Relaxation moduli of equivoluminal ring and strip samples of Solithan

Mueller, H. K.

English

NASA Technical Reports Server

ON THE RELAXATION MODULUS OF THE EQUIVOLUMINAL 
COMPOSITION OF SOLITHANE 113 
H. K. Mueller 
GALCIT SM 69-7 
This work was  supported by the 
National Aeronautics and Space Administration 
Research Grant No. NGL-05-002-005 
GALCIT 120 
and 
A i r  Force  Rocket Propulsion Laboratory 
Research and Technology Division 
Edwards, California 
A i r  Force Systems Command 
United States A i r  Force  
F04611-67-(2-0057 
Firestone Flight Sciences Laboratory 
Graduate Aeronautical Laboratories 
California Institute of Technology 
Pasadena, California 
https://ntrs.nasa.gov/search.jsp?R=19690026471 2020-03-12T05:17:34+00:00Z
INTRODUCTION 
The purpose of this note is a comparison of the relaxation moduli 
of Solithane 50/50 (equivoluminal composition of Solithane 11 3) as  
determined from relaxation tests on ring specimen and strip specimen. 
Ring specimen a r e  a particularly convenient type of specimen for 
testing the mechanical properties of elastomers under uniaxial tension. 
However, the lack of an accurate analysis of the stresses and strains 
in this geometry complicates the definition of the true s t ress  and 
strain in the ring test sections and may lead to incorrect data 
reduction. 
New data on the relaxation modulus of Solithane 50/50 has been 
obtained from tests on strip specimen and is compared to the data 
which has earlier been obtained with ring specimen [ 13 . 
schemes employed for reducing both sets of experimental data a r e  briefly 
described. 
found to be unsuited for relaxation’ tests at small strains. 
The two 
The particular ring geometry used in these experiments is 
-1- 
TESTS WITH STRIP SPECIMEN 
Strips of 0.5 in width were 
The sheet had a thickness of 0.1 
cut out of a cast  sheet of Solithane 50/50. 
in and could be easily cut with the help 
of a razor blade. 
leaving a straight test section of 4 in length in between them. 
shows the test  specimen. 
fracture at the ends of the test section because the relaxation tests were 
carried out at less than 3% strain. 
Metal tabs were then glued on either end of the strips 
Figure 1 
No special precautions were necessary to avoid 
The effect of the metal tabs on the s t ress  and strain field was 
neglected and the two quantities were calculated a s  follows 
F 
A ’  
0- = . -  
0 
where 
1 = length of specimen test section, 
A 1,’ = 
F = force, 
A = cross section of the unstretched specimen, 
0- = engineering stress. 
0 
increase in the length of the test  section, 
0 
The tests were carried out with the help of an Instron tester. The 
constant strain t: 
practice and a ramp strain history a s  shown in Figure 2 is employed, 
A strain rate of 0.5 min-’was used in order to avoid the overshoot 
encountered at higher Instron crosshead speeds when the crosshead 
motion is stopped, 
strain t: = 0.025 is equal to t = 0. 05 min in  this case. 
cannot be obtained in a truly step wise manner in 
0 
The rise time necessary to reach the constant 
0 0 
-2- 
Assuming the material to behave linearly viscoelastic [2 3, the 
s t ress  a s  a function of time is equal t o  
t 
o< t < to, - -  v(t) = R Erel dt for 
0 
t 
\ 
€ 
Ere,(-r) dt for 0 v(t) = - t > t ,  
0 
t -to 
€ 
0 where R = = strain rate, 
E = relaxation modulus, r el 
= rise time. 
The s t ress  c r ( t )  as a function of time is readily obtained from the force 
history recorded on the Instron chart and equation (lb). Equations (2a) 
or (2b) a r e  then strictly speaking integral equations for  the relaxation 
modulus. The second of these equations may, however, be approxi- 
mated by 
modulus 
thus leading to a direct relationship between s t ress  history and relaxation 
( 4) 
The geometr,cal interpretation of the approximation ,S shown in 
Figure 3. 
can simply be replaced by t. 
For times t >> to the argument of the s t ress  in equation (4) 
-3  - 
The results of relaxation tests carried out in  this manner at 
various temperatures a r e  presented in Figure 4. 
the relaxation modulus show the typical time and temperature 
dependence and can be easily shifted into a single relaxation curve. 
This curve is  shown in Figure 5 for a reference temperature of 0 C. 
The previously determined time-temperature shift factor between O°C 
and -5OC has been used to refer the data to 0 C although -5OC was 
the lowest temperature at which tests were run in this test series. " 
The segments of 
0 
0 
-4- 
TESTS WITH RING SPECIMEN 
The results of the tests with ring specimen have already been 
presented in reference 1. 
of the data reduction will be given here in order to facilitate a compari- 
son with the new data. 
Only a brief description of these tests and 
The ring specimen were cut out of cast Solithane 50/50 sheets 
of 0.1 in  thickness by means of a rotating cutting tool. 
were not of the same high quality a s  the surfaces of the cast  sheet o r  a s  
of cuts made with a razor blade. Each specimen was visually inspected 
and faulty specimen were singled out. 
The cut surfaces 
The inner and outer diameter 
of each ring were about 0.65 in and 0.75 in, respectively. ‘They were 
icdividually measured for each ring with the help of an optical comparator. 
Figure 6a shows a ring in its undeformed state. 
For testing,a ring was put over two stainless steel pins of 0.378 in 
diameter. 
cell, respectively. The initial distance 1 between the pin centers was 
such that the fiber at  the ring inside remained unstretched after placing 
the ring over the pins. The 
two straight segments of length ld a r e  considered the test sections of 
the ring. 
some A1 these test  sections a r e  stretched and a r e  essentially in a state 
of uniaxial tension if the s t resses  introduced by bending the originally 
round ring into this shape and the end affects near the pin can be 
neglected. 
is thin, that is 
These pins were attached to the Instron crosshead and load 
d 
Figure 6b shows the ring in this condition. 
As the distance between the pin centers is increased by 
This may with some confidence be assumed when the ring 
l - - < < I  Di 
0 
D 
-5- 
and when the distance between the pin centers is large compared to the 
pin diameter d, The inner and outer diameter of the original ring a r e  
denoted by Di and Do, respectively. 
be large enough, however, to not introduce too large bending stresses in the 
stretched ring which could lead to premature failure. 
The diameter of the pins has t o  
The average s t ress  in the two ring test sections is simply defined a s  
F 
2 A  I r 
( r = -  (5) 
where A r  = ring cross  section. 
Several definitions of the strain in these ring segments a r e  pos- 
An accurate stress-strain analysis of this difficult geometry is sible. 
not available and the strain is often based on the original length of some 
ring fiber. 
measure 
In terms of the central fiber one arrives at the following 
A simple analysis of the stresses and strains in the stretched 
ring by Blatz [ 3 ] leads to the following expression for  the average 
strain across the ring 
- e - 
c r  
1 3- 
2 A l  ’ 1 I t -  - C io 
4 t S o  
d 
2 2 
D. t Do - -  2n 1 s = -  
0 ‘io 
where 
-1 
= 2 ld 3. Td. ‘io 
-6- 
F o r  the values of d, Di, Do, and ld 
ship gives a value for the average strain that differs only very little 
from the one given by equation (6). Figure 7 shows the comparison 
of the two values as a function of A 1  and some other strain 
mentioned earlier this relation- 
definitions. 
Definitions (5) and (6) were used to reduce the relaxation data 
obtained by testing the ring specimen. 
calculation of the relaxation modulus a r e  the same as described for 
the strip specimen. 
is shown in Figure 8. 
determined is c0 = 0.05. 
The strain history and the 
The relaxation modulus arrived at in this manner 
The strain at which this modulus has been 
-7- 
COMPARISON OF THE DATA 
Figure 8 shows the relaxation moduli obtained by testing ring 
specimen and strip specimen. 
linearly viscoelastic up to about 7% strain and the different 
strains 6 
Solithane 50/50 may be considered 
for which these moduli have been obtained do not affect 
0 
the result. 
with strip specimen is seen to be higher at short times than the 
The relaxation modulus calculated from experiments 
modulus derived from the experimental data for the ring specimen. 
In both cases the specimen reaches its long time equilibrium at 0 C 
after about 1 minute and the rubbery modulus Er is the same for 
both sets of data. 
0 
The discrepancy at  short times is too large to be entirely caused 
by batch to batch variations in material properties. 
caused by the non-uniformity of the s t ress  and strain fields in the ring 
specimen. 
compared to the pin diameter and the circumference of the ring, 
cf. Figures 6a, b. 
definition of s t ress  (5) and strain (6) in the ring the more unrealistic 
the higher the modulus of the ring material is, i. e., lower temperatures 
o r  shorter times. 
It is believed to be 
The test sections of the ring a r e  seen to be relatively short 
The non-uniformity seems to make the simple 
It must be concluded from Figure 8 that for the small strains 
employed in s t ress  relaxation tests the ring size i s  too small in 
comparison to the pin dimensions to obtain reliable results with the 
simple definitions of s t ress  and strain (5,6). An adequate s t ress-  
strain analysis of this geometry, however, which could lead to a more 
realistic interpretation of the experimental data is not yet available. 
-8 -  
Figure 9 shows the time-temperature shift factors obtained by 
shifting both sets of data. 
of the shift factor is seen to be the same in both cases except for the 
lowest temperatures at which tests were run. 
The temperature dependence and magnitude 
-9 - 
REFERENCES 
1. Knauss, W. G. ; Mueller, H. K. ,  "The Mechanical Characteri- 
zation of Solithane 113 in  the Swollen and Unswollen State", 
GALCIT SM 67-8, California Institute of Technology, 
Pasadena, California (1967). 
2. Fliigge, W. , !!Viscoelasticity", Blaisdale Publishing Company, 
{ 1967). 
3. Blate, P. J.; George, N.; KO, W. L.; Murthy, A.;  Yoh-, J., 
"Physicomechanical Behavior of Rubberlike Materials'!, 
MATSCIT PS 64-8, California Institute of Technology, 
Pasadena, California (1964). 
-10- 
0.5 I' 
METAL TABS 
Fig. I STRIP SPECIMEN 
Fig. 2 STRAIN HISTORY 
-11- 
1 
1 
i 
1 
I 
1 
I 
I 
I 
I 
I 
i 
I 
I 
I 
I 
I 
I 
I 
I 
I 
TIME t 
Fig. 3 GEOMETRICAL INTERPRETATION OF 
EQUATION 4. 
-12. 
i 
m Q m 0 0  
1 i 1 1 1  
0 
0 
N 
0 
0 II 
0 
W 
ISd ""3 *'901 
-13- 
In 
O N  
00 
O 9  
0 
Lo 
\ 
5: 
W z a r 
t 
-I 
0 m 
LI 
0 
m 
23 
-1 
23 
n 
0 
2 
z 
I- 
X 
-1 
W 
Q= 
In 
Q, 
0 
a 
a 
i 
THICKNESS 0.1'' 
0.6 5 "4 
-0.75"- 
a )  ORIGINALLY 
0. 
0.378 'I- 
I 
b )  MOUNTED 
Fig. 6 RING SPECIMEN 
1.0 
0.8 
w 0.6 
7 
4 
E 
t- cn 
- 
064 
Od2 
I I I I I 
I 
. 
. 
Di  = 0.65 IN, 
Do= 0.75 IN. 
d = 0.378 IN. 
0 
0 0.2 0.4 0.6 0.8 I .O I .2 
EXTENSION AI, IN  
Fig.?’ COMPARISON OF VARIOUS S T R A I N  DEFINITIONS 
FOR THE RING SHOWN IN Fig.6. 
. 
W 
Z 
I: 
I- 
-I 
0 
v, 
a 
- 
Z 
0 
X 
-I 
W cr 
5 
a 
-1 7- 
8 
6 
l-4. 
0 
0 
(3 
0 
.I 
- 
2 
0 
-2 
I A STRIP SPECIMEN 
I I I I I 
-30 - 20 -10 0 IO 
TEMPERATURE, O C  
Fig. 9 COMPARISON OF THE TIME-TEMPERATURE 
SHIFT FACTOR FOR THE TWO SETS OF 
DATA 


On the relaxation modulus of the equivoluminal composition of solithane 113

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server