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Numerical Simulation of Bulging Deformation for Wide-Thick Slab Under Uneven Cooling Conditions

Abstract

In the present work, the bulging deformation of a wide-thick slab under uneven cooling conditions was studied using finite element method. The non-uniform solidification was first calculated using a 2D heat transfer model. The thermal material properties were derived based on a microsegregation model, and the water flux distribution was measured and applied to calculate the cooling boundary conditions. Based on the solidification results, a 3D bulging model was established. The 2D heat transfer model was verified by the measured shell thickness and the slab surface temperature, and the 3D bulging model was verified by the calculated maximum bulging deflections using formulas. The bulging deformation behavior of the wide-thick slab under uneven cooling condition was then determined, and the effect of uneven solidification, casting speed, and roll misalignment were investigated.

Introduction

In the continuous casting process, the bulging deformation of a solidified shell between rolls inevitably occurs under the load of ferrostatic pressure, as illustrated in Figure 1. If the tensile strain on the solidification front, caused by bulging and other factors, exceeds the critical threshold value, internal cracks occurs.[1,2,3] Additionally, a bulging deformation, especially one that occurs near the strand solidification end, significantly aggravates the slab centerline segregation.[4,5] It compresses the mushy region and promotes the solute-enriched interdendritic liquid to flow into the central part of the strand. As a result, the level of the centerline segregation is dramatically increased. Furthermore, bulging deformation decreases the equipment life owing to the greatly increased mechanical load between the strand and rolls.

Fig. 1
figure 1

Schematic of slab bulging deformation

Due to the difficulties in accurately measuring the bulging deformation in practical continuous casting,[6] only a few experimental measurements have been reported.[7,8,9,10] For this reason, various mathematical analysis methods were widely adopted by previous researchers to study this deformation behavior to provide a theoretical basis for optimizing the process parameters and to decrease the adverse effects from bulging deformation. Verma[1] and Han[2] studied the strain status at the solidification front caused by bulging deformation and other factors by using a strain analysis model. Based on the theory of a bending beam, Yoshii[11] and Sheng[12] calculated the bulging deflection of the slab, and Sheng proposed a new formula for bulging deflection with clear physical meaning, after evaluating some frequently used equations. Han[13,14] investigated the influence of bulging of the solidified shell on slab deformation in the straightening zone of the continuous caster and the influence of roll misalignment on slab bulging deformation. Based on the theory of a 2D plate, Xu[15] and Wang[16] calculated the bulging deflection of the slab, and Xu deduced an analytical model for calculating the casting withdraw resistance. All of the above works mainly adopted an analytical method. To further improve the calculation accuracy and revealing the bulging behavior more comprehensively, finite element method began to be adopted by researchers.[17,18] A series of 2D[19,20,21,22,23] or 3D[24,25,26,27,28,29,30,31,32] finite element models were then developed to quantitatively estimate the bulging deflection or strain status of the solidified shell. During these works, Lee[20] and Miki[22] clarified the origin and characteristics of unsteady bulging. Bellet[21] calculated the thermomechanical state of steel all along the continuous casting machine using a global non-steady-state approach. He[23] explained the mechanism of bulge deformation in continuous slab casting by introducing positive creep and reverse creep. Fu[26,27] studied the mechanism of slab broadening and the effect of the slab width and thickness on this deformation during continuous casting. Qin[29] investigated differences between 2D and 3D models in predicting the slab bulging deformation. Camporredondo[31] investigated the casting speed limits that the two compact-strip process (CSP) caster may reach under different casting conditions. With the developed 2.5D model, Pascon[32] studied the effect of bulging deflection and other local defects on the risk of transverse cracking in continuous casting of steel slabs.

Most of the previous works focus on the analysis of bulging deformation for a conventional slab under an even cooling condition. However, with significant increases in width and thickness, the solidification of the wide-thick slab shows obvious non-uniformity along the slab width direction due to the non-uniform water flux distribution in the secondary cooling region.[33,34] As a result, the bulging deformation behavior of the wide-thick slab will be more complicated than that of the conventional slab due to its non-uniform shell thickness and temperature distribution. However, few publications have investigated the bulging deformation of the wide-thick slab under an uneven cooling condition. In this paper, a 3D elasto-plastic and creep finite element model that considers non-uniform solidification along the slab width direction was developed using the commercial finite element software MSC.Marc. The bulging deformation behavior of the wide-thick slab under an uneven cooling condition was revealed, and the effects of non-uniform solidification, casting speed, and roll misalignment were investigated.

Finite Element Model

Simulation Parameters

In the present work, a commercial wide-thick slab continuous caster was selected as the specific research objective, as shown in Figure 2. This caster is composed of fourteen segments and divided into nine cooling zones after the primary cooling region in the mold. The bulging deformation of the wide-thick slab was calculated at three strand positions in Figure 2 (Location 1, Location 2, Location 3) located in the bending, bow, and straightening regions, respectively. More detailed parameters of the cooling zones and the continuous caster structure at the three strand positions are listed in Tables I and II, respectively.

Fig. 2
figure 2

Schematic of the wide-thick slab caster

Table I Parameters of the Cooling Zones
Table II Parameters of the Continuous Caster Structure at Three Strand Positions

The chosen steel grade is peritectic steel, and its nominal composition is 0.16 wt pct C, 0.25 wt pct Si, 0.70 wt pct Mn, 0.024 wt pct P, and 0.006 wt pct S. Due to the symmetry of the slab in both the thickness and width directions, one quarter of the commonly produced 2000 mm × 280 mm section size slab was selected for the 2D heat transfer model and the 3D bulging model. Due to the large section size of the wide-thick slab and the characteristics of the nozzle arrangement in the secondary cooling region, the water flux distribution along the slab width direction is non-uniform. According to the arranged position of nozzles in a row, the water flux distribution of zone 5 to 8 shown in Figure 2 was measured. The corresponding nozzle arrangement in zone 5 to 8 and the measured water flux distribution are, respectively, shown in Figures 3(a) and (b) (2100 mm in Figure 3(a) indicates the maximum width of the slab section size that could be produced by the present CC machine, and 2000 mm × 280 mm in Figure 3(b) is the studied slab section size in the present paper). Figure 3(b) clearly shows that the water flux distribution percentage is similar with different water and air pressures and decreasing from slab wide surface center to corner. The water flux distribution in zone 1 to 4 is assumed to be uniform because the nozzles are densely arranged (7 nozzles in a row) to achieve strong cooling intensity at the early casting stage.

Fig. 3
figure 3

(a) Nozzle arrangement and (b) the corresponding measured water flux distribution along slab width in zones 5-8

2D Heat Transfer Model

Based on some simplified assumptions,[35] a 2D heat transfer model was established to calculate the solidified shell profile and its temperature distribution, as illustrated in Figure 4. 4-node quadrilateral elements with 2 mm side length were applied to evenly mesh the calculation domain. During the calculation, the time step was set as constant of 0.2 second.

Fig. 4
figure 4

2D finite element model for solidification analysis

The solidification process of the 2D heat transfer model can be described by a transient heat conduction equation as follows:

$$ \frac{\partial }{\partial x}\left( {\lambda \frac{\partial T}{\partial x}} \right) + \frac{\partial }{\partial z}\left( {\lambda \frac{\partial T}{\partial z}} \right){ + }\rho L\frac{{\partial f_{s} }}{\partial t} = \rho c\frac{\partial T}{\partial t}, $$
(1)

where T and t are the temperature in °C and calculation time in s, respectively; and ρ, c, and λ are the temperature-dependent density, specific heat, and conductivity in kg/m3, J/(kg K), and W/(m K), respectively; L is the latent heat of steel solidification and equal to 272000[36] J/kg; fs is the solid fraction.

To acquire more accurate thermal material properties of the peritectic steel, a microsegregation model that was described in detail in the present authors’ previous work[33] was used to calculate the phase fraction evolution, as shown in Figure 5(a). Based on the phase fraction evolution, the thermal material properties of conductivity, density, and enthalpy, as shown in Figures 5(b) through (d), respectively, were calculated using the formulas proposed by Li and Thomas.[36] It should be noted that the thermal conductivity of liquid steel is magnified by a few times compared to that in the solid state for considering the improving effect of molten steel flow on heat conduction of steel.[33,34]

Fig. 5
figure 5

Evolution of the (a) phase fraction of δ-Fe(f δ ), γ-Fe(f γ ), solid(f s ) and liquid(f l ), (b) conductivity, (c) density, and (d) enthalpy with temperature

Cooling boundary conditions, such as the heat flux in the mold (qmold), the heat transfer coefficient between the strand and cooling water (h iw ), and the heat transfer coefficient of radiation (hrad) below the mold, were calculated according to the present authors’ previous work.[33] It should be noted that the measured non-uniform water flux distribution along the slab width direction, as shown in Figure 3(b), was applied to the calculation of h iw . Additionally, roll contact, as one of the main cooling phenomena below the mold (the other two cooling phenomena are sprayed cooling water and radiation), significantly influences the solidification process of the continuous casting steel. Given this, an investigation aiming to quantitatively evaluate the influence of the roll contact on the slab heat transfer and stress–strain was carried out by Xia.[37] Using a thermocouple embedded in the strand surface, the slab surface temperature variation caused by roll contact was measured. Then, the key parameters for the heat transfer process between the rolls and the strand, such as the heat transfer coefficients and the contact length, were derived using the measured data and the inverse calculation method. In the present work, the influence of roll contact on the solidification was considered according to the actual position of the rolls and the key parameters obtained by Xia. During the calculation, the 2D heat transfer model was assumed to move from meniscus along the casting direction at the casting speed. When the model was located in the roll contact region, the extracted heat flux by rolls was calculated as follows:

$$ q_{\text{con}} = h_{\text{con}} \cdot (T_{\text{surf}} - T_{\text{roll}} ), $$
(2)

where hcon is the heat transfer coefficient between the rolls and strand in W/(m2·K) and Tsurf and Troll are the temperatures of the strand surface and the roll in K, and Troll is taken as 423 K.[37]

3D Bulging Model

Based on the parameters of the continuous caster given in Table II and the calculated profile of the solidified shell at three strand positions according to the 2D heat transfer model, a 3D bulging model of the wide-thick slab over several roll pitches was then established, as illustrated in Figure 6. 6-node pentahedron elements with 3 to 5 mm sides were used to unevenly mesh the 3D bulging model. During the calculation, automatic time step was employed, and the maximum and minimum time step are, respectively, 0.01 and 0.2 second.

Fig. 6
figure 6

3D finite element model for bulging deformation analysis

Bulging deformation is not only composed of time-independent elastic deformation, but also the time-dependent inelastic deformation. An elasto-viscoplastic model[38] that characterizes the material structure evolution through the inelastic strain was used to describe the bulging deformation behavior of the casting steel:

$$ \dot{\varepsilon }_{\text{p}} = C\exp \left( {\frac{ - Q}{T}} \right)\left[ {\sigma - a_{\varepsilon } \varepsilon_{\text{p}}^{{n_{\varepsilon } }} } \right]^{n} , $$
(3)

where T is temperature, K; C = 46550 + 71400 (pct C) + 12000 (pct C)2, carbon content-dependent function, MPansm−1; Q = 44650, activation energy constant, K; a ε  = 130.5 − 5.128 × 10−3T, temperature-dependent constant MPa s-; n ε  = − 0.6289 + 1.114 × 10−3T, temperature-dependent inelastic strain exponent; n = 8.132 − 1.540 × 10−3T, temperature-dependent net stress exponent; σ is stress, MPa; εp is inelastic strain.

The elastic modulus and Poisson’s ratio were calculated using Eqs. [4] and [5], which were obtained from the numerical fitting of the experimental data by Mizukami[39] and Uehara.[40]

$$ E(T) = 968 - 2.33T + 1.9 \times 10^{ - 3} \times T^{2} - 5.18 \times 10^{ - 7} T^{3} $$
(4)
$$ \upsilon = 8.23 \times 10^{ - 5} T + 0.278, $$
(5)

where E is elastic modulus, GPa; υ is the Poisson’s ratio; T is the temperature, °C.

The yield stress of the research steel grade was measured and applied during the establishment of the 3-D bulging model, and Figure 7 shows the measured yield stress under different temperatures.

Fig. 7
figure 7

Variation of the measured yield stress with temperature

The temperature history of nodes calculated by the 2D heat transfer model was applied as a thermal load for the 3D bulging model by an interpolating method based on the strand position and the relationship of node locations between the models. The displacement of nodes on the symmetrical planes was constrained along the direction perpendicular to the symmetrical planes.

To consider the dynamic contact between the rolls and the slab, rolls were modeled as rigid bodies and rotated around their axes at casting speed. The slab was driven forward by the interfacial friction force between it and the driving rolls. The friction coefficient was chosen as 0.3[26,29,41] for the driving rolls and 0.001[29] for the driven rolls. The node to segment contact algorithm was adopted to detect the contact between the rolls and slab, and the tolerance of contact penetration was set as 0.25 mm.[41] The ferrostatic pressure was applied on the solidification front and calculated according to the vertical height from the meniscus as follows:

$$ P = \rho_{\text{l}} gH, $$
(6)

where P is the ferrostatic pressure, Pa; ρl is the steel density at liquidus temperature, kg/m3; g is the gravity acceleration, m/s2; and H is the vertical height from the meniscus, m.

Model Validation

The 2D heat transfer model and the 3D bulging deformation model were verified when the 2000 mm × 280 mm section slab was cast at 0.8 m/min.

To verify the non-uniform solidification results along the slab width direction predicted by the heat transfer model, the surface temperature and solidified shell thickness at the different positions (P0.5, P0.7) indicated in Figure 8(a) were measured using an infrared camera (A40, FLIR) and nail shooting experiments at the exit of Segs. 8 and 9 and were compared with the predicted results in Figures 8(b) and (c). Nail shooting experiment had been widely used to measure the solidified shell thickness by the previous researchers[33,34,41,42] due to its simplicity and accuracy. The principle of this experiment, which had been described in detail in the previous work of Long,[42] is that the sulfur embedded in the slot of the nail surface diffused in the mushy region after the nail is shot into the strand. The location where the diffusion of the sulfur occurs could be clearly distinguished on the macrograph of the slab, and the shell thickness could be obtained by measuring the distance between the slab surface and the location where the sulfur begins to diffuse. It can be clearly seen in Figures 8(b) and (c) that the surface temperature and shell thickness at P0.7 are obviously higher and thinner than those at P0.5 owing to the continuously declining water flux from slab center to corner, as shown in Figure 3(b). The absolute value of the relative error between the predicted and the measured results is less than 2.6 pct for temperature and 3.0 pct for shell thickness. Furthermore, predicted temperature dips of approximately 80 °C can be observed due to the roll contact, which is generally in agreement with the measured results of Xia.[37]

Fig. 8
figure 8

(a) Positions where the temperature and shell thickness were measured and comparison between the calculated and measured results of (b) temperature and (c) shell thickness

Previous researchers verified their bulging FE models by comparing the maximum bulging deflection predicted by the FE model with either the measured results[24,28,30] or the results calculated by formulas.[22] Due to the lack of measurements of the bulging deflection for the wide-thick slab under an uneven cooling condition, the latter method is adopted in the present paper.

For calculating the maximum bulging deflection, Han[2] proposed a calculation formula based on a beam model with both edges fixed, and Sheng[12] derived a calculation formula with clear physical meaning using creep law. Furthermore, a calculation equation for bulging deflection of the flat plate was employed by Toishi[22] to validate the established bulging FE model by him. These formulas are expressed as follows:

$$ \delta_{\hbox{max} } = \frac{{Pl^{4} }}{{3200E_{e} S^{3} }}\sqrt t $$
(7)
$$ \delta_{\hbox{max} } = \frac{{Pl^{4} }}{{3200ES^{3} }}\left( {1 + \sqrt t } \right) $$
(8)
$$ \delta_{ \hbox{max} } = (1/16) \times ({\text{Pl}}^{2} /{\text{ES}}^{3} ) \times \{ (12/2) - 3\upsilon S^{2} + \upsilon (b + S)^{2} \}, $$
(9)

where δmax is the maximum bulging deflection, cm; P is the ferrostatic pressure, kg/cm2; l is the roll pitch, cm; t is the time for slab to travel a pitch, min; E and Ee are the elastic modulus and equivalent elastic modulus, kg/cm2; S is the shell thickness, cm; b is the slab width, cm; and ν is the Poisson’s ratio.

Table III compares the calculated maximum bulging deflection at the mid-width of the wide-thick slab by Formulas [7] through [9] with the predicted results. It shows that the predicted results agree well with the range of maximum bulging deflection values calculated by these formulas, which verifies the reasonability of the developed FE model in this study.

Table III Comparison of the Maximum Bulging Deflection Calculated by the FE Model and the Formulas

Results and Discussion

Simulation Results

Figure 9 shows the distribution of δ x and δ z for a 2000 mm × 280 mm section slab cast at 0.8 m/min. δ x is defined to mainly show the displacement amount of the wide side of the solidified shell along X axis (perpendicular to the wide side of the solidified shell), while δ z is meant to reflect the displacement amount of the narrow side of the solidified shell along Z axis (perpendicular to the narrow side of the solidified shell). δ x and δ z could be used to quantitatively estimate the bulging deflection of the wide and narrow side of the solidified shell. Under the load of ferrostatic pressure, obvious bulging deformation of the solidified shell is observed in both the thickness and width directions. The bulging deflection in the thickness direction (δ x ) concentrates on the wide surface around the central position of the roll pitch due to the absence of rolls support there, and the maximum bulging deflection in the width direction (δ z ) appears on the narrow surface.

Fig. 9
figure 9

Distribution of bulging deflection (mm) in thickness direction (δ x ) and width direction (δ z ) at three strand positions

The bulging deflection of the wide surface in thickness direction and of the narrow surface in width direction is compared at three strand positions in Figure 10. Although the deformation resistance of the solidified shell continuously improves with the increase of shell thickness along the casting direction, the ferrostatic pressure also increases, and the roll pitch becomes longer. As a result, the bulging deflection of the wide surface shows an uptrend at the three strand positions, as indicated in Figure 10(a). Furthermore, the location where the bulging deflection reaches a maximum at each strand position is not exactly at the center of the roll pitch but instead deviates by some distance along the casting direction, due to time-dependent creep deformation. Figure 10(b) shows that the bulging deflection of the narrow surface constantly decreases from the narrow surface center to the corner at the early casting stage (Location 1). However, with the increase of shell thickness on the narrow side, this trend is reversed at Locations 2 and 3, which indicates that the constraining effect of the narrow side of the solidified shell affects the bulging deflection of the narrow surface more significantly than does the ferrostatic pressure. In addition, the bulging deflection of the narrow surface at Location 3 is larger than that at Location 2 due to the higher ferrostatic pressure and the larger roll pitch at the former.

Fig. 10
figure 10

Comparison of the bulging deflection of the (a) wide surface and (b) narrow surface at three strand positions

Figure 11(a) shows the distribution of the normal strain in three axes directions (ε xx , ε yy , ε zz ), and ε xx /ε yy /ε zz denote the strain along X/Y/Z axis direction and perpendicular to the plane of YOZ/XOZ/XOY. Due to the bulging deformation of the solidified shell, normal strains in both tensile (strain that is greater than zero) and compressive (strain that is less than zero) states can be observed, and the maximum tensile strain along each direction exhibits a tendency to concentrate on the solidification front. Internal cracks will be easily induced if the tensile strain concentrated on the solidification front caused by bulging, together with other strain caused by bending, roll misalignment, or straightening, exceeds the critical threshold value.[2]

Fig. 11
figure 11

(a) Distribution of the normal strains in axes directions on the solidification front at three strand positions, and (b) schematic of the internal cracks that often occur in a continuously cast slab

Figure 11(b) schematically illustrates the positions where internal cracks are most often observed in slabs. It can be found that the concentrated tensile strains of ε xx , ε yy , ε zz on the solidification front in Figure 11(a) are mainly located at the triple-point region, over the support rolls and near the corner, which correspond to the positions where triple-point cracks, mid-way cracks, and internal corner cracks often occur, respectively. This result is generally in agreement with but slightly differs from that obtained by Okamura[24] for a conventional slab under an even cooling condition. In the work of Okamura, the location where the maximum tensile normal strain in the thickness direction (corresponding to ε xx in the present work) concentrates on the solidification front changes with the increase of solidified shell thickness. However, this location is unchanged and is always located at the triple-point region for a wide-thick slab in the present work, as shown in Figure 11(a).

In order to quantitatively investigate the variation of ε xx /ε yy /ε zz during the continuous casting process, ε xx /ε yy /ε zz at a certain position within the concentrated distribution region in Figure 11(a) was extracted and compared in Figure 12(b), and Figure 12(a) shows the specific distribution of positions for extracting ε xx /ε yy /ε zz . It can be seen from Figure 12(b) that the concentrated tensile strain in three axes directions is almost the same at Location 1 due to the relatively small ferrostatic pressure and roll pitch. However, with the increase of ferrostatic pressure and roll pitch, the concentrated tensile strains all increase in the subsequent two strand positions, and ε xx increases much more rapidly than ε yy and ε zz . As a result, ε xx becomes the dominant concentrated tensile strain at Locations 2 and 3, which indicates that internal cracks will be more likely to form due to the bulging deformation in the triple-point region than in other regions at these two strand positions.

Fig. 12
figure 12

(a) Schematic of the positions where the tensile strain concentrated on the solidification front was extracted, and (b) the variation of the corresponding concentrated tensile strains at three strand positions

Effect of Non-uniform Solidification

Because the solidification is obviously non-uniform along the width direction for the wide-thick slab compared with the conventional one, the effect of uneven cooling on the bulging deformation was investigated. Since the water flux distribution was considered uniform in zones 1 to 4 before Location 1, the bulging deformation under even and uneven cooling conditions was calculated and compared at Locations 2 and 3. The cooling intensity under the even cooling condition was assumed to be uniform along the slab width direction and equal to the value at the slab surface center under the uneven cooling condition. It is obvious from Figure 13(a) that the bulging deflection of the wide surface in the region of 400 to 800 mm at Location 2 is larger under the uneven cooling condition than that under the even cooling condition. This occurs because in this region, the temperature and the shell thickness are higher and thinner, respectively, under the uneven cooling condition, as shown in Figure 13(b). This significantly decreases the deformation resistance of the solidified shell. With the increase of ferrostatic pressure and the roll pitch, the difference between the bulging deflection of the wide surface under the uneven cooling condition and that under the even cooling condition further increases, and the bulging deflection of the wide surface in the region of 0 to 800 mm at Location 3 is obviously larger under the uneven cooling condition than that under the even cooling condition. In addition, the reduced deformation resistance of the wide side of the solidified shell under the uneven cooling condition also contributes to the increase of bulging deflection of the narrow surface at Locations 2 and 3, as shown in Figure 13(c).

Fig. 13
figure 13

Distribution of (a) the wide surface bulging deflection (δ x ), (b) temperature, shell thickness along the slab width direction, and (c) narrow surface bulging deflection (δ z ) under even and uneven cooling conditions

Figures 14(a) and (b) show the variation of ε xx and ε zz in Figure 12(a) and the distribution of concentrated tensile strain ε yy on the solidification front along the slab width direction (the concentrated tensile strain ε yy is located over the rolls as shown in Figure 11(a)), respectively. It is obvious from Figure 14(a) that ε xx and ε zz are dramatically increased owing to the non-uniform solidification. It should also be noted that ε xx is in a compressive state at Location 2 under the even cooling condition. However, due to the non-uniform solidification, this strain state changes to tensile under the uneven cooling condition. Figure 14(b) indicates that the distribution of ε yy shows non-uniformity characteristic just like the bulging deflection of the wide surface in Figure 13(a). Furthermore, ε yy in the region of 300 to 700 mm at Location 2 and ε yy in the region of 0 to 700 mm at Location 3 are both obviously increased under the uneven cooling condition compared with that under the even cooling condition. It can be concluded from the above discussions that the risk of mid-way cracks in the range of 0 to 700 mm from slab center, triple-point cracks, and internal corner cracks induced by bulging deformation is obviously increased due to the non-uniform solidification.

Fig. 14
figure 14

Comparison of the (a) ε xx , ε zz and (b) ε yy under even and uneven cooling conditions at Locations 2 and 3

Effect of Casting Speeds

Casting speed is closely related to the productivity and quality of the continuously cast slab. To study the influence of the casting speed, the bulging deformation of the wide-thick slab under different casting speeds at Location 2 was simulated and compared in Figure 15. During the practical continuous casting process, the cooling water intensity in the secondary cooling region is enhanced with the increase of casting speed to achieve the target slab surface temperature. However, the effective cooling time for the slab is simultaneously reduced. As a result, the shell thickness becomes thinner, and the overall temperature in the solidified shell is increased, which obviously decreases its deformation resistance and thus tends to increase the bulging deformation. On the other hand, the required time for the slab to travel a roll pitch is reduced with the increase of casting speed, which reduces the creep time of the solidified shell between rolls and thus tends to decrease the bulging deformation.

Fig. 15
figure 15

Effect of casting speed on the bulging deflection of (a) wide surface, (b) narrow surface, and (c) normal strains at Location 2

As shown in Figure 15(a), under the influence of the two factors mentioned above, the maximum bulging deflection of the wide surface in Figure 15(a) firstly increases from 0.105 to 0.117 mm when the casting speed is increased from 0.7 to 0.8 m/min and decreases to 0.108 mm when the casting speed is increased to 0.9 m/min. Figure 15(b) shows the bulging deflection of the narrow surface under different casting speeds. It can be seen that the bulging deflection of the narrow surface continuously increases with the increase of casting speed, which indicates that the influence of the decreased shell thickness and the increased temperature of the solidified shell on the bulging deflection of the narrow surface is more significant compared with the influence of creep time.

Figure 15(c) compares the normal strain under different casting speeds. It can be seen that ε xx and ε zz firstly present an upward trend and then a downward trend with increasing the casting speed and reach the maximum value at 0.8 m/min. It means that the risk of inducing triple-point cracks and internal corner cracks by the bulging deformation under the casting speed of 0.8 m/min is higher than that of others. It also can be seen from Figure 15(c) that ε yy continuously increases from 0.04 pct under 0.7 m/min to 0.06 pct under 0.9 m/min, which indicates that the increase of casting speed continuously increases the risk of inducing mid-way cracks by the bulging deformation.

Effect of Roll Misalignment

Roll misalignment is mainly caused by installation error, rolls bending, and wearing, which significantly influence the slab bulging deformation. The effect of the roll misalignment on the bulging deformation of the wide-thick slab was studied at Location 2 with different roll misalignment amounts, and Figure 16 schematically shows the roll misalignment.

Fig. 16
figure 16

Schematic of roll misalignment

As shown in Figure 17(a), the bulging deflection of the wide surface is greatly increased when the roll is misaligned from its original position because the misaligned roll cannot provide effective support; the deflection presents a rapid increasing trend with an increase in the amount of roll misalignment. In addition, the difference for the bulging deflection of the narrow surface in Figure 17(b) is further enlarged with the increase of roll misalignment.

Fig. 17
figure 17

Influence of roll alignment on the bulging deflection of the (a) wide surface, (b) narrow surface, and (c) the concentrated tensile strain

Figure 17(c) shows the influence of roll misalignment on the concentrated tensile strain on solidification front in Figure 11(a). It can be seen that all the normal strains approximately linearly increase with increasing the roll misalignment. In addition, the normal strain of ε xx and ε yy increases more rapidly than that of ε zz with increasing the roll misalignment. As a result, the normal strain of ε xx and ε yy , respectively, reaches 0.66 and 0.49 pct under the roll misalignment of 0.9 mm, which is much larger than ε zz (0.1 pct). This indicates that the roll misalignment obviously increases the risk of inducing cracks on the slab solidification front, especially the triple-point cracks and mid-way cracks, by the bulging deformation of the solidified shell.

Conclusions

Based on the solidification results of the 2D heat transfer model and the structural parameters of the continuous caster, a 3D bulging model was established. The bulging deformation of the wide-thick slab under an uneven cooling condition was then simulated at three strand positions. The main conclusions are summarized as follows:

  1. 1.

    The bulging deflection of the wide surface continuously increased from Location 1 to Location 3, and the bulging deflection of the narrow surface presents an uptrend from the narrow surface center to the corner at Locations 2 and 3, which is contrary to that at Location 1.

  2. 2.

    The maximum tensile normal strains on the solidification front present a characteristic of concentrated distribution, and the positions where ε xx , ε yy , and ε zz concentrate correspond well with the region where the triple-point cracks, mid-way cracks, and internal corner cracks are often observed. With the increase of the ferrostatic pressure from Location 1 to Location 3, ε xx , ε yy , and ε zz show a rising trend, and the triple-point cracks will be more easily induced by the bulging deformation from Location 2 because ε xx becomes the dominant tensile normal strain at this strand position.

  3. 3.

    Due to the uneven solidification, bulging deflections of the wide surface and ε yy exhibit non-uniformity along the slab width direction, and both the bulging deflections and all of ε xx , ε yy , and ε zz are increased.

  4. 4.

    With increasing the casting speed from 0.7 m/min to 0.9 m/min, the bulging deflection of the wide surface and the normal strain of ε xx and ε zz firstly present an upward trend and then a downward trend, but the bulging deflection of the narrow surface and the normal strain of ε xx continuously increase.

  5. 5.

    When the roll is misaligned from its original position, bulging deflection of the wide surface greatly increases, and the difference for the bulging deflection of the narrow surface is further enlarged. With the increase of roll misalignment, all the normal strains approximately linearly increase, and the normal strain of ε xx and ε yy increases much more rapidly than that of ε zz .

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Acknowledgments

The present work is financially supported by the National Natural Science Foundation of China Nos. 51474058 and U1708259, and the Program for Liaoning Excellent Talents in University (LJQ2015036). Special thanks are due to our cooperating company for industrial trials and application.

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Correspondence to Cheng Ji.

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Manuscript submitted April 23, 2017.

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Wu, C., Ji, C. & Zhu, M. Numerical Simulation of Bulging Deformation for Wide-Thick Slab Under Uneven Cooling Conditions. Metall Mater Trans B 49, 1346–1359 (2018). https://doi.org/10.1007/s11663-018-1173-3

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Keywords

  • Bulging Deformation
  • Roll Misalignment
  • Casting Speed
  • Water Flux Distribution
  • Slab Surface Temperature