
stituents and composites is quite diver.Bio-
minerals exhibit linear elastic stress-strain plots,
whereas the biopolymer constituents are non-
linear,demonstrating either a J shape or a curve
with an inflection point.Foams are characterized
by a compressive respon containing a plastic or
crushing plateau in which the porosity is elim-
inated.Many biological materials are composites
with many components that are hierarchically
structured and can have a broad variety of con-
stitutive respons.Below,we prent some of the
structures and functionalities of biological ma-
terials with examples from current rearch.Here,
we focus on three points:(i)How high tensile
strength is achieved(biopolymers),(ii)how high
toughness is attained(composite structures),and
(iii)how bending resistance is achieved in light-
weight structures(shells with an interior foam).
Structures in Tension:Importance of Biopolymers
The ability to sustain tensile forces requires a
specific t of molecular and configurational con-
formations.The initial work performed on exten-
sion should be small,to reduce energy expenditure,
whereas the material should stiffen clo to the
breaking point,to resist failure.Thus,biopolymers,
such as collagen and viscid(catching spiral)spider
silk,have a J-shaped stress-strain curve where mo-
lecular uncoiling and unkinking occur with con-
siderable deformation under low stress.
This stiffening as the chains unfurl,straighten,
stretch,and slide past each other can be repre-
nted analytically in one,two,and three dimen-
sions.Examples are constitutive equations initially
developed for polymers by Ogden(7)and Arruda
and Boyce(8).An equation specifically propod
for tissues is given by Fung(3).A simpler for-
mulation is given here;the slope of the stress-strain
(s-e)curve increas monotonically with strain.
Thus,one considers two regimes:(i)unfurling
and straightening of polymer chains
d s
d e
ºe nðn>1Þð1Þ
and(ii)stretching of the polymer chain backbones
d s
d e
ºEð2Þ
where E is the elastic modulus of the chains.The
combined equation,after integrating Eqs.1and2,is
s=k1e n+1+H(e c)E(e–e c)(3)
Here k1is a parameter,and H is the Heaviside
function,which activates the cond term at e=
e c,where e c is a characteristic strain at which
collagen fibers are fully extended.Subquent strain
gradually becomes dominated by chain stretch-
ing.The computational results by Gautieri et al.
(9)on collagen fibrils corroborate Eq.3for n=1.
This corresponds to a quadratic relation between
1Department of Mechanical and Aerospace Engineering and
Materials Science and Engineering Program,University of
California,San Diego,La Jolla,CA92093,USA.2Department of
Nanoengineering,University of California,San Diego,La Jolla,
CA92093,USA.3Department of Materials Science and En-
gineering,National Tsing Hua University,Hsinchu30013,
Taiwan,Republic of China.
*To whom correspondence should be addresd.E-mail:
mameyers@ucsd.edu
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stress and strain (s ºe 2),which has the char-acteristic J shape.
Collagen is the most important structural bio-logical polymer,as it is the key component in many tissues (tendon,ligaments,skin,and bone),as well as in the extracellular matrix.The de-formation process is intimately connected to the different hierarchical levels,starting with the poly-peptides (0.5-nm diameter)to the tropocollagen molecules (1.5-nm diameter),then to the fibrils (~40-to 100-nm diameter),and finally to fibers (~1-to 10-m m diameter)and fascicles (>10-m m diameter).Molecular dynamics computations (9)of entire fibrils show the J -curve respon;the computational predictions are well matched to atomic force microscopy (AFM)(10),small-angle x-ray scattering (SAXS)(11),and experiments by Fratzl et al .(12),as shown in Fig.1A.The effect of hydration is also en and is of great impor-tance.The calculated density of collagen de-creas from 1.34to 1.19g/cm 3with hydration and is accompanied by a decrea in the Young ’s modulus from 3.26to 0.6GPa.
The respon of silk and spider thread is fascinating.As one of the toughest known ma-terials,silk also has high tensile strength and extensibility.It is compod of b sheet (10to 15volume %)nanocrystals [which consist of highly conrved poly-(Gly-Ala)and poly-Ala domains]embedded in a disordered matrix (13).Figure 1B shows the J -shape stress-strain curve and molecular configurations for the crystalline domains in silkworm (Bombyx mori )silk (14).Similar to collagen,the l
ow-stress region corre-sponds to uncoiling and straightening of the pro-tein strands.This region is followed by entropic unfolding of the amorphous strands and then stiffening due to load transfer to the crystalline b sheets.Despite the high strength,the major mo-lecular interactions in the b sheets are weak hy-drogen bonds.Molecular dynamics simulations,
Fig.1.Tensile stress-strain relationships in bio-polymers.(A )J -shaped curve for hydrated and dry collagen fibrils obtained from molecular dynamics (MD)simulations and AFM and SAXS studies.At low stress levels,considerable stretching occurs due to the uncrimping and unfolding of molecules;at higher stress levels,the polymer backbone stretches.Adapted from (9,12).(B )Stretching of dragline spider silk and molecular schematic of the protein fibroin.At low stress levels,entropic effects domi-nate (straightening of amorphous strands);at higher levels,the crystalline parts sustain the load.(C )Mo-lecular dynamics simulation of silk:(i)short stack and (ii)long stack of b -sheet crystals,showing that a higher pullout force is required in the short stack;for the long stack,bending stress become im-portant.Hydrogen bonds connect b -sheet crystals.Adapted from (14).(D )Egg whelk ca (bioelastomer)showing three regions:straightening of the a helices,the a helix –to –b sheet transformation,and b -sheet extension.A molecular schematic is shown.Adapted from (18).
300.00
0.2
Yield point
Entropic unfolding
MD simulations
Stick slip
Stiffening β-crystal
1
2
3456
7
000
1
2
34567
01020304050500100015002000250500
75010001250
15001750
0.4
0.6
0.8
0.010.020.030.040.05
MD wet (Gautieri et al)
SAXS (Sasaki and Odajima)AFM (Aladin et al)MD dry (Gautieri et al)
2520
151050S t r e s s (M P a )
S t r e s s
(M P a )
Strain
A
B
C
D
Strain (m/m)
Length (nm)
Length (nm)
Stick-slip deformation (robust)"brittle" fracture (fragile)
i ii
P u l l -o u t f o r c e (p N )0
0.2
0.4
Native state Unloading: reformation of α-helices
Domain 4: Extension and
alignmentof β-sheets
0.60.8
ε=0
ε4ε=0
1.0
012345Strain
S t r e s s (M P a )
E n e r g y /v o l u m e (k c a l /m o l /n m 3)
L e n g t h
I I II II III III
IV
IV
F
Domain 3: Formation of β-sheets
from random coils
ε3Domain 2: Extension of random coils
ε2
Domain 1: Unraveling of α-helices
into random coils
ε1Toughness (MD)Resilience (MD)
T=-1°C T=20°C T=40°C T=60°C T=80°C
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shown in Fig.1C,illustrate an energy dissipative stick-slip shearing of the hydrogen bonds during failure of the b sheets (14).For a stack with a height L ≤3nm (left-hand side of Fig.1C),the shear stress are more substantial than the flex-ure stress,and the hydrogen bonds contribute to the high strength obtained (1.5GPa).How-ever,if the stack of b sheets is too high (right-hand side of Fig.
1C),it undergoes bending with tensile paration between adjacent sheets.The nanoscale dimension of the b sheets allows for a ductile instead of brittle failure,resulting in high toughness values of silk.Thus,size affects the mechanical respon considerably,changing the deformation characteristics of the weak hydro-gen bonds.This has also been demonstrated in bone (15–17),where sacrificial hydrogen bonds between mineralized collagen fibrils contribute to the excellent fracture resistance.
Other biological soft materials have more complex respons,marked by discontinuities in d s /d e .This is the ca for wool,whelk eggs,silks,and spider webs.Several mechanisms are responsible for this change in slope;for instance,the transition from a -to b -keratin,entropic changes with strain (such as tho prevalent in rubber,where chain stretching and alignment decrea entropy),and others.The example of egg whelk is shown in Fig.1D (18).In this ca,there is a specific stress at which a -keratin heli-ces transform to b sheets,with an associated change in length.Upon unloading,the rever occurs,and the total reversible strain is,therefore,extensive.This stress-induced pha transforma-tion is similar to what occurs in shape-memory alloys.Thus,this material can experience sub-stantial reversible deformation (up to 80%)in a reversible fashion,when the stress is raid from 2to 5MPa,ensuring the survival of whelk eggs,which are continually swept by waves.
The examples demonstrate the distinct properties of biopolymers that allow the ma-terials to be strong and highly extensible with distinctive molecular deformation characteristics.However,many interesting biological materials are composites of flexible biopolymers and stiff minerals.The combination of the two constit-uents leads to the creation of a tough material.Imparting Toughness:Importance of Interfaces One hallmark property of most biological com-posites is that they are tough.Toughness is defined as the amount of energy a material ab-sorbs before it fails,expresd as
U ¼∫e f
s d e
ð4Þ
where U is the energy per volume absorbed,s is the stress,e is the strain,and e f is the failure strain.Tough materials show considerable plastic deformation (or permanent damage)coupled with considerable strength.This maximizes the integral expression in Eq.4.Biological com-posite materials (for example,crystalline and noncrystalline components)have a plethora of
toughening mechanisms,many of which depend on the prence of interfaces.As a crack im-pinges on an interface or discontinuity in the material,the crack can be deflected around the interface (requiring more energy to propagate than a straight crack)or can drive through it.The strength of biopolymer fibers in tension im-pedes crack opening;bridges between micro-cracks are another mechanism.The toughening mechanisms have been divided into intrinsic (ex-isting in the material ahead of crack)and extrinsic (generated during the progression of failure)cat-egories (19).Thus,toughening is accomplished by a wide variety of stratagems.We illustrate this concept for four biological materials,shown in Fig.2.
All inorganic materials contain flaws and cracks,which reduce the strength from the theo-retical value (~E /10to E /30).The maximum stress (s max )a material can sustain when a preexisting crack of length a is prent is given by the Griffith equation
s max ¼ffiffiffiffiffiffiffiffiffiffi
2g s E p a r ¼YK Ic
ffiffiffiffiffip a
p ð5Þwhere E is the Young ’s modulus,g s is the sur-face (or damage)energy,and Y is a geometric parameter.K Ic ¼Y −1ffiffiffiffiffiffiffiffiffiffi2g s E p is the fracture toughness,a materials property that express the ability to resist crack propagation.Abalone (Haliotis rufescens )nacre has a fracture tough-ness that is vastly superior to that of its major constituent,monolithic calcium carbonate,due to an ordered asmbly consisting of mineral tiles with an approximate thickness of 0.5m m and a diameter of ~10m m (Fig.2A).Additionally,this material contains organic mesolayers (parated by ~300m m)that are thought to be asonal growth bands.The tiles are connected by mineral bridges with ~50-nm diameter and are parated by organic layers,consisting of a chitin network and acidic proteins,which,when combined,have a similar thickness to the mineral bridge diame-ters.The Griffith fracture criterion (Eq.5)can be applied to predict the flaw size (a cr )at which the theoretical strength s th is achieved.With typical values for the fracture toughness (K Ic ),s th ,and E ,the critical flaw size is in the range of tens of nanometers.This led Gao et al .(20)to propo that at sufficiently small dimensions (less than the critical flaw size),materials become innsitive to flaws,and the theoretical strength (~E /30)should be achieved at the nanoscale.However,the strength of the material will be determined by fracture mechanisms operating at all hierar-chical levels.
The central micrograph in Fig.2A shows how failure occurs by tile pullout.The interdigitated structure
deflects cracks around the tiles instead of through them,thereby increasing the total length of the crack and the energy needed to fracture (increasing the toughness).Thus,we must de-termine how effectively the tiles resist pullout.Three contributions have been identified and are believed to operate synergistically (21).First,the
mineral bridges are thought to approach the
theoretical strength (10GPa),thereby strongly attaching the tiles together (22).Second,the tile surfaces have asperities that are produced during growth (23)and could produce frictional resist-ance and strain hardening (24).Third,energy is required for viscoelastic deformation (stretching and shearing)of the organic layer (25).
One important aspect on the mechanical prop-erties is the effect of alignment of the mineral crystals.The oriented tiles in nacre result in an-isotropic properties with the strength and modulus higher in the longitudinal (parallel to the organic layers)than in the transver direction.For a composite with a disperd mineral m of volume fraction V m embedded in a biopolymer (bp)matrix that has a much lower strength and Young ’s modulus than the mineral,the ratio of the lon-gitudinal (L)and transver (T)properties P (such as elastic modulus)can be expresd,in simpli-fied form,as
P L P T ¼P m
P bp
V m ð1−V m Þð6ÞThus,the longitudinal properties are much higher than the transver properties.This aniso-tropic respon is also obrved in other oriented mineralized materials,such as bone and teeth.Another tough biological material is the exo-skeleton of an arthropod.In the ca of marine animals [for instance,lobsters (26,27)and crabs (28)],the exoskeleton structure consists of layers of mineralized chitin in a Bouligand arrange-ment (successive layers at the same angle to each other,resulting in a helicoidal stacking quence and in-plane isotropy).The layers can be en-visaged as being stitched together with ductile tubules that also perform other functions,such as fluid transport and moisture regulation.The cross-ply Bouligand arrangement is effective in crack stopping;the crack cannot follow a straight path,thereby increasing the materials ’toughness.Upon being stresd,the mineral components frac-ture,but the chitin fibers can absorb the strain.Thus,the fractured region does not undergo physical paration with dispersal of fragments,and lf-healing can take place (29).Figure 2B shows the structure of the lobster (Homarus americanus )exoskeleton with the Bouligand ar-rangement of the fibers.
Bone is another example of a biological ma-terial that demonstrates high toughness.Skeletal mammalian bone is a composite of hydroxyapatite-type minerals,collagen and water.On a volu-metric basis,bone consists of ~33to 43volume %minerals,32to 44volume %organics,and 15to 25volume %water.The Young ’s modulus and strength increa,but the toughness decreas with increasing mineral volume fraction (30).Cortical (den)mammalian bone has blood ves-ls extending along the long axis of the limbs.In animals larger than rats,the vesl is encad in a circumferentially laminated structure called the osteon.Primary osteons are surrounded by hypermineralized regions,whereas condary
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(remodeled)osteons are surrounded by a cement line (also of high mineral content)(31).In mam-malian cortical bone,the following intrinsic toughening mechanisms have been identified:molecular uncoiling and intermolecular sliding of collagen,fibrillar sliding of collagen bonds,and microcracking of the mineral matrix (19).Extrinsic mechanisms are collagen fibril bridging,uncracked ligament bridging,and crack deflec-tion and twisting (19).Rarely does a limb bone snap in two with smooth fracture surfaces;the crack is often deflected orthogonal to the crack front direction.In the ca of (rehydrated)elk (Cervus elaphus )antler bone (shown in Fig.2C)(32),which has the highest toughness of any bone type by far (33),the hypermineralized re-gions around the primary osteons lead to crack
deflection,and the high amount of collagen (~60volume %)adds mechanisms of crack re-tardation and creates crack bridges behind the crack front.The toughening effect in antlers has been estimated as:crack deflection,60%;un-cracked ligament bridges,35%;and collagen as well as fibril bridging,5%(33).A particu-larly important feature in bone is that the fracture toughness increas as the crack propagates,as shown in the plot.This plot demonstrates the crack extension resistance curve,or R -curve,behavior,which is the rate of the total energy dissipated as a function of the crack size.This occurs by the activation of the extrinsic tough-ening mechanisms.In this manner,it becomes gradually more difficult to advance the crack.In human bone,the cracks are deflected and/or
twisted around the cement lines surrounding the condary osteons and also demonstrate R -curve behavior (34).
The final example illustrating how the prence of interfaces is ud to retard crack propagation is the glass a sponge (Euplectella aspergillum ).The entire structure of the V enus ’flower basket is shown in Fig.2D.Biological silica is amorphous and,within the spicules,consists of concentric layers,parated by an organic material,silicatein (35,36).The flexure strength of the spicule notably exceeds (by approximately fivefold)that of monolithic glass (37).The principal reason is the prence of interfaces,which can arrest and/or deflect the crack.
Biological materials u ingenious meth-ods to retard the progression of cracks,thereby
Abalone shell: Nacre
Mineral bridges
Lobster
Deer antler
Chitin fibril network
Human cortical bone
Mineral crystallites
Primary osteons
Subvelvet/compact Subvelvet/c
Compact Comp p act
Transition zone
Cancellous
Collagen fibrils
Deep a sponge
Skeleton
Spicules
20 mm
1 cm
Human cortical bone
Elk antler
Transver
In-plane longitudinal
ASTM valid
ASTM invalid Mesolayers A
B
C
D
0.1 mm
500 nm
500 nm ˜1 nm
˜3 nm
˜20 nm
Crack extension, ⌬a (mm)
T o u g h n e s s , J (k J m -2)
50 nm
200 nm 10 m
500 nm
2 m
1 m
200 m
300 m
˜10 m
0.01
0.1
1
10
100
0.2
0.4
0.6
500 00 nm
50 nm
Fig.2.Hierarchical structures of tough biological materials demonstrating the heterogeneous interfaces that provide crack deflection.(A )Abalone nacre showing growth layers (mesolayers),mineral bridges between mineral tiles and asperities on the surface,the fibrous chitin network that forms the backbone of the inorganic layer,and an example of crack tortuosity in which the crack must travel around the tiles instead of through them [adapted from (4,21)].(B )Lobster exoskeleton showing the twisted plywood structure of the chitin (next to the shell)and the tubules that extend from the chitin layers to the animal [adapted from (27)].(C )Antler bone image showing the hard outer sheath (cortical bone)surrounding the porous bone.The collagen fibrils are highly aligned in the growth direction,with nanocrystalline minerals disperd in and around them.The osteonal structure in a cross ction of cortical bone illustrates the boundaries where cracks perpendicular to the osteons can be directed [adapted from (33)].ASTM,American Society for Testing and Mate-rials.(D )Silica sponge and the intricate scaffold of spicules.Each spicule is a circumferenti
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increasing toughness.The methods operate at levels ranging from the nanoscale to the structur-al scale and involve interfaces to deflect cracks,bridging by ductile phas (e.g.,collagen or chitin),microcracks forming ahead of the crack,delocal-ization of damage,and others.
Lightweight Structures Resistant to Bending,Torsion,and Buckling —Shells and Foams
Resistance to flexural and torsional tractions with a prescribed deflection is a major attribute of many biological structures.The fundamental mechanics of elastic (recoverable)deflection,
as it relates to the geometrical characteristics of beams and plates,is given by two equations:The first relates the bending moment,M ,to the curvature of the beam,d 2y /dx 2(y is the deflection)
d 2y dx 2¼M
EI
ð7Þ
where I is the area moment of inertia,which de-pends on the geometry of the cross ction (I =p R 4/4,for circular ctions,where R is the ra-dius).Importantly,the curvature of a solid beam,and therefore its deflection,is inverly propor-tional to the fourth power of the radius.The c-ond equation,commonly referred to as Euler ’s buckling equation,calculates the compressive load at which global buckling of a column takes place (P cr )
P cr ¼
p 2EI ðkL Þ2
ð8Þ
where k is a constant dependent on the column-end conditions (pinned,fixed,or free),and L is the length of the column.Resistance to buck-ing can also be accomplished by increasing I .Both Eqs.7and 8predict the principal design
Longitudinal ction
Toucan beak Keratin layers
(i) Fibers
(circumferential)
Megafibrils and fibrils
Barbs
Barbules
Cortex
Cortical ridges
Foam
Rachis
Nodes
(iii) Medulloid
pith
(ii) Fibers (longitudinal)
Feather rachis
Plant-Bird of Paradi
Porcupine quills
Nodes
Rebar
Clod-cell foam
Transver
Longitudinal
Cross ction
A
B
C
D
5 mm 1 mm
1 cm 0.1 mm
5m 5 m m
1c 1 c m
1 mm
100 m
500 m
Fig.3.Low-density and stiff biological materials.The theme is a den outer layer and a low-density core,which provides a high bending strength –to –weight ratio.(A )Giant bird of paradi plant stem showing the cellular core with porous walls.(B )Porcupine quill exhibiting the den outer cortex surrounding a uniform,clod-cell foam.Taken from (42).(C )Toucan beak showing the porous
interior (bone)with a central void region [adapted from (43)].(D )Schematic view of the three major structural components of the feather rachis:(i)superficial layers of fibers,wound circumferentially around the rachis;(ii)the majority of the fibers extending parallel to the rachidial axis and through the depth of the cortex;and (iii)foam comprising gas-filled polyhedral structures.Taken from (45).
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