Linear Micropolar Elasticity Analysis of Stresses in Bones under Static Loads
We discuss the finite element modeling of porous materials such as bones using the linear micropolar elasticity. In order to solve static boundary-value problems, we developed new finite elements, which capture the micropolar behavior of the material. Developed elements were implemented in the comme...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2017
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| Цитувати: | Linear Micropolar Elasticity Analysis of Stresses in Bones under Static Loads / V.A. Eremeyev, A. Skrzat, F. Stachowicz // Проблемы прочности. — 2017. — № 4. — С. 103-114. — Бібліогр.: 43 назв. — англ. |
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irk-123456789-1736932020-12-17T01:26:04Z Linear Micropolar Elasticity Analysis of Stresses in Bones under Static Loads Eremeyev, V.A. Skrzat, A. Stachowicz, F. Научно-технический раздел We discuss the finite element modeling of porous materials such as bones using the linear micropolar elasticity. In order to solve static boundary-value problems, we developed new finite elements, which capture the micropolar behavior of the material. Developed elements were implemented in the commercial software ABAQUS. The modeling of a femur bone with and without implant under various stages of healing is discussed in details. Рассматривается моделирование таких пористых материалов, как кость, методом конечных элементов с помощью линейной микрополярной теории упругости. Для решения статических краевых задач разработаны новые конечные элементы, которые воспринимают микрополярное поведение этого материала. Разработанные элементы были реализованы в коммерческом программном комплексе ABAQUS. Рассматривается моделирование бедренной кости с имплантатом и без него на различных стадиях заживления 2017 Article Linear Micropolar Elasticity Analysis of Stresses in Bones under Static Loads / V.A. Eremeyev, A. Skrzat, F. Stachowicz // Проблемы прочности. — 2017. — № 4. — С. 103-114. — Бібліогр.: 43 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/173693 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел |
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Научно-технический раздел Научно-технический раздел Eremeyev, V.A. Skrzat, A. Stachowicz, F. Linear Micropolar Elasticity Analysis of Stresses in Bones under Static Loads Проблемы прочности |
| description |
We discuss the finite element modeling of porous materials such as bones using the linear micropolar elasticity. In order to solve static boundary-value problems, we developed new finite elements, which capture the micropolar behavior of the material. Developed elements were implemented in the commercial software ABAQUS. The modeling of a femur bone with and without implant under various stages of healing is discussed in details. |
| format |
Article |
| author |
Eremeyev, V.A. Skrzat, A. Stachowicz, F. |
| author_facet |
Eremeyev, V.A. Skrzat, A. Stachowicz, F. |
| author_sort |
Eremeyev, V.A. |
| title |
Linear Micropolar Elasticity Analysis of Stresses in Bones under Static Loads |
| title_short |
Linear Micropolar Elasticity Analysis of Stresses in Bones under Static Loads |
| title_full |
Linear Micropolar Elasticity Analysis of Stresses in Bones under Static Loads |
| title_fullStr |
Linear Micropolar Elasticity Analysis of Stresses in Bones under Static Loads |
| title_full_unstemmed |
Linear Micropolar Elasticity Analysis of Stresses in Bones under Static Loads |
| title_sort |
linear micropolar elasticity analysis of stresses in bones under static loads |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2017 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/173693 |
| citation_txt |
Linear Micropolar Elasticity Analysis of Stresses in Bones under Static Loads / V.A. Eremeyev, A. Skrzat, F. Stachowicz // Проблемы прочности. — 2017. — № 4. — С. 103-114. — Бібліогр.: 43 назв. — англ. |
| series |
Проблемы прочности |
| work_keys_str_mv |
AT eremeyevva linearmicropolarelasticityanalysisofstressesinbonesunderstaticloads AT skrzata linearmicropolarelasticityanalysisofstressesinbonesunderstaticloads AT stachowiczf linearmicropolarelasticityanalysisofstressesinbonesunderstaticloads |
| first_indexed |
2025-07-15T10:26:11Z |
| last_indexed |
2025-07-15T10:26:11Z |
| _version_ |
1837708264167440384 |
| fulltext |
UDC 539.4
Linear Micropolar Elasticity Analysis of Stresses in Bones under Static Loads
V. A. Eremeyev,
a,1
A. Skrzat,
b,2
and F. Stachowicz
b,3
a Gdansk University of Technology, Gdansk, Poland
b Rzeszów University of Technology, Rzeszów, Poland
1 eremeyev.victor@gmail.com
2 askrzat@prz.edu.pl
3 stafel@prz.edu.pl
We discuss the finite element modeling of porous materials such as bones using the linear micropolar
elasticity. In order to solve static boundary-value problems, we developed new finite elements, which
capture the micropolar behavior of the material. Developed elements were implemented in the
commercial software ABAQUS. The modeling of a femur bone with and without implant under
various stages of healing is discussed in details.
Keywords: bone, implant, Cosserat continuum, micropolar elasticity, numerical simulation,
finite element method.
Introduction. Bones constitute a particular class of materials which may demonstrate
very complex behavior including such phenomena as growth, anisotropy, porosity, piezo-
electricity, etc., see, e.g., [1–4]. Mechanics of porous materials is a quite developed branch
of mechanics. It includes various models of material behavior. Among these models the
continuum Cosserat called also micropolar medium plays an important role. For example,
the micropolar elasticity may predict size-effect observed for bones. Within the micropolar
elasticity, each material point possesses properties of a rigid body that is it has both
translational and rotational degrees of freedom. The duals for the translational and
rotational degrees of freedom are forces and couples (moments). Since the bone can be
treated as a microstructured material with such micro-elements as struts modeled as beams,
the appearance of moments seems to be quite natural for such materials. In fact, a bone can
be considered as an example of an open-cell foam. Since in foam struts there exist moments
in addition to forces, this naturally leads to the micropolar elasticity, see the original works
of Lakes and coworkers [5–9], where few experiments on porous materials including bones
are performed. Recently, the homogenization technique leading to the micropolar elasticity
in the case of bones is presented in [10–14]. In particular, the latter papers give the material
parameters of the micropolar elasticity. Let us also note that such microstructured materials
as foams, porous media, beam lattices may lead to more general models of continuum
mechanics, see for example, the strain gradient elasticity [15–21]. Let us note that for
modeling of a bone, various mechanical models were proposed, for example, media with
internal variables [22–25].
The micropolar theory may play an even more important role if considered from the
fracture mechanics point of view. Here the so-called constraint pseudo continuum Cosserat
is the most used model. First using this model one observes that the stress and moment
singularity may be quite different in comparison with the classic fracture mechanics
[26–28]. This cause a question what type of fracture criterion should be used to decide
when the crack starts propagating and what material parameters should be evaluated to
determine the material toughness. Two criteria were discussed in [27, 28] to tackle the
problem: energy release rate criterion and the maximum stress criterion proposed. They
both demonstrate similar ability in predicting the fracture while the second one looked
© V. A. EREMEYEV, A. SKRZAT, F. STACHOWICZ, 2017
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 4 103
more promising. The situation complicated even further when one considers an interface
crack in such material [29]. Finally when analyzing a moving crack an additional problem
of the stability of the crack propagating in such material arises [30–33]. Here various
effects were determined and the aforementioned criteria were used to estimate material
toughness. From the presented analysis, the fracture criterion based on the maximum stress
again demonstrated better performance. However, all those results are still to be verified
experimentally.
The aim of the paper is the implementation of developed finite elements into
ABAQUS and analysis of the static deformations of the loaded bone with an implant. The
paper is organized as follows. First, we recall the basic equations of the micropolar
elasticity. Then we introduce new tetrahedral micropolar finite element. Here we follow
previous results [34–36] for the implementation of the hexahedral finite micropolar
element. Finally, within the micropolar elasticity, we consider the modeling of a femur bone
with an implant using the micropolar finite elements.
Basic Equations of the Linear Isotropic Micropolar Elasticity. The micropolar
continuum model was introduced by Cosserat brothers more than hundred years ago [37].
For the current state-of-the-art, we refer to [38, 39]. Unlike classic elasticity, within the
micropolar elasticity, we introduce not only displacements u j but also rotations � j as
kinematic variables. Here we adopt the following convention: Latin indices take values 1,
2, 3, so i j k, , , , .�1 2 3 The equilibrium equations take the form
� ji j, ,� 0 (1)
m eji j ijk ji, ,� �� 0 (2)
where � ji and m ji are the stress and couple stress tensors, respectively, and eijk is the
Levi-Civita permutation symbol.
Hereinafter we use the Einstein summation rule over repeating indices and, for
simplicity, we neglect body forces and couples. Equations (1) and (2) are the local balances
of momentum and moment of momentum. The strain tensors in the micropolar elasticity are
given by [40, 41]
� �ij j i ijk ku e� �, , � �ij i j� , . (3)
Note that � ji and m ji , � ij and � ij are all nonsymmetric tensors, in general.
In what follows we are restricted ourselves considering isotropic micropolar materials
[38, 42, 43]. Using the modified for the micropolar elasticity the Voigt notation [34–36] the
constitutive relations transform into the following form:
{ } [ ]{ },� �M MC� (4)
where
{ } ,�
�
M
m
�
�
�
�
{ } ,�
�
�M �
�
�
�
(5)
� � � � � � � � � ��{ , , , , , , , , } ,xx yy zz xy yx yz zy xz zx
T
(6)
m m m m m m m m m mxx yy zz xy yx yz zy xz zx
T�{ , , , , , , , , } , (7)
� � � � � � � � � ��{ , , , , , , , , } ,xx yy zz xy yx yz zy xz zx
T
(8)
V. A. Eremeyev, A. Skrzat, and F. Stachowicz
104 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 4
� � � � � � � � � ��{ , , , , , , , , } ,xx yy zz xy yx yz zy xz zx
T
(9)
and stiffness matrix [ ]C consists of the following submatrices [34–36]:
[ ] ,C
A
B
�
�
��
�
��
0
0
(10)
with
[ ]A �
� �
� �
� �
�
�
�
�
�
� � � � �
� � � � �
� � � � �
� � �
� � �
� � �
� � �
� � �
� �
2
2
2
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
,
(11)
and matrix B has a similar structure:
[ ]B �
� �
� �
� �
�
�
�
�
�
�
�
�
�
�
�
�
�
� � � � �
� � � � �
� � � � �
� �
� �
� �
� �
� �
� �
�
�
�
�
�
�
�
�
�
�
�
�
�
. (12)
Here � � �, , and � � �, , are material moduli. The latter ones depend on the engineering
parameters as follows:
�
�
�
�
�
�
�
� �
�
�
�
�
�
�
�
�
�
�
!
!
!!
!
!
!
2
1
1
1
2
1 2
2
2
2
2
GN
N
G
N
N
G
,
,
!
,
� � �
� � �
� � �
� �
�
� � �
� �
�
l
l l
l
t
t b
b
2
2 2
2
2
1
2 2
2 2
( ) ,
( )( ),
( ).
"
"!
!
!
!
(13)
Linear Micropolar Elasticity Analysis of Stresses ...
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 4 105
The values of the engineering moduli for bones are given in Table 1.
4-Node Micropolar Tetrahedral Element. Following the technique used for
developing of the 8-node hexahedral micropolar element [35, 36] the 4-node micropolar
tetrahedral element has been developed. This element is suitable for domains of arbitrary
(even very sophisticated) geometry, and, e.g., for bones. The position of an arbitrary point
inside tetrahedron can be specified in Cartesian coordinates {x y z, , } or in natural
coordinates {# # # #1 2 3 4, , , }. Natural coordinates are defined as the fraction of the
volumes Vi related to the volume V, where Vi is defined as the volume of subtetrahedron
bounded by the inside point and the face Fi (Fig. 1). Natural coordinates are constrained
by the equation:
# # # #1 2 3 4
1 2 3 4
1� � � � � � � �
V
V
V
V
V
V
V
V
. (14)
So only three coordinates are independent. Mapping of natural coordinates into
Cartesian ones is given as follows:
1 1 1 1 1
1 2 3 4
1 2 3 4
1 2 3 4
x
y
z
x x x x
y y y y
z z z z
�
!!
!
!
�
�
!!
!
!
�
�
�
�
�
�
�
�
�
�
�
�
�
�
!
!
!
!
�
�
!
!
!
!
�
�
!
!
!
!
#
#
#
#
#
#
#
#
1
2
3
4
1
2
3
4
[ ]J
�
�
!
!
!
!
, (15)
where xi , yi , and zi are Cartesian coordinates of node i. The volume of tetrahedron
takes the value
V J�
1
6
det [ ]. (16)
From (15) it follows that
$
$%
x
x
i
i� ,
$
$%
y
y
i
i� ,
$
$%
z
z
i
i� , i�1 2 3 4, , , . (17)
The inverse of the Jacobian (which can be found numerically or in an explicit form) is
[ ]
...
...
...
...
J
V
a b c
a b c
a b c
a b c
� �
�
�
�
�
�
1
1 1 1
2 2 2
3 3 3
4 4 4
1
6
�
�
�
�
�
�
�
. (18)
106 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 4
T a b l e 1
Micropolar Material Data for Bones [5–9]
Parameter Symbol Value
Shear modulus (MPa) G 4 000
Poisson’s ratio � 0.25
Coupling number N & 0 5.
Characteristic length (torsion) (m) lt 0.00022
Characteristic length (bending) (m) lb 0.00045
Polar ratio " 1.5
V. A. Eremeyev, A. Skrzat, and F. Stachowicz
The first column is omitted here because it is not important from the point of view of
further derivation.
From (18) we obtain
6V
x
a
i
i
$%
$
� , 6V
y
b
i
i
$%
$
� , 6V
z
c
i
i
$%
$
� , i�1 2 3 4, , , . (19)
Natural coordinates which are assumed to be the shape functions ( fi i� # ) are used in
the interpolation of displacement field inside tetrahedron
u
v
w
u u u u
v v v v
w w w w
�
!
!
�
�
!
!
�
�
�
�
�
�
�
�
�
�
�
1 2 3 4
1 2 3 4
1 2 3 4
1#
#
#
#
2
3
4
�
!!
!
!
�
�
!!
!
!
. (20)
In order to find strains, we need displacements derivatives in terms of the Cartesian
coordinates. Applying the chain rule to shape functions one obtains:
f
f
x
f a
V
f
f
y
f b
V
x
i
i
i
i
y
i
i
i
i
,
,
,
,
� �
� �
$
$#
$#
$
$
$#
$
$#
$#
$
$
$#
6
6
f
f
z
f c
V
z
i
i
i
i
, .� �
$
$#
$#
$
$
$# 6
(21)
Combining (18) and (21) and noticing that
$
$%
f i j
i j
i
j
�
�
'
�
1
0
for
for
,
,
(22)
the shape functions derivatives can be found from the inverse of the Jacobian
[ ]
...
...
...
, , ,
, , ,
, ,
J
V
f f f
f f f
f f
x y z
x y z
x y
� �1
1 1 1
2 2 2
3 3
1
6 f
f f f
z
x y z
3
4 4 4
,
, , ,...
.
�
�
�
�
�
�
�
�
�
�
�
�
(23)
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 4 107
Fig. 1. Nodes and faces numbering.
Linear Micropolar Elasticity Analysis of Stresses ...
The stiffness matrix of tetrahedron element is given by
k B CBdVT
V
�( . (24)
Integration in (24) is made by one-point Gaussian quadrature.The matrix of shape
functions derivatives and shape functions itself B is
B
f f
f f
f
x x
y y
z
�
1 4
1 4
1
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0
, ,
, ,
,
...
...
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0
4
1 1 4 4
1 1
...
...
.
,
, ,
,
f
f f f f
f f
z
x x
y � ..
...
...
,
, ,
,
f f
f f f f
f f
y
y y
z
4 4
1 1 4 4
1 1
0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0
�
� 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0
4 4
1 1 4 4
1 1
f f
f f f f
f f f
z
x x
z
,
, ,
,
...
...
�
� �
4 4
1 4
1 4
0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
,
, ,
, ,
...
...
z
x x
y y
f
f f
f f 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0
1 4
1 4
1
f f
f f
f
z z
y y
, ,
, ,
,
...
...
x x
z z
y
f
f f
f
0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
4
1 4
1
...
...
...
,
, ,
, 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
4
1 4
1 4
f
f f
f f
y
z z
x x
,
, ,
, ,
...
...
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
.
(25)
It is noteworthy that in contrast to the in classical theory of elasticity, the micropolar
tetrahedral element is not the element of constant stress/strain. Indeed, in (25) linear shape
functions appear. In (25) the following order of strain and displacement components is
assumed:
{ } {� � � � � � � � � � � � � � � �T
xx yy zz xy yx yz zy xz zx xx yy zz xy yx y� z zy xz zx� � � },
{ } { ... }.u u v w u v wT
x y z x y z� 1 1 1 1 1 1 4 4 4 4 4 4� � � � � � (26)
Constitutive matrix C is defined in (11).
In order to solve various boundary value problems, a special 4-node micropolar finite
element has been developed in the form of UEL (user element) procedure for commercial
ABAQUS program. UEL procedure is called twice for each finite element and for each
Gaussian point. In the first call, the element stiffness is welcomed. The UEL procedure
calls UELMAT (user material procedure) necessary to obtain the relation between stress
and strain tensors (or between stress and strain increments). Another call of UEL procedure
is necessary to compute residual forces – element nodal forces resulting from element
stresses – which essential in the monitoring of convergence in nonlinear problems. This
second call may also require another call of UELMAT procedure.
108 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 4
V. A. Eremeyev, A. Skrzat, and F. Stachowicz
Linking the user element procedure with the commercial software allows one to solve
large boundary value problems in an efficient way. Unfortunately for users, the visualization
of the obtained results causes many problems, but it can be avoided by writing special
Python scripts for ABAQUS, which create several output databases available for the
postprocessor. Four output databases are necessary to show all components of strains and
stresses. Note that, in micropolar elasticity, the number of strain/stress components exceeds
the number of components (six) in the classical approach.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 4 109
Fig. 2. Femur bone an hip implant.
Fig 3. A400 lateralized stem specifications with parameters for different sizes (http://www.renovis-
surgical.com/2011/09/a400-sizes-and-specs/).
Linear Micropolar Elasticity Analysis of Stresses ...
Modeling of a Femur Bone with an Implant. Using the developed software we
discussed earlier some test problems including also some problems for bones [34–36].
Here, as an example of real FEM modeling of a complex structure, we consider a femur
bone with a hip implant (Fig. 2). A solid model of hip transplant was created using 3D
CAD tools of SolidWorks 2010 (SolidWorks Corporation, USA) on the basis of A400
lateralized specification that shown in Fig. 3 and Table 2. For the bone and implant, we
developed a fine mesh (Fig. 4). Here we pay a special attention to the implant area
increasing there the number of elements. It was almost 100,000 elements with the
maximum size of 3 mm. The value of the static load acting on the femur is 800 N. The load
is directed along an axis extending from the upper pole of the prosthesis head to the middle
110 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 4
Fig. 4. FEM mesh.
T a b l e 2
Geometry Details for Fig. 3
Size Neck
axis
distance
(mm)
Offset
(mm)
Stem
lengt
h
(mm)
Distal
stem
lengt
h
(mm)
Proximal
cross
section
M–L
width
(mm)
Proximal
cross
section
A–P
width
(mm)
Distal
cross
section
n M–L
width
(mm)
Distal
cross
section
n A–P
width
(mm)
4.50 52.4 38.5 131.0 9.0 29.1 11.5 5.5 4.2
5.25 52.9 39.3 132.3 10.5 29.7 11.8 6.3 4.5
6.00 53.5 40.0 133.6 12.0 30.4 12.0 7.0 4.7
6.75 54.0 40.8 134.9 13.5 31.0 12.3 7.8 5.0
7.50 54.6 41.5 136.2 15.0 31.7 12.5 8.5 5.3
8.25 55.2 42.5 137.6 16.5 32.3 12.8 9.3 5.6
9.00 55.8 43.0 138.9 18.0 33.0 13.1 10.0 5.8
9.75 56.3 43.8 138.9 19.5 33.7 13.3 10.8 6.1
10.50 56.9 44.5 140.2 21.0 34.4 13.6 11.5 6.4
11.20 57.1 45.3 141.5 22.5 35.1 13.8 12.5 6.6
12.00 58.1 46.0 142.9 24.0 35.7 14.1 13.0 6.9
13.50 59.3 47.5 144.1 27.0 37.0 14.7 14.5 7.4
15.00 60.5 49.0 146.8 30.0 38.4 15.2 16.0 8.0
V. A. Eremeyev, A. Skrzat, and F. Stachowicz
of the distance between the extreme lower divisions femoral condyle. The results of
calculations are given in Figs. 5 and 6. In particular, it is clear that influence of the couple
stresses is important in the vicinity of some singularities such as edges of the implant
(Fig. 6). This means that the rotations of the material particles and couple stresses play a
role in such zones.
Conclusions. In this paper, the linear isotropic micropolar elasticity was applied to
modeling of femur bones with and without implants. The analysis of bones and its
interaction with implants is a complex task from the point of view of mechanics and
numerical computations. Indeed, this problem combines both a complex geometry and
complex material properties. A bone can be considered as a porous composite consisting of
hydroxyapatite (which is mainly calcium phosphate) and collagen. For numerical analysis
in addition to developed earlier hexahedral finite micropolar element we developed new
specific tetrahedral micropolar finite element and implemented it the commercial software
ABAQUS. The results obtained within the micropolar theory are compared with the classic
linear elasticity. Numerical tests have shown that the couple stresses appear in the vicinity
of singularities, such as holes and edges of the size which is comparable to the characteristic
length parameters. These effects may be very important for highly porous materials such as
bones and bioceramics. The presenting results may be important for personalization of the
surgery. Indeed, pre-surgery planning is important to select the optimum acetabular
component, and in predicting the approximate volume of components which can be used
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 4 111
Fig. 5. Distribution of Huber–Mises stress (Pa).
Fig. 6. Distribution of Mxx couple stress (N/m).
Linear Micropolar Elasticity Analysis of Stresses ...
during the intervention. Using the developed tool one can relatively easy calculate the level
of stresses, stress concentrations and other characteristics of the mechanical behavior of a
bone with and without an implant.
Acknowledgments. The research leading to these results has received funding from
the People Program (Marie Curie Curie ITN transfer) of the European Union’s Seventh
Framework Programme for research, technological development and demonstration under
grant agreement No PITN-GA-2013-606878.
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Received 01. 09. 2017
114 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 4
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/HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. Stvoreni PDF dokumenti mogu se otvoriti Acrobat i Adobe Reader 5.0 i kasnijim verzijama.)
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/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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/UKR <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>
/ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
/RUS <FEFF04180441043f043e043b044c04370443043904420435002004340430043d043d044b04350020043d0430044104420440043e0439043a043800200434043b044f00200441043e043704340430043d0438044f00200434043e043a0443043c0435043d0442043e0432002000410064006f006200650020005000440046002c0020043c0430043a04410438043c0430043b044c043d043e0020043f043e04340445043e0434044f04490438044500200434043b044f00200432044b0441043e043a043e043a0430044704350441044204320435043d043d043e0433043e00200434043e043f0435044704300442043d043e0433043e00200432044b0432043e04340430002e002000200421043e043704340430043d043d044b04350020005000440046002d0434043e043a0443043c0435043d0442044b0020043c043e0436043d043e0020043e0442043a0440044b043204300442044c002004410020043f043e043c043e0449044c044e0020004100630072006f00620061007400200438002000410064006f00620065002000520065006100640065007200200035002e00300020043800200431043e043b043504350020043f043e04370434043d043804450020043204350440044104380439002e>
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /ConvertToCMYK
/DestinationProfileName ()
/DestinationProfileSelector /DocumentCMYK
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure false
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles false
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /DocumentCMYK
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /UseDocumentProfile
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|