The paper deals with an investigation of the transient process of heat generation and stabilization of
temperature balance inside pneumatic tires during stationary rolling. The heat generation value have been
obtained via integration of deformation over the cycles for each point of model. The function of heat generation
rate have been created in each point based on previously defined cycles of deformation and experimentally
obtained of loss modulus for materials. The temperature distribution of tire has been found within process of
thermal stabilization

Larin, Oleksiy

Petrova, Iuliia

Library of National Technical University "Kharkiv Polytechnic Institute"

Proceedings of the 5th International Conference on Nonlinear Dynamics
ND-KhPI2016
September 27-30, 2016, Kharkov, Ukraine
A  S i m u l a t i o n  o f  t h e  T r a n s i e n t  S e l f - h e a t i n g  P r o c e s s  
i n  D y n a m i c a l l y  L o a d e d  R u b b e r - b a s e d  M u l t i l a y e r s  
R e i n f o r c e d  C o m p o s i t e s
Petrova Iul iia1*, Larin Oleksiy1
Abstract
The paper deals with an investigation o f the transient process o f heat generation and stabilization of 
temperature balance inside pneumatic tires during stationary rolling. The heat generation value have been 
obtained via integration of deformation over the cycles for each point o f model. The function o f heat generation 
rate have been created in each point based on previously defined cycles o f deformation and experimentally 
obtained o f loss modulus for materials. The temperature distribution o f tire has been found within process of 
thermal stabilization.
Keywords
Heat generation, pneumatic tire, cycles of deformation, transient thermal analysis,
1National Technical University “Kharkiv Polytechnic Institute, Department of Dynamics and Strength of 
Machines, Kharkiv, Ukraine
* Corresponding author: usyanushka@gmail.com
Introduction
Elastomers are widely used in modern mechanical engineering, especially in transport. The 
most widespread are rubber elements and reinforced rubber composites. Analysis of characteristics of 
these elements is an important part in determination of operational characteristics of whole machine 
or mechanism. Typical operational conditions of the designs with elastomers and their composites are 
come with the dynamic load. It associates with two phenomena: fatigue damage accumulation and 
heat generation. The latter requires special on a attention as it can leads to self-heating of the design. 
Higher temperature in its turn can significantly change properties of material (elastic and strength 
characteristics). Moreover, it can increase the rate of ageing processes.
This research deals with investigation of heat generation in automobile pneumatic tire during 
its deformation at the stationary rolling. From practice it is known that the rolling process of 
pneumatic tires accompanied with the self-heating of its internal elements while at some points the 
temperature can rise to above 80oC and over. In addition to previously outlined the changing of 
thermal conditions in the tire leads to a changing of internal pressure and creates an additional thermal 
stress state.
Heating process of automobile tire is the result of several processes. The first one is the 
conversion of the mechanical energy of cyclic deformation of rubber-like materials into heat. The 
second is due to internal friction between the different components of complicated composite 
structures and last is external friction of tire with a road surface.
It should be noted that due to the presence of complicated strain state, curvilinear geometry and 
multilayer structure the temperature field is considerably non-uniform, that leads to changing in the 
stress-strain state of tire and can leads to formation and grow of internal defects. Thus, an actual task 
is to study the processes of formation of temperature distribution in the tire during its operation.
370
Petrova Iuliia, Larin Oleksiy.
1. The problem statement
This paper deals with theoretical modelling of the transient thermal analysis of the tire up to a 
temperature balance during it stationary rolling. The problem was solved in three stages (Fig. 1).
The first stage is solving of static problem of tire interaction with the road surface. Besides the 
tire was loaded by internal pressure of 0.25 MPa and a vertical load of the weight of the vehicle. The 
analysis of results obtained in this stage allows to find the strain cycles of one revolution in each of 
the layers.
The second phase deals with determination of the heat generation rate in each point. To achieve 
this goal the amount of energy released per cycle of deformation was determined. In addition, loss 
modules were determined previously by experiment for all materials of tire.
The third stage consists of solving of transient convective heat transfer problem. The function 
of heat generation rate was applied as the load during the calculations. This function was determined 
at the step 2 .
Figure 1. The mechanism of solving the problem of convective heat transfer
2. Finite-element modelling
In this paper the pneumatic tire 205/55R16 model is considered. During the simulations the 
main design features (Fig. 2) and material properties were taken into account (see. Table. 1).
, r ■ w H l"\ pr— tf —i
h^older
T - —^  (zone 1) r
A \  Carcass doubling 
(zone 2)
\ \  Bead
J (zone 3) /
Figure 2. Structure of pneumatic tire
371
Petrova Iuliia, Larin Oleksiy.
The main calculations have been carried out on the base of FE-modeling.
The problem is solved in three stages, the first and last of which require specific models. The 
first FE model was designed to solve the problem for determination cycles of deformation. The 8- 
nodal finite element with 3 degrees of freedom in the node was used. The second FE model is needed 
to solve the problem of convective heat transfer. The 20-nodes finite element with 1 degree of 
freedom in the node was used to create this model.
Table 1. Materials properties
Elasticity
modulus Poisson ratio Density
Heat capacity Thermal
conductivity
E, MPa u p, kg/m3 c, J/kgC0 k, W /mC0
Tread 3.00 0.49 1200 1020 0.273
Sidewall 3.74 0.49 1200 1020 0.273
Bead chafer 7.00 0.49 1200 1020 0.273
Filler 16.00 0.49 1200 1020 0.273
Rubber matrix 6 0,49 1200 1020 0.273
Carcass (cord) 77120 0,3 1020 0.273
Belt (cord) 74730 0,3 1020 0.273
FE meshes of both models had the same position in the space.
Figure 3. FE- model, a -  full FE-model of tire, b -  segment of tire
In the first stage of the analysis were carried out in the framework of multiscale approach of FE 
modeling. Its technology is described in [1]. Physical nonlinearity, which occurs due to the presence 
of large deflections and strains is accounted by using the model of Neo-Hooke [2] (which accurately 
approximates the behavior of rubber materials with about 20% strains deformation). Rubber-cord 
plies of carcass and belt have been modeled as orthotropic with averaged characteristics for each 
direction according to the generalized Hooke's law [2]. The curvilinear orthotropy of mechanical 
properties of a rubber-cord layers was realized by introduction finite-toroidal local coordinate systems 
in which axes directions repeat geometry of corresponding layer [2]
The strain cycles per revolution were defined in all layers of the tire. These calculations 
showed that the largest stresses were observed along the lines with the circumferential coordinate 50 
and 307 degrees. The zones correspond to the change from the contact region to the contactless zone. 
The largest stresses were observed in the of belt layer. Cycles of stress state (the ratio of current stress 
to ultimate tensile strength) is shown on the Fig. 4 in polar coordinates (0 ° corresponds to the tire 
contact area with the road surface).
372
Petrova Iuliia, Larin Oleksiy.
- Zciia 1 
— Zo ne 2
Degrees
b
Figure 4. Strain cycles of specific zones 
a - in the Cartesian coordinate system, b -  in polar coordinates 
zone 1 -  tire shoulder, zone 2 -zone of carcass semi-coupling, zone 3 - bead
As result of first stage, the strain cycles in each layer were obtained. Also they were divided 
into 4 categories by strain value and saved for using this data in the second phase of overall process.
The second stage is the determination of heat generation rate per unit volume. According to 
[3] it can be calculated as follows:
ai, O) ■
jO)
(1)dt
where A W  -  Hysteresis loss, Tc -  period (during one rotation), a'-J, -  stress and strain tensors 
respectively.
In the literature [3] definition of is often simplified through schematization of strain cycles. 
It is considered that stress and strains are harmonic process with the equal frequency, but with phase 
shift. Then expression of AW simplifies to
All/ = (2)
where £ — loss modulus, which for the corresponding materials is determined experimentally, by tests 
with cyclic loading; fi -  the maximum principal strain, which have been identified and saved for each 
FE previously.
The loss modulus for each material is found experimentally on the base of cyclic tension tests. 
Experiments were carried out with displacement control conditions. The plain specimen of geometry 
corresponds to standards for mechanical tests of rubber-like materials and rubber-cord composites 
ISO 527-2 1A. A displacement control load were posed i.e. fixed values of strains were set for 
specimen. The results are automatically recorded, for each test in real time and have been recorded 
each 0.1 seconds.
Hysteresis loop areas were determined by experiments (Figure 5) and loss modulus were 
found for all materials (Table 2), according to following formula
E =
AIV’
( "1>i f  
where -  hysteresis loop area which was defined in the experiment (Figure 5); 
amplitude, that was applied to the specimen.
(3) 
- strain
a
373
Petrova Iuliia, Larin Oleksiy.
$\fB innpn
a b
Figure 5. Hysteresis loops for different materials of tire layers: a -  sidewall, b -  belt
Table 2 Loss modulus of tire materials
Carcass Belt Filler Sidewall Tread
E, MPa 1.830 9.490 0.136 0.012 0.081
The last stage is presents the solution of the convective heat transfer problem.
FE model was built to solve this problem. Fig. 6, b illustrates the surfaces with different 
boundary conditions.
a b
Figure 6. FE -  model (a) and boundary conditions (3rd stage) (b)
The temperature and convective heat transfer coefficient *c, W/™‘c" were applied as
the boundary condition on the external (G2), internal (G3) surface of tire and on the place of 
interaction with the disk (G3).
So on the surface G1: 
on the surface G2: 
on the surface G3:
I To = 25 Co, hci = 16,18 W/rn2C° ;
I To = 25 Co, hC2 = 5,9 W/m2Co;
5: r„= 25 C°, hc3 = 88000 W/m2C°;
Heat generation rate Q was applied in each FE as a load that previously was expanded in 
Fourier series.
Schematic cycle of heat generation pulse according to corresponding characteristics of 
strain cycles (Fig. 7) has a form:
374
Petrova Iuliia, Larin Oleksiy.
Q =
A W - te nt
2  - t  sin —  t  E (0, t , )
o te  ( t1; t j }  i
MV- tc , n{ t  -  t2)
sm  — —  t  e  ( t 2, t 3)
(4)
2 ■ ( t3 — t2) ( t3 —12)
where -  is determined by the formula (2).
lw— y .J
0.02 0.04 0.06 0.0 0.1a 0.12
a b
Figure 7. Heat generation cycle: a -  pulse of one revolution, 
b -  Fourier decomposition (5 wheel revolutions)
As a result, we obtain the temperature distribution in the tire for different periods of time. 
According to Fig. 8 carcass zone is the most heated due to presence in it larger strains in comparison 
with other layers.
Fig. 9 shows the character of heating of different parts of the carcass (with the highest 
temperature). According to Fig. 9 the active heating of tire takes place at first 15 minutes and after 30 
minutes the temperature distribution is stabilized.
Ba
375
Petrova Iuliia, Larin Oleksiy.
c D
Figure 8. Temperature distribution in the tire: 
a -  0.15 min., b -  0.6 min., c -  12.75 min., d -  30 min.
•'-1 t,s
a B
Figure 9. Graphic of heating of tire layers: a -  carcass (contact zone), b -  carcass (bead)
Conclusions
Two FE-models of sector of pneumatic tire were created. The first one for solving of the 
structural problem (determining strain cycles) and the second for convective heat transfer problem. 
Based on the strain cycles that were previously schematized the heat generation model of the tire 
during its rolling was created. The necessary data for implementation of this model (loss modulus of 
materials) were obtained by experiments. As a result of solving this problem we have received the 
temperature distribution in the tire until the moment of stabilization of the thermal state.
References
[1] Larin O.O., Petrova Ju.A., Mateichyk V. Two-scale approach to modelling of pneumatic tyres 
Systems and means o f  motor transport. Selected Problems. — Monographia N  4. Seria: Transport. 
Rzeszow: Politechnika Pzeszowska Im. Ignacego Lukasiewicza, 2013, pp. 123-128
[2] Larin O.O., Petrova Ju.A. A sub-modeling procedure on stress cycles determination of pneumatic 
tire structural elements. NTU «KhPI» Bulletin: Series «Dynamics and Strength o f  Machines», 2014, 
No. 57, p. 37-49. [in Ukrainian]
[3] Cho J.R., Lee H.W., Jenog W.B. Jeong, K.M., Kim K.W. Numerical estimation of rolling 
resistance and temperature distribution of 3-D periodic patterned Int. J. o f  solids and structures, 2013, 
N° 50, p. 86-96.
376


A Simulation of the Transient Self-heating Process in Dynamically Loaded Rubber-based Multilayers Reinforced Composites

Electronic National Technical University &quot;Kharkiv Polytechnic Institute&quot; Institutional Repository (eNTUKhPIIR)

Proceedings of the 5th International Conference on Nonlinear DynamicsND-KhPI2016September 27-30, 2016, Kharkov, UkraineA  S i m u l a t i o n  o f  t h e  T r a n s i e n t  S e l f - h e a t i n g  P r o c e s s  i n  D y n a m i c a l l y  L o a d e d  R u b b e r - b a s e d  M u l t i l a y e r s  R e i n f o r c e d  C o m p o s i t e sPetrova Iul iia1*, Larin Oleksiy1AbstractThe paper deals with an investigation o f the transient process o f heat generation and stabilization of temperature balance inside pneumatic tires during stationary rolling. The heat generation value have been obtained via integration of deformation over the cycles for each point o f model. The function o f heat generation rate have been created in each point based on previously defined cycles o f deformation and experimentally obtained o f loss modulus for materials. The temperature distribution o f tire has been found within process of thermal stabilization.KeywordsHeat generation, pneumatic tire, cycles of deformation, transient thermal analysis,1National Technical University “Kharkiv Polytechnic Institute, Department of Dynamics and Strength of Machines, Kharkiv, Ukraine* Corresponding author: usyanushka@gmail.comIntroductionElastomers are widely used in modern mechanical engineering, especially in transport. The most widespread are rubber elements and reinforced rubber composites. Analysis of characteristics of these elements is an important part in determination of operational characteristics of whole machine or mechanism. Typical operational conditions of the designs with elastomers and their composites are come with the dynamic load. It associates with two phenomena: fatigue damage accumulation and heat generation. The latter requires special on a attention as it can leads to self-heating of the design. Higher temperature in its turn can significantly change properties of material (elastic and strength characteristics). Moreover, it can increase the rate of ageing processes.This research deals with investigation of heat generation in automobile pneumatic tire during its deformation at the stationary rolling. From practice it is known that the rolling process of pneumatic tires accompanied with the self-heating of its internal elements while at some points the temperature can rise to above 80oC and over. In addition to previously outlined the changing of thermal conditions in the tire leads to a changing of internal pressure and creates an additional thermal stress state.Heating process of automobile tire is the result of several processes. The first one is the conversion of the mechanical energy of cyclic deformation of rubber-like materials into heat. The second is due to internal friction between the different components of complicated composite structures and last is external friction of tire with a road surface.It should be noted that due to the presence of complicated strain state, curvilinear geometry and multilayer structure the temperature field is considerably non-uniform, that leads to changing in the stress-strain state of tire and can leads to formation and grow of internal defects. Thus, an actual task is to study the processes of formation of temperature distribution in the tire during its operation.370Petrova Iuliia, Larin Oleksiy.1. The problem statementThis paper deals with theoretical modelling of the transient thermal analysis of the tire up to a temperature balance during it stationary rolling. The problem was solved in three stages (Fig. 1).The first stage is solving of static problem of tire interaction with the road surface. Besides the tire was loaded by internal pressure of 0.25 MPa and a vertical load of the weight of the vehicle. The analysis of results obtained in this stage allows to find the strain cycles of one revolution in each of the layers.The second phase deals with determination of the heat generation rate in each point. To achieve this goal the amount of energy released per cycle of deformation was determined. In addition, loss modules were determined previously by experiment for all materials of tire.The third stage consists of solving of transient convective heat transfer problem. The function of heat generation rate was applied as the load during the calculations. This function was determined at the step 2 .Figure 1. The mechanism of solving the problem of convective heat transfer2. Finite-element modellingIn this paper the pneumatic tire 205/55R16 model is considered. During the simulations the main design features (Fig. 2) and material properties were taken into account (see. Table. 1)., r ■ w H l"\ pr— tf —ih^olderT - —^  (zone 1) rA \  Carcass doubling (zone 2)\ \  BeadJ (zone 3) /Figure 2. Structure of pneumatic tire371Petrova Iuliia, Larin Oleksiy.The main calculations have been carried out on the base of FE-modeling.The problem is solved in three stages, the first and last of which require specific models. The first FE model was designed to solve the problem for determination cycles of deformation. The 8- nodal finite element with 3 degrees of freedom in the node was used. The second FE model is needed to solve the problem of convective heat transfer. The 20-nodes finite element with 1 degree of freedom in the node was used to create this model.Table 1. Materials propertiesElasticitymodulus Poisson ratio DensityHeat capacity ThermalconductivityE, MPa u p, kg/m3 c, J/kgC0 k, W /mC0Tread 3.00 0.49 1200 1020 0.273Sidewall 3.74 0.49 1200 1020 0.273Bead chafer 7.00 0.49 1200 1020 0.273Filler 16.00 0.49 1200 1020 0.273Rubber matrix 6 0,49 1200 1020 0.273Carcass (cord) 77120 0,3 1020 0.273Belt (cord) 74730 0,3 1020 0.273FE meshes of both models had the same position in the space.Figure 3. FE- model, a -  full FE-model of tire, b -  segment of tireIn the first stage of the analysis were carried out in the framework of multiscale approach of FE modeling. Its technology is described in [1]. Physical nonlinearity, which occurs due to the presence of large deflections and strains is accounted by using the model of Neo-Hooke [2] (which accurately approximates the behavior of rubber materials with about 20% strains deformation). Rubber-cord plies of carcass and belt have been modeled as orthotropic with averaged characteristics for each direction according to the generalized Hooke's law [2]. The curvilinear orthotropy of mechanical properties of a rubber-cord layers was realized by introduction finite-toroidal local coordinate systems in which axes directions repeat geometry of corresponding layer [2]The strain cycles per revolution were defined in all layers of the tire. These calculations showed that the largest stresses were observed along the lines with the circumferential coordinate 50 and 307 degrees. The zones correspond to the change from the contact region to the contactless zone. The largest stresses were observed in the of belt layer. Cycles of stress state (the ratio of current stress to ultimate tensile strength) is shown on the Fig. 4 in polar coordinates (0 ° corresponds to the tire contact area with the road surface).372Petrova Iuliia, Larin Oleksiy.- Zciia 1 — Zo ne 2DegreesbFigure 4. Strain cycles of specific zones a - in the Cartesian coordinate system, b -  in polar coordinates zone 1 -  tire shoulder, zone 2 -zone of carcass semi-coupling, zone 3 - beadAs result of first stage, the strain cycles in each layer were obtained. Also they were divided into 4 categories by strain value and saved for using this data in the second phase of overall process.The second stage is the determination of heat generation rate per unit volume. According to [3] it can be calculated as follows:ai, O) ■jO)(1)dtwhere A W  -  Hysteresis loss, Tc -  period (during one rotation), a'-J, -  stress and strain tensors respectively.In the literature [3] definition of is often simplified through schematization of strain cycles. It is considered that stress and strains are harmonic process with the equal frequency, but with phase shift. Then expression of AW simplifies toAll/ = (2)where £ — loss modulus, which for the corresponding materials is determined experimentally, by tests with cyclic loading; fi -  the maximum principal strain, which have been identified and saved for each FE previously.The loss modulus for each material is found experimentally on the base of cyclic tension tests. Experiments were carried out with displacement control conditions. The plain specimen of geometry corresponds to standards for mechanical tests of rubber-like materials and rubber-cord composites ISO 527-2 1A. A displacement control load were posed i.e. fixed values of strains were set for specimen. The results are automatically recorded, for each test in real time and have been recorded each 0.1 seconds.Hysteresis loop areas were determined by experiments (Figure 5) and loss modulus were found for all materials (Table 2), according to following formulaE =AIV’( "1>i f  where -  hysteresis loop area which was defined in the experiment (Figure 5); amplitude, that was applied to the specimen.(3) - straina373Petrova Iuliia, Larin Oleksiy.$\fB innpna bFigure 5. Hysteresis loops for different materials of tire layers: a -  sidewall, b -  beltTable 2 Loss modulus of tire materialsCarcass Belt Filler Sidewall TreadE, MPa 1.830 9.490 0.136 0.012 0.081The last stage is presents the solution of the convective heat transfer problem.FE model was built to solve this problem. Fig. 6, b illustrates the surfaces with different boundary conditions.a bFigure 6. FE -  model (a) and boundary conditions (3rd stage) (b)The temperature and convective heat transfer coefficient *c, W/™‘c" were applied asthe boundary condition on the external (G2), internal (G3) surface of tire and on the place of interaction with the disk (G3).So on the surface G1: on the surface G2: on the surface G3:I To = 25 Co, hci = 16,18 W/rn2C° ;I To = 25 Co, hC2 = 5,9 W/m2Co;5: r„= 25 C°, hc3 = 88000 W/m2C°;Heat generation rate Q was applied in each FE as a load that previously was expanded in Fourier series.Schematic cycle of heat generation pulse according to corresponding characteristics of strain cycles (Fig. 7) has a form:374Petrova Iuliia, Larin Oleksiy.Q =A W - te nt2  - t  sin —  t  E (0, t , )o te  ( t1; t j }  iMV- tc , n{ t  -  t2)sm  — —  t  e  ( t 2, t 3)(4)2 ■ ( t3 — t2) ( t3 —12)where -  is determined by the formula (2).lw— y .J0.02 0.04 0.06 0.0 0.1a 0.12a bFigure 7. Heat generation cycle: a -  pulse of one revolution, b -  Fourier decomposition (5 wheel revolutions)As a result, we obtain the temperature distribution in the tire for different periods of time. According to Fig. 8 carcass zone is the most heated due to presence in it larger strains in comparison with other layers.Fig. 9 shows the character of heating of different parts of the carcass (with the highest temperature). According to Fig. 9 the active heating of tire takes place at first 15 minutes and after 30 minutes the temperature distribution is stabilized.Ba375Petrova Iuliia, Larin Oleksiy.c DFigure 8. Temperature distribution in the tire: a -  0.15 min., b -  0.6 min., c -  12.75 min., d -  30 min.•'-1 t,sa BFigure 9. Graphic of heating of tire layers: a -  carcass (contact zone), b -  carcass (bead)ConclusionsTwo FE-models of sector of pneumatic tire were created. The first one for solving of the structural problem (determining strain cycles) and the second for convective heat transfer problem. Based on the strain cycles that were previously schematized the heat generation model of the tire during its rolling was created. The necessary data for implementation of this model (loss modulus of materials) were obtained by experiments. As a result of solving this problem we have received the temperature distribution in the tire until the moment of stabilization of the thermal state.References[1] Larin O.O., Petrova Ju.A., Mateichyk V. Two-scale approach to modelling of pneumatic tyres Systems and means o f  motor transport. Selected Problems. — Monographia N  4. Seria: Transport. Rzeszow: Politechnika Pzeszowska Im. Ignacego Lukasiewicza, 2013, pp. 123-128[2] Larin O.O., Petrova Ju.A. A sub-modeling procedure on stress cycles determination of pneumatic tire structural elements. NTU «KhPI» Bulletin: Series «Dynamics and Strength o f  Machines», 2014, No. 57, p. 37-49. [in Ukrainian][3] Cho J.R., Lee H.W., Jenog W.B. Jeong, K.M., Kim K.W. Numerical estimation of rolling resistance and temperature distribution of 3-D periodic patterned Int. J. o f  solids and structures, 2013, N° 50, p. 86-96.376

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A Simulation of the Transient Self-heating Process in Dynamically Loaded Rubber-based Multilayers Reinforced Composites

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Library of National Technical University "Kharkiv Polytechnic Institute"

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A Simulation of the Transient Self-heating Process in Dynamically Loaded Rubber-based Multilayers Reinforced Composites

Abstract

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Library of National Technical University "Kharkiv Polytechnic Institute"

Electronic National Technical University &quot;Kharkiv Polytechnic Institute&quot; Institutional Repository (eNTUKhPIIR)

Electronic National Technical University "Kharkiv Polytechnic Institute" Institutional Repository (eNTUKhPIIR)