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© The Japan Society of Mechanical Engineers
Advance Publication by J-STAGE
Mechanical Engineering Reviews
DOI:10.1299/mer.15-00291
Received date : 27 May, 2015
Accepted date : 22 May, 2016
J-STAGE Advance Publication date : 30 May, 2016
© The Japan Society of Mechanical Engineers
1. Introduction
Bone tissue is subject to in vivo stresses that are static stresses due to the body weight and cyclic, dynamic stresses
due to the daily movement, and is optimized for the mechanical environments with reconstructing its structure,
generally called as “functional adaptation of bone”. Stress measurements of bone are essential for evaluating the risk of
bone fracture, the cure of bone diseases (e.g., osteoporosis), bone turnover, and the bone adaptation.
Few studies have reported the existence of residual stresses in bone tissue. The residual stress is defined as the
stress that remains in the tissue without any external forces. The stress will be one of the important factors to
understand the bone strength and bone adaptation. It is well known that living tissue, as blood vessels, is subject to
residual stresses (Fung, 1990). In blood vessel wall, compressive residual stress exists in the inner wall and tensile
stress exists in the outer wall along the circumferential direction, suggesting that the residual stresses decrease high in
vivo stress applied to the inner wall due to the blood pressure. It appears that the residual stress in living tissue plays an
important role in the mechanical strength. However, the residual stresses in the bone tissue have not been fully
investigated and elucidated.
Researches attempted to measure the strain of bone tissue by invasive procedures using strain gauges glued to the
bone surface (Al Nazer et al., 2012). For instance, large strain was obtained in vivo as 9,096 × 10 -6 on the human
medial tibia during basketball rebounding (Milgrom et al., 2000), also large strain rate was measured in vivo as -85,500
× 10-6/s at the human distal radius during forward fall from standing, landing on extended hands (Földhazy et al.,
2005). Tanaka and Adachi (1994) measured the changes in the strain gauge values attached to the surface of rabbit tibia
in situ by cutting the fibula, suggesting the existence of residual stresses in the tibia-fibula indeterminate structure.
Adachi et al. (1998) used strain gauges bonded onto the cortical surface of bovine coccygeal vertebrae aligned in the
How is residual stress/strain detected in bone tissue ?
Shigeru TADANO* and Satoshi YAMADA*
*Division of Human Mechanical Systems and Design, Faculty of Engineering, Hokkaido University
N13 W8, Kita-ku, Sapporo, Hokkaido 060-8628, Japan
E-mail: tadano@eng.hokudai.ac.jp
Abstract
Recent experimental approaches were reviewed to the residual stress/strain in bone tissue using X-ray
diffraction (XRD) techniques. After a brief introduction of the experimental methods and obtained results, we
discussed the generation mechanisms and biomechanical implications of the residual stress/strain in bone
tissue. Strain gauge approaches provided the existence of residual stresses in the bone at the whole bone level.
XRD approaches have also provided the existence of residual stresses at the tissue and mineral phase levels.
The residual strains at the mineral phase related to the degree of orientation of the HAp crystals. The
distributions of residual stress were obtained around the surface and along the radial depth of the diaphysis of
quadrupedal extremities. The correlation between the residual stress and the osteon structures was indicated
and the difference of residual stress with growth was revealed. It would appear that the residual stress state
might be generated by the indeterminate structure in the hierarchical structures of the bone tissue relating to
bone adaptation with the bone formation and reconstruction process. A long-term study is needed to better
understand the generation and biomechanical implications of residual stress in bone tissue throughout the
hierarchical structure during maturation and aging.
Key words : Biomechanics, Bone, Residual stress, Hierarchical structure, X-ray diffraction, Hydroxyapatite
crystal
© The Japan Society of Mechanical Engineers
cephalocaudal and circumferential directions and reported the changes in strains induced by removal of the end-plate
and the cancellous bone. The results suggested the existence of compressive residual stress in the cortical bone and
tensile residual stress in the cancellous bone in both the cephalocaudal and the circumferential direction. However,
strain gauge approaches have some limitations; it requires specimen destruction processes (i.e., cutting and boring) and
provides residual stresses in indeterminate structure at whole bone level.
As shown in Fig. 1, bone has a hierarchical structure, spanning from the macrostructure at several millimeters or
whole bone level, the microstructure at several hundred micrometers level, including osteons and Haversian canals in
the cortical bone, to the nanostructure at hydroxyapatite (HAp) crystals and collagen fibrils level. It is well known that
bone is usually replaced by new tissue (Currey, 2002; Fung, 1990). Since the new tissue develops under in vivo
loadings as the non-deformed state, an indeterminate structure may be generated by the difference of the deformation
between the old and new phases. The mechanical properties (e.g. elastic modulus) are also different in these phases
(Gibson et al., 2006; Rho et al, 1999). Because of such non-uniform structures, residual stress/strain will remain in the
bone tissue even without external forces being applied. Therefore, to fully understand the residual stresses in the bone
tissue, it is important to investigate the residual stress/strain related to the indeterminate structure in the hierarchical
structures.
X-ray diffraction (XRD) is an alternative promising tool to obtain measurements of the stress/strain in bone tissue,
because X-rays have nondestructive and noninvasive properties (Tadano and Giri, 2011). It is possible to obtain the
distribution of the stress/strain state in the tissue because X-rays have can pass through the tissue and can be collimated
locally. Some studies reported the deformation of hydroxyapatite (HAp) crystals in bone tissue under external loads
using X-ray diffraction techniques (e. g., Borsato and Sasaki, 1997; Almer and Stock, 2005, 2007; Fujisaki et al., 2006;
Gupta et al., 2006; Fujisaki and Tadano, 2007; Tadano et al., 2008; Akhtar et al., 2008a, 2008b, 2011; Giri et al., 2012;
Singhal et al., 2012; Yamada et al., 2013b, 2014b). It has been shown that the distance between the lattice planes of the
HAp crystals changes proportionally to the deformation of the bone tissue (Fujisaki and Tadano, 2007). The HAp
crystal strain can be calculated by the deformation of the interplanar spacing compared with a reference state (Fujisaki
et al., 2006; Tadano et al., 2008). Based on this, the residual stress/strain in the bone tissue is measured by XRD.
In this article, we review recent experimental approaches to reveal the residual stress/strain in bone tissue by the
methods based on XRD. After a brief introduction of the experimental methods and obtained results, we discuss the
generation mechanisms and biomechanical implications of the residual stress/strain in bone tissue.
2. X-ray diffraction for bone tissue
Bragg’s law, the fundamental equation for X-ray diffraction, is expressed as in Eq. (1)
(1)
2dsin
θ
=n
λ
Fig. 1
Five levels of hierarchical structure in cortical bone; (I) Macrostructure level (10 mm to several cm), or whole
bone level, consisting of cortical and trabecular bone types. (II) Mesostructure level (0.5–10 mm), or cortical bone
level. (III) Microstructure level (10–500 µm), single osteon and interstitial lamella level in cortical bone. (IV)
Sub-microstructural level, single lamella level (1–10 µm). (V) Nanostructure level (<1 µm), multiphase
nanocomposite consisting of an organic phase, an inorganic phase and water. Reprinted from Tadano and Giri
(2011), with permission from TAYLOR & FRANCIS INFORMA UK LTD.
© The Japan Society of Mechanical Engineers
where θ and d are the Bragg angle and the interplanar spacing at a specific lattice plane (hkl) of HAp crystals in bone
tissue, respectively.
The HAp crystal strain εH is defined as in Eq. (2) using the changes in the interplanar spacing of the crystals, where
the subindex 0 indicates the values at the non-strained state.
(2)
3. Residual stress/strain of mineral phase in bone tissue
Some researches attempted to obtain the stress/strain state of the mineral phase in bone by using XRD (Almer and
Stock, 2005, 2007; Giri et al., 2008; Stock et al., 2011; Hoo et al., 2011).
Almer and Stock (2005, 2007) detected the stress state of mineral phase inside a beagle fibula and a canine fibula
under in situ compressive loading by XRD with high-energy synchrotron X-rays (80.7 keV). The deviatoric stress of
mineral phase σH
x of the bone was calculated from deviatoric strains of HAp crystals (εH
x and εH
y) and the Kröner–
Eshelby model as in Eq. (3)
(3)
where x axis was the bone axis and the loading direction in the experiments. The y axis was the circumferential
direction. S1 and S2/2 were X-ray elastic compliances for a specific lattice plane of the HAp crystals in the model.
Additionally, εH
y and εH
z were assumed to be equal here.
In the experiments, the diffracted X-rays transmitted through the bones were detected, and the stress/strain in the
mineral phase was the weighted average of the X-ray pathway from the irradiated surface to the opposite surface of the
diaphysis. As a result, internal stress of the mineral phase was proportionally increase with applied tissue stress, as
shown in Fig. 2 (Almer and Stock, 2005). Under no external load, compressive residual stresses were observed in the
mineral phase inside the bone specimen along the bone axis on the order from -60 to -200 MPa. The amount of residual
stresses in the mineral phase was different between (222) planes and (002)/(004) planes. Furthermore, it was also noted
that the amount of residual stress in the mineral phase decreased with immersion time in saline solution (Almer and
Stock, 2007).
Giri et al. (2008) noted the site-specific residual strain around the small hole, called foramen, in bovine metacarpals
and femurs detected by XRD (Fig. 3). The diffracted X-rays penetrated the sliced specimens and then were detected by
an imaging plate (IP), which is a two-dimensional X-ray detector. The traveling direction of the diffracted X-rays
ε
H=d−d0
d0
σ
x
H=1
S2/ 2
ε
x
H−S1
S2/ 2 −3S1
(
ε
x
H+2
ε
y
H)
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Fig. 2
(a) Bone specimens under in situ compressive loading with irradiation of high-energy synchrotron X-rays.
Reprinted from Almer and Stock (2007), with permission from Elsevier. (b) Internal stress of the mineral phase in
a beagle fibula calculated from HAp crystal strains measured by XRD and the Kröner–Eshelby model under the
compression test. Reprinted from Almer and Stock (2005), with permission from Elsevier.
© The Japan Society of Mechanical Engineers
depends on the direction and the displacement of the lattice planes in the HAp crystals. The measurements assumed the
plane stress state and Poisson’s ratio as 0.29. Residual strains were found to exist around the foramen in the specimens
and trends of residual strains were mostly consistent with the degree of orientation of the HAp crystals, which may
reinforce the tissue around the hole (Giri et al., 2007). It may be able to explain the response behavior of bone to the
mechanical loading history near the foramen.
4. Sin2ψ method for residual stress in the surface region of the bone tissue
Todoh et al. (2000) reported to detect the residual stresses in bone tissue at the tissue level by calculating the HAp
crystal strain measured with XRD. The surface region (up to 200 µm depth) of the cortical bone specimens from bovine
femurs was examined. Tadano and Okoshi (2006) also investigated the residual stresses in rabbit tibiofibula. In these
studies, the HAp crystal strain was calculated from the displacement of the lattice plane with reference to the lattice
plane in bone powder as non-strained specimens. However, the lattice planes of the bone powder were much influenced
by the particle size (Tadano and Okoshi, 2006). The non-strained state of the bone tissue was not easy to decide.
Fig. 3
(a) A foramen specimen in a bovine femur. (b) XRD setup. (c) Degree of HAp crystal orientation for the bone axis
<cos2β> around the foramen. (d) Residual strain of the (211) planes in the HAp crystals along the bone axis
(×10-6). Reprinted from Giri et al. (2008), with permission from Elsevier.
Fig. 4
Interplanar spacing d of HAp crystals at the bone surface in cortical bone tissue under (a) non-strained and (b)
tensile loading states. The variation in d with orientation ψ normal to the lattice planes of the HAp crystals is
shown in polar coordinates, and the lengths and directions of the vectors in the diagrams show the interplanar
spacing and plane-normal direction, respectively. In the non-strained state the interplanar spacing is d0. Under
tensile loading horizontal to the surface of the specimen, the lattice plane deforms and the interplanar spacing of
the lattice planes oriented in the loading direction is the largest and that oriented normal to the surface the smallest
(d1 < d2 < d3). Reprinted from Yamada et al. (2011a), with permission from Elsevier.
© The Japan Society of Mechanical Engineers
An XRD technique based on the sin2ψ method solves the problems on the non-strained state. The angle of
inclination ψ is defined as the angle between the normal direction of the specimen surface and the diffracted lattice
plane. As suggested in Fig. 4a, in the non-strained state the interplanar spacing is d0. Under tensile loading horizontal to
the surface of the specimen, the interplanar spacing in lattice planes under tensile stress in the ψ = 90° direction is
larger than that in the ψ = 0° direction (Fig. 4b). The relation between d and ψ is affected by the intensity of the stress.
Based on this, residual stress is measured from the variation of the interplanar spacing of HAp crystals without a
comparison against non-strained samples or information on the interplanar spacing in the stress direction (Yamada and
Tadano, 2010).
A coordinate system is fixed at the bone surface, and the x, y, and z axes correspond to the bone axis,
circumferential, and radial directions, respectively. The ψ is defined as the angle in the x-z plane. Cortical bone in the
diaphysis of the extremities is considered as an orthotropic compound. It is assumed that the residual stress state in the
measurement region is a plane stress, because the X-rays generated by X-ray tubes only penetrate up to 100 µm into the
specimens and only the outermost region is measured. The relationship between bone tissue stress σB and HAp crystal
strain εH in the bone is assumed to be described by Eq. (4)
(4)
where E* and ν* are X-ray elastic constant and X-ray Poisson's ratio respectively, and these are the elastic properties
between the bone tissue stress and the HAp crystal strain. The bone tissue stress σB in the bone axis is calculated with
the relationship between εH in the ψ direction and sin2ψ as in Eq. (5).
(5)
The bone tissue stress σx in the bone axis is described with θ and ψ using Eqs. (1) and (2) as in Eq. (6)
(6)
where K is the stress constant.
Therefore, the residual stress σB
x in the bone axis is calculated from the variation of the interplanar spacing of HAp
crystals in the x-z plane by using X-ray diffraction. The calculated stress is the deviatoric stress because the hydrostatic
deformation resulting from hydrostatic stress is not detected. The residual stress σB
y in the circumferential direction is
also calculated from the variation of the interplanar spacing of HAp crystals in the y-z plane. To measure
∂(2θ)/∂(sin2ψ), rotating and tilting the sample is required with a conventional X-ray diffraction instrument and a
scintillation counter. To determine K, a four-point bending test of a thin bone specimen with X-ray irradiation was
conducted and the relationship between applied bone tissue stress σB and ∂(2θ)/∂(sin2ψ) was measured as a calibration
(Yamada and Tadano, 2010; Yamada et al., 2011a; Yamada and Tadano, 2013a).
Using this method, adult bovine femurs were examined (Fig. 5a, Yamada and Tadano, 2010). The diaphyseal
specimens were 50 mm long in the bone axis. The bone marrow and the soft tissue around the surfaces were removed,
and the specimens were air-dried. The X-ray diffraction profile from the (211) planes of HAp crystals, which included
(112) and (300) planes, was detected by using an X-ray diffractometer with characteristic Mo-Kα X-rays (λ = 0.071
nm). The residual stresses were measured along the bone axial and circumferential directions around the specimens.
Figure 5b shows an example of residual stress distributions in the bone axial and the circumferential directions in
ε
x
H
ε
y
H
ε
z
H
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=
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E*−
ν
*
E*−
ν
*
E*
−
ν
*
E*
1
E*−
ν
*
E*
−
ν
*
E*−
ν
*
E*
1
E*
(
)
*
*
*
*
*
*
*
*
+
,
-
-
-
-
-
-
-
-
σ
x
B
σ
y
B
σ
z
B
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σ
x
B
=E
*
1+
ν
*
∂
ε
ψ
H
∂(sin
2
ψ
)
σ
x
B=−E*
2(1+
ν
*)
π
180 cot
θ
0
∂(2
θ
)
∂(sin2
ψ
)=K∂(2
θ
)
∂(sin2
ψ
)
© The Japan Society of Mechanical Engineers
the middle section of the femoral diaphysis. The residual stresses in the bone axial direction were everywhere tensile.
The stresses in the circumferential direction were smaller than in the bone axial direction and some positions were close
to zero. After the measurements, a diaphysis specimen was cut 2 mm above, in the upward direction from, where the
stress measurements were made. Here the residual stress after the cutting was lower than before the cutting. It suggests
that residual stress was released and that a new equilibrium was established in the bone, and that there is residual stress
in the bone axial direction at the surface of the bovine femoral diaphysis.
Furthermore, diaphyseal surface of the extremities from adult rabbits were examined (Fig. 6a, Yamada et al.,
2011a). Three femur, three tibia/fibula, three humerus, and three radius/ulna specimens were used in the experiment.
The specimens were 60 mm long in the bone axial direction. The bone marrow and the soft tissue around the surfaces
were removed and the specimens were kept in saline until just before the X-ray measurements. The residual stresses
along the bone axis were measured at the bone surface of the anterior and posterior positions in each extremity using an
X-ray diffractometer with characteristic Mo-Kα X-rays. The specimens were air dried after the X-ray
Fig. 5
(a) Diaphyseal specimens of bovine femurs. (b) Residual stresses in the middle section of a diaphysis of the bone
axial and the circumferential directions. After the same specimen was cut 2 mm above, in the upward direction
from, where the stress measurements were made. A=anterior, AL=lateral anterior, L=lateral, LP= lateral posterior,
P=posterior, PM=medial posterior, M=medial, and MA=medial anterior. Reprinted from Yamada and Tadano
(2010), with permission from ASME.
Fig. 6
(a) Diaphyseal specimens of rabbit extremities. Measurement positions were anterior and posterior at the center of
the diaphyses. (b) Typical microscopic images at the X-ray measurement positions in the hindlimb bones of the
same limb. The dashed lines show a depth from the surface of about 100 µm, and the black arrows indicate osteons
included wholly or partly in the region. Reprinted from Yamada et al. (2011a), with permission from Elsevier.
© The Japan Society of Mechanical Engineers
measurements and then they were cut out, to observe the microstructures at the X-ray measurement position in the
transverse cross-section of the specimens (Fig. 6b). The number of osteons in the region was counted and the osteon
population density (OPD) was calculated. The osteons were quantified to secondary osteons (Haversian systems) and
primary osteons.
Figure 7a shows the distribution of residual stress in the rabbit limb bones. Tensile residual stresses were also
observed at the bone surface. The hindlimb bones were subject to tensile residual stress 1.4 times higher than that in the
forelimb bones. In the femur and humerus, the OPD in the anterior positions were larger than in the posterior positions.
In the tibia, the OPD in the posterior position was larger than that in the anterior position. Overall, more osteon
structures were observed in the positions subjected to higher residual stresses. As shown in Fig. 7b, there was a
statistically positive correlation between the residual stress and the OPD (r = 0.55, P < 0.01).
5. XRD-IP system based on the sin2ψ method
The sin2ψ method required a complicated experimental setup, long irradiation time, and limitations of the sample
size in the longitudinal direction because rotating and tilting the sample is required to obtain the distribution of the HAp
crystal deformation. The method could not be directly applied to the measurements of residual stress distribution in situ
and in vivo. To profoundly enhance the investigating of residual stress distribution in bone tissue, it is necessary to
develop an improved method that features a simple setup without limitations on the sample size and shape.
Imaging plate (IP) can obtain the distribution of the diffracted X-rays from the HAp crystals with only one
irradiation. The distribution of the HAp crystal deformation is then calculated from the XRD pattern detected via the
IP. A measurement system using an XRD technique with the IP set in the reflection side (XRD-IP system) has been
proposed for obtaining the surface distribution of residual stress in the diaphysis of extremities without limitations of
the sample size and long irradiation time (Yamada et al., 2014a).
Figure 8 shows the measurement setup of the XRD-IP system. The diaphysis specimen is irradiated with
characteristic X-rays and the resulting diffracted X-rays are detected by the IP in the reflection side. The x’-y’-z’
coordinate system is defined as the X-ray coordinate system and the incident X-rays enter the specimen along the
y’-axis. The x’-axis corresponds to the x-axis and the angle between the y’- and y-axes is set to 7.25°, which is the
Bragg angle θ0 of the (211) lattice planes in the HAp crystals in a non-strained state. The X’-Y’-Z’ coordinate system is
fixed at the IP surface and the Y’-axis corresponds to the y’-axis. The 1-, 2-, and 3-axes are the principal axes and the
3-axis corresponds to the z-axis. The residual stress state in the measurement region is assumed as a plane stress. The
angle φ is defined as the angle between the 1- and x-axes.
The XRD pattern, which was a portion of the Debye ring of (211) lattice planes in HAp crystals, was detected by
the IP. The direction of εH
β was slightly inclined at an angle of Δφ from the x-z plane toward the y-axis with increasing
β (Fig. 8b). Because Δφ was presumed as negligibly small, cos2(φ+Δφ) and sin2(φ+Δφ) were approximated as cos2φ and
sin2φ, respectively. Then, the bone tissue stress σx in the bone axis is calculated based on Eq. (6). Therefore, the
residual stress is calculated via the portion of the Debye ring of (211) lattice plane in HAp crystals detected by the IP in
the reflection side.
Fig. 7
(a) Distribution of the residual stresses in the rabbit limb bones. A=anterior and P=posterior. (b) Relationship
between the residual stress and the osteon population density in the observed images (r = 0.55, p < 0.01).
Reprinted from Yamada et al. (2011a), with permission from Elsevier.
© The Japan Society of Mechanical Engineers
6. Residual stress distribution along the radial depth of cortical cylinders by synchrotron
The residual stress distribution in the deeper region of the diaphysis of extremities is not measured using the
previous methods based on the sin2ψ method. An alternative method for measurement of residual stresses in deeper
regions of the diaphysis has been proposed using synchrotron white X-rays (Yamada et al., 2011b).
A coordinate system is fixed at each measurement location inside the bone and the x, y, and z axes are defined as
the principal axes. It is assumed that the diaphysis is not subject to residual stress in the radial direction (σB
z = 0). Under
the assumptions, Eq. (6) is satisfied.
The energy-dispersive X-ray diffraction with synchrotron white X-rays is used to measure the d–ψ relationship.
The relationship between the wavelength λ and energy W of X-rays is expressed as in Eq. (7), where h is Planck’s
constant and c is the speed of light, and the relationship between d and W is expressed as in Eq. (8).
(7)
(8)
The d is measured from the energy W of the diffracted X-rays when θ is fixed. Therefore, the σx at each
measurement locations inside the cortical bone is calculated form Eq. (9).
(9)
To understand the force equilibrium of residual stresses inside the bone, the residual stress distribution along the
radial depth of bovine femurs was investigated using the method (Yamada et al., 2011b). Furthermore, to investigate
the effects of growth on the distribution, different animal ages were examined (Yamada and Tadano, 2013a).
Diaphyseal specimens were obtained from less-than-one-month-old (Group Y) and two-year-old (Group M) bovine
femurs. The bone marrow and the soft tissue around the surfaces were removed and the specimens were air-dried. The
synchrotron white X-ray diffraction was performed at the BL28B2 beamline of SPring-8, a large synchrotron radiation
facility managed by RIKEN and JASRI. The diaphysis specimens from Group Y were measured at 0.5-mm intervals
from the outer surface to the deeper region of the specimens at four positions: anterior, posterior, lateral, and medial.
The Group M samples were measured at 1-mm intervals.
Figure 9 shows typical distributions of interplanar spacing of the HAp crystals and residual stress along the bone
axis from the outer surface region to the deeper region in a diaphysis specimen from Group M. A trend was observed in
the specimens from Group M toward tensile residual stresses around the surface region and compressive residual
λ
=hc /W
d=hc
2Wsin
θ
σ
x
B=E*
d0(1+
ν
*)
hc
2sin
θ
∂(1 / W
ψ
)
∂(sin2
ψ
)
Fig. 8
(a) Measurement setup of the XRD-IP system. (b) The relationship among x-y-z, x’-y’-z’, and 1-2-3 coordinate
systems and the direction of εH
β. The 1-, 2-, and 3-axes are the principal axes and the 3-axis corresponds to the
z-axis. The angle φ is defined as the angle between the 1- and x-axes. Reprinted from Yamada et al. (2014a), with
permission of Springer.
© The Japan Society of Mechanical Engineers
stresses inside the diaphysis. In the measurements, it was assumed that the diaphysis specimen in the radial direction
was not subject to residual stresses. The distribution of interplanar spacings closely corresponded to the distribution of
residual stresses. It suggests that even when the results included the radial component, this component may have little
effect on the measured values. Although the amount of surface residual stresses were smaller than the results from the
measurements with characteristic X-rays generated by X-ray tubes, it may be affected by the difference of the measured
volume at the surface region.
Figure 10a shows typical distributions of residual stress at the anterior part in the diaphysis specimens from Group
Y and Group M. In Group Y, residual stresses did not vary with the depth from the surface. There was no significant
difference between the surface and deeper regions in Group Y. On the other hand, in the Group M, there was a strong
significant difference between the residual stresses measured at the surface region and in the deeper region (Fig. 10b).
The value of residual stresses had a positive statistical correlation with the cortical thickness.
7. Limitations of the experiments
Some studies used diaphyseal specimens taken from whole bones. To evaluate the effects of the cutting on the
measured stress state, strain gauges glued on the surface of the mid-diaphysis in a thawed whole femur of a mature
bovine and the diaphysis specimen with 60 mm long was taken from it. The strain changing in the bone axis were
smaller than 50µ. It suggests that the cutting process may have less impact on the residual stress in the measurements
Fig. 9
(a) Radial distribution of the interplanar spacing d(00 2) of the (002) lattice plane in the bone axial direction from the
outer surface to the inside of the diaphysis specimen at the four parts: anterior (A), posterior (P), lateral (L), and
medial (M). (b) Typical distribution of the residual stress in the bone axis from the outer surface to the inside of
the diaphysis specimen at the four parts. Reprinted from Yamada et al. (2011b).
Fig. 10
(a) Residual stresses between the two groups at the anterior position. Each value indicates the average of five
specimens, and the errors corresponding to their standard deviations. (b) Comparisons of residual stresses
between the surface and deeper regions of the diaphysis specimens and the averages of cross-sectional area. The
inner region in Group M is located at the 3-mm depth from the surface, and that in Group Y is located at the
1.5-mm depth. Reprinted from Yamada and Tadano (2013a), with permission from Elsevier.
© The Japan Society of Mechanical Engineers
(Yamada and Tadano, 2013a). Although the preparation may release some amount of residual stress from in vivo whole
bone, the residual stresses that occurred in these specimens were detected at the tissue level.
In some studies, air-dried specimens were used. The effects of air-drying process might have less impact on the
measured deviatoric stresses, although there may be hydrostatic compressive deformation related with the sample
volume changes due to the air-drying. In deed, the trend that there were tensile residual stresses around the surface
region of the mature bovine femurs (Yamada and Tadano, 2010, 2013a) corresponded to the results of the study with
the limb bones of adult rabbits that kept saline solution just before the measurements (Yamada et al., 2011a). However,
Almer and Stock (2007) and Tung et al. (2013) noted the effects of hydration/dehydration on the residual stress in the
mineral phase. A further study of these effects should be conducted.
Synchrotron X-rays are able to pass through the thick bone specimen like diaphyses and are useful to detect the
HAp crystal deformation state inside the bone. Further, the method is also applied to the collagen phase (Almer and
Stock, 2005, 2007; Gupta et al., 2006; Dong et al., 2011). However, Singhal et al. (2011) and Tung et al. (2013) pointed
out that the residual stress/strain in the mineral phase in bone tissue markedly changed with increasing high-energy
X-ray doses, because of the damage at the HAp-collagen interface in the tissue. To study the bone residual stresses, it is
important to minimize the X-ray energy and irradiation time.
8. Generation mechanisms and biomechanical implications of bone residual stress
Strain gauge approaches provided the existence of residual stresses in the bone at the whole bone level. XRD
approaches have also provided the existence of residual stresses at the tissue and mineral phase levels. It was
investigated that the residual strains at the mineral phase related to the degree of orientation of the HAp crystals (Giri et
al., 2008). It also has been attempted to indicate the distributions of residual stresses around the surface and along the
radial depth of diaphysis of extremities. The positive correlation between the residual stress and the osteon structures
was indicated (Yamada et al., 2011a). Furthermore, the difference of residual stress with growth was indicated
(Yamada and Tadano, 2013a). It would appear that the residual stress state might be generated by the indeterminate
structure in the hierarchical structures of the bone tissue relating to bone adaptation with the bone formation and
reconstruction process.
The young bones were not subjected to residual stresses, whereas larger residual stresses were observed in the
mature bones (Yamada and Tadano, 2013a). In addition, the mature bones showed a trend toward tensile residual
stresses at the surface region and compressive residual stresses in the deeper region. The results suggest that residual
stress is generated during growth, satisfying the equilibrium of forces between the surface and the deeper regions of the
diaphysis. The differences in residual stress may be related to mechanical loads applied to bone tissue in vivo during
bone formation or reconstruction. In vivo mechanical loading from body weight increases with growth. Because new
tissue develops in a nondeformed state under in vivo loading, tensile stresses may occur in the new surface growth and
compressive stresses may remain in the older, deeper bone tissue even after any external loadings have been removed.
Based on this hypothesis, although the young bones develop at a rapid pace, the large residual stresses may not be
generated in the young bones. In deed, bone residual stress/strain at the whole bone level detected in the strain gauge
measurements provides the existence of the indeterminate structure at tibia-fibula and cortical-cancellous bone level.
However, because bone tissue has a hierarchical structure, the hypothesis may be quiet simplified. Residual stress
showed the positive correlation with osteon structures (Yamada et al., 2011a). The population density and the geometry
of osteon structures were correlated with the compression/tension mechanical environments in vivo (e.g., Skedros et al.,
2009). The collagen-lamellar organization and mineral crystallite orientation also differ in these regions. In this
perspective, these nonuniform structures of the tissue derived from the osteon formation and the internal organization
of these entities down to the nanostructural level may explain the spatial differences in residual stress. Furthermore, it
is well known that the mechanical and physicochemical properties of bone tissue depend substantially on age (e.g.,
Raghavan et al., 2012). Such heterogeneity among these properties may produce a locally indeterminate structure in the
tissue.
The residual stress being nonuniform may be related to nonuniformities in the mechanical environments in vivo
and the resulting functional adaptation of the bone tissue. The residual stress in the bone may be regarded as an
epiphenomenon and may be a circumstantial finding of the adapted state. Giri et al. (2008) investigated that the trends
of residual strain in mineral phases around the foramen were mostly consistent with the degree of orientation of the
HAp crystals that reinforced the tissue. Adachi et al. (1998) discussed that bone residual stresses may work to allow a
© The Japan Society of Mechanical Engineers
stress state to become more uniform. The existence of residual stresses might have the potential to characterize the
equilibrium state of mechanical bone adaptation by remodeling. The high tensile residual stresses measured at the bone
surface of extremities (e.g., Yamada and Tadano, 2010) might work to reinforce the bone tissue in compressive and
bending loadings in vivo. In deed, the surface region of the extremities is under sever mechanical environments.
However, the limb bones used in the studies are subjected to too complex loadings (e.g. compression, bending, and
torsion) (e.g., Skedros et al., 2009). Gautier et al. (2000) measured the maximum strain in the bone axial direction in
sheep tibia during locomotion using strain gauges; and here the anterior part was subject to tensile and the posterior
part to compressive stresses, and further, there were also the effects of the torsion in the strains. The measured residual
stresses may not be directly related to the tension/compression strain in vivo. To understand the biomechanical
implications of bone residual stress, in vivo stress measurements are quite important. The investigations of the residual
stresses in the bones that are subject to simple in vivo loadings might be helpful. Furthermore, comparing the residual
stresses of intact and adapted bones would provide more direct data for an elucidation of the biomechanical aspects of
the bone residual stress.
A long-term study is needed to better understand the generation and biomechanical implications of residual stress
in bone tissue throughout the hierarchical structure during maturation and aging.
Acknowledgments
The authors would like to thank Professor Masahiro Todoh, Professor Kazuhiro Fujisaki, and Dr. Bijay Giri for
scientific and technical supports. They are members of the Laboratory of Biomechanical Design, Division of Human
Mechanical Systems and Design, Faculty of Engineering, Hokkaido University. This work was supported by JSPS
Grant-in-Aid for Scientific Research (A) Grant No. 24240068, and No. 15H02207.
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