5.4 Result and Discussion
5.4.1 Results with Fuzzy Parameters for Steel Material
Several authors including Dixit and Dixit [1996] and Li and Kobayashi [1982] used the experimental results of Shida and Awazuhara [1973] and Al-Salehi et al. [1973]
for validation of their models. In this work, the roll force and roll torque values are compared with the experimental results of Shida and Awazuhara [1973] and Al- Salehi et al. [1973] and finite element results of Dixit and Dixit [1996].
The inherent variations in the experimental investigation causes scatter in the experimental data and a large deviation with the FEM results. Due to this, Dixit and Dixit [1996] considered the material and the process parameters as fuzzy parameters. Also, it is observed experimentally and analytically that the value of the friction coefficient varies in the roll gap instead of being constant [Rooyen and Backofen, 1957; Al-Salehi et al., 1973; Lim and Lenard, 1984 and Rao and Lee, 1989]. Thus, Dixit and Dixit [1996] considered the input parameters i.e. initial values of yield strength
( )
σY 0, strain hardening coefficients b and n, and coefficient of friction f to be four fuzzy parameters which are characterized by their membership functions. Standard methods are available in the textbook for constructing the membership function. Dixit and Dixit considered a linear triangular membership function. In this process of determining the fuzzy parameters, parameters are divided into three subsets such as low, most likely and high. A triangular membership function is given by0
( )
0
x l
x l l x m
x m l
h x m x h
h m
x h µ
≤ ′
− ′ ′ ≤ ≤
− ′
= ′ − ≤ ≤ ′
′ − ≥ ′
(5.16)
where
2( ) 2( )
0 2( )
m m l m m l
l m m l
− − > −
′ = ≤ −
and
2( )
h′ = +m h m−
Here, l, m and h are fuzzy subsets of a parameter respectively. The value of µ is 0.5 at x l= and h. The value of µ is 1 at x m= .
Table 5.2. Fuzzy input parameters for two steels, Steel 1 with h1 = 1 mm and Steel 2 with h2 = 0.5 mm
Low
(l) Most likely
(m) High
(h) Parameters
Steel 1 Steel 2 Steel 1 Steel 2 Steel 1 Steel 2
( )
σY 0, MPa 291.6 322.2 324 358 356.4 393.8f 0.06 0.06 0.08 0.08 0.14 0.14
b 0.0416 0.0352 0.052 0.044 0.0624 0.0528
n 0.236 0.24 0.295 0.3 0.354 0.36
The values of the fuzzy parameters for two different materials are shown in Table 5.2. Dixit and Dixit assumed that the variations in yield stress and hardening coefficients may go up to ±10% and ±20%, respectively. While updating the parameters, the equivalent Coulomb’s coefficient of friction has been taken as 0.14.
The finite element model with the fuzzy parameters provides the roll force and roll torque as a band bounded by the upper and lower limit curves for each membership grade. Dixit and Dixit [1996] observed that almost all experimental data is contained in the band of 0.5 membership grade. However, it is noted that the experimental values of roll torque lies only in the upper half of the band unlike the experimental values of roll force. This indicates that the model of Dixit and Dixit [1996] is unable to provide the same order of accuracy for roll force and roll torque even after updating of the material and process parameters. In the present analysis, corresponding to the upper limit of 0.5 membership grade which corresponds to the high estimate of the parameters, the finite element computations are carried out by
the present model and are compared with the model of Dixit and Dixit [1996] as well as the experimental results of Shida and Awazuhara [1973] for roll force and roll torque. Figure 5.5 and 5.6 shows the comparison for roll force and roll torque prediction respectively.
Figure 5.5. Comparison of FEM and experimental results for roll force (Steel, R h/ 1= 65)
Figure 5.6. Comparison of FEM and experimental results for roll torque (Steel, R h/ 1= 65)
It is observed from Figs. 5.5 and 5.6 that the results obtained by the model of Dixit and Dixit [1996] is in a good agreement with the experimental roll torque values but over predicts the roll force. On the other hand, the present model predicts both the roll force and roll torque with a close agreement with that of experimental results.
The only difference in the present model and the model of Dixit and Dixit is the method of obtaining the pressure. Thus, the method proposed in this work for the computation of pressure seems to be better.
The roll pressure distribution for a typical case of 24% reduction is compared with that of Dixit and Dixit [1996]. The experimental roll pressure distribution for steel is not reported in the literature. Figure 5.7 shows the pressure distribution for the two models after updating the parameters. Although, the profile of roll pressure distribution is similar in both the models, it is observed that Dixit and Dixit model predicts higher values of the non-dimensionalized roll pressure as compared to the present model.
Figure 5.7. Comparison of roll pressure distribution for a typical case (Steel, R h/ 1= 65)
The accuracy of the present model is further assessed by carrying out the computations for the second set of material and process parameters for different reductions. Like previous set, the material to be rolled is considered as steel but with different initial strip thickness (h1=0.5mm). For this set, the results are not reported in the paper of Dixit and Dixit [1996]. The results presented in Fig. 5.8 and 5.9 have been generated by running the code developed by Dixit [1997] for comparison with the present model. Figure 5.8 shows that after updating the material and process parameters, Dixit’s model over predicts the roll force by a large amount when compared with the experimental results of Shida and Awazuhara [1973]. On the other hand, the present model shows good agreement with the experimental results of Shida and Awazuhara [1973]. Figure 5.9 shows the comparison for roll torque prediction. Considering the scatter present in the experimental results of roll torque, it is concluded that roll torque predicted by the present model is in fair agreement with the experimental values.
Figure 5.8. Comparison of FEM and experimental results for roll force (Steel, R h/ 1= 130)
Figure 5.9. Comparison of FEM and experimental results for roll torque (Steel, R h/ 1 = 130)
Figure 5.10. Comparison of roll pressure distribution for a typical case (Steel, R h/ 1 = 130)
Figure 5.10 shows the roll pressure distribution for a second set of material and process parameter (h1=0.5 mm). Similar to the results reported in Fig. 5.7, the roll pressure distribution obtained by the present model and by the model of Dixit [1997]
differ in magnitude for the second set of material and process parameters.
Figures 5.5-5.6 and Figs. 5.8-5.9 show that present model and Dixit and Dixit model differs largely for roll force prediction. This is due to the method of obtaining pressure in the present model. However, there is a lesser deviation in both the models for roll torque prediction. This may be attributed to the method of roll torque computation. In both the models, roll torque is computed by dividing the total power with the angular velocity (VR)of the roll, where the total power consist of power required for plastic deformation, power required to overcome friction at the roll- work interface and power required due to front and back tension (Eqs. 3.21-3.24).
The pressure plays a minor role here.