=Paper=
{{Paper
|id=Vol-3038/paper21
|storemode=property
|title=Modeling of Heat Transfer and Deformation Processes in Biomaterials with Fractal Structure
|pdfUrl=https://ceur-ws.org/Vol-3038/short8.pdf
|volume=Vol-3038
|authors=Volodymyr Shymanskyi,Yaroslav Sokolovskyy,Olha Mokrytska,Iryna Boretska,Oleksiy Sinkevych,Volodymyr Kryshtapovych
|dblpUrl=https://dblp.org/rec/conf/iddm/ShymanskyiSMBSK21
}}
==Modeling of Heat Transfer and Deformation Processes in Biomaterials with Fractal Structure==
https://ceur-ws.org/Vol-3038/short8.pdf
Modeling of Heat Transfer and Deformation Processes in
Biomaterials with Fractal Structure
Volodymyr Shymanskyia, Yaroslav Sokolovskyyb, Olha Mokrytskaс, Iryna Boretskaс,
Oleksiy Sinkevychс and Volodymyr Kryshtapovychd
a
Department of Artificial Intelligence, Lviv Polytechnic National University, Lviv, Ukraine
b
Department of Computer-Aided Design Systems, Lviv Polytechnic National University, Lviv, Ukraine
c
Department of Information Technologies, National Forestry University of Ukraine, Lviv, Ukraine
d
Department of Automation and Computer-Integrated Technologies, National Forestry University of Ukraine,
Lviv, Ukraine
Abstract
The theoretical foundations of the study of deformation-relaxation and heat exchange
processes in biomaterials with a fractal structure based on the synthesis of the basic laws of
thermodynamics of nonequilibrium processes and mechanics of a continuous medium were
further developed. A new physical and mathematical model of interconnected deformable
and heat exchange processes in biomaterial with taking into account its fractal structure
depending on viscoelastic characteristics was synthesized. To take into account the fractal
structure of the material the mathematical apparatus of integro-differentiation of fractional
order was used. For the partial case, the numerical values of the simulated physical
characteristics were obtained. The obtained results were analysed and conclusions were made
about the distribution and influence of interconnected heat exchange and deformation
relaxation processes in biomaterials with taking into account their fractal structure.
Keywords 1
Fractal structure, biomaterial, deformation-relaxation processes, heat transfer, total energy
1. Introduction
In general uneven fuel expansion cannot occur freely in a solid body and causes thermal (thermal,
temperature) stresses. Knowledge of the magnitude and nature of the thermal stresses effect is
necessary for a comprehensive analysis of the structural strength. The action of mechanical stresses
from external forces and in combination with thermal loads can cause cracks and destruction of
structures that have been made of materials with a complex structure [1-5].
With the rapid appearance of stresses due to the action of a sharp gradient of an unsteady
temperature field some materials become brittle and cannot withstand thermal shock. Repeated
application of thermal loads leads to an overload of structural elements [6].
In the general case, a change in body temperature occurs not only due to the supply of heat from
external sources but also due to the deformation process itself. Thermomechanical effects of a
different kind are important in deformations of a body that are flowing at a finite speed: the formation
and movement of heat flow inside the body, the appearance of bound elastic and heatwaves in it, and
thermoelastic energy dissipation [7, 8].
Most real physical processes cannot be described using the basic principles and concepts of the
mechanics of a continuous medium. This requires the involvement of non-traditional approaches, in
IDDM’2021: 4rd International Conference on Informatics & Data-Driven Medicine, November 19–21, 2021, Valencia, Spain
EMAIL: vshymanskiy@gmail.com (V. Shymanskyi); sokolowskyyyar@yahoo.com (Ya. Sokolovskyy); mokrytska@nltu.edu.ua
(O. Mokrytska); iryna.boretska@gmail.com (I. Boretska); oleksiy1694@gmail.com (O. Sinkevych); kvi9249926@gmail.com
(V. Kryshtapovych)
ORCID: 0000-0002-7100-3263 (V. Shymanskyi); 0000-0003-4866-2575 (Ya. Sokolovskyy); 0000-0002-2887-9585 (O. Mokrytska);
0000-0002-6767-104X (I. Boretska); 0000-0001-6651-5494 (O. Sinkevych); 0000-0002-0542-9178 (V. Kryshtapovych)
©️ 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
particular, the fractality of the environment in which these processes take place. Superslow diffusion-
type processes, deformation-relaxation processes in the middle with obvious memory effects can be
brought to such processes. To describe them in the proper way the accordingly modified laws are
used, which require the implementation of the fractional integro-differential mathematical apparatus
[9-11].
Fractional mathematical analysis has a long history, but despite this, it is a rapidly developing area
of modern analysis and has extremely rich content. At present, there is not a single area of classical
analysis that has not been touched by fractional analysis, which is closely related to various issues of
the theory of functions, integral and differential equations, mechanics, equations of mathematical
physics, etc. The mathematical language of operators of fractional integro-differentiation is
indispensable for describing and researching physical fractal systems, stochastic transfer processes,
the study of mechanics, biology, probability theory and other applied sciences [12-15].
It is necessary to take into account the internal structure and properties of materials during
constructing mechanical and mathematical models. It is very important to build mathematical models
that describe the stress-strain state and thermoelasticity in media that “do not fit into the traditional
approach” of known and developed standard mathematical models of specific branches of mechanics
of continuous media. So, for example, the use of classical models of elastic bodies deformation in
bio/nanomechanics without taking into account the complicated structure of biomaterials does not
allow to obtain reliable results and sharply narrows the scope of the problems being solved [16-21].
At the same time to obtain the results of higher accuracy by improving and complicating the
algorithms for the numerical implementation of the tasks is almost impossible.
Thus, to obtain adequate results for modeling the behavior of real physical objects, it is necessary
to change the approach when building a mathematical model. Construction of basic equations of
mechanics using the apparatus of fractional calculus to describe the behaviour of real heterogeneous
materials is one of the promising areas for further improvement and development of mechanical and
mathematical models [22-24, 9, 12].
Thus, the study of the thermomechanical characteristics of porous biomaterials, which are
characterized by the complex nature of the spatial structure, existing memory effect, self-organization
and deterministic chaos properties, is an urgent scientific task.
2. Production of a problem
Many definitions of derivatives and fractional integrals are known today, in particular: Caputo,
Marshaud, Weil and others. All these operators of fractional integro-differentiation have a rather
cumbersome construction and can take various forms is the actions on which do not always coincide.
On the other side, it is impossible to firmly assert which of the approaches is advisable to study and
which is not since there is no unambiguous definition of derivatives of non-integer order. The
mathematical apparatus of integro-differentiation of fractional order was used in the construction of
mathematical models of the studied physical processes occurring in biomaterials with a fractal
structure. In particular, the integral and the derivative of the function f ( x, y, z ) by the variable x in
the Caputo’s sense can be written as follows [10, 12-15, 22]:
1 f ( , y, z ) d
x
1
Dx f
(1 ) a 1
( x )
, (1)
1 f ( , y, z ) d
x
1
Ix f , (2)
( ) a 1
( x )
where , N , 0 1, ( ) x e dx - gamma function.
1 x
0
2.1. Determination of internal and free energy of deformation and
heat exchange processes
The purpose of this section was to build theoretical foundations and develop methods for the
synthesis of deformation-relaxation processes in biomaterials with a fractal structure during their heat
treatment. Biomaterial, in particular bone, is considered to be an open thermodynamic system with a
variable volume [1, 2, 6].
Let's select an infinitesimal element of material in the Cartesian coordinate system. Its state is
characterized by a system of independent variables S , ij ,U ij , or equivalent system T , ij ,U ij , where
S is the entropy of a unit of mass; T - absolute temperature; ij - components of the strain tensor;
- density of the material; - time; J J ijk - tensor of the third rank which characterizes the
corresponding thermodynamic flow and satisfies the balance equation:
divJ 0 (3)
The relationship between deformations and displacements that take into account the fractal
structure of the biomaterial is recorded using modified Cauchy relations:
ij
1
ui, j u j ,i (4)
2
ui , j Dxj ui (5)
The thermodynamic potential of the described system in case of temperature change is free energy.
The full differential of free energy can be written in the form [25]:
dF dF0 dF1 , (6)
dF 0 SdT , (7)
dF1 ij d ij ij dij , (8)
where, ij - components of the stress tensor; ij - components of the damage tensor; ij -
corresponding potential. Thus, the full differential of free energy can be written as [25]:
dU TdS ij d ij lm dlm (9)
For small deviations of the system from the equilibrium state, the free energy, expressed in terms
of temperature, stress tensors, deformations and damages, is a complete differential and can be
represented as:
F0 1 2 F0
F T , ij , lm F0 ij ij lm (10)
ij 2 ij lm
At the initial moment, the equilibrium state is characterized by the initial conditions:
S S 0 , T T0 , ij 0, ij 0. (11)
For the free energy function, the Taylor series expansion according to the degrees of deformation
invariants is valid. Restricted to members not higher than the second order, we obtain:
F0 F 1 2 F0 2
F F0 T , r0 I1 0 I 2 I1 (12)
I1 I 2 2 I1 2
I1 ii ; I 2 ij ij (13)
Given the generalization of Hooke's law for materials with a fractal structure in the presence of
viscoelastic deformations:
dT dV
F cv ij ij 1 ij2 ij 2 3 (14)
dT0 2 3 V0
2 dV
ij 2 1 ij ij 3 2 P ij (15)
3 V0
dV
S cv
dT
ij 2 3 P ij ij 1 ij2 (16)
dT0 T V0 T T 2 3
where - is determined based on measuring the pressure and the correspondence of the change in
specific volume in the region of elastic deformations at a constant temperature.
2.2. Synthesis of a model of deformation and heat exchange
processes in a biomaterial with a fractal structure
To obtain a system of equations that describe the interrelated deformation-relaxation and heat
exchange processes in biomaterials with a fractal structure it is necessary to obtain the equation of
energy and motion balance. According to the first law of thermodynamics the total energy U * of a
volume V limited by a surface is determined by the ratio [4, 5]:
1 2
U * UdV V dV (17)
V
2V
where V - velocity, U - the density of internal energy per unit volume of deformed biomaterial.
The work of external forces over some time with taking into account the available memory effect
can be described [25]:
A I FV dV n Vn d (18)
V
n i cosnxi (19)
where n is the vector of normal stresses to the surface; n - normal to the surface .
The total flow of thermal energy over some time due to the action of an internal source q and heat
input is determined by the ratio [25]:
Q* I qdV S n d (20)
V
Sn Si cosnxi (21)
In the case of internal heat sources, we obtain:
U * I FV dV S n n Vn d (22)
V
When considering an infinitesimal period, ie 0 , we can write the following relationship:
V
2
D U
FV dV Sn nVn d 0 (23)
V 2
Applying Green's formula and taking into account the arbitrariness of the volume we get:
DU J i ,i i Vi F i V (24)
Given that the components of the strain tensor and strain rates are interrelated, the expression for
the increase in internal energy per unit mass and unit time depending on the temperature and strain
fields can be written:
DU J i ,i ij D eij (25)
Using (9) the equation of entropy balance will be written in the form
TD S DU ij D eij ij D eij (26)
Given (25) the equation of state will be written:
TD S J i ,i ij J kij , k (27)
Using the specific heat of the material of the relationship (26) can be written as:
dV (28)
cv D T TD ii 2 3 P J i ,i ij J kij ,k
T V0
From relation (27) we obtain the definition of thermodynamic forces of energy transfer:
TD S J i J i X i J ijk X jki (29)
T
X i i ,i (30)
T
Thus, the equations combining thermodynamic flows will look like this:
J ij Tij , ji (31)
Ti ,i
J i LTT (32)
T
Taking into account the above relations, the physical-mathematical model of interconnected heat
exchange and deformation processes in biomaterials with a fractal structure during their heat
treatment is synthesized.
ij ,i D2 ui F (33)
Tij , ji D T TD kk G T , P (34)
cv T
1
D P aP 2 P Tij , ji PD kk GT , P (35)
C P
dV
G T , P 3 2 3P (36)
V0
To relations (33) - (36) are also added (15), (16), kinematic equations (31) - (32) and geometric
(4). Thus, these equations form a physic-mathematical model for calculating the interconnected heat
transfer and deformation processes in biomaterials with a fractal structure from the time of their heat
treatment.
3. Obtained results
The biomaterials tend to expand during the heat treatment process. This fact encourages the
appearance of stresses in the material. Let us consider a numerical experiment for the obtained model,
which is designed to investigate the effect of heat load and the degree of fractality of the material on
the maximum modulus of stress values in it [26, 27].
A numerical experiment [28] is given for bone material with the initial value of temperature
T0 36.6 0C . The bone material was taken as a sample with the following geometric dimensions
x 0; l , where l 0.15 - the length of the sample. The interaction of the studied biomaterial with
external thermal loads is given by the boundary conditions, in particular, the impermeability condition
is set at the left boundary x 0 , and the interaction with the medium that has a temperature
t c 60 0C is set at the right boundary x l [29, 30, 11]. The hypothesis of a natural stress state, the
absence of stresses, deformations and displacements in the material at the initial time is accepted
( 11 0 0; 11 0 0; u 0 0 ). Also, the material is not subject to any mechanical stress [31,
32].
Fig. 1 shows the dynamics of temperature change in bone material during 2 hours exposed to heat
flux, the nature of which is described above. The curve corresponding to the parameter 1 shows
the dynamics of temperature change calculated using the traditional approach. The curves
corresponding to the parameters 0.95 and 0.9 represent the dynamics of temperature
change, calculated using the apparatus of integro-differentiation of fractional order, i.e. take into
account the fractal structure of the medium. Fig. 2 shows the dynamics of normal stress 11 over 2
hours, which occurs in the material due to the action of heat fluxes.
Figure 1: Changing of the temperature at the right boundary x l on the sample depending on
time and fractional degree of material
Figure 2: Changing of the stress component 11 at the right boundary x l on the sample
depending on time and fractional degree of material
Analysing the behaviour of the curves in Fig. 1 and Fig. 2 can be concluded that the rate of its
heating increases with the increasing degree of fractality of the material. In particular, the maximum
difference between the temperatures of the material with taking into account the fractal structure
0.95 and without 1 is equal 0.78 оС , and for the material with the degree of fractality
0.9 - is 2.27 оС .
The temperature changing in the material leads to stresses in it. The higher in the absolute value
stresses appear when the faster temperature changes. In particular, the maximum difference between
the components of the normal stress 11 of the material with taking into account the fractal structure
0.95 and without 1 is equal 0.21KPa , and for a material with a degree of fractality
0.9 - is 1.07 KPa . It can also be noted that with increasing degree of fractality of the material
increases the value of residual stresses that have a significant impact on the development of stresses in
the material when applying to it repeated thermal or mechanical loads.
4. Conclusions
The use of the mathematical apparatus of integro-differentiation of fractional order allows creating
a new basis for the further development of the theoretical foundations of the study of deformation-
relaxation and heat exchange processes in biomaterials with a fractal structure. The synthesized
physical-mathematical model of interconnected deformable and heat-exchange processes allows to
describe the rheological behaviour of biomaterial in the process during its heat treatment with taking
into account available memory effects, self-organization and deterministic chaos depending on
viscoelastic characteristics.
After analysing the obtained numerical results for partial cases, we can conclude that in heat
treatment processes biomaterials with a higher degree of fractality heat up faster, which leads to the
presence of higher absolute values of stresses. This fact leads to the accumulation of residual stresses,
which significantly affect the development of stresses in the material when applying to it repeated
thermal or mechanical loads.
5. References
[1] Attar M.M., Haghpanahi M., Shahverdi H., et al. Thermo-mechanical analysis of soft tissue in
local hyperthermia treatment, J. Mech. Sci. Technol. 30 (3), 2016, pp. 1459–1469.
[2] Behnke R., Kaliske M., Kluppel M. Thermo-mechanical analysis of cyclically loaded particle-
reinforced elastomer components: experiment and finite element simulation. Rubber Chem.
Technol. 89, 2016, pp. 154–176.
[3] Garcia-Gonzalez D., Zaera R., Arias A. A hyperelastic-thermoviscoplastic constitutive model for
semi-crystalline polymers: application to PEEK under dynamic loading conditions. Int. J. Plast.
88, 2017, pp. 27–52.
[4] Gudimetla M.R., Doghri I. A finite strain thermodynamically-based constitutive framework
coupling viscoelasticity and viscoplasticity with application to glassy polymers. Int. J. Plast. 98,
2017, pp. 197–216.
[5] Hendy M.H., Amin M.M., Ezzat M.A. Two-dimensional problem for thermoviscoelastic
materials with fractional order heat transfer, J. Therm. Stresses (3), 2019, pp. 1–18.
[6] Johlitz M., Dippel B., Lion A. Dissipative heating of elastomers: a new modelling approach
based on finite and coupled thermomechanics. Continuum Mech. Thermodyn. 28, 2016,
pp. 1111–1125.
[7] Majeed A ., Zeeshan A ., Alamri S.Z., et al. Heat transfer analysis in ferromagnetic viscoelastic
fluid flow over a stretching sheet with suction, Neural Comput. Appl. 30 (6), 2018, pp. 1947–
1955.
[8] Yu C., Kang G., Chen K. A hygro-thermo-mechanical coupled cyclic constitutive model for
polymers with considering glass transition. Int. J. Plast. 89, 2017a, pp. 29–65.
[9] Ionescu, C., Lopes, A., Copot, D., Machado, J.T., Bates, J.: The role of fractional calculus in
modeling biological phenomena: A review. Communications in Nonlinear Science and
Numerical Simulation 51, pp. 141-159 (2017). DOI 10.1016/j.cnsns.2017.04.001.
[10] Karniadakis G.E. (ed.): Handbook of Fractional Calculus with Applications: Numerical Methods,
De Gruyter Reference, vol. 3. De Gruyter, Berlin (2019). DOI 10.1515/9783110571684.
[11] Povstenko Y. Time-fractional radial heat conduction in a cylinder and associated thermal
stresses. Arch. Appl. Mech. 82(3), 2012, pp. 345–362.
[12] Kilbas A., Srivastava H. M., Trujillo J. J. Theory and Applications of Fractional Differential
Equations. Elsevier, 2006.
[13] Kochubei A.N. General fractional calculus. In: A. Kochubei, Yu. Luchko (Eds.), Handbook of
Fractional Calculus with Applications. Volume 1: Basic Theory, De Gruyter: Berlin, 2019,
pp. 111–126.
[14] Machado, J., Kiryakova, V.: The chronicles of fractional calculus. Fractional Calculus and
Applied Analysis 20(2), 307{336 (2017). DOI 10.1515/fca-2017-0017.
[15] Podlubny I. Fractional Differential Equations. Academic Press, SanDiego, 1999.
[16] Ayyildiz M., Cinoglu S., Basdogan C. Effect of normal compression on the shear modulus of soft
tissue in rheological measurements, J. Mech. Behav. Biomed. Mater. 49, 2015, pp. 235–243.
[17] Capilnasiu A., Bilston L., Sinkus R., et al. Nonlinear viscoelastic constitutive model for bovine
liver tissue, Biomech. Model. Mechanobiol. 2, 2020, pp. 1–22.
[18] Dadkhah E., Shiri B., Ghaffarzadeh H., Baleanu D. Viscoelastic dampers in structural buildings
and numerical solution with spline collocation methods, J. Appl. Math. Comput. 63, 2020,
pp. 29–57.
[19] Djordjevic V.D., Jaric J., Fabry B., Fredberg J.J., Stamenovic D. Fractional derivatives embody
essential features of cell rheological behaviour, Ann. Biome. Eng. 31, 2003, pp. 692–699.
[20] Khiem V.N., Itskov M. An averaging based tube model for deformation induced anisotropic
stress softening of filled elastomers. Int. J. Plast. 90, 2017, pp. 96–115.
[21] Wu F., Liu J.F., Wang, An improved Maxwell creep model for rock based on variable-order
fractional derivatives, Environ. Earth Sci. 73, 2015, pp. 6965–6971.
[22] Luchko Yu. Operational Calculus for the general fractional derivatives with the Sonine kernels.
Fract. Calc. Appl. Anal. 2021, 24, pp. 338–375.
[23] Nadzharyan T.A., Sorokin V.V., Stepanov G.V., Bogolyubov A.N., Kramarenko E.Yu., A
fractional calculus approach to modeling rheological behavior of soft magnetic elastomers,
Polymer 92, 2016, pp. 179–188.
[24] Xu X., Xu D. Legendre wavelets method for approximate solution of fractional-order differential
equations under multi-point boundary conditions, Int. J. Comput. Math. 95, 2018, pp. 998–1014.
[25] Sokolovskii Ya.I. Relationship between deformation-relaxation and heat-and-mass-transfer
processes in the drying of capillary-porous bodies. International Applied Mechanics, 1998, 34(7),
pp. 694–699.
[26] Sokolovskyy Y., Levkovych M., Sokolovskyy I. The study of heat transfer and stress-strain state
of a material, taking into account its fractal structure. Mathematical Modeling and Computing,
vol. 7(2), 2020, pp. 400-409.
[27] Sokolovskyy Ya., Nechepurenko A., Sokolovskyy I., Mokrytska O. Software and algorithmic
support for finite-element analysis of anisotropic heat-and-mass transfer using parallel and cloud
technologies. Communications in Computer and Information Science (CCIS), Springer
Computer Science, 2020, pp. 143-159.
[28] Oguza C., Sezer M. Chelyshkov collocation method for a class of mixed functional
integrodifferential equations. Appl. Math. comput. 259, 2015, pp. 943-954.
[29] Li X., Wu B. Approximate analytical solutions of nonlocal fractional boundary value problems,
Appl. Math. Model. 39, 2015, pp. 1717–1724.
[30] Plagge J., Kluppel M. A physically based model of stress softening and hysteresis of filled rubber
including rate-and temperature dependency. Int. J. Plast. 89, 2017, pp. 173–196.
[31] Shymanskyi V., Sokolovskyy Y. Variational Formulation of Viscoelastic Problem in
Biomaterials with Fractal Structure. 3rd International Conference on Informatics & Data-Driven
Medicine, IDDM-2020, Växjö, Sweden, November 19 - 21, 2020, pp. 360–369.
[32] Shymanskyi V., Sokolovskyy Ya. Variational Formulation Of The Stress-Strain Problem In
Capillary-Porous Materials With Fractal Structure. XV International Scientific and Technical
Conference on Computer Science and Information Technologies- CSIT2020, Vol.1, 23-26
September, 2020, pp. 200-203.