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