Korea-Australia Rheology Journal, 27(4), 259-266 (November 2015)
DOI: 10.1007/s13367-015-0026-8 www.springer.com/13367
Fang Tian
1
, Yingshe Luo
1,
*
, Shuiping Yin
1
, Hong Wang
2
and Chun Cao
3
1
College of Civil Engineering and Mechanics, Central South University of Forestry and Technology,
Changsha 410004, P.R. China
2
College of Science, Central South University of Forestry and Technology, Changsha 410004, P.R. China
3
China North Vehicle Research Institute, Beijing 100000, P.R. China
(Received December 15, 2014; final revision received October 8, 2015; accepted October 17, 2015)
The experimental research of dynamic viscoelastic properties of polyvinyl chloride was conducted by the dynamic mechanical analysis method in this paper. And the fitting equation of dynamic modulus of polymers has been presented. Based on the time-aging time equivalent principle, horizontal shift factor and vertical shift factor of aging time are carried out, which proposes a novel method for the research on time-aging time equivalent analysis of dynamic mechanical properties of polymers during physical aging.
Keywords : aging, polyvinyl chloride, dynamical viscoelasticity
Polymers have been widely applied in many fields such as aerospace, machinery manufacturing, building energy and biological engineering due to their high strength, light weight, corrosion resistance and good glass-forming ability (Ma, 2012; Tian et al ., 2013). Compared with metals and ceramics, one of the most important mechanical properties of polymers is their viscoelastic properties, which means that the stress response of polymers depends not only on the strain but also the changing rate of strain, that is, the mechanical properties of polymers are time-dependent. There is a characteristic time scalar exiting in polymers, which is different to natural time scalar and sensitive to environmental factors, such as temperature, aging, stress, strain and so on (Zhao et al ., 2008). So, viscoelastic solids usually exhibit complicate mechanical behaviors.
Previous researchers have made lots effort to establish relationships between the characteristic time scalar of polymers and various influence factors. Based on the free volume theory, first proposed by Flory et al . (Knauss and
Emri, 1987), Williams et al . (Williams et al ., 1955) introduced the temperature shift factor and established the time-temperature equivalent function (WLF function) by supposing that the effect of temperature on viscoelastic properties of polymers is equivalent to the horizontal shift of time axis and all the viscoelastic curves with different temperatures keep in the same shape. Based on WLF function, Luo et al . (2001) further considered the effect of stress and established the time-temperature-stress equivalent function, which makes it possible to predict the vis-
#
This paper is based on work presented at the 6th Pacific Rim
Conference on Rheology, held in the University of Melbourne,
Australia from 20 th
to 25 th
July 2014.
*Corresponding author; E-mail: lys0258@sina.com
© 2015 The Korean Society of Rheology and Springer coelastic behavior polymers with low-stress level and long-time load by shift-factor introduced superposition method. Based on experimental research, Plazek (Plazek et al ., 1995) pointed out that the mechanical behaviors of many polymers are not consistent with the WLF function for viscoelastic modulus curves with temperature change not only their position but also their shape, which is known as the complexity of thermal rheology. Thus, in this paper, considering the change of position and shape of modulus curves, the horizontal shift factor and the vertical shift factor are introduced at the same time.
With the passage of time, reversible physical changes and irreversible chemical changes occur continuously in the interior of polymers, which affects many mechanical properties of polymers, performances of polymer-based composites and so on. The occurrence of aging causes deterioration of mechanical properties of polymers, leading to its premature failure that not only makes the economy suffered huge losses, leading to a waste of resources, but also causes serious environmental pollutions due to the decomposition of materials (Liu et al ., 2012). Struik made lots of experimental researches on 40 different kinds of polymers and pointed out that viscoelastic mechanical behaviors of polymers are affected by physical aging and then established the time-aging time superposition principle based on his experiments (Struik, 1987a; 1987b; 1989a;
1989b). Based on Struik’s research, Shaukat discussed the practical application of the time-aging time superposition principle (Montes et al ., 2006). Joshi (Awasthi and Joshi,
2009; Joshi, 2014) analyzed the relationship between real time and aging time by the effective time approach and further predicted the long-term creep behavior of polymers.
However, previous researches on the time-aging time equivalent principle mainly focused on conditions of static load, which failed to take dynamic loads and multi-temperature tests into consideration (Bandyopadhyay et al ., pISSN 1226-119X eISSN 2093-7660 259

2010; Chen et al ., 2010). Yet, in fact, structures made by polymers and polymer-based composites are usually subject to linear and nonlinear dynamic loads in their working environments accompanied by physical aging (He et al .,
2007; Zhang et al ., 2007) such as rotating tires, transferring gears and damping rubbers and so on (Guo, 2002).
So, the importance of research on dynamic viscoelasticity of polymers during physical aging can never be exaggerated.
This paper makes an experimental research on effects of physical aging on linear dynamic viscoelastic mechanical behaviors of polymers with different testing temperatures by dynamic mechanical thermal instrument. Furthermore, the application of time-aging time superposition on dynamic conditions is discussed in this paper.
A proper function to describe relationships between dynamic modulus and load frequency is the foundation of research on dynamic viscoelastic properties of polymers with different aging time. Presently, study about relationships between dynamic viscoelastic properties and load frequencies is still not enough, while research methods for static viscoelastic properties are relatively mature. Struik used the three-parameter Kohlrausch model to describe short-term creep curves of polymers accompanied by physical aging and made a comparison of fitting curves and relevant experimental data, which proved that the three-parameter Kohlrausch model is suitable to describe short-term creep curves of polymers. The fitting function of creep compliance is as follows:
S
( log t ; S
0
,
γ
,
β
; t e
)
= S
0
exp ⎝
⎛ log t ⎞
γ ⎠
β
(1) where S is the compliance; t is the test time; S
0
,
γ
,
β
are fitting parameters, t e
is the aging time. For creep compliance of polymers is a function of the aging time, fitting parameters S
0
,
γ
,
β
are related to the aging time t e
.
Based on the fitting function of creep compliance, Eq.
(1), supposing
β
= 1 for simplicity and replacing the test time t with the test frequency f , the fitting function of dynamic modulus can be written as follows:
E
( log f ; S
0
,
γ
; t e
)
= S
0
exp
⎛
⎝ log f
γ
⎞
⎠
R ( log f ) = E ( log f ; S
R
, γ
R
; t eref
)
(2) where E is the dynamic modulus, f is the test frequency.
For further discussion, a reference aging time should be chosen out based on experiments. The dynamic modulus curve of polymer sample undergoing the reference aging time is called the reference modulus curve, which is given by:
(3)
Fang Tian, Yingshe Luo, Shuiping Yin, Hong Wang and Chun Cao where t eref
is the reference aging time; S
R
,
γ
R
are fitting parameters of reference modulus curve.
Shift factors can be obtained by analyzing relationships between the reference modulus curve R (log f ) and dynamic modulus curves S with different aging time, which makes it possible to predict dynamic modulus of polymers with an arbitrary aging time t e
based on the reference modulus curve R .
Taking horizontal shift into consideration, the function is given by:
E
( log f ; S
0
,
γ
; t e
)
=
( t log f
) (4) where a t
is the aging-time horizontal shift factor.
When considering horizontal shift and vertical shift of modulus curves at the same time, the function is given by:
E ( log f ; S
0
, γ ; t e
) = b t
( t log f ) (5) where b t
is the aging-time vertical shift factor.
Putting Eqs. (2) and (3) into Eq. (5), the dynamic modulus function considering the aging-time horizontal shift factor and the aging-time vertical shift factor is given by:
E ( log f ; S
0
, γ ; t e
) = S
0
exp
⎝
⎛ log t ⎞
γ ⎠
= b t
S
R
exp
⎛
⎝ a t log f
γ
R
⎞
⎠
.
(6)
Then the aging-time horizontal shift factor and the agingtime vertical shift factor are separately given by: a t
=
γ
R
, b t
=
S
-----
S
0
R
.
(7)
(8)
Experimental samples of polyvinyl chloride (PVC) were cut into strips with dimension of 50 mm × 5 mm × 1 mm.
In order to eliminate the effects of thermal history, test samples were put into an air convection oven at 90 o
C for
30 minutes before suffered by thermal physical aging at
55 o
C for various periods.
Frequency sweep tests were conducted by dynamic mechanical analyzer, GABO EPLEXOR 500N. Strain controlling method was adopted in all tests of this paper.
The maximum static strain is 1% and the maximum dynamic strain is 0.05%. In frequency sweep tests, the frequency scan range was from 0.01 Hz to 100 Hz, and nine constant temperature levels (15 o
C, 55 o
C,
65 o
C, 70 o
C, 75 o
C, 80 o
C, 85 o
C, 35 o
C, 45 o
C) were considered. Values of storage modulus and loss modulus with different test temperatures and load frequencies were recorded by frequency sweep tests.
260 Korea-Australia Rheology J., 27(4), 2015

Dynamic viscoelastic properties of polyvinyl chloride with physical aging
Fig. 1.
(Color online) Test data and fit curves of storage modulus E' of PVC with different aging time.
Experimental data of storage modulus and loss modulus of PVC samples with different temperature levels (15 o
C,
35 o
C, 45 o
C, 55 o
C, 65 o
C, 70 o
C, 75 o
C, 80 o
C, 85 o
C) and different aging time (20 d, 40 d, 60 d) were separately shown in Figs. 1 and 2 and fitted by dynamic modulus function,
Eq. (2).
The experimental data of storage modulus are shown in
Fig. 1 as hollow points, and solid lines are corresponding
Korea-Australia Rheology J., 27(4), 2015 fitting curves of the dynamic modulus function, Eq. (2), wherein parameters of this fitting function are shown in
Figs. 2 and 3. As shown in Fig. 1a-c, the modulus change with aging time depends on the test temperature rather than the test frequency, that is, the modulus changing with aging time keep the same variation tendency at different test frequency but different variation tendency for different test temperature. For that, taking the fixed frequency f
= 10 Hz as an example, the modulus changes with aging time at different test temperature are shown in Fig. 1d-f. It
261

Fang Tian, Yingshe Luo, Shuiping Yin, Hong Wang and Chun Cao
Fig. 2.
(Color online) Fit parameter S
0
′ of storage modulus E' .
Fig. 3.
(Color online) Fit parameter
γ ′
of storage modulus E' .
can be seen that with increasing aging time periods when the test temperature is lower than the aging temperature
55 o
C, values of storage modulus decrease, and fitting parameters S
0
' and
γ
' present the same tendency; when the test temperature is equal to the aging temperature 55 o
C, values of the storage modulus do not change much; when the temperature is higher than the aging temperature 55 o
C, values of storage modulus and values of fitting parameters
S
0
' and
γ
' show increasing trends as a whole.
Experimental data of loss modulus of PVC samples are shown in Fig. 4 as hollow points, and solid lines are corresponding fitting curves of the dynamic modulus function, Eq. (2), wherein parameters of this fitting function are shown in Figs. 5 and 6. It can be seen that when the test temperature is lower than or equal to the aging temperature 55 o
C, fitting parameters S
0
'' and
γ
'' increase with increasing test temperature; when the test temperature is between 65 and 70 o
C, there is a peak value of the fitting parameter S
0
' , and values of fitting parameters
γ
' change discontinuously around temperature 65 o
C. In addition, the minimum value of parameter
γ
'' appears at 65 o
C, while
262
Fig. 4.
(Color online) Test data and fit curves of loss modulus E'' of PVC with different aging time.
the maximum value is at 70 o
C during the test temperature range; When the test temperature is higher than or equal to 75 o
C, fitting parameters S
0
'' and
γ
'' are decreased with increasing test temperature.
Figs. 1 and 4 show that the dynamic modulus fitting function, Eq. (2) presented in this paper is a proper mechanical model to simulate storage modulus curves and loss modulus curves of PVC samples. In addition, experimental results show that fitting parameters of the function, Eq. (2) are closely related to test temperatures.
Korea-Australia Rheology J., 27(4), 2015

Dynamic viscoelastic properties of polyvinyl chloride with physical aging
Fig. 5.
(Color online) Fit parameter S
0
″ of loss modulus E'' .
Fig. 6.
(Color online) Fit parameter γ 0
of loss modulus E'' .
Taking 20 days as the reference aging time and putting fitting parameters S
0
' and
γ
' of storage modulus (shown in
Figs. 2 and 3) into the aging-time horizontal shift factor function, Eq. (7) and the aging-time vertical shift factor function, Eq. (8), values of the aging-time horizontal shift factor a t
' (shown in Table 1) and the aging-time vertical shift factor b t
' (shown in Table 2) of storage modulus curves of samples with different test temperatures and aging time periods can be carried out. According to Table
1 and Table 2, values of the horizontal shift factor a t
' and the vertical shift factor b t
' of storage modulus of samples with the reference aging time are equal to 1. When the test temperature is lower than the aging temperature 55 o
C, the aging-time horizontal shift factor a t
' would be greater than or equal to 1 (only when t e
= t eref
= 20 d), and the agingtime vertical shift factor b t
' would be less than or equal to
1 (only when t e
= t eref
= 20 d), which indicates that with increasing aging time periods storage modulus curves of
PVC samples would generate a shift horizontally to right and vertically to downwards relative to reference storage modulus curves, which means that effects of aging on storage modulus equal to increasing frequency and temperature; when the test temperature is above the aging temperature 55 o
C, the aging-time horizontal shift factor a t
' would be less than or equal to 1 (only when t e
= t eref
= 20 d), and the aging-time vertical shift factor b t
' would be greater than or equal to 1(only when t e
= t eref
= 20 d), which suggests that with increasing aging time periods storage modulus curves of PVC samples would make a shift horizontally to left and vertically to upwards relative to their corresponding reference storage modulus curves, which implies that effects of aging equal to decreasing frequency and temperature. Therefore, values of the agingtime horizontal shift factor a t
' and the aging-time vertical shift factor b t
' of PVC samples are closely related to test temperatures that greatly affect the movement of molecular chains.
Based on the aging-time shift factor (shown in Tables 1 and 2), experimental data of storage modulus (shown in
Fig. 3) are shifted horizontally and vertically (shown as hollow dots in Fig. 7). Taking fitting parameters S
0
' and
γ
' of samples with the reference aging time into the dynamic
Table 1.
Aging-time horizontal shifting factors a ′ t of storage modulus E' .
a
′ t
15 o
C 35 o
C 45 o
C 55 o
C 65 o
C 70 o
C 75 o
C 80 o
C 85 o
C
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1.1060 1.1372 1.0390 0.9788 0.9160 0.8917 0.9478 0.9597 0.9482
1.1590 1.1442 1.1800 1.0169 0.8221 0.8131 0.9531 0.9789 0.9384
Table 2.
Aging-time vertical shifting factors b ′ t of storage modulus E' .
b
′ t
15 o
C 35 o
C 45 o
C 55 o
C 65 o
C 70 o
C 75 o
C 80 o
C 85 o
C
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
0.9734 0.9831 0.9798 0.9829 1.0434 1.1801 1.1910 1.2353 1.1956
0.9629 0.9626 0.9542 0.9899 1.1073 1.2980 1.2388 1.2076 1.2332
Korea-Australia Rheology J., 27(4), 2015 263

Fang Tian, Yingshe Luo, Shuiping Yin, Hong Wang and Chun Cao
Fig. 7.
(Color online) Aging-time shifting curves of storage modulus E' of PVC ( t eref
= 20 d ).
modulus function, Eq. (2), the reference storage modulus curves is derived (shown as solid lines in Fig. 7). Fig. 7 shows that reference fitting curves of storage modulus can be used to describe the variation of storage modulus of
PVC samples along with the load frequency by using the aging-time horizontal shift factor a t
' and the aging-time vertical shift factor b t
' .
Taking fitting parameters S
0
'' and
γ
'' of loss modulus
(shown in Figs. 5 and 6) into the aging-time horizontal shift factor function, Eq. (7) and the aging-time vertical shift factor function, Eq. (8), values of the aging-time horizontal shift factor a t
'' (shown in Table 3) and the agingtime vertical shift factor b t
'' (shown in Table 4) of loss modulus curves of samples with different test temperatures and aging time periods can be carried out. Values of the horizontal shift factor a t
'' and the vertical shift factor b t
'' of loss modulus of samples with the reference aging time are equal to 1. When the value of the aging-time horizontal shift factor a t
'' is greater than 1, it’s suggested that loss modulus curve generate a shift to right related to their corresponding reference loss modulus curves; when the value is less than 1, there would be a shift to left. When the value of the vertical shift factor b t
'' is greater than 1, it’s indicated that loss modulus curves generate a shift to upward related to their corresponding reference loss modulus, and when the value is less than 1, there would be a shift to downward. Thus, when the test temperature is above on or equals to 70 o
C, values of the horizontal shift factor a t
'' are less than (except for t e
= 60 d, T = 65 o
C) or equal to (only when t e
= t eref
= 20 d) 1, and values of the vertical shift factor b t
'' are greater than or equals to (only when t e
= t eref
= 20 d) 1, which proves that with increasing aging-time periods the loss modulus of PVC samples would shift horizontally to left and vertically to upward, indicating that effects of aging on loss modulus equal to decreasing frequency and temperature; when the test temperature is lower than 70 o
C, the aging-time shift factors of loss modulus are greatly affected by temperatures.
Based on the aging-time shift factor (shown in Tables 3 and 4), experimental data of loss modulus (shown in Fig.
Table 3.
Aging-time horizontal shifting factors a t
″ of loss modulus E''.
a t
″
15 o
C 35 o
C 45 o
C 55 o
C 65 o
C 70 o
C 75 o
C 80 o
C 85 o
C
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
0.5945 1.2204 1.0153 1.0469 1.0010 0.6949 0.9220 0.9509 1.0193
0.7288 1.1958 1.2081 1.0089 0.9472 0.4563 0.9072 0.9554 0.9873
Table 4.
Aging-time vertical shifting factors b t
″ of loss modulus E'' .
b t
″ 15 o
C 35 o
C 45 o
C 55 o
C 65 o
C 70 o
C 75 o
C 80 o
C 85 o
C
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
0.9812 0.9879 0.9992 0.9965 0.9273 1.0005 1.0883 1.1238 1.0018
1.0223 1.0654 1.1091 0.9358 0.8807 1.0176 1.1257 1.1405 1.0885
264 Korea-Australia Rheology J., 27(4), 2015

Dynamic viscoelastic properties of polyvinyl chloride with physical aging
Fig. 8.
(Color online) Aging-time shifting curves of loss modulus E'' of PVC ( t eref
= 20 d ).
8) are shifted horizontally and vertically (shown as hollow dots in Fig. 8). Taking fitting parameters S
0
'' and
γ
'' of samples with the reference aging time into the dynamic modulus function, Eq. (2), reference curves off loss modulus are derived (shown as solid line in Fig. 8). From this figure, it can be seen that reference curves of loss modulus can describe the variation of loss modulus of PVC samples along with the load frequency by using the aging-time horizontal shift factor a t
'' and the aging-time vertical shift factor b t
'' .
This paper presents a dynamic modulus fitting function of polymers, which is in good agreement with the experimental data. It can be used for both storage modulus and loss modulus calculations of PVC. Based on the dynamic modulus fitting function, this paper introduces the aging time horizontal shift factor and vertical shift factor and establish the equivalent relationship of dynamic modulus of PVC with different aging time. And the effects of test temperatures on shift factors are also discussed in this paper. A new method for research on time-aging time equivalent principle of dynamic mechanic behaviors of polymers is presented.
This work was supported by the National Natural Science Foundation (No.11072270), the Education Department of Hunan Province (No.12C0464) and the Postgraduate
Innovation Funding of Central South University of Forestry and Technology (No. CX2013B33).
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