Mechanical properties of PA and GFRPA with controlled crystallinity
Few papers consider the effects of both fiber volume fraction and crystallinity when estimating the mechanical properties of a material. Figure 4 shows the mechanical properties obtained from bending tests performed on each PA and GFRPA specimen. Both strength and modulus increased with increasing crystallinity and fiber volume fraction such that the materials’ bending behavior changed from ductile to brittle.
Effect of crystallinity on the creep behavior of PA and GFRPA
Previous studies Sakai and Somiya (2006, 2009, 2011, 2012), Sakai et al. (2007, 2011a, 2011b, 2015) and Tanks et al. (2016) have detailed creep behavior as a function of time, temperature, fiber volume fraction, physical aging, and crystallinity. These reports discussed the application of the TTSP and proposed a master curve of creep compliance showing the effect of each influencing variable. In the current study, creep tests were performed at elevated temperatures to assess the effects of crystallinity on the creep behavior of PA and GFRPA.
To evaluate the effect of temperature on creep phenomena, creep compliance curves were generated at each test temperature. The creep compliance curves for PA (crystallinity: χ = 32%) are presented in Fig. 5a, which shows that higher test temperatures and longer creep test durations resulted in higher creep compliance. A TTSP master curve of creep compliance is shown in Fig. 5b for a reference temperature of 40 °C. The master curve was obtained by replacing the realtime t for each shifted curve with the physical time t’ at the reference temperature T_{0}. The amount that each individual curve needed to shift to create the master curve is the time–temperature shift factor. Shift factors for PA are plotted as an Arrheniustype plot in Fig. 6. This curve is a straight line at temperatures above 40 °C, indicating that creep phenomena comply with the TTSP in Arrhenius mode. Specimens of PA with other degrees of crystallinity and GFRPA exhibited the same trends as those in Figs. 5 and 6.
To confirm the effects of crystallinity on creep behavior, Fig. 7a shows all of the master curves for PA. These data show that less crystallinity and longer creep test duration resulted in higher creep compliance. Master curves of crystallinity were generated by shifting individual curves horizontally and vertically until they overlapped with the reference curve. Figure 7b shows the grand master curve for crystallinity. These data show that creep compliance curves of PA specimens with different crystallinity are components of the grand master curve. Furthermore, the shape of the grand master curve is the same as those of creep compliance curves obtained from specimens of varying crystallinity. This shows that creep behavior was the same regardless of crystallinity. The shift factors along the horizontal and vertical directions are shown in Fig. 8. Vertical shifts show a change in modulus while horizontal shifts are indicative of time retardation effects. Crystalline domains delayed creep behavior. It is thought that this delay was due to decreasing molecular mobility with increasing crystallinity and consequent increases in viscosity. The same trends were observed with GFRPA, for which the creep deformation of materials with arbitrary crystallinity could be estimated using only the master curve and shift factors for each fiber volume fraction.
The effects of fiber volume fraction on the creep behavior of PA and GFRPA
Next, the effects of fiber volume fraction on creep behavior were estimated for PA and GFRPA specimens at a fixed crystallinity of χ = 45% using the TTSP in Arrhenius mode. Master curves of PA and GFRPA were generated at elevated temperatures with TTSF of PA resin, as indicated in Fig. 9a, by shifting creep compliance curves horizontally until they overlapped. Figure 9a shows that lower fiber volume fractions and increased creep test duration yielded higher creep compliance. To confirm this effect, a grand master curve for fiber volume fraction, shown in Fig. 9b, was generated by shifting individual curves horizontally and vertically until they overlapped with a reference master curve at V_{f} = 0%. These data show that the creep behavior of GFRPA was the same as that of PA and that, furthermore, the creep behaviors of PA and GFRPA depended on the behavior of the PA resin. The horizontal and vertical shift factors for fiber volume fraction are shown in Fig. 10. Vertical shifts were due to changes in modulus with increasing fiber content. Horizontal shifts were due to time retardation effects.
Longterm estimation of creep deformation accounting for the effects of crystallinity and fiber volume fraction
Above, we discussed the separate influences of crystallinity and fiber volume fraction on creep behavior. Two effects were observed with both variables: changes in material modulus, and a retardation effect on time. We then tried to elucidate the relationship between the two variables. The shapes of the grand master curves in Figs. 7b and 9b are similar, so we tried to superimpose these into a single, greatgrand master curve. To generate such a curve requires that a suitable reference curve be chosen. To do this, one must consider the relationship between the two grand master curves at χ = 32% and V_{f} = 0%. The crystallinity of specimens used to generate the grand master curve for fiber volume fraction was χ = 45%. This curve was then shifted both horizontally and vertically, using the modulus and time shift factors for crystallinity, to create a grand master curve for crystallinity. Thus, we obtained the greatgrand master curve shown in Fig. 11. The greatgrand master curve is smooth, indicating that the effects of crystallinity and fiber volume fraction on creep behavior were similar. Furthermore, these data indicate that creep behavior can be controlled by adjusting the crystallinity and/or fiber volume fraction in accordance with the greatgrand master curve and associated shift factors.
Estimation of creep deformation
Creep deformation under the influence of different variables can be estimated using only the greatgrand master curve and associated shift factors. First, however, one must obtain an approximate equation of the master curve. An approximate equation, determined through a Prony series by Schapery (Schapery 1969; Schapery 1997; Lou and Schapery 1971), is shown in Eq. (4).
$$ {D}_c(t)={C}_e+\sum \limits_{i=1}^N{C}_i\left(1{e}^{\frac{t}{\tau_i}}\right) $$
(4)
C_{e}: initial creep compliance, C_{i}: relaxation modulus, τ_{i}: relaxation time
Creep deformation can be estimated using Eq. (4). However, the data used to generate the greatgrand master curve in this study were better fitted by Eq. (5), which is the same equation as that representing the Maxwell model. Despite the good fit with the experimental data, the Maxwell model is not applicable since it does not have any physical meaning pertaining to creep functions.
$$ {D}_c(t)={C}_e+{C}_1\exp \left(\frac{t}{\tau_1}\right) $$
(5)
To estimate creep behavior, the effects of variables that influence shift factors were built into Eq. (1). The shift factors were multiplied by the test duration, t, since they exhibited retardation effects on time. Equation (6) calculates creep compliance taking into account the effects of time, temperature, and fiber volume fraction.
$$ {D}_C\left(t,T,{V}_f\right)={D}_C\left(t\times {a}_{TR}\times {a}_{Tt^{\hbox{'}}}\right)+{a}_{TD_C} $$
(6)

a_{TR}: Timetemperature shift factor,

a_{Tt’}: Time shift factor for fiber content,

a_{TDc}: Modulus shift factor for fiber content,

V_{f}: Fiber volume fraction
Equation (7) calculates creep compliance taking into account the effects of time, temperature, and crystallinity.
$$ {D}_C\left(t,T,C\right)={D}_C\left(t\times {a}_{CTR}\times {a}_{cTt^{\hbox{'}}}\right)+{a}_{cTD_C} $$
(7)
These equations show that creep analyses incorporating the greatgrand master curve equation and shift factors for each of the influencing variables are useful for estimating creep behaviors. In addition, the suitability of the TTSP in these analyses was validated by the observed effects of fiber volume fraction and crystallinity.
Creep deformation under arbitrary conditions can be estimated using only the greatgrand master curve and associated shift factors, as shown in the block diagram of Fig. 12. The process for estimating behaviors is the reverse procedure of creating a greatgrand master curve. Experimental values and estimated values for GFRPA (V_{f} = 7%, χ = 34%) are compared in Fig. 13. The test conditions were 50 °C with a creep time of 1000 min and 14 MPa of applied stress. Note that the estimated data are in agreement with the experimental data. Of course, there is some difference between estimated and experimental data. Usually, experimental data has a dispersion as 5 ~ 10%. Considering this dispersion, it can be said that our method can estimate long term creep behavior.
This study proposes a new means of estimating material properties based on the application of the timetemperature superposition principle, taking into account the effects of fiber volume fraction and crystallinity. We also showed that our estimates of creep deformation fitted well with experimental results. Based on these findings, it should be possible to control the creep deformation of a polymer material or fiber composite by controlling the fiber volume fraction and crystallinity.