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Computational homogenisation approaches using high resolution images and finite element (FE) modelling have been extensively employed to evaluate the anisotropic elastic properties of trabecular bone. The aim of this study was to extend its application to characterise the macroscopic yield behaviour of trabecular bone. Twenty trabecular bone samples were scanned using a micro-computed tomography device, converted to voxelised FE meshes and subjected to 160 load cases each (to define a homogenised multiaxial yield surface which represents several possible strain combinations). Simulations were carried out using a parallel code developed in-house. The nonlinear algorithms included both geometrical and material nonlinearities. The study found that for tension-tension and compression-compression regimes in normal strain space, the yield strains have an isotropic behaviour. However, in the tension-compression quadrants, pure shear and combined normal-shear planes, the macroscopic strain norms at yield have a relatively large variation. Also, our treatment of clockwise and counter-clockwise shears as separate loading cases showed that the differences in these two directions cannot be ignored. A quadric yield surface, used to evaluate the goodness of fit, showed that an isotropic criterion adequately represents yield in strain space though errors with orthotropic and anisotropic criteria are slightly smaller. Consequently, although the isotropic yield surface presents itself as the most suitable assumption, it may not work well for all load cases. This work provides a comprehensive assessment of material symmetries of trabecular bone at the macroscale and describes in detail its macroscopic yield and its underlying microscopic mechanics.
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Research Paper
Evaluating the macroscopic yield behaviour
of trabecular bone using a nonlinear
homogenisation approach
Francesc Levrero-Florencio
a,
n
, Lee Margetts
b
, Erika Sales
a
, Shuqiao Xie
a
,
Krishnagoud Manda
a
, Pankaj Pankaj
a
a
Institute for Bioengineering, School of Engineering, The University of Edinburgh, Faraday Building, King's Buildings,
EH9 3JG Edinburgh, United Kingdom
b
School of Mechanical, Aerospace and Civil Engineering, University of Manchester, Oxford Road, M13 9PL Manchester,
United Kingdom
article info
Article history:
Received 3 September 2015
Received in revised form
28 March 2016
Accepted 6 April 2016
Available online 13 April 2016
Keywords:
Finite Elements
Anisotropic Material
Yield Surface
Multiscale Modelling
Trabecular Bone
abstract
Computational homogenisation approaches using high resolution images and nite element
(FE) modelling have been extensively employed to evaluate the anisotropic elastic properties
of trabecular bone. The aim of this study was to extend its application to characterise the
macroscopic yield behaviour of trabecular bone. Twenty trabecular bone samples were
scanned using a micro-computed tomography device, converted to voxelised FE meshes and
subjected to 160 load cases each (to dene a homogenised multiaxial yield surface which
represents several possible strain combinations). Simulations were carried out using a
parallel code developed in-house. The nonlinear algorithms included both geometrical and
material nonlinearities. The study found that for tension-tension and compression-
compression regimes in normal strain space, the yield strains have an isotropic behaviour.
However, in the tension-compression quadrants, pure shear and combined normal-shear
planes, the macroscopic strain norms at yield have a relatively large variation. Also, our
treatment of clockwise and counter-clockwise shears as separate loading cases showed that
the differences in these two directions cannot be ignored. A quadric yield surface, used to
evaluate the goodness of t, showed that an isotropic criterion adequately represents yield
in strain space though errors with orthotropic and anisotropic criteria are slightly smaller.
Consequently, although the isotropic yield surface presents itself as the most suitable
assumption, it may not work well for all load cases. This work provides a comprehensive
assessment of material symmetries of trabecular bone at the macroscale and describes in
detail its macroscopic yield and its underlying microscopic mechanics.
& 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY
license (http://creativecommons.org/licenses/by/4.0/).
http://dx.doi.org/10.1016/j.jmbbm.2016.04.008
1751-6161/& 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/).
Abbreviations: (FE), Finite element; (CT), Computed tomography; (MIL), Mean interceptlength; (RVE), Representative volume
element; (MPI), Message passing interface
n
Corresponding author.
E-mail address: f.levrero-orencio@ed.ac.uk (F. Levrero-Florencio).
journal of the mechanical behavior of biomedical materials 61 (2016) 384–396
1. Introduction
Exponential growth of older population implies that problems
associated with deteriorated mechanical capabilities of bone
need urgent attention. Computational modelling to examine
the mechanical response of musculoskeletal systems
requires the mechanical behaviour of bone to be dened
satisfactorily (Pankaj, 2013). A continuum description of bone
that can be related to its microstructure and includes its
anisotropy and its yield behaviour will go a long way in
predicting failure of bone and bone-implant systems.
The macroscopic elastic behaviour of bone has been
mostly modelled using isotropic linear elasticity. Often, bone
macroscopic properties are assumed to be homogeneous with
separate elastic properties being assigned to cortical and
trabecular bone (Completo et al., 2009; Conlisk et al., 2015).
Sometimes, subject specic macroscopic elastic properties
are assigned using computed tomography (CT) scans, which
permit inhomogeneity in the material properties on the basis
of CT attenuations (Helgason et al., 2008; Schileo et al., 2008;
Tassani et al., 2011). That said, since CT attenuations can only
provide scalar values, assumption of isotropy needs to be
made. However, it is well recognised that the macroscopic
behaviour of bone is not isotropic. For trabecular bone, which
resembles open cell foams, the anisotropy is largely a con-
sequence of its anisotropic microarchitecture (Odgaard et al.,
1997; Turner et al., 1990). An ultrasonic approach proposed by
van Buskirk et al. (1981) was shown to provide a good
approximation of nine orthotropic elastic constants if a
heterogeneity correction were included. In general, experi-
mental mechanical techniques are unable to provide the
complete stiffness tensor at the resolution required for
modelling (Odgaard et al., 1989).
Image based computational approaches have been success-
fully applied for the evaluation of the macroscopic stiffness
tensors (Donaldson et al., 2011; van Rietbergen et al., 1995). In
these, micro-CT (or micro-magnetic resonance imaging) scans
of bone are converted into high resolution 3D nite element
(FE) meshes, with a detailed geometry of its microstructure.
The solid phase (or bone tissue) is assigned isotropic elastic
properties and the volume element (VE) is then computation-
ally subjected to six strain/stress states (three normal and
three shear). The response enables evaluation of the full
macroscopic elastic stiffness tensor using the standard
mechanics methodology (van Rietbergen et al., 1996). Previous
studies have extensively employed these homogenisation
approaches, and relationships between stiffness and micro-
architectural indices (volume fraction and fabric tensor) have
also been established (Cowin, 1986; Turner and Cowin, 1987;
Turner et al., 1990; Zysset and Curnier, 1995).
While modelling bone as an elastic material may be
adequate for a few applications, a signicant proportion of
applications requires evaluation of post-elastic response, e.g.
to evaluate implant loosening resulting in its failure. Many
studies still continue to employ elastic analyses to predict
arbitrarily post-elastic behaviour (Falcinelli et al., 2014).
Both stress- and strain-based criteria have been used to
describe the macroscopic yield surface of bone (Keaveny
et al., 1994; Keller, 1994; Kopperdahl and Keaveny, 1998). In
recent years a consensus appears to be emerging that strain-
based criteria are easier to apply as trabecular bone behaviour
in this space is more isotropic and density independent
than in stress space (Bayraktar et al., 2004; Chang et al., 1999;
Pankaj and Donaldson, 2013). There is also now some evi-
dence to suggest that failure of bone is strain-controlled
rather than stress-controlled (Nalla et al., 2003). However,
there is little consensus on the yield criterion that may be
suitable for this cellular material.
Homogenisation techniques, using micro-CT images and FE
analyses, that have been successful in the elastic domain,
require huge computational resources in the plastic regime
for a number of reasons: nonlinear homogenisation requires a
large number of load cases (unlike the linear elastic regime
which only requires six); nonlinear simulations require con-
siderably more computational effort; and to capture nonlinear
phenomena FE meshes need to be ner. As a consequence,
nonlinear homogenisation to obtain the macroscopic yield
criterion of bone requires high performance computing and
has been attempted only by a few previous studies (Bayraktar
et al., 2004; Sanyal et al., 2015; Wolfram et al., 2012). All these
studies used a simple bilinear criterion to represent the solid
phase of bone. Wolfram et al. (2012) used a limited number of
load cases which can lead to loss of information on physiolo-
gically possible complex load cases, while both Sanyal et al.
(2015) and Wolfram et al. (2012) made a priori assumptions with
regard to macroscopic yield surface symmetries; the former
assumed it to be transverse isotropic and the latter orthotropic.
Nanoindentation experiments on bone suggest that the
solid phase of bone has a pressure-dependent yield surface
(i.e. its yielding depends on hydrostatic stress), which arises
because of bone's cohesive-frictional behaviour (Tai et al.,
2006). Due to this reason, bone tissue (or the solid phase) can
be modelled using classical criteria, such as Mohr-Coulomb or
Drucker-Prager (Carnelli et al., 2010; Tai et al., 2006).
On the macroscale, high density bone is prone to tissue
yielding, while low density bone is likely to fail via a mixture
of large deformation failure mechanisms and tissue yielding
(Bevill et al., 2006; Morgan et al., 2004; Stolken and Kinney,
2003). At the microscale, total strains can be large and a small
strain approximation may be invalid. It is important to note
that local yielding or buckling may not imply simultaneous
yielding of the homogenised structure; the latter results from
a signicantly compromised stress carrying capacity.
The aim of this study is to characterise the macroscopic
yield surface of trabecular bone by using a numerical homo-
genisation approach, derived from multiscale theory (de
Souza Neto et al., 2015; Kruch and Chaboche, 2011;
McDowell, 2010): using high resolution FE meshes obtained
from micro-CT images; applying a range of load cases which
adequately describes the multiaxial behaviour of bone at the
macroscale (including complex normal and shear load com-
binations); incorporating both geometrical and material non-
linearities; and with a validated pressure sensitive yield
criterion for the solid phase. We consider a range of trabe-
cular bone densities and also examine the efcacy of quadric
surfaces as representatives for its macroscopic yield surface.
journal of the mechanical behavior of biomedical materials 61 (2016) 384–396 385
2. Material and methods
The tensorial notation used in this study largely follows the
notation used by Schwiedrzik et al. (2013).Arst-order tensor
(or vector) is denoted by a lowercase bold letter (e.g. m), a
second-order tensor is denoted by an uppercase bold letter
(e.g. A) and a fourth-order tensor is denoted by a double-
barred uppercase letter (e.g. A). The tensor operations which
will appear throughout the text are: single contraction of two
second-order tensors, e.g. AB (or A
ik
B
kj
in indicial notation);
double contraction of two second-order tensors, e.g.
A : B ðA
ij
B
ij
Þ; double contraction of a fourth-order tensor and
a second-order tensor, e.g. A : B ðA
ijkl
B
kl
Þ; double contraction
of a second-order tensor and a fourth-order tensor, e.g.
B : A ðB
ij
A
ijkl
Þ; tensor product of two rst-order tensors, e.g.
m m ðm
i
m
j
Þ; tensor product of two second-order tensors,
e.g. A B A
ij
B
kl

; and the symmetric tensor product of two
second-order tensors, e.g. A
B ð
1
2
A
ik
B
jl
þ A
il
B
jk

Þ .
2.1. Imaging and nite element meshing
Ten trabecular bone specimens were extracted from bovine
trochanters and femoral heads (young cattle, o2.5 years old).
Coarse micro-CT images of the central femoral head and
trochanter regions were taken for three specimens prior to
coring. All subsequent samples were then cored with respect
to the visually ascertained trabecular directions. The extracted
cylindrical specimens had a diameter of 10.7 mm and a length
of 29.9 mm. Diamond-tipped cores (Starlite Industries, Rose-
mont PA, USA) were used in the extraction of the specimens
and the top and bottom edges of the cores were cut with a
slow speed saw (Isomet 1000, Buehler, Düsseldorf, Germany)
by using a diamond wafering blade designed for bone. All
these procedures were performed under constant irrigation to
avoid excessive abrasion and overheating.
The specimens were submerged into phosphate buffered
saline and scanned using micro-CT (Skyscan 1172, Bruker,
Zaventern, Belgium) with a resolution of 17.22 mm. The scan-
ning parameters were 94 kV, 136 mA and 200 ms integration
time; 4 scans in 720 equiangular radial positions which were
averaged. The grey scale images were binarised with an
automatic thresholding script (Gomez et al., 2013).
Twenty virtual cubes of 5 mm length were extracted from
the scanned and segmented cylinders. The Mean Intercept
Length (MIL) fabric tensor (Harrigan and Mann, 1984) was
evaluated using BoneJ (Doube et al., 2010) and then used to
align the coordinate axes of the images with the eigenvectors
of the fabric. This approach has been employed in a recent
study (Wolfram et al., 2012). After the 5 mm cubes were
cropped, the alignment was rechecked to ensure that there
was no misalignment larger than 81 (Sanyal et al., 2015;
Wolfram et al., 2012). MIL is known to approximate the elastic
orthotropic directions of trabecular bone (Odgaard, 1997). By
undertaking such alignment there was an expectation that
these axes may also represent the orthotropic directions of
the yield criterion if the criterion was orthotropic. The bone
tissue volume (solid phase) over total volume ratios (BV/TV)
had a range from 13.7% to 30.3%. The degree of anisotropy of
these cubes, which is the ratio between the largest and
smallest eigenvalues of the fabric tensor, ranged from 1.52
to 3.86. A 5 mm length Volume Element (VE) which has been
previously considered appropriate to capture the features of
trabecular bone (Harrigan et al., 1988; Sanyal et al., 2015; van
Rietbergen et al., 1995) was employed for all simulations.
2.2. Constitutive model and computational procedure
The twenty specimens were meshed using a voxelised mesh,
where every voxel corresponds to a trilinear hexahedron,
with an in-house developed script that meshes in parallel
using the Message Passing Interface (MPI) (Forum, 1994). The
meshing procedure was performed using 30 cores on a cluster
at The University of Edinburgh, which is called Eddie
(Edinburgh Compute and Data Facilities, ECDF). The largest
Fig. 1 FE meshes of the most porous sample (13.7% BV/TV) (left) and of the densest sample (30.3% BV/TV) (right). Both
samples are cubes of 5 mm edge length.
journal of the mechanical behavior of biomedical materials 61 (2016) 384–396386
mesh had around 9 million nodes. FE models for two of the
twenty specimens are shown in Fig. 1.
The solid phase was modelled as an isotropic elastoplastic
material. It is important to mention that trabecular bone at
the tissue level is actually transverse isotropic or orthotropic
(Hellmich et al., 2004; Malandrino et al., 2012; Wolfram et al.,
2010). However, as pointed out by Cowin (1997), there is little
to no error in assuming tissue isotropy. This is because the
trabecula are composed of laminated material about their
axes, which implies transverse isotropy or orthotropy; since
the axis of the trabecula is the same as the loading axis, a
beam made of orthotropic material can be reduced to a beam
made of isotropic material.
The elastic regime was modelled using Hencky hyperelas-
ticity, which restricts this material model to isotropic, with a
Poisson's ratio of 0.3 and a Young's Modulus of 12700 MPa
(Wolfram et al., 2012). A quadric yield surface (Schwiedrzik
et al., 2013) given by
G τðÞ¼
ffiffiffiffiffiffiffiffiffiffiffi
τ:G:τ
p
þ G:τ 1 ¼0 ð1Þ
was employed, where τ is the Kirchhoff stress and G and G
are, respectively, a fourth-order tensor and a second-order
tensor dened by
G ¼ζ
0
G
2
0
I IðÞþζ
0
þ 1

G
2
0
I
I

ð2Þ
and
G ¼
1
2
1
s
þ
0
1
s
0

I ð3Þ
where
G
0
¼
s
þ
0
þ s
0
2s
þ
0
s
0
ð4Þ
and I is the second-order unit tensor, s
þ
0
and s
0
are the
tensile and compressive yield stresses, respectively, and ζ
0
is
an interaction parameter.
Eq. (1) approximates a Drucker-Prager criterion when
ζ
0
¼0.49. Recent studies using indentation tests on bone
tissue have suggested that the solid phase of bone can be
represented using a Drucker-Prager type criterion (Carnelli
et al., 2010; Tai et al., 2006). Uniaxial yield strains of 0.41% in
tension and 0.83% in compression (Bayraktar and Keaveny,
2004) were converted to yield stresses by simply multiplying
them by the Young's modulus of the solid phase (Schwiedrzik
et al., 2015). Although there have been some experimental
studies that have evaluated hardening of the extracellular
matrix (Luczynski et al., 2015; Schwiedrzik et al., 2014), there
is no agreement on the solid phase hardening behaviour of
trabecular bone. Some studies have assumed linear hard-
ening (Bayraktar and Keaveny, 2004; Bevill et al., 2006). In this
study perfect plasticity was assumed (Carnelli et al., 2010),
though small hardening (0.02% of the Young's modulus) was
included to aid prevention of loss of ellipticity. In order to
ensure global convergence of the Newton-Closest Point Pro-
jection Method (Newton-CPPM) scheme, a line search proce-
dure was implemented as in the primal-CPPM algorithm
proposed by Perez-Foguet and Armero (2002).
Each cubic specimen was subjected to 160 strain-controlled
load cases as described in Table 1. The boundary conditions
used to constrain the VE were kinematic uniform boundary
conditions applied as described by Wang et al. (2009).Itis
recognised that these boundary conditions provide an upper
bound for trabecular bone stiffness and also for yield
(Panyasantisuk et al., 2015; Wang et al., 2009). The simulations
were run on a Cray XC30 supercomputer hosted by ARCHER,
the UK National Supercomputing Service. The analyses were
carried out with an in-house parallel implicit nite strain
solver, developed within the context of ParaFEM (Margetts,
2002; Smith et al., 2014), which uses an Updated Lagrangian
formulation. This code uses MPI to perform the parallelisation
(Smith and Margetts, 2003, 2006). The high scalability of the
code has already been demonstrated in previous work (Levrero
Florencio et al., 2015; Margetts et al., 2015). Each of the 160
simulations per sample took approximately 12 min using 1920
cores; therefore the total number of core hours employed in
this study is approximately 1.2 million.
A Newton-Raphson scheme was used as the solution
tracking technique and a preconditioned conjugate gradient
solver was used to solve the resulting linear algebraic sys-
tems. They are fast and if there are any convergence pro-
blems, they arise from the same origin (e.g. due to loss of
positive deniteness of the stiffness matrix). However, con-
vergence problems were only encountered in few of the
porous samples (in 20 out of 3200 simulations) and can be
related to a limit point or large-deformation related failure
mechanisms (Bevill et al., 2006; de Souza Neto et al., 2008).
Table 1 Description of the load cases undertaken. Clockwise and counter-clockwise shear are differentiated by the sign of
the off-diagonal terms of the homogeneous strain: clockwise corresponds to positive sign and counter-clockwise
corresponds to negative sign. If T and C represent tension and compression respectively in normal strain space and
clockwise and counter-clockwise shear in shear strain space, then the quadrants comprise of C-C, T-T, C-T and T-C; and
octants comprise of C-C-C, C-C-T, C-T-C, C-T-T, T-C-C, T-C-T, T-T-C and T-T-T.
Type of analysis Number of analyses
Uniaxial normal 3 tensile and 3 compressive 6
Uniaxial shear 3 clockwise and 3 counter-clockwise 6
Biaxial normal 3 planes, 1 analysis per quadrant 12
Triaxial normal 8 octants, 1 analysis per octant 8
Biaxial shear 3 planes, 1 analysis per quadrant 12
Triaxial shear 8 octants, 1 analysis per octant 8
Biaxial normal-shear 9 planes, 1 analysis per quadrant 36
Biaxial normal-shear with different ratios 9 planes, 2 analysis per quadrant 72
Total 160
journal of the mechanical behavior of biomedical materials 61 (2016) 384–396 387
These points were marked differently in the gures and
included in the tting procedure as yield points.
The initial load increment size corresponded to 0.1%
macroscopic strain norm and could decrease to a minimum
of 0.001% if global convergence was not achieved in larger
load increments.
3. Theory and calculation
3.1. Denition of the macroscopic yield points
The yield points were described in the plane where the
abscissa is the Frobenius norm of the applied macroscopic
strain, which corresponds to the Green-Lagrange strain, and
the ordinate is the Frobenius norm of the homogenised
Second Piola-Kirchhoff stress (Fig. 2), described by
S
hom
¼
1
V
0
X
nel
i ¼ 1
X
nip
j ¼ 1
w
i
detJ
ij
S
ij
ð5Þ
where there is no summation implied on repeated indices, V
0
is the initial volume of the VE, nel is the number of elements
in the FE simulation, nip is the number of Gauss integration
points in a trilinear hexahedra, J is the Jacobian, S is the
Second-Piola Kirchhoff stress and w are the weights corre-
sponding to the specic Gauss integration point.
The 0.2% criterion was used to dene the yield points
(Wolfram et al., 2012), as shown in Fig. 2, and the elastic slope
was obtained from the rst two load increments, which were
always fully elastic.
3.2. Formulation for macroscopic yield surface
Although a key aim of this study was to assess how different
bone samples yield when subjected to the wide array of 160
load cases, we also examined the macroscopic yield surface
t using a quadric surface. The choice was based on its
simplicity, because it has been previously related to the fabric
tensor of bone (Cowin, 1986; Wolfram et al., 2012), and
because it is a smooth surface.
The quadric yield surface is described in strain space as
Y EðÞ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E : F : E
p
þ F : E1 ¼0 ð6Þ
where E is the Green-Lagrange strain tensor, F and F are used
to dene the shape, directionality and eccentricity of the yield
surface. F is a fourth-order tensor, which has major and
minor symmetries (F
ijkl
¼F
klij
and F
ijkl
¼F
jikl
¼F
ijlk
¼F
jilk
),
allowing it to be dened on a symmetric matrix space
(Sym
6
), by 21 coefcients (Mehrabadi and Cowin, 1990). As
stated in Schwiedrzik et al. (2013), convexity of Eq. (6) is
ensured with positive semi-deniteness of F. F is a symmetric
second-order tensor, thus being described by 6 coefcients.
3.3. Different symmetries of the yield surface
Three different cases were investigated: isotropy, orthotropy
and full anisotropy. Details about isotropic and orthotropic
formulations can be found in (Schwiedrzik et al., 2013); only
anisotropy is discussed in the following.
In the case of full anisotropy, or triclinic symmetry, the
material can have different shear yield strains clockwise and
counter-clockwise. For an anisotropic quadric, normal strains
can interact with shear strains and shear strains can interact
amongst themselves (Theocaris, 1992; Tsai and Wu, 1971).
This means that in the triclinic case, F and F have 21 and
6 independent coefcients respectively.
By performing uniaxial strain load cases, several coef-
cients of F and all the coefcients of F can be determined. For
F, the coefcients are
F
ij
¼
1
2
1
ε
þ
ij
1
ε
ij

if i ¼ j
1
4
1
ε
þ
ij
1
ε
ij

if ia j
i; j ¼1; 2; 3
8
>
>
>
<
>
>
>
:
ð7Þ
In the case of F, the six diagonal coefcients of the
projection of F onto Sym
6
are
F
ijij
¼
ε
þ
ij
þε
ij
2ε
þ
ij
ε
ij

2
if i ¼ j
1
2
ε
þ
ij
þε
ij
2ε
þ
ij
ε
ij

2
if ia j
i; j ¼1; 2; 3
8
>
>
>
>
<
>
>
>
>
:
ð8Þ
The 15 remaining parameters to be determined corre-
spond to three normal strain interaction parameters, three
shear strain interaction parameters and nine normal-shear
strain interaction parameters. These parameters have
expressions in the coefcients of F which are related to the
previously stated diagonal coefcients, as shown in Table 2.
These, together with the six uniaxial normal strains and
six uniaxial shear strains, add up to a total of 27 parameters.
Calculating the determinant of 1 1 and 2 2 principal
minors of the projection of F onto Sym
6
allows establishment
Fig. 2 Determination of the yield points by using the 0.2%
criterion for the tensile and compressive uniaxial load cases
of one sample. As it can be seen, the tensile and
compressive uniaxial cases have the same elastic slope
(dashed red line), as expected.
journal of the mechanical behavior of biomedical materials 61 (2016) 384–396388
of basic restrictions on some of the coefcients to ensure that
F is positive semi-denite, which are
ε
7
ij
Z 0;jζ
kl
jr1; i; j ¼ 1; 2; 3; k; l ¼ 1; 2; ; 6 ð9Þ
The remaining restrictions on the coefcients are not
expressed analytically but checked after the minimisation
procedure to ensure positive semi-deniteness of F. For every
symmetry case and for every sample, the eigenvalues of the
projection of F onto Sym
6
were checked to ensure they were
non-negative.
3.4. Evaluating goodness of t
The macroscopic yield envelope was tted by using a mini-
misation procedure in MATLAB (Mathworks, Natick MA, USA).
To evaluate the goodness of t, the error was evaluated as
Fitting error ¼
1
N
X
N
i ¼ 1
J E
fitted
E
FE
J
J E
FE
J
ð10Þ
where is the Frobenius norm of the corresponding macro-
scopic strain and N is the cardinality of a specic set of load
cases. This error was evaluated for four different sets: for all
the load cases; for the load cases which entirely lie on the
normal strain space; for the load cases which entirely lie on
the shear strain space; and for the strain cases that have one
component in the normal strain space and one component in
the shear strain space (which from now onwards will be
referred to as combined normal and shear strain space).
4. Results
4.1. Macroscopic yield strains
Macroscopic yield points in strain space for all twenty
considered samples in the normal-normal and shear-shear
planes are shown in Fig. 3. These represent 36 of the 160 load
cases analysed for each sample, and no projections have
been made, i.e. the yield strains only contain out-of-plane
components equal to zero. Results show that the macroscopic
yield surface of bone has a higher yield strain in compression
than in tension in the normal strain space. This is expected
due to the characteristics of its solid phase.
It can be seen that the tensile quadrant displays quasi-
uniform macroscopic yield strains across samples (upper
right quadrant of Fig. 3a, b, c). The compressive yield strains
have some variability across samples, as can be seen from the
spread of yield points in the lower left quadrant of Fig. 3a, b,
c. The largest variation in the normal-normal planes is in the
tensile-compressive quadrants (upper left and lower right
quadrants, Fig. 3a, b, c).
Macroscopic yield strains in the shear-shear planes show
a large variation of yield strains for different specimens
(Fig. 3d, e, f). It can also be observed that the shear yield
strains of bone are different in clockwise and counter-
clockwise directions, with these absolute differences ranging
from 0.0034% to 0.4463%. A statistical comparison between
these yield strains was performed for all pure shear cases
with a paired t-test. This test suggests that paired clockwise
and counter-clockwise shear yield strains are statistically
different (p o0.01).
Macroscopic uniaxial (tensile, compressive and shear)
yield strains were related to BV/TV and fabric through multi-
linear regressions performed in log space. No relationship
between yield strains and BV/TV and fabric was found. Only
compressive uniaxial yield strains were mildly related to BV/
TV and fabric (R
2
¼0.44, p-0).
In order to examine macroscopic yield strains in tension-
tension, compression-compression and tension-compression
regimes, we evaluated the mean of the macroscopic Green
Lagrange strain norm for each of the above three regimes, as
shown in Fig. 4. As expected, the mean of the norms is the
lowest for tension-tension, highest for compression-
compression and in between for tension-compression. Fig. 4
also shows the standard deviation in the evaluated norms. It
can be observed that the deviation is relatively small for the
tension-tension regime, higher for compression-compression
regime and the highest for tension-compression regime.
4.2. Solid phase strains
We examined strains at the microscale (solid phase strains).
Under uniaxial macroscopic tension, more localised strains
were found to occur at the solid phase level, and there were
mostly no compressive solid phase strains anywhere in the
specimens. However, under uniaxial macroscopic compres-
sion, the compressive solid phase strains were more diffused
and found to occur throughout the geometry. Further, under
macroscopic compression, large tensile solid phase strains
Table 2 Interaction coefcients for the anisotropic
quadric.
Coefcient
Normal interaction
F
1122
¼ζ
12
ε
þ
11
þε
11
2ε
þ
11
ε
11

ε
þ
22
þε
22
2ε
þ
22
ε
22

F
1133
¼ζ
13
ε
þ
11
þε
11
2ε
þ
11
ε
11

ε
þ
33
þε
33
2ε
þ
33
ε
33

F
2233
¼ζ
23
ε
þ
22
þε
22
2ε
þ
22
ε
22

ε
þ
33
þε
33
2ε
þ
33
ε
33

Shear interaction
F
1213
¼ζ
45
ε
þ
12
þε
12
2ε
þ
12
ε
12

ε
þ
13
þε
13
2ε
þ
13
ε
13

F
1223
¼ζ
46
ε
þ
12
þε
12
2ε
þ
12
ε
12

ε
þ
23
þε
23
2ε
þ
23
ε
23

F
1323
¼ζ
56
ε
þ
13
þε
13
2ε
þ
13
ε
13

ε
þ
23
þε
23
2ε
þ
23
ε
23

Normal-shear interaction
F
1112
¼ζ
14
ε
þ
11
þε
11
2ε
þ
11
ε
11

ε
þ
12
þε
12
2ε
þ
12
ε
12

F
1113
¼ζ
15
ε
þ
11
þε
11
2ε
þ
11
ε
11

ε
þ
13
þε
13
2ε
þ
13
ε
13

F
1123
¼ζ
16
ε
þ
11
þε
11
2ε
þ
11
ε
11

ε
þ
23
þε
23
2ε
þ
23
ε
23

F
2212
¼ζ
24
ε
þ
22
þε
22
2ε
þ
22
ε
22

ε
þ
12
þε
12
2ε
þ
12
ε
12

F
2213
¼ζ
25
ε
þ
22
þε
22
2ε
þ
22
ε
22

ε
þ
13
þε
13
2ε
þ
13
ε
13

F
2223
¼ζ
26
ε
þ
22
þε
22
2ε
þ
22
ε
22

ε
þ
23
þε
23
2ε
þ
23
ε
23

F
3312
¼ζ
34
ε
þ
33
þε
33
2ε
þ
33
ε
33

ε
þ
12
þε
12
2ε
þ
12
ε
12

F
3313
¼ζ
35
ε
þ
33
þε
33
2ε
þ
33
ε
33

ε
þ
13
þε
13
2ε
þ
13
ε
13

F
3323
¼ζ
36
ε
þ
33
þε
33
2ε
þ
33
ε
33

ε
þ
23
þε
23
2ε
þ
23
ε
23

journal of the mechanical behavior of biomedical materials 61 (2016) 384–396 389
were found to arise, due to bending and buckling of trabecu-
lae. Fig. 5 and Fig. 6 show this for a slice from one typical
porous sample.
Fig. 3 Macroscopic yield points of the 20 specimens in normal strain planes (a, b, c) and in shear strain planes (d, e, f). In the
shear strain planes, clockwise shear is represented as positive and counter-clockwise shear is represented as negative.
Density of the samples is indicated by the colour-bar. Yield points obtained for a few cases from the loss of positive
deniteness of the stiffness matrix are marked with an empty circle.
Fig. 4 Bar plot of the mean of the macroscopic Green
Lagrange strain norms for tensile cases (i.e. cases in the
normal strain space where all strain components are
positive), compressive cases (i.e. cases in the normal strain
space where all strain components are negative) and
tensile-compressive cases (i.e. cases in the normal strain
space where one component is positive and one component
is negative). The error bars correspond to the standard
deviation of these values.
Fig. 5 Distribution of the Green-Lagrange solid phase strain
component E
11
for a 0.5x5
5 mm slice of bone under
macroscopic uniaxial tension. Direction 1 is in the direction
denoted by the arrows.
journal of the mechanical behavior of biomedical materials 61 (2016) 384–396390
4.3. Macroscopic yield surface and tting errors
A macroscopic yield surface was tted for each of the 20
samples by using isotropic, orthotropic and fully anisotropic
quadric yield surfaces, using all of the 160 load cases. The
parameters for the anisotropic surface have been added as a
statistical evaluation in Table 3. Plots for the two samples
with the highest and lowest densities are shown in Fig. 7. The
lower density sample shows a higher level of anisotropy in
comparison to the higher density sample. Further, if con-
secutive macroscopic yield points of the porous sample are
joined up, then the homogenised yield envelope does not
always remain entirely convex in some of the planes (e.g.
Fig. 7i, k).
Mean tting errors (Eq. (10) considering all samples and all
strain cases are shown in Fig. 8. It can be seen that the
isotropic assumption leads to the highest (11%) error,
followed by the orthotropic (10%) and the anisotropic
(8%) assumptions. The standard deviation of the tting
errors is also shown in the gure and it can be seen that the
error variation across densities with the isotropic assumption
is similar to the orthotropic case; the anisotropic assumption
produces the smaller variations across samples.
In general, the tting errors were not found to correlate
with bone density (Fig. 9). In normal strain space (Fig. 9a), the
assumption of an isotropic quadric led to consistently higher
errors, while the orthotropic and anisotropic assumptions
resulted in smaller tting errors. In the shear strain space
(Fig. 9b), in the combined normal and shear strain space
(Fig. 9c), and in the general strain space (Fig. 9d), the
assumption of anisotropic quadric had the smallest errors.
A mild trend of errors decreasing with increasing density was
observed for the isotropic and orthotropic assumptions in
shear strain space. In the general strain space, the errors and
the error differences between assumptions tend to reduce
with increasing density.
5. Discussion
Our study shows that the macroscopic yield surface of bone
in normal strain space is fairly uniform across a wide range of
samples; this conrms ndings of previous research
(Bayraktar et al., 2004; Lambers et al., 2014; Pankaj and
Donaldson, 2013).
Our results also demonstrate that the full three-
dimensional macroscopic yield behaviour of trabecular bone
can be reasonably well described using the isotropic quadric
yield surface, though orthotropic and anisotropic surfaces
lead to smaller errors. This is in agreement with previous
studies in which the strain space yield surface was reported
to be isotropic (Bayraktar et al., 2004) and more recently
transversely isotropic (Sanyal et al., 2015) and orthotropic
(Wolfram et al., 2012). However, unlike the above studies, our
Fig. 6 Distribution of the Green-Lagrange solid phase strain
under macroscopic uniaxial compression for a 0.5x5
5mm
slice of bone. Tensile component E
22
(top) and compressive
component E
11
(bottom). Direction 1 is in the direction
denoted by the arrows and direction 2 is the orthogonal in-
plane direction.
Table 3 Statistical evaluation of the parameters of the
anisotropic surface.
Coefcient Value (mean 7 standard deviation)
F
1111
18391.875092.2
F
1122
7801.171056.9
F
1133
8492.171021.2
F
1112
155.57866.8
F
1113
306.671883.35
F
1123
150.07684.1
F
2222
17510.172826.5
F
2233
8644.471713.2
F
2212
83.47800.1
F
2213
84.07744.1
F
2223
495.571619.2
F
3333
20689.074299.2
F
3312
12.27628.9
F
3313
260.171616.6
F
3323
538.971787.2
F
1212
4557.37688.8
F
1213
228.27374.2
F
1223
287.87482.6
F
1313
5016.17724.8
F
1323
3.57427.7
F
2323
5200. 47844.0
F
11
52.6714.3
F
22
52.078.2
F
33
60.9712.5
F
12
0.472.5
F
13
0.875.1
F
23
1.874.6
journal of the mechanical behavior of biomedical materials 61 (2016) 384–396 391
treatment of clockwise and counter-clockwise shears as
separate loading cases showed that the differences in the
two directions are signicant. This is probably because
trabeculae are not symmetrically aligned with respect to the
axes of the material, which in this case were assessed
through the eigenvectors of the MIL fabric tensor. It is
important to note that the assumption of identical
macroscopic yield points in clockwise and counter-
clockwise directions restricts the system to orthotropy at
best (Theocaris, 1992; Tsai and Wu, 1971). We also observed
predominance of tensile solid phase strains in pure macro-
scopic shear, which is consistent with Sanyal et al. (2012).
Multilinear regressions suggest that uniaxial yield strains
are not correlated with BV/TV and fabric (Matsuura et al.,
Fig. 7 Macroscopic yield points for the densest and most porous samples and their corresponding isotropic, orthotropic and
anisotropic tted quadric surfaces in the normal strain planes (a, b, c), shear strain planes (d, e, f), and combined normal and
shear strain planes (g o). Yield points obtained for a few cases from the loss of positive deniteness of the stiffness matrix are
marked with an empty circle.
journal of the mechanical behavior of biomedical materials 61 (2016) 384–396392
2008; Morgan and Keaveny, 2001; Panyasantisuk et al., 2015).
Only a mild dependence was found for the uniaxial compres-
sive yield strains (R
2
¼0.44, p-0), with a positive slope for
density and a negative slope for fabric, which suggests that
long trabeculae, i.e. associated with a high fabric eigenvalue,
have lower macroscopic yield strain, as suggested by
Matsuura et al. (2008). Since no clear relationship between a
fabric tensor and the yield strains was found, use of an
isotropic yield surface formulation in strain space is more
practical for real applications.
In uniaxial macroscopic tension, solid phase strains are
almost exclusively tensile and independent of density
(Bayraktar and Keaveny, 2004; Lambers et al., 2014). The
highly oriented structure of trabecular bone results in the
yield strains at the solid phase and at the macroscale being
very similar in tension. When trabecular bone is loaded in
macroscopic compression, yield mechanisms are different: in
this case, yielding at the solid phase was found to occur both
due to tension (arising from bending and buckling of trabe-
culae) and compression. As expected, we found considerable
tensile strains in trabeculae for low density samples as has
been previously reported (Bevill et al., 2006; Morgan et al.,
2004; Stolken and Kinney, 2003). This density dependence
results in macroscopic yield variation being displayed via a
small spread of yield points in the compression-compression
quadrants (lower left quadrants of Fig. 3ac), as shown by the
mild relationship between compressive uniaxial yield strains
and density and fabric. This also implies that solid phase
uniaxial yield strain asymmetry is not fully maintained at the
macroscale and generally reduces with increased porosity
and increased fabric eigenvalues. Our results are consistent
with the experimental results of Lambers et al. (2014) in the
sense that the number of microscopic yielded sites in macro-
scopic compression and in macroscopic tension are similar in
number, but in macroscopic tension, the microscopic yield
zones have more localised strains, which could be related to
microcrack propagation.
Fig. 8 Bar plot of the mean of the tting errors across all
samples for the isotropic quadric, orthotropic quadric and
anisotropic quadric. All the strain cases are taken into
account. The error bars correspond to the standard deviation
of these values.
Fig. 9 Fitting errors as described in eq. 20 for the normal strain space (a), shear strain space (b), combined normal and shear
strain space (c) and in general strain space (d).
journal of the mechanical behavior of biomedical materials 61 (2016) 384–396 393
The few previous studies that have evaluated the macro-
scale yield surface of bone from its microstructure have all
used a bilinear criterion, with yield strain asymmetry, to dene
the solid phase of bone with a reduced stiffness beyond
dened tissue yield values (Bayraktar et al., 2004; Sanyal
et al., 2015; Wolfram et al., 2012). In our study, the solid phase
of bone was modelled using a Drucker-Prager type criterion
which has been validated via experimental studies (Carnelli
et al., 2010; Tai et al., 2006). Our study considered yield points
arising from 160 different load cases. Some recent studies have
been limited to 17 load cases (Panyasantisuk et al., 2015;
Wolfram et al., 2012). In order to compare our results we
considered 17 strain cases similar to those in the cited studies.
We found the errors to be 11.4% for the isotropic case and
10.3% for the orthotropic case for the 17 cases. The errors,
taking into account all 160 cases, are 11.2% for the isotropic
case, 10.4% for the orthotropic case and 7.8% for the aniso-
tropic case. In other words, the 17 load cases lead to errors of a
similar magnitude to those obtained using all 160 load cases.
To further examine the effect of the considered load cases on
the tting error, we considered a single sample with different
load cases. We evaluated the tting errors considering all
normal load cases proposed by Wolfram et al. (2012) (14 cases)
and all our shear cases (26 cases). The errors were 18.0%, 16.3%
and 7.6% for isotropic, orthotropic and anisotropic assump-
tions respectively. With the 17 load cases mentioned pre-
viously, the errors reduce to 9.8% and 6.5% for isotropic and
orthotropic assumptions respectively for this sample. With all
the 160 load cases, the errors are 13.3%, 12.9% and 9.2% for
isotropic, orthotropic and anisotropic assumptions respec-
tively. This illustrates that shear cases contribute importantly
to anisotropy, and that the tting errors clearly depend on the
considered load cases, which illustrates the importance of
examining a large range of complex load cases. With respect to
the combined normal and shear strain spaces, the shear
component of the macroscopic yield strain is often found to
increase when there is a compressive normal component,
indicating typical cellular solid behaviour (Fenech and
Keaveny, 1999; Gibson and Ashby, 1997).
Our study has a number of limitations. We chose to use a
quadric due to its simplicity, because it has been used in
previous studies, because it requires fewer parameters than
higher order criteria, and because it is a smooth surface, not
requiring multiple plastic multipliers. Although our primary
aim was to examine the effect of material symmetry assump-
tions on the macroscopic yield surface, the tting errors
clearly depend on the shape of the chosen surface and on
the considered load cases. While we examined full anisotropy
with a quadric surface, a previous study employed a higher
order polynomial surface, a quartic, but restricted it to
transverse isotropy (Sanyal et al., 2015); a restriction that is
not fully supported by our results.
Although the solid phase constitutive law has been vali-
dated (Carnelli et al., 2010; Tai et al., 2006), there is no
experimental validation for the macroscopic yield surfaces. In
fact,suchavalidationisimpossibleassamplestestedonce
cannot be retested and it is not possible to obtain numerous (or
even two) identical samples. There have, however, been some
attempts to on trabecular bone wherein the loading cases have
been limited to triaxial compression (Keaveny et al., 1999;
Rincon-Kohli and Zysset, 2009). Thus, the range of complex
strain cases we have tested can only be performed numerically.
We considered homogeneous tissue elastic properties
while some previous studies have assessed the effect of
heterogeneous mineral density on the macroscopic stiffness
of bone (Blanchard et al., 2013; Renders et al., 2008). Renders
et al. (2008) found a decrease of 21% in apparent stiffness
when considering mineral heterogeneity. However the effect
of heterogeneities at the solid phase on nite element models
with geometrical nonlinearities is still unclear. Since we
wanted to be able to compare our macroscopic yield strains
with previously published results, we kept our solid phase
elastic properties as homogeneous.
The study assumes that the solid phase of bone can be
modelled as a plastic material, which is not entirely true as
high localised strains can cause microcracks and eventual
fracture; effects that full plasticity based models may not be
able to capture. Furthermore, we did not consider hardening
(Carnelli et al., 2010) because there is no agreement on the
hardening law at the scale of the solid phase we are
considering. However, previous experimental and theoretical
studies such as Schwiedrzik et al. (2014), Luczynski et al.
(2015) and Fritsch et al. (2009) showed that the extracellular
matrix of bone has a hardening behaviour after yield.
6. Conclusions
Trabecular bone has fairly uniform macroscopic yield beha-
viour across samples in normal strain space. Thus, modelling
it by using strain-based plasticity makes sense. For tension-
tension and compression-compression quadrants in normal
strain space, the strain norm at yield shows little variation,
indicating an isotropic behaviour in these regimes.
In the tension-compression quadrants, pure shear and
combined normal-shear planes, the macroscopic strain
norms at yield have a relatively large variation, indicating a
possible absence of isotropy. Further, differences in yield
strain values in clockwise and counter-clockwise shear may
indicate a possible anisotropy for the macroscopic behaviour
of trabecular bone. However, due to the difculties of for-
mulating a non-isotropic closed-form yield surface in strain
space due to the weak relationships between fabric and yield
strains, and due to the small difference in tting errors
between isotropic and orthotropic or anisotropic considera-
tions, an isotropic criterion presents itself as the most
suitable approximation. However, for some load cases, con-
siderable differences between the closed-form yield criterion
and the actual yield strain may arise.
With respect to the yield surface, an eccentric-ellipsoid
may adequately represent the macroscopic yield surface of
bone as the tting errors for all the considered symmetries
are reasonably small. However, it is important to be mindful
of the asymmetry in shear yield strains and that in the
normal-shear load cases, the quadric may not be able to
represent the macroscopic yield behaviour of trabecular bone.
This work provides a comprehensive assessment of mate-
rial symmetries of trabecular bone at the macroscale and
describes in detail its macroscopic yield and its underlying
microscopic mechanics.
journal of the mechanical behavior of biomedical materials 61 (2016) 384–396394
Acknowledgements
The authors would like to thank EPSRC for providing access to
ARCHER, UK National Supercomputing Service, through pro-
ject e398, Modelling nonlinear micromechanical behaviour of
bone, and funding through the grant EP/K036939/1. We also
would like to acknowledge The Hartree Centre, Science and
Technology Facilities Council for access to Blue Joule through
project HCP010 and funding from BBSRC through the grant BB/
K006029/1. Additionally, the rst author is grateful for the
Principal's Career Development PhD Scholarship of the Uni-
versity of Edinburgh.
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journal of the mechanical behavior of biomedical materials 61 (2016) 384–396396
... Principal strain rather than stress was used for the comparison [27][28][29] and the strain limits of N0.2% (tensile) and b−0.2% (compressive) for maximum and minimum principal strains, respectively [30] were employed. These values are close to the upper limit of physiological strain [30]. ...
... Strain limits of 0.2% and − 0.2% were selected for maximum and minimum principal strain, although the limits are close to the reported peak of physiological strain but far less than apparent yield strain of bone [27,28,31]. Therefore, the high percentage of bone volume greater than the preselected strain range does not necessarily imply bone yield. ...
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Background In the UK around 10% of hip and knee arthroplasties are revision operations. At revision total knee arthroplasty (rTKA), bone loss management is critical to achieving a stable bone-implant construct. Though tritanium cones have been used to manage bone defects in rTKA, their biomechanical performance with varying defects remains unknown. Methods Uncontained tibial bone defects at four anatomic locations, with varying depths and widths (Type T2A and T2B) were investigated computationally in a composite tibia which was subjected to four loading scenarios. The ability of the tritanium cone to replace the tibial bone defect was examined using the outcome measures of bone strain distribution and interface micromotions. Results It was found that anterior and lateral defects do not significantly alter the strain distribution compared with intact bone. For medial defects, strain distribution is sensitive to defect width; while strain distributions for posterior defects are associated with defect width and depth. In general, micromotions at the bone-implant interface are small and are primarily influenced by defect depth. Conclusions Our models show that the cone is an acceptable choice for bone defect management in rTKA. Since all observed micromotions were small, successful osteointegration would be expected in all types of uncontained defects considered in this study. Tritanium cones safely accommodate uncontained tibial defects up to 10 mm deep and extending up to 9 mm from the centre of the cone. Medial and posteriorly based defects managed with symmetric cones display the greatest bone strains and asymmetric cones may be useful in this context.
... As marked recently [195], there is nothing more fundamental in nonlinear mechanics of anisotropic materials with tension-compression asymmetry than the constitutive framework [191][192][193][194] developed in the early 1980s. Furthermore, such constitutive framework was used in [196][197][198][199][200][201][202][203][204][205][206][207] for modeling of the natural and artificial materials, however, without referencing sources [191][192][193][194]. ...
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... As marked recently [195], there is nothing more fundamental in nonlinear mechanics of anisotropic materials with tension-compression asymmetry than the constitutive framework [191][192][193][194] developed in the early 1980s. Furthermore, such constitutive framework was used in [196][197][198][199][200][201][202][203][204][205][206][207] for modeling of the natural and artificial materials, however, without referencing sources [191][192][193][194]. ...
... As marked recently [195], there is nothing more fundamental in nonlinear mechanics of anisotropic materials with tension-compression asymmetry than the constitutive framework [191][192][193][194] developed in the early 1980s. Furthermore, such constitutive framework was used in [196][197][198][199][200][201][202][203][204][205][206][207] for modeling of the natural and artificial materials, however, without referencing sources [191][192][193][194]. ...
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Viruses are a large group of pathogens that have been identified to infect animals, plants, bacteria and even other viruses. The 2019 novel coronavirus SARS-CoV-2 remains a constant threat to the human population. Viruses are biological objects with nanometric dimensions (typically from a few tens to several hundreds of nanometers). They are considered as the biomolecular substances composed of genetic materials (RNA or DNA), protecting capsid proteins and sometimes also of envelopes. The study of viruses and virus-like structures has been analyzed using models and methods of nonlinear mechanics. In this regard, quantum, molecular and continuum descriptions in virus mechanics have been considered. Application of single molecule manipulation techniques, such as, atomic force microcopy, optical tweezers and magnetic tweezers has been discussed for a determination of the mechanical properties of viruses. Particular attention has been given to continuum damage–healing mechanics of viruses, proteins and virus-like structures. Also, constitutive modeling of viruses at large strains is presented. Nonlinear elasticity, plastic deformation, creep behavior, environmentally induced swelling (or shrinkage) and piezoelectric response of viruses were taken into account. The environment is considered to be made up of those parts of the universal system with which the virus interacts. Integrating a constitutive framework into ABAQUS, ANSYS and in-house developed software has been discussed. Link between virus structure, environment, infectivity and virus mechanics may be useful to predict the response and destructuration of viruses taking into account the influence of different environmental factors. Computational analysis using such link may be helpful to give a clear understanding of how neutralizing antibodies and T cells interact with the 2019 novel coronavirus SARS-CoV-2.
... By using large models, it is not useful to represent and calculate the cancellous bone geometrically. Approaches to homogenize the trabecular structure are necessary without losing information about the realistic material properties and the dependence on bone quality [1,2,19,26,50]. A first approach has been explored by Cowin [3] using a single second rank tensor (fabric tensor). An alternative model for the anisotropic elasticity based on fabric tensors was presented by Zysset and Curnier [51]. ...
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In biomechanics, large finite element models with macroscopic representation of several bones or joints are necessary to analyze implant failure mechanisms. In order to handle large simulation models of human bone, it is crucial to homogenize the trabecular structure regarding the mechanical behavior without losing information about the realistic material properties. Accordingly, morphology and fabric measurements of 60 vertebral cancellous bone samples from three osteoporotic lumbar spines were performed on the basis of X-ray microtomography (μCT) images to determine anisotropic elastic parameters as a function of bone density in the area of pedicle screw anchorage. The fabric tensor was mapped in cubic bone volumes by a 3D mean-intercept-length method. Fabric measurements resulted in a high degree of anisotropy (DA = 0.554). For the Young’s and shear moduli as a function of bone volume fraction (BV/TV, bone volume/total volume), an individually fit function was determined and high correlations were found (97.3 ≤ R2 ≤ 99.1,p < 0.005). The results suggest that the mathematical formulation for the relationship between anisotropic elastic constants and BV/TV is applicable to current μCT data of cancellous bone in the osteoporotic lumbar spine. In combination with the obtained results and findings, the developed routine allows determination of elastic constants of osteoporotic lumbar spine. Based on this, the elastic constants determined using homogenization theory can enable efficient investigation of human bone using finite element analysis (FEA). Cancellous Bone with Fabric Tensor Ellipsoid representing anisotropy and principal axis (colored coordinate system) of given trabecular structure
... and the Jaumann stress derivative [116] . Second, the tensor relationship defined by Eq. (14) was used in [120][121][122][123][124][125][126][127] for modeling of the biological tissues, however, without referencing sources [117][118][119]. Third, the present model reflects the characteristic features of the anisotropic bimodular materials [128,129]. ...
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... 35 The use of strain-based criteria rather than stress-based criteria accommodates material anisotropy and bone volume ratio (or porosity)-while yield stresses for bone are both anisotropy and porosity dependent, yield strains are largely uniform. 26,36 The partitioning of the ROIs may have affected the interpretation of results and conclusions may be different if different ROIs were used. ...
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At the time of medial opening wedge high tibial osteotomy (HTO) to realign the lower limb and offload medial compartment knee osteoarthritis, unwanted fractures can propagate from the osteotomy apex. The aim of this study was to use finite element (FE) analysis to determine the effect of hinge location and apical drill holes on cortical stresses and strains in HTO. A monoplanar medial opening wedge HTO was created above the tibial tuberosity in a composite tibia. Using the FE method, intact lateral hinges of different widths were considered (5, 7.5 and 10mm). Additional apical drill holes (2, 4 and 6mm diameters) were then incorporated into the 10mm hinge model. The primary outcome measure was maximum principal strain in the cortical bone surrounding the hinge axis. Secondary outcomes included the force required for osteotomy opening, minimum principal strain and mean cortical bone stresses (maximum principal/minimum principal/von Mises). Larger intact hinges (10mm) were associated with higher cortical bone maximum principal strain and stress, lower minimum principal strain/stress and required greater force to open. Lateral cortex strain concentrations were present in all scenarios, but extended to the joint surface with the 10mm hinge. Apical drill holes reduced the mean cortical bone maximum principal strain adjacent to the hinge axis: 2mm hole 6% reduction; 4mm 35% reduction; 6mm 55% reduction. Incorporating a 4mm apical drill hole centred 10mm from the intact lateral cortex maintains a cortical bone hinge, minimises cortical bone strains and reduces the force required to open the HTO thus improving control. This article is protected by copyright. All rights reserved.
Chapter
Devices for orthopedic trauma treatment need to satisfy three key clinical requirements and consequent mechanical demands arising from them: they must support fracture healing; they must not fail during the healing period; and they should not loosen or cause patient discomfort. This chapter discusses the biomechanics and design issues associated with fulfilling the above demands by commonly used devices: external fixators, plates and screws, and intramedullary nails. In most cases, healing occurs via callus formation, which is supported by an appropriate amount of interfragmentary motion between fractured bone fragments. Delayed healing can cause failure of the device due to fatigue. Healing depends on the mechanical behavior of the bone–device construct, which has been extensively investigated. The problem of device loosening, particularly in poor-quality bone, has received relatively less attention, and this chapter considers its evaluation in some detail. This chapter also shows that the demands on the device change as healing progresses and device choice and configuration needs to consider healing in preoperative planning. It also discusses the issues associated with replicating the in vivo boundary conditions in laboratory experiments and numerical simulation. Recent studies on modeling of loosening due to cyclic loading are also discussed.
Chapter
Bone is a complex, hierarchically organized organ system whose composition and structure are closely related to, and in many ways controlled by, the functional demands made upon it. Bone tissue is constantly undergoing turnover via coordinated activities by osteoblasts, osteoclasts, osteocytes, and their precursors. Through this process of bone remodeling, the bone organ system can respond relatively quickly to changes in metabolic and mechanical needs. The concepts presented in this chapter provide a framework for elucidation of the biological and biomechanical mechanisms underlying the close relationship between form and function in bone.
Research Proposal
Full-text available
This research aimed to identifies the effect of a functional training program on some physical variables(Back flexibility, back strength, Range of motion , the trunk flexion , Range of motion the trunk extension , pain Feeling , Balance) and bone mineral density (BMD (L5-L4) , BMC (L5-L4), Femur bone density , BMD. F.N , Density of hip bone , hip bone BMD.Tro , neck BMC F.N, rotation BMC.Tro) in those injured with lumbar arthritis. The researcher used the experimental approach using an experimental group and a control group, using pre and post measurements, The sample of the research was chosen purposely , the sample consisted of (12) males (15:20) years , and were divided into two groups , the experimental group consisting of (6) injured and applied to them the functional training program, and control group consisting of (6) injured who were treated with the hospital’s treatment program . The researcher recommend with The importance of periodic medical examination regularly to observe any changes in the anatomical and morphological aspects of the spinal column and early detection of any shortcomings or injuries that may be exposed to the person , Guided by The functional training program and the generalization of its use in centers and Rehabilitation institutions and hospitals , Spreading cultural awareness of community members towards the early detection of low back pain cases to prevent the progression of infection to advanced grades , Continue to perform exercises and exercise, which works to maintain muscle strength and flexibility even after the completion of the proposed program.
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In this paper we discuss parallel processing by Finite Element Methods for nonlinear problems in engineering in general. In particular we focus on two completely different algorithms for the solution of large (say > 1 million degrees of freedom) problems in elastoplasticity. Our intention is to show that these two algorithms, and many others, can be readily parallelised using a subroutine library callable from Fortran. The particular library is written in MPI but this is not a restriction in principle. The authors have pursued an exclusively iterative technique for solution of the simultaneous algebraic equations. When allied with an element-by-element strategy the solution algorithms all have at their core only matrix-vector products and vector-vector operations which means that implicit plasticity calculations can be parallelised with not much more effort than explicit ones. This has been demonstrated by the parallelisation of all the algorithm types in Smith and Griffiths, amounting to some 10 explicit, implicit and eigenvalue computations. Of course, the parallel algorithms may not be optimal for the solution of any particular problem in elastoplasticity or elsewhere, but efficiency considerations are addressed. The aim is generality and portability of the approach to shared or distributed memory systems and to clusters of PCs. Finally, the solution of a “coupled” problem in geomechanics is reported which involves Biot-type coupling between solid and fluid phases of the geomaterials. The example described concerns aspects of processing of toxic waste derived from nuclear energy production.