5th International DAAAM Baltic Conference
"INDUSTRIAL ENGINEERING – ADDING INNOVATION CAPACITY OF LABOUR
FORCE AND ENTREPRENEURS"
20–22 April 2006, Tallinn, Estonia
METHOD OF FORECASTING OF MECHANICAL PROPERTIES
AND DURATIONS OF SERVICE LIFE OF DETAILS FROM RUBBER
Martinovs, A. & Gonca, V.
Abstract: Wide application in various
areas of techniques of rubber details
demands skill to predict service life of
these details. Existing methods of
forecasting of service life of rubber
products are based on measurement of
changes of mechanical modules of rubber
in current of their operation. Authors have
experimentally
established
stochastic
dependence between mechanical and
electric/ optical characteristics of rubber
materials. The mathematical model of
rubber which allows to deduce analytical
dependence, between mechanical and
electric
(dielectric
permeability)
characteristics of rubber during its ageing
is offered. Specifications and the
equipment, allowing the express train by a
method to define dielectric permeability of
rubber directly in a product at any moment
of its operation are developed. It has
allowed to develop a simple method of
definition of service life of a product from
the rubber, based on measurement of its
dielectric permeability (or specific
resistance, or infrared ray transmission
capacity), during any period of its
operation.
Key words: rubber, dielectric permeability,
mechanical properties, artificial aging,
service life.
rubber and forecast how will they
change during ageing as well as predict
the lifespan of the rubber product. In
this work are analysed publications of
the last six years from journals „Rubber
Chemistry and Technology” and
„Caoutchouc and Rubber” (Kautshuk i
rezina; in Russian), are investigated
articles of K.T.Gillen [ 1 ], M.Celina [ 1 ],
W.V.Mars
[ 2 ],
V.Gubanov
[ 3 ],
V.Druzinin [ 4 ] and others authors, are
studied functioning standards [ 5 ]. Is
established, that in forecasting of
service life of details from rubber are
used only mechanical testings, what are
frequently united with artificial ageing.
Non- mechanical measurements, which
are easy automatizable, were not used
in forecasting methods. The objective
of the Thesis is to develop expressmethod, which is based on the electric
(dielectric
permeability,
specific
resistance) or optical (infrared ray
transmission capacity) measurements,
for determination and forecasting of
mechanical properties and lifespan of
elastomers (rubber).
2. MATHEMATICAL MODEL
In base of method of forecasting use
improved variant of model, given in
literature [6 ], in which de-structuring and
structuring processes are included. In
this model complicated structure of
rubber is substituted with platelike
structure. Ageing of every platelike
element is characterised by cross-link
yield γ:
1. INTRODUCTION
Rubber products have very wide use
in many different economic sectors. In
order to solve specific engineering and
technical tasks, it is necessary to
determine mechanical properties of the
279
N t 
,
(1)
N max
where N(t)- number of links in time
moment t, which carry tension load,
Nmax - maximal number of these links
(t=0). Changes of cross-link yield dγ in
time dt are characterized by equation:
3. OBTAIN INITIAL DATA TO
FORECAST
 t  
In order to forecast mechanical
properties and lifespan of rubber in the
beginning it is necessary to obtain initial
data. To do this it is proposed to take
samples of new rubber and rubber aged 24,
72 and 168 hours at 1000C in air [7]. For
these samples with different levels of
ageing elongation characteristics are
produced [8 ]. For approximation of tensile
strain under constant elongation w
characteristic use function
0  q  d ,
(5)
where σ 0 – mechanical stress on crosssectional area at unstrained conditions;
 – deformation; q, d – material
constants, which are determined with
method of the least square. For each
sample is measured breaking time in
elongation t *, length of the work area of
the sample put on tension test in
unstrained conditions l 0 , average rate of
elongation w, with which work area
border marks separate. Temperature T,
at which tension tests are performed,
ageing temperature T n and ageing time
t n are taken. It is experimentally
established that the biggest relative
measurement errors are related to
determination of breaking time t * and
parameter q of elastic properties. This is
why measurement error of these
characteristics Δq and Δt * further are
used in forecasting method. Parameter d
is absolutely accurate because error of
this
parameter
is
included
in
measurement error of parameter q.
Relative measurement error for other
characteristics – l 0 , w, T, T n , t n , is much
smaller compared to measurement
errors of t * un q, therefore average
values of these characteristics are used
in forecasting method, leaving errors of
measurement out. In experiments is
used rubber 2H-1-MБC-C 2 ГOCT 733890. It is characterised by initial data to
forecast: tn0=0; tn1=(1·24·3600)600s;
tn3=(3·24·3600)600s;
Qd t 
 3


Q t 
x W t 
 k T t 
 s


 t 
k T t 
d t     t   e
 a  1   t   e
  Z   dt




(2)
where Qd – de-structuring energy
(energy needed to tear link), J; Q s –
structuring energy (energy needed to
have a new link); W – elastic potential
(energy of deformation received by one
volume unit) of given platelike element,
J/m 3 ; T – absolute temperature, K; x –
distance between centres of contiguous
atoms in unstrained conditions, m; k –
Boltzmann constant; a – structuring
constant; Z=6 – number of the closest
contiguous atoms in cubic structure;
=110 13 Hz – frequency of thermal
oscillations of atoms; t – time, s;
x 3 W/- energy mechanically delivered
to one link. Given differential equation
is characterized by conditions at the
beginning, at the moment of mechanical
loading and at the end:
if t=0, then γ=1;
if t=t 0 , then γ=γ 0 ;
(3)
if t=t*, then γ=0.
If during ageing anisotropic state within
rubber are not developed (ageing in not
deformed state), then it is possible to
include in model the differential
equation with cross-link yield , time t
and relative dielectric permeability :
d
    t   dt .
(4)
2
Coefficient  is material constant,
which characterizes electrical properties
of rubber and is linked to electrical
field frequency ratio, which is used in
dielectric permeability measurements.
If during ageing relative dielectric
permeability  of rubber increases, then
>0, if  decreases, then <0.
280
If functional correlations between
mechanical parameters q, d, t * and
electrical or optical are known, then
electrical or optical measures can be
used for determination of mechanical
parameters of given type of rubber.
Authors have experimentally established
stochastic dependence between mechanical
and electric/ optical characteristics of
rubber materials. Examples for rubber with
basic content: natural caoutchouc –
55.46%, filler K354- 27.73% (produced in
“Baltijas gumijas fabrika” in Riga) see in
Figure 1- 3. Methodology and results of
investigations are given in literature [9].
Infrared ray transmission capacity is
measured with Perkin Elmer FT-IR
Spectrometer Spectrum 1000.
tn7=(7·24·3600)600s;
q0=3.530.45MPa;
q1=4.170.30MPa;
q3=4.740.50MPa;
q7=5.430.50MPa;
d0=0.70; d1=0.69; d3=0.64; d7=0.64;
t0*=28.42.8s;
t1*=23.53.2s;
t3*=21.63.5s;
t7*=19.53.5s;
l0=20.00.3mm;
w=2.1560.055mm/s;
T=(20+273)1K; Tn=(100+273)1K.
4. DETERMINATION OF
STRUCTURING ENERGY, DESTRUCTURING ENERGY AND
MATERIAL CONSTANTS
Fig. 1. Correlation between parameter q
of elastic properties and relative
dielectric permeability
Forecasting method requires knowing
values of de-structuring energy Qd,
structuring energy Qs, atomic structure
constant x and structuring constant a.
Assuming, that during ageing the values of
these parameters are constant. To
determine their values will use correlation
(2), where 4 cases of elongation under
constant rate and 3 cases of artificial
ageing without mechanical load can be
written down as equation system:
Fig. 2. Correlation between parameter q
and specific resistance
Qd
 3


x W t 
Q
 k T
 s 

 t 
k T
0   i     t   e
 a  1   t   e   Z   dt
0



ti*
(6)
Q
Q
 d
 s 

 j   j 1     t   e kTn  a  1   t   e kTn   Z   dt

tn j 1 

tn
j
(7)
where i=0, 1, 2, 3; j=1, 2, 3. Equations
(6) describe processes of elongation at
constant speed to failure for:
1) new rubber (i=0, 0 =1);
2) rubber, which has been aged for 24
hours at elevated temperature (i=1);
3) rubber, which has been aged for 72
hours at elevated temperature (i=2);
Fig. 3. Correlation between parameter d
of elastic properties and infrared
transmission capacity at 2915cm -1
281
4) rubber, which has been aged for 168
hours at elevated temperature (i=3);
In this processes cross-link yield
changes from i (before elongation test)
to =0 (time of failure). t i * designates
time in which sample is destroyed.
Equations (7) describe process of
rubber ageing at elevated temperature
Tn without mechanical load in period of
time from:
5) t n0 =0 to t n1 =24 hours. During this
process cross-link yield changes
0 =1 to 1 (j=1);
6) t n1 =24 to t n2 =72 hours. Cross-link
yield changes from 1 to 2 ; (j=2);
7) t n2 =72 to t n3 =168 hours. Cross-link
yield changes from 2 to 3 (j=3);
Set
1.
2.
3.
4.
5.
6.
7.
8.
9.
q,
t*
q
t*
q+Δq
t*+Δt*
q-Δq
t*-Δt*
q+Δq
t*-Δt*
q-Δq
t*+Δt*
q
t*+Δt*
q
t*-Δt*
q+Δq
t*
q-Δq
t*
Qd
10-19J
2.28
Qs
10-19J
2.33
10-10m
5.638
9.7
2.05
2.06
4.633
6.4
2.29
2.35
6.322
7.2
2.25
2.30
5.692
5.2
x
and using equations (5), (8), (9) can get
d 1
q  wt 
W t  

(10)
 .
d  1  l0 
System of equations (6) – (7) has 7
independent equations with 7 unknown
quantities Qd, Qs, x, a, 1, 2, 3. The
solution of such equation system is found
with numerical methods. Results for
different initial data sets are given in Table
1. Average values from parameters q and t*
(see 1 row from Table 1) establish highest
probability of service life. Two another sets
of parameters Qd, Qs, x and a set minimal
and maximal values of forecasting service
life.
5. FORECASTING OF LIFESPAN
AND MAXIMUM DEFORMATION
IN CREEP UNDER CONSTANT
TENSILE LOAD
a
Creep (under constant tensile load
P) process can be described by equation
(2), which in this case looks like
following:



0   0      t   e
0 

tm
2.34
2.35
5.712
9.8
2.28
2.29
5.340
8.3
2.27
2.30
5.968
3.7
2.25
2.30
5.355
8.7
2.31
2.35
5.970
9.1
 a  1    t    e

Qs
k T


  Z   dt


(11)
where 0 – cross-link yield before
loading; t m – maximum lifespan of
sample in creep. Process of creep is
divided into two parts: 1) sample is
loaded until assigned P value under
constant elongation speed w, time
t=[0...t0]; 2) creep as a result of constant
tensile load P, time tt0.
For approximation of deformation 
from moment t 0 , when sample takes full
load P, is recommended to use function:
c
(12)
   0  1  b  t  t0  ,


where 0 – deformation relevant to the
moment in time t0; c, d – constants
characterizing samples under creep test,
which are determined with method of the
least
square.
Example
of
creep
characteristic for rubber 2H-1-MБC-C 2
ГOCT 7338-90 is given in Figure 4.
Table 1. The solutions of equation
system (6) – (7) in dependence on
values of parameters q and t *
Elastic potential W for rubber in
equations (6) in general case is found as
follows:

W    0  d ,
Qd
x3 W ( t )
 k T
 t 
(8)
0
where d- deformation change in time dt. If
the speed of elongation w is constant, then
wt
(9)

l0
282
maximum deformation of rubber 2H-1MБC-C 2 ГOCT 7338-90 are shown in
Figure 5 and 6. Rhombus show the
values of highest probability of lifespan
and maximum deformation, small lines–
range of forecasted error interval;
uninterrupted
line
characterizes
experimental values, ideal congruence
between forecasted and experimental
results. As shown, all values of lifespan
and maximum deformation are within
forecasted error interval. Given method
fits for any size solid. Creep test is not
sustained for sample studied. Creep test is
not obligatory to be realized until the
destruction of sample. By improvement of
tensile tests quality and determination
precision of cross-sectional area, is
possible to scale down forecasting
quantity inaccuracy interval.
Fig. 4. Creep characteristic until sample
destruction
Maximum lifespan in creep t m is
calculated from equation (11); maximal
deformation m – from equation (12), by
replacing t with t m . Mechanical stress 0
in creep under constant tensile load P is
constant (in time tt0). It is calculated:
P
0 
,
(13)
F0
where F0 - cross-sectional area of the
work area of the sample at unstrained
conditions (perpendicular to direction
of elongation). Using equations (5), (8)
(9), (10), (12), get the expression to
calculate elastic potential in random
moment in time tt0 for creep:
q 0
W t  

d  1  q



d 1
d

  0   0
 q
1
d

  b  t   0

 q
1
c
 d l0
 
 w
.
(14)
To calculate cross-link yield 0 before
loading, use equation (6) and tension
test [8] data for given ageing levels
rubber. If functional correlations
between mechanical parameters 0, q, d,
t * and electrical or optical are known,
then yield test is not necessary,
mechanical parameters of given type of
rubber can be determined with electrical
or optical measures.
6. COMPARISON AND ANALYSIS
OF NUMERICAL AND
EXPERIMENTAL RESULTS
Fig. 5. Comparison of forecasted and
experimentally determined lifespan of
rubber sample in creep
Fig. 6. Comparison of forecasted and
experimentally determined values of
maximum deformation of rubber sample
in creep under constant tensile load
The comparison of numerical and
experimental results of lifespan and
283
7. CONCLUSION
Method for forecasting mechanical
properties and lifespan of elastomers in
creep under constant one-way tensile load
is developed and experimentally tested. All
experimental values of lifespan and
maximum deformation are within
forecasted error interval. Electrical and
optical measurements can be used to obtain
initial mechanical parameters necessary for
forecasting method. At the same time use
of electrical/ optical parameters helps
increase accuracy of determination of
mechanical parameters and consequently
improve the quality of forecast.
5.
6.
This work has been partly supported by the
European Social Fund within the National
Programme “Support for the carrying out
doctoral study programm’s and postdoctoral researches” project “Support for
the development of doctoral studies at Riga
Technical University”.
7.
8.
8. REFERENCES
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Sandia
National
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Livermore, California 94550 Sandia
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as a Predictor of Fatigue Life Under
Multiaxial
Conditions.
Rubber
Chemistry and Technology, 2002, 75.
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Zinatne, 1985, 46, 33- 36. (Губанов
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durability of rubber details at large
9.
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(Дружинин
В.А.
Оценка
длительной прочности резиновых
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ISO
11346:2004.
Rubber,
vulcanized
or
thermoplastic.
Estimation
of
life-time
and
maximum temperature of use.
Martinovs A., Gonca V. A rubber fall
model for yield deformation. In
International Conference on bionics
and prosthetics, biomechanics and
mechanics, mechatronics and robotics,
Varna, 2004, 4, 47- 50.
DIN 53508 Prüfung von Kautschuk
und Elastomeren. Künstliche Alterung,
Oktober 1993.
DIN 53504 Prüfung von Kautschuk
und Elastomeren. Bestimmung von
Reißfestigkeit,
Zugfestigkeit,
Reißdehnung und Spannungswerten im
Zugversuch, Mai 1994.
Martinovs A., Timmerberg J., Gonca
V. Research of relevances between
mechanical and electrical parameters of
rubber. For publishing in Riga TU
Scientific Proceedings accepted work,
Riga, RTU, 2005, p.8.
Andris Martinovs, lecturer, doctoral
studies
Riga Technical University, Institute of
Mechanics; Rezekne Higher Education
Institution
Adress: Atbrivosanas aleja 76, LV 4600,
Rezekne, Latvia
e-mail: andris@ru.lv
Vladimir Gonca, professor, Dr.sc.ing.
Riga Technical University, Institute of
Mechanics
Adress: Ezermalas Str. 6 - 305, LV-1014,
Riga, Latvia
e-mail: Vladimirs.Gonca@rtu.lv
284