Chapter 2. Polymer Modeling
2.1 Experiments
2.1.1 Compression Test
The polymer (Bisphenol A) in the form of powder provided from Ad-group Taiwan was filled into a pre-designed stainless mold for fabricating the cylindrical specimens. In the beginning, the mold was putted into a vacuum oven and heated from room temperature to 75 Co within 50 minutes. During this process, the polymer was changed from powder state to liquid state with very high viscosity and its volume decreased due to gas disappearance, then, some powder was replenished until the desired amount of polymer was reached. In the next 8 minutes, the temperature was raised to 95 Co and then kept for 130 minutes. At the same time, the polymer was also degassing in the vacuum oven. After degassing for a period of time, the polymer was overflowed on the mold easily and we should open the door of the vacuum oven and scrape the polymer to retreat to cavities by using a thin plate.
It was to be noted that this scraping process must be finished as soon as possible (within 5 minutes) to avoid a large drop of oven temperature. After repeating the degassing and scraping process 6 times within the 130 minute, the temperature was held on 90 Co for 30 minutes to perform the curing process and raised to 145 Co within 10 minutes and maintained 60 minutes to carry out the post curing process.
After the curing and post-curing processes, the specimens were removed from the mold with care. In order to have parallel and smooth loading surfaces, all specimens were polished using a polishing machine with 25.0µ aluminum oxide powers. After polishing, the final dimensions for the specimens are 10 mm in height and 12mm in diameter as shown in Fig. 2.1(a). To demonstrate the strain rate effect on the polymer, compression tests were performed on the cylindrical specimens using hydraulic MTS machine at three different strain rates, 10-4, 10-2 and 1/s. Back to back strain gages were adhered on the specimens for the strain measurement during compression tests. Fig. 2.2 demonstrates the experimental setup for the compression tests. The stress history was obtained from the load cell and the associated strain history was measured from the strain gages mounted on the specimens. During the tests, both stress and strain signals were recorded by LabView together with PC computer. All results of compression test were shown in Fig. 2.3 and the Young’s modulus of the polymer was determined as 3.4 GPa.
2.1.2 Tensile Test
For measuring the Poisson’s ratio of the polymer, tensile tests were carried out on the coupon specimens, with the dimensions as shown in Fig. 2.1(b), fabricated in the same manner as described early excepted that the designed mode is different.
Two strain gages were mounted on the centers of the specimens. One was in the axial direction and the other was in the lateral direction to measure the axial and
transverse strains, respectively. The tensile test was implemented on a hydraulic MTS system at 10-4/s strain rate and the result was shown in Fig. 2.4. According to this result, the Poisson’s ratio of the polymer was evaluated as 0.37.
2.1.3 Measurement of Coefficient of Thermal Expansion
In the analysis of thermal residual stress effect, the coefficient of thermal expansion (CTE) of the matrix was measured first. A simple method [25-27] has been applied to finish this measurement in which the EA-06-062TT-120 strain gage was chosen and the adhesive M-bond 610 was used for its high operation temperature.
The EA-06-062TT-120 strain gage has two pieces of electrical resistance on a unit, one is an axial field and the other is transverse. Therefore, axial and transverse deformations of a specimen can be measured at the same time. Based on the strain gage technique, when the gage was subjected to a biaxial strain field, as shown in Fig.
2.5, the following relation was found
t t a
a F
R F
R = ε + ε
∆ (2.1.1)
where
R = original gage resistance F = axial gage factor a
F = transverse gage factor t
ε = axial strain field a
ε = transverse strain field t
Define the transverse sensitivity coefficient K as
a t
F
K= F (2.1.2)
If the strain gage was mounted on a specimen with Poisson’s ratio ν and the 0 specimen was under a uniaxial loading, the strain fields can be represented as
a 0 t =−ν ε
ε (2.1.3) Substituting eqns (2.1.2) and (2.1.3) into eqn (2.1.1) yields
(
0)
a g aa 1 K F
R F
R = −ν ε = ε
∆ (2.1.4)
where Fg =Fa
(
1−ν0K)
is the well-known gage factor and the measured strain can be represented asg
a F
R
∆R
=
ε (2.1.5) Since eqn (2.1.4) can be applied only if the specimen was subjected to a uniaxial stress field and the transverse strain field was due to the Poisson’s ratio effect only, on the measurement of CTE, the matrix under thermal expansion was within a biaxial strain field and eqn (2.1.4) can not be followed directly. Therefore, the transverse sensitivity must be embraced to correct the gage results. With the assistance of measured strains at the axial and transverse direction, ε and mx ε , my the corrected strains ε and x ε are given by [25] y
( ) ( )
2
my mx
0
x 1 K
K K
1
−
ε
− ε ν
= −
ε (2.1.6)
( ) ( )
2 mx my
0
y 1 K
K K
1
−
ε
− ε ν
= −
ε (2.1.7)
It can be shown that in the current analysis, the strain of isotropic test material with Poisson’s ratio equal to 0.37 under the same measured strain εmx =εmy will be about 2 % error if the correction equations (2.1.6) and (2.1.7) are not applied. It’s a slight effect so the correction hasn’t been done here.
It was noted that when the gage was mounted on a stress free specimen and underwent temperature change, we can not say the gage signal was fully induced by the specimen deformation but also affected by the thermal effect. To cancel the thermal effect on the electrical resistance, a half-bridge circuit as shown in Fig. 2.6
was applied [26]. There are two materials in the system, one is the test material and the other is the reference material. The CTE of the test material is unknown but known for the reference material. Since
T
r r x
x ∆
ε
−
= ε α
−
α (2.1.8) and
( )
∆ −∆ +∆ −∆
= +
∆
4 4 3
3 2
2 1
1 2 2 1
2 1
R R R
R R
R R
R R
R R V R
E (2.1.9)
where α is the CTE of the test material at measured direction, x α is the CTE of r the reference material, ε is the thermal strain from the test material and x ε is the r thermal strain from the reference material. By means of eqns (2.1.8) and (2.1.9), the thermal effect on the electrical resistance can be eliminated skillfully and the CTE of the test material can be determined.
The experimental system was placed in a programmable-control vacuum oven where the test material is a 30×20×2 mm3 thin plate and the reference material is a titanium silicate material with very small CTE (here we assume it is equal to zero).
A thermal couple was adhered on the reference material to record the history of temperature change but not on the test material due to the limitation of specimen size.
Because of low heating and cooling rate (about 19 0C/hr and 22 0C/hr, respectively), it can be assumed that the test and reference material possess the same temperature during heating and cooling processes so the temperature signal of the reference material can present the temperature of the test material, too. According to the final result shown in Fig. 2.7, the CTE of the matrix is about 5.9×10−5 o/ C from the average of heating and cooling slopes.