3.3.2. Long-Term Startup Shear with *t*<sup>0</sup> = 100 s

The predictions of the trapezoidal-loop experiments with maximum shear rates of 0.1 s−<sup>1</sup> , 1 s−<sup>1</sup> , and 5 s−<sup>1</sup> , *t*<sup>0</sup> = 100 s and *t*<sup>1</sup> = 200 s are shown in Figure 12, where the modified calculation of the experiment with 0.1 s−<sup>1</sup> is also given simultaneously. It is obvious that all three trapezoidal-loop experiments and calculations do not exhibit strain strengthening behavior due to the large *t*0. Moreover, both a slight and a severe shear-weakening behavior occur at maximum shear rates of 1 s−<sup>1</sup> and 5 s−<sup>1</sup> , respectively, during long-term shear. The calculations of the RS model reaffirm the shear-weakening phenomenon of the melt.

According to the experimental and calculated loops, both shear strain strengthening and stress overshoot are time-dependent viscoelastic behaviors. Shear strain strengthening may not be evidently affected by the shear-weakening behavior of viscoelastic fluid due to short-term startup shear, which could be more suitable for illuminating the elastic behavior of fluid; moreover, stress overshoot could contain some shear-weakening effect due to long-term shear sometimes at a shear rate, which complicates stress overshoot.

**Figure 12.** Predictions of the trapezoidal-loop experiments with the maximum shear rates of 0.1 s−<sup>1</sup> , 1 s−<sup>1</sup> and 5 s−<sup>1</sup> , *t*<sup>0</sup> = 100 s and *t*<sup>1</sup> = 200 s. Solid line is the calculation by the RS model, and dashed line is the modification with *f* = 0.899 in Table 4.

#### *3.4. f Effect*

In Sections 3.2 and 3.3, the influences of possible sample deviations were analyzed using the changing spectrum method, and the modified calculations with *f* in Table 4 by the RS model are in good agreement with the loop experiments. The effects of four *f*-parameters on the original relaxation spectrum in Table 2 are shown in Figure 13, accompanied by the present experimental G0 and G" in Figure 2 and the previous [31]. The calculations of four changed spectra show an approximate parallel property to the original spectrum. The modified G0 and G" for the previous frequency sweep experiment with *f* = 1.314 are acceptable although there are some deviations. It is unclear whether the difference between the present G0 and G" and the previous is caused by the sample, nonetheless, the changing spectrum method is effective.

**Figure 13.** Calculated frequency sweep in small amplitude oscillatory shear mode using some spectra with *f* in Table 4, the present G0 and G" experimental data (square symbol) and the reported G0 and G" [31] (triangle symbol). (**a**) G0 , and (**b**) G".

Eight groups of experiments have been modified with *f* in Table 4, in which the largest *f* is 1.314 for discerning the difference between two frequency sweep experiments. Another *f* varies from 0.758 to 1.145. Therefore, the relative error between the present frequency sweep experiment and the other experiments is about 10–30%, and the definition of relative error is |*f* − 1|/1 × 100. If the error is regarded as sample deviation, the error includes not only the deviation between different batches of the molded sheets but one batch.

#### *3.5. Discussion on the Shear Weakening Behavior*

It can be assumed that the shear weakening phenomenon of a viscoelastic fluid is the result of a temporal, irreversible change of the structure in the fluid, which can lead to changes in the viscoelastic property during the shearing process. Then, shear weakening behavior cannot be described by the constitutive equation without considering the structure effect on the change in viscoelasticity. In order to understand the structure effect of longtime shearing on the change in linear viscoelasticity, a composite experiment including both the trapezoidal-loop and the dynamic frequency sweep should be conducted. For example, we first conduct the trapezoidal loop experiment with a maximum shear rate of 5 s−<sup>1</sup> , *t*<sup>0</sup> = 100 s, and *t*<sup>1</sup> = 200 s, then, simultaneously, we carry out the dynamic frequency sweep at a small strain of 5%. However, the composite cannot be carried out, since the ARES rheometer has a limitation in which the dynamic test must be conducted before the transient or the steady shear test. The startup position of dynamic sweep is fixed for the ARES rheometer, and the position at the end of the loop experiment is unlike that required in the dynamic sweep. No alternative method was found to solve the position problem and the scheme was abandoned.

Another approach to qualitatively research the influence of nonlinear large strain shear on the change of linear viscoelasticity for the LDPE(Q200) melt at 150 ◦C was then adopted, i.e., both the nonlinear and linear shear experiments are accomplished in dynamic mode. A time sweep of 300 s at the large strain of 300% with an angle frequency of 1 rad/s was carried out first, and then, the frequency sweep in small amplitude oscillation shear at a 5% strain was immediately conducted. The experimental G0 and G" data in two-time sweep experiments at the large strain of 300% are shown in Figure 14, which decrease with time due to the nonlinear shear effect. The G0 and G" in two small-amplitude oscillation shear experiments obtained upon the cessation of the time sweep of 300 s at the large strain in Figure 15 are reduced by comparing with the normal G0 and G". The decrease in G0 and G" in Figures 14 and 15 could be the manifestation of shear weakening phenomenon. The linear viscoelasticity of the melt could be changed by the shear weakening effect in shearing.

**Figure 14.** Dynamic time sweeps at large strain, *γ* = 300%, in dynamic shear mode for the LDPE melt at 150 ◦C (*ω* = 1 rad/s).

**Figure 15.** Dynamic frequency sweep experiments at = 5% for the LDPE melt at 150 ◦C, which are obtained after the time sweeps at the large strain in Figure 14, and the fit on the frequency sweep experiments.

To understand the qualitative influence of the altered linear viscoelasticity on the calculation of the loop experiment, a new relaxation spectrum of the LDPE melt was fitted to the G0 and G" obtained using the shear weakening process in Figure 14 and given in Table 5; the calculated linear viscoelastic property with the new spectrum is shown in Figure 15. Assuming that the nonlinear characteristic of the LDPE melt is not influenced by the shear weakening behavior, the triangular loop experiment with the maximum shear rate of 5 s−<sup>1</sup> and *t*<sup>0</sup> of 100 s was calculated again with the spectrum in Table 5 and shown in Figure 8d by a dashed line. The new spectrum evidently reduces the calculated stress of the triangular loop, indicating the qualitative effect of the change of linear viscoelasticity on the description of the shear weakening behavior of the viscoelastic fluid.

The linear viscoelasticity or the relaxation spectrum is unchanged in the characterization of the viscoelastic property of a fluid using the RS model. Thus, the RS model is unsuitable for describing the shear weakening flow behavior of viscoelastic fluid caused possibly by the change of microstructure under large-strain shear. Some structuralized viscoelastic theories [40,42,45,46] may be available in characterizing the shear weakening behavior of the viscoelastic fluid. It is worth noting that the shear weakening in the triangular loop experiment with the maximum shear rate of 5 s−<sup>1</sup> and *t*<sup>0</sup> of 100 s in Figure 8d was enveloped by the calculations with both the original spectrum in Table 2 and the new spectrum in Table 5, which indicates that the structure-dependent spectrum approach [40] may be valid in describing the shear weakening behavior.

One crucial issue is whether the shear weakening behavior of the LDPE melt is the manifestation of experimental errors, e.g., edge fracture and loss of material from the gap. Some authors [16,22,47] have emphasized the effect of edge fracture on the shear weakening in the step rate experiment. However, shear weakening may be induced by other factors, such as the shear-structure effect stated above. Previously, shear modification or shear refinement of polythene [48] was reported from an industry viewpoint, and shear modification refers to the reduction in viscoelasticity in shearing, similar to the shear weakening behavior. Rudin [49] (1983) published a review on shear modification, and the authors in [50,51] introduced the possible mechanisms of shear modification. Therefore, the edge fracture effect is still an issue for shear weakening behavior.

#### **4. Conclusions**

Five types of viscoelastic behaviors specified in the triangular-loop experiments of the LDPE (Q200) melt reported previously [35] were predicted in this study using the Rivlin–Sawyers (RS) model. The viscoelastic properties of the melt were characterized

using both the frequency sweep in the small-amplitude oscillation shear experiment and the steady shear viscosities. Type I–IV viscoelastic behaviors can be predicted by the RS model, although some errors appear in the experimental work. Type V flow cannot be predicted, which shows shear-weakening behavior. Most of the calculations agree with the experiments, indicating that both the previous experiments and present theoretical analyses are reasonable.

The trapezoidal-loop experiments of the melt in [35] were also predicted. The shear strain strengthening behavior occurring during the short-term startup shear process in the trapezoidal-loop experiment can be described well by the RS model, which is caused by both the growth of strain and shear rate effect and is similar to that in the triangular-loop experiment. The shear-weakening behavior in the trapezoidal-loop experiment also cannot be described by the model.

Differences between some of the calculations and experiments could be caused by sample deviation, which can be discerned through the changing spectrum approach. The modified calculations with the changing spectrum approach show agreement with the experiments. The large relative error in the experiment is about 10–30%.

The shear weakening behavior of the melt is an issue that can be studied further from two perspectives. One is to study the influence of edge fracture, and the other is to study the effect of changed linear viscoelastic property induced by shear, which could lead to a reasonable description of the shear weakening behavior by theoretical model if the structure effect is real.

**Funding:** This research was partly funded by the National Natural Science Foundation of China, grant number 10402024.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data that supports the findings of this study are available within the article.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**

