Volume 2013 Issue No.9, Dt. 11 September 2013 Time: 1h04m. PM.
Investigations
on the Relationship of Life Time durability of Rubber under Applied Stress
Narayana.
L. Kotcherlakota
Polymers and General
Physics Labs, Shivjai University, Kolhapur-416004.
{Original write up on 20 April 1991 by Narayana. K.
L. signed}
{Retd. Prof.of
Physics, SU, Kolhapur} 17-11-10, Narasimha Ashram,
Official Colony,
Maharanipeta.P.O. Visakhapatnam -530002,
Mobile No.
9491902867 & 9542717723
Key
Words: Styrene Rubber, Nitril Rubber, logarithmic lifetime, with and
without Carbon Black.
A B S T R A C T
Experimental data of lifetimes versus the
applied stresses on the Styrene based and the Nitril based Rubbers reported by
A. Tager (1978) have been critically examined to establish the correct
relationship between lifetime against stress. The stress applied varies from about 0.5 to 10
kg/mm2.
It is found that while Nitrile based Rubber are susceptible to more
fracture the styrene based rubber can withstand stress but exhibit a peaked
lifetime behavior. This is evident from the calculated formula that logarithmic
lifetime = 45.68 exp(-1.608*σ). Deviations from Bartnev and Zhukov equations
thus arise.
From a macromechanical view it is
noted that Styrene based rubbers have a different bond formations which is
responsible for the peaked durability lifetime of the rubber. Relationship with
chemical constitution of Rubber and the structure of the vulcanizate is under
investigation. The new formula given envisages however a distinct role of the
super-molecular orientations in this type of rubbers and thus seeks to account
for the pre-exponential constant 45.68, a characteristic of the material.
----------------------------------------
DETAILS
(Dated 20 April 1991 by Narayana. K.
L. signed)
We use here σ and τ as the stress and relaxation
time respectively.
1. For Rubber Buna-N without Carbon Black
σ kgf/mm2 0.2 0.6 0.8 0.9 1.2 1.5 1.8 2
log(τ) secs
5.75 5.2 4.5
4.2 3.8 3.2
2.6 2.1
Fig.1
log(τ) 3.345
1.2637 1.5 5.5
σ 1.655
0.2101 1.3 1.95
Difference 1.69 1.4712
-0.45 4.2 10
Correlation
coefficient
Pearson’s r= 0.9855
One tail- significance =0
Standard error of
mean difference =0.4652
T score =
3.6327 with 9 diff … Two
tail-significance 0.0055
Valid cases 10:
Missing cases 0.
2 For Buna-Styrene Carbon Black Rubber
σ (kgf/mm2)
1.3 1.4
1.5 1.6 1.65
1.7 1.75 1.8
1.9 1.95
log(τ) secs 5.5 4.5
4.25 3.8 3.6
3.4 2.9 2.2 1.8
1.5
Horizontal axis is σ and Vertical axis is log(τ)
Buna- Styrene Rubber with Carbon Black fill KLN12 scatter plot is given
in see Fig.2.
Y=a* exp(b*x) y= 73.32 *
exp(-1.909769*x)
Natural Rubber
a. Crystallize when extended.
300kgf/cm2
Group-I Polychloroprene 270kgf/cm2
Butyl Rubber 200 kgf/cm2
b. Do not Crystallize when extended
BUTOC Rubber ckb
10kgf/cm2
Group-II TYRE 1E ckh-u 10kgf/cm2
Nitrile ckc-30 14kgf/cm2
------------------------------------------------------------
For Rubber 0.001
kgf load σ =6.0895E+10 erg.cm-3
i.e. 6.0895E+10
gm.cm2.sec-2.cm-3
erg= dyne.cm so 6.0895E+10 dyne.cm-2
6.0895E=10 gm.cm-1.sec^-2
6.0895E+07
kg f= 6.11E+06 mm.sec-2
Stress= load/area
= kgf/area = dyne/cm2
Variation of σ = 10 kgf/mm2to
160kgf/mm2
10kgf. mm-2
= 10kgf * 100 cm-2
3 Fitting
Exponential
Same as for Fig.2
just repeated
σ kgf/mm2 1.3
1.4 1.5 1.6
1.65 1.7 1.75
1.8 1.9 1.95
log(τ) secs 5.5 4.5
4.25 3.8 3.6 3.4
2.9 2.2 1.8
1.5
Page 1
y=A*exp(b*x) : y= 73.32055* exp(-1.905769)*x
log(τ) = 45.68091*
exp(-1.608991 * σ)
============================================
Dependent Variable
Independent Variables in the model σ
Variable B Std.error t
score 2-tail Sig
Intercept 13.1552 0.601 21.8606 0
σ -5.9276 0.361 -16.4198 0
=========================================
Analysis of Variance
Source
SS DF MS
F Prob
Regression
13.9581 1 13.9581 269.6111
0
Residual
0.4141 8 0.0518
Total
14.3723 9
R-Squared = 0.971
R-Squared adjusted for DF=0.968
τ= τ0 *exp(- α * σ )
log(τ) =7.27 + 1.409 * σ – 2.2522* σ2
|
log(τ) = 13.1552 - 5.9276
* σ
log(τ)= log(τ0) - α * σ ; implies α = 5.9276.
σ = 1.65 log(τ)=3.37466 observed 3.60
σ = 1.8 log(τ)=2.48552 observed 2.2
σ = 1.4 log(τ)=4.85656 observed 4.5
σ = 1.3 log(τ)= 5.44932) observed 5.5
Page 2
Y= y0* exp(-k*x) with k=0.9 and dk= 0.01 statpal correlation file
kln 112
Log(τ) σ
|
log(τ) 1 -0.9855
|
10 10
|
0 0
|
σ -0.9855 1
|
10
10
|
0 0
|
STATPAL 4.0 DESCRIPTIUVE STATISTICS
Statistics for variable log(τ)
Mean
3.345 Std. dev 1.2637
Std. Error 0.3996
Range 4 Minimum
1.5 Maximum 5.5
Valid Cases
10: Missing Cases: 0
Statistics for variable σ
Mean 1.655 Std. dev 0.2101
Std. Error 0.0664
Range 0.65 Minimum
1.3 Maximum 1.95
Valid
Cases 10: Missing Cases: 0
For entire table , chi-square =90 with DF Significance=0.231
A total of 100 (or 100%) of the cells have expected frequency
less than 5.
4 Rubber without Carbon Black
(Dt.
20th April 1991).
Horizontal axis σ
Vertical axis log(τsq)
log(τ) secs 5.75
5.2 4.5 4.2
3.8 3.2 2.6
2.1
σ kgf/mm2 0.2
0.6 0.8
0.9 1.2 1.5
1.8 2
τarcs1 all terms are
-999999 (M)
τsq 2.397915 2.28035
2.12132 2.04939 1.94935
1.788854 1.612451 1.449137
d(log(τ)/d(σ) = - alpha;
0.80-1.5 = -7 Horizontal
4.5 -3.20 = 1.3 Vertical
alpha= 1.8571428
dy/dx= m = - 2.0259
log(τ)= A - σ * alpha
log(τ)= -2.0259* σ + 6.1979 where the last term is log(τ0)
σ = 0.8 log(τ)=4.57718
σ =1.5 log(τ)=3.15905
Depended Variable : ΤSQ
Regression log(τsq) Vs σ
|
Independent Variables in the model : σ
|
Variable B Std
Error t – score 2-tail Sig
|
Intercept 2.5461 0.0305 83.5909 0
|
σ 0.5244 0.0241 -21.762 0
|
Analysis of Variance
|
Source SS DF MS F
Prob
|
Regressio 0.7302 1 0.7302 473.5833 0
|
Residual 0.0093 6 0.0015
|
Total 0.7395 7
|
Data of σ Vs log(τsq) plotted in Fig.4.
5 Without Carbon Black Rubber Buna- N
σ kgf/mm2 0.2
0.6 0.8 0.9
1.2 1.5 1.8
log(τ)
Secs 5.75
5.2 4.5 4
3.8 3.2 2.6
dlog(τ)/d(σ)
= - alpha
0.80 – 1.5
= -0.7 Horizontal
4.5
- 3.20 = 1.3
Vertical
alpha= 1.8571428
log(τ)= A
- σ * alpha
dy/dx=m = -2.0259
log(τ) = alpha * σ + log(τ0)
log(τ)= -2.0259 * σ + 6.1979
σ = 0.8 log(τ)=
4.57718
σ = 1.5 log(τ)=
3.15905
Dependant Variable log(τ)
|
Independent Variable in the
model : σ
|
Variable B Std.Error t-score
2-tail Sig
|
Intercept 6.1979 0.0965 64.2105 0
|
σ -2.0259 0.0764 -26.5277
0
|
Analysis of Variance
|
Source SS DF MS
F Prob
|
Regression 10.8968 1
10.8968 703.7199 0
|
Residual 0.0929 6 0.0155
|
Total
10.9897 7
|
R-squarred = 0.992
R-squared
adjusted for DF =0.99
6 For Rubber with Carbon Black BUNA- N
log(τ)
5.5
4.5 4.25
3.8 3.6 3.4 2.9 2.2
1.8 1.5
σ 1.3 1.4 1.5 1.6 1.65 1.7 1.75
1.8 1.9 1.95
log(τsq) 2.345207
2.12132 2.061552 1.949358
1.897366 1.843908
1.702938
1.483239
1.34164
1.224744
Fig.6
Dependent Variable : log(τsq)
|
Independent Variables in the
model : σ
|
Variable B Std Error t
- score 2-tail Sig
|
Intercept 4.5464 0.2234 20.3467 0
|
σ -1.6612 0.134 -12.3928 0
|
Analysis of Variance
|
Source SS DF MS F Prob
|
Regression 1.0962 1 1.0962 153.5812 0
|
Residual 0.0571 8 0.0071
|
Total 1.1533 9
|
7 Without Carbon Black Rubber BUNA-N
σ 1.3
1.4 1.5 1.6 1.65 1.7 1.75 1.8 1.9 1.95
unspecified 1.3 1.3866
1.4733 1.56 1.6466 1.69
1.7333 1.7766 1.8633 1.95
Fig.7
Fig.7
DISCUSSION AND RESULTS ANALYSIS
Relaxation behavior of a polymer material obeys the generalized equation
of visco-elastic body
dσ/dt
= E*dε/dt - σ / τr
where σ is stress, ε is strain i.e. uni-axial compression, t
is the time, E is the modulus of elasticity and τr is relaxation time at temperature T a function
of σ .
The dependence of τr
on σ and T is given by equation,
τr = τ0 * exp( ( U0 - γr * σ ) / (kT)
where U0
and γr parameters determining the relaxation
properties of the relevant solid body, τ0 is pre-exponential factor, and k is molar
gas constant.
Then
log(τ)= log(τ0)
+ (UO - γ r * σ) / (kT)
where log(τ0) not a constant of the method
depends on strain.
Let -12 = -12 + ( UO - γ r * σ)/(kT) then UO= γ
r* σ
UO/γ r = 13.2
kcal/mole/a0-23 cm3 = σ ;
and (13.2
erg. cm-3 / 1.4394506E+13) = 9.17016E+10 erg.cm-3
i.e. 9.17016E+10
dyne.cm-2.
The expressions of the curve fittings are given below for the quadratic equations obtained for all the seven sets of polymer data.
log(τ) Vs σ
Fig.1 Y
=6.19703-2.02386*X-8.96194E-04*X2 BUNA-N without Carbon black
log(τ) Vs σ
Fig.2 Y
=7.27195+1.40828*X-2.25197*X2BUNA-S without Carbon black
Fig.3 same as for Fig.2
log(τsq) Vs σ
Fig.4 Y =2.47878-0.36447*X-0.07052*X2 log(τsq) without Carbon black
log(τsq) Vs σ
Fig.4 Y =2.47878-0.36447*X-0.07052*X2 log(τsq) without Carbon black
log(τ) Vs σ
Fig.5 Y =5.89653-1.52972*X-0.18034*X2
BUNA-N without Carbon black
log(τ) Vs σ
Fig.6 Y =0.81969+2.98566*X-1.42648*X2 BUNA – N with C parabolic!
unspecified Vs σ
Fig.7 Y =0.34247+0.56418*X+0.1302*X2 gives an unusual straight line.
BUNA-Styrene
yields a larger constant value 7.27195 while the other coefficients of X and X2
are very less compared to BUNA-N without Carbon in both cases. A repeat case of
log (τ) versus σ in the trial of Fig.6 gives parabolic expression.
ACKNOWLEDGEMENT
I am indebted to the staff of the
Chemical Technology Institute in Matunga, Mumbai and as well their associated
laboratories of the Polymer Science, visited by me in the years 1991 to 1993, who
permitted me to carry out the investigations at their laboratories and obtain
the given results. I stayed at my father’s(Prof K R Rao of AU, Visakhapatnam)
student, G. Gurunadham’s (relative of Dr. M. Gourinadha Shastri of Hindu
College, Muslipatam) residence, in campus, in IIT Powai, Mumbai. His children were very cordial to me and
served me excellent food and breakfast. Several staff members of Polymer Labs,
etc. were very cordial to me there during my stay in Mumbai due to the good
will and the God devoted member of staff, and Spectroscopist, G. Gurunadham, of
IIT, Powai, Mumbai.
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