The high prevalence of musculoskeletal injuries, reaching 24% in the
structure of temporary disability, and the high incidence of femoral fractures
indicate the relevance of studying the mechanism of their formation [1].
However, most scientific works describe the mechanism of fracture formation in
one type of deformation. The criteria for establishing the mechanism of damage
formation obtained in such works are used in forensic medical examinations
using the method of loose analogy. In real life, such simple conditions of
loading the femur are rare, therefore the morphology of a real fracture has
multiple additional signs that are not typical for one type of deformation,
which raises reasonable doubts among the expert about his correctness in
establishing the mechanism of injury and requires the development of new
methods for establishing the mechanism of injury.
Our previous experiments on mathematical modeling using the finite
element analysis method, which is widely used to solve problems in the
mechanics of a deformable solid in the science of “strength of materials”, of
femur fractures allowed us to validate a solid parametric model of the femur
created from computed tomography data of real people and apply it to study the
mechanism of formation of femoral fractures in various conditions and
circumstances [2].
The purpose of this work is to establish the possibility of using
the finite element analysis method to study complex stress states during a
femur fracture with subsequent visualization of the results.
During the preliminary part of the study, literature sources were
analyzed that examined the morphology of fractures of the diaphysis of long
tubular bones caused by the impact of blunt objects. This analysis was based on
data obtained from experiments on biomannequins or during practical
observations (342 experiments on 200 biomannequins, 56 experiments on lower
limb bone samples, 116 expert observations) [3]. The femoral loading conditions
and failure modes used in the model were taken from these studies to verify the
validity of the model. In the experiments, impact on biomannequins was carried
out with a horizontal position of the body on a metal surface and a blow with a
blunt hard object on the anterior surface of the diaphysis of a long tubular
bone, while the distal part of the femur was fixed bone and pressure was
applied to the pelvis to simulate the load of rotation of the upper femur
clockwise or counterclockwise. The impact force causing fractures in the
experiments ranged from 1800 N to 1900 N [3].
To test the possibility of mathematical modeling of the process of
formation of a fracture of the femur, the finite element method (FEM) was used,
implemented using the ANSYS LS-DYNA software environment, which is a popular
program for finite element analysis, developed by Ansys inc., which is used to
solve linear problems. and nonlinear dynamic problems of mechanics of
deformable solids, including fracture analysis (ANSYS inc.,
https://www.ansys.com)
[4]. In this work, ANSYS LS-DYNA was used to reproduce experiments
conducted on biomannequins and during practical observations under impact
impacts. To recreate the loading conditions of the hip model, fastening was
performed in the area of the articular surfaces with a rigid base and the
formation of an elastic substrate simulating the Winkler base. The finite
element mesh was automatically generated using 5 mm Solid elements to simulate
the volumetric stress-strain state.
The movement of the bone along the Z axis
is limited in the proximal (upper) part of the bone (hip joint area) and in the
distal (lower) part of the bone (knee joint area). The preload was modeled by
applying a force of 100 N to the surface of the femoral head and directed
in the first experiment along the Y axis, and in the second – against the Y
axis. The force was applied at an angle of 90 degrees to the axis of the
bone of the femur model in the projection of the middle third of the anterior
surface of the femur and was implemented a simulated cylindrical steel indenter
with a radius of curvature of 30 mm and a length of 180 mm. Moreover, its
movements are limited in all directions except vertical. The impact was modeled
at an impact speed of 18 m per second. The following influence conditions are
specified between the parts of the model:
-
between spongy
substance, compact substance and muscle tissue – conditions of inextricable
connection, which is achieved by constructing a conformal mesh of finite elements
on these parts;
-
between the muscle
tissue, the punch and the base – contact with friction.
Considering that when conducting
experiments on biomannequins and in practical observations, the properties of
bone and soft tissues were not studied, averaged mechanical properties of
tissues from various literature sources (see Tables 1 and 2) [5-8]. Such tissue
characteristics described well the behavior of materials in previous
experiments [2]. The properties of bone tissue components are described by material
models – isotropic elastoplastic for spongy substance and anisotropic elastic
for compact substance. The theory of maximum principal stress (Tresca theory)
was adopted as a destruction model.
Table 1. Mechanical properties of a compact bone tissue
|
Density
|
2,000
kg/m3
|
|
Young’s modulus X direction
|
12 GPa
|
|
Young’s modulus Y direction
|
12 GPa
|
|
Young’s modulus Z direction
|
20 GPa
|
|
Poisson’s ratio XY
|
0.38
|
|
Poisson’s ratio YZ
|
0.22
|
|
Poisson’s ratio XZ
|
0.22
|
|
Shear
modulus XY
|
4.5
GPa
|
|
Shear modulus YZ
|
5.6
GPa
|
|
Shear modulus XZ
|
5.6
GPa
|
|
Compressive
ultimate strength
|
0.205
GPa
|
|
Tensile
ultimate strength
|
0.133
GPa
|
|
Maximum tensile stress
|
52 MPa
|
|
Maximum shear stress
|
65 MPa
|
Table 2. Mechanical properties of sponge bone tissue
|
Density
|
127
kg/m3
|
|
Young's
modulus
|
0.38
MPa
|
|
Poisson’s
ratio
|
0.33
|
|
Bulk modulus
|
0.37255
MPa
|
|
Shear
modulus
|
0.14286
MPa
|
|
Compressive
ultimate strength
|
6.23
MPa
|
|
Tensile
ultimate strength
|
8.4
MPa
|
|
Maximum tensile stress
|
8.4
MPa
|
|
Maximum
shear stress
|
7.4
MPa
|
|
Yield
strength
|
1.75
MPa
|
|
Tangent
modulus
|
41.8
MPa
|
Due to the fact that
in our model, the destruction of soft tissues was not studied, and the physical
properties of soft tissues were generally used only as a spacer between the
indenter and the base, the mechanical properties of soft tissues were taken
from the properties of the ballistic gel. The Johnson-Cook model was adopted as
a fracture model, taking into account changes in strength criteria depending on
loading rate and temperature (see Table 3):
Table 3. Mechanical properties of soft tissuess
|
Density
|
1.030 kg/m3
|
|
Young’s
modulus
|
59.862
kPa
|
|
Poisson’s
ratio
|
0.4956
|
|
Bulk modulus
|
29 kPa
|
|
Shear
modulus
|
20 kPa
|
|
Johnson Cook failure
|
|
|
Damage constant D1
|
−
0.13549
|
|
Damage constant D2
|
0.6015
|
|
Damage constant D3
|
0.25892
|
|
Damage constant D4
|
0.030127
|
|
Damage constant D5
|
0
|
|
Melting
temperature
|
20°C
|
|
Reference
strain rate (/sec)
|
1
|
Visualization of the research results was obtained in the ANSYS
LS-DYNA postprocessor.
In experiment 1, when preloading in the
proximal part of the femur with a value of 100 N along the Y axis, as a
result of modeling an impact perpendicular to the longitudinal axis of the
femur, an oblique transverse asymmetrical fracture line was formed in the
conditional middle of the femoral diaphysis with chipping of elements in the
fracture line and the beginning of the formation of a fracture line in the area
of minimum thickness of the compact bone substance in the injured area proximal
to the site of application of the traumatic force of the femur on the opposite
side of the load. Maximum equivalent voltages reached 52.958 MPa (Fig. 1-3).
When analyzing the principal stress vectors (Fig. 3), a dynamic distribution of
compressive (blue arrows) and tensile (red arrows) stresses is noted, changing
their location and magnitude depending on the stage of bone destruction. At the
initial stage of loading, asymmetric stresses arose in the proximal part of the
bone, indicating a helical deformation of the bone, not exceeding the limits of
structural destruction that occur during preloading (Fig. 4).
Fig.
1. Visualization of the final stage of modeling a fracture of the femoral
diaphysis in a complex stress state with preload in the proximal part of the
femur along the Y axis. Von Mises analysis.
Fig.
2. Dynamic visualization of modeling a fracture of the femoral diaphysis in a
complex stress state with preload in the proximal part of the femur along the Y
axis. Von Mises analysis.
Fig.
3. Dynamic visualization of modeling a fracture of the femoral diaphysis in a
complex stress state with preload in the proximal part of the femur along the Y
axis. Analysis using principal stress vectors.
Fig. 4.
Visualization of internal stresses in the proximal part during torsional
deformation of the upper part of the femur with preload in the proximal part of
the femur along the Y axis.
In experiment 2, with a preload of
100 N in the proximal part of the femur directed in the opposite direction
of the Y axis, as a result of modeling the impact perpendicular to the
longitudinal axis of the femur, an oblique transverse asymmetrical fracture
line was formed in the conditional middle of the femoral diaphysis with
chipping of elements in the fracture line and the beginning of formation the
fracture line in the area of minimum thickness of the compact bone substance in
the injured area proximal to the site of application of the traumatic force of
the femur on the opposite side of the load. The maximum equivalent stresses
reached 59.218 MPa (Fig. 5-7). When analyzing the principal stress vectors
(Fig. 7), a dynamic distribution of compressive (blue arrows) and tensile (red
arrows) stresses is noted, changing their location and magnitude depending on
the stage of bone destruction. At the initial stage of loading, asymmetric
stresses arose in the proximal part of the bone, indicating a helical
deformation of the bone, not exceeding the limits of structural destruction
that occur during preloading (Fig. 8).
Fig.
5. Visualization of the final stage of modeling a fracture of the femoral
diaphysis in a complex stress state with preload in the proximal part of the
femur in the opposite direction of the Y axis. Von Mises analysis.
Fig.
6. Dynamic visualization of modeling a fracture of the femoral diaphysis in a
complex stress state with preload in the proximal part of the femur in the
opposite direction of the Y axis. Von Mises analysis
.
Fig.
7. Dynamic visualization of modeling a fracture of the femoral diaphysis in a
complex stress state with preload in the proximal part of the femur in the
opposite direction of the Y axis. Analysis by principal stress vectors.
Fig.
8. Visualization of internal stresses in the proximal part during torsional
deformation of the upper part of the femur with preload in the proximal part of
the femur in the opposite direction of the Y axis.
To validate the mathematical
model, we compared the morphologies of fractures that occur under similar
conditions on biomannequins and in practical observations, in which the
fracture line arose in the area of action of a traumatic object with the
formation of an oblique-transverse asymmetric fracture line with typical signs
of bone tissue stretching on the opposite side from the site of impact. In some
observations, an additional helical fracture line was formed on the proximal
part of the bone, which is closer to the area of bone rotation [3, 10] (Fig.
9-11).
Fig. 9. Image of the fracture line of the femoral diaphysis at the
site of local impact trauma.
Fig.
10. Image of the fracture surface of the femoral diaphysis at the site of local
impact trauma with pronounced asymmetry.
Fig. 11. Image of a helical fracture line of
the proximal part of the femoral diaphysis outside the site of local impact
trauma.
In the described natural experiments, the properties of bone tissue
were not studied; only the absence of any pathology of bone tissue was noted.
Also, in experiments on biomannequins and practical observations, the change in
the thickness of the compact bone plate along the diaphysis of the femur in the
area of traumatic influence was not studied.
When analyzing experimental data and practical observations, we
assumed that the formation of a helical fracture line in the proximal part of
the femur occurs when the stresses of the bone tissue exceed its strength
characteristics. Therefore, it was decided to conduct an additional experiment
by applying more force to the proximal femur under preload and removing the
rigid support without changing the remaining boundary conditions.
In experiment 3, with a preload in the proximal part of the femur of
300 N along the Y axis as a result of modeling the impact perpendicular to
the longitudinal axis of the femur in the conditional middle of the diaphysis
of the femur, an oblique-transverse asymmetrical fracture line has formed with
chipping of elements in the fracture line and the beginning of the formation of
a fracture line in the area of minimum thickness of the compact bone substance
in the injured area proximal to the place of application of the traumatic force
of the femur on the opposite side from the loading and a helical fracture line
in the proximal part of the femoral diaphysis . The maximum equivalent stresses
reached 119.95 MPa (Fig. 12-14).
Fig.
12. Visualization of the final stage of modeling a fracture of the femoral
diaphysis in a complex stress state with a preload increased to 300 N in
the proximal part of the femur along the Y axis. Von Mises analysis.
Fig.
13. Dynamic visualization of modeling a fracture of the femoral diaphysis in a
complex stress state with a preload increased to 300 N in the proximal
part of the femur along the Y axis. Von Mises analysis.
Fig.
14. Dynamic visualization of modeling a fracture of the femoral diaphysis in a
complex stress state with preload increased to 300 N in the proximal part of
the femur along the Y axis. Analysis using principal stress
vectors.
When comparing the morphological features
of fractures and their modeling in our experimental study, we found a
correspondence between the localization and nature of fractures. However, the
variability of the morphological features of femoral shaft fractures in our
experimental study using mathematical modeling had a pattern depending on:
•
direction of the action
vector of the traumatic force and the magnitude of the preload,
•
places of application
of traumatic force (local fracture occurs in the area application of a
traumatic force with a rupture area on the opposite side, a helical fracture
occurs with significant rotational deformation closer to the rotation area),
•
thickness of the
compact plate in the area of traumatic impact on the opposite side from the
site of application of the traumatic force.
Finite element analysis allows us to visualize and predict the
stresses that arise in bone tissue under a combination of impact and rotation
(complex stress state). The data obtained during modeling are confirmed by the
results of original field experiments and practical observations.
Using mathematical modeling, the dependence of the morphological
characteristics of fractures on the place of application of the traumatic force
on the anterior surface of the thigh, the thickness of the compact plate, and
the place of application of rotating loads was revealed. When a combination of
impact and rotation effects occurs in a local fracture, a combination of signs
of angular deformation with asymmetry of the fracture plane, characteristic of
a helical fracture, appears. As rotation of the femur increases, an additional
helical fracture line appears closer to the point of rotation.
The experiments performed show the possibility of using the finite
element method in forensic medicine, which makes it possible to reliably
predict the process of destruction of biological objects under various types of
mechanical impact and visualize its results at any stage of the experiment with
a large amount of displayed data. In the future, it is possible to use this
information to solve the inverse problem – determining the trace evidence
properties of a traumatic weapon based on the morphological picture of
destruction. This confirms the high efficiency of finite element analysis in
forensic medicine.
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