Application of the Micropolar Theory to the Strength Analysis of Bioceramic Materials for Bone Reconstruction
The application of the linear micropolar theory to the strength analysis of bioceramic materials for bone reconstruction is described. Micropolar elasticity allows better results to be obtained for microstructural and singular domains as compared to the classical theory of elasticity. The fundamenta...
Saved in:
| Published in: | Проблемы прочности |
|---|---|
| Date: | 2016 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут проблем міцності ім. Г.С. Писаренко НАН України
2016
|
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/173513 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Application of the Micropolar Theory to the Strength Analysis of Bioceramic Materials for Bone Reconstruction / V.A. Eremeyev, A. Skrzat, A. Vinakurava // Проблемы прочности. — 2016. — № 4. — С. 119-128. — Бібліогр.: 28 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859851886021050368 |
|---|---|
| author | Eremeyev, V.A. Skrzat, A. Vinakurava, A. |
| author_facet | Eremeyev, V.A. Skrzat, A. Vinakurava, A. |
| citation_txt | Application of the Micropolar Theory to the Strength Analysis of Bioceramic Materials for Bone Reconstruction / V.A. Eremeyev, A. Skrzat, A. Vinakurava // Проблемы прочности. — 2016. — № 4. — С. 119-128. — Бібліогр.: 28 назв. — англ. |
| collection | DSpace DC |
| container_title | Проблемы прочности |
| description | The application of the linear micropolar theory to the strength analysis of bioceramic materials for bone reconstruction is described. Micropolar elasticity allows better results to be obtained for microstructural and singular domains as compared to the classical theory of elasticity. The fundamental equations of the Cosserat continuum are cited. The description of FEM implementation of micropolar elasticity is given. The results of solving selected 3D test problems are presented. Comparison of classical and micropolar solutions is discussed.
The research received funding from the People Program (Marie 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.
|
| first_indexed | 2025-12-07T15:42:02Z |
| format | Article |
| fulltext |
UDC 539. 4
Application of the Micropolar Theory to the Strength Analysis of Bioceramic
Materials for Bone Reconstruction
V. A. Eremeyev, A. Skrzat, and A. Vinakurava
Rzeszów University of Technology, Rzeszów, Poland
The application of the linear micropolar theory to the strength analysis of bioceramic materials for
bone reconstruction is described. Micropolar elasticity allows better results to be obtained for
microstructural and singular domains as compared to the classical theory of elasticity. The
fundamental equations of the Cosserat continuum are cited. The description of FEM implementation
of micropolar elasticity is given. The results of solving selected 3D test problems are presented.
Comparison of classical and micropolar solutions is discussed.
Keywords: Cosserat continuum, micropolar elasticity, stress concentration, finite element
analysis, porous media, bones.
Introduction. Bones, bones with implants, and their modeling within the framework
of continuum mechanics constitute a very challenging problem due to the complex
microstructure, material properties, physical and chemical processes [1, 2]. Among many
problems related to the bone modeling, the bone reconstruction with porous implants is one
of the most important from the practical and theoretical points of view [3–8].
Mesoporous biomaterials allow for the regeneration of a bone tissue in the treatment
of various pathologies [9]. Porosity necessary for the bone regeneration has been shown by
Kuboki using solid and porous particles of hydroxyapatite for BMP-2 delivery. It is
important that any new bone was not formed on solid particles, while in the porous
scaffolds direct osteogenesis occurred [10]. The analysis of such materials is an elaborate
procedure since their microstructure should be taken into account.
Reliable numerical analysis of bones and biomaterials is a very complicated task
mostly because of their complex geometry. In practice, only small representative volume
elements can be examined. Indeed, a trabecular bone can be considered a porous solid with
a highly nonuniform microstructure (Fig. 1). For the bone modeling, different mechanical
models were proposed. Among them, poroelasticity [3, 4], media with internal variables
[5–8] should be mentioned. In some cases a bone can be treated as an open foam-like
structure or a system of beams. Since in foam struts or in beams not only forces but also
moments exist, this naturally leads to the model of the Cosserat continuum, also termed
micropolar elasticity [11, 12]. Within the micropolar media model normal and tangential
surface tractions are considered, but unlike classical elasticity we also take into account
moment interactions and microrotations of material particles. So, we may replace the real
bone structure with an equivalent homogeneous micropolar solid (Fig. 1). This replacement
may significantly reduce the computational efforts with sufficient accuracy in comparison
with the real bone structure or beam lattice. The idea of applying the Cosserat continuum
model to the bone modeling relates to the original studies by Lakes and coworkers [13–16]
where a few experiments on porous materials, including bones, were performed. Recently,
in [17–19] the homogenization technique leading to micropolar elasticity as applied to
bones has been developed. In the latter publications the material parameters of micropolar
elasticity are derived. It should also be noted that such materials, as foams, porous media,
beam lattices may bring to more general models of continuum mechanics [20–23].
© V. A. EREMEYEV, A. SKRZAT, A. VINAKURAVA, 2016
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 4 119
This study covers the following issues. The basic equations of linear isotropic
micropolar elasticity are first recalled. Emphasis is given to the constitutive relations for the
bone modeling. In the second section, the finite element technique is presented, which
would be used for the calculations of a few problems in the next section. Here the stress
concentration problems in the vicinity of notches and holes of various sizes would be
analyzed.
Basic Equations of Micropolar Elasticity. Within micropolar continuum mechanics,
apart from the field of translations, the local rotations of material particles of a deformable
body and their moment interactions are taken into account as the additional degrees of
freedom. Following Eringen [11], the strain measures are introduced, being the linear
stretch tensor and bending-torsion tensor, also termed the wryness tensor
� �ij j i ijk ku e� �, , (1)
� �ij i j� , , (2)
where u j is the vector of translations, � j is the vector of microrotations, eijk is the
Levi-Civita symbol, i j k, , , , .�1 2 3 Hereinafter we use the Einstein summation rule over
repeating indices. Due to the contribution of microrotations, the tensors � ij and � ij are
asymmetric. In the micropolar theory, the stress measures are the stress tensor � ji (force
per unit area) and coupled stress tensor m ji (moment per unit area).
In the absence of body forces and moments, the equilibrium equations of the Cosserat
continuum take the form
� ji j, ,� 0 (3)
m eji j imn mn, .� �� 0 (4)
It is worthy of emphasis that the stress tensors are also asymmetric and strains (1), (2)
and (3), (4) are energy-related.
Further the Voigt notation is applied. So the micropolar stress and strain tensors have
the following generalized form:
{ } ,�
�
M
m
�
�
�
�
{ } ,�
�
�M �
�
�
�
(5)
where the vectors are introduced
� � � � � � � � � ��{ , , , , , , , , } ,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)
V. A. Eremeyev, A. Skrzat, and A. Vinakurava
120 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 4
Fig. 1. Real bone structure, the beam lattice, and homogeneous micropolar media representations.
� � � � � � � � � ��{ , , , , , , , , } ,xx yy zz xy yx yz zy xz zx
T
(8)
� � � � � � � � � ��{ , , , , , , , , } .xx yy zz xy yx yz zy xz zx
T
(9)
With this notation, the constitutive equations of the Cosserat medium are defined as
{ } [ ]{ },� �M MC� (10)
where the stiffness matrix [ ]C consists of the following submatrices:
[ ] .C
A
B
�
�
��
�
��
0
0
(11)
In the isotropic case the matrix [ ]A is given by the formula [11, 12, 24, 25]:
A�
� �
� �
� �
�
�
�
�
�
�
� � � � �
� � � � �
� � � � �
� � �
� � �
� � �
� � �
� � �
� � �
2
2
2
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
,
(12)
and the matrix [ ]B has a similar structure:
[ ]B �
� �
� �
� �
�
�
�
�
�
�
�
�
�
�
�
�
�
� � � � �
� � � � �
� � � � �
� �
� �
� �
� �
� �
� �
�
�
�
�
�
�
�
�
�
�
�
�
�
. (13)
In (12) and (13) only nonzero elements � � �, , are shown up, and � � �, , are the
elastic moduli discussed below. Six engineering moduli for the isotropic case and their
values for bones [13–16] are cited in Table 1. These values were found through predicted
relations between material properties and specimen sizes within the micropolar elasticity
size-effect. Thus, there are theoretically two characteristic length parameters for torsion and
bending, which can be determined from experimental data.
Application of the Micropolar Theory to Strength Analysis ...
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 4 121
The first three parameters are related to � � �, , as
G
N
�
�
�
� �
�
�
�
�
�
�
�
�
�
2
2
2 2
2
2
� �
�
�
� � �
�
� �
,
,
( )
.
(14)
Solving the system of equations (14) for � � �, , , we obtain
�
�
�
�
�
�
�
� �
�
�
�
�
�
�
!
!
�
�
�
�
�
��
�
�
2
1
1
1
2
1 2
2
2
2
2
GN
N
G
N
N
G
,
,
.�
�
(15)
The next three material moduli � � �, , depend on l lt b, , " via relations
l
l
t
b
2
2
2
2 2
�
�
�
�
�
�
�
� �
�
�
�
�
�
�
�
� �
� �
�
� �
� �
� � �
,
( )
,
."
(16)
Solving the system of equations (16) in terms of � � �, , , we find
� � �
� � �
� � �
� � �
� � �
� �
l
l l
l
t
t b
b
2
2 2
2
2 1
2 2
2 2
( )( ) ,
( )( ),
( ).
" "�
��
�
�
(17)
122 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 4
T a b l e 1
Micropolar Material Characteristics for Bones
Characteristic Symbol Value
Shear modulus (MPa) G 4000
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 A. Vinakurava
The 8-Node Hybrid Micropolar Isoparametric Element. For strength analysis
based on micropolar elasticity, the 8-node hexahedral element is constructed. The element
is represented in the Cartesian and natural coordinates (Fig. 2).
For the interpolation of displacements and microrotations the same shape functions
are used. The ith shape function is defined as
N i i i i( , , ) ( )( )( ).$ % � $$ %% ��� � � �
1
8
1 1 1 (18)
The stiffness matrix of the element is given by
k D CDdVT
V
�& , (19)
where the matrix C is defined in (11)–(13) and matrix D is the matrix of shape function
derivatives viz. the relation between the Erringen strains and nodal displacements and
microrotations,
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
xx
yy
zz
xy
yx
yz
zy
xz
zx
xx
yy
zz
xy
yx
yz
zy
xz
zx�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
f f
f f
x x
y y
1 8
1 8
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
1 8
1 1 8 8
1
f f
f f f f
f
z z
x x
y
, ,
, ,
,
...
...
0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0
1 8 8
1 1 8 8
1
� �
�
f f f
f f f f
f
y
y y
z
...
...
,
, ,
, f f f
f f f f
f
z
x x
z
1 8 8
1 1 8 8
1
0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0
...
...
,
, ,
,
�
� �
f f f
f f
f
z
x x
y
1 8 8
1 8
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 0 0 0
8
1 8
1 8
f
f f
f f
y
z z
y y
,
, ,
, ,
...
...
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 8 1
1 8
1
f f
f f
f
x
z z
, ,
, ,
,
...
...
y y
z z
x
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
8
1 8
1 0 8f x,
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
u
u
u
u
u
u
x
y
z
x
y
z
x
y
z
x
y
z
1
1
1
1
1
1
8
8
8
8
8
8
�
�
�
�
�
�
...
�
�
�
�
�
�
�
�
�
��
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
��
�
�
�
�
�
�
�
�
�
�
�
.
(20)
Integration in (19) is over the element volume presented as the loop of the Gaussian
points. The 2 2 2' ' Gaussian quadrature is used.
Test Problems. The finite element method program elaborated for solving the 3D
problems of micropolar elasticity has to be verified and validated in several test ones. The
two examined problems, including singularities are presented in Fig. 3. This choice is
stemming from more pronounced micropolar properties in a singular case [26–28]. Test
problems are solved for different a, b, d , and h dimensions and mesh densities. Two types
of loads are considered: constant pressure (replaced with kinematically equivalent forces)
or displacement applied to the right-hand lateral (edge) surface (kinematic load).
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 4 123
Application of the Micropolar Theory to Strength Analysis ...
As is known, classical elasticity is a special case of micropolar elasticity with fixed
microrotations (equal zero). The results of solving test problems using micropolar software,
wherein all microrotations are fixed, are compared with those obtained through commercial
software. An alternative opportunity to obtain classical elasticity solutions is the employment
of only two first material characteristics from Table 1, if all remaining data are taken as
zero.
In the test problems the effect of mesh density on the body response was investigated.
The symmetry of obtained solutions was checked (problems can be solved more efficiently
in terms of double symmetry).
As an example, the following dimensions: L� 0.03 m and H� 0.01 m are given in
Fig. 4. The other parameters were varied, e.g., over the range of 0.001–0.0005 m for the
diameter d and of 0.0003–0.001 m for the minor axis a of the ellipse. Such dimensions
are chosen for evaluating the impact of the characteristic length in the micropolar media.
Finite element meshes (Fig. 4) are generated with commercial software (ANSYS or
ABAQUS). The maximum size of test problems is 120 thousand degrees of freedom. As
mentioned above, micropolar elasticity provides the same solutions as the classical theory
of elasticity if all microrotations are fixed or if only two material characteristics are used
(shear or Young’s modulus and Poisson’s ratio). Solutions obtained with commercial
software (classical theory of elasticity) and elaborated software are compared in Figs. 5 and
6. As can be seen, stress distributions and magnitudes (not shown for micropolar software)
converge excellently. It proves that the results, obtained with elaborated software in terms of
micropolar elasticity and reduced to the classical one, are reliable. The results of micropolar
analysis are processed with a specially developed OpenGL library-based program.
As expected, the whole family of test problems solved so far confirmed that the
coupled stresses appear only in the vicinity of singularities. In other cases the solutions
obtained in terms of micropolar and classical elasticity are comparable. The distribution of
the mzz stresses (axis z is normal to the plate) in the plate with a sharp notch is given in
Fig. 7. In this test problem the magnitude of coupled stresses (all components) is a few
orders lower than, e.g., the magnitude of a longitudinal stress component.
124 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 4
Fig 2. The 8-node hexahedral element.
Fig. 3. Examined test problems.
V. A. Eremeyev, A. Skrzat, and A. Vinakurava
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 4 125
Fig. 4. Examples of coarse and fine meshes.
Fig. 5. Comparison of von Mises stresses (Pa) obtained with commercial (top) and elaborated
software (bottom).
Application of the Micropolar Theory to Strength Analysis ...
Conclusions. Micropolar elasticity as one of the models for the materials with a
complex microstructure is used for the bone modeling. The micropolar, elasticity is a very
effective tool for solving the singular problems and the problems for microstructure-
containing domains, like in foams and bones. For numerical analysis specific finite elements
were constructed and implemented in commercial software. An excellent convergence has
126 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 4
Fig. 6. Comparison of longitudinal stresses (Pa) obtained with commercial (top) and elaborated
software (bottom).
Fig. 7. Distribution of the mzz stresses in the plate with a sharp notch.
V. A. Eremeyev, A. Skrzat, and A. Vinakurava
been achieved for all test problems. The results obtained within micropolar elasticity are
compared with classical linear one. Numerical tests demonstrated that coupled stresses
appeared in the vicinity of singularities. So the effect of micropolar properties is related to
the notches and holes with the size of the same order as the characteristic length parameters
of micropolar elasticity.
Acknowledgments. The research received funding from the People Program (Marie
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.
1. S. Cowin (Ed.), Bone Mechanics Handbook, CRC Press, Boca Raton, FL (2001).
2. Y. H. An and R. A. Draughn (Eds.), Mechanical Testing of Bone and the
Bone–Implant Interface, CRC Press, Boca Raton, FL (2000).
3. S. C. Cowin and D. H. Hegedus, “Bone remodeling I: theory of adaptive elasticity,” J.
Elasticity, 6, No. 3, 313–326 (1976).
4. D. H. Hegedus and S. C. Cowin, “Bone remodeling II: small strain adaptive
elasticity,” J. Elasticity, 6, No. 4, 337–352 (1976).
5. T. Lekszycki and F. dell’Isola, “A mixture model with evolving mass densities for
describing synthesis and resorption phenomena in bones reconstructed with bio-
resorbable materials,” Z. Angew. Math. Mech, 92, No. 6, 426–444 (2012).
6. I. Giorgio, U. Andreaus, D. Scerrato, and F. dell’Isola, “A visco-poroelastic model of
functional adaptation in bones reconstructed with bio-resorbable materials,” Biomech.
Model. Mechanobiol., 15, No. 5, 1325–1343 (2016).
7. I. Giorgio, U. Andreaus, D. Scerrato, and P. Braidotti, “Modeling of a non-local
stimulus for bone remodeling process under cyclic load: Application to a dental
implant using a bioresorbable porous material,” Math. Mech. Solids (2016), DOI:
10.1177/1081286516644867.
8. I. Giorgio, U. Andreaus, T. Lekszycki, and A. Della Corte, “The influence of different
geometries of matrix/scaffold on the remodeling process of a bone and bio-
resorbable material mixture with voids,” Math. Mech. Solids (2015), DOI: 10.1177/
1081286515616052.
9. M. Vallet-Regi, I. Izquierdo-Barba, and M. Colilla, “Structure and functionalization of
mesoporous bioceramics for bone tissue regeneration and local drug delivery,” Phil.
Trans. R. Soc. A, 370, 1400–1421 (2012).
10. V. Karageorgiou and D. Kaplan, “Porosity of 3D biomaterial scaffolds and
osteogenesis,” Biomaterials, 26, 5474–5491 (2005).
11. A. C. Eringen, Microcontinuum Field Theories: I. Foundations and Solids, Springer-
Verlag, New York (1999).
12. V. A. Eremeyev, L. P. Lebedev, and H. Altenbach, Foundations of Micropolar
Mechanics, Springer, Berlin (2013).
13. J. F. C. Yang and R. S. Lakes, “Experimental study of micropolar and couple stress
elasticity in compact bone in bending,” J. Biomech., 15, No. 2, 91–98 (1982).
14. H. C. Park and R. S. Lakes, “Cosserat micromechanics of human bone: strain
redistribution by a hydration sensitive constituent,” J. Biomech., 19, No. 5, 385–397
(1986).
15. R. S. Lakes, “Experimental microelasticity of two porous solids,” Int. J. Solids Struct.,
22, No. 1, 55–63 (1986).
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 4 127
Application of the Micropolar Theory to Strength Analysis ...
16. R. S. Lakes, “Experimental micro mechanics methods for conventional and negative
Poisson’s ratio cellular solids as Cosserat continua,” J. Eng. Mater. Technol., 113,
No. 1, 148–155 (1991).
17. I. Goda, M. Assidi, S. Belouettar, and J. F. Ganghoffer, “A micropolar anisotropic
constitutive model of cancellous bone from discrete homogenization,” J. Mech.
Behav. Biomed. Mater., 16, 87–108 (2012).
18. I. Goda, M. Assidi, and J. F. Ganghoffer, “A 3D elastic micropolar model of vertebral
trabecular bone from lattice homogenization of the bone microstructure,” Biomech.
Model. Mechan., 13, No. 1, 53–83 (2014).
19. I. Goda and J. F. Ganghoffer, “Identification of couple-stress moduli of vertebral
trabecular bone based on the 3D internal architectures,” J. Mech. Behav. Biomed.
Mater., 51, 99–118 (2015).
20. F. dell’Isola, D. Steigmann, and A. Della Corte, “Synthesis of fibrous complex
structures: designing microstructure to deliver targeted macroscale response,” Appl.
Mech. Rev., 67, No. 6, 060804-060804-21 (2016).
21. F. dell’Isola, I. Giorgio, M. Pawlikowski, and N. L. Rizzi, “Large deformations of
planar extensible beams and pantographic lattices: heuristic homogenization,
experimental and numerical examples of equilibrium,” Proc. Roy. Soc. A, 472, Issue
2185 (2016), DOI: 10.1098/rspa.2015.0790.
22. D. Scerrato, I. Giorgio, N. L. Rizzi, “Three-dimensional instabilities of pantographic
sheets with parabolic lattices: numerical investigations,” Z. Angew. Math. Phys., 67,
No. 3, 1–19 (2016).
23. D. Scerrato, I. A. Zhurba Eremeeva, T. Lekszycki, and N. L. Rizzi, “On the effect of
shear stiffness on the plane deformation of linear second gradient pantographic
sheets,” Z. Angew. Math. Mech. (2016), DOI: 10.1002/zamm.201600066.
24. V. A. Eremeyev and W. Pietraszkiewicz, “Material symmetry group of the non-linear
polar-elastic continuum,” Int. J. Solids Struct., 49, No. 14, 1993–2005 (2012).
25. V. A. Eremeyev and W. Pietraszkiewicz, “Material symmetry group and constitutive
equations of micropolar anisotropic elastic solids,” Math. Mech. Solids, 21, No. 2,
210–221 (2016).
26. T. C. Kennedy and J. B. Kim, “Dynamic stress concentrations in micropolar elastic
materials,” Comput. Struct., 45, No. 1, 53–60 (1992).
27. P. Kaloni and T. Ariman, “Stress concentration effects in micropolar elasticity,” Z.
Angew. Math. Phys., 18, No. 1, 136–141 (1967).
28. A. R. Khoei, S. Yadegari, and S. O. R. Biabanaki, “3D finite element modeling of
shear band localization via the micro-polar Cosserat continuum theory,” Comp. Mater.
Sci., 49, No. 4, 720–733 (2010).
Received 10. 08. 2016
128 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 4
V. A. Eremeyev, A. Skrzat, and A. Vinakurava
<<
/ASCII85EncodePages false
/AllowTransparency false
/AutoPositionEPSFiles true
/AutoRotatePages /None
/Binding /Left
/CalGrayProfile (Dot Gain 20%)
/CalRGBProfile (sRGB IEC61966-2.1)
/CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2)
/sRGBProfile (sRGB IEC61966-2.1)
/CannotEmbedFontPolicy /Error
/CompatibilityLevel 1.4
/CompressObjects /Tags
/CompressPages true
/ConvertImagesToIndexed true
/PassThroughJPEGImages true
/CreateJobTicket false
/DefaultRenderingIntent /Default
/DetectBlends true
/DetectCurves 0.0000
/ColorConversionStrategy /CMYK
/DoThumbnails false
/EmbedAllFonts true
/EmbedOpenType false
/ParseICCProfilesInComments true
/EmbedJobOptions true
/DSCReportingLevel 0
/EmitDSCWarnings false
/EndPage -1
/ImageMemory 1048576
/LockDistillerParams false
/MaxSubsetPct 100
/Optimize true
/OPM 1
/ParseDSCComments true
/ParseDSCCommentsForDocInfo true
/PreserveCopyPage true
/PreserveDICMYKValues true
/PreserveEPSInfo true
/PreserveFlatness true
/PreserveHalftoneInfo false
/PreserveOPIComments false
/PreserveOverprintSettings true
/StartPage 1
/SubsetFonts true
/TransferFunctionInfo /Apply
/UCRandBGInfo /Preserve
/UsePrologue false
/ColorSettingsFile ()
/AlwaysEmbed [ true
]
/NeverEmbed [ true
]
/AntiAliasColorImages false
/CropColorImages true
/ColorImageMinResolution 300
/ColorImageMinResolutionPolicy /OK
/DownsampleColorImages true
/ColorImageDownsampleType /Bicubic
/ColorImageResolution 1200
/ColorImageDepth -1
/ColorImageMinDownsampleDepth 1
/ColorImageDownsampleThreshold 1.50000
/EncodeColorImages false
/ColorImageFilter /DCTEncode
/AutoFilterColorImages true
/ColorImageAutoFilterStrategy /JPEG
/ColorACSImageDict <<
/QFactor 0.15
/HSamples [1 1 1 1] /VSamples [1 1 1 1]
>>
/ColorImageDict <<
/QFactor 0.15
/HSamples [1 1 1 1] /VSamples [1 1 1 1]
>>
/JPEG2000ColorACSImageDict <<
/TileWidth 256
/TileHeight 256
/Quality 30
>>
/JPEG2000ColorImageDict <<
/TileWidth 256
/TileHeight 256
/Quality 30
>>
/AntiAliasGrayImages false
/CropGrayImages true
/GrayImageMinResolution 300
/GrayImageMinResolutionPolicy /OK
/DownsampleGrayImages true
/GrayImageDownsampleType /Bicubic
/GrayImageResolution 1200
/GrayImageDepth -1
/GrayImageMinDownsampleDepth 2
/GrayImageDownsampleThreshold 1.50000
/EncodeGrayImages false
/GrayImageFilter /DCTEncode
/AutoFilterGrayImages true
/GrayImageAutoFilterStrategy /JPEG
/GrayACSImageDict <<
/QFactor 0.15
/HSamples [1 1 1 1] /VSamples [1 1 1 1]
>>
/GrayImageDict <<
/QFactor 0.15
/HSamples [1 1 1 1] /VSamples [1 1 1 1]
>>
/JPEG2000GrayACSImageDict <<
/TileWidth 256
/TileHeight 256
/Quality 30
>>
/JPEG2000GrayImageDict <<
/TileWidth 256
/TileHeight 256
/Quality 30
>>
/AntiAliasMonoImages false
/CropMonoImages true
/MonoImageMinResolution 1200
/MonoImageMinResolutionPolicy /OK
/DownsampleMonoImages true
/MonoImageDownsampleType /Bicubic
/MonoImageResolution 1200
/MonoImageDepth -1
/MonoImageDownsampleThreshold 1.50000
/EncodeMonoImages false
/MonoImageFilter /CCITTFaxEncode
/MonoImageDict <<
/K -1
>>
/AllowPSXObjects false
/CheckCompliance [
/None
]
/PDFX1aCheck false
/PDFX3Check false
/PDFXCompliantPDFOnly false
/PDFXNoTrimBoxError true
/PDFXTrimBoxToMediaBoxOffset [
0.00000
0.00000
0.00000
0.00000
]
/PDFXSetBleedBoxToMediaBox true
/PDFXBleedBoxToTrimBoxOffset [
0.00000
0.00000
0.00000
0.00000
]
/PDFXOutputIntentProfile (None)
/PDFXOutputConditionIdentifier ()
/PDFXOutputCondition ()
/PDFXRegistryName ()
/PDFXTrapped /False
/CreateJDFFile false
/Description <<
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
/BGR <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>
/CHS <FEFF4f7f75288fd94e9b8bbe5b9a521b5efa7684002000410064006f006200650020005000440046002065876863900275284e8e9ad88d2891cf76845370524d53705237300260a853ef4ee54f7f75280020004100630072006f0062006100740020548c002000410064006f00620065002000520065006100640065007200200035002e003000204ee553ca66f49ad87248672c676562535f00521b5efa768400200050004400460020658768633002>
/CHT <FEFF4f7f752890194e9b8a2d7f6e5efa7acb7684002000410064006f006200650020005000440046002065874ef69069752865bc9ad854c18cea76845370524d5370523786557406300260a853ef4ee54f7f75280020004100630072006f0062006100740020548c002000410064006f00620065002000520065006100640065007200200035002e003000204ee553ca66f49ad87248672c4f86958b555f5df25efa7acb76840020005000440046002065874ef63002>
/CZE <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>
/DAN <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>
/DEU <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>
/ESP <FEFF005500740069006c0069006300650020006500730074006100200063006f006e0066006900670075007200610063006900f3006e0020007000610072006100200063007200650061007200200064006f00630075006d0065006e0074006f00730020005000440046002000640065002000410064006f0062006500200061006400650063007500610064006f00730020007000610072006100200069006d0070007200650073006900f3006e0020007000720065002d0065006400690074006f007200690061006c00200064006500200061006c00740061002000630061006c0069006400610064002e002000530065002000700075006500640065006e00200061006200720069007200200064006f00630075006d0065006e0074006f00730020005000440046002000630072006500610064006f007300200063006f006e0020004100630072006f006200610074002c002000410064006f00620065002000520065006100640065007200200035002e003000200079002000760065007200730069006f006e0065007300200070006f00730074006500720069006f007200650073002e>
/ETI <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>
/FRA <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>
/GRE <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>
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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.)
/HUN <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>
/ITA <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>
/JPN <FEFF9ad854c18cea306a30d730ea30d730ec30b951fa529b7528002000410064006f0062006500200050004400460020658766f8306e4f5c6210306b4f7f75283057307e305930023053306e8a2d5b9a30674f5c62103055308c305f0020005000440046002030d530a130a430eb306f3001004100630072006f0062006100740020304a30883073002000410064006f00620065002000520065006100640065007200200035002e003000204ee5964d3067958b304f30533068304c3067304d307e305930023053306e8a2d5b9a306b306f30d530a930f330c8306e57cb30818fbc307f304c5fc59808306730593002>
/KOR <FEFFc7740020c124c815c7440020c0acc6a9d558c5ec0020ace0d488c9c80020c2dcd5d80020c778c1c4c5d00020ac00c7a50020c801d569d55c002000410064006f0062006500200050004400460020bb38c11cb97c0020c791c131d569b2c8b2e4002e0020c774b807ac8c0020c791c131b41c00200050004400460020bb38c11cb2940020004100630072006f0062006100740020bc0f002000410064006f00620065002000520065006100640065007200200035002e00300020c774c0c1c5d0c11c0020c5f40020c2180020c788c2b5b2c8b2e4002e>
/LTH <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>
/LVI <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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.)
/NOR <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>
/POL <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>
/PTB <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>
/RUM <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>
/RUS <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>
/SKY <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>
/SLV <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>
/SUO <FEFF004b00e40079007400e40020006e00e40069007400e4002000610073006500740075006b007300690061002c0020006b0075006e0020006c0075006f00740020006c00e400680069006e006e00e4002000760061006100740069007600610061006e0020007000610069006e006100740075006b00730065006e002000760061006c006d0069007300740065006c00750074007900f6006800f6006e00200073006f00700069007600690061002000410064006f0062006500200050004400460020002d0064006f006b0075006d0065006e007400740065006a0061002e0020004c0075006f0064007500740020005000440046002d0064006f006b0075006d0065006e00740069007400200076006f0069006400610061006e0020006100760061007400610020004100630072006f0062006100740069006c006c00610020006a0061002000410064006f00620065002000520065006100640065007200200035002e0030003a006c006c00610020006a006100200075007500640065006d006d0069006c006c0061002e>
/SVE <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>
/TUR <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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.)
>>
/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
|
| id | nasplib_isofts_kiev_ua-123456789-173513 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0556-171X |
| language | English |
| last_indexed | 2025-12-07T15:42:02Z |
| publishDate | 2016 |
| publisher | Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| record_format | dspace |
| spelling | Eremeyev, V.A. Skrzat, A. Vinakurava, A. 2020-12-07T20:32:06Z 2020-12-07T20:32:06Z 2016 Application of the Micropolar Theory to the Strength Analysis of Bioceramic Materials for Bone Reconstruction / V.A. Eremeyev, A. Skrzat, A. Vinakurava // Проблемы прочности. — 2016. — № 4. — С. 119-128. — Бібліогр.: 28 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/173513 539. 4 The application of the linear micropolar theory to the strength analysis of bioceramic materials for bone reconstruction is described. Micropolar elasticity allows better results to be obtained for microstructural and singular domains as compared to the classical theory of elasticity. The fundamental equations of the Cosserat continuum are cited. The description of FEM implementation of micropolar elasticity is given. The results of solving selected 3D test problems are presented. Comparison of classical and micropolar solutions is discussed. The research received funding from the People Program (Marie 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. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Application of the Micropolar Theory to the Strength Analysis of Bioceramic Materials for Bone Reconstruction Использование микрополярной теории при расчете прочности биокерамических материалов для костной пластики Article published earlier |
| spellingShingle | Application of the Micropolar Theory to the Strength Analysis of Bioceramic Materials for Bone Reconstruction Eremeyev, V.A. Skrzat, A. Vinakurava, A. Научно-технический раздел |
| title | Application of the Micropolar Theory to the Strength Analysis of Bioceramic Materials for Bone Reconstruction |
| title_alt | Использование микрополярной теории при расчете прочности биокерамических материалов для костной пластики |
| title_full | Application of the Micropolar Theory to the Strength Analysis of Bioceramic Materials for Bone Reconstruction |
| title_fullStr | Application of the Micropolar Theory to the Strength Analysis of Bioceramic Materials for Bone Reconstruction |
| title_full_unstemmed | Application of the Micropolar Theory to the Strength Analysis of Bioceramic Materials for Bone Reconstruction |
| title_short | Application of the Micropolar Theory to the Strength Analysis of Bioceramic Materials for Bone Reconstruction |
| title_sort | application of the micropolar theory to the strength analysis of bioceramic materials for bone reconstruction |
| topic | Научно-технический раздел |
| topic_facet | Научно-технический раздел |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/173513 |
| work_keys_str_mv | AT eremeyevva applicationofthemicropolartheorytothestrengthanalysisofbioceramicmaterialsforbonereconstruction AT skrzata applicationofthemicropolartheorytothestrengthanalysisofbioceramicmaterialsforbonereconstruction AT vinakuravaa applicationofthemicropolartheorytothestrengthanalysisofbioceramicmaterialsforbonereconstruction AT eremeyevva ispolʹzovaniemikropolârnoiteoriiprirasčetepročnostibiokeramičeskihmaterialovdlâkostnoiplastiki AT skrzata ispolʹzovaniemikropolârnoiteoriiprirasčetepročnostibiokeramičeskihmaterialovdlâkostnoiplastiki AT vinakuravaa ispolʹzovaniemikropolârnoiteoriiprirasčetepročnostibiokeramičeskihmaterialovdlâkostnoiplastiki |