Fatigue strength analysis of bimetallic sleeve roll by simulation of local slip accumulation at shrink-fit interface caused by roll rotation

Abstract

Next generation rolls such as super-cermet rolls and all-ceramic rolls can be only manufactured as sleeve rolls, although the circumferential slippage appears at the shrink-fit interface. In this study, the fatigue strength of the sleeve roll is evaluated by applying the load shifting method on the fixed roll to realize the local slip accumulation during roll rotation. The simulation shows that the fatigue-inducing stress amplitude remains constant although the accumulated slip amount increases. Based on those results, the fatigue strength of standard rolling rolls is estimated considering the slip defect. The defect dimension can be characterized by the root area parameter and the value $\sqrt{\mathrm{area}}$ =1254 $\upmu m$ can be estimated from smaller roll experimental results and the previous report for larger diameter sleeve rolls. The results show that in the absence of slip damage, the fatigue strength of sleeve rolls is not much lower than that of solid rolls without shrink-fit.

Appendices

Appendix A: Load shifting method to realize the relative interfacial displacement ${{\varvec{u}}}_{{\varvec{\theta}}}^{{\varvec{P}}\left(0\right)\sim {\varvec{P}}\left(\boldsymbol{\varphi }\right)}\left({\varvec{\theta}}\right)$ and average interfacial displacement ${{\varvec{u}}}_{{\varvec{\theta}}, {\varvec{a}}{\varvec{v}}{\varvec{e}}.}^{{\varvec{P}}\left(0\right)\sim {\varvec{P}}\left(\boldsymbol{\varphi }\right)}\left({\varvec{\theta}}\right)$

Fig. 10

The roll rotation can be replaced by discrete load shifting by the angle ${\varphi }_{0}$ on the fixed roll

Fig. 11

Definition of interfacial displacement ${u}_{\theta }^{P(0)\sim P(\varphi )}\left(\theta \right)$ due to the load shifting $P(0)\sim P(\varphi )$

Figure 10 illustrates the load shifting method where the roll rotation is expressed by the load shifting on the fixed roll surface [17,18,19,20,21]. Assume the roll subjected to the concentrated rolling load P. As shown in Fig. 10, the continuous roll rotation can be expressed by the discrete load shifting with a constant interval ${\varphi }_{0}$. The most suitable value of ${\varphi }_{0}$ can be chosen to reduce the computational time without loosening the accuracy. From the comparison among the results ${\varphi }_{0}=0.25^\circ \sim 12^\circ$, the load shift angle ${\varphi }_{0}=4^\circ$ is adopted in the following discussion since the relative error between ${\varphi }_{0}=0.25^\circ$ and ${\varphi }_{0}=4^\circ$ is less than a few percent. In the following, both forces are denoted by P.

The relative displacement accumulation between the sleeve and shaft may represent the interfacial slip. In Fig. 11, the relative displacement ${u}_{\theta }^{P\left(0\right)\sim P\left(\varphi \right)}\left(\theta \right)$ due to the load shifting $P\left(0\right)\sim P\left(\varphi \right)$ is defined between the sleeve and shaft when the load moves from the angle $\varphi =0$ to $\varphi =\varphi$. Here, notation $\varphi$ denotes the angle where the load is shifting and notation $\theta$ denotes the position where the displacement is evaluated. The load $P\left(\varphi \right)$ is defined as the pair of forces acting at $\varphi =\varphi$ and $\varphi =\varphi +\pi$. The notation ${u}_{\theta }^{P\left(0\right)\sim P\left(\varphi \right)}\left(\theta \right)$ means the relative displacement ${u}_{\theta }\left(\theta \right)$ at $\theta =\theta$ when the pair of loads are applied at $\varphi =0$ to $\varphi =\varphi$ and $\varphi =\pi$ to $\varphi =\varphi +\pi$. Since the relative displacement ${u}_{\theta }\left(\theta \right)$ varies depending on $\theta$, the average displacement ${u}_{\theta ,ave.}^{P\left(0\right)\sim P\left(\varphi \right)}$ can be defined in Equation (A1).

$${u}_{\theta ,ave.}^{P\left(0\right)\sim P\left(\varphi \right)}=\frac{1}{2\pi }{\int }_{0}^{2\pi }{u}_{\theta }^{P\left(0\right)\sim P\left(\varphi \right)}\left(\theta \right)d\theta$$

(A1)

Appendix B: Estimation of slip defect dimension in standard sleeve roll in Fig. 1B

Figure 12(a) illustrates the real roll at the central cross section in Fig. 1B in comparison with Fig. 12(b) the miniature roll to verify the slippage experimentally [20, 48]. The miniature roll’s diameter is about 1/10 of the real roll. As shown in Fig. 12(b), the miniature roll consists of the sleeve, the outer shaft and the inner shaft. The inner and outer shafts are fixed by key so that the interfacial slippage between the outer shaft and the sleeve shrink-fitted can be prevented.

Figure 13 shows an example of the defect observed on the sleeve surface after slippage. The sleeve is cut along the cross section at the AA′ and BB′ to identify the defect dimensions. Figure 14 illustrates the three dimensional shape of the defect approximated by an ellipsoid. As can be expressed ${\left(x/a\right)}^{2}+{\left(y/b\right)}^{2}+{\left(z/c\right)}^{2}=1, a=1000 \mathrm{\mu m}, b=250 \mathrm{\mu m}, c=4000\upmu m$, the stress concentration can be ${K}_{t}=1.14$ [48]. In this study, the fatigue strength reduction is evaluated by using the parameter $\sqrt{\mathrm{area}}$, which is the square root of the projected area of the defect onto a plane perpendicular to the maximum principal stress [49]. Then, the miniature roll’s defect can be characterized by $\sqrt{\mathrm{area}}=\sqrt{(\pi ab)/2}=627\upmu m$ from the defect geometry $a=1000 \mathrm{\mu m} , b=250 \mathrm{\mu m}$ in Fig. 14.

 

Fig. 12

Schematic illustration for (a) Real roll and (b) Miniature roll

Fig. 13

Example of defect formed by the slippage on the sleeve surface to identify defect dimension observed in the miniature roll in Fig. 12  when the shrink-fitting ratio $\delta /d=0.21\times {10}^{-3}$

Fig. 14

Ellipsoidal plow defect geometry in Fig. 12 approximated by the equation $(x/a{)}^{2}+(y/b{)}^{2}+(z/c{)}^{2}=1,\hspace{0.33em}a=1,\hspace{0.33em}b=0.25,\hspace{0.33em}c=8.0$ with the stress concentration factor ${K}_{t}=1.14$

On the other hand, the defect depth ${b}^{^{\prime}}=$ 1 mm was reported after slip in the hot rough rolling sleeve roll whose body diameter $D=$ 1150 mm although the detail geometry is unknown [5, 6]. As shown in Fig. 3A, in this study, the real roll diameter $D=700$ mm is considered, and the depth of the defect can be a bit smaller than ${b}^{^{\prime}}=$ 1 mm although it can be larger than the defect depth $b=0.25 \mathrm{mm}$ of the miniature roll$.$ Assume that the similar shape of the defect in Fig. 13 and Fig. 14 is formed due to the sllipage in the real roll. By assuming double sizes of the defect of the miniature roll, $\sqrt{\mathrm{area}}$ dimension in the real roll can be $\sqrt{\mathrm{area}}=\sqrt{\pi (2a)(2b)/2}$=$627\times 2=$ 1254 $\upmu m$. Here $a=$ 1000 $\upmu m$ and $b=$ 250 $\upmu m$ is the defect dimension of the miniature roll.

Table 2 Test work roll specifications used in the miniature roll experiment in Fig. 12(b)

Table 3 Conditions of the miniature roll experiment in Fig. 12(b)

Table 2 shows the specifications of the test work roll. The diameter of the test roll is about 1/10 of the diameter of the real roll. Table 3 shows the experimental conditions. Roll A denotes the roll without the shrink-fitting ratio of $\delta /d=0$ and roll B denotes the roll with the shrink-fitting ratio of $\delta /d=0.21\times {10}^{-3}$. In the experiment, the work roll is cooled with water at room temperature to prevent the change in shrink-fitting rate due to the temperature rise caused by friction due to the load. When the steady rotation speed reached 106 rpm or 212 rpm, a load of 1 ton is applied to ensure that the temperature change of the roll surface is within $5\mathrm{^\circ{\rm C} }$ or less during the experiment by a contact thermometer.

Appendix C: Fatigue strength analysis results of bimetallic solid roll by simulation of cyclic loading caused by roll rotation

 

Fig. 15

Conventional bimetallic solid roll considered previously (mm)

Fig. 16

Stress amplitude ${\sigma }_{a}$ versus mean stress ${\sigma }_{m}$ diagram to compare the fatigue failure risk at three critical points in the solid bimetallic roll in Fig. 15

Figure 15 shows a bimetallic solid roll whose fatigue strength was considered in the previous paper [50, 51]. In the solid bimetallic roll, it was reported that the failures happened as the debonding at HSS/DCI boundary due to the stress ${\sigma }_{\mathrm{r}}$ as well as the roll center facture [5, 52]. Therefore, the risk of fatigue failure was evaluated at those critical points focusing on the stress ${\sigma }_{\mathrm{r}}$. Figure 16 illustrates the stress amplitude versus mean stress diagram (${\sigma }_{\mathrm{a}}$-${\sigma }_{\mathrm{m}}$ diagram) focusing on the fatigue limit under large compressive alternative loading ${\sigma }_{\mathrm{m}}$≤$0$[50]. In this evaluation, the repeated maximum and minimum stress ${\sigma }_{\mathrm{r}}$ was considered during the roll rotation as the driving force causing the internal fatigue failure. Figure 17 illustrates those three critical points denoted by ${\left.{\mathrm{B}}_{0}^{270}\right|}_{\mathrm{Rolled steel}}$, ${\left.{\mathrm{B}}_{750}^{270}\right|}_{\mathrm{Backup roll}}$, and ${\mathrm{C}}_{0}^{0}$.

 

Fig. 17

Illustration of three critical points denoted by ${\left.{\mathrm{B}}_{0}^{270}\right|}_{\mathrm{Rolled steel}}$, ${\left.{\mathrm{B}}_{750}^{270}\right|}_{\mathrm{Backup roll}}$, and ${\mathrm{C}}_{0}^{0}$ where fatigue risk should be evaluated based on the analysis and experience