The locus of the Mises yield criterion is a circle on the π plane. The dodecagon between the circumscribed hexagon (TSS yield locus) and the inscribed hexagon (Tresca yield locus) can be used as a linear approximation to the Mises circle. In Figure 1, suppose that there is a moving point E on the line segment BF, and a new yield edge can be obtained by connecting B ′ E . Set the angle ∠ F B ′ E between B ′ F and B ′ E to θ , then it can be seen that when θ = 0 ∘ , it corresponds to Tresca yield locus B ′ F and when θ = 30 ∘ , it corresponds to the TSS yield locus B ′ B .

In Figure 1, the radius of the Mises circle satisfies O B ′ = O D = 6 / 3 σ s ( σ s is the yield strength). So, according to the geometric relationship, O F = σ s / 2 , B ′ F = σ s / 6 .

Set the origin of the coordinate system at the point B ′ , as shown in Figure 2, the slope of B ′ E is k ( θ ) = tan ⁡ θ , 0 ° ≤ θ ≤ 30 ° . (1)

From the mean value theorem of integral (Zhang et al., 2020), the mean slope k ¯ of B ′ E can be obtained as k ¯ = 1 ( π 6 − 0 ) ∫ 0 π 6 tan ⁡ θ d θ = 0.275 (2)

Let k ¯ = tan ⁡ θ , the undetermined value of θ and the length of B ′ E at this time can be obtained θ = arctan k ¯ = 0.268 (3) B ′ E = B ′ F cos ⁡ θ = 0.423 σ s (4)

Based on the above formulas, we can get F E = B ′ E 2 − B ′ F 2 = 0.110 σ s . (5)

The modulus length of the deviation vector on the Mises yield locus is O D = O B ′ = 6 / 3 σ s . (6)

The modulus length of the deviation vector of the newly proposed yield criterion is O E = 0.817 σ s . (7)

So, the error between them is Δ = ( 0.817 − 6 3 ) / 6 3 = 0.0617 % . (8)

It can be seen that the vector modulus length of the newly proposed yield criterion is just a little higher than that of the Mises yield criterion, which means that the point E should be locates in between B and D, and near D, as shown in Figure 2.

The expressions of the lines A ′ E and B ′ E are established in the following section. Figure 3 shows the projection of the principal stress component σ 1 on the π plane. Considering the projection relationship and the plain deformation condition, the three principal stress components σ 1 , σ 2 , σ 3 at point E can be calculated by { σ 1 = O E × 3 2 cos 30 ∘ = 1.1554 σ s ; σ 3 = 0 ; σ 2 = σ 1 + σ 3 2 = 0.57776 σ s . (9)

It can be assumed here that the A ′ E line can be written in the following form σ 1 − a 1 σ 2 − a 2 σ 3 − c = 0 (10)

When a material yields, there have c = σ s , a 1 + a 2 = 1 . Substituting Eq. 9 into Eq. 10, it produces a 1 = 0.268 , a 2 = 0.732 . (11)

Substituting Eq. 11 into Eq. 10, the expression of A ′ E can be derived as σ 1 − 0.268 σ 2 − 0.732 σ 3 = σ s , σ 2 ≤ 1 2 ( σ 1 + σ 3 ) . (12)

In the same way, the equation of the locus B ′ E can be determined as 0.732 σ 1 + 0.268 σ 2 − σ 3 = σ s , σ 2 ≥ 1 2 ( σ 1 + σ 3 ) . (13)

Eqs 12, 13 are the mathematical expressions of the new yield criterion, which is a linear combination of the principal stress components. Because the slope of the locus side B ′ E of this criterion is the mean value of the slopes of the Tresca and TSS loci, it can be called as the average slope yield criterion.

It can be seen from Figure 2 that the present yield locus intersects the Mises circle, and the vertex angles are as follows: { ∠ F B ′ E = ∠ θ = 15.355 ∘ ; ∠ O B ′ E = 60 ∘ + 15.355 ∘ = 75.355 ∘ ; ∠ O E B ′ = 180 ∘ − 30 ∘ − 75.355 ∘ = 74.645 ∘ ; 2 ∠ O B ′ E = 150.71 ∘ ,   2 ∠ O E B ′ = 149.29 ∘ . (14)

Figure 1 and Eq. 14 show that the present yield locus is an equilateral and non-equiangular dodecagon which intersects the Mises circle with twelve intersection points. Of which, six vertices of the locus are on the Mises circle, and the vertex angle is 150.71 ° . The other six vertices are located outside the Mises circle, with a distance of 0.000503 σ s and the vertex angle is 149.29 ° . Except for these, it is known that each side length of the present dodecagon is equal to 0.423 σ s .

Suppose that the stress component σ i j satisfies f ( σ i j ) = 0 and the strain rate component ε ˙ i j satisfies the flow rule of plastic deformation, then we can have (Kobayashi et al., 1989) ε ˙ i j = d λ ∂ f ∂ σ i j = d λ σ i j ′ . (15)

Assuming that there are λ ≥ 0 , μ ≥ 0 , then from Eqs 12, 15, it results in ε ˙ 1 : ε ˙ 2 : ε ˙ 3 = 1 : ( − 0.268 ) : ( − 0.732 ) = λ : ( − 0.268 ) λ : ( − 0.732 ) λ . (16)

Similarly, according to Eqs 13, 15, it results in ε ˙ 1 : ε ˙ 2 : ε ˙ 3 = 0.732 : 0.268 : ( − 1 ) = 0.732 μ : 0.268 μ : ( − μ ) . (17)

The linear combination of the results obtained from the above two formulas gives ε ˙ 1 : ε ˙ 2 : ε ˙ 3 = ( λ + 0.732 μ ) : 0.268 ( μ − λ ) : ( − 0.732 λ − μ ) . (18)

Take ε ˙ 1 = λ + 0.732 μ , then ε ˙ 2 = 0.268 ( μ − λ ) , ε ˙ 3 = − ( 0.732 λ + μ ) . (19)

Since ε ˙ max = ε ˙ 1 , ε ˙ min = ε ˙ 3 , it results in ε ˙ max − ε ˙ min = 1.732 ( λ + μ ) , ( λ + μ ) = 1000 1732 ( ε ˙ max − ε ˙ min ) . (20)

At the vertex E, we notice that σ 2 = ( σ 1 + σ 3 ) / 2 , so it can be obtained from Eqs 12, 13 that σ 1 − σ 3 = 2000 1732 σ s . (21)

From Eqs 20, 21, the specific plastic power calculated by D ( ε ˙ i j ) = σ 1 ε ˙ 1 + σ 2 ε ˙ 2 + σ 3 ε ˙ 3 (Hosford and Caddell, 2011) based on the present yield criterion can be obtained as D ( ε ˙ i j ) = σ 1 ε ˙ 1 + σ 1 + σ 3 2 ε ˙ 2 + σ 3 ε ˙ 3                     = 0.8660 ( σ 1 − σ 3 ) ( μ + λ )                     = 1732 2000 × 2000 1732 σ s × 1000 1732 ( ε ˙ max − ε ˙ min )                     = 1000 1732 σ s ( ε ˙ max − ε ˙ min )                     = 0.577 σ s ( ε ˙ max − ε ˙ min ) . (22)

By using the same method, the specific plastic power based on the Tresca yield criterion and the TSS yield criterion can be written as follows (Kobayashi et al., 1989) D ( ε ˙ i j ) Tresca = 2 3 σ s ( ε ˙ max − ε ˙ min ) , (23) D ( ε ˙ i j ) TSS = 1 2 σ s ( ε ˙ max − ε ˙ min ) . (24)

Comparing Eq. 22 with Eqs 23, 24, the relative errors of the specific plastic power among the present one, the Tresca one, and the TSS one are { Δ Tresca = ( 0.577 − 1 2 ) / 1 2 = 15.40 %; Δ TSS = ( 0.577 − 2 3 ) / 2 3 = − 13.45 % . (25)

The relative errors in Eq. 25 indicate that the calculated value of using the mean slope yield criterion is 15.40 % higher than the calculated value based on the Tresca one, but is lower 13.45 % than that based on the TSS one.

When the principal stress components satisfy σ 1 ≥ σ 2 ≥ σ 3 , Lode proposed a stress characteristic parameter to reflect the difference among different yield criteria. The parameter is (Lode, 1926): μ = 2 σ 2 − σ 1 − σ 3 σ s . (26)

Substituting Eq. 26 into the Tresca criterion, Mises criterion, TSS yield criterion as well as the AS yield criterion, the following expressions in terms of the Lode characteristic parameter can be derived as follows:

Tresca: σ 1 − σ 3 σ s = 1 (27)

Mises: σ 1 − σ 3 σ s = 2 3 + μ 2 (28)

TSS: σ 1 − σ 3 σ s = { 4 + μ 3 , − 1 ≤ μ ≤ 0 ; 4 − μ 3 , 0 ≤ μ ≤ 1. (29)

AS: σ 1 − σ 3 σ s = { 2000 + 268 μ 1732 , − 1 ≤ μ ≤ 0 ; 2000 − 268 μ 1732 , 0 ≤ μ ≤ 1. (30)

Based on the above expressions, a comparison can be carried out with the experimental data from Ref. (Lode, 1926; Lessells and Macgregor, 1940; Naghdi et al., 1957; Maxey, 1974), shown in Figure 4.

It can be seen from Figure 4 that the Tresca yield criterion predicts the lower bound of the experimental data, while the TSS yield criterion predicts the upper bound. The present calculated value is lie between those based on the TSS criterion and the Tresca criterion, and closes to that based on the Mises yield criterion. It can be seen that the predictive results based on the AS yield criterion correlate well with the experimental data, and can bring in the reasonable intermediate results since both the average error and the maximum error are very small, just 3.21 and 10.66% respectively.

As shown in Figure 5, h 0 is half thickness of the workpiece on the inlet section, h 1 is half thickness of the workpiece on the outlet section, R is the radius of the roller, point O is the geometrical center of the roller, v 0 is the initial velocity on the inlet section, and v 1 is the final velocity on the outlet section. θ is the contact angle, α is the neutral angle, x is the length of the neutral point from the entrance of the deformation zone, and h x is half thickness of the workpiece in the deformation zone. It should be noted that when the tangential velocity of the rolled workpiece at the neutral point is equal to the tangential velocity of the roller, and the tangential velocity discontinuity and friction power at this time are both zero. According to the geometric relationship in Figure 5, the main deformation parameters can be expressed as (Li et al., 2017) { z = h x = R + h 1 − [ R 2 − ( l − x ) 2 ] 1 / 2 ; z = h α = R + h 1 − R ⁡ cos ⁡ α . (31) where, h x is the half thickness at the position x and h α is the corresponding thickness at the angle α . l − x = R ⁡ sin ⁡ α , d x = − R ⁡ cos ⁡ α d α (32) h x ′ = − tan ⁡ α ,   h x ″ = ( R ⁡ cos 3 ⁡ α ) − 1 (33) { y = b x = b 0 + Δ b l x = b 1 − Δ b l R ⁡ sin ⁡ α ; b x ′ = Δ b l . (34)

From Eqs 31–34, the boundary conditions are as follows { x = 0 , α = θ :   h x = h α = h θ = h 0 , b x = b α = b θ = b 0 , h x ′ = − tan ⁡ θ , b x ′ = − Δ b l R ⁡ cos ⁡ θ ; x = l , α = 0 :   h x = h α = h 1 , b x = b α = b 1 ,  h x ′ = 0 , b x ′ = 0. (35) where, b x is the half of plate width along the x and Δ b = b 1 − b 0 is the spread between the final width and the initial width.

This article assumes that the horizontal velocity component v x in the deformation zone increases gradually according to the Logistic function from the inlet to the outlet, that is, it has the following functional form v x = c 1 c 3 e c 2 x c 1 + c 3 ( e c 2 x − 1 ) . (36) where, c 1 , c 2 , c 3 are undetermined parameters, which are determined by the boundary conditions on the entrance and exit sections: x = 0 , v x = v 0 ;   x = l / 2 ,  v x = ( v 0 + v 1 ) / 2 ;   x = l ,  v x = v 1 . So, the following velocity field can be established { v x = ( v 0 + v 1 ) 1 + ( v 1 v 0 ) 1 − 2 x l ; v y = b x ′ ⋅ v x = Δ b l ⋅ ( v 0 + v 1 ) 1 + ( v 1 v 0 ) 1 − 2 x l ; v z = v x h x ′ h x z = ( v 0 + v 1 ) 1 + ( v 1 v 0 ) 1 − 2 x l ⋅ h x ′ h x z . (37)

According to the volume constant condition, i.e. v 0 h 0 b 0 = v 1 h 1 b 1 , v 1 / v 0 = h 0 b 0 / h 1 b 1 = ξ , then the velocity field can be simplified into the following form: { v x = ( 1 + ξ ) v 0 1 + ξ 1 − 2 x l ; v y = b x ′ ⋅ v x = Δ b l ⋅ ( 1 + ξ ) v 0 1 + ξ 1 − 2 x l ; v z = v x h x ′ h x z = ( 1 + ξ ) v 0 1 + ξ 1 − 2 x l ⋅ h x ′ h x z . (38) where, v x , v y and v z are called the velocity components in the rolling direction, the spreading direction and the reduction direction, respectively. According to the geometric equation, we can get { ε ˙ x = d v x d x = [ 2 v 0 ( 1 + ξ ) ln ⁡ ξ ] ξ ( 1 − 2 x l ) l [ 1 + ξ ( 1 − 2 x l ) ] 2 = ε ˙ max ; ε ˙ y = 0 ; ε ˙ z = − ε ˙ max = − [ 2 v 0 ( 1 + ξ ) ln ⁡ ξ ] ξ ( 1 − 2 x l ) l [ 1 + ξ ( 1 − 2 x l ) ] 2 = ε ˙ min . (39) where, ε ˙ x , ε ˙ y , and ε ˙ z are the strain rate components in the rolling direction, the spreading direction and the reduction direction, respectively. The volume rate per second denoted by U in the deformation zone can be written as follows: U = v x h x b x = v n h n b n = v R ⁡ cos ⁡ α n ( R + h 1 − R ⁡ cos ⁡ α n ) ( b 1 − Δ b l R ⁡ sin ⁡ α n ) = v 1 h 1 b 1 . (40)

In Eqs 38, 39, due to ε ˙ x + ε ˙ z = 0 ; x = 0 , v x = v 0 ; x = l , v x = v 1 ; z = 0 , v z = 0 ; z = h x , v z = − v x ⁡ tan ⁡ α , the velocity field meets the kinematically admissible condition rigorously and can be used as a basis for the following energy calculation.

It is known that the specific plastic power based on the Mises yield criterion is nonlinear, which is difficult to be integrated to obtain the internal deformation power. For solving this problem, a new method called the replacement method of specific plastic power is proposed here. This method is to replace the specific plastic power of the Mises yield criterion by the present specific plastic power based on the AS yield criterion. Therefore, the internal deformation power N d (Sezek et al., 2008) can be calculated by N d = 4 ∫ 0 l ∫ 0 b m ∫ 0 h x 0.577 σ s ( ε ˙ max − ε ˙ min ) d x d y d z       = 4 ∫ 0 l ∫ 0 b m ∫ 0 h x 0.577 σ s ⋅ 2 ε ˙ max d x d y d z       = 4.616 σ s ∫ 0 l ∫ 0 b m ∫ 0 h x [ 2 v 0 ( 1 + ξ ) ln ⁡ ξ ] ξ ( 1 − 2 x l ) l [ 1 + ξ ( 1 − 2 x l ) ] 2 d x d y d z       = 4.616 σ s U ⋅ ln ⁡ ξ . (41)

According to the roller surface equation d S = 1 + h x ′ 2 d x d y = sec ⁡ α d x d y and Eq. 36, the friction power consumed on the contact surface between the roller and the workpiece can be calculated by (Sezek et al., 2008) N f = 4 m k ∫ 0 l ∫ 0 b x Δ v f 1 + ( h x ′ ) 2 d x d y (42) Δ v f = v R − v x 1 + ( h x ′ ) 2 = v R − v x ⁡ sec ⁡ α (43) where, m is the friction factor, Δ v f is the velocity discontinuity. The shear yield strength is k = σ s / 3 . According to the collinear characteristic between the friction shear stress τ f and the velocity discontinuity Δ v f , the shear power can be calculated by N f = 4 ∫ 0 l ∫ 0 b x τ f ⋅ Δ v f d S       = 4 ∫ 0 l ∫ 0 b x ( τ f x Δ v x + τ f y Δ v y + τ f z Δ v z ) 1 + ( h x ′ ) 2 d x d y       = 4 m k ∫ 0 l ∫ 0 b x ( Δ v x ⁡ cos ⁡ α + Δ v y ⁡ cos ⁡ β + Δ v z ⁡ cos ⁡ γ ⁡ sec ⁡ α ) d x d y . (44)

From Figure 5, the direction cosines between Δ v f and the coordinate axis are respectively { cos ⁡ α = ± [ R 2 − ( l − x ) 2 ] 1 / 2 / R ; cos ⁡ β = 0 ; cos ⁡ γ = ± ( l − x ) / R = sin ⁡ α . (45)

Substituting Eq. 45 into Eq. 44, we can get N f = 4 m k [ ∫ 0 l ∫ 0 b x ( v R ⁡ cos ⁡ α − v x ) cos ⁡ α ⁡ sec ⁡ α d x d y + ∫ 0 l ∫ 0 b x ( v R ⁡ sin ⁡ α − v x ⁡ tan ⁡ α ) cos ⁡ γ ⁡ sec ⁡ α d x d y ]       = 4 m k [ Δ b R 2 v R l ( 1 − 2 ⁡ cos ⁡ α n + cos ⁡ θ ) + R v R b 1 ( θ − 2 α n ) + R U h m ln tan 2 ( α n 2 + π 4 ) tan ( θ 2 + π 4 ) ]        = 4 m k b 1 [ R v R ( θ − 2 α n ) + R U b 1 h m ln tan 2 ( α n 2 + π 4 ) tan ( θ 2 + π 4 ) ] . (46) where, h m is the average value of h x .

From Eqs 35, 37, we can see that: on the exit section x = l , h x ′ = b x ′ = 0 ; v y | x = l = v z | x = l = 0 is on the exit section. Therefore, we can have | v ¯ z | x = 0 = v ¯ y 2 + v ¯ z 2 | x = 0 (47)

Since there is no shear power consumed on the outlet section, the shear power N s consumed on the inlet section is equal to the total shear power. Therefore, we have N s = 4 ∫ 0 h 0 ∫ 0 b 0 v ¯ y 2 + v ¯ z 2 d y d z      = 4 k ∫ 0 h 0 ∫ 0 b 0 v 0 Δ b l 1 + l 2 ⁡ tan 2 ⁡ θ 4 Δ b 2 d y d z      = 2 k U l 4 Δ b 2 + l 2 ⁡ tan 2 ⁡ θ . (48)

The total power Φ can be calculated by Φ = N d + N s + N f .(49)

Therefore, by adding Eqs 41, 46, 48, it results in Φ = 4.616 σ s U ⋅ ln ⁡ ξ + 2 k U Δ h l + 4 m k b 1 [ R v R ( θ − 2 α n ) + R U b 1 h m ln tan 2 ( α n 2 + π 4 ) tan ( θ 2 + π 4 ) ] .(50)

From Eqs 40, 41, 46, 48, one can obtain that d U d α n = v R b 1 R ( sin ⁡ 2 α n − R + h 1 R sin ⁡ α n ) − Δ b l R 2 v R ( R + h 1 R cos ⁡ 2 α n + sin ⁡ 2 α n − cos 3 α n ) = N (51) d N d d α n = 4.616 σ s N ⋅ ln ⁡ ξ (52) d N s d α n = 2 k N l 4 Δ b 2 + l 2 ⁡ tan 2 ⁡ θ (53) d N f d α n = 4 m k [ N R h m ln tan 2 ( π 4 + a n 2 ) tan ( π 4 + θ 2 ) + 2 U R h m ⁡ cos ⁡ a n − 2 v R b 1 R ] . (54) Therefore, we can have d Φ d α n = d N d d α n + d N s d α n + d N f d α n . (55)

When the total power Φ achieves the minimum value, the theoretical expression of the friction factor m can be obtained by solving Eq. 55. Therefore, we have m = − N ( 4.616 σ s ⁡ ln ⁡ ξ + 2 k l 4 Δ b 2 + l 2 ⁡ tan 2 ⁡ θ ) 4 k [ N R h m ln tan 2 ( π 4 + a n 2 ) tan ( π 4 + θ 2 ) + 2 U R h m ⁡ cos ⁡ a n − 2 v R b 1 R ] .(56)

Substituting a neutral angle α n into Eq. 56, then the friction factor m can be obtained. Therefore, we can obtain the analytical solution of rolling torque M , rolling force F and stress state coefficient n σ : M = R 2 v R Φ min ; F = M χ 2 R Δ h ; n σ = F 4 b l k (57) where, the arm coefficient χ can refer to the literature (Harris, 2014).

From the above derivation, it can be seen that the present new method of using a linear yield criterion to analyze a new velocity field can solve the problem of integral difficulty due to the nonlinear Mises yield criterion and can provide the possibility of acquiring an analytical model of rolling force for multi-component alloys. It also can be seen that such a model has a clear functional relationship, which is convenient to be used on-line in the process parameter design and the thickness control of a rolled plate.

An on-site rolling experiment was carried out in a domestic factory. The roller diameter is 1120mm, and the billet size is 320 mm × 2050 mm × 3250 mm. The measured rolling velocities for the 2nd–6th passes are 1.64, 1.66, 1.68, 1.82, and 1.97 m/s respectively. The arm coefficient χ is respectively taken as 0.49, 0.50, 0.51, 0.49, and 0.50; the corresponding rolling temperatures are 944.56°C, 933.49°C, 922.97°C, 924.68°C, and 932.11°C, respectively. The outlet thickness and the rolling force for each pass can be accurately acquired by thickness measurement. The deformation resistance of the multicomponent alloys in the experiment adopted in the present paper is (Zhang, 2018) σ s = 3583 . 195 e − 2.233 × 10 − 3 T ⋅ ε 0.42437 ⋅ ε ˙ − 0.3486 × 10 − 3 T + 0.46339 ; T = t + 273. (58) where, ε is effectiveness strain, ε ˙ is effectiveness strain rate, t is deformation temperature, T is Kelvin temperature.

The rolling torque and rolling force for the above passes can be predicted by Eq. 57, which is shown in Table 1.

In Table 1, Δ 1 denotes the relative error between the calculated rolling force F A and the experimental one F M . It can be seen that the maximum error is 12.8%. Δ 2 denotes the relative error between the calculated rolling torque M A and the experimental one M M . As shown in Table 1, the maximum error is 12.5%. Therefore, both the two maximum errors are within the engineering allowable value of 15% (Wang et al., 2019). It should be noted here that it is better to further give the comparison between the present results based on the present yield criterion and those based on the other yield criteria, which can further reflect the superiority of the present yield criterion. However, due to the complexity of giving a set of formulas based on the other yield criteria synchronously, it is inconvenient to do this work at this time. In the future, we will work on this aspect systematically.

At the same time, the rolling force model obeys the bounce equation (Sun et al., 2013) H f = S 0 + E = S 0 + P K (59) where, H f is the final thickness of the rolled product, S 0 is the no-load roll gap which can be determined by the regression of experimental data of measured rolling force and measured exit thickness in Table 1, E is the bouncing amount of the rolling mill, P is the rolling force, and K is the stiffness coefficient of the rolling mill, K = 10700 kN/mm is for the present rolling mill.

Based on Eq. 59, the comparison between the predicted value of outlet thickness and the measured value for each pass can be shown in Table 2.

In Table 2, Δ H is the absolute error between the measured exit thickness 2 h 1 and the predicted exit thickness H f . The average error remains at 0.25mm, and the maximum error does not exceed 0.50mm, which meets the allowable deviation of GB/T 709-2019. Therefore, it can be concluded that the prediction accuracy is high. Also, it can be concluded that due to the high precision of the rolling force model, the precision of the outlet thickness is also high, which can provide scientific guidance for rolling process optimization.