Cast-rolling force model of multi-roll solid–liquid cast-rolling bonding process for fabricating metal cladding materials

Ce JI, Huagui HUANG

a College of Mechanical and Vehicle Engineering, Taiyuan University of Technology, Taiyuan 030024, China

b Engineering Research Center of Advanced Metal Composites Forming Technology and Equipment of Ministry of Education, Taiyuan University of Technology, Taiyuan 030024, China

c National Key Laboratory of Metal Forming Technology and Heavy Equipment, Taiyuan University of Technology,Taiyuan 030024, China

d College of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, China

e National Engineering Research Center for Equipment and Technology of Cold Strip Rolling, Yanshan University,Qinhuangdao 066004, China

KEYWORDS

Abstract Based on twin-roll casting technology and multi-roll groove rolling technology,a Multi-Roll Solid-Liquid Cast-Rolling Bonding (MRSLCRB) process was proposed to fabricate Cu/steel cladding bars, which processes the advantages of short flow and high-efficiency.However, it is a typical 3-D thermal-fluid-mechanics coupled problem,and determining cast-rolling force is difficult during the equipment design.Therefore,the geometrical evolution of the cast-rolling area was studied, laying the foundation to establish contact boundary equations and analyze mechanical schematics and metal flow.Then, a 3-D steady-state thermal-fluid coupled simulation model,including casting roll, substrate bar, and cladding metal, was established.The Kissing Point (KP)height, average outlet temperature, and process window were predicted, and simulation results of the three-roll layout indicate that the KP distribution along the circumferential direction can be considered uniform.Hence, the engineering cast-rolling force model was derived based on the differential element method and plane deformation hypothesis.The accuracy was verified by the 3-D finite element model,and the influences of process layouts and technological parameters on the castrolling force were analyzed.Through the indirect multi-field coupled analysis method, the temperature–pressure evolution and reasonable process window can be predicted,which provides a significant basis for guiding equipment design and improving product quality.

1.Introduction

With the rapid development of modern industries, traditional metals become difficult to fulfill the performance requirements of state-of-the-art applications.Therefore, investigations of novel materials such as laminated metal cladding materials that consist of two or more dissimilar metals have been motivated due to the excellent features obtained from each metal component.1–3The metal cladding material is a kind of specially laminated metal cladding material.Typical products,such as Cu/steel cladding bar, Al/Ti cladding pipe, and brass/Cu cladding stranded wire, are key materials in aerospace, petrochemical, and rail transit.Super-long highperformance metal cladding materials require a seamless cladding layer as well as a metallurgical bonding interface.The cross-section of the product varies according to service purpose,but most have a round cross-section.4Therefore,the performance uniformity of cladding thickness, mechanical property,and bonding interface is very important.Metal cladding materials can save precious metal resources, reduce production costs, and improve product performance.With the development of China’s wire and cable industry and railway industry,metal cladding materials are bound to become popular products and have huge market prospects.5Hence,efficient continuous forming technologies and performance control methods have become the industry’s difficulty and international research hotspot.

Numerous solid–solid phase bonding technologies have been employed to fabricate metal cladding materials, which made tremendous contributions to industry development.Yao et al.established a finite element model to study the liquid impact forming process of bimetallic cladding tubes, and the optimal loading parameters were obtained by selecting the mold side length, clamping speed, and initial internal pressure as variables.6Falahati Naqibi et al.fabricated Al/Cu cladding pipes using a friction stir additive manufacturing process.7The 3D thermo-mechanical simulation results show that, although the rotational-to-traverse speed ratio with a good approximation can predict the heat input, it is not a precise measure to predict the occurrence of defects in the cross-section.Tomczak et al.indicated that the three roller skew rolling can be successfully used in the production of bimetallic cladding rods, but proper fastening of the two materials depends on the billet geometrical parameters, and the quality of bimetallic cladding rods depends on the heating method used.8Xu and Li Y studied the spin-bonding process of 20/316L cladding pipes by theoretical mechanical calculation, and the minimum radial rotary pressure and maximum radial rotary pressure were obtained to optimize the design of the spinning device.9

For the solid–solid phase bonding method, the substrate and cladding need to be preprocessed and pre-assembled in advance,which requires high straightness roundness and coaxiality accuracy.Besides, if the pre-assembled billets are heated without protective atmospheres, the surfaces of the materials being joined together will be oxidized,which hinders the bonding process and adversely affects the physical and chemical properties.Therefore,it is challenging for fabricating products with long sizes and small sections by solid–solid phase bonding method.In contrast, the solid–liquid phase bonding methods are expected to realize flexible cladding and continuous forming,and subsequent plastic deformation helps to solve defects,such as shrinkages or porosities, and improve the interfacial bonding effect.For example, Jiang et al.first prepared brass cladding pure copper stranded wire billets with a diameter of 8.5 mm through solid–liquid continuous casting bonding technology, then the composite billets were then drawn.10The results showed that the composite billet has a good surface quality and metallurgical bonding interface.Besides, the plastic deformation of the pure copper wire reduced triangular arc gaps between the pure copper wires, and the triangular arc gaps were fully filled at 50%.Even for liquid–liquid phase bonding methods,plastic deformation is often required to regulate the properties of the components and interfaces.11Therefore, solid–liquid phase bonding methods with plastic deformation and liquid–liquid phase bonding methods followed by plastic deformation have become important development directions.

As a green manufacturing technique,the Twin-Roll Casting(TRC) process has been successfully applied to fabricate laminated metal cladding strips using solid substrate and liquid cladding.12The TRC process combines rapid solidification and rolling deformation,and the cladding metal changes from a liquid state to a solid state during forming.Therefore,in the cast-rolling area, the interface evolution process between substrate and cladding includes three stages:solid–liquid interface,solid-semisolid interface, and solid–solid interface.13The ratio of the three stages is determined by the Kissing Point (KP),which determines the cast-rolling force and further influences the product quality.The high temperature and liquid flow at the solid–liquid interface are favorable to the initial physical contact and infiltration, and the plastic deformation at the solid–solid interface is favorable to the interface bonding and cladding mechanical properties.14Therefore, interfacial metallurgical bonding can be achieved with most of the component metal combinations,such as Cu/Al,steel/Al,Ti/Al,and Ti/Cu.Interfacial bonding strength and component metal properties can be further improved by subsequent annealing or rolling.15,16Furthermore,external energy field assisted forming technology has become a key research direction for performance improvement.For example, Xu et al.investigated the effects of pulsed electric field and electromagnetic oscillation field on interfacial bonding strength during fabricating Ti/Al clad sheets by horizontal TRC process.17The results indicate that the oscillation effect reduces the temperature gradient at the contact area between the strip and melt and makes the newly formed crystal nuclei fall off the surface of the strip,which is conducive to forming more metallurgical bonding regions and improving interfacial bonding strength.

Recently,Ji et al.proposed a Multi-Roll Solid-Liquid Cast-Rolling Bonding (MRSLCRB) process for fabricating metal cladding materials with round sections,based on the TRC process and groove rolling technology.18The influence of the process layout, including twin-roll layout, three-roll layout, and four-roll layout, on the circumferential performance uniformity, was discussed.Different from fabricating laminated metal cladding strips by the TRC process,the MRSLCRB process is a typical 3-D deformation problem.When the substrate is a hollow pipe,the substrate pipe might be flattened or wrinkled under high temperature and large cast-rolling force.Then,lateral ears of cladding metal or rolling block would happen,and the continuous forming process would be terminated.Therefore, the structure-performance control has become a key bottleneck restricting the development of the MRSLCRB process.However, the influences of the process layouts and technical parameters on the temperature field, flow field, KP circumferential distribution, and plastic deformation flow,are not clear yet.The cast-rolling force is closely related to the KP and deformation temperature.The KP determines the geometrical parameters of the deformation zone, and the deformation temperature determines the deformation resistance.Hence, the calculation and control of the KP position and cast-rolling force need to be solved urgently.

Whether the preparation of laminated metal cladding strips or metal cladding materials, the evolution in the cast-rolling area is difficult to be observed experimentally due to the narrow space and block of the experiment.Hence,numerical simulation became one of the main investigation methods.Stolbchenko et al.proposed a 2-D finite-element simulation based on the ANSYS software, and the dependences of the deformation strain and outlet temperature of the cladding strip on the main technological process parameters were obtained.19Park applied the 2-D rigid-thermoviscoplastic finite-element method to analyze fabricating laminated metal cladding strips.The occurrence of buckling and thickening in the cladding and incomplete solidification in the substrate were predicted as potential problems of the processes.20Chen et al.built a 2-D model by the ANSYS software ignoring the effect of casting rolls to find out the temperature distribution in the castrolling area as well as at the Al/steel interface.21It indicates that asymmetrical and lower heat transport caused by the cladding strip makes the process window narrow.Li LX et al.established a cell model of Cu-Al intermetallic compounds based on the experimental results, and calculated the mechanical and electronic properties by first principles.The results show that vibration can improve the comprehensive properties of the Cu-Al interface by changing the intermetallic compound phase and proportion.22Although numerical simulation provides a visualization method, the thermal-fluid-mechanics multi-field coupled problem is still a research difficulty to be solved directly.

For the TRC processes fabricating single metal strips, the location of KP determines the cast-rolling force and further influences the inner and surface quality of cast strips.Cao GM et al.discussed the influence of the key technical parameters on the KP through the 3-D finite element method coupled with turbulent flow and temperature fields.23The results indicate that determining the optimal location of KP is very important for establishing stable online control models.Similarly, Kim et al.indicated that the cast-rolling force played a significant role in maintaining good contact between the roll surface and the strip, which resulted in a high cooling rate and enabled the fabrication of fine microstructures.24Lee et al.indicated that modeling twin-roll casting focused on the cast-rolling force had the merits of predicting mechanical interaction between the rolls and solid strips, in addition to temperature distribution, which can be used effectively to reduce the experimental trial-and-error procedures.25Cao XL et al.established a sample-based stochastic model based on Latin hypercube sampling to evaluate the influence of various uncertain parameters on the KP height and the castrolling force,and a polynomial mathematical model combined with numerical simulation is built with a full factorial design approach.26For the TRC processes fabricating laminated metal cladding strips, most studies of the cast-rolling force model are based on the plane deformation hypothesis.Besides,the cladding deformation resistance varies greatly due to the physical state change, and the deformation resistance of the solidification zone can be ignored because it is quite small compared to that of the deformation zone.27However, different from the single metal forming, sufficient cast-rolling force is required to achieve interfacial metallurgical bonding during fabricating laminated metal cladding materials.28Therefore,the cast-rolling force has become one of the research focuses,which not only provides a significant basis for the equipment design, but also lays the foundation to improve the product quality.

In this paper, theoretical and simulation studies were carried out to analyze the thermal-fluid-mechanics multi-field coupled problem of the MRSLCRB process.First, a 3-D thermal-fluid coupled simulation model was built to analyze the influence of the technical parameters on the KP height and average outlet temperature.Subsequently,geometric characteristics of the cast-rolling area were studied to analyze the section evolution, and the mechanical schematics and metal flow were discussed.Then, the MRSLCRB process was reasonably simplified and the engineering cast-rolling force model was established, and a 3-D finite element model was built to verify the accuracy.Finally, the influences of process layouts and technical parameters on the cast-rolling force were analyzed based on the simulation results and engineering castrolling force model.

2.MRSLCRB process and investigation methods

2.1.Technological principle of MRSLCRB process

The MRSLCRB process was proposed to fabricate metal cladding materials with round sections through the solid–liquid flexible forming, as shown in Fig.1.It would have significant advantages, such as low energy consumption, high wall thickness accuracy, and high product performance uniformity.There are mainly three layouts, including the twin-roll layout,three-roll layout, and four-roll layout.Correspondingly, the round groove consists of two,three,or four arcs.With increasing the casting roll number,the linear velocity and cooling rate of the MRSLCRB process are expected to be more uniform along the circumferential direction.Besides, heat transfer,mass transfer, solidification, and deformation are expected to be more uniform and stable.

As the number of casting rolls increases, the number of round grooves uniformly divided increases.Therefore,the biggest limiting factors are the cooling system and transmission system of casting rolls.When the diameter of metal cladding material is small, the cooling system and transmission system between different casting rolls are easy to interfere with each other, which limits the increase of casting roll number.Furthermore, since the cladding metal is cast in a liquid state,the groove formed by casting rolls needs to be strictly closed.Otherwise, it will cause liquid leakage or lateral ears.Therefore, the groove is round.However, how to ensure that there is no assembly gap between the casting rolls is a challenge.During equipment manufacturing, casting rolls need to be matched to eliminate assembly gaps.In addition, sufficient preload force is required to avoid the displacement of the groove during the MRSLCRB process.Meanwhile, the KP position cannot be measured online due to high temperature and shielding.It is very important to establish the quantitative relationship between cast-rolling force and KP height to inversely calculate KP height based on the measured cast-rolling force.Therefore, the cast-rolling force calculation model is also the basis of product quality control.

Fig.1 Schematic diagram of MRSLCRB process for fabricating metal cladding materials with round cross-sections.

In the MRSLCRB process,the geometric characteristics of the cast-rolling area are different from the traditional TRC process, as shown in Fig.1.Some common names are defined to describe the differences.The casting roll number N represents different process layouts.The demarcation angle θ0refers to the demarcation boundary between the casting rolls.It is θ0=±90° when N = 2, θ0=±60° when N = 3, and θ0=±45° when N = 4.Nominal roll radius R0refers to the distance between the casting roll axis and the rolling axis.Therefore, nominal cast-rolling velocity vNCastrefers to the product of the angular velocity ω and the nominal roll radius R0.Groove radius r0refers to the radius of the round groove formed by all casting rolls and substrate radius rsis the radius of the substrate metal.To deeply understand the evolution process from the inlet section to the outlet section,the geometric characteristics, mechanical diagram, and metal flow are analyzed, which lays a theoretical foundation for mechanical analysis.

2.2.Thermal-fluid coupled simulation

Cu/steel cladding bars were selected as the target product.The substrate material is Q345 bar with a density of 7820 kg/m3,specific heat of 500 J/(kg?K), and thermal conductivity of 40 W/(m?K).The cladding material is industrial pure copper,with a density of 7930 kg/m3, specific heat of 490 J/(kg?K),and thermal conductivity of 167 W/(m?K) in the liquid state,and with a density of 8920 kg/m3,specific heat of 385 J/(kg?K),and thermal conductivity of 399 W/(m?K) in the solid state.The material of the casting roll sleeve is 42CrMo, with a density of 7850 kg/m3, specific heat of 540 J/(kg?K), and thermal conductivity of 50 W/(m?K).The melting temperature of Cu is 1083 °C, and the latent heat of solidification is 205 kJ/kg.The viscosity of liquid Cu is 3.41 × 10-3kg/(m?s).

The main technological parameters of the MRSLCRB process include molten pool height H, nominal cast-rolling velocity vNCast, cladding casting temperature TCast, substrate preheating temperature TSub, and substrate radius rs.To obtain a reasonable process window and shorten the process development cycle, a steady-state 3-D thermal-fluid coupled simulation model was developed based on the Fluent software.

Fig.2 shows the geometric model and the mesh model of the three-roll layout.The simulation model includes casting rolls, substrate, and cladding.The symmetry and local grid refinement were considered, and the total number of grid elements is about 680000.The parameters and values used in the simulation are shown in Table 1.The solidification/melting model is used to track the liquid–solid front.29During the modeling process,casting rolls are uniformly rotated and considered as rigid bodies.Liquid Cu and semisolid Cu are regarded as incompressible Newtonian fluids.Besides, the deformation of the substrate is neglected.

Simulation results of typical working conditions show that the KP heights of Symmetry I and Symmetry II are basically the same.Hence, it can be approximated that the KP height is evenly distributed along the circumferential direction.Thus,the influences of the technical parameters on the KP height HKPand the average outlet temperature TOutcan be obtained.Then,the cladding deformation resistance δ at the cast-rolling area outlet can be calculated based on the strain rate ˙ε,according to the constitutive model Eq.(1).30Finally,the cast-rolling force can be calculated based on these key parameters.

Fig.2 Schematic diagram of geometric model and mesh model.

where T is the deformation temperature; σ is the flow stress.

For the cast-rolling area,there are two types of heat input,namely, the heat input of cladding and the heat input of substrate.The heat input of cladding mainly depends on the nominal cast-rolling velocity, inlet area, and casting temperature,among which the inlet area depends on the cladding thickness and the molten pool height.The heat input of the substrate depends on the nominal cast-rolling velocity,the substrate preheating temperature,and the substrate radius.The output heat is usually only for the cladding and can be divided into two categories,namely,the cooling heat transfer of the casting roll and the cooling heat transfer of the substrate.According to the conservation of energy,the temperature field in the cast-rolling area ultimately depends on the combined effect of heat input and heat output.

2.3.Engineering cast-rolling force model

According to thermal-fluid coupled simulation results, the KP height can be regarded as evenly distributed along the circumferential direction.Therefore, when only the deformation behavior of the solid cladding below the KP is considered,the solid–solid rolling bonding stage is regarded as a pure wall reduction follow-up mandrel rolling process.Based on the differential element method and plane deformation hypothesis,the engineering cast-rolling force model for the MRSLCRB process was derived.

The KP height is equal to the deformation area height.To verify the accuracy of the engineering cast-rolling force model, a 3-D finite element model of the solid–solid roll bonding process under the KP was established based on the DEFORM software.The geometric parameters and the analysis range are shown in Table 2, where the nominal roll radius is 125 mm, the groove radius is 12.5 mm, the substrate radius is 10 mm, and the nominal cast-rolling velocity is 3.5 m/min.

Considering the distribution periodicity and geometric structure symmetry, the finite element simulation model includes two types.One is the half-roll simulation model, that is, half of a single cast-rolling unit was taken for analysis, as shown in Fig.3(a).The other is the one-roll simulation model,that is, a complete single cast-rolling unit was taken for analysis,as shown in Fig.3(b).The rotation velocity of the casting roll and the translational velocity of the substrate were calculated according to the nominal cast-rolling velocity.The basic assumptions are as follows:

(1) The casting roll, conformal side dam, and substrate are rigid bodies, while the cladding is the deformed body.

(2) The material equivalent deformation resistance of the cladding is simplified as the average value according to the average outlet temperature.

2.4.Experimental verification

Fig.4(a)shows the MRSLCRB experimental equipment of the three-roll layout,and the casting rollers are shown in Fig.4(b).Based on thermal-fluid coupled simulation results, a reasonable process window was predicted and the preparation experiments of Cu/steel cladding bars were conducted.Before the experiment, the equipment was preheated, and the surface of the steel bar was mechanically polished and cleaned.The industrial pure copper ingot was melted in an inert atmosphere and held at casting temperature.During the experiment, the cast-rolling force was measured by pressure sensors.Fig.4(c)shows the grooves and experiments on three different substrate metal radius, rs= 8, 9, 10 mm, were carried out.

Table 1 Parameters used in thermal-fluid coupled simulation.

Table 2 Parameters used in 3-D finite element simulation.

Fig.3 Schematic diagram of finite element model.

Fig.4 MRSLCRB equipment.

3.Characteristic analysis of cast-rolling area

3.1.Geometrical evolution of cast-rolling area

In the MRSLCRB process,the geometric characteristics of the cast-rolling area are complex.Hence, the evolution process of the cast-rolling area from the inlet section to the outlet section was analyzed by taking the three-roll layout as an example.

In the three-roll layout, the geometrical characteristics of the outlet section of the cast-rolling area, when the molten pool height H=0 mm,are shown in Fig.5.The outlet section is a round pass with O as the center and r0as the groove radius.Point F is located on the groove.The angle θ is formed by the Line OF and the symmetric Line OG.The projection of Point F onto the X-axis is Point A.According to the geometric relationship of Point F, the horizontal width W(θ,0)= r0sin θ,vertical height LV(θ,0)= r0cos θ, groove rotation radius Rθ= R0–r0cos θ, and linear velocity vθ= ωRθ.

The maximum groove rotation radius RMax=R0–r0cos θ0,the minimum groove rotation radius RMin=R0–r0.Hence,the average groove rotation radius RAve=(RMax–RMin)/2.The molten pool height is a key technological parameter, and it is a technological setting value.The deformation area height is equal to the KP height HKP,which is the control core.However,it depends on the structural parameters and various technical parameters.

Fig.5 Outlet section geometric parameters of cast-rolling area.

The geometric characteristics of the inlet section are shown in Fig.6.With the increase of molten pool height H,the difference between the inlet section and the outlet section becomes larger.Points O′, A′, F′, and G′can be obtained by projecting the points on the outlet section to the inlet section.A′F′extends along the vertical direction and intersects with the inlet section at Point E′.The axis line of the casting roll is BC, and Point E′is used to make the radial section of the casting roll.According to the geometric relationship of Point E′, the horizontal width W(θ,H)= r0sin θ, and vertical height LV(θ,H)is expressed as

The central angle β(θ,H)and contact arc length M(θ,H)at Point E′on the radial section of the casting roll are expressed as

Fig.6 Inlet section geometric parameters of cast-rolling area.

The vertical difference between the inlet section profile and the outlet section profile ΔLV(θ,H)is expressed as

The central angle α, radial distance LR(α,H), and radial difference ΔLR(α,H)of Point E′on the inlet section can be expressed as

In particular, Fig.6 indicates that the inlet section is composed of arcs and straight lines, and there is a demarcation central angle α0between arcs and straight lines.Therefore,the arcs in the central angle range of 0°–α0depend on the groove structure and can be solved by geometric relations.The straight lines within the central angle range α0–θ0depend on the conformal side dam and can be solved by linear interpolation.

In addition, when the casting roll number and structure parameters of casting rolls are different, the demarcation central angle α0of the inlet section is related to the demarcation angle θ0, which can be expressed as

The radial distance LR(α0,H)and radial difference ΔLR(α0,H)at the demarcation central angle α0are expressed as

At the demarcation angle θ0, the radial distance LR(θ0,H)= LR(α0,H)cos(θ0–α0), and radial difference ΔLR(θ0,H)= LR(θ0,H)–r.Therefore, the radial distance LR(α,H)of the inlet section within the range of 0°–θ0at the molten pool height H can be expressed as

According to Eq.(12), the geometrical evolution of the cast-rolling area can be investigated.When the KP is considered to be evenly distributed along the circumferential direction, the section where the KP is located is parallel to the inlet section.Hence, the calculation method of geometric parameters is the same.In addition, the projection of the contact area between the cast-rolling area and the casting roll on the Z-axis equals the molten pool height H.

3.2.Mechanical schematics and metal flow

According to the groove rolling conditions, the reduction of the cladding along the circumferential direction is ununiform when assuming the KP height is uniformly distributed.At the top of the groove, the reduction and the elongation reach the maximum.At the demarcation boundary θ0,the reduction and the elongation reach the minimum.The outlet section is shown in Fig.7(a), and the wall thickness of the cladding is uniformly distributed along the circumferential direction.Hence, it is equivalent to circular ring deformation with an equal wall thickness under internal and external compression.Diagrams of principal strain and principal stress are shown in Fig.7(b),compressing along the radial direction(r),tangential direction (θ), and extending along the axial direction (l).

The middle section of the deformation area is shown in Fig.8(a), and the groove is formed by the casting rolls and conformal side dam.Due to the circumferential geometric nonuniformity, there may be two extending directions when the cladding is compressed, namely circumferential flow and longitudinal extension.The cladding is in a 3-D deformation state,which is equivalent to the circular ring deformation with a non-equal wall thickness under internal and external compression.Diagrams of principal strain and principal stress are shown in Fig.8(b), compressing along the radial direction(r),extending along the tangential direction(θ)and axial direction(l).In particular,when located at the demarcation boundary, the principal strain and principal stress are in the same state as those in Fig.7(b) due to the symmetry.

Fig.7 Schematic diagram of mechanics in outlet section.

The purpose of the groove parameter optimization is to improve the deformation uniformity along the circumferential direction.Therefore, the cladding is forced to flow as far as possible along the longitudinal (axial) direction and as little as possible along the circumferential (tangential) direction.Due to the compression nonuniformity, the cladding at the top of the groove flows longitudinally with a large elongation.To ensure the deformation continuity, the adjacent positions are forced to extend longitudinally with the same elongation.Therefore, the actual elongation of each part of the cladding should be averaged when the end condition is ignored.This uniformity occurs not only at the outlet section but also in the whole deformation area.The velocity difference at each section is very small compared with the total deformation velocity.Hence, it can be approximately considered that the velocity on the section is the same, which is usually known as the flat section assumption.

In the MRSLCRB process, the movement of the substrate depends on the friction stress between the substrate and the cladding, that is, it is following.At the inlet of the castrolling area, there is physical contact between the substrate and the cladding, which is mainly manifested as sliding friction.At the outlet of the cast-rolling area, the substrate and the cladding move synchronously due to the metallurgical bonding,and there is no relative sliding.When there is no plastic deformation of the substrate,the velocity of the substrate is constant.Therefore, the velocity of the substrate is always greater than or equal to that of the cladding.From the inlet section to the outlet section, the friction coefficient between the substrate and the cladding is in a state of the continuous increase until they are completely bonded.

Fig.8 Schematic diagram of mechanics in middle section.

4.Influence of technical parameters on KP height and average outlet temperature

4.1.Influence of molten pool height

In the single variable condition, the influence of molten pool height on KP height and strain is shown in Fig.9(a), and the influence on average outlet temperature and cladding deformation resistance at the outlet section is shown in Fig.9(b).As the molten pool height increases, the cladding volume increases,and the areas contacted with the casting roll and the substrate increase.Hence, the contact time becomes longer.According to the energy conservation principle, the KP height increases because the input heat is constant and the output heat increases.The average outlet temperature decreases so that the cladding deformation resistance increases.For the analyzed range,the dependence of the KP height and average outlet temperature on the molten pool height is linear.The fitted functions are expressed as Eqs.(13) and (14), respectively.

4.2.Influence of nominal cast-rolling velocity

Fig.9 Influence of molten pool height.

Fig.10 Influence of nominal cast-rolling velocity.

In the single variable condition, the influence of nominal castrolling velocity on KP height and strain is shown in Fig.10(a),and the influence on average outlet temperature and cladding deformation resistance at the outlet section is shown in Fig.10(b).As the nominal cast-rolling velocity increases, the contact time becomes shorter, and the output heat decreases.Therefore, the KP height and strain decrease.The average outlet temperature increases, thus the deformation resistance decreases.For the analyzed range, the dependence of the KP height and average outlet temperature on the nominal castrolling velocity is linear.The fitted functions are expressed as Eqs.(15) and (16), respectively.

4.3.Influence of cladding casting temperature

In the single variable condition,the influence of cladding casting temperature on KP height and strain is shown in Fig.11(a),and the influence on average outlet temperature and cladding deformation resistance at the outlet section is shown in Fig.11(b).As the cladding casting temperature increases, the input heat of the cladding increases.Therefore, the KP height and strain decrease according to the energy conservation.The average outlet temperature increases, and the deformation resistance decreases.

For the analyzed range, the dependence of the KP height and average outlet temperature on the cladding casting temperature is linear.The fitted functions are expressed as Eqs.(17) and (18), respectively.The adjustable range of cladding casting temperature is usually small.The effect on KP height and average outlet temperature is not significant,but the effect on undercooling degree and liquid metal fluidity is significant.When the cladding casting temperature is low, the undercooling degree is small and it is conducive to improving the nominal cast-rolling velocity.However, the liquid metal fluidity becomes worse and it is not conducive to improving the castrolling velocity.Therefore, comprehensive consideration should be taken according to the process requirements.

4.4.Influence of substrate preheating temperature

Fig.11 Influence of cladding casting temperature.

In the single variable condition,the influence of substrate preheating temperature on KP height and strain is shown in Fig.12(a),and the influence on average outlet temperature and cladding deformation resistance at the outlet section is shown in Fig.12(b).The substrate preheating temperature affects the temperature distribution of the cast-rolling area by influencing the heat transfer between the cladding and the substrate.As the substrate preheating temperature increases, the input heat of the substrate increases.Therefore,the KP height and strain decrease according to the energy conservation.The average outlet temperature increases, and the deformation resistance decreases.

For the analyzed range, the dependence of the KP height and average outlet temperature on the substrate preheating temperature is linear.The fitted functions are expressed as Eqs.(19)and(20),respectively.High substrate preheating temperature is advantageous to solid–liquid interfacial diffusion.However, when the substrate preheating temperature is too high,the substrate strength is low,and significant deformation is easy to occur.When the substrate is a hollow pipe,it may be flattened.Furthermore, it may be broken or fused when the melting point of the substrate is similar to or lower than that of the cladding.

4.5.Influence of substrate radius

In the single variable condition, the influence of substrate radius on KP height and strain is shown in Fig.13(a), and the influence on average outlet temperature and cladding deformation resistance at the outlet section is shown in Fig.13(b).When the groove radius is constant, the input heat of the substrate increases with the increase of the substrate radius.Meanwhile, the wall thickness of the cladding decreases, and the input heat of the cladding decreases.However, the decreased degree of heat input of the cladding is significantly greater than the increased degree of heat input of the substrate.Besides, the ability of the substrate to absorb the heat of the cladding is enhanced.Therefore, the KP height and strain increase.The average outlet temperature decreases,and the deformation resistance increases.For the analyzed range, the dependence of the KP height and average outlet temperature on the substrate radius is nonlinear.The fitted functions are expressed as Eqs.(21) and (22), respectively.

Fig.12 Influence of substrate preheating temperature.

Fig.13 Influence of substrate radius.

4.6.Process window prediction model

The KP height and the average outlet temperature are mainly affected by the molten pool height,nominal cast-rolling velocity,cladding casting temperature,substrate preheating temperature, and substrate radius.Therefore, the prediction models can be expressed as

According to the simulation results, the influence of each process parameter shows a monotonic change in the single variable condition.To comprehensively consider the influence of various parameters under multivariable conditions,the prediction model can be approximately constructed in

where CHand CTare the correction coefficients of KP height and average outlet temperature, respectively.

The basic condition was selected as H = 30 mm,vNCast= 3.5 m/min, TCast= 1120 °C, TSub= 25 °C, and rs= 10 mm, and the correction coefficients CHand CTcan be obtained by submitting parameters into Eqs.(25) and(26).Hence, the prediction models are expressed as

The prediction models are in good agreement with the basic conditions used for fitting,and the maximum error is less than 5%.To further verify the accuracy of the prediction model,the comparison was conducted when H = 30 mm, vNCast= 4 m/min,TSub= 25°C, rs=9 mm,and the cladding casting temperature as a variable.The simulation results agree well with the calculation results, as shown in Fig.14.However, it needs to point out that the greater the difference between the calculation condition and the basic condition, the greater the deviation will be.

For a specific product, the adjustable range of technical parameters is usually small.Therefore, the prediction models can be used for qualitative analysis and semi-quantitative evaluation, to obtain a reasonable process window and significantly shorten the process development cycle.

5.Engineering cast-rolling force model

5.1.Basic assumptions

For the MRSLCRB process, the force on the cladding comes from the casting rolls, conformal side dam, and substrate.The cast-rolling force of the casting rolls is the main parameter for equipment design.Furthermore, there is a significant difference between the deformation resistance of the substrate and the cladding.Hence, the solid–solid rolling bonding process below the KP can be regarded as the pipe rolling process with a moving mandrel.The basic assumptions are as follows:

(1) The MRSLCRB process is stable,and a one-third model is recommended for the three-roll layout.

(2) The KP height is uniformly distributed along the circumferential direction.The influence of liquid cladding is ignored, and only the deformation of solid cladding below the KP is considered.

(3) The cladding is an ideal rigid plastic body,and the casting roll and the substrate are undeformed rigid bodies.

(4) The cladding thickness is usually small.Therefore, the circumferential flow of the cladding is ignored, and the plastic deformation of the cladding under the action of casting rolls is regarded as plane deformation.

Fig.14 Comparison of simulation results with calculation results.

Fig.15 Schematic diagram of element division.

5.2.Differential element division

The inlet section of the deformation area,where the KP height is HKP, is shown in Fig.15.Due to the geometric symmetry,one-half of the section can be taken for analysis.For the three-roll layout mode, the demarcation angle θ0= 60°.The contact interval between the cladding and the casting roll along the direction of the X-axis is 0–x0, which is divided into longitudinal element strips N with the width of Δx=r0(sin θi–sin θi–1).The intersection of the adjacent element strips is called the element interface.The central angle of element interfaces on the outlet section is represented by θi(i = 1,2,???,n),and the corresponding central angle on the inlet section is represented by αi(i = 1,2,???,n).The geometric relationship is as

where S(θi,HKP)is the vertical thickness of cladding metal in KP section; S(θi,H0)is the vertical thickness of cladding metal in outlet section.

When xi> rssin θ0:

The critical bite condition can be expressed as

where μRis the critical friction coefficient between cladding metal and rolls.

According to Eq.(34),the maximum height of the deformation area is expressed as

5.3.Unit pressure equation

The element body is selected from the element strip, and the forces acting on the element body are shown in Fig.16.Projection the external force to the Z-axis, and the sum is zero.Therefore, the force balance differential equation can be written as

where σzis the force in the Z direction; S(θi,z)is the width of element body at Z section; pRis the rolling force applied by rolls;τSis the frictional force between cladding metal and substrate metal.

Fig.16 Schematic diagram of force of element body in deformation area.

When similar terms are combined and higher-order terms are ignored, Eq.(36) is expressed as

where τRis the frictional force between cladding metal and casting roll.

Among them, the frictional force τRbetween cladding metal and casting roll is ‘‘+τR” in the backward slip zone and ‘‘–τR” in the forward slip zone.The frictional force τSbetween cladding metal and substrate metal is always ‘‘+τS”.

To facilitate engineering calculation,the following assumptions are made:

(1) The yield condition is approximately written as pR–σz=Kf= 1.15σs, where σsis the deformation resistance of cladding metal.

(2) The corresponding contact arcs are replaced by straight lines and expressed as

(3) The pSis the force of the substrate on the cladding, and

the pRis the force of the casting roll on the cladding.The

pSand pRhave the same projection along the Y-axis:

where βzis the horizontal contained angle of the element body.

(4) The deformation area is mainly sliding friction, τR=μRPRand τS=μSPS,where μRis the friction coefficient between cladding metal and casting roll and μSis the friction coefficient between cladding metal and substrate metal.

By substituting the above assumptions into Eq.(37)

Separate the variables and Eq.(41) can be expressed as

Considering the uniformity of heat and mass transfer along the circumferential direction, it can be approximated that the deformation resistance only changes along the height direction of the molten pool.Therefore, from the inlet section HKPto the outlet section H0, and the cladding temperature gradually decreases, while the deformation resistance of the material gradually increases.

Therefore, the unit pressure equation in the forward slip zone Eq.(44) can be simplified as

The unit pressure equation in the backward slide zone Eq.(46) can be simplified as

5.4.Average unit pressure

For the MRSLCRB process,the cast-rolling force of half casting roll is approximately equal to the sum of the cast-rolling forces on all strip elements.Therefore, the cast-rolling force PKPof one casting roll can be expressed as

where piis the average unit pressure of the contact surface between the i-th element and the casting roll, N/mm2; Aiis the projection area of the contact surface of the i-th element with the casting roll, mm2.

The projected area Aiis expressed as Ai= HKPΔx.The average unit pressure is equal to the total deformation force divided by the vertical projection of the contact area:

Substitute Eqs.(49) and (50) into Eq.(52), it can be expressed as

The deformation of cladding is relatively small.Therefore,in engineering calculation, it can be approximated as

5.5.Influence of process layouts

The simulated and calculated results of the one-roll cast-rolling force in different process layout modes are shown in Fig.17(a).The maximum relative error between the simulated and calculated results is about 15%,which is in good agreement.For the engineering calculation model of cast-rolling force,the change of process layout mode is mainly reflected in the difference of demarcation angle θ0.Hence,Eq.(53)can be applied to calculate the cast-rolling force under different process layouts.

Fig.17 One-roll cast-rolling force comparison between simulation results and calculation results.

When the MRSLCRB process is arranged in the three-roll layout, the comparisons between the simulated results (represented by Symbol S) and the calculated results (represented by Symbol C) are shown in Fig.17(b).When the deformation area height is constant,the cast-rolling force increases with the increase of equivalent deformation resistance.When the equivalent deformation resistance is constant, the cast-rolling force increases with the increase of deformation area height.The comparison results show that the simulation results are in good agreement with the calculation results,and the maximum relative error is about 10%.It is proved that the engineering calculation model of the cast-rolling force can meet the requirements of engineering application, and can be used to estimate the cast-rolling force in the design stage of the MRSLCRB equipment.

The cast-rolling force simulation results of the half-roll simulation model in different process layouts are shown in Fig.18.The groove is distributed along the circumferential direction,and the wall thickness of the cladding is gradually uniform and thinned.Therefore, the force in the X-axis and Y-axis directions is larger,while that in the Z-axis direction is smaller.In addition, at the same mesh density, the cast-rolling force fluctuation tends to be gentle and the value in three directions decreases with the increase of the casting roll number.The main reason is that the contact area between the groove and the cladding decreases.

The cast-rolling force in the X-axis direction is closely related to the circumferential deformation flow of the cladding metal.Fig.19 shows the metal plastic flow and velocity isosurface cloud of the cladding metal under three process layouts.As can be seen from Fig.19, with the increase of the number of casting rolls, the circumferential flow trend of the cladding metal slows down and is gradually uniform along the rolling direction.Therefore, the velocity isosurface cloud gradually flattens out.Meanwhile, with the increase of the casting roll number, the force in the X-axis direction gradually decreases,as shown in Fig.18.Therefore,the cladding flows to the rolling direction as much as possible,and to the circumferential direction as little as possible.Namely, the deformation is mainly concentrated in the YZ plane, and the circumferential deformation becomes uniform.

The cast-rolling force simulation results of the one-roll simulation model in different process layouts are shown in Fig.20.At the same mesh density, with the increase of the casting roll number, the forces in three directions decrease and the fluctuation tends to be gentle.By comparing Fig.18 and Fig.20, it can be seen that the cast-rolling forces in the Y-axis and Z-axis directions of the one-roll simulation model and the half-roll simulation model are approximately proportional.For the casting roll, the groove is distributed symmetrically along the X-axis direction and the cast-rolling forces on both sides are opposite.Hence, for the one-roll simulation model, the forces in the X-axis direction cancel each other.The cast-rolling force is mainly distributed in the YZ plane,which is the same as the hypothesis.

5.6.Influence of technological parameters

Fig.18 Cast-rolling force simulation results of half-roll simulation model under different layouts.

The influence of various technical parameters on cast-rolling force can be calculated according to the thermal-flow coupling simulation results and engineering cast-rolling force model, as shown in Fig.21.Each process parameter directly affects the temperature field,thus changing the KP height and the average outlet temperature and determining the strain and material deformation resistance, and finally influencing the castrolling force.

Figs.21(a) and (e) show that increasing the molten pool height and substrate radius can significantly increase the cast-rolling force in the single variable condition, mainly because both of them change the heat input and heat output as well as the geometric structure of the cast-rolling area.Figs.21(b), (c), and (d) show that increasing nominal castrolling velocity, cladding casting temperature, and substrate preheating temperature can effectively reduce cast-rolling force in the single variable condition, mainly because of increasing heat input.

Fig.19 Metal plastic flow and velocity isosurface surface under different process layouts.

Therefore, the temperature–pressure state of the bonding interface can be adjusted by adjusting the technical parameters when the product specifications are fixed, which helps achieve process stability control and improve product quality.

5.7.Experimental validation

Based on the reasonable process window obtained by simulation results, preparation experiments of fabricating Cu/steel cladding bars were conducted on the MRSLCRB equipment.The casting rolls were preheated to get to a steady state as quickly as possible.Three kinds of Cu/steel cladding bars with different cladding ratios were successfully prepared with the molten pool height of 30 mm, nominal cast-rolling velocity of 4.5 m/min, casting temperature of 1120 °C, and groove diameter of 25 mm.The wall thickness of the cladding metal is uniform, and there are no macroscopic holes or cracks.From the macroscopic point of view, the bonding interface is continuous and uniform, which preliminarily achieves the expected goal.The sections of prepared Cu/steel cladding bars were segmented by wire cutting, as shown in Fig.22.When there is atmosphere protection, interfacial bonding between the substrate and the cladding can be realized after solid–liquid flexible bonding.Therefore, no separation between the substrate and the cladding occurred under the constraint of the bonding interface.The microstructure of bonded interface is shown in Fig.22(b).There are no obvious microscopic defects.The Energy Dispersive Spectroscopy (EDS) surface scanning result in Fig.22(c) and the EDS line scanning result in Fig.22(d) indicate that the thickness of the diffusion layers is about 1.4 μm.

Fig.20 Cast-rolling force simulation results of one-roll simulation model under different process layouts.

During the experiments, average outlet temperature and cast-rolling force were measured.Comparison results between calculation and experiment are shown in Fig.23.The maximum relative errors of the average outlet temperature and the cast-rolling force are about 5% and 15%, respectively.The temperature field has a significant influence on the KP height and material deformation resistance, and they are closely related to the cast-rolling force.Therefore, the maximum relative error of the cast-rolling force is larger than that of the average outlet temperature.It needs to point out that the test error exists because the product length was limited by the experiment.In general, the experimental results are in agreement with the simulation results, which verifies the accuracy of the analysis.

5.8.Product defect control through cast-rolling force

Usually, the typical lateral ear is inevitable for groove rolling.In the traditional pipe rolling process,the groove is not closed,and the lateral ear is a kind of typical defect, which is formed when the metal is overfilled in the finished groove and the excess metal is squeezed into the roll gap.For the MRSLCRB process, although the groove is closed, the lateral ear can also occur when there is an assembly gap caused by the machining process, as shown in Fig.24(a).The KP height determines the degree of deformation, which in turn affects the lateral ear.When the KP is low, the lateral ear is thin and discontinuous,and it is easy to clean.But when the KP is high,the lateral ear is thick and continuous,and clearing becomes difficult.Therefore,the machining accuracy of casting rolls is very important.To prevent displacement of the groove during the MRSLCRB process, sufficient preload force should be provided.Therefore, the calculation of cast-rolling force is very important for setting preload force.After ensuring machining accuracy and providing sufficient preload force, Cu/steel cladding bars with good cross sections can be prepared, as shown in Fig.24(b).

Furthermore, the KP height is the core of product quality control in the MRSLCRB process, and it is closely related to the cast-rolling force.However, the KP position cannot be measured online due to high temperature and shielding.Hence, it is very important to establish the quantitative relationship between cast-rolling force and KP height to inversely calculate KP height based on the measured cast-rolling force,and then lay the foundation for product quality control.When the substrate metal is a hollow pipe, the MRSLCRB process can be used to prepare bimetallic cladding pipes.For example,Cu/Ti cladding pipe has excellent corrosion resistance of Ti and excellent thermal conductivity of Cu, which can be used as ideal heat exchange pipes for key equipment.

For the MRSLCRB process, the forming mechanism of fabricating bimetallic cladding bars and bimetallic cladding pipes is the same.However, due to the hollow structure, the substrate pipe is prone to be folded and flattened due to large deformation or unequal deformation, as shown in Figs.25(a)and (b), respectively.It will not only cause quality problems but also lead to the interruption of production.The higher the KP, the greater the deformation and the greater the castrolling force.The lower the KP, the smaller the deformation and the smaller the cast-rolling force.Before the experiment,the KP height and cast-rolling force can be predicted according to the preset process parameters, and a reasonable process window can be given.During the experiment, the KP height can be verified indirectly by adjusting the process parameters to control the cast-rolling force.Therefore, through controlling KP height and cast-rolling force, Cu/Ti cladding pipes with round cross-sections were prepared, as shown in Fig.25(c).However, the quantitative relationship between the substrate instability condition and the interfacial bonding strength is still to be further studied, which is critical to developing more cladding specifications and component matching.

Fig.21 Influence of technical parameters on cast-rolling force.

Fig.22 Section of Cu/steel cladding bars after segmentation.

Fig.23 Comparison between calculation and experiment of three-roll layout.

Fig.24 Cu/steel cladding bar section and typical product defects.

Fig.25 Cu/Ti cladding pipe section and typical product defects.

When the substrate metal is the core stranded wire, the MRSLCRB process can be used to prepare the metal cladding core stranded wire, such as brass cladding copper stranded wires.Fig.26 shows the possible states of fabricating brass cladding copper stranded wires by the MRSLCRB process,including interface bonding, interface separation, overfilling of cladding metal, underfilling of cladding metal, extrusion deformation of stranded wires, internal gap of stranded wire,etc.The interaction between substrate metal and cladding metal is determined by the cast-rolling force.The specimen in the cast-rolling area was obtained by emergency stop and quick cooling.The macroscopic section evolution at equal intervals along the rolling direction is shown in Fig.27.

In Section I-I, there are obvious approximately circular casting shrinkage holes, which are caused by the solidification of the cladding metal after the casting is stopped.Because no tension is applied in the experimental process, the core stranded wire is prone to loose.Besides, the cladding metal would flow to the local gap between stranded wires.

In Section II-II, under the small certain deformation, the shrinkage cavity is elliptic or long and has a closing trend.At the same time, the filled cladding metal between stranded wires continues to flow to the depth of the gap under the action of rolling force.

Similarly, in Sections III-III and IV-IV, the shrinkage cavity gradually disappears with the densification deformation.The cladding metal between stranded wires continues to flow to the depth of the gap under the action of rolling force,while the internal gap between stranded wires gradually shrinks.

When reaching Section V-V, a continuous interface is formed between the cladding metal and the stranded wires,and the internal gap between the stranded wires mainly exists in two states.One is the closed boundary, the other is the closed gap.

Fig.26 Section of brass cladding copper stranded wire.

Fig.27 Cross section evolution of cast-rolling area of brass cladding pure copper stranded wire.

In conclusion, for the MRSLCRB process, both the cladding metal and the substrate metal may undergo significant plastic deformation.The KP height is different with different process parameters, that is, the cast-rolling force is different.Therefore, by controlling the KP height and the cast-rolling force, the microstructure of the cladding metal, the bonding effect of the interface,and the geometrical structure of the substrate metal can be controlled.

6.Conclusions

(1) Thermal-fluid coupled simulation results of the threeroll layout indicate that the KP distribution along the circumferential direction can be considered uniform,and an average can be used for analysis.In the single variable condition, the influence of molten pool height,nominal cast-rolling velocity, cladding casting temperature, and substrate preheating temperature on the KP height and the average outlet temperature is linear,and the influence of the substrate radius is nonlinear.Each process parameter shows a monotonic change.

(2) Based on the differential element method and plane deformation hypothesis, the engineering cast-rolling force model was derived and the accuracy was verified by the 3-D finite element model.For the analyzed range,the maximum relative error at different process layouts is about 15%,and that for the three-roll layout at different deformation area heights and equivalent deformation resistances is about 10%.

(3) Each process parameter directly affects the temperature field,thus changing the KP height and the average outlet temperature, and determining the strain and material deformation resistance, and finally influencing the castrolling force.Increasing the molten pool height and substrate radius can significantly increase the cast-rolling force because both of them change the heat input and heat output as well as the geometric structure of the cast-rolling area.Increasing nominal cast-rolling velocity,cladding casting temperature,and substrate preheating temperature can effectively reduce the cast-rolling force, mainly because of increasing heat input.

(4) The 3-D steady-state thermal-fluid coupling simulation model can be used to predict the KP height and average outlet temperature.Then engineering cast-rolling force model can be used to calculate the cast-rolling force.Through the indirect multi-field coupled analysis method,the temperature–pressure evolution and process window can be predicted,which not only provides a significant basis for the equipment design but also lays the foundation to improve product quality.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was co-supported by the National Key Research and Development Program, China (No.2018YFA0707300),the National Natural Science Foundation of China (Nos.51974278 and 52205406), China Post Doctoral Science Foundation (No.2023M732572), the Key Science and Technology Project of Shanxi Province, China (No.20191102009), and the Fundamental Research Program of Shanxi Province,China (No.202203021212289).