Modelling the three-dimensional elastic constants of parallel-fibred and lamellar bone

Abstract

The complex hierarchical structure of lamellar bone makes understanding structure–mechanical function relations, very difficult. We approach the problem by first using the relatively simple structure of parallel-fibred bone to construct a mathematical model for calculating Young's moduli in three-dimensions. Parallel-fibred bone is composed essentially of arrays of mineralized collagen fibrils, which are also the basic structural motif of the individual lamellae of lamellar bone. Parallel-fibred bone structure has orthotropic symmetry. As the sizes and shapes of crystals in bone are not well known, the model is also used to compare the cases of platelet-, ribbon- and sheet-reinforced composites. The far more complicated rotated plywood structure of lamellar bone results in the loss of the orthotropic symmetry of individual lamellae. The mathematical model used circumvents this problem by sub-dividing the lamellar unit into a thin lamella, thick lamella, transition zone between them, and the recently observed back-flip lamella. Each of these is regarded as having orthotropic symmetry. After the calculation of their Young's moduli they are rotated in space in accordance with the rotated plywood model, and then the segments are combined to present the overall modulus values in three-dimensions. The calculated trends compare well with the trends in microhardness values measured for circumferential lamellar bone. Microhardness values are, as yet, the only measurements available for direct comparison. Although the model is not directly applicable to osteonal bone, which is composed of many hollow cylinders of lamellar bone, the range of calculated modulus values and the trends observed for off-axis calculations, compare well with measured values. 1998 Chapman & Hall

Full text

0022—2461 ( 1998 Chapman & Hall 1497
JOURNAL OF MATERIALS SCIENCE: 33 (1998) 1497– 1509
Modelling the three-dimensional elastic constants
of parallel-fibred and lamellar bone
U. AKIVA, H. D. WAGNER, S. WEINER*
Department of Materials and Interfaces, and
*
Department of Structural Biology, The
Weizmann Institute of Science, Rehovot 76100, Israel
E-mail. cpwagner@wis.weizmann.ac.il
The complex hierarchical structure of lamellar bone makes understanding
structure—mechanical function relations, very difficult. We approach the problem by first
using the relatively simple structure of parallel-fibred bone to construct a mathematical
model for calculating Young’s moduli in three-dimensions. Parallel-fibred bone is composed
essentially of arrays of mineralized collagen fibrils, which are also the basic structural motif
of the individual lamellae of lamellar bone. Parallel-fibred bone structure has orthotropic
symmetry. As the sizes and shapes of crystals in bone are not well known, the model is also
used to compare the cases of platelet-, ribbon- and sheet-reinforced composites. The far
more complicated rotated plywood structure of lamellar bone results in the loss of the
orthotropic symmetry of individual lamellae. The mathematical model used circumvents
this problem by sub-dividing the lamellar unit into a thin lamella, thick lamella, transition
zone between them, and the recently observed ‘‘back-flip’’ lamella. Each of these is regarded
as having orthotropic symmetry. After the calculation of their Young’s moduli they are
rotated in space in accordance with the rotated plywood model, and then the segments are
combined to present the overall modulus values in three-dimensions. The calculated trends
compare well with the trends in microhardness values measured for circumferential lamellar
bone. Microhardness values are, as yet, the only measurements available for direct
comparison. Although the model is not directly applicable to osteonal bone, which is
composed of many hollow cylinders of lamellar bone, the range of calculated modulus
values and the trends observed for off-axis calculations, compare well with measured
values. 
1998 Chapman & Hall
1. Introduction
Relating the mechanical properties of bone to its
structure is a very challenging task. Bone occurs in
several different forms or types and has a complex
hierarchical structure [1, 2]. Furthermore, bone struc-
ture, and, in particular, mineral content, changes with
increasing age of the bone, and this too has a direct
effect on mechanical properties [1, 3, 4]. Bone struc-
ture—mechanical relations have, therefore, to be re-
solved for well-defined bone types of a specific age.
Here we address this question first for the relatively
simple parallel-fibred bone, and then use this as
a basis for addressing the far more complex structure,
rotated plywood, that is present in lamellar bone [5].
We approach this by using our current knowledge of
the structure as a basis for constructing a mathematical
model of the material properties of the bone type, and
then compare the calculated results with the few appro-
priate data sets of experimentally measured values.
This approach has been used in several other
studied. Recent examples are the studies of Sasaki et
al. [6], Wagner and Weiner [7] and Currey et al. [8].
Sasaki et al. [6] used fibrolamellar bone which is
composed of both lamellar and parallel-fibred bone.
They measured the preferred orientations of the c-axes
of the crystals and introduced this information into
their model. They, like many earlier studies, assumed
that the crystals are needle-shaped. Direct observa-
tions [9—13] as well as the modelling studies of Wag-
ner and Weiner [7] and Currey et al. [8], both show
that most, if not all, the crystals are plate-shaped and
that bone should be regarded as a platelet-reinforced
composite material. The structure of the narwhal tusk
studied by Currey et al. [8] is very poorly known, and
this limits the extent to which its structure can be
related to measured mechanical properties.
Parallel-fibred bone is essentially an array of paral-
lel mineralized collagen fibrils [14]. The plate-shaped
crystals within each fibril are arranged in layers that
traverse the collagen fibril [15]. The structure of
lamellar bone has been the subject of many studies,
and was reviewed by Weiner and Traub [2]. Its
basic plywood motif has been well documented
[16—19] and a well substantiated structural model for
collagen fibril organization was proposed by Giraud-
Guille [17]. In this model each lamella is composed of
arrays of parallel fibrils. The orientations of the fibril
arrays in adjacent lamellae are different, thus forming

Figure 1 Scanning electron micrograph of the fracture surface of rat
tibia lamellar bone (published previous [22]) showing the lamellar
units (between white bars), each of which is composed of a thin
lamella (1), a transition zone (2), a thick lamella (3) and a ‘‘back-flip’’
lamella’’ (4). We deliberately use a micrograph published previously
[22], to show that the ‘‘back-flip’’ lamella is present, but we did not,
at the time, note its significance. For more information, see [53].
a plywood-like motif [16—19]. Note that not all
investigators concur with this view of the lamellar
structure [20]. Weiner et al. [5] observed that the
crystal layers are parallel to the lamellar boundary
plane in the thin lamellae, but are rotated relative to
the lamellar boundary plane in the thick lamella. They
therefore called this structure ‘‘rotated plywood’’. This
constituted the basis for a model that attempted to
relate structure to material properties by Wagner and
Weiner [7]. Our understanding of the rotated ply-
wood structure has subsequently been further refined
to include more information on the structure of the
transition zone between the thick and thin lamellae,
the presence of a so-called ‘‘back-flip’’ lamella and
semi-quantitative measurements of the plywood angle
[21, 22]. The basic features of the structure are shown
in Fig. 1.
The models proposed by Sasaki et al. [6], Wagner
and Weiner [7], Currey et al. [8] and indeed the one
proposed here, all predict the variations in elastic
modulus with changing orientations of the specimen
tested in relation to the long axis of the bone. This is
a fairly stringent test. A serious problem, however, is
the paucity of appropriate experimental measure-
ments of this type. Reilly and Burstein [23], Bonfield
and Grynpas [24] and recently Turner et al. [25]
collected such data for a variety of lamellar bones
and fibrolamellar bone. The lamellar bones used were
osteonal, which complicates the geometry enormously
due to the cylindrical nature of the osteon and its
central blood vessel. Fibrolamellar bone is a mixture
of two bone types [16], making its interpreta-
tions equivocal. Ziv et al. [15] performed a detailed
microhardness study on circumferential lamellar bone
(which is essentially composed of parallel lamellar
units) probed in many different orientations with re-
spect to the rotated plywood structure. Microhard-
ness is a convenient tool for doing this, but unfortu-
nately it is difficult to relate hardness values to most
other material properties of bone. The trends, how-
ever, should be the same. Evans et al. [26] made an
empirical study of microhardness—Young’s modulus
relations using many different bones. Although a cor-
relation is observed, we hesitate to use this to convert
microhardness values to elastic moduli because the
samples were not oriented relative to their structures
and many different structures are all lumped together.
Here we present a more refined micromechanical
model for the three-dimensional Young’s modulus of
lamellar bone. The present model follows, at first, the
logic of our previous model [7], that is, the focus rests
first on the single lamella (parallel-fibred bone in
essence), then on bone viewed as an assembly of lamel-
lae. However, regarding the latter, the approach taken
here is now fully three-dimensional and eliminates
the need for the arbitrary factor, n, used in our pre-
vious model [7] to account for the weakening effect
induced by the rotation of the crystal layers in the
thick lamellae.
1.1. Bone structure: comments on the key
features incorporated into the model
The basic building block of bone is a mineralized
collagen fibril [27], which is composed of thin, plate-
shaped crystals of carbonated apatite (dahllite) ar-
ranged in parallel layers within the type I collagen
fibril [9]. The crystal thicknesses, as measured by
small-angle X-ray scattering are about 1.5—3.0 nm [12,
29]. The crystals themselves, once extracted, are very
irregular in shape, but are on average 50 nm long by
25 nm wide [11, 13]. Very little, however, is known
about their in vivo dimensions. As extraction almost
certainly causes breakage, they may be much larger, as
has been suggested by Boyde [30] who studied ion-
beam thinned dentine. We also know very little about
the nature of crystal—crystal contacts in vivo, and
about crystals that may be located between collagen
fibrils. The structure of the type I collagen fibril, to
a large extent, dictates the organization of the crystals
inside the fibril [31]. Although crystals initially form
in the gap zones [31—33], they rapidly fill the gap
zones, and then spread into the overlap zones [34]
ultimately to form extended layers. Following the
model for the type I collagen fibril based on
stoichiometric measurements of cross-links [35, 36],
the layers of crystals are separated by four layers of
close-packed triple helical molecules. The distance
separating crystal layers is about 4—5 nm [2], based
on a fairly prominent 4 nm reflection present in X-ray
diffraction patterns of type I collagen [37] and from
direct measurements [38]. Note too that the propor-
tions of collagen, crystals and water vary in normal
bone from species to species, with the age of the animal,
and within the bone as a function of remodelling.
This has an important effect on almost all material
properties [3, 4]. The relative proportions of collagen,
1498

TABLE I Structural and material parameters of lamellar bone used for the calculations (refer to Figs 1 and 2 for notations)
Structural parameters Reference
Average crystal platelet dimensions 50]25]2nm3 [1, 3]
Distance between crystal layers: d"2 nm [2, 37]
corresponds to 4 nm spacing in structure (d"2nm
and 2]1/2 crystal thickness of 2 nm)
Lateral space between crystals u"0.1 nm (parallel to collagen fibril axis) —
(assumed to be very small)
v"0.1 nm (perpendicular to collagen fibril axis) —
Calculated (Equation 3) platelet volume fraction
(corresponds to 67% ash weight) »
1
"0.5 [15, 39]
Average lamellar unit 3.2 lm [21]
thickness (rat)
Average thick lamella 1.8 lm [21]
thickness (rat)
Average thin lamella thicknes 0.4 lm! [21]
Average transition zone 0.4 lm! [21]
(between thin and thick lamellae) thickness (rat)
Average thickness of the ‘‘back-flip’’ lamellae 0.6 lm [21]
w
1
and w
2
angles
Thin lamella w
1
"0 w
2
"0 [5]
Transition zone calculated —
Thick lamella w
1
"90° w
2
"70" [21]
Back-flip lamella w
1
"120° w
2
"90° [21]
Material stiffness constants
Young’s modulus of mineral E
1
"114 GPa [40]
Shear modulus of mineral G
1
"44.5 GPa [40]
Poisson ratio of mineral m
1
"0.30 —
(calculated for isotropic platelets)
Young’s modulus of collagen E
.
"1.5 GPa [4]
Shear modulus of collagen G
.
"1 GPa [41]
Collagen’s Poisson’s ratio m
.
"0.38 [39]
(corresponds to m
12
"0.35 for the overall
Poisson’s ratio of the collagen—mineral composite
calculated using the rule of mixtures)
! These two zones cannot be differentiated in scanning electron micrographs, so we arbitrarily divided them into two equal halves.
" No data are available for w
2
. Based only on impressions from scanning electron micrographs.
crystals and water more or less follow a regular pat-
tern [39].
Bundles of parallel mineralized collagen fibrils can
be arranged in a variety of ways to form different bone
types and associated tissues such as cementum, den-
tine and mineralized tendon. In this study we focus on
two such bone types: the parallel-fibred bone and
lamellar bone. In terms of the model, we regard paral-
lel-fibred bone as being composed of an array of
mineralized collagen fibrils, aligned not only along
their fibril axes, but also in terms of their crystal layers.
This highly ordered three-dimensional structure
is clearly idealized, and in vivo there is almost cer-
tainly less order particularly in the azimuthal orienta-
tions of the crystal layers [15]. We also use an
idealized rotated plywood structure for modelling
lamellar bone.
An analysis of the basic features of the rotated
plywood structure showed that much of the variation
within the structure can be described in terms of two
angles, the plywood angle, w
1
, and the rotation angle,
w
2
[7]. This relatively simple situation arises because
most of the mineralized collagen fibrils are in planes
parallel to the boundaries between lamellae. The ply-
wood angle, w
1
, describes the extent of offset of the
collagen fibril axes from one layer of collagen fibrils to
the next. The rotation angle, w
2
, is the extent of rota-
tion about the fibril axis from one layer to the next.
Wagner and Weiner [7] arbitrarily defined w
1
"0
when the fibrils are perpendicular to long axes of the
bone, and w
2
"0 when the crystal layers are parallel
to the lamellar boundary plane. Weiner et al. [21]
have measured w
1
angles between associated layers
using decalcified and vitrified thin sections of the rat
femur lamellar bone cut approximately parallel to the
lamellar boundary plane. They proposed a model in
which the w
1
values of a lamellar unit increase in
discrete steps of roughly 30°, such that the thick
lamella fibrils are roughly orthogonal to those of the
thin lamella; a result consistent with other observa-
tions [16, 17]. There also appears to be an additional
lamella in which the w
1
angle is 120°. This ‘‘back-flip’’
lamella borders on the lamellar boundary. In order to
simplify this situation, we propose to use the term
‘‘lamellar unit’’ to include the traditional thin and
thick lamellae, as well as the transition zone between
them and the 120° ‘‘back-flip’’ lamella. The structure
of an individual lamellar unit can be seen in Fig. 1.
Most of the features described above are incorporated
into the theoretical model. The specific values used
are, for the most part, for the midshaft of a rat tibia or
femur and are listed in Table I.
1499

2. Theoretical model
2.1. Elastic constants of parallel-fibred bone
Parallel-fibred bone and an individual lamella of
lamellar bone are here regarded as having the same
structures.
2.1.1. Platelet and ribbon reinforcement
The structure of the collagen—crystal composite shows
that it should be modelled as a unidirectional plane-
parallel (UPP) platelet-reinforced composite, rather
than a fibre composite. UPP platelet ribbon- or sheet-
reinforced composites (where a ribbon has one of its
dimensions much larger than a platelet and a sheet has
both dimensions much larger than a platelet) have
three orthogonal planes of symmetry and, thus, their
elastic behaviour is characteristic of three-dimensional
orthotropic materials, which possess nine independent
elastic constants [42, 43].
Very little theoretical or experimental work has
been performed on the elastic properties of platelet- or
ribbon-reinforced composites, apart from some results
proposed by Halpin [44], a theoretical model by
Padawer and Beecher [45], and a model by Lusis et al.
[46]. Related work by Nielsen [47, 48] is also avail-
able. Some elastic constants (like Young’s moduli)
have been studied more than others, and therefore
theoretical models for some constants may not be
available. In the following it is assumed that the ‘‘1’’
direction is parallel to the long dimension of the plate-
let or the ribbon, the ‘‘2’’ direction is perpendicular to
the former, within the plane of the platelet or ribbon,
and the ‘‘3’’ direction is the out-of-plane direction. The
‘‘1’’ direction in vivo is also parallel to the axis of the
collagen fibril [10]. We now review the models avail-
able for each of the nine elastic constants for the UPP
platelet- and ribbon-reinforced composites.
2.1.1.1. ¸ongitudinal ½oung’s modulus, E
1
. According
to the Halpin—Tsai model [44], the longitudinal (par-
allel to the long axis of the platelet or ribbon) modulus
of a UPP composite is given by
E
1
"E
.
1#AB»
1
1!B»
1
(1)
where
B"
E
1
/E
.
!1
E
1
/E
.
#A
(2)
and where E
.
is the matrix modulus, E
1
is the platelet
(or ribbon) modulus, »
1
is the volume fraction of
platelet reinforcement, given by
»
1
"
A
¸
¸#u
BA
¼
¼#v
BA
¹
¹#d
B
"
A
1#
u
¸
BA
1#
v
¼
BA
1#
d
¹
B
~1
(3)
where ¸ is the platelet length, ¼ is the platelet width
and ¹ is its thickness, and u, v and d are the longitudi-
nal, transverse in-plane and transverse out-of-plane
Figure 2 Schematic illustrations of the arrangements of highly sty-
lized platelet-shaped crystals in parallel-fibred bone introducing the
geometrical parameters and the defined (1, 2, 3) set of principal axes
constituting the model for this specific bone type.
distances between platelets, respectively (Fig. 2). In
particular, for ribbon-reinforced composites (uP0
and ¸PR)
»
1
"
A
1#
v
¼
BA
1#
d
¹
B
~1
The parameter A in Equations 1 and 2 is given by
A"2
A
¸
¹
B
#40»
10
1
(4)
For typical platelet dimensions and platelets volume
fraction in bone (Table I) 40»
10
1
;2(¸/¹). Thus, for
a UPP platelet-reinforced ‘‘boney’’ composite
A+2(¸/¹). In the case of ribbon-reinforced com-
posites, ¸PR, thus APR and (from Equation 2)
BP0 whereas ABP(E
1
/E
.
)!1. Equation 1, there-
fore, transforms into the rule of mixtures
E
1
"E
.
(1!»
1
)#E
1
»
1
(5)
Other models for Young’s modulus, E
1
, of platelet-
reinforced composites were proposed by Padawer and
Beecher [45] and by Lusis et al. [46]. Both have
similar expressions as follows
E
1
"E
.
(1!»
1
)#lE
1
»
1
(6)
in which, for the Padawer and Beecher (PB) model,
l"1![tanh(n)/n], whereas for the Lusis et al.
(LWX) model, l"1![ln(n#1)/n]. In both cases
n"
¸
¹
A
G
.
E
1
B
1@2
A
»
1
1!»
1
B
1@2
(7)
where G
.
is the matrix shear modulus, and the other
parameters were previously defined. Here also, as for
the Halpin and Tsai (HT) scheme (Equation 1), the
rule of mixtures is obtained for ribbon-reinforced
composites, because ¸PR, nPR and lP1in
both cases. The LWX scheme is essentially a modified
version of the PB model, as it takes into account
1500

platelet—platelet interactions at high platelet content.
The PB and LWX schemes are based on Cox’s early
model [9] but they take into account the platelet-like
geometry of the reinforcement. Available experimental
data (for glass, aluminium diboride and silicon carbide
platelets in PB [45] and for phlogopite and muscovite
mica flakes in LWX [46] fall short of all schemes due
to variability in the geometrical and the structural
parameters, but the general trends are quite satisfac-
tory. Thus, three different expressions are available for
the calculation of E
1
.
2.1.1.2. ¹ransverse in-plane ½oung’s modulus E
2
.
Three different expressions are also available for the
transverse in-plane Young’s modulus, E
2
, based on the
HT, PB, and LWX schemes. The expressions are the
same as Equations 1 and 5, but the aspect ratio ¸/¹ is
now replaced by ¼/¹ in Equations 4 and 7. Regard-
ing Equation 4, one has again the simplification
40»
10
1
;2(¼/¹).
2.1.1.3. ¹ransverse out-of-plane ½oung’s modulus, E
3
.
The HT model does not offer an expression for the
transverse out-of-plane Young’s modulus, E
3
. PB,
LWX models are not relevant here because they deal
with the modulus reduction factor, l, in the plane of
reinforcement only. We thus suggest the following
approximation for E
3
. By interchanging axes 2 and 3,
and 1 and 3, one obtains aspect ratios (¹/¼) and
(¹/¸), respectively. As the length and width of an
individual platelet are of the same order of magnitude,
and are both much larger than its thickness (Table I),
the aspect ratio for E
3
, to be used in Equations 4 and
7, is assumed to be A+2¹/(¼#¸). As ¹ gets small-
er with respect to ¼ and ¸, the value of A approaches
zero, as indeed was suggested by Nielsen [48] for
ribbons. For A"0, Equation 1 becomes the inverse
rule of mixtures
1
E
3
"
»
1
E
1
#
1!»
1
E
.
(8)
2.1.1.4. Shear moduli and Poisson’s ratios. For the
shear moduli, G
ij
, and the Poisson’s ratios m
ij
(i, j"1, 2, 3 and iOj) the HT equations may be used.
Table II lists the values of A used in the HT equations,
for the shear moduli and the Poisson’s ratios in the
cases of platelet or ribbon reinforcement, as suggested
by Halpin [44] and Nielsen [48]. Note from Table II
that no expressions for A exist for G
31
and m
31
in the
case of platelets, and for m
12
, m
31
in the case of ribbons.
For these missing values the following approaches will
be taken. Because for ribbon reinforcement
A
G
31
"A
G
23
(Table II), we will assume that the same
identity is valid for platelet reinforcement. Moreover,
because we have A
m
12
"R for platelet reinforcement,
the same will be assumed for ribbon reinforcement.
Finally, for both platelet and ribbon reinforcements
we will arbitrarily assume that A
m
31
"A
m
23
"0.
2.1.2. Sheet reinforcement
Sheet reinforcement is the limiting case of very wide
ribbons (large aspect ratio ¼/¹ ), as discussed by
TABLE II Literature values of the parameter A in Equations
1 and 2 for the principal elastic constants of platelet [44] and ribbon
[47, 48] reinforced composites
Elastic constant A (platelets) A (ribbons)
G
12
A
¸#¼
2¹ B
1.73
A
¼
¹ B
1.73
G
23
1
4!3m
.
0
G
31
— 0
m
12
R —
m
23
00
m
31
——
Nielsen [47, 48]. Elastic constants for this case, as
suggested by Nielsen are
E
1
"E
2
"E
1
»
1
#E
.
(1!»
1
);
1
E
3
"
»
1
E
1
#
1!»
1
E
.
(9)
G
12
"G
1
»
1
#G
.
(1!»
1
);
1
G
23
"
1
G
31
"
»
1
G
1
#
1!»
1
G
.
(10)
m
12
"m
1
»
1
#m
.
(1!»
1
);
1
m
23
"
1
m
31
"
»
1
m
1
#
1!»
1
m
.
(11)
where the volume fraction, »
1
, now becomes, as uP0,
vP0 and ¸PR, ¼PR
»
1
"
A
1#
d
¹
B
~1
(12)
2.2. Elastic constants of lamellar bone
The calculation of Young’s modulus of lamellar bone
must take into account the fact that its basic repeating
structure, the lamellar unit, is composed of stacked
sets of ordered lamellae comprising a thin lamella,
a transition zone, a thick lamella and a fourth 120°
‘‘back-flip’’ lamella. Furthermore, each lamella has
a different orientation in space as defined by (
1
and
(
2
angles [7]. The relative orientation in space, and
thickness, of each lamella are important for accurate
numerical estimations of the overall Young’s modulus
of lamellar bone, using the following approach. We
assume [5, 7] that the principal axes of the thin
lamella are aligned with the principal axes of bone
(Fig. 3) and, thus, the stiffness constants of the thin
lamella in the major axes of bone (i.e. bone axis,
tangential axis, and radial axis) are merely those of an
individual lamella in its principal axes. We will first
transform the principal elastic moduli of each of the
remaining lamellae in a lamellar unit into their corres-
ponding values along the major axes of bone, then use
a rule of mixtures approach to calculate the overall
values of the moduli of bone along its principal axes.
For the purpose of transforming the elastic moduli
of the thick lamella and the ‘‘back-flip’’ lamella
1501

Figure 3 Schematic illustration (top) of the principal axes of a
long bone. BA, bone axis; TA, tangential axis; RA, radial axis. The
lower schematic illustration shows the orientations of the crystals
and the collagen fibrils in the thin lamella in relation to the 3 princi-
pal axes.
each along its principal axes (i.e. 1, 2 and 3 axes of
individual lamellae as defined above) into their elastic
moduli along the major axes of bone, we suggest the
following method.
We consider a rectangular system of coordinates
(x, y, z) with axes oriented arbitrarily with respect to
the principal directions of elasticity (1, 2, 3) of an or-
thotropic body. Following Lekhnitskii [42], we have
a general expression for Young’s moduli E
,,
, shear
moduli G
,-
and Poisson’s ratio m
,-
with k, l"x, y, z
along the arbitrary system of coordinates (x, y, z), in
terms of the cosine directors a
i
, b
i
,c
i
(i"1, 2, 3), the
corresponding Young’s moduli, E
ii
, and the other elas-
tic constants in the principal axes, m
ij
, (Poisson’s ra-
tios), and G
ij
(the shear moduli), as follows
(k, l"x, y, z, i, j"1, 2, 3 and k and l are linked to
i and j, respectively, such that k (or l)"x implies i (or
j)"1, k (or l)"y implies i (or j)"2, k (or l)"z
implies i (or j)"3)
E
kk
"
A
a
4
i
E
1
#
b
4
i
E
2
#
c
4
i
E
3
#a
2
i
b
2
i
I
12
#b
2
i
c
2
i
I
23
#c
2
i
a
2
i
I
31
B
~1
(13)
where
I
mn
"
1
G
mn
!2
m
mn
E
mm
(m, n"1, 2, 3 mOn) (14)
1
G
kl
"4
A
a
2
i
a
2
j
E
1
#
b
2
i
b
2
j
E
2
#
c
2
i
c
2
j
E
3
B
!8
A
m
12
E
1
a
i
a
j
b
i
b
j
#
m
23
E
2
b
i
b
j
c
i
c
j
#
m
31
E
3
c
i
c
j
a
i
a
j
B
#
(a
i
b
j
#a
j
b
i
)2
G
12
#
(b
i
c
j
#b
j
c
i
)2
G
23
#
(c
i
a
j
#c
j
a
i
)2
G
31
(kOl) (15)
m
kl
E
ii
"
C
m
12
E
1
(a
2
i
b
2
j
#a
2
j
b
2
i
)#
m
23
E
2
(b
2
i
c
2
j
#b
2
j
c
2
i
)
#
m
31
E
3
(c
2
i
a
2
j
#c
2
j
a
2
i
)
D
!
A
a
2
i
a
2
j
E
1
#
b
2
i
b
2
j
E
2
#
c
2
i
c
2
j
E
3
#
a
i
a
j
b
i
b
j
G
12
#
b
i
b
j
c
i
c
j
G
23
#
c
i
c
j
a
i
a
j
G
31
B
(kOl) (16)
The cosine directors a
i
, b
i
, c
i
, i"1, 2, 3 determine the
position and orientation of the arbitrary (x, y, z) sys-
tem with respect to the (1, 2, 3) system, via the trans-
formation
A
x
y
z
B
"
A
a
1
b
1
c
1
a
2
b
2
c
2
a
3
b
3
c
3
BA
1
2
3
B
or x"T·1 (17)
where T is the matrix of cosine directors. Thus,
a
1
"cos(x, 1), a
2
"cos(y, 1), etc. It results from
Equations 13—17 that to calculate the nine elastic
moduli in the new coordinates system (x, y, z) one
needs to know: (a) the nine direction cosines a
i
, b
i
, c
i
,
(i"1, 2, 3) and (b) the nine elastic constants (Young’s
and shear moduli, and Poisson’s ratios) in the princi-
pal axes. Expressions for the nine elastic constants in
the principal axes were discussed in the section on
single lamellae. We now deal with the determination
of direction cosines.
For the calculation of the direction cosines the
following approach can be adopted. It is well known,
since Euler, that the relative orientations of two coor-
dinate systems may be specified by a set of three
independent angles [50]. There is a large number
of possible sequential choices for these three angles,
and we adopt the following transformations from
principal (or body) coordinates of the thin lamella
to spatial (or arbitrary) coordinates of thick or
‘‘back-flip’’ lamellae, based on the specific require-
ments in our studies of bone micromechanics [9].
Starting with the principal axes typical of a thin
lamella, and using the right-handed coordinate system
described in Fig. 4, one first rotates the initial (princi-
pal) system of axes 123 by an angle w
1
clockwise about
the 3 axis, and the new coordinate system is labelled
ngf. Next, the intermediate axes, ngf, are rotated
about the n axis counterclockwise by an angle w
2
to
produce another intermediate set, the n@g@f@ axes. Fi-
nally, the n@g@f@ axes are rotated counterclockwise by
an angle w
3
about the f@ axis to produce the desired
xyz system of axes. Fig. 4 illustrates the various stages
of the sequence. The elements of the complete trans-
formation T are obtained as the triple product of the
1502

T"T
3
T
2
T
1
"
A
cos w
1
cos w
3
#sin w
1
cos w
2
sin w
3
cos w
1
cos w
2
sin w
3
!sin w
1
cos w
3
sin w
2
sin w
3
sin w
1
cos w
2
cos w
3
!cos w
1
sin w
3
cos w
1
cos w
2
cos w
3
#sin w
1
sin w
3
sin w
2
cos w
3
!sin w
1
sin w
2
!cos w
1
sin w
2
cos w
2
B
(19)
Figure 4 Rotations defining the Eulerian angles. (a) First rotation,
w
1
, clockwise about axis 3, (b) second rotation, w
2
, counterclockwise
about axis n; (c) third rotation, w
3
, counterclockwise about axis f@.
individual rotations:
¹
1
"
A
cos w
1
!sin w
1
0
sin w
1
cos w
1
0
001
B
¹
2
"
A
10 0
0 cos w
2
sin w
2
0 !sin w
2
cos w
2
B
¹
3
"
A
cos w
3
sin w
3
0
!sin w
3
cos w
3
0
001
B
(18)
and thus, using Equation 17, one has x"T
·
1 where
By identification of the elements of matrix T
using Equations 17 and 19, the cosine directors a
i
, b
i
,
and c
i
are now defined in terms of the elementary
rotations, which are convenient from an experimental
viewpoint.
As discussed, much of the variation within the struc-
ture of lamellar bone can be described in terms of the
plywood angle w
1
which describes the extent of offset
of the collagen fibril axes from the thin lamella to
other lamella in a given lamellar unit and the rotation
angle w
2
which describes the extent of rotation about
the fibril axis relative to fibrils in the thin lamella.
These two elementary rotations are modelled exactly
by the first two rotations of the Eulerian transforma-
tion presented above. No third Eulerian angle w
3
is
needed in this particular case in order to complete the
transformation of the principal coordinates of the thin
lamella into those of the thick or the ‘‘back-flip’’
lamella. Thus, w
3
"0°. Therefore, the matrix for the
complete transformation from thin lamella to thick or
‘‘back-flip’’ lamellae is
¹"
A
cos w
1
!sin w
1
0
sin w
1
cos t
2
cos w
1
cos w
2
sin w
2
!sin w
1
sin w
2
!cos w
1
sin w
2
cos w
2
B
(20)
However, as the principal axes of the thin lamella are
those which correspond (Fig. 3) to the major axes of
bone, we are interested in the inverse transformation,
namely, transforming the principal elastic constants of
the thick and the ‘‘back-flip’’ lamellae into those in the
principal axes of the thin lamella. Thus, the desired
matrix of direction cosines should be that of the in-
verse transformation, which is actually the transposi-
tion of the matrix derived above, as we are dealing
with an orthogonal transformation [50]. Thus, the
matrix of direction cosines for the transformation of
principal axes from thick or ‘‘back-flip’’ lamellae to
thin lamella is
¹~1"
A
cos w
1
sin w
1
cos w
2
!sin w
1
sin w
2
!sin w
1
cos w
1
cos w
2
!cos w
1
sin w
2
0 sin w
2
cos w
2
B
(21)
To conclude, a method is available for calculating the
elastic moduli of the thick and ‘‘back-flip’’ lamellae in
the principal axes of the thin lamella which corres-
pond to the major axes of bone. For simplicity, the
elastic moduli of the intermediate lamellae (i.e. the
transition zone) in the major axes of bone are assumed
here to be the mathematical average of the corres-
ponding moduli of the thin and thick lamellae. More
accurate calculations should take into account the
exact contributions of the lamellae constituting the
transition zone but this awaits the determination of
the values of the rotational angle, w
2
, for each of these
lamellae which has not, as yet, been measured.
Assuming the thicknesses of the four types of lamel-
lae to be ¹ 5)*#,, ¹ 5)*/, ¹ 53!/4 and ¹ "& with a total
lamellar unit thickness of ¹"¹ 5)*#,#¹ 5)*/#
¹ 53!/4#¹"&, the elastic moduli of bone in its major
axes may be assumed to be given by
D
B0/%
kl
"
1
¹
(¹ 5)*/D
5)*/
kl
#¹ 5)*#,D
5)*#,
kl
#¹ 53!/4D
53!/4
kl
#¹ "&D
"&
kl
(22)
1503

where D
B0/%
kl
stands for the overall elastic moduli E, G
and m of bone in its principal axes k, l(k, l"
BA, TA, RA), and D
5)*/
kl
, D
5)*#,
kl
, D
53!/4
kl
, D
"&
kl
stand for the
elastic constants of the thin lamellae, thick lamellae,
the transition zone and the ‘‘back-flip’’ lamellae, res-
pectively, in the same major axes of bone.
2.3. Young’s modulus of bone in an
arbitrary direction
Knowing the principal elastic constants of parallel
fibred bone (or lamellar bone), we are now also inter-
ested in calculating their Young’s modulus in any
direction in space. It is more convenient to do so by
expressing E along an arbitrary direction in space in
terms of spherical coordinates. One possibility is to
rotate the rigid framework of principal axes of parallel
fibred bone (or the rigid framework of principal axes
of lamellar bone) such that the ‘‘1’’ axis (or the bone
axis) would point into that arbitrary direction. This is
performed (Fig. 5) by first rotating the framework by
an angle, u, clockwise about the ‘‘1’’ axis (or the bone
axis) and then by an angle, h, clockwise about the new
‘‘3’’ axis (or the new radial axis). The matrix of trans-
formation is, thus
A
cos h !sin h 0
sin h cos h 0
001
BA
10 0
0 cos u !sin u
0 sin u cos u
B
"
A
cos h !sin h cos u sin h sin 6
sin h cos h cos u !cos h sin u
0 sin u cos u
B
(23)
Thus, the direction cosines a
1
, b
1
, c
1
needed for the
calculation of the Young’s modulus of parallel fibred
bone in an arbitrary direction in space defined by the
spherical coordinates h, u (Fig. 5), using Equation 13,
are now a
1
"cos h, b
1
"!sin h cos u, c
1
"sin h
sin u. Substituting a
1
, b
1
, c
1
into Equation 13 one gets
E(h, u)"
A
cos4 h
E
1
#
sin4 h cos4 u
E
2
#
sin4 h sin4 u
E
3
#cos2 h sin2 h cos2 uI
12
#sin4 h sin2 u cos2 uI
23
#sin2 h cos2 h sin2 uI
31
B
~1
(24)
where
I
ij
"
1
G
ij
!2
m
ij
E
ii
(i, j"1, 2, 3 iOj) (25)
Thus, Young’s modulus of parallel fibred bone in the
space defined by the 1—2, 2—3, and 3—1 planes
(0°)h)90°,0°)u)90°) is given as a function of
its elastic constants in its principal directions and the
spherical coordinates, h, u, as defined above. Equation
24 is valid for the calculation of the three-dimensional
Young’s modulus of lamellar bone in the space defined
by the BA—TA. TA—RA, and RA—BA, as well, with the
bone axis (BA), tangential axis (TA), and radial axis
Figure 5 Definition of direction in space for parallel-fibred bone (or
for lamellar bone) by spherical coordinates. (a) First rotation, u,
clockwise about the ‘‘1’’ axis (or the bone axis); (b) second rotation,
h, clockwise about the ‘‘3’’ axis (or the radial axis). The subsequent
rotations result in the ‘‘1’’ axis (or the bone axis) pointing to the
specific direction in space defined by h and u with regard to its
orientation in the original system of coordinates. Letting
0°)h)90° and 0°)u)90° the whole three-dimensional space
defined by the 1—2, 2—3, and 3—1 planes of parallel-fibred bone (or
the BA—TA, TA—RA, and RA—BA planes of lamellar bone) is
covered.
(RA) replacing ‘‘1’’, ‘‘2’’, and ‘‘3’’ axes of parallel fibred
bone, respectively (Fig. 5).
3. Strategy
The elastic constants of platelet-reinforced parallel-
fibred bone in its principal axes are first calculated
using the HT, PB and LWX models with the para-
meters listed in Table II. Next Young’s modulus of
platelet-reinforced parallel-fibred bone in three-di-
mensional space is calculated using Equation 24. For
this purpose, only the HT model is used as it contains
expressions for all nine elastic constants which are
necessary for three-dimensional calculations. For
comparison, Young’s moduli of platelet-reinforced
parallel-fibred bone in the 1—2 plane are calculated for
the HT, PB and LWX models, and the results for the
cases of platelet, ribbon and sheet reinforcement are
compared.
The calculated elastic constants in the principal
axes of a single UPP platelet-reinforced lamella are
used to calculate (using Equations 13—16) the stiffness
constants of individual thin, thick, and ‘‘back-flip’’
lamellae in the major axes of bone. The orientations of
the latter two lamellae relative to the thin lamella in
terms of w
1
and w
2
are (Table I) w
1
"90° and
w
2
"70°, w
1
"120° and w
2
"90°, respectively. The
values of the stiffness constants of the transition zone
in the major axes of bone are taken as the average
1504

value of the respective thick and thin moduli. Using
the relative lamellae thicknesses listed in Table I, the
overall elastic constants of lamellar bone in its major
axes are calculated using Equation 22. The three-di-
mensional Young’s modulus of lamellar bone is cal-
culated by introducing the latter results into Equation
24. The model for rotated plywood lamellar bone
structure is then examined with respect to several
variables and is compared to existing experimental
data.
4. Results
4.1. Parallel-fibred bone
Fig. 6 shows the three-dimensional Young’s modulus
of parallel-fibred bone using the HT model for platelet
reinforcement. A highly anisotropic structure is re-
vealed with values of E
1
"33.6 GPa, E
2
"24.3 GPa
and E
3
"3.0 GPa for the Young’s moduli in the prin-
cipal axes of parallel-fibred bone. Young’s modulus of
parallel-fibred bone in the 1—2 plane (u"0), as cal-
culated using the three different models is reported in
Fig. 7. The HT, PB and LWX models all yield similar
trends for the relation between stiffness and angle h.
These similarities are expected because the assump-
tion of orthotropy is made in all models. The differ-
ences in three curves are due only to the different
expressions for E
1
and E
2
used in Equation 24 One
striking feature is the presence of a local maximum at
a value of hO0° or 90° for u"0° (1—2 plane). This
maximum progressively disappears (Fig. 6) as u in-
creases. As discussed by Wagner and Weiner [7], it
can be shown [51, 52] that for u"0°, E(h, u) has
a maximum different from both E
1
and E
2
for some
values of h other than 0° and 90° if
G
12
'
E
1
2(1#m
12
)
(26)
and, similarly, E(h, u) has a minimum different from
both E
1
and E
2
for some values of h other than 0° and
Figure 6 Three-dimensional Young’s moduli of parallel-fibred bone
(HT model) in the space defined by the 1—2 plane (u"0°), 1—3
plane (u"90°), nd 2—3 plane (h"90°).
Figure 7 Young’s modulus of parallel-fibred bone in the 1—2 plane
(u"0°) as calculated by the three different models (HT, PB, LWX).
90° if
G
12
(
E
1
2[(E
1
/E
2
)#m
12
]
(27)
The extremum (minimum or maximum) occurs at the
values of h which satisfy
tan2 h"
(2/E
1
)#(2m
12
/E
1
)!(1/G
12
)
(2/E
2
)#(2m
12
/E
1
)!(1/G
12
)
(28)
Thus, for example, using the HT model, the following
elastic constants are calculated for parallel-fibred
bone in the 12 plane (u"0°): E
1
"33.6 GPa,
E
2
"24.3 GPa, G
12
"19.8 GPa and m
12
"0.34. Sub-
stituting these values into Equations 26 and 27 a value
of 12.5 GPa for the right-hand side of Equation 26 and
a value of 9.8 GPa for the right-hand side of Equation
27 are obtained. This fulfils the conditions of Equation
26 but not of Equation 27, implying that E will have
a maximum between 0° and 90°. For Equation 28, this
maximum is calculated to occur at h+37°.
The cases of ribbon and sheet reinforcement as
compared to platelet reinforcement in parallel-fibred
bone are shown in Fig. 8. Ribbon reinforcement en-
hances the Young’s modulus of parallel-fibred bone in
the longitudinal direction of the reinforcement from
33.6 GPa (platelet reinforcement) to 57.5 GPa. The
Young’s modulus in the other two directions remain
the same as for platelets. For the case of sheet rein-
forcement E
1
"E
2
"57.5 GPa, whereas E
3
does not
change. Note, that in the mid-range of the h angle
(around 30°(h(60°) Young’s modulus shows an
intermediate value for ribbon reinforcement, whereas
for sheets, as for platelets, there is a local maximum of
the Young’s modulus in this range.
4.2. Lamellar bone
The three-dimensional Young’s modulus of lamellar
bone as calculated using the HT model for the structural
1505

Figure 8 Young’s modulus in the 1—2 plane (u"0°) of platelet,
ribbon- and sheet-reinforced parallel-fibred bone (HT model).
Figure 9 Three-dimensional Young’s moduli of lamellar bone as
calculated for the material and structural parameters listed in
Table I in the space defined by the bone axis—tangential axis plane
(u"0°), bone axis—radial axis plane (u"90°), and tangential
axis—radial axis plane (h"90°).
parameters listed in Table I versus the load
direction in space, as defined by the spherical coordi-
nates h, u, is shown in Fig. 9. The structure is clearly
anisotropic. The calculated Young’s modulus values
in the three orthogonal directions are shown in
Table III. Young’s modulus of bone in the bone
axis—tangential axis plane, as determined using the
Halpin—Tsai, Padawer and Beecher and the Lusis et
al. models, is shown in Fig. 10. All three models show
a smoothly decreasing trend in Young’s modulus from
the bone axis value to the tangential axis. Note that
the HT and LWX models give approximately the
Figure 10 Young’s modulus of lamellar bone in the bone
axis—tangential axis plane (u"0°) as calculated by the three differ-
ent models (HT, PB, LWX) for the material and structural para-
meters listed in Table I.
same values (26—27 GPa and &9 GPa) for the
Young’s modulus in the bone axis and tangential axis,
respectively, whereas the LWX model gives lower re-
sults (&23 GPa and &8 GPa).
Table III also lists values for the Young’s modulus
in the bone axis and tangential axis (both calculated
by the LWX model), and in the radial axis (calculated
by the HT model) for various values of the rotational
angle, (
2
, of the thick and ‘‘back-flip’’ lamellae. For
each case, results for platelet, ribbon and sheet rein-
forcement are compared. Results for the specific case
in which the ‘‘back-flip’’ lamella in a lamellar unit is
absent, are also listed. Values of all other parameters
(Table I) were kept constant. Note that (
2
values of
the rotated plywood structure are not known, and
hence it is of interest to examine the sensitivity of the
model to variations of (
2
. The case in which indi-
vidual platelets adhere along their width dimension to
form long bands, the longitudinal direction of which is
the former width dimension of platelets (‘‘transversally
formed ribbons’’), which is also structurally possible, is
examined as well. Here, ¼PR (instead of ¸ in Equa-
tions 4 and 7) and, thus, the rule of mixtures is ob-
tained by all three models for E
2
but not for E
1
in an
individual lamella. In addition, A
G
12
"(¸/¹ )1.73
(Table II). All the other values of A for the remaining
elastic moduli for ribbons (Table II) do not change.
Table III shows that variations in (
2
affect the
Young’s modulus values in the radial axis direction in
particular. Furthermore, if the ‘‘back-flip’’ lamella is
removed, the modulus values are much less isotropic.
Table III also shows that the moduli for sheets and
longitudinally formed ribbons as compared to plate-
lets are extremely high in the bone axis (BA) direction,
whereas for transversally formed ribbons, values of the
Young’s modulus in the bone axis are much closer to
those of platelets.
1506

TABLE III Data for Young’s modulus of lamellar bone in the bone axis, tangential axis and radial axis directions for various values of the
rotational angle, (
2
, and for the case in which the ‘‘back-flip’’ lamella is absent: (a) using parameters defined in Table I; (b) as in (a), but
(
2
"50° for the thick lamella; (c) as in (a), but (
2
"30° for the ‘‘back-flip’’ lamella; (d) as in (a), but assuming no ‘‘back-flip’’ lamella present
(a) (b) (c) (d)
Lamella (
1
"90°, (
2
"70° (
1
"90°, (
2
"50° (
1
"90°, (
2
"70° (
1
"90°, (
2
‘‘Back-flip’’ (
1
"120°, (
2
"90° (
1
"120°, (
2
"90° (
1
"120°, (
2
"30° No ‘‘back-flip’’
lamella
PRR*SPRSPRSPR
E
1
(GPa) 22.9 41.2 24.6 48.4 22.9 41.2 48.4 25.3 44.1 51.3 26.2 48.8
E
2
(GPa) 8.3 13.6 6.5 13.6 9.5 14.6 14.8 10.0 15.1 15.4 9.2 15.9
E
3
(GPa) 13.6 12.8 24.4 24.5 9.2 8.6 16.1 10.7 9.8 14.4 12.2 11.1
P"platelet reinforcement; R"ribbon reinforcement; S"sheet reinforcement; R*"transversally formed ribbons (see text).
5. Discussion
The models presented here are based on our current
understanding of the parallel-fibred and rotated ply-
wood structures and their materials properties. The
calculated values reflect the accuracies and inaccur-
acies both in our understanding of the in vivo struc-
tures and in the construction of the mathematical
model, and of course do not take into account the
considerably natural variation that exists in all bones.
With so many unknowns, a fairly stringent way to test
the model is to compare it to measured values of
Young’s modulus in different directions via-a-vis the
bone axial directions, following the approach of Bon-
field and Grynpas [24] and Reilly and Burstein [23].
Unfortunately, here too difficulties are encountered.
No measured modulus values of pure parallel-fibred
bone exist. All the measurements made are of
fibrolamellar (also known as plexiform bone), which is
a mixture of both parallel-fibred and lamellar bone. In
the case of lamellar bone, our model is only relevant to
a planar array of lamellae, and not to the far more
complicated situation of osteonal bone, which is for
the most part, a series of cylinders made up of folded
lamellae [14]. No measurements of the Young’s
modulus of planar lamellar bone have been made to
date, because of the relatively large specimen size
needed. More or less planar arrays of circumferential
lamellar bone are present in certain bones, but these
are almost inevitably from small animals, and thus the
volumes available for modulus measurements, are too
small.
We have recently attempted partially to resolve this
problem by making microhardness measurements on
both pure parallel-fibred bone and planar lamellar
arrays at different angles relative to the bone axes. The
results are reported in Ziv et al. [15]. Although micro-
hardness values cannot be directly related to Young’s
modulus, an approximate linear relation for bone has
been observed26, 53, 54, 55. Ziv et al. [15] measured the
microhardness values of parallel-fibred bovine bone in
a zone close to the periosteum where lamellar bone is
absent. Measurements were made along the three
principal axes. It was found that each has a different
value and that the differences are large (compared to
lamellar bone), indicating a high degree of anisotropy.
These trends are well reproduced in the model for
platelets (cf. Figs 6 and 9 paying particular attention
to the scales). Ziv et al. [15] also measured intermedi-
ate off-axis angles in order to determine whether or
not the off-axis peak in Young’s modulus predicted by
Wagner and Weiner [7] is present. No such peak was
found. Interestingly, the intermediate values measured
are thus more consistent with the crystals being pres-
ent in the form of ribbons rather than platelets (follow-
ing Fig. 8) and it would be most interesting to try to
evaluate this possibility by obtaining better in vivo
information on crystal dimensions. Figs 11 and 12
compare two of the trends observed by Ziv et al. [15]
for lamellar bone with the calculated (LWX model,
platelet reinforcement, parameters as listed in Table I)
modulus values for the same axial directions. The
Figure 11 Trends in the microhardness measurements for the ‘‘a’’
set of planes [15] and in the Young’s modulus of the corresponding
directions in the bone axis—tangential axis plane, as calculated by
the LWX model for platelet reinforced lamellar bone (material and
structural parameters as listed in Table I).
1507

Figure 12 Trends in microhardness measurements for the ‘‘c’’ set of
planes [15] and in the Young’s modulus of the corresponding
directions in the bone axis—radial axis plane, as calculated by the
LWX model for platelet-reinforced lamellar bone (material and
structural parameters as listed in Table I).
trends are remarkably similar, providing strong sup-
port for the validity of the model.
A comparison of the calculated values of Youngs’
modulus with the measured modulus values for bulk
fibrolamellar and osteonal bone are informative, des-
pite the fact that they are not directly comparable. In
general, the well-known measured anisotropy for
fibrolamellar and osteonal bone [1] (higher values
of Young’s modulus in the bone axis direction relative
to the radial and tangential directions) is consistently
reproduced in our model for platelets, ribbons and
sheets. Care must be taken when comparing the abso-
lute values and trends measured with the calculated
values for the reasons outlined above. We will restrict
the discussion to osteonal bone measurements, as at
least here we are dealing with only one structural type
(as opposed to fibrolamellar bone). The relevant data
sets of Reilly and Burstein [23] and Hasegawa et al.
[56] range in Young’s modulus values along the bone
axis to the tangential or radial directions from 35 GPa
to about 8 GPa. Our model using the structural para-
meters in Table I conforms well to this range for the
cases of platelets and ribbons, but not sheets
(Table III). The trends observed for off-axis measure-
ments of Young’s modulus of osteonal bone (Reilly
and Burstein [23], Pidaparti et al. [25]) show a
uniform decrease from the bone axis direction to the
tangential/radial directions. Our model reproduces
this trend well, but also predicts that the tangential
and radial directions of planar lamellar sheets will
have quite different absolute values (from Fig. 9). Such
measurements, when available, could also provide in-
direct information on whether or not the dominant in
vivo crystal shape of a given lamellar bone is in the
form of platelets or ribbons, as these have (Table III)
quite different tangential and radial values.
6. Conclusion
Relating bone structure, and, in particular, the lamel-
lar structure common in mammalian bones, to mech-
anical properties, is indeed a very challenging task.
The difficulties are not only due to the complex hier-
archical structure and the necessity for measuring
mechanical properties of very small specimens, but
also to the considerable structural variation that is an
inherent feature of biological materials. A flexible
mechanical model is, therefore, an important means of
overcoming some of these difficulties. A model of this
type may also be useful for better understanding com-
plex synthetic composite materials, and could, in par-
ticular, provide insights into the relative importance of
the various structural features in terms of bulk mater-
ial properties.
Acknowledgements
This study was funded by US Public Health Service
grant (DE 06954) from the NIDR to S. W. S. W. holds
the I.W. Abel Professorial Chair of Structural Biology.
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Received 19 August
and accepted 20 August 1997
1509