=Paper=
{{Paper
|id=Vol-2488/paper11
|storemode=property
|title=Mathematical Models and Analysis of Deformation Processes in Biomaterials with Fractal Structure
|pdfUrl=https://ceur-ws.org/Vol-2488/paper11.pdf
|volume=Vol-2488
|authors=Yaroslav Sokolovskyy,Maryana Levkovych,Olha Mokrytska,Svitlana Yatsyshyn,Yaroslav Kaspryshyn,Christine Strauss
|dblpUrl=https://dblp.org/rec/conf/iddm/SokolovskyyLMYK19
}}
==Mathematical Models and Analysis of Deformation Processes in Biomaterials with Fractal Structure==
https://ceur-ws.org/Vol-2488/paper11.pdf
Mathematical Models and Analysis of Deformation
Processes in Biomaterials with Fractal Structure
Yaroslav Sokolovskyy1[0000-0003-4866-2575], Maryana Levkovych1[0000-0002-0119-3954]
Olha Mokrytska1[0000-0002-2887-9585] Svitlana Yatsyshyn1[0000-0001-5200-4837]
Yaroslav Kaspryshyn1[0000-0003-1118-9059] and Christine Strauss2[0000-0003-0276-3610]
1
Department of Information Technologies, Ukrainian National Forestry University, UNFU
Lviv, UKRAINE
sokolowskyyyar@yahoo.com, maryana.levkovych@gmail.com,
mokrytska@nltu.edu.ua, svitlana0981@gmail.com
kaspryshyn.ya@gmail.com
2
Department of Electronic Business, University of Vienna
Vienna, AUSTRIA
christine.strauss@univie.ac.at
Abstract. Mathematical models of heat and mass transfer and deformation pro-
cesses of biomaterials are investigated, taking into account such properties as
memory-effect (eriditarity), self-organization, deterministic chaos, heterogenei-
ty of structure, variability of rheological properties. The obtained results of
numerical modelling of non-isothermal moisture transfer and deformation of
biomaterials taking into account fractal structure make it possible to estimate –
based on the type of material and its thermo-mechanical characteristics – the
residual deformation of the material. A mathematical rheological model of two-
dimensional visco-elastic deformation of biomaterials with regard to memory-
effect and self-organization is constructed, which is described using equilibrium
equations with fractional order. The relation between the two-dimensional
stress-deformation state of biomaterials for the rheological models of Maxwell,
Kelvin and Voigt, which are presented in the integral form, was obtained. The
aspects of the algorithm of numerical implementation of two-dimensional
mathematical model of visco-elastic deformation in fractured media are pre-
sented. The method of splitting fractional-differential parameters of models was
adapted, which was used in the problems of identification of non-integer pa-
rameters of models. The results of the identification and numerical implementa-
tion of the mathematical model of heat and mass transfer processes of biophysi-
cal materials are considered, taking into account the fractal structure.
Keywords: eriditarity, biophysical process, non-integer integro-differentiation
apparatus.
Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution
4.0 International (CC BY 4.0)
2019 IDDM Workshops.
2
1 Introduction
The biophysical process is often characterized by the simultaneous influence on the
material of several factors - load, moisture and temperature. Changing at least one of
them in the biomaterial, to which belong transplant, implants, joints, medical silicon,
leads to formation of deformation and its transition from one type to the other, result-
ing in the total or partial restoration of the original physical state. This ability of the
material characterizes the presence of the memory-effect, which is based on residual
deformations. In addition to residual memory, biomaterials are characterized by sto-
chastic heterogeneity of the structure and significant variability of rheological proper-
ties. To investigate the above-mentioned properties in biophysical materials, as well
as deterministic chaos, the complex nature of spatial correlations and self-
organization is possible by formal means of fractional integro-differential operators
[1, 10]. This approach provides the basis for the development of mathematical models
of non-equilibrium biophysical processes with a fractal structure. At present, a very
small number of works [7, 8] is devoted to the development of algorithms and soft-
ware for studying the processes of deformation and heat-moisture transfer, taking into
account the memory-effect-properties and self-organization of materials, which al-
lows us to estimate the residual and elastic stress values.
2 Problem formulation
The mathematical rheological model of two-dimensional visco-elastic deformation of
biomaterials, taking into account eriditarity (memory-effect) and self-organization, is
described using equilibrium equations with a fractional order 0 1 in spa-
tial coordinates x1 and x 2 :
~ 22 ~ 2 12 ~ 2
C11 R11 11 R11 C12 R12 R
12 2C
33 R33 R33 0, (1)
x1 x1 x 2
~ 22 ~ 1 12 ~ 1
C 21 R21 11 R21 C 22 R22 R
22 2C
33 R33 R33 0. (2)
x 2 x 2 x1
~
where Rij , Rij are the corresponding values of the integrals:
T 1,T 2 k 1,2
t t
Rij t z, T ,U dz Rij , Rij t z, T ,U
~
dz Rij ,
0 0 x k
t
T 3 t
T 3
0 R t z , T , U
dz
~2
R , R t z , T , U
~1
dz R33 ,
x 2 x1
ij 33 ij
0
Rij are relaxation kernels of fractional-differential models, which are dependent on
time t , temperature T and moisture U ;
T
11, 22 , 12 is a deformation vec-
3
tor, components of which are dependent on time t and spatial variables x1 and x2 ,
t , x1 , x2 D, D 0,T~ 0, l1 0, l2 , T T 1 , T 2 , T 3 T is deformation
vector, components of which are dependent on temperature variations T and mois-
ture content U :
T 1 11T 11U , T 2 22T 22U , T 3 0,
11, 22 , 11, 22 are coefficients of thermal expansion and moisture-condition
shrinkage; Cij are components of the elastic tensor of an orthotropic body:
E E
, C33 ,
E11 E 22
C11 , C12 2 11 , C 21 1 22 , C 22
1 1 2 1 1 2 1 1 2 1 1 2
where is shear modulus in plane, E11, E22 , Е12 are Young’s moduli, 1 , 2 are
Poison’s ratios.
Let us set the following boundary conditions:
ij x 0 0, ij x l 0, (3)
j j j
and initial conditions, respectively:
ij t 0 0. (4)
The relationship between the components of stress T 11, 22 , 12 and
strain T 11, 22 , 12 for two-dimensional fractional-differential rheological
models, respectively, can be written as follows:
Voigt’s model
11
11 Dt 11 T 1 2 11 Dt 22 T 2
1 1 2 1 1 2 (5)
2
D D ,
t 11 T1
t 22 T2
1 2 22
22 Dt 11 T 1 Dt 22 T 2
1 1 2 1 1 2 (6)
2 D 11 T 1 D 22 T 2 ,
t
t
12 Dt 12 T 3 Dt 12 T 3 , (7)
is elastic modulus of an elastic element of a Voigt’s body, 0 1 ;
Kelvin’s model
4
1
11 Dt 11
11
11 T 1 2 11 22 T 2
1 2 1 1 2 1 1 2 (8)
2
1 2
1 2
Dt 11 T 1 Dt 22 T 2 ,
1 22
22 Dt 22 1 22 11 T 1
1 2 1 1 2 1 1 2 22 T 2 (9)
2
1 2
D Dt 22 T 2 ,
1 2 t 11 T 1
1
12 Dt 12 12 T 3 1 2 D , (10)
1 2 1 2 t 12 T 3
where 1 is elastic modulus of an elastic element of a Voigt’s body, 2 is elastic
modulus of an elastic element, , are fractional derivatives and 0 , 1 ;
Maxwell’s model
11 Dt 11
11
11 T 1 2 11 22 T 2
1 1 2 1 1 2 (11)
2 2
D D ,
t 11 T1
t 22 T2
1 2
22 Dt 22 11 T 1 22 22 T 2
1 1 2 1 1 2 (12)
2 2 Dt 11 T 1 Dt 22 T 2 ,
12 Dt 12 12 T 3 2 Dt 12 T 3 , (13)
where 2 is elastic modulus of an elastic element for Maxwell’s model,,
0 1.
If we put 0, 1 in relations (5) - (7), we get the classical two-dimensional
Voigt’s model in the case of orthotropy. Relations (8) - (13) will describe the classi-
cal Maxwell’s and Kelvin’s models at fractal values 1, 1 .
For the integral representation of relations (5) - (13), we consider the properties of
fractional derivatives [5], the definition of fractional derivative , 0 1 :
t
Dt f t Dt t f d .
1
1 0
(14)
as well as the Laplace transform method [9].
5
Thus, the relations describing the relationship between stress and strain (5) - (13)
can be rewritten after the corresponding transformations in the integral form.
Two-dimensional fractional-differential Voigt’s model:
t
Dt t [ p1 11 T 1 p2 22 T 2 ]d
Cii
1 0
ii
2
t
Dt t [ 11 T 1 22 T 2 ]d ,
1 0
(15)
t t
Dt t 12 T 3 d Dt t 12 T 3 d .
12
1 0 1 0
(16)
Two-dimensional fractional-differential Kelvin’s and Maxwell’s models:
t
ii Сi G t A G t [ p1 11 T 1 2 BDt 11 T 1 ]d
0
t
A G t [ p2 22 T 2 2 BDt 22 T 2 ]d ,
0 (17)
t
12 C3G t A G t 2C33 12 T 3 BDt 12 T 3 d ,
0 (18)
t 1 E , At , Kelvin 1 2
, Kelvin
G t t B 1 2
1
t E , , Maxwell , Maxwell
where , 2 ,
1 2
, Kelvin
1
A
1
, Maxwell
for Maxwell’s and Kelvin’s models –
i 1 p1 C11 , p2 C12 i 1 p1 1, p2 2
; .
i 2 p1 C21 , p2 C22 for Voigt’s model – i 2 p1 1 , p2 1 .
3 A numerical method for the realization of two-dimensional
fractional-differential rheological models
To implement the numerical method, we introduce the space-time grid in the
region D :
6
t ,h ,h {t k , x1( n ) , x2( m) : x1( n ) n 1h1 , x2( m) m 1h2 , t k kt , n 1,..., N ;
1 2
~ (19)
l1 l2 T
h1 ; m 1,..., M ; h2 ; k 0,1,..., K ; }.
N 1 M 1 K
Given the Riemann-Liouville formula [10], the difference approximation of a frac-
tional derivative 0 1 by coordinates x1 , x2 can be written as follows
[4, 5]:
u u n1 u n , h x
i ( n1) xi ( n ) , i 1,2, (20)
2 hi
i
xi x
i(n)
where is a Gamma function.
Given (20), the finite-difference approximation of the system of differential equa-
tions (1) - (2) will take the form:
C11
2 h1
~
R11 11k ( n 1,m ) 11k ( n ,m ) C11R11
C12
2 h1
R12 22
k
k ~
( n 1,m ) 22( n ,m ) C12 R12
(21)
2C33
2 h2
~
R332 12k ( n ,m1) 12k ( n ,m ) 2C33R332 0,
R21 11k ( n ,m1) 11k ( n ,m ) C21R21
C21 ~
2 h2
R22 22( n ,m 1) 22( n ,m ) C 22 R22
C22 ~
k k
(22)
2 h2
2C33
2 h1
1
R33 12k ( n1,m ) 12k ( n,m ) 2C33R33
~1
0.
The boundary and initial conditions (3) - (4) in the finite-difference form are writ-
ten, respectively:
11, 22,12k 0, 11, 22,12 N ,m 0, 11, 22,12n,1 0, 11, 22,12n,M 0,
k k k
(23)
1, m
11, 22,120n,m 0. (24)
7
4 Splitting two-dimensional fractional-differential kernels
Let a force act on a wooden specimen along the axis OX. Then from the components
of the stress tensor 11 0 , and 22 0 .
Since it is known that the average stress is specified by the formula:
~
1
11 22 , (25)
3
then in this case ~ 1 11 .
3
The stress tensor deviator then takes the form:
2 0
Sij t ij ~ ij 3 11 , (26)
0 1 11
3
1, i j;
where ij is the Kronecker symbol.
0, i j.
The strain tensor deviator will take the form:
2 11 22 0
eij t ij e~ ij 3 , (27)
0 1
3 22 11
11 22 .
~ 1 1
where e
3 3
The displacement equation [6] takes the form:
S ij t t
eij t зс t S ij d ,
1
2 2 0
(28)
where зс t is shear creep kernel, is shear modulus.
The equation of volumetric deformation (strain) is written accordingly [11]:
~t 1 t
t t ~ d ,
об (25)
B B 0
8
where об t is volumetric creep kernel, B is volumetric modulus of elasticity
associated with the longitudinal modulus of elasticity 11 and elastic Poisson's ratio
0 by the formula B 11 .
31 2 0
Based on the creep data of stretched or compressed specimens, using the measured
values of longitudinal 11 t and transverse 22 t deformations, we can construct
functions of longitudinal 11 t and transverse 22 t creep.
We write the equations of the processes of longitudinal and transverse deformation
of a wooden specimen that is stretched by stress 11 t [11]:
1
t
11t
11 t 11t 11 d , (30)
11 0
0 t
22 t
11 t 12 t 11 d , (31)
11 0
where 0 0 is the value of the elastic Poisson ratio.
For the component e11 from (27) we have:
t
e11 t 11 22 2 1 11 t 11 t 11 d
2
3 3 11 0
2 0
t
11 t 12 t 11 d (32)
3 11 0
21 0 t
11 t 11 t 0 12 t 11 d .
2
311 0
311
From (26) it is known that S11 2
3
11 . Then 11 3 2 S11 and we get:
1 0 11t 0 12 t
t
e11t S11t S11 d
11 0
1 0 (33)
1 0
t
S11t зс t S11 d ,
11 0
where t 11t 012 t .
1 0
зс
The creep equation in the case of stretching will take the form:
9
1 2 0
t t
11 t t d
11 t 12 t 11 d . (34)
11 11
11 11
0 0
~ , then
Since 11 3
1 2 0 ~ 1 2
t
3 t 11 t 0 12 t 3~ t d
11
0 11
11
31 2 0 ~ t 2 0 12 t ~
t
(35)
t 11 t d
11 1 2 0
0
31 2 0 ~
t
t об t ~ t d ,
11
0
where об
11 t 2 0 12 t
.
1 2 0
For fractional-differential models of the shear and volumetric creep kernel, it can
be written for each model as follows:
for Voigt’s model
зс F t
t
1
( E ,
11
t
2 1 0
2 1 1 2
(36)
11 2
0 E , t ),
2 1 1 2
об F t
t 1 ( E 11
t
,
2 1 2 0
2 1 1 2
(37)
11 2
2 0 E , t ),
2 1 1 2
for Kelvin’s model
зс K t
1 2 t 1 ( E 111 2 t
,
21 2 1 0 21 2 1 1 2
(38)
0 E , 2 11 1 2 t ),
21 2 1 1 2
10
об K t
1 2 t 1 ( E 11 1 2 t
,
21 2 1 2 0 21 2 1 1 2
(39)
2 11 1 2
2 0 E , t ),
21 2 1 1 2
5 Numerical implementation of a mathematical model
Considering the previous studies [2, 3] regarding the identification of fractional-
differential parameters of models, we present the identification results for the rheolog-
ical Maxwell model (see Fig. 1).
Fig. 1. Identification of fractional-differential parameters of the Maxwell model.
In Fig. 2, the deformation change for a sample of biomaterial (modulus of elasticity
E 16,1GPa ) [12] was investigated using a Kelvin rheological model taking into
account the fractal structure of the medium without taking it into account. Such stud-
ies have shown that by decreasing the fractional-differential parameter , the defor-
mation functions increase more slowly, and at a value 0,1 , the deformation curve
acquires the form in which the deformation of the material is smallest. Thus, it is pos-
sible to trace the relationship between the fractal parameters of the model and the
process of deformation change.
11
Fig. 2. Change of the deformations of the fractional-differential Kelvin’s model.
Our results show that the difference between the stress curves with the fractal
structure and without taking into account for more solid types of biomaterials does not
exceed 16.7%, whereas the difference between the stress curves for materials with a
lower density lies between 19.6 and 24.0%.
6 Conclusions
Two-dimensional mathematical models of deformation processes of biomaterials have
been constructed, which make it possible to take into account the fractal struc-ture of
a material depending on the initial values of temperature and moisture content,
thermo-mechanical characteristics of anisotropy, different types of material. An algo-
rithm for numerical implementation of two-dimensional mathematical models of vis-
co-elastic deformation of biomaterials has been developed, which allows calculating
the components of the stress-strain state of a material taking into account the effects
of memory and self-organization.
Adaptation has been carried out of the method of splitting fractional-differential
creep kernels, which makes it possible to determine the functions of volumetric and
shear creep according to the experimental data of one-dimensional models of visco-
elastic deformation, to identify fractional-differential parameters of models taking
12
into account the fractal structure of the medium and to estimate the values of elastic
and residual stresses of biomaterials. Presented are the results of the numerical im-
plementation of the mathematical model, taking into account the heterogeneity of the
structure of biomaterials, self-organization and memory-effect.
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