Abstract—-
The
purposes
of
this
article
were
to
obtain
mechanical
properties
of
the
dry
femur
cortical
bone
samples
through
a
tensile
load
and
stress
concentration
factor
approach
and
to
provide
simulations
to
p
r
e
d
i
c
t
e
x
p
e
r
i
m
e
n
t
a
l
b
e
h
a
v
i
o
r
s
b
a
s
e
d
o
n
m
a
n
i
p
u
l
a
t
i
o
n
s
o
f
c
e
r
t
a
i
n
p
r
o
p
e
r
t
i
e
s
a
n
d
p
a
r
a
m
e
t
e
r
s
o
f
t
h
e
b
i
o
m
a
t
e
r
i
a
l
.
S
i
n
c
e
b
o
n
e
s
a
m
p
l
e
s
h
a
v
e
c
h
a
r
a
c
t
e
r
i
s
t
i
c
s
a
n
d
g
e
o
m
e
t
r
i
e
s
,
t
h
e
d
e
v
e
l
o
p
m
e
n
t
o
f
a
m
a
t
h
e
m
a
t
i
c
a
l
m
o
d
e
l
w
a
s
n
e
c
e
s
s
a
r
y
t
o
d
e
s
c
r
i
b
e
t
h
e
c
o
m
b
i
n
a
t
i
o
n
o
f
s
t
r
e
s
s
e
s
i
n
t
e
r
a
c
t
i
n
g
i
n
t
h
e
b
o
n
e
w
h
e
n
a
t
e
n
s
i
o
n
l
o
a
d
i
s
applied.
The
samples
have
average
diameters
and
lengths
of
0.5
and
2
inches
respectively
and
were
tested
using
a
10
kN
Universal
Tensile
Machine
to
determine
mechanical
properties
such
as
yield
and
ultimate
stress,
young
module,
fracture,
among
others.
Several
simulations
were
conducted
to
evaluate
failure
criteria
like
“Von
Mises”,
“Tresca”
and
“Tsai-Wu”.
Finally,
was
concluded
that
83%
of
the
data
obtained
from
the
22
samples
observed
in
the
“Stress-Strain”
charts
showed
a
directly
proportional
relationship.
Keywords:
mechanical
properties,
stress-strain
curve,
stress
concentration
factor,
failure
criteria
simulation
Simulation
of
Combined
Stresses
and
Stress
Concentration
Factor
Effects
on
a
Femur
Cortical
Bones
I
S
S
N
-
E
:
2
6
9
7
-
3
6
5
0
Velez.
Simulation of Combined Stresses and Stress Concentration Factor Effects on a Femur Cortical Bones
Alex
J.
Velez-Cruz
https://orcid.org/0000-0002-9289-5256
alvelez@pupr.edu
Polytechnic
University
of
Puerto
Rico
San
Juan,
Puerto
Rico
Resumen—
Los
propósitos
de
este
artículo
fueron
obtener
propiedades
mecánicas
de
muestras
secas
en
hueso
fémur
a
través
de
fuerzas
en
tensión
y
del
factor
de
concentración
de
esfuerzos
y
proveer
simulaciones
para
predecir
comportamientos
experimentales
basados
en
manipulaciones
de
ciertos
parámetros
y
propiedades.
Dado
que
las
muestras
tienen
geometrías
características,
fue
necesario
desarrollar
un
modelo
matemático
para
describir
las
combinaciones
de
esfuerzos
que
interaccionaban
en
el
hueso
cuando
se
aplica
una
carga
de
tensión.
Las
muestras
tienen
diámetros
promedios
y
longitudes
de
0.5
y
2
pulgadas
respectivamente
y
fueron
evaluadas
utilizando
una
Maquina
de
Tensión
Universal
para
determinar
propiedades
mecánicas
como
esfuerzos
ultimo
y
de
fluencia,
módulo
de
elasticidad,
entre
otras.
Varias
simulaciones
fueron
ejecutadas
para
evaluar
criterio
de
fallas
tales
como
“Esfuerzo
Von
Mises”,
“Tresca”
y
“Tsai-Wu”.
Finalmente,
se
concluyó
que
83%
de
los
datos
obtenidos
de
22
muestras
observadas
en
gráficas
“Esfuerzo-Desplazamiento”
mostraron
una
relación
directamente
proporcional.
Palabras
clave:
propiedades
mecánicas,
curva
de
esfuerzo
y
desplazamiento,
factor
de
concentración
de
esfuerzo,
simulación
de
criterio
de
falla
Simulación
de
los
Efectos
de
Combinaciones
de
Esfuerzo
y
Factor
de
Concentración
de
Esfuerzo
en
Hueso
Femoral
Cortical
8
Recibido(16/04/2022),
Aceptado(15/05/2022)
Minerva
Journal
Vol.3, Issue. 8, (pp. 8-19)
https://doi.org/10.47460/minerva.v3i8.60
I.Introduction
The
characterization
of
biomaterials
is
necessary
to
determine
the
mechanical,
chemical,
and
electrical,
among
other
interesting
properties
of
the
material
[1].
The
mechanical
property
is
obtained
from
a
mechanical
destructive
test,
called
the
tensile
test.
Basically,
it
is
when
a
pulling
force
or
tension
is
applied
to
materials
until
it
fails
or
breaks,
providing
information
about
the
Yield
Strength
,
Ultimate
Tensile
Strength
,
Ductility
(D),
Young's
modulus
(E),
and
Poisson's
ratio
(ν)
of
the
material
[2]
[3]
[4].
Bone
is
composed
of
three
different
types
of
bones,
cortical,
trabecular
(cancellous),
and
marrow
bones.
C
o
r
t
i
c
a
l
b
o
n
e
i
s
d
e
n
s
e
a
n
d
s
o
l
i
d
a
n
d
s
u
r
r
o
u
n
d
s
t
h
e
m
a
r
r
o
w
s
p
a
c
e
,
w
h
e
r
e
a
s
t
r
a
b
e
c
u
l
a
r
b
o
n
e
i
s
c
o
m
p
o
s
e
d
o
f
a
honeycomb-like
network
of
trabecular
plates
and
rods
interspersed
in
the
bone
marrow
compartment
[5].
Those
bones
are
separated
into
two
main
elements,
the
cellular
component,
and
an
extracellular
matrix.
The
matrix,
which
is
responsible
for
the
mechanical
strength
of
the
bone
tissue,
is
formed
by
an
organic
and
a
mineral
phase,
but
a
liquid
component
is
also
present
[6].
By
weight,
bone
contains
approximately
60%
mineral,
10%
water,
and
about
30%
collagenous
matrix.
The
mineral
component
influences
the
stiffness
of
the
b
o
n
e
,
w
h
e
r
e
a
s
t
h
e
c
o
l
l
a
g
e
n
n
e
t
w
o
r
k
c
o
n
t
r
i
b
u
t
e
s
s
i
g
n
i
f
i
c
a
n
t
l
y
t
o
i
t
s
f
r
a
c
t
u
r
e
p
r
o
p
e
r
t
i
e
s
.
Typically,
engineers
consider
three
basic
tasks
when
biomaterials
are
being
evaluated.
The
first,
understanding
the
properties
of
the
materials
(strength,
fatigue,
among
others);
second,
the
analysis
of
the
response
of
the
study
material
when
is
subject
to
external
loads
(Free
Body
Diagrams)
and
third;
the
determination
of
the
w
e
a
k
e
s
t
a
r
e
a
s
o
f
t
h
e
m
a
t
e
r
i
a
l
(
s
t
r
e
s
s
c
o
n
c
e
n
t
r
a
t
i
o
n
f
a
c
t
o
r
s
)
[
7
]
.
T
h
e
i
n
t
e
n
d
e
d
r
e
s
e
a
r
c
h
i
s
o
r
i
e
n
t
e
d
t
o
o
b
t
a
i
n
the
stress-strain
relation
of
dry
canine
cadaveric
cortical
bone
samples
using
the
stress
concentration
factor
rrrrr
analysis.
The
Kscf
is
the
ratio
of
the
highest
stress
(σmax)
to
reference
stress
(σref)
of
the
gross
cross-section.
This
experimental
factor
shall
be
considered
as
part
of
the
engineering
analysis
on
the
stress-strain
curve
since
the
mechanical
properties
of
the
biomaterial
can
be
affected
directly.
During
this
research
will
be
seen
a
combination
of
the
stresses
interacting
on
bone
samples
when
they
are
subject
to
axial
loads.
Based
on
the
destructive
tensile
test
and
the
stress
concentration
factor
approach,
it
is
expected
to
see
normal,
bending
and
shear
stresses
influencing
the
behavior
of
the
stress-strain
curve.
Therefore,
the
target
of
this
research
will
be
focused
to
perform
a
simulation
through
a
Computer
Aid
Drafting
(CAD)
tool
(CREO
Parametric)
with
the
intent
to
use
a
failure
criterion
(Von
Miss
Stress,
Tresca,
Tsai-Wu,
etc.)
to
determine
their
critical
values
before
a
fracture
occurs
and
to
compare
those
values
among
them.
Also,
will
serve
to
model
the
interaction
of
the
principal
stresses
and
the
effect
that
those
stresses
have
on
the
behavior
of
the
stress-strain
curve.
The
new
stress-strain
curve
obtained
from
the
simulation
will
be
compared
against
the
experimental
curve
obtained
from
the
tensile
tests.
Finally,
error
calculations
are
documented
to
analyze,
describe,
and
predict
the
accuracy
and
precision
of
the
proposed
model
as
well
as
how
well
is
behaving.
9
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Journal
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9
7
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3
6
5
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Velez.
Simulation of Combined Stresses and Stress Concentration Factor Effects on a Femur Cortical Bones
Vol.3, Issue. 8, (pp. 8-19)
II.
DEVELOPMENT
A.
Mechanical
Properties
of
the
Bone
The
diverse
forms
and
geometries
of
cortical
and
trabecular
bones
result
in
different
mechanical
properties.
T
h
e
m
e
c
h
a
n
i
c
a
l
p
r
o
p
e
r
t
i
e
s
o
f
t
h
e
b
o
n
e
v
a
r
y
a
c
c
o
r
d
i
n
g
t
o
s
p
e
c
i
e
s
,
s
i
z
e
,
a
g
e
,
a
m
o
n
g
o
t
h
e
r
c
h
a
r
a
c
t
e
r
i
s
t
i
c
s
a
n
d
parameters.
The
mineral
content
in
a
bone
shows
little
changes
with
increasing
age,
and
this
behavior
is
o
b
s
e
r
v
e
d
i
n
i
t
s
s
t
i
f
f
n
e
s
s
.
I
n
c
o
n
t
r
a
s
t
,
t
h
e
e
n
e
r
g
y
a
b
s
o
r
b
e
d
(
t
o
u
g
h
n
e
s
s
)
d
u
r
i
n
g
t
h
e
f
r
a
c
t
u
r
e
o
f
b
o
n
e
d
e
c
r
e
a
s
e
s
s
i
g
n
i
f
i
c
a
n
t
l
y
w
i
t
h
i
n
c
r
e
a
s
i
n
g
a
g
e
,
w
h
i
c
h
c
o
n
t
r
i
b
u
t
e
s
t
o
a
n
i
n
v
e
r
s
e
p
r
o
p
o
r
t
i
o
n
a
l
r
e
l
a
t
i
o
n
s
h
i
p
.
T
h
e
m
i
n
e
r
a
l
p
h
a
s
e
most
likely
imparts
stiffness
to
the
bone,
whereas
the
collagen
network
contributes
significantly
to
its
fracture
properties
[6].
Cortical
bone
is
an
anisotropic
material,
meaning
that
its
mechanical
properties
vary
according
to
the
direction
of
load.
The
strength
and
tensile/compressive
moduli
of
cortical
bone
along
the
longitudinal
direction
are
greater
than
those
along
the
radial
and
circumferential
directions.
Nowadays,
minor
fluctuations
in
mechanical
properties
have
been
observed
in
the
radial
versus
circumferential
direction,
recommending
that
cortical
bone
can
be
considered
as
a
transversely
isotropic
material.
When
samples
receive
tension
along
the
longitudinal
direction,
cortical
bone
shows
a
bilinear
stress-strain
response
in
which
a
distinct
yield
point
separates
a
linearly
elastic
region
and
a
region
of
linear
hardening
that
ends
abruptly
at
a
fracture
strain
of
less
than
3.
Cortical
bone
specimens
loaded
in
the
transverse
direction
fail
in
a
more
brittle
manner
compared
with
those
loaded
in
the
longitudinal
direction
[8].
B.
Stress-Strain
Curve
The
following
diagram
has
the
intent
to
provide
detailed
background
information
regarding
the
behavior
of
the
femur
bone
material
when
is
subject
to
tension
loads.
10
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Journal
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Fig.
1.
Stress-Strain
Diagram
[9]
In
Fig.
1
above,
point
A
represents
the
proportional
limit,
which
the
slope
of
this
line
is
better
known
as
the
Young’s
Modulus.
For
segment
AB,
the
material
may
still
be
elastic
in
the
sense
that
the
deformations
are
completely
recovered
when
the
load
is
removed,
and
this
point
B
is
called
the
elastic
limit
or
yield
point.
Point
A
and
segment
AB
are
part
of
the
elastic
region,
which
is
governed
by
Hook’s
law.
Beyond
point
B,
Velez.
Simulation of Combined Stresses and Stress Concentration Factor Effects on a Femur Cortical Bones
Vol.3, Issue. 8, (pp. 8-19)
C.
Principal
Stresses
The
engineering
measures
of
stress
(σ)
and
strain
(ε)
are
determined
from
the
measured
load
(P)
and
deflection
(δ)
using
the
original
specimen
cross-sectional
area
and
length
as:
11
Minerva
Journal
ISSN-E:
2697-3650
W
h
e
n
t
h
e
s
t
r
e
s
s
(
σ
)
i
s
p
l
o
t
t
e
d
a
g
a
i
n
s
t
t
h
e
s
t
r
a
i
n
(
ε
)
,
a
n
e
n
g
i
n
e
e
r
i
n
g
s
t
r
e
s
s
-
s
t
r
a
i
n
c
u
r
v
e
s
u
c
h
a
s
t
h
a
t
s
h
o
w
n
i
n
Fig.
1
is
obtained
[10].
In
the
early
phase
of
the
stress-strain
curve,
various
materials
obey
Hooke’s
law
to
a
reasonable
approximation,
so
that
stress
is
proportional
to
strain
with
the
constant
of
proportionality
being
the
modulus
of
elasticity
or
Young’s
modulus
(E)
[11]:
Since
bones
vary
in
geometry,
a
representation
of
the
mathematical
model
is
needed
to
explain
the
physical
phenomena
occurring
during
the
experimental
tests.
Therefore,
the
following
equation
related
to
the
c
o
m
b
i
n
a
t
i
o
n
o
f
p
r
i
n
c
i
p
a
l
s
t
r
e
s
s
e
s
(
σ
t
)
(
n
o
r
m
a
l
,
b
e
n
d
i
n
g
,
a
n
d
s
h
e
a
r
)
w
i
l
l
b
e
b
r
i
e
f
l
y
d
i
s
c
u
s
s
e
d
a
n
d
f
u
r
t
h
e
r
implemented
in
the
proposed
cortical
bone
model.
The
kscf,
σn,
σb
and
τ
are
provided
as
follows:
where
stress
concentration
factor
(kscf)
is
the
ratio
of
the
highest
stress
(σmax)
to
reference
stress
(σref)
of
the
gross
cross-section:
where
F
is
the
applied
normal
force
and
A
is
the
cross-sectional
area
of
the
specimen:
where
F
is
the
applied
normal
force,
r
is
radio,
and
I
is
the
inertia
moment
A
is
the
cross-sectional
area
of
the
specimen:
and
assuming
a
cylindrical
hollow
element,
τ,
can
be
approximated
as
described
in
(7),
where
V
is
the
shear
stress
value
and
A
is
the
cross-sectional
area
of
the
specimen.
Thus,
substituting
(4),
(5),
(6)
and
(7)
into
(3),
the
final
combined
stresses
equation
is
represented
as
follows:
Velez.
Simulation of Combined Stresses and Stress Concentration Factor Effects on a Femur Cortical Bones
Vol.3, Issue. 8, (pp. 8-19)
D
.
P
r
i
n
c
i
p
a
l
S
t
r
a
i
n
s
On
the
other
hand,
principal
strains
(maximum
and
minimum
normal
strains)
shall
be
considered
as
part
of
the
cortical
bone
behavior,
which
is
obtained
from
differentiating
axial
,
and
lateral
with
respect
to
θ.
Then,
the
general
equation
for
the
total
principal
strains
present
in
the
experiment
is
provided
as
follows:
12
Minerva
Journal
I
S
S
N
-
E
:
2
6
9
7
-
3
6
5
0
where
and
are
obtained
from
the
following
plane
stress-strain
formula
[12]:
where
shear
strain
is
related
with
the
orientation
of
the
planes
of
the
principal
strains
as
follows:
According
to
the
experiment
provided
there
is
no
variation
in
the
stress-strain
planes,
therefore,
θ
=
0,
which
makes
for
all
data
collected
from
the
experimental
tests.
Furthermore,
no
shear
strains
will
be
acting
o
n
t
h
e
p
l
a
n
e
s
o
f
t
h
e
p
r
i
n
c
i
p
a
l
s
t
r
a
i
n
s
,
w
h
i
c
h
m
e
a
n
s
t
h
a
t
(
1
0
)
w
i
l
l
b
e
e
l
i
m
i
n
a
t
e
d
f
r
o
m
t
h
e
p
r
i
n
c
i
p
a
l
s
t
r
e
s
s
analysis
when
simulation
take
place.
Nevertheless,
substituting
(9)
into
(8),
the
combined
principal
strains
equation
is
represented
as
follow:
Finally,
incorporating
(8)
and
(12)
respectively
(principal
stresses
and
strains)
into
the
mathematical
model,
predictions
can
be
made
through
a
simulation
to
determine
the
main
objectives
of
this
research,
mechanical
properties,
and
failure
criteria.
E.
Failure
Criteria
Failure
principles
are
used
to
determine
and
predict
if
a
material
will
fail
under
certain
circumstances,
including
loads,
engineering
parameters,
mechanical
properties,
among
others.
From
a
mechanics
of
material
perspective,
exits
basic
failure
criteria.
For
purpose
of
this
research,
the
Distortion
Energy
(Von
Mises),
Tresca
(Maximum
Shear
Stress)
and
Tsai-Wu
(Failure
Index)
criteria
will
be
used
to
evaluate
the
fracture
point
of
the
biomaterial.
According
to
[13],
Von
Mises’s
theory,
a
ductile
solid
will
yield
when
the
distortion
energy
density
reaches
a
critical
value
for
that
material.
Since
this
should
be
true
for
the
uniaxial
stress
state
also,
the
critical
value
of
the
distortional
energy
can
be
estimated
from
the
uniaxial
test.
Based
on
this
experiment,
the
analysis
is
considered
as
two–dimensional
plane
stress
state,
which
indicated
that
,
simplifying
the
principal
Von
Mises
stress
equation
as
follows:
Velez.
Simulation of Combined Stresses and Stress Concentration Factor Effects on a Femur Cortical Bones
Vol.3, Issue. 8, (pp. 8-19)
A
reasonable
safety
of
factor
(SOF)
for
engineering
analysis
should
be
greater
than
1.
The
inverse
of
the
SOF
is
failure
index
(Tsau-Wu),
which
means
that
Failure
Index
simulations
with
less
than
1
are
not
going
to
fail.
III.
METHODOLOGY
The
bones
are
considered
biological
materials;
therefore,
health
and
safety
must
be
present
throughout
the
research
to
avoid
and
minimize
any
type
of
risk
to
researchers.
The
research
conducted
has
a
combination
of
m
e
t
h
o
d
o
l
o
g
i
e
s
s
i
n
c
e
e
x
p
e
r
i
m
e
n
t
a
l
t
e
s
t
s
,
p
r
o
c
e
d
u
r
e
s
,
e
q
u
i
p
m
e
n
t
p
r
o
t
o
c
o
l
s
a
n
d
o
b
s
e
r
v
a
t
i
o
n
s
w
e
r
e
c
o
n
d
u
c
t
e
d
with
the
intent
to
obtain
representative
real
scenarios
regarding
the
behavior
of
the
bones
when
they
critical
axial
loads
are
applied.
In
general
terms,
this
research
has
focused
in
quantitative
and
experimental
methodologies.
T
h
e
c
a
n
i
n
e
c
o
r
t
i
c
a
l
b
o
n
e
s
a
m
p
l
e
s
w
e
r
e
c
o
l
l
e
c
t
e
d
b
y
a
v
e
t
e
r
i
n
a
r
i
a
n
a
n
d
t
e
m
p
o
r
a
r
i
l
y
s
t
o
r
e
d
i
n
a
f
r
e
e
z
e
r
a
t
5
0
degrees
Fahrenheit.
The
humerus
and
femur
dry
bone
samples
were
extracted
from
adults
and
young
dogs
of
mid
and
large
sizes
respectively.
The
samples
were
transported
in
a
foam
cooler
covered
with
ice
from
the
Veterinarian
Hospital
to
the
Bioimpedance
Laboratory
at
the
Polytechnic
University
of
Puerto
Rico.
Then,
samples
were
cleaned
with
water
at
room
temperature
(72
degrees
Fahrenheit)
to
remove
leftover
tissues
such
as
muscles,
tendons,
and
ligaments.
To
preserve
the
dry
samples
in
good
shape
were
placed
at
the
Bioimpedance
Freezer
at
50
degrees
Fahrenheit
and
relative
humidity
of
55%.
This
sample
preparation
of
the
cortical
bones
was
divided
into
4
different
batches
and
each
batch
into
3
groups
taking
into
consideration
the
following
elements:
type
of
bone,
aging,
and
bone
size,
and
including
their
associated
parameters
such
as
pin
diameter,
overall
length,
thickness,
eccentricity,
and
external
and
internal
diameter.
The
first
group
was
identified
with
dry
femur
samples
for
adults’
mid-size
bones.
Those
samples
were
cut
into
two
sections
with
lengths
of
two
and
one
inches
respectively.
The
samples
two
inches
long
were
used
for
a
destructive
tensile
test
while
the
one
inch
was
labeled
and
stored
within
the
freezer
to
be
used
later
for
the
Bioimpedance
test.
The
final
samples
for
this
group
have
pin
diameter,
overall
length,
thickness,
and
external
and
internal
diameter
dimensions
of
0.10,
2.0,
0.10,
0.20,
0.5,
and
0.40
inches
respectively.
The
second
group
was
identified
with
dry
humerus
samples
for
adults’
mid-size
bones.
Those
samples
were
cut
into
two
sections
with
lengths
of
two
and
one
inches
respectively.
The
samples
two
inches
long
were
used
for
a
destructive
tensile
test
while
the
one
inch
was
labeled
and
stored
within
the
freezer
to
be
used
later
for
the
Bioimpedance
test.
The
final
samples
for
this
group
have
pin
diameter,
overall
length,
thickness,
and
external
and
internal
diameter
dimensions
of
0.10,
2.0,
0.10,
0.20,
0.5,
and
0.40
inches
respectively.
F
o
r
m
a
x
i
m
u
m
s
h
e
a
r
s
t
r
e
s
s
t
h
e
o
r
y
,
t
h
e
m
a
t
e
r
i
a
l
y
i
e
l
d
s
w
h
e
n
t
h
e
m
a
x
i
m
u
m
s
h
e
a
r
s
t
r
e
s
s
a
t
a
p
o
i
n
t
e
q
u
a
l
s
t
h
e
critical
shear
stress
value
for
that
material.
Since
this
should
be
true
for
uniaxial
stress
state,
we
can
use
the
results
from
uniaxial
tension
test
to
determine
the
maximum
allowable
shear
stress.
The
stress
state
in
a
t
e
n
s
i
l
e
s
p
e
c
i
m
e
n
a
t
t
h
e
p
o
i
n
t
o
f
y
i
e
l
d
i
n
g
i
s
g
i
v
e
n
b
y
:
.
T
h
e
m
a
x
i
m
u
m
s
h
e
a
r
s
t
r
e
s
s
i
s
calculated
as
[13]:
13
Minerva
Journal
ISSN-E:
2697-3650
The
Tsai-Wu
failure
criterion
is
one
of
the
first
failure
criteria
studied
by
scientists
to
evaluate
factor
of
safety
f
o
r
c
o
m
p
o
s
i
t
e
m
a
t
e
r
i
a
l
s
.
T
h
i
s
f
a
i
l
u
r
e
c
r
i
t
e
r
i
o
n
t
a
k
e
s
i
n
t
o
c
o
n
s
i
d
e
r
a
t
i
o
n
t
h
e
t
o
t
a
l
s
t
r
a
i
n
e
n
e
r
g
y
i
n
t
e
r
a
c
t
i
n
g
i
n
t
h
e
specimen.
Based
on
the
experiment
proposed,
it
can
be
assumed
a
bi-dimensional
plane
stress,
which
simplifies
the
equation
to
the
following:
Velez.
Simulation of Combined Stresses and Stress Concentration Factor Effects on a Femur Cortical Bones
Vol.3, Issue. 8, (pp. 8-19)
The
third
group
was
identified
with
dry
femur
samples
for
adults’
mid-size
bones.
Those
samples
were
cut
into
two
sections
with
lengths
of
two
and
one
inches
respectively.
The
samples
two
inches
long
were
used
for
a
destructive
tensile
test
while
the
one
inch
was
labeled
and
stored
within
the
freezer
to
be
used
later
for
the
Bioimpedance
test.
The
final
samples
for
this
group
have
pin
diameter,
overall
length,
thickness,
and
external
and
internal
diameter
dimensions
of
0.10,
1.5,
0.10,
0.20,
0.5,
and
0.40
inches
respectively.
The
fourth
group
w
a
s
i
d
e
n
t
i
f
i
e
d
w
i
t
h
d
r
y
f
e
m
u
r
s
a
m
p
l
e
s
f
o
r
y
o
u
n
g
l
a
r
g
e
-
s
i
z
e
b
o
n
e
s
.
T
h
o
s
e
s
a
m
p
l
e
s
w
e
r
e
c
u
t
i
n
t
o
t
w
o
s
e
c
t
i
o
n
s
with
lengths
of
two
and
one
inches
respectively.
The
samples
two
inches
long
were
used
for
a
destructive
t
e
n
s
i
l
e
t
e
s
t
w
h
i
l
e
t
h
e
o
n
e
i
n
c
h
w
a
s
l
a
b
e
l
e
d
a
n
d
s
t
o
r
e
d
w
i
t
h
i
n
t
h
e
f
r
e
e
z
e
r
t
o
b
e
u
s
e
d
l
a
t
e
r
f
o
r
B
i
o
i
m
p
e
d
a
n
c
e
t
e
s
t
s
.
T
h
e
f
i
n
a
l
s
a
m
p
l
e
s
f
o
r
t
h
i
s
g
r
o
u
p
h
a
v
e
p
i
n
d
i
a
m
e
t
e
r
,
l
e
n
g
t
h
,
t
h
i
c
k
n
e
s
s
,
a
n
d
e
x
t
e
r
n
a
l
a
n
d
i
n
t
e
r
n
a
l
d
i
a
m
e
t
e
r
dimensions
of
0.10,
3.0,
0.13,
0.25,
0.65,
and
0.52
inches
respectively.
The
samples
were
cut
and
machined
using
a
low-speed
diamond
saw.
The
diamond
saw
blade
was
immersed
in
a
saline
bath
to
minimize
the
heat
created
from
friction,
which
has
been
shown
to
significantly
affect
the
material
properties,
specifically,
the
plasticity
of
the
cortical
bone.
Then,
the
sections
cut
from
the
humerus,
and
femur
were
placed
in
a
bone
chuck
and
a
transverse
cut
was
made
along
the
axis
of
interest.
Great
care
was
taken
when
placing
the
bone
sections
in
a
bone
chuck
to
ensure
the
axis
of
interest
coincided
with
the
axis
of
cutting.
The
cylindrical
cortical
bone
samples
were
placed
in
a
custom
bone
chuck,
and
a
drill
hole
was
made
to
remove
the
cancellous/trabecular
bone
from
those
samples.
Also,
additional
cuts
were
made
to
level
the
uncut
side
and,
if
necessary,
trim
the
ends
to
fit
on
the
accessories
manufactured,
and
customized
for
the
universal
tensile
test
machine
since
the
existing
and
available
grips
are
for
metal
specimens
use
only.
Therefore,
aluminum
accessories
(grips)
were
designed
to
minimize
the
crushing
effect
on
the
bone
ends
when
a
tension
load
is
applied.
Once
the
grips
were
manufactured
according
to
technical
specifications,
the
initial
calibration
and
setup
were
done
properly
as
presented
in
Fig.
2
below.
14
Minerva
Journal
ISSN-E:
2697-3650
Fig.
2.Tensile
Test
Calibration
and
Setup
Prior
to
conducting
the
destructive
tests,
familiarization
and
practice
with
the
tensile
test
machine
and
its
software
were
required
[14].
Also,
safety,
biohazard
and
waste
disposal,
and
biomaterials
handling
trainings
were
taken
to
comply
with
US
federal
codes,
standards,
and
regulations.
The
universal
tensile
machine
used
to
perform
the
test
was
a
brand
Applied
Test
Systems,
model
910,
a
double-column
and
rated
at
10
kN.
The
machine
pre-set
values
for
pre-load
cell,
velocity
and
displacement
were
100
lbs,
0.15
in/min,
and
0.05
inches
respectively.
Velez.
Simulation of Combined Stresses and Stress Concentration Factor Effects on a Femur Cortical Bones
Vol.3, Issue. 8, (pp. 8-19)
IV.
RESULTS
The
experimental
tests
of
the
dry
cortical
bones
were
divided
into
two
different
types
of
charts,
the
first
one
“Load
vs
Displacement”
and
the
other
“Stress
vs
Strain”.
The
Fig.
3
(a)
represent
the
rough
data
obtained
from
the
tensile
tests
while
Fig.
3
(b)
represents
the
relations
between
principal
stresses
and
strains,
including
the
s
t
r
e
s
s
c
o
n
c
e
n
t
r
a
t
i
o
n
f
a
c
t
o
r
c
o
e
f
f
i
c
i
e
n
t
.
T
h
e
c
o
m
p
a
r
i
s
o
n
s
m
a
d
e
b
e
t
w
e
e
n
f
i
g
u
r
e
s
3
,
5
a
n
d
7
a
r
e
n
e
c
e
s
s
a
r
y
t
o
determine
the
variations
between
rough
data
and
math
models,
considering
the
load
and
displacement
and
p
r
i
n
c
i
p
a
l
s
t
r
e
s
s
e
s
a
n
d
s
t
r
a
i
n
s
p
a
r
a
m
e
t
e
r
s
.
T
h
e
u
l
t
i
m
a
t
e
s
t
r
e
n
g
t
h
v
a
l
u
e
s
f
r
o
m
“
b
a
t
c
h
1
,
s
a
m
p
l
e
s
”
o
b
t
a
i
n
e
d
f
r
o
m
f
i
g
u
r
e
s
3
(
b
)
,
4
(
b
)
a
n
d
5
(
b
)
a
r
e
1
1
(
1
,
5
9
5
)
,
3
9
(
5
,
6
5
6
)
a
n
d
2
7
(
3
,
9
1
6
)
M
P
a
(
P
s
i
)
r
e
s
p
e
c
t
i
v
e
l
y
.
T
h
e
p
r
i
n
c
i
p
a
l
s
t
r
a
i
n
s
values
on
figures
3
(b),
4
(b)
and
5(b)
were
obtained
from
ultimate
strength
points
such
as
0.0011,
0.0062
and
0.0023
respectively.
Also,
it
has
been
noticed
that
beyond
these
ultimate
strength
values,
the
fracture
process
begins.
The
Young’s
modulus
for
3
(b),
4
(b)
and
5(b)
were
obtained
as
follows:
10
(1,450),
6,290
(912,287)
and
11,739
(1,702,598)
MPa
(Psi).
The
charts
provided
in
figures
3
(b),
4
(b)
and
5(b)
show
a
directly
proportional
relationship
among
the
samples.
A
statistics
analysis
was
performed
through
Minitab
for
all
samples
presented
in
this
article
to
evaluate
the
standard
deviations,
normal
distribution,
among
other
parameters.
For
descriptive
analysis,
it
was
assumed
confidence
level
and
error
(α)
of
95%
and
5%
respectively.
The
total
“Load”
data
analyzed
for
batch
1-sample
1
was
39
readings
and
their
mean,
standard
deviation
and
variance
were
0.03819,
0.02280
and
0.00052
respectively.
The
histogram
of
this
sample
provided
in
Fig.
9
showed
a
normal
distribution
pattern.
The
22
samples
were
analyzed
and
have
been
observed
that
other
21
samples
showed
similar
trend
to
the
one
observed
in
Fig.
9.
The
Fig.
6
shows
the
fracture
detail
in
the
samples.
It
was
demonstrated
that
samples
failed
due
to
stress
concentration
factor
with
a
combination
of
longitudinal
and
oblique
fracture
effects.
The
presence
of
normal
and
bending
stresses
along
the
samples
validate
that
mathematical
model
serve
to
describe
the
physical
phenomena
encounter
in
the
experimental
tests.
On
Fig
7,
the
oblique
fractures
predominate
against
the
longitudinal,
indicating
that
more
presence
of
bending
was
affecting
the
samples.
Subsequently,
it
has
been
noticed
that
samples
with
non-uniform
geometries
played
a
big
role
in
the
engineering
analysis.
In
order
to
calibrate
the
model
in
CREO
Parametric,
the
software
request
the
entry
of
certain
parameters
and
material
properties
such
as
Young’s
modulus,
Poisson's
ratio,
yield
strength
and
shear
stiffness,
which
according
to
[12]
were
initially
assumed
as
6.964
x
108
psi,
0.4,
4,352
psi
and
2.487
x
108
psi
respectively:
Based
on
experimental
fracture
points,
which
are
the
same
than
ultimate
tensile
strength
value
obtained
from
Fig.
3
(b),
4
(b)
and
5(b)
(1,595;
5,656
and
3,916
Psi
respectively),
and
CREO
simulation
values
obtained
from
Fig.
8
(a),
which
indicated
a
Von
Mises
stress
value
of
4,770.94
Psi,
a
comparison
was
made
between
Fig.
4
(b)
and
8
(a)
to
determine
the
percentage
of
error
between
the
model
and
the
physical
experiment.
The
corresponding
percent
of
error
was
15.64%.
Since
there
is
no
validated
standard
or
average
values
to
works
with
canine
cortical
bones
and
the
initial
values
used
in
CREO
are
slightly
higher
because
corresponded
to
human
cortical
bones,
the
percent
of
error
might
be
affected.
15
Minerva
Journal
I
S
S
N
-
E
:
2
6
9
7
-
3
6
5
0
Velez.
Simulation of Combined Stresses and Stress Concentration Factor Effects on a Femur Cortical Bones
Vol.3, Issue. 8, (pp. 8-19)
Fig.
3.
(a)
Load
vs.
Displacement
(Batch
1,
Sample
1)
(b)
Principal
Stress
vs.
Strain
(Batch
1,
Sample
1)
16
Minerva
Journal
ISSN-E:
2697-3650
Velez.
Simulation of Combined Stresses and Stress Concentration Factor Effects on a Femur Cortical Bones
Vol.3, Issue. 8, (pp. 8-19)
Fig.
5.
(a)
Load
vs.
Displacement
(Batch
1,
Samples
4
&
5)
(b)
Principal
Stress
vs.
Strain
(Batch
1,
Samples
4
&
5)
17
Minerva
Journal
I
S
S
N
-
E
:
2
6
9
7
-
3
6
5
0
Fig.
6
(a)
Longitudinal
Fracture
on
Batch
3,
Sample
1
(b)
Longitudinal
Fracture
on
Batch
3,
Sample
2
(c)
Longitudinal
Fracture
on
Batch
3,
Sample
3
Fig.
7.
Tensile
Tests
Results
(a)
Close
View
of
the
Oblique
Fracture
on
Batch
1,
Sample
1
(b)
Oblique
Fracture
on
Batch
3,
Sample
1
(c)
Oblique
Fracture
on
Batch
3,
Sample
4
Velez.
Simulation of Combined Stresses and Stress Concentration Factor Effects on a Femur Cortical Bones
Vol.3, Issue. 8, (pp. 8-19)
Fig.
8.
Failure
Criteria
Simulation:
(a)
Von
Mises
Stress;
(b)
Failure
Index;
(c)
Maximum
Shear
Stress
18
Minerva
Journal
ISSN-E:
2697-3650
Fig.
9.
Histogram
of
the
Load-(Batch
1-Sample
1)
CONCLUSIONS
A
comprehensive
methodology
for
destructive
mechanical
tests
were
established
for
preparing
and
testing
canine
cadaveric
cortical
femur
bone
samples
when
are
subject
to
axial
loads
and
stress
concentration
factors
to
obtain
mechanical
properties
of
the
biomaterial.
It
was
concluded
that
83%
of
the
data
obtained
from
the
22
samples
observed
from
“Stress-Strain”
charts
showed
a
directly
proportional
relationship.
The
ultimate
tensile
strength
(σult)
values
of
the
dry
cortical
femoral
bones
are
equal
to
yield
strength
(σy)
values,
implying
that
bones
samples
behaved
as
a
brittle
material.
Due
to
the
small
anisotropy
of
bone
material
analyzed
(2
in
length
x
0.5
in
diameter)
the
eccentricity
distance
was
relatively
small,
that
might
influence
the
conclusions
of
the
study.
The
standard
deviations
per
batches
were
considered
within
acceptable
parameters.
The
experimental
results
on
cortical
bones
are
closer
to
the
predictions
made
by
the
simulation
to
evaluate
failure
criteria,
considering
a
percentage
of
error
of
15.64
%.
Velez.
Simulation of Combined Stresses and Stress Concentration Factor Effects on a Femur Cortical Bones
Vol.3, Issue. 8, (pp. 8-19)
19
Minerva
Journal
ISSN-E:
2697-3650
ACKNOWLEDGEMENT
I
want
to
thank
Dr.
Luis
Thomas
Ramos
from
“San
Francisco
de
Asis”
Veterinarian
Hospital
to
support
this
research
by
providing
the
femur
and
humerus
bone
samples.
Also,
I
want
to
thank
Dr.
Julio
Noriega
Motta,
Mechanical
Engineering
Department
Head
of
the
Polytechnic
University
of
Puerto
Rico
to
support
with
the
mechanical
tensile
machine
and
manufacturing
engineering
laboratories.
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