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Three-dimensional thermo-mechanical finite element simulation of ribbed strip rolling with friction variation
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Finite Elements in Analysis and Design 40 (2004) 1139 – 1155 www.elsevier.com/locate/!nel
Three-dimensional thermo-mechanical !nite element simulation of ribbed strip rolling with friction variation Z.Y. Jianga;∗ , A.K. Tieua , C. Lua , W.H. Suna , X.M. Zhangb , X.H. Liub , G.D. Wangb , X.L. Zhaoc a
School of Mechanical, Materials and Mechatronic Engineering, University of Wollongong, Northelds Avenue, Wollongong NSW 2522, Australia b State Key Laboratory of Rolling Technology and Automation, Northeastern University, Shenyang, Liaoning 110004, People’s Republic of China c Hot Strip Mill, Benxi Iron and Steel Company, Benxi, Liaoning 114001, People’s Republic of China Received 15 September 2002; received in revised form 30 May 2003; accepted 19 July 2003
Abstract This paper presents a three-dimensional thermo-mechanical simulation of the rolling of ribbed strip combined with a friction variation model in the roll bite. The theoretical analysis is based on the utilization of the !nite element 9ow formulation to characterize the material 9ow, employing the friction variation model, to predict the metal 9ow, the distribution of temperature and stress and to estimate roll separating force and rib height. The numerical predictions were veri!ed by the experiments performed in a hot strip mill. When the friction variation is considered in the thermo-mechanical model, the computed values are in good agreement with the experimental values, and the numerical model is stable. ? 2003 Elsevier B.V. All rights reserved. Keywords: Friction variation; Thermo-mechanical rigid visco-plastic FEM; Ribbed strip; Hot rolling simulation
1. Introduction Ribbed strip rolling is typically an inhomogeneous deformation process. Shown in Fig. 1 (h is the rib height, and E1 the distance between two ribs) is a typical example of a ribbed strip, which that involves a signi!cant local residual deformation. Strip deformation involved in the rolling process can be characterized in three di=erent zones [1]. Metals in zone II are compressed with some of the volume forming the rib, and the other 9ows in longitudinal direction. Some of the metals compressed ∗
Corresponding author. Tel.: +61-2-42214545; fax: 61-2-42213101. E-mail address: [email protected] (Z.Y. Jiang).
0168-874X/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.!nel.2003.08.004
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Nomenclature B BI bx c D E1 g h h2 henv hroll hwater K1 K2 k k k1 , k 2 l M m m1 N N Ne P q˙ qn qf S T T1 T2 Tenv Troll Twater V vx ; v y ; v z VR Vg Zi I
appropriate spatial derivatives of the shape functions element average matrix width of strip in the deformation zone speci!c heat function of strain and temperature distance between two ribs compressible factor rib height thickness of ribbed strip heat convection coeKcient thermal resistance due to scale and roughness heat convection coeKcient in water constant for forward slip zone constant for backward slip zone shear yield stress thermal conductivity coeKcient very small positive constant projected length of the deformation zone number of elements strain rate sensitivity index friction factor shape function number of nodes number of Gauss points of an element roll separating force heat generation rate heat 9ux surface heat generation rate due to friction a closed surface conversion tension temperature ambient temperature roll temperature cooling water temperature volume velocity component in x, y and z directions tangential velocity of roll relative velocity between the surfaces of the strip and rolls z coordinate of node on rib top at exit equivalent stress
Z.Y. Jiang et al. / Finite Elements in Analysis and Design 40 (2004) 1139 – 1155
! s z I˙ I ˙V f ! !1
1141
Stefan–Boltzmann constant 9ow stress yield stress stress along the thickness direction equivalent strain rate emissivity equivalent strain volumetric strain rate frictional shear stress density angular position in roll bite parameter parameter inclined angle of the rib
A
A E1 h III
II I
A-A
Fig. 1. Cross-section and plan view of the ribbed strip.
in zone III 9ow to zones II and I to increase the rib height, and the others 9ow longitudinally. The spread is small due to the restriction of the inclined surfaces of the roll grooves. Therefore, the metal in zone III has a local large deformation, that in zone I has little deformation, and zone II is an intensive deformation transition zone. Due to its unique features, this type of strips has been used widely in various engineering !elds, such as in electrical equipment, the automobile industry, o=shore platform construction, ship structures and thermal transfer industry. One of the important application of this strip, for example, is steel pipe piles in civil construction, where they are made into internal spiral pipes with the ribs providing a much stronger bond between the pipe and the concrete than that of the smooth-surfaced ones [1,2]. Thus it leads to a signi!cant savings of both the steel pipes and concrete. The application of this type of strips is therefore expected to increase signi!cantly in civil construction [2], for example, in o=shore platform structures, tall buildings and so on. Moreover, it has been veri!ed that the higher the rib height, the stronger the bond [1].
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There was a report about the cold rolling of ribbed strip [3], but the rib height is very small (strip thickness is 1:0 mm, rib height 0.04 –0:05 mm). The rigid plastic/visco-plastic !nite element method (FEM) has been used in the past in the analysis of strip rolling [4–9], shape rolling [10,11] and slab/edge rolling [12–14]. A model for hot rolling of ribbed strip has also been developed by Jiang et al. [15,16]. However, no other investigation, as far as authors aware, has yet dealt with the full three-dimensional thermo-mechanical analysis of the rolling of ribbed strip with friction variation consideration. In hot rolled strip, there are two di=erent surfaces of the strip that a=ect the accuracy of simulation. The traditional approach is to assume that the frictional force in the roll bite is proportional to the normal force, with the friction coeKcient remaining constant in the roll bite. However, this will incur a loss of accuracy in the roll gap model, and a=ect the thickness, shape and 9atness of the rolled materials. In order to understand the friction mechanism in metal rolling processes, many researchers have been made in both experiments and theoretical modelling [17–20]. Lenard [21] and Liu et al. [22] have shown that the friction varies in the roll bite, and it a=ects the rolling force and the accuracy of the on-line models. Lenard et al. [23] has also investigated the coeKcient of friction variations with reduction. Therefore, the FEM modelling should consider the friction variation in the roll bite as it has a signi!cant e=ect on the accuracy and stability of the model and can improve the practical control accuracy of ribbed strip rolling. So an analysis of the thermo-mechanical process of ribbed strip rolling with friction variation models is employed in this study. This paper presents a full three-dimensional !nite element simulation of the ribbed strip rolling. The approach considers the thermal phenomenon simultaneously with the solution of the velocity !eld and a friction variation model at the strip-roll interface. Predictions of roll separating force and rib height agree well with experimental data. Results of this numerical simulation show that the simulation of the ribbed strip with friction variation is e=ective and stable. 2. FEM formulation The analysis of the ribbed strip rolling makes use of the !nite element method based on the 9ow formulation model for slightly compressible materials. The main advantage of this approach over the conventional 9ow formulation following the Levy–Mises rigid-plastic constitutive equations is the possibility of obtaining the stress–strain rate relationship directly by [24] ij = II˙ 23 ˙ij + $ij g1 − 29 ˙v ; (1) where $ij is the Kronecker delta, and g a small positive constant that may vary from 0.01 to 0.0001, which is a factor conferring a small degree of compressibility. The e=ective stress, , I and the ˙ e=ective strain rate, , I are de!ned, respectively, by (2) I = 32 ij ij + gm2 ; where ij is the deviatoric stress tensor, m the hydrostatic stress. The equivalent strain rate is written as I˙ = 23 ˙ij ˙ij + g1 ˙2v ; (3)
Z.Y. Jiang et al. / Finite Elements in Analysis and Design 40 (2004) 1139 – 1155
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Frictional shear stress (MPa)
K1 m1 k kVa=0.001 1 = 0.001 kVa=0.1 1 = 0.1 kVa=10.0 1 = 10.0
P'P section (k1 = 0.001) 0
o' o''
-K 2 m 1 k Entry
Neutral point
Exit
Roll bite
Fig. 2. Distribution of the frictional shear stress model.
where ˙ij = ˙ij − $ij ˙v =3 is the deviatoric strain rate tensor and ˙v = ˙x + ˙y + ˙z is the volumetric strain rate. The slightly compressible method in a full 3-D rigid visco-plastic FEM has been employed to solve the rolling of ribbed strip problem. According to the variational principle [25], the real velocity !eld must minimize the following functional: 1 ˙ &= I I dv + f QVf ds ± T1 v ds = (p + (f + (t ; (4) m+1 v sf sv where the !rst term on the right hand side is the work rate of plastic deformation ((p ), I the equivalent stress, I˙ the equivalent strain rate and m is the strain rate sensitivity index. The second term on the right hand side is the work rate of friction ((f ); QVf is the relative slip velocity on the interface of the rolled material and the rolls where the frictional shear stress f is applied. The friction variation model shows the friction varies along the contact length of the deformation zone. In this study, the frictional shear stress model [1] is modi!ed as follows: Vg m1 s f = ± Ki √ ; (5) 3 V 2 + k2 g
1
where m1 is friction factor; s is yield stress; Ki is a coeKcient describing the changes of frictional shear stress in the deformation zone with K1 used in the forward zone and K2 in the backward zone; k1 is a small positive constant. The relative slip velocity, Vg , between the rolled material and the roll is given by (6) Vg = (vx sec − VR )2 + vy2 + k22 ; where vx and vy are the velocity components in the x and y directions, respectively, is the angular position of the node, VR is the tangential velocity of the roll, k2 is a small positive constant and the distribution of these frictional shear stress models is shown in Fig. 2. If k1 = 0:0 and K1 = K2 = 1:0, Eq. (5) results in a constant friction model. The neutral point in the roll bite varies for ribbed strip
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rolling. QVf is also the relative slip velocity along the the same as Vg . The third term on the right hand side of Eq. (4) is the and v is the velocity of the cross section with tension. “+” the back tension. Assuming the material is rigid–viscoplastic and obeys
roll–slab contact interface and its value is work rate ((t ) of tension. T1 is the tension Here “−” indicates the front tension, and the following constitutive relation:
s = DI˙ m ;
(7)
where D is a function of strain and temperature and m is the strain rate sensitivity index. The procedure of the standard 3-D rigid visco-plastic !nite element discretization with slightly compressible materials is as follows: a control volume, V , is discretised into M eight-node hexahedron elements linked through N nodes. In each element, the velocity distribution u can be represented in vectorial form by u = N T C;
(8)
where C is the nodal velocity vector and N is the shape function matrix of the element. The strain rate tensor is then obtained from ”˙ = BC;
(9)
where matrix B contains appropriate spatial derivatives of the shape functions. Expressing I˙ in terms of the nodal velocities, it can be written in matrix form as I˙ = ”˙T Z ”˙ + g1 (C T ”) ˙ 2 = CT B T ZBC + g1 CT B T C T CBC; (10) where [C] = [1 1 1 0 0 0]T and the correspondent matrix Z is as follows: 4 − 29 − 29 0 0 0 9 2 4 −9 − 29 0 0 0 9 2 4 0 0 0 − 9 − 29 9 : [Z] = 1 0 0 0 0 0 3 0 0 0 0 13 0 0
0
0
0
0
(11)
1 3
In the present work, the slight compressibility is enforced in terms of elements. Consequently, an element average matrix BI is used to express the volumetric strain. BI is evaluated by Ne 1 I B= Bi ; Ne i=1
(12)
where Ne is the number of Gauss points of an element and Bi is the B matrix at Gauss point i. Eq. (11) is rewritten as I (13) I˙ = CT B T ZBC + g1 CT BI T C T C BC:
Z.Y. Jiang et al. / Finite Elements in Analysis and Design 40 (2004) 1139 – 1155
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Thus, the functional Equation (4) can be approximated by the sum of the functional at element level, which results in e e 1 i I dV ( = s CT B T ZBC + g1 CT BI T C T C BC &≈ m + 1 i V i=1 i=1 cos m1 s T T √ 0 + C N N C − 2vRT C N T + vR2 + k22 − k2 dS 3 Sfi sin ± T1 vi dS :
Sti
(14)
The solution for the nodal velocity vector is obtained as it makes the above functional minimum [25]. This requires 9& = 0: 9C The resulting governing equation is nonlinear and can be written [16] as e 1 s m 1 s Z + CC T dV + N T N dS C i I g i 2 2 V S 3(v + k ) f i=1 g
+
e i=1
−
Sfi
(15)
1
cos m 1 s 0 vR N T T1 dS dS ± = 0: Sti 3(vg2 + k12 ) sin
(16)
It can be solved by the modi!ed Newton–Raphson iteration: vj = vj−1 + *Qvj " # $ % 92 & 9& − Qvj = ; T 9v9v j−1 9v j−1
(17)
where * is the deceleration coeKcient, and −92 &=9v9Tv [16] is # e " 92 & s 1 9s s T − = − Z+ 2 Z CC Z dV 9C9CT 9I˙ I˙ I˙ I˙ Ve i=1
+
e i=1
Sfi
m1 s 3(vg2 + k12 )
Ni N j T F Fj dS; vg2 + kl2 i
(18)
where Fi = {−vy sec i ; vx sec i − VR }:
(19)
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The roll separating force can be calculated by Eq. (20): l bx z dy d x; P= 0
(20)
0
where l is the projected length of the deformation zone; bx is the strip width in the deformation zone; z is the stress along the thickness direction. The rib height can be determined from Eq. (21): h = Zi − 0:5h2 ;
(21)
where Zi is the z coordinate of node on rib top at exit and h2 the thickness of the ribbed strip. 3. Thermal analysis The equation governing the heat-conduction in a continuous medium with an arbitrary volume, V , bounded by a closed surface, S, can be expressed as [26] q˙ 1 dT2 ∇ 2 T2 + = ; (22) k 1 dt where 1 = k =( · c), q˙ is the heat generation rate; , c are density and speci!c heat respectively; k is a thermal conductivity coeKcient, and T2 is the temperature. q˙ = ij ˙ij ;
= 0:85 − 0:95;
(23)
where is a parameter which gives the fraction of the plastic deformation energy which is dissipated as heat. Applying the classical Galerkin method, the heat transfer Equation (22) can be written in matrix form at elemental level as T T ˙ ˙ kMM T2 dV + cNN T 2 dV − I N I dV − qn N dSq = 0; (24) Vm
Ve
Sqe
Ve
where M = ∇N , qn is the heat 9ux along the boundary surface Sq , and T2 the nodal temperature vector. The later is interpolated using identical shape functions as those utilized for the velocity !eld, since the numerical model of the deforming workpiece uses the same !nite element mesh for thermal as for the mechanical analysis. Eq. (24) can be expressed in abbreviated form as KT 2 + C1 T˙ 2 = Q;
(25)
where K is the heat conduction matrix, C1 the heat capacity matrix and Q the heat 9ux vector. The heat 9ux vector Q in the above equation has several components and can be expressed as follows: 4 4 Q= (I )N I˙ dV + ! (Tenv − T2 )N dS + henv (Tenv − T2 )N dS Ve
+
SRm
e SRad
hroll (Troll − T2 )N dS +
e SConv
SFer
qf N dS;
(26)
where Tenv and Troll are the ambient and roll temperatures, respectively. The !rst term on the right side is the contribution of the net heat generated by plastic deformation inside the ribbed strip, while other terms are due to the heat exchanged between the strip and the ambient environment or rolls.
Z.Y. Jiang et al. / Finite Elements in Analysis and Design 40 (2004) 1139 – 1155
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Therefore, the second term on the right side is the result of the heat radiation to the environment, where ! is the Stefan–Boltzmann constant and is the emissivity. The third term on the right side is the contribution of the heat convection between the surface of the strip and the environment, where henv is the heat convection coeKcient. The fourth term on the right side describes the heat transferred between the ribbed strip and the roll through their contact interface. The last term on the right side is the contribution of the heat generated by friction along the roll–ribbed strip interface, qf being the surface heat generation rate due to friction: qf = f |Vg |;
(27)
where f is the friction shear stress at the roll–ribbed strip, and |Vg | the relative velocity between the two regions. The major boundary conditions to be applied in the !nite element thermal analysis of the process are the following: (1) When the rolled ribbed strip is placed between stands, the surface loses heat by radiation and convection: 4 Q1 = (Tenv − T24 )N dS + henv (Tenv − T2 )N dS: (28) e SRad
e SConv
(2) When the rolled ribbed strip is water-cooled between stands, the working temperature is much higher than the boiling point of the water and therefore !lm boiling takes place: Q2 = hwater (Twater − T2 )N dS: (29) e SConv
(3) When the ribbed strip is being rolled, the surface in contact with the rolls is cooled, while conversely the rolls are heated: Q3 = hroll (Troll − T2 )N dS; (30) e SConv
where hroll is the thermal resistance due to scale and roughness which is responsible for promoting a temperature discontinuity across the roll–ribbed strip interface. Furthermore, it is also assumed that the initial temperature of the ribbed strip at the exit of the furnace is uniform throughout its volume. The integration of Eq. (24) requires the utilization of the following time-stepping scheme: T2(t+Qt) = T2t + Qt[(1 − !)T˙ 2t + !T˙ 2(t+Qt) ];
(31)
where t and Qt are time and its increment, and ! a parameter varying between 0 and 1. Details on implementation of the computer algorithm to include the coupling of the solution of stress equilibrium equations with heat balance in the !nite element model is given in Ref. [1]. 4. Simulation conditions Process data, the mechanical and thermal properties of the material utilised in the theoretical analysis of the ribbed strip rolling is shown in Table 1. Initial strip thickness is 7.2–8:0 mm. The
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Table 1 Process data, the mechanical and thermal properties of the material Material Chemical composition Density Speci!c heat Stefan–Boltzmann constant Surface emissivity Thermal conductivity Ambient temperature Equipment Diameter of upper roll Diameter of down roll Rolling speed Flow stress model [27]
Carbon steel (235 MPa) 0.16 C; 0.07 Si; 0.49 Mn 7800 kg=m3 670 J=(kg K) 5:675 W=(m2 K 2 ) 0.8 32 W=(m K) 20◦ C 740:85 mm 749:55 mm 3.9 –5:4 m=s & ' 0:244 0:13 2 = 4098 exp −3:5T I I˙ 1000
Table 2 Velocity boundary conditions for the rolling of ribbed strip No. 1 2 3 4 5 6
Location ABCDEFGHIJKL A B K L AA B B RSR S , etc. MM X X Inclined contact surface, OO P P, etc.
Velocity vx = v1 (unknown), vy = 0, vz = 0 vx = v0 (unknown), vy = 0, vz = 0 vy = 0 vz = −vx tan vz = vx tan vz = −vx tan , vy = −vx tan =tan !1 + VR sin =tan !1
slab was heated to 1200◦ C in furnace, and rolled in the roughing mills and !nishing mills. Based on the production conditions, the rolling of ribbed strip was carried out on the 6th !nishing stand, and the rolling temperature is 850 –900◦ C at this stand. The initial temperature !eld of the ribbed strip before entering the 6th stand is determined by the thermal computation from the furnace to the !nishing mills. The boundary conditions are as shown in Table 2. It should be noted that if the ribs do not !ll up the grooves of the roll, the surfaces of the PP Q Q and TT U U are free and do not satisfy the relationship vz = −vx · tg . !1 is the inclined angle of the rib. Considering the characteristic of the rolling of ribbed strip, the spread is small due to the restriction of the inclined surfaces of the roll grooves. Therefore, the strip deformation on middle part of the width of strip is similar. The analysis is carried out for two types of ribs: in one model the edge plane KLL K is a free surface (see Fig. 3) which considers the deformation of the side rib with free spread and in the other model the plane KLL K is not a free surface and is constrained by the surrounding ribs. Discretization of the control volume was conducted by 432 eight-nodded isoparametric linear elements and 665 nodes, as shown in Fig. 3. In this !gure, ABB A represents the symmetrical surface, KLL K the free surface, MNWX the exit section and M N W X the entry section. The rolls were assumed as rigid, and friction variation conditions were applied to the contact interface between the ribbed strip and the rolls. The parameters for the model of friction variation were determined
Z.Y. Jiang et al. / Finite Elements in Analysis and Design 40 (2004) 1139 – 1155 z(v z(vzz))
1149
Entry
B'
K'
N'
O'
P'
Q' R'
S' T' U' V'
W'
y(v y(vy)y)
L'
A' Q
P M'
N
O B
D C
E R F
X' T U H S I V G
W K
J
M
X A
x(vx)x) x(v
Exit
L
Fig. 3. FE mesh for simulating calculation of ribbed strip.
based on the rolling conditions and the comparison of the calculated results with measured values. For this study, m1 = 0:35 and K1 = 1:0, K2 = 1:0 and k1 = 0:1. In the coupled simulation, the initial temperature !eld and velocity !eld were given. For a given time increment Qt, the temperature !eld was calculated !rst, and the 9ow stress (see Table 1, 9ow stress model) was obtained based on the present calculated temperature !eld, and the velocity !eld was then calculated according to Eq. (17). This iterative calculation continues and stops when the increments of the velocity !eld and temperature !eld respectively reach a very small value [1]. In the uncoupled thermal (isothermal) analysis, the initial temperature is set a uniform !eld and a certain value. However, the calculated velocity !eld is based on the calculated true temperature !eld at present iteration step for coupled thermal analysis. 5. Results The theoretical predictions obtained with the thermo-mechanical !nite element model presented in the previous sections are discussed and veri!ed against experimental data obtained from a steel company; comparison is made for the average surface temperature of the ribbed strip, the roll separating force and rib height. The temperature at the middle of the rib top surface was measured via an infrared thermal instrument made by Land Co. of England after the ribbed strip exits from the !nishing mill. The measured accuracy is less than 1◦ C. 5.1. Distribution of temperature Fig. 4 is the distribution of temperature of the cross section at the exit for reductions 18.5 and 25.7% as the friction variation model was used. It can be seen that the temperature at the centre of
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Fig. 4. Theoretical distribution of temperature for the cross section at exit. Reduction: (a) 18:5% and (b) 25:7%.
Table 3 Comparison of the calculated and measured mean surface temperatures Reduction (%) Calculated (◦ C) Measured (◦ C)
18.50 859 860
20.65 879.5 880
23.46 873.5 875
25.70 892 890
the strip in the thickness direction is the highest. However, the temperature at the top surface of ribs is much lower, and the lowest is at the side of the rib. The temperature at the top of CDEF rib (see Fig. 3) is lower than that at the top of GHIJ rib (see Fig. 3). The temperature di=erence between the two ribs shows a trend expected since for reduction 25.7%, rib CDEF (h = 2:14 mm) is higher than GHIJ (h = 1:89 mm), o=ering more surface area for heat loss to the roll. The calculated mean surface temperatures of the ribbed strip at the exit of the roll bite for di=erent reductions are in good agreement with the measured surface temperature of the ribbed strip obtained under the industrial conditions, as shown in Table 3. 5.2. Distribution of stress The distribution of equivalent stress (unit in MPa) at various cross sections is illustrated in Fig. 5 (x is a distance of the cross section to the entry section) for reduction 25.7%. The equivalent stress increases from the entry section to the 0:5l cross section. However, it decreases from the 0:75l cross section to exit section. The change of the isopleth of the equivalent stress is much larger at the sides of ribs than that at other parts. As mentioned before, the stress distribution across the width of the ribbed strip is non-uniform and leads to di=erence in material 9ow behaviour. It can clearly be seen that the highest compressive state of stress and the highest gradient of stresses occur near to the rib side as a result of the signi!cant local residual deformation. The shape of rib shows that the rib top does not contact the roll (if the rib top touches the roll, the shape of rib must be same as the groove of roll). The shape of the equivalent stress at the side (the right hand part in Fig. 5)
Z.Y. Jiang et al. / Finite Elements in Analysis and Design 40 (2004) 1139 – 1155
1151
Fig. 5. Theoretical distribution of equivalent stress.
is di=erent from that between the ribs, because of the e=ect of the lower temperature at the side of ribbed strip. 5.3. Roll separating force To validate the simulation, the roll separating force was also calculated and compared with industrial rolling conditions. In the seven 4-high hot tandem mills, the ribbed strip 6:0 mm thick and 1050 mm wide with 30 ribs was rolled. The roll separating forces were measured by load cells. The distance between two ribs (E 1 ) is 34 mm and rib shape used in the simulation is the same as the rolled strip in plant. The geometry of the rolled strip is the same as that of the numerical model. The predictions were obtained from !nite element models with and without the thermal e=ect. In industrial practice, the friction factor is determined by matching the measured rolling load (from load cells) and the rolling force calculated from the slab method. In this calculation, the friction variation condition was applied (Eq. (5)), m1 = 0:35, K1 = 1:0, K2 = 1:0 and k1 = 0:1, the friction factor is close to the practical rolling of hot strip 0.3– 0.4 [28].
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Roll separating forces (kN)
20000
Non-coupled 16000
Coupled Measured
12000 8000 4000 0
5
10
15 20 25 Reduction (%)
30
35
Fig. 6. Comparison of the calculated roll separating forces and experimental values.
2.5 Measured
Rib height (mm)
2
Calculated 1.5 1 0.5 0 0
5
10
15 20 Reduction (%)
25
30
35
Fig. 7. Comparison of the calculated rib height and measured values.
Fig. 6 shows the comparison of the theoretical predictions of the roll separating forces and the experimental values. It can be seen that the roll separating forces obtained from the coupled thermo-mechanical !nite element model are lower than that obtained from the non-coupled model (isothermal analysis a uniform temperature is set at a given value), and for coupled thermo-mechanical analysis, the roll separating force is in good agreement with the experimental values in the rolling of ribbed strip. It is also seen that the relationship between the rolling force and reduction is non-linear because of the experimental errors. The relationship between non-coupled calculated rolling force and reduction is non-linear, but the coupled results are closer to being linear, which indicates that the coupled calculation considering the thermal e=ect can improve the simulation results. 5.4. Rib height A comparison of the theoretical prediction of the rib height and the plant measured values is carried out as shown in Fig. 7. The calculated and measured rib height is an average of middle rib (see Fig. 3, such as rib CDEF). Rib heights of the middle ribs (28 ribs out of a total of 30 ribs)
Z.Y. Jiang et al. / Finite Elements in Analysis and Design 40 (2004) 1139 – 1155
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2.2
Rib height (mm)
2 1.8
KK1=K2=1.0 1 = K2 =1.0 KK1=1.2, K2=0.8 1 = 1.2, K 2 =0.8 KK1=1.2, K2=1.0 1 = 1.2, K 2 =1.0 KK1=1.0, K2=0.8 1 = 1.0, K 2 =0.8 Measured Measured
1.6 1.4 1.2 1 0.2
0.3
0.4 Friction factor
0.5
0.6
Fig. 8. E=ect of friction variation on rib height.
were measured after rolling [1], and the measurements were repeated three times for each rib. The relative change for the measured rib height of these ribs is about 3%. Therefore, results of the central rib were chosen to compare with the computation values. Due to the limitation of the load capacity of the 6th !nishing mill and production/customer requirements, the maximum reduction is 25.7% in practice, and therefore, the range of conditions and measured results are limited. It can be seen that in this simulation conditions the calculated value of the rib height is close to the experimental value. The prediction of the rib height can be extended to other plates and ribs. Fig. 8 indicates the e=ect of friction variation on rib height when reduction is 23.46% and m1 = 0:35. It can be seen that the calculated rib height is close to the measured value for friction factor 0.3– 0.4, and the rib height increases with friction factor. It can also be seen that the rib height for K1 = 1:0, K2 = 1:0 is close to that when K1 = 1:2, K2 = 0:8, but the rib height increases faster when K1 = 1:2, K2 = 1:0. The rib height with friction variation model K1 = 1:2, K2 = 0:8 is greater than that with K1 = 1:0, K2 = 0:8, which means that the rib height increases with larger friction in the forward slip zone.
6. Conclusions In the present investigation, three-dimensional rolling of ribbed strip is modelled by using a coupled thermo-mechanical three-dimensional !nite element model with friction variation consideration. The major innovation of this work is the simulation of the local residual deformation of the ribbed strip rolling with the combined thermal and deformation analysis. The approach provides the possibility of estimating the temperature and the distribution of the stress and deformation throughout the deformed process of ribbed strip. The rib height increases when the friction factor and friction in the forward slip zone increase. So the rib height can be increased with a higher friction condition. The calculated evolution of the stress !eld also veri!es the local non-uniform characteristic of the material 9ow resulting from the rib formation. The predicted roll separating force with coupled thermo-mechanical model and the rib height are in agreement with the experimental values.
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Acknowledgements This work is supported by the Australian Research Council (ARC) IREX Award No. X00106473 and University of Wollongong Start Up Grant. References [1] Z.Y. Jiang, Forming Processes, Theory of the strip with longitudinal ribs and its application, Ph.D. Thesis, Department of Metal Forming, Northeastern University, 1995. [2] S. Masude, T. Hirazawa, I. Taniguchi, Development of hot-rolled strip with longitudinal ribs and application to the spiral pipe, Nippon Kokan Technical Report 107 (1985) 31–37. [3] F. Schroller, O. Pawelski, W. Rasp, Analysis of an improved width-9ow concept for the production of ribbed strips by cold rolling, Steel Res. 70 (3) (1999) 110–114. [4] S. Kobayashi, S.I. Oh, T. Altan, Metal Forming and the Finite-Element Method, Oxford University Press, New York, 1989. [5] X.H. Liu, Rigid-Plastic FEM and its Application in Steel Rolling, Metallurgical Industry Press, Beijing, 1994. [6] K. Osakda, J. Nakano, K. Mori, Finite element method for rigid-plastic analysis of metal forming—formulation for !nite deformation, Int. J. Mech. Sci. 24 (8) (1982) 459–468. [7] K. Mori, K. Osakada, T. Oda, Simulation of plane-strain rolling by the rigid-plastic !nite element method, Int. J. Mech. Sci. 24 (9) (1982) 519–527. [8] Z.Y. Jiang, A.K. Tieu, A simulation of three-dimensional metal forming processes by rigid plastic !nite element method, J. Mater. Process. Technol. 112 (1) (2001) 144–155. [9] K. Mori, C.C. Wang, K. Osakada, Inclusion of elastic deformation in rigid-plastic !nite element analysis, Int. J. Mech. Sci. 38 (6) (1996) 621–631. [10] J. Yanagimoto, M. Kiuchi, General purpose FEM simulator for the three-dimensional deformation analysis of strip, bar/wire and shape rolling processes, in: Proceedings of the 7th International Conference on Steel Rolling, 1998, pp. 278–283. [11] J.Z. Xu, Y.C. Miao, X.H. Liu, G.D. Wang, G.R. Bai, Rigid-plastic FEM analysis of metal forming for rolling shape slab, Chinese J. Mech. Eng. 36 (9) (2000) 11–15. [12] T. Nikaido, K. Naoi, T. Shibata, K. Kondo, K. Osakada, K. Mori, Numerical simulation of width spread of “Dog-Bone” slab in non-steady horizontal rolling, J. Japanese Soc. Technol. Plasticity 25 (277) (1984) 129–135. [13] S.W. Xiong, X.H. Liu, G.D. Wang, Q. Zhang, A three-dimensional !nite element simulation of the vertical-horizontal rolling process in the width reduction of slab, J. Mater. Process. Technol. 101 (1–3) (2000) 146–151. [14] S.H. Xiong, X.H. Liu, G.D. Wang, Q. Zhang, Simulation of vertical–horizontal rolling process during width reduction by full three-dimensional rigid-plastic !nite element method, J. Mater. Eng. Perform. 6 (6) (1997) 757–765. [15] Z.Y. Jiang, S.H. Xiong, X.H. Liu, G.D. Wang, Q. Zhang, 3-D rigid-plastic FEM analysis of the rolling of a strip with local residual deformation, J. Mater. Process. Technol. 79 (1–3) (1998) 109–112. [16] Z.Y. Jiang, X.L. Liu, X.H. Liu, G.D. Wang, Analysis of ribbed-strip rolling by rigid-viscoplastic FEM, Int. J. Mech. Sci. 42 (2000) 693–703. [17] G.T. Van Rooyen, W.A. Backofen, Friction in cold rolling, J. Iron Steel Inst. 186 (1957) 235–244. [18] L-S Lim, J.G. Lenard, Study of friction in cold strip rolling, J. Eng. Mater. Technol. 106 (1984) 139–146. [19] D-F Chang, N. Marsault, M.R.D. Wilson, Lubrication of strip rolling in the low-speed mixed regime, Tribol. Trans. 39 (1996) 407–415. [20] Z.L. Qiu, W.Y.D. Yuen, A.K. Tieu, Mixed-!lm lubrication theory and tension e=ects on metal rolling processes, Trans. ASME J. Tribol. 121 (1999) 908–915. [21] J.G. Lenard, Friction and forward slip in cold strip rolling, Tribol. Trans. 35 (3) (1992) 423–428. [22] Y.J. Liu, A.K. Tieu, D.D. Wang, W.Y.D. Yuen, Friction measurement in cold rolling, J. Mater. Process. Technol. 111 (1–3) (2001) 142–145. [23] J.G. Lenard, S. Zhang, A study of friction during the lubricated cold rolling of an aluminum alloy, J. Mater. Process. Technol. 72 (2) (1997) 293–301.
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[24] K. Mori, K. Osakada, Simulation of three-dimensional rolling by the rigid-plastic !nite element method, in: Proceedings of International Conference on Numerical Methods in Industrial Forming Processes, Pineridge Press, Swansea, 1982, pp. 747–756. [25] Z. Zhao G.D. Wang, Modern Plastic Forming Mechanics, Northeastern University Press, Shenyang, 1987. [26] J.L. Chenot, P. Montmitonnet, P. Buessler, F. Fau, Finite element computation of spread in hot 9at and shape rolling with a steady state approach, Eng. Comput. 8 (3) (1991) 245–255. [27] S.W. Xiang, G.D. Wang, Q. Zhang, X.F. Zhang, G.L. Wu, A new model for deformation resistance, J. Iron Steel 28 (5) (1993) 21–26. [28] J.G. Lenard, M. Pietrzyk, L. Cser, Mathematical and physical simulation of properties of hot rolled products, Elsevier Science Ltd, Kidlington, 1999.
Finite Elements in Analysis and Design 40 (2004) 1139 – 1155 www.elsevier.com/locate/!nel
Three-dimensional thermo-mechanical !nite element simulation of ribbed strip rolling with friction variation Z.Y. Jianga;∗ , A.K. Tieua , C. Lua , W.H. Suna , X.M. Zhangb , X.H. Liub , G.D. Wangb , X.L. Zhaoc a
School of Mechanical, Materials and Mechatronic Engineering, University of Wollongong, Northelds Avenue, Wollongong NSW 2522, Australia b State Key Laboratory of Rolling Technology and Automation, Northeastern University, Shenyang, Liaoning 110004, People’s Republic of China c Hot Strip Mill, Benxi Iron and Steel Company, Benxi, Liaoning 114001, People’s Republic of China Received 15 September 2002; received in revised form 30 May 2003; accepted 19 July 2003
Abstract This paper presents a three-dimensional thermo-mechanical simulation of the rolling of ribbed strip combined with a friction variation model in the roll bite. The theoretical analysis is based on the utilization of the !nite element 9ow formulation to characterize the material 9ow, employing the friction variation model, to predict the metal 9ow, the distribution of temperature and stress and to estimate roll separating force and rib height. The numerical predictions were veri!ed by the experiments performed in a hot strip mill. When the friction variation is considered in the thermo-mechanical model, the computed values are in good agreement with the experimental values, and the numerical model is stable. ? 2003 Elsevier B.V. All rights reserved. Keywords: Friction variation; Thermo-mechanical rigid visco-plastic FEM; Ribbed strip; Hot rolling simulation
1. Introduction Ribbed strip rolling is typically an inhomogeneous deformation process. Shown in Fig. 1 (h is the rib height, and E1 the distance between two ribs) is a typical example of a ribbed strip, which that involves a signi!cant local residual deformation. Strip deformation involved in the rolling process can be characterized in three di=erent zones [1]. Metals in zone II are compressed with some of the volume forming the rib, and the other 9ows in longitudinal direction. Some of the metals compressed ∗
Corresponding author. Tel.: +61-2-42214545; fax: 61-2-42213101. E-mail address: [email protected] (Z.Y. Jiang).
0168-874X/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.!nel.2003.08.004
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Nomenclature B BI bx c D E1 g h h2 henv hroll hwater K1 K2 k k k1 , k 2 l M m m1 N N Ne P q˙ qn qf S T T1 T2 Tenv Troll Twater V vx ; v y ; v z VR Vg Zi I
appropriate spatial derivatives of the shape functions element average matrix width of strip in the deformation zone speci!c heat function of strain and temperature distance between two ribs compressible factor rib height thickness of ribbed strip heat convection coeKcient thermal resistance due to scale and roughness heat convection coeKcient in water constant for forward slip zone constant for backward slip zone shear yield stress thermal conductivity coeKcient very small positive constant projected length of the deformation zone number of elements strain rate sensitivity index friction factor shape function number of nodes number of Gauss points of an element roll separating force heat generation rate heat 9ux surface heat generation rate due to friction a closed surface conversion tension temperature ambient temperature roll temperature cooling water temperature volume velocity component in x, y and z directions tangential velocity of roll relative velocity between the surfaces of the strip and rolls z coordinate of node on rib top at exit equivalent stress
Z.Y. Jiang et al. / Finite Elements in Analysis and Design 40 (2004) 1139 – 1155
! s z I˙ I ˙V f ! !1
1141
Stefan–Boltzmann constant 9ow stress yield stress stress along the thickness direction equivalent strain rate emissivity equivalent strain volumetric strain rate frictional shear stress density angular position in roll bite parameter parameter inclined angle of the rib
A
A E1 h III
II I
A-A
Fig. 1. Cross-section and plan view of the ribbed strip.
in zone III 9ow to zones II and I to increase the rib height, and the others 9ow longitudinally. The spread is small due to the restriction of the inclined surfaces of the roll grooves. Therefore, the metal in zone III has a local large deformation, that in zone I has little deformation, and zone II is an intensive deformation transition zone. Due to its unique features, this type of strips has been used widely in various engineering !elds, such as in electrical equipment, the automobile industry, o=shore platform construction, ship structures and thermal transfer industry. One of the important application of this strip, for example, is steel pipe piles in civil construction, where they are made into internal spiral pipes with the ribs providing a much stronger bond between the pipe and the concrete than that of the smooth-surfaced ones [1,2]. Thus it leads to a signi!cant savings of both the steel pipes and concrete. The application of this type of strips is therefore expected to increase signi!cantly in civil construction [2], for example, in o=shore platform structures, tall buildings and so on. Moreover, it has been veri!ed that the higher the rib height, the stronger the bond [1].
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There was a report about the cold rolling of ribbed strip [3], but the rib height is very small (strip thickness is 1:0 mm, rib height 0.04 –0:05 mm). The rigid plastic/visco-plastic !nite element method (FEM) has been used in the past in the analysis of strip rolling [4–9], shape rolling [10,11] and slab/edge rolling [12–14]. A model for hot rolling of ribbed strip has also been developed by Jiang et al. [15,16]. However, no other investigation, as far as authors aware, has yet dealt with the full three-dimensional thermo-mechanical analysis of the rolling of ribbed strip with friction variation consideration. In hot rolled strip, there are two di=erent surfaces of the strip that a=ect the accuracy of simulation. The traditional approach is to assume that the frictional force in the roll bite is proportional to the normal force, with the friction coeKcient remaining constant in the roll bite. However, this will incur a loss of accuracy in the roll gap model, and a=ect the thickness, shape and 9atness of the rolled materials. In order to understand the friction mechanism in metal rolling processes, many researchers have been made in both experiments and theoretical modelling [17–20]. Lenard [21] and Liu et al. [22] have shown that the friction varies in the roll bite, and it a=ects the rolling force and the accuracy of the on-line models. Lenard et al. [23] has also investigated the coeKcient of friction variations with reduction. Therefore, the FEM modelling should consider the friction variation in the roll bite as it has a signi!cant e=ect on the accuracy and stability of the model and can improve the practical control accuracy of ribbed strip rolling. So an analysis of the thermo-mechanical process of ribbed strip rolling with friction variation models is employed in this study. This paper presents a full three-dimensional !nite element simulation of the ribbed strip rolling. The approach considers the thermal phenomenon simultaneously with the solution of the velocity !eld and a friction variation model at the strip-roll interface. Predictions of roll separating force and rib height agree well with experimental data. Results of this numerical simulation show that the simulation of the ribbed strip with friction variation is e=ective and stable. 2. FEM formulation The analysis of the ribbed strip rolling makes use of the !nite element method based on the 9ow formulation model for slightly compressible materials. The main advantage of this approach over the conventional 9ow formulation following the Levy–Mises rigid-plastic constitutive equations is the possibility of obtaining the stress–strain rate relationship directly by [24] ij = II˙ 23 ˙ij + $ij g1 − 29 ˙v ; (1) where $ij is the Kronecker delta, and g a small positive constant that may vary from 0.01 to 0.0001, which is a factor conferring a small degree of compressibility. The e=ective stress, , I and the ˙ e=ective strain rate, , I are de!ned, respectively, by (2) I = 32 ij ij + gm2 ; where ij is the deviatoric stress tensor, m the hydrostatic stress. The equivalent strain rate is written as I˙ = 23 ˙ij ˙ij + g1 ˙2v ; (3)
Z.Y. Jiang et al. / Finite Elements in Analysis and Design 40 (2004) 1139 – 1155
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Frictional shear stress (MPa)
K1 m1 k kVa=0.001 1 = 0.001 kVa=0.1 1 = 0.1 kVa=10.0 1 = 10.0
P'P section (k1 = 0.001) 0
o' o''
-K 2 m 1 k Entry
Neutral point
Exit
Roll bite
Fig. 2. Distribution of the frictional shear stress model.
where ˙ij = ˙ij − $ij ˙v =3 is the deviatoric strain rate tensor and ˙v = ˙x + ˙y + ˙z is the volumetric strain rate. The slightly compressible method in a full 3-D rigid visco-plastic FEM has been employed to solve the rolling of ribbed strip problem. According to the variational principle [25], the real velocity !eld must minimize the following functional: 1 ˙ &= I I dv + f QVf ds ± T1 v ds = (p + (f + (t ; (4) m+1 v sf sv where the !rst term on the right hand side is the work rate of plastic deformation ((p ), I the equivalent stress, I˙ the equivalent strain rate and m is the strain rate sensitivity index. The second term on the right hand side is the work rate of friction ((f ); QVf is the relative slip velocity on the interface of the rolled material and the rolls where the frictional shear stress f is applied. The friction variation model shows the friction varies along the contact length of the deformation zone. In this study, the frictional shear stress model [1] is modi!ed as follows: Vg m1 s f = ± Ki √ ; (5) 3 V 2 + k2 g
1
where m1 is friction factor; s is yield stress; Ki is a coeKcient describing the changes of frictional shear stress in the deformation zone with K1 used in the forward zone and K2 in the backward zone; k1 is a small positive constant. The relative slip velocity, Vg , between the rolled material and the roll is given by (6) Vg = (vx sec − VR )2 + vy2 + k22 ; where vx and vy are the velocity components in the x and y directions, respectively, is the angular position of the node, VR is the tangential velocity of the roll, k2 is a small positive constant and the distribution of these frictional shear stress models is shown in Fig. 2. If k1 = 0:0 and K1 = K2 = 1:0, Eq. (5) results in a constant friction model. The neutral point in the roll bite varies for ribbed strip
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rolling. QVf is also the relative slip velocity along the the same as Vg . The third term on the right hand side of Eq. (4) is the and v is the velocity of the cross section with tension. “+” the back tension. Assuming the material is rigid–viscoplastic and obeys
roll–slab contact interface and its value is work rate ((t ) of tension. T1 is the tension Here “−” indicates the front tension, and the following constitutive relation:
s = DI˙ m ;
(7)
where D is a function of strain and temperature and m is the strain rate sensitivity index. The procedure of the standard 3-D rigid visco-plastic !nite element discretization with slightly compressible materials is as follows: a control volume, V , is discretised into M eight-node hexahedron elements linked through N nodes. In each element, the velocity distribution u can be represented in vectorial form by u = N T C;
(8)
where C is the nodal velocity vector and N is the shape function matrix of the element. The strain rate tensor is then obtained from ”˙ = BC;
(9)
where matrix B contains appropriate spatial derivatives of the shape functions. Expressing I˙ in terms of the nodal velocities, it can be written in matrix form as I˙ = ”˙T Z ”˙ + g1 (C T ”) ˙ 2 = CT B T ZBC + g1 CT B T C T CBC; (10) where [C] = [1 1 1 0 0 0]T and the correspondent matrix Z is as follows: 4 − 29 − 29 0 0 0 9 2 4 −9 − 29 0 0 0 9 2 4 0 0 0 − 9 − 29 9 : [Z] = 1 0 0 0 0 0 3 0 0 0 0 13 0 0
0
0
0
0
(11)
1 3
In the present work, the slight compressibility is enforced in terms of elements. Consequently, an element average matrix BI is used to express the volumetric strain. BI is evaluated by Ne 1 I B= Bi ; Ne i=1
(12)
where Ne is the number of Gauss points of an element and Bi is the B matrix at Gauss point i. Eq. (11) is rewritten as I (13) I˙ = CT B T ZBC + g1 CT BI T C T C BC:
Z.Y. Jiang et al. / Finite Elements in Analysis and Design 40 (2004) 1139 – 1155
1145
Thus, the functional Equation (4) can be approximated by the sum of the functional at element level, which results in e e 1 i I dV ( = s CT B T ZBC + g1 CT BI T C T C BC &≈ m + 1 i V i=1 i=1 cos m1 s T T √ 0 + C N N C − 2vRT C N T + vR2 + k22 − k2 dS 3 Sfi sin ± T1 vi dS :
Sti
(14)
The solution for the nodal velocity vector is obtained as it makes the above functional minimum [25]. This requires 9& = 0: 9C The resulting governing equation is nonlinear and can be written [16] as e 1 s m 1 s Z + CC T dV + N T N dS C i I g i 2 2 V S 3(v + k ) f i=1 g
+
e i=1
−
Sfi
(15)
1
cos m 1 s 0 vR N T T1 dS dS ± = 0: Sti 3(vg2 + k12 ) sin
(16)
It can be solved by the modi!ed Newton–Raphson iteration: vj = vj−1 + *Qvj " # $ % 92 & 9& − Qvj = ; T 9v9v j−1 9v j−1
(17)
where * is the deceleration coeKcient, and −92 &=9v9Tv [16] is # e " 92 & s 1 9s s T − = − Z+ 2 Z CC Z dV 9C9CT 9I˙ I˙ I˙ I˙ Ve i=1
+
e i=1
Sfi
m1 s 3(vg2 + k12 )
Ni N j T F Fj dS; vg2 + kl2 i
(18)
where Fi = {−vy sec i ; vx sec i − VR }:
(19)
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The roll separating force can be calculated by Eq. (20): l bx z dy d x; P= 0
(20)
0
where l is the projected length of the deformation zone; bx is the strip width in the deformation zone; z is the stress along the thickness direction. The rib height can be determined from Eq. (21): h = Zi − 0:5h2 ;
(21)
where Zi is the z coordinate of node on rib top at exit and h2 the thickness of the ribbed strip. 3. Thermal analysis The equation governing the heat-conduction in a continuous medium with an arbitrary volume, V , bounded by a closed surface, S, can be expressed as [26] q˙ 1 dT2 ∇ 2 T2 + = ; (22) k 1 dt where 1 = k =( · c), q˙ is the heat generation rate; , c are density and speci!c heat respectively; k is a thermal conductivity coeKcient, and T2 is the temperature. q˙ = ij ˙ij ;
= 0:85 − 0:95;
(23)
where is a parameter which gives the fraction of the plastic deformation energy which is dissipated as heat. Applying the classical Galerkin method, the heat transfer Equation (22) can be written in matrix form at elemental level as T T ˙ ˙ kMM T2 dV + cNN T 2 dV − I N I dV − qn N dSq = 0; (24) Vm
Ve
Sqe
Ve
where M = ∇N , qn is the heat 9ux along the boundary surface Sq , and T2 the nodal temperature vector. The later is interpolated using identical shape functions as those utilized for the velocity !eld, since the numerical model of the deforming workpiece uses the same !nite element mesh for thermal as for the mechanical analysis. Eq. (24) can be expressed in abbreviated form as KT 2 + C1 T˙ 2 = Q;
(25)
where K is the heat conduction matrix, C1 the heat capacity matrix and Q the heat 9ux vector. The heat 9ux vector Q in the above equation has several components and can be expressed as follows: 4 4 Q= (I )N I˙ dV + ! (Tenv − T2 )N dS + henv (Tenv − T2 )N dS Ve
+
SRm
e SRad
hroll (Troll − T2 )N dS +
e SConv
SFer
qf N dS;
(26)
where Tenv and Troll are the ambient and roll temperatures, respectively. The !rst term on the right side is the contribution of the net heat generated by plastic deformation inside the ribbed strip, while other terms are due to the heat exchanged between the strip and the ambient environment or rolls.
Z.Y. Jiang et al. / Finite Elements in Analysis and Design 40 (2004) 1139 – 1155
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Therefore, the second term on the right side is the result of the heat radiation to the environment, where ! is the Stefan–Boltzmann constant and is the emissivity. The third term on the right side is the contribution of the heat convection between the surface of the strip and the environment, where henv is the heat convection coeKcient. The fourth term on the right side describes the heat transferred between the ribbed strip and the roll through their contact interface. The last term on the right side is the contribution of the heat generated by friction along the roll–ribbed strip interface, qf being the surface heat generation rate due to friction: qf = f |Vg |;
(27)
where f is the friction shear stress at the roll–ribbed strip, and |Vg | the relative velocity between the two regions. The major boundary conditions to be applied in the !nite element thermal analysis of the process are the following: (1) When the rolled ribbed strip is placed between stands, the surface loses heat by radiation and convection: 4 Q1 = (Tenv − T24 )N dS + henv (Tenv − T2 )N dS: (28) e SRad
e SConv
(2) When the rolled ribbed strip is water-cooled between stands, the working temperature is much higher than the boiling point of the water and therefore !lm boiling takes place: Q2 = hwater (Twater − T2 )N dS: (29) e SConv
(3) When the ribbed strip is being rolled, the surface in contact with the rolls is cooled, while conversely the rolls are heated: Q3 = hroll (Troll − T2 )N dS; (30) e SConv
where hroll is the thermal resistance due to scale and roughness which is responsible for promoting a temperature discontinuity across the roll–ribbed strip interface. Furthermore, it is also assumed that the initial temperature of the ribbed strip at the exit of the furnace is uniform throughout its volume. The integration of Eq. (24) requires the utilization of the following time-stepping scheme: T2(t+Qt) = T2t + Qt[(1 − !)T˙ 2t + !T˙ 2(t+Qt) ];
(31)
where t and Qt are time and its increment, and ! a parameter varying between 0 and 1. Details on implementation of the computer algorithm to include the coupling of the solution of stress equilibrium equations with heat balance in the !nite element model is given in Ref. [1]. 4. Simulation conditions Process data, the mechanical and thermal properties of the material utilised in the theoretical analysis of the ribbed strip rolling is shown in Table 1. Initial strip thickness is 7.2–8:0 mm. The
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Z.Y. Jiang et al. / Finite Elements in Analysis and Design 40 (2004) 1139 – 1155
Table 1 Process data, the mechanical and thermal properties of the material Material Chemical composition Density Speci!c heat Stefan–Boltzmann constant Surface emissivity Thermal conductivity Ambient temperature Equipment Diameter of upper roll Diameter of down roll Rolling speed Flow stress model [27]
Carbon steel (235 MPa) 0.16 C; 0.07 Si; 0.49 Mn 7800 kg=m3 670 J=(kg K) 5:675 W=(m2 K 2 ) 0.8 32 W=(m K) 20◦ C 740:85 mm 749:55 mm 3.9 –5:4 m=s & ' 0:244 0:13 2 = 4098 exp −3:5T I I˙ 1000
Table 2 Velocity boundary conditions for the rolling of ribbed strip No. 1 2 3 4 5 6
Location ABCDEFGHIJKL A B K L AA B B RSR S , etc. MM X X Inclined contact surface, OO P P, etc.
Velocity vx = v1 (unknown), vy = 0, vz = 0 vx = v0 (unknown), vy = 0, vz = 0 vy = 0 vz = −vx tan vz = vx tan vz = −vx tan , vy = −vx tan =tan !1 + VR sin =tan !1
slab was heated to 1200◦ C in furnace, and rolled in the roughing mills and !nishing mills. Based on the production conditions, the rolling of ribbed strip was carried out on the 6th !nishing stand, and the rolling temperature is 850 –900◦ C at this stand. The initial temperature !eld of the ribbed strip before entering the 6th stand is determined by the thermal computation from the furnace to the !nishing mills. The boundary conditions are as shown in Table 2. It should be noted that if the ribs do not !ll up the grooves of the roll, the surfaces of the PP Q Q and TT U U are free and do not satisfy the relationship vz = −vx · tg . !1 is the inclined angle of the rib. Considering the characteristic of the rolling of ribbed strip, the spread is small due to the restriction of the inclined surfaces of the roll grooves. Therefore, the strip deformation on middle part of the width of strip is similar. The analysis is carried out for two types of ribs: in one model the edge plane KLL K is a free surface (see Fig. 3) which considers the deformation of the side rib with free spread and in the other model the plane KLL K is not a free surface and is constrained by the surrounding ribs. Discretization of the control volume was conducted by 432 eight-nodded isoparametric linear elements and 665 nodes, as shown in Fig. 3. In this !gure, ABB A represents the symmetrical surface, KLL K the free surface, MNWX the exit section and M N W X the entry section. The rolls were assumed as rigid, and friction variation conditions were applied to the contact interface between the ribbed strip and the rolls. The parameters for the model of friction variation were determined
Z.Y. Jiang et al. / Finite Elements in Analysis and Design 40 (2004) 1139 – 1155 z(v z(vzz))
1149
Entry
B'
K'
N'
O'
P'
Q' R'
S' T' U' V'
W'
y(v y(vy)y)
L'
A' Q
P M'
N
O B
D C
E R F
X' T U H S I V G
W K
J
M
X A
x(vx)x) x(v
Exit
L
Fig. 3. FE mesh for simulating calculation of ribbed strip.
based on the rolling conditions and the comparison of the calculated results with measured values. For this study, m1 = 0:35 and K1 = 1:0, K2 = 1:0 and k1 = 0:1. In the coupled simulation, the initial temperature !eld and velocity !eld were given. For a given time increment Qt, the temperature !eld was calculated !rst, and the 9ow stress (see Table 1, 9ow stress model) was obtained based on the present calculated temperature !eld, and the velocity !eld was then calculated according to Eq. (17). This iterative calculation continues and stops when the increments of the velocity !eld and temperature !eld respectively reach a very small value [1]. In the uncoupled thermal (isothermal) analysis, the initial temperature is set a uniform !eld and a certain value. However, the calculated velocity !eld is based on the calculated true temperature !eld at present iteration step for coupled thermal analysis. 5. Results The theoretical predictions obtained with the thermo-mechanical !nite element model presented in the previous sections are discussed and veri!ed against experimental data obtained from a steel company; comparison is made for the average surface temperature of the ribbed strip, the roll separating force and rib height. The temperature at the middle of the rib top surface was measured via an infrared thermal instrument made by Land Co. of England after the ribbed strip exits from the !nishing mill. The measured accuracy is less than 1◦ C. 5.1. Distribution of temperature Fig. 4 is the distribution of temperature of the cross section at the exit for reductions 18.5 and 25.7% as the friction variation model was used. It can be seen that the temperature at the centre of
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Fig. 4. Theoretical distribution of temperature for the cross section at exit. Reduction: (a) 18:5% and (b) 25:7%.
Table 3 Comparison of the calculated and measured mean surface temperatures Reduction (%) Calculated (◦ C) Measured (◦ C)
18.50 859 860
20.65 879.5 880
23.46 873.5 875
25.70 892 890
the strip in the thickness direction is the highest. However, the temperature at the top surface of ribs is much lower, and the lowest is at the side of the rib. The temperature at the top of CDEF rib (see Fig. 3) is lower than that at the top of GHIJ rib (see Fig. 3). The temperature di=erence between the two ribs shows a trend expected since for reduction 25.7%, rib CDEF (h = 2:14 mm) is higher than GHIJ (h = 1:89 mm), o=ering more surface area for heat loss to the roll. The calculated mean surface temperatures of the ribbed strip at the exit of the roll bite for di=erent reductions are in good agreement with the measured surface temperature of the ribbed strip obtained under the industrial conditions, as shown in Table 3. 5.2. Distribution of stress The distribution of equivalent stress (unit in MPa) at various cross sections is illustrated in Fig. 5 (x is a distance of the cross section to the entry section) for reduction 25.7%. The equivalent stress increases from the entry section to the 0:5l cross section. However, it decreases from the 0:75l cross section to exit section. The change of the isopleth of the equivalent stress is much larger at the sides of ribs than that at other parts. As mentioned before, the stress distribution across the width of the ribbed strip is non-uniform and leads to di=erence in material 9ow behaviour. It can clearly be seen that the highest compressive state of stress and the highest gradient of stresses occur near to the rib side as a result of the signi!cant local residual deformation. The shape of rib shows that the rib top does not contact the roll (if the rib top touches the roll, the shape of rib must be same as the groove of roll). The shape of the equivalent stress at the side (the right hand part in Fig. 5)
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Fig. 5. Theoretical distribution of equivalent stress.
is di=erent from that between the ribs, because of the e=ect of the lower temperature at the side of ribbed strip. 5.3. Roll separating force To validate the simulation, the roll separating force was also calculated and compared with industrial rolling conditions. In the seven 4-high hot tandem mills, the ribbed strip 6:0 mm thick and 1050 mm wide with 30 ribs was rolled. The roll separating forces were measured by load cells. The distance between two ribs (E 1 ) is 34 mm and rib shape used in the simulation is the same as the rolled strip in plant. The geometry of the rolled strip is the same as that of the numerical model. The predictions were obtained from !nite element models with and without the thermal e=ect. In industrial practice, the friction factor is determined by matching the measured rolling load (from load cells) and the rolling force calculated from the slab method. In this calculation, the friction variation condition was applied (Eq. (5)), m1 = 0:35, K1 = 1:0, K2 = 1:0 and k1 = 0:1, the friction factor is close to the practical rolling of hot strip 0.3– 0.4 [28].
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Roll separating forces (kN)
20000
Non-coupled 16000
Coupled Measured
12000 8000 4000 0
5
10
15 20 25 Reduction (%)
30
35
Fig. 6. Comparison of the calculated roll separating forces and experimental values.
2.5 Measured
Rib height (mm)
2
Calculated 1.5 1 0.5 0 0
5
10
15 20 Reduction (%)
25
30
35
Fig. 7. Comparison of the calculated rib height and measured values.
Fig. 6 shows the comparison of the theoretical predictions of the roll separating forces and the experimental values. It can be seen that the roll separating forces obtained from the coupled thermo-mechanical !nite element model are lower than that obtained from the non-coupled model (isothermal analysis a uniform temperature is set at a given value), and for coupled thermo-mechanical analysis, the roll separating force is in good agreement with the experimental values in the rolling of ribbed strip. It is also seen that the relationship between the rolling force and reduction is non-linear because of the experimental errors. The relationship between non-coupled calculated rolling force and reduction is non-linear, but the coupled results are closer to being linear, which indicates that the coupled calculation considering the thermal e=ect can improve the simulation results. 5.4. Rib height A comparison of the theoretical prediction of the rib height and the plant measured values is carried out as shown in Fig. 7. The calculated and measured rib height is an average of middle rib (see Fig. 3, such as rib CDEF). Rib heights of the middle ribs (28 ribs out of a total of 30 ribs)
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2.2
Rib height (mm)
2 1.8
KK1=K2=1.0 1 = K2 =1.0 KK1=1.2, K2=0.8 1 = 1.2, K 2 =0.8 KK1=1.2, K2=1.0 1 = 1.2, K 2 =1.0 KK1=1.0, K2=0.8 1 = 1.0, K 2 =0.8 Measured Measured
1.6 1.4 1.2 1 0.2
0.3
0.4 Friction factor
0.5
0.6
Fig. 8. E=ect of friction variation on rib height.
were measured after rolling [1], and the measurements were repeated three times for each rib. The relative change for the measured rib height of these ribs is about 3%. Therefore, results of the central rib were chosen to compare with the computation values. Due to the limitation of the load capacity of the 6th !nishing mill and production/customer requirements, the maximum reduction is 25.7% in practice, and therefore, the range of conditions and measured results are limited. It can be seen that in this simulation conditions the calculated value of the rib height is close to the experimental value. The prediction of the rib height can be extended to other plates and ribs. Fig. 8 indicates the e=ect of friction variation on rib height when reduction is 23.46% and m1 = 0:35. It can be seen that the calculated rib height is close to the measured value for friction factor 0.3– 0.4, and the rib height increases with friction factor. It can also be seen that the rib height for K1 = 1:0, K2 = 1:0 is close to that when K1 = 1:2, K2 = 0:8, but the rib height increases faster when K1 = 1:2, K2 = 1:0. The rib height with friction variation model K1 = 1:2, K2 = 0:8 is greater than that with K1 = 1:0, K2 = 0:8, which means that the rib height increases with larger friction in the forward slip zone.
6. Conclusions In the present investigation, three-dimensional rolling of ribbed strip is modelled by using a coupled thermo-mechanical three-dimensional !nite element model with friction variation consideration. The major innovation of this work is the simulation of the local residual deformation of the ribbed strip rolling with the combined thermal and deformation analysis. The approach provides the possibility of estimating the temperature and the distribution of the stress and deformation throughout the deformed process of ribbed strip. The rib height increases when the friction factor and friction in the forward slip zone increase. So the rib height can be increased with a higher friction condition. The calculated evolution of the stress !eld also veri!es the local non-uniform characteristic of the material 9ow resulting from the rib formation. The predicted roll separating force with coupled thermo-mechanical model and the rib height are in agreement with the experimental values.
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