1
Nanofibril
-
mediated Fracture Resistance of Bone
Authors:
Ottman A. Tertuliano
1
, Bryce W. Edwards
2
, Lucas
R.
Meza
3
, Vikram
S.
Deshpande
4
,
Julia R. Greer
2
Affiliations:
1
Mechanical Engineering, Stanford University, Stanford,
CA 94305
, United States
2
Division of Engineering and Applied Sciences, California Institute of Technology
Pasadena, CA
91125, United States
3
Mechanical Engineering, University of Washington, Seattle,
WA 98115
4
Department of Engineering, University of
Cambridge
, Cambridge CB2 1TN, United Kingdom
*Ottman A. Tertuliano
Email:
ottmant@stanford.edu
Keywords
Fracture, Bone,
Toughening mechanisms, fatigue, micro
-
nanoscale mechanics
Abstract
:
Natural hard composites like human bone possess a combination of strength and toughness that
exceeds that of their constituents and of many engineered composites. This augmentation is
attributed to their complex hierarchical structure, spanning multiple le
ngth scales; in bone,
characteristic dimensions range from nanoscale fibrils to microscale lamellae to mesoscale osteons
and macroscale organs. The mechanical properties of bone have been studied, with the
understanding that the isolated microstructure at
micro
-
and nano
-
scales gives rise to superior
strength compared to that of whole tissue, and the tissue possesses an amplified toughness relative
to that of its nanoscale constituents. Nanoscale toughening mechanisms of bone are not adequately
understood a
t sample dimensions that allow for isolating salient microstructural features, because
of the challenge of performing fracture experimen
ts on small
-
sized samples.
We developed an in
-
situ three
-
point bend experimental methodology that probes site
-
specific
f
racture behavior of micron
-
sized specimens of hard material. Using this, we quantify crack
initiation and growth toughness of human trabecular bone with sharp fatigue pre
-
cracks and blunt
notches. Our findings indicate that bone with fatigue cracks is two
times tougher than that with
blunt cracks. In
-
situ data
-
correlated electron microscopy videos reveal this behavior arises from
crack
-
bridging by nanoscale fibril structure. The results reveal a transition between fibril
-
bridging
(~1μm) and crack deflectio
n/twist (~500μm) as a function of length
-
scale, and quantitatively
demonstrate hierarchy
-
induced toughening in a complex material. This versatile approach enables
quantifying the relationship between toughness and microstructure in various complex material
systems and provides direct insight for designing biomimetic composites.
2
Introduction
Strength and toughness are classically described as mutually exclusive properties of engineering
materials
1
. Hard natural materials, such as nacre and bone, have demonstrated an exceptional
combination of strength and toughness, i.e., damage tolerance, that arises from their hierarchical
structu
re which originates from the nanometer length scale
2
. Bone is macroscopically
characterized as the
dense
cortical bone, composed of ~200 μm
-
diameter osteons, or the trabecular
bone architecture composed of similarly sized, curved beams (Fig 1a,b)
3
. The osteons and
trabeculae are comprised of 3
-
5 μm thick lamellar structures (Fig 1c,d) that are comprised of 50
-
200 nm
-
diameter mineralized collage
n fibrils
4
(Fig 1e). Formed via a cell mediated process, these
nanoscale fibrils are primarily composed of collagen and calcium phosphate
-
based bioapatite
nanocrystals.
The microstructural origin of
strength and deformation at the bone organ level (e.g., femur, tibia,
etc...) has been studied over the past four decades, and its bending and compressive strength
increases with mineral content
5,6
. More refined descriptions of bone deformation have been
developed using
in situ
X
-
ray uniaxial tensile experiments on bovine
femur
(
150
μ
m
x 50 μm x
3
mm
)
that attribute its post
-
yield deformation to the sliding of mineralized collagen fibrils with
respect to
each other
7
. Experiments on whole organs offer insights into their overall
properties but
are not capable of capturing or isolating the effects of microstructural constituents within bone on
its mechanical response upon loading because of its complex, multi
-
scale hierarchy. Micro
-
and
nano
-
mechanical compression experiments on in
dividual lamellae cylinders extracted from
trabeculae of human femoral bone have enabled site
-
and microstructure
-
specific measurements
of bone strength and revealed a “smaller is stronger” trend when
sample dimensions ranged from
5
000 nm down to the 250
nm
-
wide collagen fibrils
8,9
. This size effect was attributed to the scarcity
of failure
-
initiating surface defects (e.g., pores and interfac
es) in small
-
scale samples relative to
macroscale tissue.
The complex bioapatite
-
collagen hierarchy has been credited with improving bone toughness
10
,
yet no experiments that unambiguously demonstrate and quantify the contribution and the role of
individual constituents within the hierarchical microstructure of bone to its crack growth
resistance have been reported.
Hydrated
micropillar bone compressi
on experiments
have shown
evidence of fibril kink bands that result in axial splitting and fracture
11
.
In sit
u
tension studies on
~1 mm bone during atomic force microscopy(AFM) suggest that non
-
collagenous proteins
(NCPs) in interlamellar regions of osteonal bone result in an elastic mismatch with lamellae and
provide preferred path for crack propagation
12
.
Fracture studies on dry and wet bone specimens
with nominal dimensions above 5 mm have measured single value toughness based on collagen
orientation
13
and reported toughening mechanisms that are active at various length scales: from
diffuse
inter
-
lamellar microcracking at the microscale to uncracked
-
ligament bridging and crack
deflection/twist at larger length scales
13,14
.
Nanoscale toughening mechanisms, such as the
uncoiling of collagen molecules and the bridging
of crack surfaces by fibrils (i.e., fibril
-
bridging), have been proposed but ne
ver experimentally
isolated or quantified during crack growth
13,15,16
. These mechanisms are hypothesized to
collectively incite crack growth resistance in bone, manifested by requiring a greater driving force
to extend a pre
-
existing crack through the material, typically described as a rising “R
-
curve”, a
phenomenon that remains to be experimentally demonstrated
at the lamellar scale
10
. Identifying
3
and quantifying the contributions of individual and coupled
elements within the hierarchical
microstructure of bone at each relevant length scale, from ~100 nm
-
diameter mineralized collagen
fibrils to ~5 μm
-
sized lamellae to 200 μm
-
sized trabeculae and osteons, is critical to enable
formulation of crack growth resi
stance in bone as a function of relevant microstructural
components within its hierarchical construct.
The dearth of experiments on small
-
scale fracture stems from the fracture toughness experiments
being classically standardized for the macroscale, i.e.
for samples dimensions much larger than the
characteristic length scale of the material being tested. In contrast to the now
-
ubiquitous
experiments on measuring compressive strengths of nano
-
and micro
-
sized materials, fracture
experiments have not been pu
rsued for
micron
-
sized
specimens
17
–
20
because 1) linear elastic and
elastic
-
plastic fracture mechanics (LEFM, EPFM) place restrictions on the minimum specimen
size and 2) the practiced standards f
or fracture experiments that provide valid measurements of
toughness at the macroscale pose significant fabrication and experimental difficulties when
adapted for the microscale
21
. Microscale methods based on notched beam geometries have been
reported, but the asymmetric loading conditions on the notch tip of individual cantilevers and
premature failure of clamped bea
ms at the elastically
-
fixed boundary conditions limit viability of
these techniques, particularly for heterogeneous materials
22
.
We
developed an experimental
methodology that enables conducting
in
-
situ
three
-
point bending
fracture experiments on free
-
standing micron
-
sized beams, similar to the ASTM standardized
macroscale single edged notched (SEN
B
) bend experiments
20
. We validated this methodology
using single crystalline silicon samples with the same dimensions as a standard to reveal fracture
toughness of 0.94 MPa m
1/2
. We utilized this methodology to directly
observe and quantify crack
initiation in 10 μm
-
wide specimens extracted from
the lamellar structure of
individual human
trabeculae, as well as to measure fracture toughness during micro
-
scale stable crack
growth
. We
place the measured crack growth toughne
ss of bone lamellae in the context of bridging zone
model
23
, whereby we attribute the energy dissipated during crack growth to bridging of the crack
surfaces by the underlying nanoscale fibril structure. We observe a ~7 times lower toughening
rate in the crack growth resistance of the microscale trabecular lamell
ae up to ~1 μm crack
extensions compared to that of the previously reported macroscale cortical bone specimens at 500
μm and larger crack extensions
10
. We attribute this crack growth resistance behavior to the
transition from fibril bridging dominated fracture at the microscale to the crack deflection/twist
dominated fracture observed at larger crack
lengths.
Results
Performing
three
-
point
bend fracture experiments at the microscale
Figure 1 illustrates the fabrication process where we used a dual beam Ga
+
focused ion beam
-
scanning electron microscope (FIB
-
SEM) to excise site
-
specific fracture specimens from the
lamellar structure of trabecular bone (Fig 1a) with nominal dimensions of 40 μm
in span (S), 10
μm in width (W), and 5 μm in thickness (B) (Fig 1b)
. These dimensions were carefully
determined to meet the plain strain small
-
scale yielding conditions
necessary to measure valid
fracture toughness in accordance with ASTM
20
(see Methods). Using a nano
-
manipulation
needle in the FIB
-
SEM, we transferred the samples onto pre
-
fabricated parallel silicon supports
to create a free
-
standing microscale beam bending geometry
(Fig 1d)
.
A total of 13 bone
specimens were created: 3 beams were fabricated without notches as controls to characterize the
4
bending strength and deformation behavior of bone at the lamellae length scale,
5 beams were
fabricated with FIB
-
produced through
-
n
otches of length
푎
!
~
0
.
4
푊
, and 5 beams with slightly
shorter FIB
-
produced through
-
notches were subsequently fatigue pre
-
cracked into realistically
sharp cracks to a similar length
(
푎
!
~
0
.
4
푊
) as the notched beams (Fig 1d). The fatigue pre
-
cracking c
onditions were in accordance with ASTM standards; the specimens were loaded
cyclically at a frequency of 100 Hz with the nominal applied stress not exceeding 75% of the
bending yield stress of bone
measured in the present study (585 ±19 MPa)
, until the des
ired
crack length of
0
.
4
푊
was reached, which corresponds to ~10
4
cycles (Fig S7)
.
The specimens
were prepared such that the nominal orientation of collagen fibrils was parallel to the span of the
specimens, as shown by transmission electron microscopy (
TEM) in Figure 1e, to experimentally
mimic the physiologically relevant “breaking” or crack
-
arresting orientation of bone
10,13
.
Additional details on sample preparation is provided in Meth
ods.
We conducted
in situ
3
-
point bending fracture experiments in an SEM using a nanoindenter
equipped with a diamond rounded wedge tip to apply the axial load along the notch (Fig 1d). To
validate this microscale methodology in the context of LEFM, we pe
rformed the experiments on
5 single crystalline silicon beams prepared similarly to the bone samples. The initial notches
were fabricated to orient the cracks along the {110} planes and to travel in the <100> directions
of the silicon. The load vs load
-
lin
e displacement data was acquired at a rate of 100 Hz by the
nanomechanical module and corrected post
-
process by subtracting the contribution of
indentation into the specimens by the three contact points in the bending setup (Fig S1). Our
experiments reveal
ed that the silicon beams respond linear elastically up to catastrophic failure at
a displacement of 106.9 ± 10.1 nm and a critical load,
푃
#
,
of 3.28 ± 0.28 mN (Supplementary
Video 1 and Fig S2). Fig. 2a contains a plot of the mode
-
I crack opening stress
intensity factor,
퐾
$
, as a function of load
-
line displacement. For a macroscale 3
-
point bending fracture specimen,
the stress intensity factor is linearly proportional to the applied load,
푃
,
as
퐾
$
=
%&
'
(
!
"
푓
.
)
(
/
, where
푓
.
)
(
/
is a dimensionless
function of the fracture specimen geometry
20
(see Supplementary Information). We calculate the crack initiation toughness of silicon,
퐾
$#
,
to
0.94 ± 0.08 MPa m
1/2
, defined as the
stress intensity at which failure occurs. This value is within
5% of the typical toughness of Si reported in larger fracture specimens
24
–
26
.
Deformation and fracture response of lamella
r
bone
To characterize the deformation of lamellar bone, we first performed 3
-
point bending
experiments on unnotched bone specimens. Figure 2b shows maximum tensile stress versus
strain data during bending, calculated as
휎
=
*
+
%&
'
(
"
and
휖
=
6
,(
&
"
from the me
asured load
-
line
displacement,
푢
, and load, P, for 3 bone specimens, one of which was “glued” to the supports
using a platinum organometallic injection in the FIB (square data symbols). The data indicates
that all specimen deformed linear elastically to f
ailure, with an elastic modulus of 20 ±2 GPa,
which was calculated by linear fitting of the middle third of the stress versus strain data
(Supplementary Video 2). This modulus is in agreement with the 20
-
27 GPa reported for
trabecular and cortical bone loa
ded in tension and compression along the mineralized fibril axis
in micrometer
-
sized specimens, and along the osteon axis in millimeter specimens
9,10
. We
measure an average failure strength of 585 ± 19 GPa, which is ~70% higher than the 50
-
350
MPa range of bending strengths reported for millimeter
-
sized bone specimens from a variety of
species
6
. Those authors attribute the spread in strength to the extensive porosity in bones at
5
millimeter and larger length scales. The bending
strength of lamellar bone in micron
-
sized
specimen is expected to be higher than in larger, more porous specimen
8,9
.
The predominantly br
ittle behavior of micron
-
sized bone specimens allows applying LEFM to
calculate fracture toughness based on the data generated by fracture experiments on the notched
and fatigue pre
-
crack
ed
specimens. Figure 2 contains representative plots of
퐾
$
as a fun
ction of
load
-
line displacement of the notched Fig. 2d) and
fatigue
pre
-
cracked (Fig. 2f) bone specimens,
both generally exhibiting regions of linear behavior up to a displacement of 500
-
700 nm (region
I
-
II Fig 2d,f) followed by inelastic deformation (reg
ion II
-
III Fig 2d,f) and failure a
t
displacements greater than 1 μ
m.
The
in
-
situ
video frames confirm that slight deviations from
linearity in the initial stress intensity data, up to the displacements of ~300 nm and prior to crack
initiation, are caused b
y the imperfect initial contact between the indenter tip and the sample, as
well as between the sample and support surfaces (Supplemental Video 3). To distinguish the real
crack initiation and extension events from these settling events during the experime
nt, we
superimposed an oscillation at a frequency of 45 Hz to the applied load that resulted in a 2 nm
amplitude displacement to measure the instantaneous contact stiffness of the beam, which
produces a more precise measure of modulus
27
; further details are available
in the Supplementary
Information and Fig S2. The data in Figure 2d and the corresponding video frames throughout
the experiment, shown in Figure 2e, convey that the stress intensity in the notched specimens
increased linearly until the initiation of a c
rack at a
퐾
$#
of 0.54 ± 0.15 MPa m
1/2
followed by
stable crack growth
.
The
in situ
SEM images show sample morphology evolution during bending
up to crack initiation (Fig 2eI
-
II),
and the insets contain magnified images of the notch tip after
the emergence of the crack, revealing its trajectory to be straight, without any deflections, an
d
propagating through the height of the specimen until catastrophic failure (Fig 2eII
-
III,
Supplementary Video 3). Figure 2f shows that in the
fatigue
pre
-
cracked bone specimens, the
initial linear increase in the stress intensity with load
-
line displaceme
nt occurred up to a
displacement of 700 nm, during which we observed the separation of the fatigue pre
-
cracked
surfaces shown in the second panel of Figure 2g. Cracks subsequently initiated a
퐾
$#
of
1.08±0.06
MPa m
1/2
and continued to propagate stably t
hrough the specimen (Figure 2gII
-
III)
until catastrophic failure (Fig 2gIII, Supplementary Video 4).
To characterize the toughness of bone during crack propagation, we plotted the crack growth
resistance of the
fatigue
pre
-
cracked bone specimens as a func
tion of crack extension (R
-
curve)
in Figure 2c by calculating the nonlinear elastic J
-
integral. The J
-
integral accounts for 1) the
linear elastic contribution to toughness in
plane
strain,
퐽
-.
=
퐾
$
+
(
1
−
휈
+
)
/
퐸
, where
휈
is the
Poisson’s ratio (~0.3 for
bone
28
) and
퐸
is the elastic modulus, and 2) the inelastic contributions
to toughness, i.e. the plastic work done during crack extension, which we calculated by
incrementally integrating the area under the load vs load
-
line displacement data durin
g crack
growth, as described by ASTM (Supplementary Information, Fig S7). We calculated crack
extension from the increase in compliance caused by crack growth: the measured dynamic
stiffness at crack initiation was converted to compliance and related the i
nitial crack length using
a standard effective compliance calibration method (Supplementary Information)
14,20
.
Figure 2c
shows the crack growth resistance of the
fatig
ue
pre
-
cracked specimens, characterized by their
stable crack growth over 2 μm and up to a 3 times increase in toughness from a mean 40.6 ± 4.8
J/m
2
at crack initiation up to ~120 J/m
2
at catastrophic failure for the toughness specimen.
6
Discussion
Valida
tion of three
-
point bend fracture methodology
Isolating and quantifying the damage tolerance of bone at the fibril and lamellae length scales
remains a substantial challenge due to a lack of microscale experimental fracture work. Jaya, et
al
22
evaluated the efficacy of the few proposed methodologies aimed at measuring the toughness
of micron
-
sized materials, with the most prevalent being the notched cantilever
fracture
geometry. Using this approach, these authors reported an average
퐾
$#
of 0
.76 MPa m
1/2
for single
crystalline silicon
beams with dimensions (20 x 5 x 5 μm )
22
, a result in the lower range of 0.7
-
1.3 MPa reported in liter
ature
24,26
.
Using the microscale 3
-
point bending
fracture methodology
developed in this work, we measured a
퐾
$#
of 0.94 ±0.08 MPa m
1/2
for silicon, consistent with
the ~1 MPa m
1/2
reported from 3
-
point bend and double
-
cantilever fracture experiments
conducted on macroscale single crystalline silicon
24
–
26
. This improvement in the accuracy of
toughness measurements offered by the microscale 3
-
point bend fracture methodology developed
in this work stems from the elimination of the inherent fracture mode
-
mixity that exists in other
experiments, i.
e. cantilever geometry
22
, because the notch is located directly on the load
-
line of
the wedge tip (Fig 1d), resulting in a symmetric crack opening
(Supplementary Videos 1,3,4) and
a mode
-
I measure of fracture toughness,
퐾
$#
.
The fracture toughness of silicon was measured from specimens with notch radii,
휌
≤
125
푛푚
.
(Fig S8a) rather than ones containing atomically sharp cracks,
휌
→
0, typically
required by
LEFM for measuring a true
퐾
$#
. It has been experimentally shown that as the crack tip radius in
brittle materials decreases below some material dependent crack tip radius, the apparent
toughness also decreases and converges to
퐾
$#
29,30
. The agreement between the experiment
ally
measured toughness of micro
-
sized silicon in this work and that measured for atomically
-
sharp
pre
-
cracked bulk
-
scale silicon reported in literature
24,26
suggests that the close
-
to
-
pure mode
-
I
loading conditions and the relatively small notch radii enabled by the methodology developed in
this work allo
w for measuring valid fracture toughness of micro
-
sized materials using established
macroscale standards.
Nanoscale fibrils enhance crack initiation toughness
Using this microscale methodology, we measured a 2
-
times higher crack initiation toughness of
퐾
$#
=
1.08 ±.06 MPa m
1/2
in the bone specimens containing atomically sharp
fatigue
pre
-
cracks
than the
퐾
$#
= 0.54 ± 0.15 MPa m
1/2
characteristic of those with FIB milled notches (
휌
~
80
푛푚
)
.
This result is surprising because it is counter to
both, the empirical relationship between the
crack tip radius and the apparent toughness in brittle materials
29,30
, suggesting an additional
contribution to the measured toughness.
Figure 2g and Supplementary
Video 5 reveal
mineralized collagen fibrils bridging the
fatigue
pre
-
crack surfaces
of the specimens during
linear elastic loading, prior to crack initiation. In fiber
-
reinforced brittle
-
matrix engineering
composites, fracture typically first occurs via matrix cracking, debonding of the fiber
-
matrix
interface from the matrix, and fiber pu
llout from the matrix material, as illustrated in Figure
s
3
b,
c. Crack initiation toughness is associated with the first event, i.e. matrix cracking because it
represents the onset of permanent damage
31
. This suggests t
he intact mineralized collagen fibrils
in the
fatigue
pre
-
cracked bone specimen observed here provide additional toughening by
shielding the crack tip from remotely applied stress
32
. This mechanism h
as been previously
deemed negligible in toughening large cracks in cortical bone (
푎
/
~
1
푚푚
), in part because the
fibril diameters range from 50 to 200 nm
33,34
, but it is kn
own that large cracks grow from
7
smaller
(
푎
~
1
휇푚
)
, fibril
-
bridged microcracks that are initiated by cyclic loading, i.e., fatigue
35
,
suggesting the effect of nanofibril toughening at lar
ge cracks lengths may not be negligible.
The 2
-
times higher crack initiation
toughness
of
fatigue
pre
-
cracked bone specimens relative to
notched result is counter to well
-
established notions in fracture mechanics that the effective
toughness of a blunt n
otch is significantly higher than a sharp crack due to the reduction in the
stress levels at the notch root compared to those at the tip of a sharp crack
36
. However, these
notions are typically valid for continuum materials which do not display a strongly stochastic
response. The failure of brittle ceramics with stochastic response is typically characterized by
Weibull statistics and here in an attempt to r
ationalize this counterintuitive behavior we combine
Weibull statistics with the widely used RKR fracture model
37
, which postulates that fracture
occurs when the stress over a characteristic length scale
푟
ahead of the crack
-
tip attains a critical
value
휎
/
. We performed Finite
Element (FE) tension applied to a plate with edge
-
cracks of
varying root radii
휌
as shown in Figure S5a. Combining the RKR model with Weibull statistics
we defined the probability of survival as
푃
0
=
exp
H
−
1
2
3
"
∫
.
4
""
4
#
/
5
푑
A
L
, where
휎
++
is the local
tensile
stress distribution ahead of the notch and
푚
is the Weibull exponent that parametrises the
stochastic nature of the response (higher
푚
implies a more deterministic response). The critical
stress intensity
퐾
$#
is defined as loading corresponding to
푃
0
=
푒
6
1
. Additional information on
the simulations can be found in the Supplementary Information. Simulation suggest that for a
material with the Weibull modulus of bone (
푚
=
3.3
-
5.7)
38
–
40
, the
fatigue
pre
-
cracked
specimens, i.e., those with
7
3
≪
1
,
can achieve
퐾
$#
values nominally s
imilar to those of the
notched specimens, whose
7
3
<
1
(Fig
S5
b
). Because the experiments show a 2
-
times higher
measured
퐾
$#
in
fatigue
pre
-
cracked specimens, it is likely that the experimentally observed
fibril
-
bridging in the
fatigue
pre
-
cracked specim
ens primarily contribute to their higher crack
initiation toughness
, but a contribution from a statistical size
-
effect as described here is not
negligible
. Our experiments demonstrate that mineralized collagen fibrils with nanoscale
dimensions serve to sup
press fracture initiation by increasing the critical driving force needed to
grow physiologically relevant microscale cracks formed via cyclic fatigue loading in bone.
Nanofibril
-
bridged crack growth in bone lamellae
The macroscale toughness of bone is primarily defined by its resistance to crack growth, not just
to crack initiation, because of its hierarchical structure. The macroscale crack growth resistance
of bone has
been mainly attributed to the deflection and tw
isting of cracks at the so
-
called
cement lines, i.e., the highly mineralized interfaces with dimensions on the order of 5 μm in
cortical bone
10,13
;
at the microscopic lamellar length scale
s, stable crack growth and mechanisms
that enable it have yet to be experimentally observed and quantified. This represent the main
focus of this work.
Supplementary Video 5 and Figure 2c show stable crack growth up to an
ASTM
-
valid 1.2 μm of vertical disp
lacement in lamellae bone specimens with realistically sharp,
fatigue
-
generated pre
-
cracks. The video also shows the active role of the fibrils between the two
surfaces that bound the crack during
its growth. Electron microscopy images reveal the presence
of multiple fibrils, with an average diameter of 51 ± 10 nm
(measured from n=12 fibrils)
,
protruding from one fracture surface of the beam (Fig.
3
b), with the complementary fracture
surface containing multiple holes with an average diameter of 53 ± 10 nm (
Fig.
3c
,
S9,
measured
8
from n=56
). These post
-
mortem images of bone fracture surfaces indicate that a “bridging zone”
that contains nanometer
-
sized fibrils behind the crack tip.
In engineering composites designed to leverage fiber bridging for toughening, the change in
fracture energy
during crack growth,
Δ
퐽
,
primarily arises from the fiber
-
matrix debonding and
fiber pullout as a function of crack opening displacement,
훿
, as shown in the schematic in Fig
3d
23,41,42
. To determine the increase in fracture energy as a function of crack opening
di
splacement,
Δ
퐽
(
훿
)
, we correlate
퐽
(
Δ
푎
)
,
shown in Figure 2c, with the corresponding δ obtained
from the
in situ
video frames using an image processing procedure described in Methods
(Supplementary Video 5), and plot it in Figure 3e. This plot conveys
an incr
ease in toughness of
50
-
70 J/m
2
in the
fatigue
pre
-
cracked bone specimens during
a bridging zone opening of ~ 250
nm
, which is over twice the crack initiation toughness
~40.6
J/m
2
.
Given the fractographic
evidence and the over two fold increase in toughness during crack growth, we hypothesize that
fibril bridging is the main toughening mechanism at small crack length scales (~1 μm crack
growth and ~250 nm crack opening).
To probe t
he hypothesis of the fibril bridging
as the
primary
toughening
mechanism,
we
place
the
measured
toughening data in the context of a bridging zone model that accounts for fibril
debonding and pullout, in which the change in the fracture energy is given by
Δ
퐽
(
훿
)
=
∫
휎
(
훿
)
푑훿
, where
휎
(
훿
)
is the stress in the bridging zone (Fig 3d)
43
.
We express the
debonding contribution to
휎
(
훿
)
as
휎
8-9!:8
(
훿
)
=
.
;
1
6
;
/
R
+
<
=
$
>
$
S
%
"
훿
%
"
,
where
휙
,
푅
?
, and
퐸
?
are
the fibril area fraction, radius, and elastic modulus, respectively, and
휏
is the fiber
-
matrix
interfacial frictional stress
31,44
. Integrating the stress gives the debonding energy as
Δ
퐽
8-9!:8@:A
(
훿
)
=
R
2
3
S
R
휙
1
−
휙
S
X
2
휏
퐸
?
푅
?
Y
1
+
훿
*
+
(1)
Given the exact natur
e of chemical bonding between the effective fibril and matrix in the present
experiment is
not well characterized
,
and the initial energy dissipation is likely a result of
frictional sliding along the debond zone
23
, or ‘frictional debonding’ here for brevity.
T
he
physical meaning of Eq. 1 is that before fibril failure, the energy dissipated through frictional
debonding monotonically increases with crack opening displacement and the fiber
-
matrix
interfacial friction.
If we assume a fiber subsequently fails at the top of this frictionally debonded
zone, the maximum contribution form pullout can be calculated. T
h
e pullout stress can be
expressed using the Hutchinson and Jensen model as
휎
B,..!,C
(
훿
)
=
휎
/
−
+
;<
>
훿
, where
휎
/
is the
maximum tensile stress supported by the bridging fibrils. The physical limit occurs when fibrils
are completely pulled out of
the matrix (Fig 3d) and cannot dissipate additional energy, i.e.,
휎
B,..!,C
(
훿
)
= 0 at
훿
=
4
#
>
$
+
;<
. The pullout energy can then be represented as:
Δ
퐽
B,..!,C
(
훿
)
=
⎩
⎪
⎨
⎪
⎧
휎
/
훿
−
휙휏
푅
?
훿
+
for
훿
≤
휎
/
푅
?
2
휙휏
휎
/
+
푅
?
4
휙휏
for
훿
>
휎
/
푅
?
2
휙휏
(2)
9
For crack openings below
훿
≤
4
#
>
$
+
;<
, the energy dissipation increases to its maximum value,
Δ
퐽
B,..!,C
=
4
#
"
>
$
D
;<
, and remains at that limit for greater crack opening displacements. Full details
on the derivations of Eqs. 1 and 2 and the bridging model can be found in the Supplementary
Information. The physical behavior described by Eqs. 1 and 2 is in agreement with
the bridging
models in various fiber reinforced composite systems, monotonically increases during debonding
and exhibit a limit due pullout at large crack openings
23,42,45
.
To evaluate the efficacy of this model, we substitute the experimentally measured
푅
?
~
25
푛푚
(Fig 3a),
휙
?
~
0
.
15
(Fig S9),
퐸
?
~
20
.
5
퐺푃푎
and
휎
/
~
585
푀푃푎
(Fig 2b) into Eqs. 1 and 2 and plot
the energy dissipated by the fibrils as a function of crac
k opening displacement in the
experiments. The plot in Figure 3e contains this data and places it in the context of a bridging
mechanism property space delineated by the contours of dissipated energy for distinct failure
mechanisms calculated based on Eqs.
1 and 2:
frictional
debonding
-
only
(teal region, bottom
right),
fibril
pullout
-
only
(light green region, center), and
the combination of fibril bridging and
non
-
bridging mechanisms not quantified in this work such as uncoiling of collagen (yellow
region,
top left)
15
. We observe that the measured fracture energy in all fatigue pre
-
cracke
d
specimens initially exceeds the energy dissipated from debonding
-
only, which
suggest
that
most
of the
fibrils
likely debonded
from
the matrix during and/or after the fatigue loading. At larger
crack openings,
훿
>
50
푛푚
, we observe an increase in fractur
e energy by 50 J/m
2
, with fracture
mechanism remaining within the pullout domain; 2 out of 5 specimens had a 70 J/m
2
increase in
fracture energy, which places them in the upper limit of the region defined by the fibril bridging
model. The ability and the
efficacy of fiber bridging to toughen a composite depends on the
relative strength of the composite constituents compared to the strength of the fiber
-
matrix
interface
,
휎
/
/
휏
23,46,47
. The two different limiting toughness values measured here, 50 and 70
J/m
2
, can potentially be explained by a change in the ratio of tensile to inte
rfacial stress in the
bridging zone,
휎
/
/
휏
, which controls the crack opening displacement and sets a limit for fracture
energy of the pullout (Eq. 2). It is reasonable to approximate an upper bound of the interfacial
frictional stress,
휏
=
154
푀푃푎
,
as t
he critically resolved shear stress of uniaxially compressed
micropillars extracted from the same bone samples
8
because shearing in bone is attributed to the
sliding of fibrils through the matrix
7,9
, similar to the debo
nding and pullout during crack growth
proposed here. The pullout domain in Figure
3e contains contours of constant
휎
/
/
휏
ratios
ranging from
2.
6
to 3.8, attained by holding
휏
constant at its upper limit of 154 MPa and
changing
휎
/
. The experimental data
of the lower toughness specimens aligns
well
with the
4
#
<
=
2
.
6
contour
, while
the tougher specimens
agree with
the
4
#
<
~
3
.
5
푡표
3
.
8
contour.
In the context of the fibril bridging model, the difference in the measured toughness of R
-
curves
of lamella
e bone could be related to the distribution of fibril strengths in the specimens, which is
common in fiber reinforced composites
42,45
. For all fatigue pre
-
cracked specimens, the
experimental toughness measurements, post
-
mortem fracture images, and bridging mechanism
map all quantitatively demonstrate that bridging of crack surfaces by nanoscale mineralized
collagen fibrils is the domina
nt toughening mechanism in the microscale lamellae structure of
bone and must be accounted for when describing the toughness of bone across its complex
hierarchy. A similar mechanism is expected to be active in the notched specimens once the crack
10
has
init
iated
,
however pre
-
cracking enables larger initial crack mouth openings
for
accurate
measurements for the analysis performed here.
The experiments here are performed
in vacuo
;
the
results and model
should be considered
in the
context of deformation of hydrated bone
.
The f
ibril bridging observed and measured
here
in
dehydrated bone is qualitatively similar to that which has been observed in fractography of
hydrated bone fracture experiments
13,34
.
Dehydration increases
collagen
stiffness
and
causes a
loss of
viscoelastic
ity
48
; m
icropillar compression on
rehydrated
bone lamellae have shown the
plastic strain
-
to
-
failure is a factor of 4 higher
than
that
in
dry
bone, while the
yield stress is a
factor of two lower
9,11
.
It is
reasonable
that
stable crack growth
in hydrated bone
lamellae
would
be mediated by mineralized collagen
fibril plastic strain
and extrafibrillar sliding
at lower loads
,
relative to dry bone
,
prior to
failure
.
This could
lead to larger crack opening
displacements and a
higher toughness
(Eqs. 1 and 2)
than measured in the present
dry bone study.
To put our results
in the context of single mineralized collagen fibril pullout experiments in environmental SEM
49
,
we estimate the fracture energy contribution per mineralized collagen fibril as
훾
=
Δ
퐽
/
푛
. Here,
푛
is the number of collagen fibrils in the given crack area given by
휙
(
Δ
푎퐵
)
/
(
휋
푅
+
)
. We calculate
훾
in our experiments to range 0.13 to 0.18 J/m
2
per fibril. This is lower but consistent with of
results of pullout experiments of antler mineralized colla
gen fibril which report
훾
~ 0.2 J/m
2
.
Although
in vacuo,
t
he present study
quantifies
fibril bridging
toughening
that is likely
amplified
under hydrated conditions
;
a
dedicated study to quantify the
effect
hydrate
d
mineralized
collagen
during
crack growth
is necessary
.
Conclusion and Outlook
Enhanced toughness through hierarchy
It is generally accepted that toughening mechanisms in hierarchical hard biomaterials exist and
operate at multiple length scales but quantitative
description of the rising R
-
curve across the
levels of hierarchy in bone had not been determined
50
.
Figure 4 shows a juxtaposition of our
experimental data of crack growth resistance in the 1 μm crack extensio
n regime to the
analogous R
-
curve for 500 μm of crack extension extracted from Ref.
10
.
In this c
ontext, an
effective stress intensity during crack growth,
퐾
E
, is calculated from the J
-
integral using the
mode
-
I elastic equivalence given by
퐾
E
=
(
퐽퐸
′
)
1
/
+
, where
퐸
′
is the
plane
strain elastic modulus.
The data is also fit to a phenomenological power
law of the form
퐾
E
~
Δ
푎
G
, similar to the fit
performed with the J
-
integral of metals
20
(Methods), where
훽
describes a phenomenological
toughening rate exponent with respect to c
rack growth. Figure 4 shows the large crack length
toughness of bone characterized by
훽
=
0
.
94
, where crack deflection/twist at stiff interfaces of
osteons contribute most to the R
-
curve of bone. By extrapolating the macro
-
scale R
-
curve using
the power law
fit down to small crack extensions relevant to the present work (dashed line in Fig.
5), we observe that it significantly underestimates the R
-
curve of micro
-
scale bone. We
discovered that by isolating the fibril microstructure in micron
-
sized beams, its t
oughness at 1
μm of crack extension is ~20 times higher than that predicted from macroscale fracture
experiments. Extrapolating the microscale R
-
curve from present study, defined by
훽
=
0
.
14
,
shows an intersection with the macroscale experiments a crack ext
ensions of ~50 μm. This
length scale is consistent with the up to 100 μm of crack extension necessary to measure the
toughening effects of crack deflection in macroscopic fracture experiments
10
. When compared to
11
nacre, which is also hierarchical but less complex than bone, and to a palladium
-
based metallic
glass, a material without hierarchy and highes
t reported toughness, Figure 4 clearly conveys that
bone at the macroscale has a uniquely high toughening rate exponent
훽
. Our microscale results
show this high toughening rate at macroscale in bone is not scale invariant, and that it is likely a
result o
f a transition from a lower toughening rate arising from fibril bridging active at the nano
-
and microscale to prevent inherently brittle and catastrophic failure. This
suggest
the multiscale
hierarchy in bone is designed such that the cyclic
physiological loading like walking or running
that leads to microcracking
35
exposes nanoscale fibrils and enables them provide nanoscale
resistance to critical loading and prevent catastrophic
macroscale fracture.
Outlook
Mimicking natural materials to achieve similar damage tolerance in engineered materials has
gained substantial traction over the past decade
51
–
53
. The main biomimetic pursuits have focused
on designing
and incorporating multi
-
scale material hierarchy, starting from nanoscale up to the
millimeter scale, to produce bulk engineering materials
54
–
56
; probing how structural hierarchy
quantitatively contribute to a desired set of properties ha
s remained elusive due experimental
challenges at the nanoscale. Our microscale fracture methodology, performed here on silicon and
human bone, provides a versatile a platform to quantitatively describe the toughness of materials
as function of hierarchy a
nd salient microstructural features. The results from the present
approach quantitatively revealed that nanoscale fibrils in a quasi
-
brittle human bone prevent
catastrophic failure at small length scales, giving rise to other toughening mechanisms observed
in macroscale bone fracture.
Looking beyond biomaterials, this platform enables an experimental
approach for understanding the properties of synthetic nanomaterials and those generated by
burgeoning hie
rarchical processing techniques
57
.
12
A
cknowledgements
The authors thank
Bill Johnson, Katherine Faber, Carlos Portela, Xiaoxing Xia, and Eric Luo for
helpful discussions.
The authors thank
Matthew Sullivan and Carol Garland
for
assistance with
experiments and instruments. The authors gratefully acknowledge the facilities and infrastructure
provided by the Kavli Nanoscience Institute at Caltech.
O
.
A
.
T would like
to thank the National
Science F
oundation for financial support
throu
gh the Graduate Research Fellowship Program
(NSF GFRP)
.
J.R.G. gratefully acknowledges financial support from the U.S. Department of
Basic Energy Sciences under Grant DE
-
SC0006599
.
Data
availability.
The data that supports the findings of this study is available from the
corresponding author upon reasonable request.