EP4721890A1 - Rolling load distribution calculation method, rolling load calculation method, contact arc length calculation method, and rolling method - Google Patents

Abstract

A method of calculating the rolling load distribution of a rolling mill 10, the method comprising: calculating the contact arc length of at least one of the elastic recovery region and the elastic reduction region based on Hooke's law under the plane-strain condition of a rolled material 1, the force balance within the roll bite of the rolling mill 10, and the rolling-direction stress distribution approximated by a function dependent on the position coordinate in the rolling direction of the rolling mill 10; and calculating the rolling load distribution of at least one of the elastic recovery region and the elastic reduction region based on the contact arc length.

Description

TECHNICAL FIELD
• This disclosure relates to a rolling load distribution calculation method, a rolling load calculation method, a contact arc length calculation method, and a rolling method.
BACKGROUND
• The rolling load in a rolling mill determines the strip thickness and shape on the exit side of a rolling stand. Therefore, accurately predicting the rolling load is important for stable operation and quality assurance.
• Various methods have been proposed for accurately predicting and calculating the rolling load (for example, PTL 1 and NPL 1).
• For example, the method described in NPL 1 is generally used for predicting and calculating the rolling load in the elastic region. Here, the term "elastic region" is used as a collective term referring to both the elastic recovery region and the elastic reduction region.
• FIG. 4 is a schematic diagram of the elastic recovery region and the elastic reduction region. The elastic recovery region is the elastic region on the exit side of the rolling mill. The elastic reduction region is the elastic region on the entry side of the rolling mill. The elastic region is a region where the material being rolled is in contact with the rolling rolls but undergoes elastic deformation without plastic deformation.
CITATION LIST Patent Literature
• PTL 1: JP 2006-55881 A
Non-patent Literature
• NPL 1: " Theory and Practice of Strip Rolling", The Iron and Steel Institute of Japan, pp. 41-43.
SUMMARY (Technical Problem)
• The method described in NPL 1 calculates the load distribution in the elastic region and the contribution of the elastic region to the rolling load by neglecting the frictional stress in the elastic region and by approximating the rolling-direction stress in the elastic region as constant. Here, the contribution of the elastic region to the rolling load corresponds to the integral of the load distribution in the elastic region.
• NPL 1 describes a method for calculating the rolling load distribution in the elastic recovery region using the following equation. $$p_0=\frac{E}{1-{\nu }^2}\frac{h_0^e-h}{h_0^e}$$
• NPL 1 also describes a method for calculating the rolling load distribution in the elastic reduction region using the following equation. $$p_1=\frac{E}{1-{\nu }^2}\frac{h_1^e-h}{h_1^e}$$
• Furthermore, NPL 1 describes a method for calculating the contribution of the elastic recovery region to the rolling load by integrating the rolling load distribution in the elastic recovery region. It also describes a method for calculating the contribution of the elastic reduction region to the rolling load by integrating the rolling load distribution in the elastic reduction region.
• Moreover, NPL 1 describes a method for calculating the contact arc length of the elastic recovery region using the following equation. $$l_0^e=\sqrt{R\prime \frac{1-{\nu }^2}{E}{h}_0^e\left(q_0^e+k_0\right)}$$
• NPL 1 also describes a method for calculating the contact arc length of the plastic region using the following equation. $$l_p=\sqrt{R\prime \left(h_1^e-h_0^e+\frac{1-{\nu }^2}{E}\left(h_0^e\left(q_0^e+k_0\right)-h_1^e\left(q_1^e+k_1\right)\right)\right)}$$
• NPL 1 also describes a method for calculating the contact arc length of the elastic reduction region using the following equation. $$l_1^e=\sqrt{R\prime \left(h_1^e-h_0^e+\frac{1-{\nu }^2}{E}{h}_0^e\left(q_0^e+k_0\right)\right)}-l_p$$
• By these calculations, NPL 1 obtains the load distribution and the contact arc length in the elastic region. Therefore, by using models such as the Bland & Ford model for the plastic region, NPL 1 can obtain the load distribution for the entire contact region.
• However, as described above, the methods described in NPL 1 neglect the influence of frictional stress in the elastic region. Thus, under rolling conditions where frictional stress increases, the approximation error increases. Examples of rolling conditions where frictional stress increases include rolling with dull rolls having high roll surface roughness and rolling of high-strength steel.
• In response to such issues, NPL 1 describes a method using a rigorous theoretical equation based on Airy's stress function. However, this method requires calculating the numerical solution of differential equations to obtain the load distribution in the elastic region, and therefore has a drawback in that it is not suitable for online calculation by process computers.
• PTL 1 describes a method for improving the rolling load prediction accuracy during rolling with dull rolls by introducing a correction parameter into a theoretical equation for determining the roll flattening radius. However, this method requires pre-determining the correction parameter for various rolling conditions, and furthermore, there is a problem that the physical meaning of the correction parameter is unclear.
• The purpose of this disclosure is to calculate, at high speed and with high accuracy, the rolling load distribution in the elastic region or the contribution of the elastic region to the rolling load, even under rolling conditions where frictional stress increases.
(Solution to Problem)
1. [1] A rolling load distribution calculation method, which is a method of calculating rolling load distribution of a rolling mill, the method comprising:
   • calculating a contact arc length of either or both of an elastic recovery region and an elastic reduction region based on Hooke's law under a plane-strain condition of a rolled material, a force balance within a roll bite of the rolling mill, and a rolling-direction stress distribution approximated by a function dependent on a position coordinate in the rolling direction of the rolling mill; and
   • calculating rolling load distribution of either or both of the elastic recovery region and the elastic reduction region based on the contact arc length.
2. [2] The rolling load distribution calculation method according to [1], wherein, when calculating the rolling load distribution, the rolling load distribution of the elastic recovery region is calculated based on the following Equation (1), and the rolling load distribution of the elastic reduction region is calculated based on the following Equation (2),
   [Math. 6] $$p_0\left(x\right)=E_0\left\{a_{0}x-\left(e^{a_{0}x}-1\right)\left(1-a_0{l}_0^e\right)\right\}$$
   [Math. 7] $$p_1\left(x\right)=E_1\left[a_1\left(l-x\right)-\left(e^{a_1\left(l-x\right)}-1\right)\left\{1-a_1\left(l-l_0^e\right)\right\}\right]$$ where, in Equation (1), the relationships of the following Equations (3) and (4) apply:
   [Math. 8] $$E_0=E\prime \frac{h_0^e}{2{\mu }^2{\nu }^{\prime 2}R\prime }$$
   [Math. 9] $$a_0=\frac{2\mathit{\mu \nu }\prime }{h_0^e}$$ where, in Equation (2), the relationships of the following Equations (5) to (7) apply:
   [Math. 10] $$E_1=E\prime \frac{h_1^e}{2{\mu }^2{\nu }^{\prime 2}R\prime }$$
   [Math. 11] $$a_1=\frac{2\mathit{\mu \nu }\prime }{h_1^e}$$
   [Math. 12] $$l=l_o^e+\sqrt{R\prime \left(h_1^e-h_0^e\right)+{l_0^e}^2}$$ where, in Equations (1) to (7),
   • p₀(x) represents the rolling load distribution of the elastic recovery region,
   • p₁(x) represents the rolling load distribution of the elastic reduction region,
   • x represents the position coordinate in the rolling direction,
     [Math. 13]
   • $l_o^e$ represents the contact arc length of the elastic recovery region,
     [Math. 14]
   • $h_0^e$ represents a strip thickness at an exit of the roll bite,
     [Math. 15]
   • $h_1^e$ represents a strip thickness at an entry of the roll bite,
   • E' represents a Young's modulus of the rolled material under the plane-strain condition,
   • µ represents a friction coefficient,
   • v' represents a Poisson's ratio of the rolled material under the plane-strain condition, and
   • R' represents a flattened roll radius.
3. [3] The rolling load distribution calculation method according to [1] or [2], wherein, when calculating the contact arc length, the contact arc length of the elastic recovery region is calculated based on the following Equation (8), and the contact arc length of the elastic reduction region is calculated based on the following Equation (9),
   [Math. 16] $$l_0^e=\sqrt{\frac{\left(q_0^e+k_o\right)h_0^{e}R\text{'}}{E\text{'}}{\beta }_0}\left(8\right)$$
   [Math. 17] $$l_1^e=\frac{3h_1^e}{4\mu }{\beta }_1$$ where, in Equation (8), the relationships of the following Equations (10) to (13) apply:
   [Math. 18] $${\beta }_0=2\sqrt{\frac{-p}{3}}\mathit{cos}\left(\frac{1}{3}{\mathit{cos}}^{-1}\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)-\frac{2\pi }{3}\right)+\frac{\eta }{3}$$
   [Math. 19] $$p=\frac{-3\nu \prime -{\eta }^2}{3}$$
   [Math. 20] $$q=\frac{\eta }{27}\left(-2{\eta }^2-9\nu \prime +27\right)$$ [Math. 21] $$\eta =\frac{3}{4\mu }\sqrt{\frac{E\prime h_0^e}{\left(q_0^e+k_0\right)R\prime }}$$ where, in Equation (9), the relationships of the following Equations (14) to (20) apply:
   [Math. 22] $${\beta }_1=2\sqrt{\frac{-p}{3}}\mathit{cos}\left(\frac{1}{3}{\mathit{cos}}^{-1}\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)-\frac{4\pi }{3}\right)-\frac{b}{3}$$
   [Math. 23] $$p=\frac{3c-b^2}{3}$$
   [Math. 24] $$q=\frac{1}{27}\left(2b^3-9\mathit{bc}+27d\right)$$
   [Math. 25] $$d=-\frac{16{\mu }^{2}R\prime }{9h_1^e}\frac{q_1^e+k_1}{E\prime }$$
   [Math. 26] $$c=\frac{8\mu }{3h_1^e}\left(l-l_0^e+\nu \prime \frac{2\mathit{\mu R}\prime }{3}\frac{q_1^e+k_1}{E\prime }\right)$$
   [Math. 27] $$b=-1-\frac{8\mu }{3h_1^e}\left(l-l_0^e\right)$$
   [Math. 28] $$l=l_0^e+\sqrt{R\prime \left(h_1^e-h_0^e\right)+{l_0^e}^2}$$ where, in Equations (8) to (20):
   • [Math. 29]
     $l_o^e$ represents the contact arc length of the elastic recovery region,
   • [Math. 30]
     • $l_1^e$ represents the contact arc length of the elastic reduction region,
     • E' represents a Young's modulus of the rolled material under the plane-strain condition,
   • [Math. 31]
     $q_0^e$ represents a tension at an exit of the roll bite,
   • [Math. 32]
     • $q_1^e$ represents a tension at an entry of the roll bite,
     • k₀ represents a yield stress at the exit of the roll bite,
     • k₁ represents a yield stress at the entry of the roll bite,
   • [Math. 33]
     $h_0^e$ represents a strip thickness at the exit of the roll bite,
   • [Math. 34]
     • $h_1^e$ represents a strip thickness at the entry of the roll bite,
     • R' represents a flattened roll radius,
     • v' represents a Poisson's ratio of the rolled material under the plane-strain condition,
     • µ represents a friction coefficient, and
     • E' represents a Young's modulus of the rolled material under the plane-strain condition.
4. [4] A rolling load calculation method, which is a method of calculating a rolling load of a rolling mill, the method comprising:
   • calculating a contact arc length of either or both of an elastic recovery region and an elastic reduction region based on Hooke's law under a plane-strain condition of a rolled material, a force balance within a roll bite of the rolling mill, and a rolling-direction stress distribution approximated by a function dependent on a position coordinate in the rolling direction of the rolling mill; and
   • calculating contribution to rolling load from either or both of the elastic recovery region and the elastic reduction region based on the contact arc length.
5. [5] The rolling load calculation method according to [4], wherein, when calculating the contribution to the rolling load, the contribution to the rolling load from the elastic recovery region is calculated based on the following Equation (1), and the contribution to the rolling load from the elastic reduction region is calculated based on the following Equation (2),
   [Math. 35] $$p_0^e=\frac{E_0}{a_0}\left\{1-\frac{{a_0}^2{l}_0^{e^2}}{2}-e^{a_0{l}_0^e}\left(1-a_0{l}_0^e\right)\right\}$$
   [Math. 36] $$p_1^e=\frac{E_1}{a_1}\left[\frac{{a_1}^2{l}_1^{e^2}}{2}+\left(1+a_1{l}_1^e-e^{a_1{l}_1^e}\right)\left\{1-a_1\left(l-l_0^e\right)\right\}\right]$$ where, in Equation (1), the relationships of the following Equations (3) and (4) apply:
   [Math. 37] $$E_0=E\prime \frac{h_0^e}{2{\mu }^2{\nu }^{\prime 2}R\prime }$$
   [Math. 38] $$a_0=\frac{2\mathit{\mu \nu }\prime }{h_0^e}$$ where, in Equation (2), the relationships of the following Equations (5) to (7) apply:
   [Math. 39] $$E_1=E\prime \frac{h_1^e}{2{\mu }^2{\nu }^{\prime 2}R\prime }$$
   [Math. 40] $$a_1=\frac{2\mathit{\mu \nu }\prime }{h_1^e}$$ [Math. 41] $$l=l_o^e+\sqrt{R\prime \left(h_1^e-h_0^e\right)+l_0^{e^2}}$$ where, in Equations (1) to (7):
   • [Math. 42]
     $p_0^e$ represents the contribution of the elastic recovery region to the rolling load,
   • [Math. 43]
     $p_1^e$ represents the contribution of the elastic reduction region to the rolling load,
   • [Math. 44]
     $l_o^e$ represents the contact arc length of the elastic recovery region,
   • [Math. 45]
     $l_1^e$ represents the contact arc length of the elastic reduction region,
   • [Math. 46]
     $h_0^e$ represents a strip thickness at an exit of the roll bite,
   • [Math. 47]
     • $h_1^e$ represents a strip thickness at an entry of the roll bite,
     • E' represents a Young's modulus of the rolled material under the plane-strain condition,
     • µ represents a friction coefficient,
     • v' represents a Poisson's ratio of the rolled material under the plane-strain condition, and
     • R' represents a flattened roll radius.
6. [6] The rolling load calculation method according to [4] or [5], wherein, when calculating the contact arc length, the contact arc length of the elastic recovery region is calculated based on the following Equation (8), and the contact arc length of the elastic reduction region is calculated based on the following Equation (9),
   [Math. 48] $$l_0^e=\sqrt{\frac{\left(q_0^e+k_0\right)h_0^{e}R\prime }{E\prime }}{\beta }_0$$
   [Math. 49] $$l_1^e=\frac{3h_1^e}{4\mu }{\beta }_1$$ where, in Equation (8), the relationships of the following Equations (10) to (13) apply:
   [Math. 50] $${\beta }_0=2\sqrt{\frac{-p}{3}}\mathit{cos}\left(\frac{1}{3}{\mathit{cos}}^{-1}\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)-\frac{2\pi }{3}\right)+\frac{\eta }{3}$$
   [Math. 51] $$p=\frac{-3\nu \prime -{\eta }^2}{3}$$
   [Math. 52] $$q=\frac{\eta }{27}\left(-2{\eta }^2-9\nu \prime +27\right)$$
   [Math. 53] $$\eta =\frac{3}{4\mu }\sqrt{\frac{E\prime h_0^e}{\left(q_0^e+k_0\right)R\prime }}$$ where, in Equation (9), the relationships of the following Equations (14) to (20) apply:
   [Math. 54] $${\beta }_1=2\sqrt{\frac{-p}{3}}\mathit{cos}\left(\frac{1}{3}{\mathit{cos}}^{-1}\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)-\frac{4\pi }{3}\right)-\frac{b}{3}$$
   [Math. 55] $$p=\frac{3c-b^2}{3}$$
   [Math. 56] $$q=\frac{1}{27}\left(2b^3-9\mathit{bc}+27d\right)$$
   [Math. 57] $$d=-\frac{16{\mu }^{2}R\prime }{9h_1^e}\frac{q_1^e+k_1}{E\prime }$$
   [Math. 58] $$c=\frac{8\mu }{3h_1^e}\left(l-l_0^e+\nu \prime \frac{2\mathit{\mu R}\prime }{3}\frac{q_1^e+k_1}{E\prime }\right)$$
   [Math. 59] $$b=-1-\frac{8\mu }{3h_1^e}\left(l-l_0^e\right)$$
   [Math. 60] $$l=l_0^e+\sqrt{R\prime \left(h_1^e-h_0^e\right)+l_0^{e^2}}$$ where, in Equations (8) to (20):
   • [Math. 61]
     $l_o^e$ represents the contact arc length of the elastic recovery region,
   • [Math. 62]
     • $l_1^e$ represents the contact arc length of the elastic reduction region,
     • E' represents a Young's modulus of the rolled material under the plane-strain condition,
   • [Math. 63]
     $q_0^e$ represents a tension at an exit of the roll bite,
   • [Math. 64]
     • $q_1^e$ represents a tension at an entry of the roll bite,
     • ko represents a yield stress at the exit of the roll bite,
     • k₁ represents a yield stress at the entry of the roll bite,
   • [Math. 65]
     $h_0^e$ represents a strip thickness at the exit of the roll bite,
   • [Math. 66]
     • $h_1^e$ represents a strip thickness at the entry of the roll bite,
     • R' represents a flattened roll radius,
     • v' represents a Poisson's ratio of the rolled material under the plane-strain condition,
     • µ represents a friction coefficient, and
     • E' represents a Young's modulus of the rolled material under the plane-strain condition.
7. [7] A contact arc length calculation method, which is a method of calculating a contact arc length of a rolling mill, the method comprising:
   calculating a contact arc length of either or both of an elastic recovery region and an elastic reduction region based on Hooke's law under a plane-strain condition of a rolled material, a force balance within a roll bite of the rolling mill, and a rolling-direction stress distribution approximated by a function dependent on a position coordinate in the rolling direction of the rolling mill.
8. [8] The contact arc length calculation method according to [7], wherein, when calculating the contact arc length, the contact arc length of the elastic recovery region is calculated based on the following Equation (1), and the contact arc length of the elastic reduction region is calculated based on the following Equation (2),
   [Math. 67] $$l_0^e=\sqrt{\frac{\left(q_0^e+k_0\right)h_0^{e}R\prime }{E\prime }}{\beta }_0$$
   [Math. 68] $$l_1^e=\frac{3h_1^e}{4\mu }{\beta }_1$$ where, in Equation (1), the relationships of the following Equations (3) to (6) apply:
   [Math. 69] $${\beta }_0=2\sqrt{\frac{-p}{3}}\mathit{cos}\left(\frac{1}{3}{\mathit{cos}}^{-1}\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)-\frac{2\pi }{3}\right)+\frac{\eta }{3}$$
   [Math. 70] $$p=\frac{-3\nu \prime -{\eta }^2}{3}$$
   [Math. 71] $$q=\frac{\eta }{27}\left(-2{\eta }^2-9\nu \prime +27\right)$$
   [Math. 72] $$\eta =\frac{3}{4\mu }\sqrt{\frac{E\prime h_0^e}{\left(q_0^e+k_0\right)R\prime }}$$ where, in Equation (2), the relationships of the following Equations (7) to (13) apply:
   [Math. 73] $${\beta }_1=2\sqrt{\frac{-p}{3}}\mathit{cos}\left(\frac{1}{3}{\mathit{cos}}^{-1}\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)-\frac{4\pi }{3}\right)-\frac{b}{3}$$
   [Math. 74] $$p=\frac{3c-b^2}{3}$$
   [Math. 75] $$q=\frac{1}{27}\left(2b^3-9\mathit{bc}+27d\right)$$
   [Math. 76] $$d=-\frac{16{\mu }^{2}R\prime }{9h_1^e}\frac{q_1^e+k_1}{E\prime }$$
   [Math. 77] $$c=\frac{8\mu }{3h_1^e}\left(l-l_0^e+\nu \prime \frac{2\mathit{\mu R}\prime }{3}\frac{q_1^e+k_1}{E\prime }\right)$$
   [Math. 78] $$b=-1-\frac{8\mu }{3h_1^e}\left(l-l_0^e\right)$$
   [Math. 79] $$l=l_0^e+\sqrt{R\prime \left(h_1^e-h_0^e\right)+l_0^{e^2}}$$ where, in Equations (1) to (13):
   • [Math. 80]
     $l_o^e$ represents the contact arc length of the elastic recovery region,
   • [Math. 81]
     • $l_1^e$ represents the contact arc length of the elastic reduction region,
     • E' represents a Young's modulus of the rolled material under the plane-strain condition,
   • [Math. 82]
     $q_0^e$ represents a tension at an exit of the roll bite,
   • [Math. 83]
     • $q_1^e$ represents a tension at an entry of the roll bite,
     • ko represents a yield stress at the exit of the roll bite,
     • k₁ represents a yield stress at the entry of the roll bite,
   • [Math. 84]
     $h_0^e$ represents a strip thickness at the exit of the roll bite,
   • [Math. 85]
     • $h_1^e$ represents a strip thickness at the entry of the roll bite,
     • R' represents a flattened roll radius,
     • v' represents a Poisson's ratio of the rolled material under the plane-strain condition,
     • µ represents a friction coefficient, and
     • E' represents a Young's modulus of the rolled material under the plane-strain condition.
9. [9] A rolling method, comprising changing a roll gap based on the rolling load distribution calculated in a manner reflecting a current lubrication condition using the rolling load distribution calculation method according to any one of [1] to [3].
10. [10] A rolling method, comprising changing a roll gap based on the rolling load calculated in a manner reflecting a current lubrication condition using the rolling load calculation method according to any one of [4] to [6].
(Advantageous Effect)
• According to the methods of the present disclosure, it is possible to calculate, at high speed and with high accuracy, the rolling load distribution in the elastic region or the contribution of the elastic region to the rolling load, even under rolling conditions where frictional stress increases.
BRIEF DESCRIPTION OF THE DRAWINGS
• In the accompanying drawings:
• FIG. 1 is a diagram illustrating an example of rolling mill equipment to which the rolling load distribution calculation method, the rolling load calculation method, and the contact arc length calculation method according to an embodiment of the present disclosure are applied;
• FIG. 2 is a diagram illustrating the results of calculating the rolling load distribution using the methods of the present disclosure and the results of calculating the rolling load distribution using a conventional method;
• FIG. 3 is a table comparing the results obtained using the methods of the present disclosure with the results obtained using conventional methods; and
• FIG. 4 is a schematic diagram of the elastic recovery region and the elastic reduction region.
DETAILED DESCRIPTION
• The following describes an embodiment of the present disclosure with reference to the drawings.
• FIG. 1 is a diagram illustrating an example of rolling mill equipment to which the rolling load distribution calculation method, the rolling load calculation method, and the contact arc length calculation method according to the embodiment of the present disclosure are applied.
• The rolling mill equipment includes rolling mills 10-1 to 10-5, a control unit 20, a process computer 30, and an online computer 40.
• When there is no need to distinguish among the rolling mills 10-1 to 10-5, they may simply be referred to as rolling mill 10. In FIG. 1, five rolling mills 10 are indicated as rolling mills 10-1 to 10-5, but this is merely one example. The number of rolling mills 10 may be one or more in any arbitrary number.
• The rolling mill 10 is equipped with rolling rolls. The rolling mill 10 rolls a material 1 using the rolling rolls. The rolled material 1 is, for example, a steel sheet. The rolled material 1 may be a non-ferrous metal such as aluminum or titanium.
• The control unit 20 is a device that controls the rolling mill 10. The control unit 20 controls the roll gap of the rolling rolls of the rolling mill 10. The control unit 20 also controls the roll speed of the rolling rolls of the rolling mill 10.
• The process computer 30 is a general-purpose computer such as a workstation or a personal computer. Alternatively, the process computer 30 may be a dedicated computer configured to function as the process computer 30 of the rolling mill equipment illustrated in FIG. 1.
• The process computer 30 executes the rolling load distribution calculation method, the rolling load calculation method, and the contact arc length calculation method according to the present disclosure. Accordingly, the process computer 30 can calculate the rolling load distribution in at least one of the elastic recovery region and the elastic reduction region. The process computer 30 can also calculate the contribution to the rolling load from at least one of the elastic recovery region and the elastic reduction region. Furthermore, the process computer 30 can calculate the contact arc length of at least one of the elastic recovery region and the elastic reduction region.
• Here, the elastic recovery region is the elastic region on the exit side of the rolling mill 10. The elastic reduction region is the elastic region on the entry side of the rolling mill 10. In the present embodiment, the term "elastic region" is used as a collective term referring to both the elastic recovery region and the elastic reduction region.
• Details of the calculation of the rolling load distribution, the calculation of the contribution to the rolling load, and the calculation of the contact arc length performed by the process computer 30 will be described later.
• Based on the calculated rolling load distribution, the calculated contribution to the rolling load, and the calculated contact arc length, the process computer 30 controls the control unit 20, thereby controlling the roll gap of the rolling rolls of the rolling mill 10 and controlling the roll speed of the rolling rolls of the rolling mill 10.
• The online computer 40 is a general-purpose computer such as a workstation or a personal computer. Alternatively, the online computer 40 may be a dedicated computer configured to function as the online computer 40 of the rolling mill equipment illustrated in FIG. 1.
• The online computer 40 calculates, by back-calculation, the friction coefficient and other parameters based on rolling performance data such as the rolling load of the rolling mill 10, the thickness of the rolled material 1, and the tension of the rolled material 1, as well as preset data such as the roll diameter of the rolling rolls of the rolling mill 10.
• By calculating the friction coefficient and other parameters using preset data such as the roll diameter of the rolling rolls of the rolling mill 10, which have been set based on the calculation results of the process computer 30, the online computer 40 can calculate the friction coefficient stably and at high speed.
• Furthermore, the online computer 40 can feed the calculation results of the friction coefficient and other parameters back to the process computer 30. The process computer 30 can use the calculation results of the friction coefficient and other parameters fed back from the online computer 40 for calculations such as rolling load distribution calculation.
(Calculation of Contact Arc Length)
• A description will now be given of the processing performed by the process computer 30 to calculate the contact arc length of the rolling mill 10.
• The process computer 30 calculates the contact arc length of at least one of the elastic recovery region and the elastic reduction region based on Hooke's law under the plane-strain condition of the rolled material 1, the force balance within the roll bite of the rolling mill 10, and the rolling-direction stress distribution approximated by a function dependent on the position coordinate in the rolling direction of the rolling mill 10.
• The process computer 30 calculates the contact arc length of the elastic recovery region based on the following Equation (1), and calculates the contact arc length of the elastic reduction region based on the following Equation (2).
  [Math. 86] $$l_0^e=\sqrt{\frac{\left(q_0^e+k_0\right)h_0^{e}R\prime }{E\prime }}{\beta }_0$$
  [Math. 87] $$l_1^e=\frac{3h_1^e}{4\mu }{\beta }_1$$ where, in Equation (1), the relationships of the following Equations (3) to (6) apply:
  [Math. 88] $${\beta }_0=2\sqrt{\frac{-p}{3}}\mathit{cos}\left(\frac{1}{3}{\mathit{cos}}^{-1}\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)-\frac{2\pi }{3}\right)+\frac{\eta }{3}$$
  [Math. 89] $$p=\frac{-3\nu \prime -{\eta }^2}{3}$$
  [Math. 90] $$q=\frac{\eta }{27}\left(-2{\eta }^2-9\nu \prime +27\right)$$
  [Math. 91] $$\eta =\frac{3}{4\mu }\sqrt{\frac{E\prime h_0^e}{\left(q_0^e+k_0\right)R\prime }}$$ where, in Equation (2), the relationships of the following Equations (7) to (13) apply:
  [Math. 92] $${\beta }_1=2\sqrt{\frac{-p}{3}}\mathit{cos}\left(\frac{1}{3}{\mathit{cos}}^{-1}\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)-\frac{4\pi }{3}\right)-\frac{b}{3}$$
  [Math. 93] $$p=\frac{3c-b^2}{3}$$
  [Math. 94] $$q=\frac{1}{27}\left(2b^3-9\mathit{bc}+27d\right)$$
  [Math. 95] $$d=-\frac{16{\mu }^{2}R\prime }{9h_1^e}\frac{q_1^e+k_1}{E\prime }$$
  [Math. 96] $$c=\frac{8\mu }{3h_1^e}\left(l-l_0^e+\nu \prime \frac{2\mathit{\mu R}\prime }{3}\frac{q_1^e+k_1}{E\prime }\right)$$
  [Math. 97] $$b=-1-\frac{8\mu }{3h_1^e}\left(l-l_0^e\right)$$
  [Math. 98] $$l=l_0^e+\sqrt{R\prime \left(h_1^e-h_0^e\right)+{l_0^e}^2}$$ where, in Equations (1) to (13):
• [Math. 99]
  $l_o^e$ represents the contact arc length of the elastic recovery region
• [Math. 100]
  • $l_1^e$ represents the contact arc length of the elastic reduction region,
  • E' represents the Young's modulus of the rolled material under the plane-strain condition,
• [Math. 101]
  $q_0^e$ represents the tension at the exit of the roll bite,
• [Math. 102]
  • $q_1^e$ represents the tension at the entry of the roll bite,
  • ko represents the yield stress at the exit of the roll bite,
  • k₁ represents the yield stress at the entry of the roll bite,
• [Math. 103]
  $h_0^e$ represents the strip thickness at the exit of the roll bite,
• [Math. 104]
  • $h_1^e$ represents the strip thickness at the entry of the roll bite,
  • R' represents the flattened roll radius,
  • v' represents the Poisson's ratio of the rolled material under the plane-strain condition,
  • µ represents the friction coefficient, and
  • E' represents the Young's modulus of the rolled material under the plane-strain condition.
<Approximation by Linear Polynomial>
• Next, a description will be given of the processing performed when the process computer 30 approximates the rolling-direction stress in the elastic recovery region using a linear polynomial with respect to the position coordinate in the rolling direction and calculates the contact arc length in the elastic recovery region. Note that the position coordinate in the rolling direction is defined such that the exit of the roll bite is zero.
• The process computer 30 approximates the rolling-direction stress q using a linear polynomial with respect to the position coordinate in the rolling direction by the following Equation (14).
  [Math. 105] $$q=q_0^e+x\dot{q}\left(x_0,q_0^e\right)$$ where, in Equation (14), xo represents the exit-side yield point.
• Based on the force balance within the roll bite of the rolling mill 10, the following Equation (15) is obtained.
  [Math. 106] $$\dot{q}\left(x_0,q_0^e\right)=\frac{\left(p\left(x_0\right)-q_0^e\right)h\left(x_0\right)+2\tau \left(x_0\right)}{h\left(x_0\right)}$$
• In the above Equation (15), it is assumed that the strip thickness becomes minimum at the exit-side yield point, as expressed by the following Equation (16).
  [Math. 107] $$\dot{h}\left(x_0\right)=0$$
• Further, the above Equation (15) is approximated as in the following Equation (17).
  [Math. 108] $$\tau \left(x_0\right)\simeq \mu \left(q_0^e+k_0\right)$$
• By substituting the assumption of Equation (16) and the approximation of Equation (17) into Equation (15), the following Equation (18) is obtained.
  [Math. 109] $$q=q_0^e+\frac{2\tau \left(x_0\right)}{h\left(x_0\right)}x\simeq q_0^e+2\mu \frac{q_0^e+k_0}{h_0^e}x$$
• Accordingly, at the exit-side yield point, the following Equation (19) is obtained.
  [Math. 110] $$p\left(x_0\right)\simeq E\prime \frac{h_0^e-h\left(x_0\right)}{h_0^e}+2\mathit{\mu \nu }\prime \frac{q_0^e+k_0}{h_0^e}x$$
• On the other hand, the yield condition at the exit-side yield point can be expressed by the following Equation (20).
  [Math. 111] $$p\left(x_0\right)=q\left(x_0\right)+k_0\simeq q_0^e+k_0+2\mu \frac{q_0^e+k_0}{h_0^e}{x}_0$$
• The strip thickness h(x) is expressed by the following Equation (21).
  [Math. 112] $$h\left(x\right)=h_0^e+\frac{x\left(x-2x_0\right)}{R\prime }$$
• Accordingly, at the exit-side yield point, the following approximate relationships given by Equations (22) and (23) are obtained.
  [Math. 113] $$E\prime \frac{{x_0}^2}{h_0^{e}R\prime }-2\mu \left(1-\nu \prime \right)\frac{q_0^e+k_0}{h_0^e}{x}_0-\left(q_0^e+k_0\right)=0$$
  [Math. 114] $${x_0}^2-2\mu \left(1-\nu \prime \right)\frac{q_0^e+k_0}{E\prime }R\prime x_0-\frac{q_0^e+k_0}{E\prime }{h}_0^{e}R\prime =0$$
• Solving the above Equation (23) for x0 results in the following Equation (24).
  [Math. 115] $$l_0^e=x_0=\mu \left(1-\nu \prime \right)\frac{q_0^e+k_0}{E\prime }R+\sqrt{{\left(\mu \left(1-\nu \prime \right)\frac{q_0^e+k_0}{E\prime }R\prime \right)}^2+\frac{q_0^e+k_0}{E\prime }{h}_0^{e}R\prime }$$
• The process computer 30 can calculate the contact arc length in the elastic recovery region according to the above Equation (24).
• If the friction coefficient µ is set to 0 in the above Equation (24), the result coincides with the conventional result in which friction is neglected.
• Next, a description will be given of the processing performed when the process computer 30 approximates the rolling-direction stress in the elastic reduction region using a linear polynomial with respect to the position coordinate in the rolling direction and calculates the contact arc length in the elastic reduction region.
• The process computer 30 approximates the rolling-direction stress q using a linear polynomial with respect to the position coordinate in the rolling direction by the following Equation (25).
  [Math. 116] $$q=q_1^e+\left(x-l\right)\dot{q}\left(x_1,q_1^e\right)$$ where, in Equation (25), x₁ represents the entry-side yield point. Further, 1 represents the contact arc length.
• Based on the force balance within the roll bite of the rolling mill 10, an approximate equation of the following Equation (26) is obtained.
  [Math. 117] $$\left(x-l\right)\dot{q}\left(x_1,q_1^e\right)\simeq -2\mu \left(x-l\right)\frac{q_1^e+k_1}{h_1^e}$$
• On the other hand, when x is approximately equal to l, the following Equations (27) and (28) are obtained.
  [Math. 118] $$h\left(x\right)\simeq h_1^e+\left(x-l\right)\dot{h}\left(l\right)=h_1^e+2\frac{l-x_0}{R\prime }\left(x-l\right)$$
  [Math. 119] $$p\left(x_1\right)=q\left(x_1\right)+k_1$$
• Accordingly, the following Equation (29) is obtained.
  [Math. 120] $$p\left(x_1\right)\simeq -2E\prime \frac{l-x_0}{h_1^{e}R\prime }\left(x_1-l\right)-2\mathit{\mu \nu }\prime \left(x_1-l\right)\frac{q_1^e+k_1}{h_1^e}$$
• In the same manner as in the case of the elastic recovery region, based on the relationship at the boundary between the elastic region and the plastic region, the following Equation (30) is obtained.
  [Math. 121] $$q_1^e-2\mu \left(x_1-l\right)\frac{q_1^e+k_1}{h_1^e}+k_1=-2E\prime \frac{l-x_0}{h_1^{e}R\prime }\left(x_1-l\right)-2\mathit{\mu \nu }\prime \left(x_1-l\right)\frac{q_1^e+k_1}{h_1^e}$$
• Based on the above Equation (30), the contact arc length in the elastic reduction region is obtained by the following Equation (31).
  [Math. 122] $$l_1^e=l-x_1=\frac{\left(q_1^e+k_1\right)h_1^e}{2E\prime \sqrt{\frac{h_1^e-h_0^e}{R\prime }+{\left(\frac{l_0^e}{R\prime }\right)}^2}-2\mu \left(1-\nu \prime \right)\left(q_1^e+k_1\right)}$$
• The process computer 30 can calculate the contact arc length in the elastic reduction region according to the above Equation (31).
<Approximation by Cubic Polynomial>
• Next, a description will be given of the processing performed when the process computer 30 approximates the rolling-direction stress in the elastic recovery region using a cubic polynomial with respect to the position coordinate in the rolling direction and calculates the contact arc length in the elastic recovery region.
• The process computer 30 approximates the rolling-direction stress q in the elastic recovery region by the following Equation (32).
  [Math. 123] $$q\left(x\right)=q_0^e+{\displaystyle {\int }_0^x\dot{q}\left(\xi \right)\mathit{d\xi }\simeq q_0^e+\frac{2\mu }{h_0^e}{\int }_0^{x}p\left(\xi \right)\mathit{d\xi }}$$
• Here, the rolling load distribution p in the elastic recovery region is approximated by a quadratic polynomial of the following Equation (33).
  [Math. 124] $$p\left(x\right)=\frac{x\left(2x_0-x\right)}{{x_0}^2}p\left(x_0\right)$$
• The above Equation (33) selects a quadratic function, but this is just an example, and other functions may also be used. For example, a linear function or an elliptic function may be used for the approximation. The subsequent discussion proceeds in essentially the same manner regardless of the function selected.
• In this case, the representation of the stress q in the rolling direction by a cubic polynomial is obtained by the following Equation (34).
  [Math. 125] $$q\left(x\right)\simeq q_0^e+\frac{2\mu }{h_0^e}{\displaystyle {\int }_0^x\frac{\xi \left(2x_0-\xi \right)}{{x_0}^2}p\left(x_0\right)\mathit{d\xi }=q_0^e+\frac{2\mu }{h_0^e}\frac{x_0{x}^2-\frac{x^3}{3}}{{x_0}^2}p\left(x_0\right)}$$
• Further, the value at the exit-side yield point is obtained by the following Equation (35).
  [Math. 126] $$q\left(x_0\right)=q_0^e+\frac{4\mu }{3h_0^e}p\left(x_0\right)x_0$$
• On the other hand, since p(x₀) = q(x₀) + k₀, substituting this into the right side of the above Equation (35) and solving for q(x₀) yields a rational function representation concerning the variable x₀ as expressed in the following Equation (36).
  [Math. 127] $$q\left(x_0\right)=q_0^e+\frac{x_0}{\frac{3h_0^e}{4\mu }-x_0}\left(q_0^e+k_0\right)$$
• Accordingly, the following Equations (37) to (39) are obtained. [Math. 128] $$E\prime \frac{{x_0}^2}{h_{0}R\prime }+\nu \prime \left(q\left(x_0\right)-q_0^e\right)=q\left(x_0\right)+k_0$$
  [Math. 129] $$\frac{E\prime }{\left(q_0^e+k_0\right)h_{0}R\prime }{x_0}^2+\left(\nu \prime -1\right)\frac{x_0}{\frac{3h_0^e}{4\mu }-x_0}-1=0$$
  [Math. 130] $$-\frac{E\prime }{\left(q_0^e+k_0\right)h_{0}R\prime }{x_0}^3+\frac{3h_0^e}{4\mu }\frac{E\prime }{\left(q_0^e+k_0\right)h_{0}R\prime }{x_0}^2+\nu \prime x_0-\frac{3h_0^e}{4\mu }=0$$
• Here, β₀ and η are defined as in the following Equations (40) and (41).
  [Math. 131] $${\beta }_0=\sqrt{\frac{E\prime }{\left(q_0^e+k_0\right)h_0^{e}R\prime }}{x}_0$$
  [Math. 132] $$\eta =\frac{3}{4\mu }\sqrt{\frac{E\prime h_0^e}{\left(q_0^e+k_0\right)R\prime }}$$
• Accordingly, the cubic equation of the above Equation (39) can be expressed by the following Equation (42) as a cubic equation concerning the variable β₀.
  [Math. 133] $${{\beta }_0}^3-{{\mathit{\eta \beta }}_0}^2-\nu \prime {\beta }_0+\eta =0$$
• When the above Equation (42) has three real solutions, p and q are defined as in the following Equations (43) and (44).
  [Math. 134] $$p=\frac{3\nu \prime +{\eta }^2}{9}$$
  [Math. 135] $$q=\frac{\eta }{54}\left(2{\eta }^2+9\nu \prime -27\right)$$
• Accordingly, the solution of the above Equation (42) can be expressed by the following Equation (45).
  [Math. 136] $${\beta }_0=2\sqrt{\frac{-p}{3}}\mathit{cos}\left(\frac{1}{3}{\mathit{cos}}^{-1}\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)-\frac{2\pi }{3}\right)+\frac{\eta }{3}$$
• Therefore, the process computer 30 can calculate the contact arc length in the elastic recovery region according to the following Equation (46).
  [Math. 137] $$l_0^e=\sqrt{\frac{\left(q_0^e+k_0\right)h_0^{e}R\prime }{E\prime }}{\beta }_0$$
• Next, a description will be given of the processing performed when the process computer 30 approximates the rolling-direction stress in the elastic reduction region using a cubic polynomial with respect to the position coordinate in the rolling direction and calculates the contact arc length in the elastic reduction region.
• In the elastic reduction region, the process computer 30 performs calculations similar to those in the elastic recovery region and obtains the value at the entry-side yield point by the following Equation (47).
  [Math. 138] $$q\left(x_1\right)=q_1^e+\frac{4\mu }{3h_1^e}p\left(x_1\right)\left(l-x_1\right)$$
• On the other hand, since the rolling load distribution p(x₁) = q(x₁) + k₁, substituting this into the above Equation (47) and solving for p(x₁) results in the following Equation (48).
  [Math. 139] $$p\left(x_1\right)=\frac{q_1^e+k_1}{1-\frac{4\mu }{3h_1^e}\left(l-x_1\right)}$$
• Also, the rolling load distribution p(x₁) can be expressed by the following Equation (49).
  [Math. 140] $$p\left(x_1\right)=E\prime \frac{h_1^e-h\left(x_1\right)}{h_1^e}+\nu \prime \left(p\left(x_1\right)-q_1^e-k_1\right)$$
• Accordingly, the following Equation (50) is obtained.
  [Math. 141] $$\begin{array}{l}{\left\{\frac{4\mu }{3h_1^e}\left(l-x_1\right)\right\}}^3-\left\{1+\frac{8\mu }{3h_1^e}\left(l-x_0\right)\right\}{\left\{\frac{4\mu }{3h_1^e}\left(l-x_1\right)\right\}}^2+\frac{8\mu }{3h_1^e}\left(l-x_0+\nu \prime \frac{2\mathit{\mu R}\prime }{3}\frac{q_1^e+k_1}{E\prime }\right)\frac{4\mu }{3h_1^e}\left(l-x_1\right)\\ -\frac{16{\mu }^{2}R\prime }{9h_1^e}\frac{q_1^e+k_1}{E\prime }=0\end{array}$$
• Here, β₁, d, c, and b are defined as in the following Equations (51) to (54).
  [Math. 142] $${\beta }_1=\frac{4\mu }{3h_1^e}\left(l-x_1\right)$$
  [Math. 143] $$d=-\frac{16{\mu }^{2}R\prime }{9h_1^e}\frac{q_1^e+k_1}{E\prime }$$
  [Math. 144] $$c=\frac{8\mu }{3h_1^e}\left(l-x_0+\nu \prime \frac{2\mathit{\mu R}\prime }{3}\frac{q_1^e+k_1}{E\prime }\right)$$
  [Math. 145] $$b=-1-\frac{8\mu }{3h_1^e}\left(l-x_0\right)$$
• Further, p and q are expressed by the following Equations (55) and (56).
  [Math. 146] $$p=\frac{3c-b^2}{3}$$
  [Math. 147] $$q=\frac{1}{27}\left(2b^3-9\mathit{bc}+27d\right)$$
• Accordingly, β₁ is expressed by the following Equation (57).
  [Math. 148] $${\beta }_1=2\sqrt{\frac{-p}{3}}\mathit{cos}\left(\frac{1}{3}{\mathit{cos}}^{-1}\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)-\frac{4\pi }{3}\right)-\frac{b}{3}$$
• Therefore, based on the definition of β₁, the process computer 30 can calculate the contact arc length in the elastic reduction region according to the following Equation (58).
  [Math. 149] $$l_1^e=\frac{3h_1^e}{4\mu }{\beta }_1$$
(Calculation of Rolling Load Distribution)
• A description will now be given of the processing performed by the process computer 30 to calculate the rolling load distribution of the rolling mill 10.
• The process computer 30 calculates the rolling load distribution of at least one of the elastic recovery region and the elastic reduction region based on the calculated contact arc length.
• The process computer 30 calculates the rolling load distribution of the elastic recovery region based on the following Equation (59), and calculates the rolling load distribution of the elastic reduction region based on the following Equation (60).
  [Math. 150] $$p_0\left(x\right)=E_0\left\{a_{0}x-\left(e^{a_{0}x}-1\right)\left(1-a_0{l}_0^e\right)\right\}$$
  [Math. 151] $$p_1\left(x\right)=E_1\left[a_1\left(l-x\right)-\left(e^{a_1\left(l-x\right)}-1\right)\left\{1-a_1\left(l-l_0^e\right)\right\}\right]$$ where, in Equation (59), the relationships of the following Equations (61) and (62) apply:
  [Math. 152] $$E_0=E\prime \frac{h_0^e}{2{\mu }^2\nu {\prime }^{2}R\prime }$$
  [Math. 153] $$a_0=\frac{2\mathit{\mu \nu }\prime }{h_0^e}$$ where, in Equation (60), the relationships of the following Equations (63) to (65) apply:
  [Math. 154] $$E_1=E\prime \frac{h_1^e}{2{\mu }^2\nu {\prime }^{2}R\prime }$$
  [Math. 155] $$a_1=\frac{2\mathit{\mu \nu }\prime }{h_1^e}$$
  [Math. 156] $$l=l_o^e+\sqrt{R\prime \left(h_1^e-h_0^e\right)+l_0^{e^2}}$$ where, in Equations (59) to (65),
• p₀(x) represents the rolling load distribution of the elastic recovery region,
• p₁(x) represents the rolling load distribution of the elastic reduction region,
• x represents the position coordinate in the rolling direction,
  [Math. 157]
• $l_o^e$ represents the contact arc length of the elastic recovery region,
  [Math. 158]
• $h_0^e$ represents the strip thickness at the exit of the roll bite,
  [Math. 159]
• $h_1^e$ represents the strip thickness at the entry of the roll bite,
• E' represents the Young's modulus of the rolled material under the plane-strain condition,
• µ represents the friction coefficient,
• v' represents the Poisson's ratio of the rolled material under the plane-strain condition, and
• R' represents the flattened roll radius.
<Explanation Using Differential Equation for Rolling Load Distribution>
• Next, the calculation of the rolling load distribution will be described using differential equations. First, the calculation of the rolling load distribution in the elastic recovery region will be described.
• In the elastic recovery region, the relationship expressed by the following Equation (66) can be obtained as a differential equation concerning the rolling load.
  [Math. 160] $$\frac{\mathit{dp}}{\mathit{dx}}\simeq -E\prime \frac{1}{h_0^e}\frac{\mathit{dh}}{\mathit{dx}}+2\mathit{\mu \nu }\prime \frac{p}{h_0^e}=-2E\prime \frac{x-x_0}{h_0^{e}R\prime }+2\mathit{\mu \nu }\prime \frac{p}{h_0^e}$$
• Here, E₀ and a₀ are defined as in the following Equations (67) and (68).
  [Math. 161] $$E_0=E\prime \frac{h_0^e}{2{\mu }^2\nu {\prime }^{2}R\prime }$$
  [Math. 162] $$a_0=\frac{2\mathit{\mu \nu }\prime }{h_0^e}$$
• Accordingly, the above Equation (66) can be expressed by the following Equation (69).
  [Math. 163] $$\frac{\mathit{dp}}{\mathit{dx}}\simeq -E_0{a_0}^2\left(x-x_0\right)+a_{0}p$$
• The initial value problem for the above Equation (69) (p(0) = 0) can be solved analytically, and its solution is expressed by the following Equation (70).
  [Math. 164] $$p_0\left(x\right)=E_0\left\{a_{0}x-\left(e^{a_{0}x}-1\right)\left(1-a_0{x}_0\right)\right\}$$
• Next, the calculation of the rolling load distribution in the elastic reduction region will be described.
• In the elastic reduction region, the relationship expressed by the following Equation (71) can be obtained as a differential equation concerning the rolling load.
  [Math. 165] $$\frac{\mathit{dp}}{\mathit{dx}}\simeq -E\prime \frac{1}{h_1^e}\frac{\mathit{dh}}{\mathit{dx}}-2\mathit{\mu \nu }\prime \frac{p}{h_1^e}=-2E\prime \frac{x-x_0}{h_1^{e}R\prime }-2\mathit{\mu \nu }\prime \frac{p}{h_1^e}$$
• Here, E₁, a₁, and z are defined as in the following Equations (72) to (74).
  [Math. 166] $$E_1=E\prime \frac{h_1^e}{2{\mu }^2\nu {\prime }^{2}R\prime }$$
  [Math. 167] $$a_1=\frac{2\mathit{\mu \nu }\prime }{h_1^e}$$
  [Math. 168] $$z=l-x$$
• Accordingly, the above Equation (71) can be expressed by the following Equation (75).
  [Math. 169] $$\frac{\mathit{dp}}{\mathit{dz}}\simeq -E_1{a_1}^2\left(z-\left(l-x_0\right)\right)+a_{1}p$$
• The initial value problem for the above Equation (75) (p(0) = 0) can be solved analytically, and its solution is expressed by the following Equation (76).
  [Math. 170] $$p_1\left(x\right)=E_1\left[a_1\left(l-x\right)-\left(e^{a_1\left(l-x\right)}-1\right)\left\{1-a_1\left(l-x_0\right)\right\}\right]$$
• Further, based on the approximate solution p₀(x) of the rolling load distribution in the elastic recovery region, q(x) in the elastic recovery region can be approximated by the following Equation (77).
  [Math. 171] $$q\left(x\right)\simeq q_0\left(x\right)=q_0^e+\frac{E_0}{\nu \prime }\left\{\frac{{\left(a_{0}x\right)}^2}{2}-{a_0}^2{x}_{0}x+a_{0}x-\left(e^{a_{0}x}-1\right)\left(1-a_0{x}_0\right)\right\}$$
• In addition, noting that p₀(x₀) = q₀(x₀) + k₀ at the exit-side yield point, a transcendental equation of the following Equation (78) is obtained.
  [Math. 172] $$f\left(x_0\right)=\left(1-\frac{1}{\nu \prime }\right)\left\{a_0{x}_0-\left(e^{a_0{x}_0}-1\right)\left(1-a_0{x}_0\right)\right\}+\frac{{\left(a_0{x}_0\right)}^2}{2\nu \prime }-\frac{q_0^e+k_0}{E_0}=0$$
• [Math. 173]
• Although the contact arc length in the elastic recovery region can be obtained by numerically solving the above Equation (78), a method based on an approximate value $l_0^e$ of the contact arc length obtained by another approach is described below.
• [Math. 174]
• First, for example, when the contact arc length in the elastic recovery region obtained by the cubic-polynomial approximation is Taylor-expanded around $l_0^e$ , the following Equation (79) is obtained.
  [Math. 175] $$f\left(x_0\right)=f\left(l_0^e\right)+f\prime \left(l_0^e\right)\left(x_0-l_0^e\right)+\frac{1}{2}f"\left(l_0^e\right){\left(x_0-l_0^e\right)}^2+\cdots$$
• By truncating the above Equation (79) at an appropriate order and solving for x₀, the contact arc length in the elastic recovery region can be obtained. For example, when truncated at the first-order term, the following Equation (80) is obtained.
  [Math. 176] $$x_0=l_0^e-\frac{f\left(l_0^e\right)}{f\prime \left(l_0^e\right)}$$
• A similar calculation can be carried out for the elastic reduction region.
(Calculation of Contribution to Rolling Load)
• A description will now be given of the processing performed by the process computer 30 to calculate the rolling load of the rolling mill 10.
• The process computer 30 calculates the contribution to the rolling load from at least one of the elastic recovery region and the elastic reduction region, based on the calculated contact arc length.
• The process computer 30 calculates the contribution of the elastic recovery region to the rolling load based on the following Equation (81), and calculates the contribution of the elastic reduction region to the rolling load based on the following Equation (82).
  [Math. 177] $$p_0^e=\frac{E_0}{a_0}\left\{1-\frac{{a_0}^2{l}_0^{e^2}}{2}-e^{a_0{l}_0^e}\left(1-a_0{l}_0^e\right)\right\}$$
  [Math. 178] $$p_1^e=\frac{E_1}{a_1}\left[\frac{{a_1}^2{l}_1^{e^2}}{2}+\left(1+a_1{l}_1^e-e^{a_1{l}_1^e}\right)\left\{1-a_1\left(l-l_0^e\right)\right\}\right]$$ where, in Equation (81), the relationships of the following Equations (83) and (84) apply:
  [Math. 179] $$E_0=E\prime \frac{h_0^e}{2{\mu }^2{\nu }^{\prime 2}R\prime }$$
  [Math. 180] $$a_0=\frac{2\mathit{\mu \nu }\prime }{h_0^e}$$ where, in Equation (82), the relationships of the following Equations (85) to (87) apply:
  [Math. 181] $$E_1=E\prime \frac{h_1^e}{2{\mu }^2{\nu }^{\prime 2}R\prime }$$
  [Math. 182] $$a_1=\frac{2\mathit{\mu \nu }\prime }{h_1^e}$$
  [Math. 183] $$l=l_o^e+\sqrt{R\prime \left(h_1^e-h_0^e\right)+{l_0^e}^2}$$ where, in Equations (81) to (87),
• [Math. 184]
  $p_0^e$ represents the contribution of the elastic recovery region to the rolling load,
• [Math. 185]
  $p_1^e$ represents the contribution of the elastic reduction region to the rolling load,
• [Math. 186]
  $l_o^e$ represents the contact arc length of the elastic recovery region,
• [Math. 187]
  $l_1^e$ represents the contact arc length of the elastic reduction region,
• [Math. 188]
  $h_0^e$ represents the strip thickness at the exit of the roll bite,
• [Math. 189]
  • $h_1^e$ represents the strip thickness at the entry of the roll bite,
  • E' represents the Young's modulus of the rolled material under the plane-strain condition,
  • µ represents the friction coefficient,
  • v' represents the Poisson's ratio of the rolled material under the plane-strain condition, and
  • R' represents the flattened roll radius.
<Explanation Using Definite Integral for Contribution to Rolling Load>
• The contribution of the elastic recovery region to the rolling load can be expressed as an elementary function as in the following Equation (88), by the definite integral of the p₀(x) over the interval corresponding to the elastic recovery region.
  [Math. 190] $$p_0^e={\displaystyle {\int }_0^{l_0^e}{p}_0\left(x\right)\mathit{dx}=\frac{E_0}{a_0}\left\{1-\frac{{a_0}^2{l_0^e}^2}{2}-e^{a_0{l}_0^e}\left(1-a_0{l}_0^e\right)\right\}}$$
• Similarly, the contribution of the elastic reduction region to the rolling load can be expressed as an elementary function as in the following Equation (89), by the definite integral of the pi(x) over the interval corresponding to the elastic reduction region.
  [Math. 191] $$p_1^e={\displaystyle {\int }_{l-l_1^e}^l{p}_1\left(x\right)\mathit{dx}=\frac{E_1}{a_1}\left[\frac{{a_1}^2{l_1^e}^2}{2}+\left(1+a_1{l}_1^e-e^{a_1{l}_1^e}\right)\left\{1-a_1\left(l-l_0^e\right)\right\}\right]}$$
• As described above, the rolling load distribution calculation method, the rolling load calculation method, and the contact arc length calculation method according to the present embodiment calculate the contact arc length of at least one of the elastic recovery region and the elastic reduction region based on Hooke's law under the plane-strain condition of the rolled material 1, the force balance within the roll bite of the rolling mill 10, and a rolling-direction stress distribution approximated as a function dependent on the position coordinate in the rolling direction of the rolling mill 10. The rolling load distribution of at least one of the elastic recovery region and the elastic reduction region is then calculated based on the calculated contact arc length. Furthermore, the contribution of at least one of the elastic recovery region and the elastic reduction region to the rolling load is calculated based on the calculated contact arc length. Since the rolling-direction stress distribution is approximated by a function dependent on the position coordinate in the rolling direction in order to account for the influence of frictional stress in the elastic region, the calculation error can be reduced under rolling conditions where frictional stress increases. Examples of rolling conditions where frictional stress increases include rolling with dull rolls having high roll surface roughness and rolling of high-strength steel.
• In addition, the methods according to the present embodiment provide analytical approximate solutions similar to the methods in NPL 1 that neglect the influence of frictional stress in the elastic region. Therefore, unlike the method based on Airy's stress function described in NPL 1, the methods according to the present embodiment can calculate the rolling load distribution and the contribution of the elastic region to the rolling load without a calculation process involving numerical solutions of differential equations. Accordingly, the methods according to the present embodiment are suitable for online calculation in the process computer 30. Moreover, the methods according to the present embodiment perform calculation using a known theoretical equation for determining the roll flattening radius without introducing a correction parameter as in PTL 1. Therefore, the methods according to the present embodiment have the advantage of eliminating the need to pre-calculate a correction parameter.
• Furthermore, because the rolling-direction stress distribution is approximated by a function dependent on the position coordinate in the rolling direction in order to account for the influence of frictional stress in the elastic region, the methods according to the present embodiment can take the influence of frictional stress in the elastic region into account during the calculation of the rolling load distribution.
• Moreover, the methods according to the present embodiment provide analytical approximate solutions for the rolling load distribution in the elastic region and also provide analytical approximate solutions for the contact arc length in the elastic and plastic regions. Therefore, by combining the methods according to the present embodiment with a model that provides analytical approximate solutions in the plastic region (such as the Bland & Ford model), the rolling load distribution can be calculated at high speed. Accordingly, for example, the method can be used for applications that visualize rolling conditions in real time by inversely calculating parameters such as the friction coefficient and the deformation resistance based on the rolling load and the forward slip, using actual rolling data.
• The methods according to the present embodiment can also be combined with a model that provides numerical, rather than analytical, solutions in the plastic region (such as numerical solutions to Karman's differential equation). In this case, instead of performing convergence calculations such as the Newton-Raphson method to obtain the contact arc length in the elastic region, the contact arc length in the elastic region calculated by the method of the present embodiment may be used, thereby speeding up the rolling load calculation.
• Thus, the methods according to the present embodiment can calculate, at high speed and with high accuracy, the rolling load distribution in the elastic region or the contribution of the elastic region to the rolling load, even under rolling conditions where frictional stress increases.
• When calculating the rolling load distribution using the rolling load distribution calculation method according to the present embodiment, the calculation may be performed in a manner that reflects the current lubrication condition. In this case, the actual rolling load may be acquired, and the friction coefficient may be varied using the rolling load distribution calculation method of the present embodiment to estimate the friction coefficient that makes the estimated load match the actual load. The estimated friction coefficient may then be regarded as a friction coefficient reflecting the current lubrication condition and applied to subsequent rolling load estimations. In doing so, the friction coefficient may be set based on tables and model formulas derived from operating factors. The roll gap may also be adjusted based on the rolling load distribution calculated in this manner. Likewise, when calculating the rolling load using the rolling load calculation method of the present embodiment, the calculation may similarly reflect the current lubrication condition, and the roll gap may be adjusted based on the calculation results.
(EXAMPLES)
• FIG. 2 illustrates the results of calculating the rolling load distribution using the methods of the present disclosure and the results of calculating the rolling load distribution using a conventional method. In FIG. 2, the models for the friction coefficient and deformation resistance are fixed. FIG. 2 illustrates the results obtained by applying the methods of the present embodiment to the calculation of the rolling load distribution assuming rolling with dull rolls.
• In FIG. 2, "BlandAndFord" represents the result of calculating the rolling load distribution using the conventional method. "BlandAndFord" calculates the rolling load distribution in the elastic region using a known method and calculates the rolling load distribution in the plastic region using the known Bland & Ford method.
• In FIG. 2, "BlandAndFord1" represents the result obtained using the method of the present embodiment, where the rolling-direction stress q is approximated by a linear polynomial to calculate the contact arc length of the elastic region, and then the rolling load distribution of the elastic region is calculated based on the calculated contact arc length. Note that "BlandAndFord1" uses the known Bland & Ford method to calculate the rolling load distribution in the plastic region.
• In FIG. 2, "BlandAndFord3" represents the result obtained using the method of the present embodiment, where the rolling-direction stress q is approximated by a cubic polynomial to calculate the contact arc length of the elastic region, and then the rolling load distribution of the elastic region is calculated based on the calculated contact arc length. Note that "BlandAndFord3" uses the known Bland & Ford method to calculate the rolling load distribution in the plastic region.
• In FIG. 2, "Karman" represents the result of calculating the rolling load distribution as a rigorous numerical solution using Karman's equation.
• Referring to FIG. 2, it can be confirmed that the rolling load distribution calculated using "BlandAndFord1" and "BlandAndFord3," which apply the methods of the present embodiment, is almost identical to the result of "Karman," which represents a rigorous numerical solution.
• FIG. 3 is a table comparing the results obtained using the methods of the present embodiment with the results obtained using conventional methods.
• In FIG. 3, "Bland&Ford" and "Bland&Sims" represent the results obtained using conventional methods.
• In FIG. 3, "Bland&Ford1" and "Bland&Sims1" represent the results obtained by calculating the rolling-direction stress q using a linear polynomial approximation in the method of the present embodiment. Likewise, "Bland&Ford3" and "Bland&Sims3" represent the results obtained by calculating the rolling-direction stress q using a cubic polynomial approximation in the method of the present embodiment.
• In FIG. 3, "Karman" represents the result obtained by calculating a rigorous numerical solution. In FIG. 3, "Karman3" represents the result obtained by calculating the rolling-direction stress q in the elastic region using a cubic polynomial approximation.
• Referring to FIG. 3, the calculation time required for calculating the rolling load using the method of the present embodiment is no more than one one-hundredth of that required for "Karman", which calculates a rigorous numerical solution. Furthermore, the rolling load calculated using the method of the present embodiment is almost identical to the rolling load calculated by "Karman", which calculates a rigorous numerical solution.
• The present disclosure is not limited to the embodiment described above. For example, multiple blocks illustrated in the block diagram may be integrated, or one block may be divided into multiple blocks. Instead of executing the plurality of steps illustrated in the flowchart sequentially in accordance with the description, the steps may be executed in parallel or in a different order depending on the processing capability of the device executing the steps or as necessary. Various modifications may be made without departing from the spirit of the present disclosure.
• For example, although the process computer 30 and the online computer 40 are separate devices in FIG. 1, they may be implemented as a single device. In such a case, the process computer 30 may have the functions of the online computer 40.
• For example, the method according to the present disclosure can also be applied when using Orowan's theory, which takes the influence of shear stress into account during the calculation of the plastic region.
REFERENCE SIGNS LIST

1

Rolled material

10

Rolling mill

20

Control unit

30

Process computer

40

Online computer

Claims (10)

1. A rolling load distribution calculation method, which is a method of calculating rolling load distribution of a rolling mill, the method comprising:

   calculating a contact arc length of either or both of an elastic recovery region and an elastic reduction region based on Hooke's law under a plane-strain condition of a rolled material, a force balance within a roll bite of the rolling mill, and a rolling-direction stress distribution approximated by a function dependent on a position coordinate in the rolling direction of the rolling mill; and

   calculating rolling load distribution of either or both of the elastic recovery region and the elastic reduction region based on the contact arc length.
2. The rolling load distribution calculation method according to claim 1, wherein, when calculating the rolling load distribution, the rolling load distribution of the elastic recovery region is calculated based on the following Equation (1), and the rolling load distribution of the elastic reduction region is calculated based on the following Equation (2),
   [Math. 1] $$p_0\left(x\right)=E_0\left\{a_{0}x-\left(e^{a_{0}x}-1\right)\left(1-a_0{l}_0^e\right)\right\}$$
   [Math. 2] $$p_1\left(x\right)=E_1\left[a_1\left(l-x\right)-\left(e^{a_1\left(l-x\right)}-1\right)\left\{1-a_1\left(l-l_0^e\right)\right\}\right]$$ where, in Equation (1), the relationships of the following Equations (3) and (4) apply:
   [Math. 3] $$E_0=E\prime \frac{h_0^e}{2{\mu }^2\nu {\prime }^{2}R\prime }$$
   [Math. 4] $$a_0=\frac{2\mathit{\mu \nu }\prime }{h_0^e}$$ where, in Equation (2), the relationships of the following Equations (5) to (7) apply:
   [Math. 5] $$E_1=E\prime \frac{h_1^e}{2{\mu }^2\nu {\prime }^{2}R\prime }$$
   [Math. 6] $$a_1=\frac{2\mathit{\mu \nu }\prime }{h_1^e}$$
   [Math. 7] $$l=l_o^e+\sqrt{R\prime \left(h_1^e-h_0^e\right)+{l_0^e}^2}$$ where, in Equations (1) to (7),

   p₀(x) represents the rolling load distribution of the elastic recovery region,

   p₁(x) represents the rolling load distribution of the elastic reduction region,

   x represents the position coordinate in the rolling direction,
   [Math. 8]

   $l_o^e$ represents the contact arc length of the elastic recovery region,
   [Math. 9]

   $h_0^e$ represents a strip thickness at an exit of the roll bite,
   [Math. 10]

   $h_1^e$ represents a strip thickness at an entry of the roll bite,

   E' represents a Young's modulus of the rolled material under the plane-strain condition,

   µ represents a friction coefficient,

   v' represents a Poisson's ratio of the rolled material under the plane-strain condition, and

   R' represents a flattened roll radius.
3. The rolling load distribution calculation method according to claim 1 or 2, wherein, when calculating the contact arc length, the contact arc length of the elastic recovery region is calculated based on the following Equation (8), and the contact arc length of the elastic reduction region is calculated based on the following Equation (9),
   [Math. 11] $$l_0^e=\sqrt{\frac{\left(q_0^e+k_0\right)h_0^{e}R\prime }{E\prime }}{\beta }_0$$
   [Math. 12] $$l_1^e=\frac{3h_1^e}{4\mu }{\beta }_1$$ where, in Equation (8), the relationships of the following Equations (10) to (13) apply:
   [Math. 13] $${\beta }_0=2\sqrt{\frac{-p}{3}}\mathit{cos}\left(\frac{1}{3}{\mathit{cos}}^{-1}\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)-\frac{2\pi }{3}\right)+\frac{\eta }{3}$$
   [Math. 14] $$p=\frac{-3\nu \prime -{\eta }^2}{3}$$
   [Math. 15] $$q=\frac{\eta }{27}\left(-2{\eta }^2-9\nu \prime +27\right)$$
   [Math. 16] $$\eta =\frac{3}{4\mu }\sqrt{\frac{E\prime h_0^e}{\left(q_0^e+k_0\right)R\prime }}$$ where, in Equation (9), the relationships of the following Equations (14) to (20) apply:
   [Math. 17] $${\beta }_1=2\sqrt{\frac{-p}{3}}\mathit{cos}\left(\frac{1}{3}{\mathit{cos}}^{-1}\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)-\frac{4\pi }{3}\right)-\frac{b}{3}$$
   [Math. 18] $$p=\frac{3c-b^2}{3}$$
   [Math. 19] $$q=\frac{1}{27}\left(2b^3-9\mathit{bc}+27d\right)$$
   [Math. 20] $$d=-\frac{16{\mu }^{2}R\prime }{9h_1^e}\frac{q_1^e+k_1}{E\prime }$$
   [Math. 21] $$c=\frac{8\mu }{3h_1^e}\left(l-l_0^e+\nu \prime \frac{2\mathit{\mu R}\prime }{3}\frac{q_1^e+k_1}{E\prime }\right)$$
   [Math. 22] $$b=-1-\frac{8\mu }{3h_1^e}\left(l-l_0^e\right)$$
   [Math. 23] $$l=l_0^e+\sqrt{R\prime \left(h_1^e-h_0^e\right)+{l_0^e}^2}$$ where, in Equations (8) to (20):

   [Math. 24]
   $l_o^e$ represents the contact arc length of the elastic recovery region,

   [Math. 25]
   $l_1^e$ represents the contact arc length of the elastic reduction region, E' represents a Young's modulus of the rolled material under the plane-strain condition,

   [Math. 26]
   $q_0^e$ represents a tension at an exit of the roll bite,

   [Math. 27]

   $q_1^e$ represents a tension at an entry of the roll bite,

   k₀ represents a yield stress at the exit of the roll bite,

   k₁ represents a yield stress at the entry of the roll bite,

   [Math. 28]
   $h_0^e$ represents a strip thickness at the exit of the roll bite,

   [Math. 29]

   $h_1^e$ represents a strip thickness at the entry of the roll bite,

   R' represents a flattened roll radius,

   v' represents a Poisson's ratio of the rolled material under the plane-strain condition,

   µ represents a friction coefficient, and

   E' represents a Young's modulus of the rolled material under the plane-strain condition.
4. A rolling load calculation method, which is a method of calculating a rolling load of a rolling mill, the method comprising:

   calculating a contact arc length of either or both of an elastic recovery region and an elastic reduction region based on Hooke's law under a plane-strain condition of a rolled material, a force balance within a roll bite of the rolling mill, and a rolling-direction stress distribution approximated by a function dependent on a position coordinate in the rolling direction of the rolling mill; and

   calculating contribution to rolling load from either or both of the elastic recovery region and the elastic reduction region based on the contact arc length.
5. The rolling load calculation method according to claim 4, wherein, when calculating the contribution to the rolling load, the contribution to the rolling load from the elastic recovery region is calculated based on the following Equation (1), and the contribution to the rolling load from the elastic reduction region is calculated based on the following Equation (2),
   [Math. 30] $$p_0^e=\frac{E_0}{a_0}\left\{1-\frac{{a_0}^2{l_0^e}^2}{2}-e^{a_0{l}_0^e}\left(1-a_0{l}_0^e\right)\right\}$$
   [Math. 31] $$p_1^e=\frac{E_1}{a_1}\left[\frac{{a_1}^2{l_1^e}^2}{2}+\left(1+a_1{l}_1^e-e^{a_1{l}_1^e}\right)\left\{1-a_1\left(l-l_0^e\right)\right\}\right]$$ where, in Equation (1), the relationships of the following Equations (3) and (4) apply:
   [Math. 32] $$E_0=E\prime \frac{h_0^e}{2{\mu }^2\nu {\prime }^{2}R\prime }$$
   [Math. 33] $$a_0=\frac{2\mathit{\mu \nu }\prime }{h_0^e}$$ where, in Equation (2), the relationships of the following Equations (5) to (7) apply:
   [Math. 34] $$E_1=E\prime \frac{h_1^e}{2{\mu }^2\nu {\prime }^{2}R\prime }$$
   [Math. 35] $$a_1=\frac{2\mathit{\mu \nu }\prime }{h_1^e}$$
   [Math. 36] $$l=l_o^e+\sqrt{R\prime \left(h_1^e-h_0^e\right)+l_0^{e^2}}$$ where, in Equations (1) to (7):

   [Math. 37]
   $p_0^e$ represents the contribution of the elastic recovery region to the rolling load,

   [Math. 38]
   $p_1^e$ represents the contribution of the elastic reduction region to the rolling load,

   [Math. 39]
   $l_o^e$ represents the contact arc length of the elastic recovery region,

   [Math. 40]
   $l_1^e$ represents the contact arc length of the elastic reduction region,

   [Math. 41]
   $h_0^e$ represents a strip thickness at an exit of the roll bite,

   [Math. 42]

   $h_1^e$ represents a strip thickness at an entry of the roll bite,

   E' represents a Young's modulus of the rolled material under the plane-strain condition,

   µ represents a friction coefficient,

   v' represents a Poisson's ratio of the rolled material under the plane-strain condition, and

   R' represents a flattened roll radius.
6. The rolling load calculation method according to claim 4 or 5, wherein, when calculating the contact arc length, the contact arc length of the elastic recovery region is calculated based on the following Equation (8), and the contact arc length of the elastic reduction region is calculated based on the following Equation (9),
   [Math. 43] $$l_0^e=\sqrt{\frac{\left(q_0^e+k_0\right)h_0^{e}R\prime }{E\prime }}{\beta }_0$$
   [Math. 44] $$l_1^e=\frac{3h_1^e}{4\mu }{\beta }_1$$ where, in Equation (8), the relationships of the following Equations (10) to (13) apply:
   [Math. 45] $${\beta }_0=2\sqrt{\frac{-p}{3}}\mathit{cos}\left(\frac{1}{3}{\mathit{cos}}^{-1}\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)-\frac{2\pi }{3}\right)+\frac{\eta }{3}$$
   [Math. 46] $$p=\frac{-3\nu \prime -{\eta }^2}{3}$$
   [Math. 47] $$q=\frac{\eta }{27}\left(-2{\eta }^2-9\nu \prime +27\right)$$
   [Math. 48] $$\eta =\frac{3}{4\mu }\sqrt{\frac{E\prime h_0^e}{\left(q_0^e+k_0\right)R\prime }}$$ where, in Equation (9), the relationships of the following Equations (14) to (20) apply:
   [Math. 49] $${\beta }_1=2\sqrt{\frac{-p}{3}}\mathit{cos}\left(\frac{1}{3}{\mathit{cos}}^{-1}\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)-\frac{4\pi }{3}\right)-\frac{b}{3}$$
   [Math. 50] $$p=\frac{3c-b^2}{3}$$
   [Math. 51] $$q=\frac{1}{27}\left(2b^3-9\mathit{bc}+27d\right)$$
   [Math. 52] $$d=-\frac{16{\mu }^{2}R\prime }{9h_1^e}\frac{q_1^e+k_1}{E\prime }$$
   [Math. 53] $$c=\frac{8\mu }{3h_1^e}\left(l-l_0^e+\nu \prime \frac{2\mathit{\mu R}\prime }{3}\frac{q_1^e+k_1}{E\prime }\right)$$
   [Math. 54] $$b=-1-\frac{8\mu }{3h_1^e}\left(l-l_0^e\right)$$
   [Math. 55] $$l=l_0^e+\sqrt{R\prime \left(h_1^e-h_0^e\right)+{l_0^e}^2}$$ where, in Equations (8) to (20):

   [Math. 56]
   $l_o^e$ represents the contact arc length of the elastic recovery region,

   [Math. 57]

   $l_1^e$ represents the contact arc length of the elastic reduction region,

   E' represents a Young's modulus of the rolled material under the plane-strain condition,

   [Math. 58]
   $q_0^e$ represents a tension at an exit of the roll bite,

   [Math. 59]

   $q_1^e$ represents a tension at an entry of the roll bite,

   ko represents a yield stress at the exit of the roll bite,

   k₁ represents a yield stress at the entry of the roll bite,

   [Math. 60]
   $h_0^e$ represents a strip thickness at the exit of the roll bite,

   [Math. 61]

   $h_1^e$ represents a strip thickness at the entry of the roll bite,

   R' represents a flattened roll radius,

   v' represents a Poisson's ratio of the rolled material under the plane-strain condition,

   µ represents a friction coefficient, and

   E' represents a Young's modulus of the rolled material under the plane-strain condition.
7. A contact arc length calculation method, which is a method of calculating a contact arc length of a rolling mill, the method comprising:
   calculating a contact arc length of either or both of an elastic recovery region and an elastic reduction region based on Hooke's law under a plane-strain condition of a rolled material, a force balance within a roll bite of the rolling mill, and a rolling-direction stress distribution approximated by a function dependent on a position coordinate in the rolling direction of the rolling mill.
8. The contact arc length calculation method according to claim 7, wherein, when calculating the contact arc length, the contact arc length of the elastic recovery region is calculated based on the following Equation (1), and the contact arc length of the elastic reduction region is calculated based on the following Equation (2),
   [Math. 62] $$l_0^e=\sqrt{\frac{\left(q_0^e+k_0\right)h_0^{e}R\prime }{E\prime }}{\beta }_0$$
   [Math. 63] $$l_1^e=\frac{3h_1^e}{4\mu }{\beta }_1$$ where, in Equation (1), the relationships of the following Equations (3) to (6) apply:
   [Math. 64] $${\beta }_0=2\sqrt{\frac{-p}{3}}\mathit{cos}\left(\frac{1}{3}{\mathit{cos}}^{-1}\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)-\frac{2\pi }{3}\right)+\frac{\eta }{3}$$
   [Math. 65] $$p=\frac{-3\nu \prime -{\eta }^2}{3}$$
   [Math. 66] $$q=\frac{\eta }{27}\left(-2{\eta }^2-9\nu \prime +27\right)$$
   [Math. 67] $$\eta =\frac{3}{4\mu }\sqrt{\frac{E\prime h_0^e}{\left(q_0^e+k_0\right)R\prime }}$$ where, in Equation (2), the relationships of the following Equations (7) to (13) apply:
   [Math. 68] $${\beta }_1=2\sqrt{\frac{-p}{3}}\mathit{cos}\left(\frac{1}{3}{\mathit{cos}}^{-1}\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)-\frac{4\pi }{3}\right)-\frac{b}{3}$$
   [Math. 69] $$p=\frac{3c-b^2}{3}$$
   [Math. 70] $$q=\frac{1}{27}\left(2b^3-9\mathit{bc}+27d\right)$$
   [Math. 71] $$d=-\frac{16{\mu }^{2}R\prime }{9h_1^e}\frac{q_1^e+k_1}{E\prime }$$
   [Math. 72] $$c=\frac{8\mu }{3h_1^e}\left(l-l_0^e+\nu \prime \frac{2\mathit{\mu R}\prime }{3}\frac{q_1^e+k_1}{E\prime }\right)$$
   [Math. 73] $$b=-1-\frac{8\mu }{3h_1^e}\left(l-l_0^e\right)$$
   [Math. 74] $$l=l_0^e+\sqrt{R\prime \left(h_1^e-h_0^e\right)+l_0^{e^2}}$$ where, in Equations (1) to (13):

   [Math. 75]
   $l_o^e$ represents the contact arc length of the elastic recovery region,

   [Math. 76]

   $l_1^e$ represents the contact arc length of the elastic reduction region,

   E' represents a Young's modulus of the rolled material under the plane-strain condition,

   [Math. 77]
   $q_0^e$ represents a tension at an exit of the roll bite,

   [Math. 78]

   $q_1^e$ represents a tension at an entry of the roll bite,

   k₀ represents a yield stress at the exit of the roll bite,

   k₁ represents a yield stress at the entry of the roll bite,

   [Math. 79]
   $h_0^e$ represents a strip thickness at the exit of the roll bite,

   [Math. 80]

   $h_1^e$ represents a strip thickness at the entry of the roll bite,

   R' represents a flattened roll radius,

   v' represents a Poisson's ratio of the rolled material under the plane-strain condition,

   µ represents a friction coefficient, and

   E' represents a Young's modulus of the rolled material under the plane-strain condition.
9. A rolling method, comprising changing a roll gap based on the rolling load distribution calculated in a manner reflecting a current lubrication condition using the rolling load distribution calculation method according to any one of claims 1 to 3.
10. A rolling method, comprising changing a roll gap based on the rolling load calculated in a manner reflecting a current lubrication condition using the rolling load calculation method according to any one of claims 4 to 6.