Aluminum clad material manufacturing method

Abstract

To provide a manufacturing method of a clad material capable of preventing falling of a skin material.SOLUTION: A manufacturing method of an aluminum clad material by joining by a hot-rolling roll includes: estimating a difference in elongation amount in a rolling direction between a core material and a skin material generated in a joining step by numerical analysis of estimating the elongation of unknown actual hot rolling; assuming that a length in the rolling direction of the core material is not changed; when joining by previously determining a length of the skin material after the joining step so as to become almost the same as a length in the rolling direction of the core material on the basis of a length in the rolling direction of the core material, acquiring a correction factor capable of being approximated to an elongation amount of known actual hot rolling by using an elongation amount estimation value obtained by the same method as the numerical analysis; and when manufacturing the aluminum clad material, determining a length in the rolling direction of the skin material before supplying it to the joining step on the basis of the elongation amount estimation value of the skin material obtained by the same method as the numerical analysis on the basis of the correction factor.SELECTED DRAWING: Figure 1

Description

Technical Field

[0001] The present invention relates to a method for manufacturing an aluminum clad material.

Background Art

[0002] A brazing sheet, which is an example of an aluminum clad material and is used as a constituent material of a heat exchanger, is formed by laminating skin materials on one or both sides of a core material made of aluminum or an aluminum alloy. The core material is obtained by cutting a slab ingot made of aluminum or an aluminum alloy into a specified thickness and shape. Also, the skin material is obtained by hot rolling a slab ingot to a specified thickness and cutting it to an appropriate length. Examples of the skin material include a brazing material and a sacrificial anode material. When manufacturing this type of aluminum clad material, an aluminum alloy skin material having a composition different from that of the core material is integrated with the core material by roll rolling. When manufacturing an aluminum clad material by rolling, if the skin material is more likely to elongate than the core material, the skin material protrudes from the end of the core material during rolling. If this protruding portion is long, the protruding portion falls off the conveyance path of the rolling device, which causes a problem of hindering production.

[0003] In hot rolling of a clad material, in the initial stage, a bonding step of bonding the interfaces of the stacked core material and skin material is performed, and then it proceeds to hot rolling for reducing the plate thickness. Until the interfaces are bonded, since the effect of the core material and the skin material restraining each other's deformation is small, a difference in elongation amount due to the strength difference is likely to occur. Therefore, a low-strength skin material that has elongated more than a high-strength core material protrudes from the rolling direction end face of the core material, and in the subsequent hot rolling for reducing the plate thickness, the protruding skin material may fall off, which may cause production troubles.

[0004] For example, when the material strength of the skin material is lower than that of the core material, the skin material stretches more than the core material. At this time, while the material slides (shifts) at the unjoined interface, the skin material protrudes from the core material. In particular, the material is likely to protrude at the front and rear ends in the rolling direction, and the protruding part may fall off starting from the point where the protruding material hits the edge part of the core material, which may interfere with production. Also, when the low-strength skin material stretches more than the core material, a large difference occurs in the amount of elongation in the rolling direction between the skin material and the core material. The difference in the amount of elongation between the skin material and the core material directly becomes the difference in the amount of change in the thickness of each layer. That is, when the difference in elongation between the skin material and the core material is large, a change in the clad ratio occurs, and the desired clad ratio cannot be obtained. As described above, since the front and rear ends in the rolling direction of the low-strength skin material are likely to protrude, the change in the clad ratio at the front and rear ends is particularly likely to be large. As a result, there is a problem that removal work is required to cut off the front and rear ends where the desired clad ratio cannot be obtained, significantly inhibiting productivity.

[0005] In the technologies described in Patent Document 1 and Patent Document 2 below, a method of suppressing the displacement between the skin material and the core material by installing a net-like object or foil at the interface between the skin material and the core material is disclosed.

Prior Art Documents

Patent Documents

[0006]

Patent Document 1

Patent Document 2

Summary of the Invention

Problems to be Solved by the Invention

[0007] However, in the technologies described in these Patent Documents 1 and 2, the presence of the net-like body or foil installed at the interface becomes an inclusion, causing poor quality of the clad material. Therefore, there is a problem that it cannot be applied to the production of high-quality clad materials in recent years. Furthermore, there are concerns about the labor and cost for separately preparing the net-like body and the sheet. Also, a method of suppressing the displacement and elongation difference at the interface by welding the outer peripheral portion of the interface between the heartwood and the skin material is known, but the cost of the welding process is incurred.

[0008] An object of the present invention is to provide a method for manufacturing an aluminum clad material that can suppress the elongation and protrusion of the skin material from the end of the heartwood during the manufacture of the clad material and prevent the skin material from falling off on the end side of the heartwood without performing welding or the like. Another object of the present invention is to provide a technique capable of efficiently manufacturing an aluminum clad material having an appropriate clad ratio by reducing the amount of cut-off of the skin material by suppressing the amount of the skin material protruding from the end of the heartwood.

Means for Solving the Problems

[0009] (1) The method for manufacturing an aluminum clad material of the present invention joins a laminate in which two or more aluminum or aluminum alloy materials are stacked in a joining step using a hot rolling roll, and continuously rolls the joined body obtained by this joining using the hot rolling roll to manufacture an aluminum clad material in which at least a heartwood and a skin material are joined. In this method, the difference in the elongation amount in the rolling direction between the heartwood and the skin material generated in the joining step is predicted by numerical analysis for predicting the elongation of unknown actual hot rolling. Assuming that the length of the heartwood in the rolling direction does not change, when joining by previously determining the length of the skin material in the rolling direction after the joining step to be substantially equal to the length of the heartwood in the rolling direction with reference to the length of the heartwood in the rolling direction, a correction coefficient that can be approximated using the elongation prediction value obtained by the same method as the numerical analysis with respect to the known actual hot rolling elongation amount is obtained. When manufacturing the aluminum clad material, the rolling direction length of the skin material before being subjected to the joining step is determined based on the elongation prediction value of the skin material obtained by the same method as the numerical analysis based on the correction coefficient.

[0010] (2) In the method for manufacturing an aluminum clad material according to the present invention, when obtaining a correction coefficient α such that the predicted elongation value obtained by the numerical analysis can approximate the measured elongation value obtained by the actual hot rolling, the following formula (1) is used so that the relationship between the actual elongation of the core material and the actual elongation of the skin material by the actual hot rolling and the predicted elongation value approximates each other. It is preferable to obtain the correction coefficient α that minimizes the sum of squared residuals represented by.

[0011]

Number

[0012] However, in formula (1), i represents the sample number, xi represents the predicted value of the corresponding sample number, and yi represents the actual value of the corresponding sample number.

[0013] (3) In the method for manufacturing an aluminum clad material according to the present invention, when obtaining correction coefficients a, b, and c such that the predicted elongation value obtained by the numerical analysis can approximate the measured elongation value obtained by the actual hot rolling, the following formula (2) is used so that the relationship between the actual elongation of the core material and the actual elongation of the skin material by the actual hot rolling and the predicted elongation value approximates each other. It is preferable to obtain the correction coefficients a, b, and c that minimize the sum of squared residuals represented by.

[0014]

Number

[0015] However, in formula (2), i represents the sample number, xi represents the predicted value of the corresponding sample number, yi represents the actual value of the corresponding sample number, and a, b, and c represent correction coefficients.

Advantages of the Invention

[0016] According to the present invention, even when a clad material is manufactured by hot rolling using a skin material with different elongation and changing length for a core material with an unchanged length in the rolling direction in the joining process before hot rolling, a method for manufacturing an aluminum clad material can be provided that suppresses the phenomenon of the skin material extending and protruding from the core material end and prevents the skin material from falling off on the core material end side. Further, according to the present invention, by suppressing the amount of the skin material protruding from the core material end, a method for manufacturing an aluminum clad material can be provided that reduces the amount of trimmed skin material and efficiently manufactures an aluminum clad material with an appropriate clad ratio.

Brief Description of the Drawings

[0017]

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Embodiments for Carrying Out the Invention

[0018] Hereinafter, an example of an embodiment of the present invention will be described in detail with reference to the accompanying drawings. Note that the drawings used in the following description may show, for the sake of clarity of the features, the characteristic parts enlarged for convenience. FIG. 1 shows an example of the material arrangement employed in the first rolling pass when implementing the method for manufacturing a clad material according to the present invention. Between hot rolling rolls (work rolls) 1 and 2 spaced apart vertically, a clad material 6 having a core material 3 and skin materials 4 and 5 is arranged. FIG. 1 shows a side view of the hot rolling rolls 1 and 2, and shows a side view state in which the conveyance direction of the clad material 6 to be rolled by the hot rolling rolls 1 and 2 extends in the left-right direction. The core material 3 is obtained by cutting a slab ingot made of aluminum or an aluminum alloy into a specified thickness and shape. Also, the skin materials 4 and 5 are obtained by hot rolling a slab ingot to a specified thickness and cutting it to an appropriate length. Examples include brazing materials and sacrificial anode materials.

[0019] In this embodiment, it is assumed that the high-temperature deformation resistance (MPa) of the core material 3 is greater than that of the skin materials 4 and 5, and it is assumed that the high-temperature deformation resistance (MPa) of the skin material 4 is smaller than that of the skin material 5. Based on these assumptions and the prerequisite conditions, the case of forming a clad material by hot rolling will be described. Note that in the state shown in FIG. 1, the core material 3 and the skin materials 4 and 5 are installed so as to be conveyable along a conveyance path (not shown in the figure) arranged before and after (in the left-right direction in FIG. 1) the hot rolling rolls 1 and 2. FIG. 1 shows the state before the start of rolling, and in this state before the start of rolling, the core material 3 and the skin materials 4 and 5 are shown in an unjoined stacked state before being joined.

[0020] The arrangement shown in FIG. 1 (starting pattern 1) shows the case where the hot rolling rolls 1 and 2 are arranged near the center in the length direction of the clad material 6 (near the center in the length direction of the core material 3). In the case of the starting pattern 1 shown in Fig. 1, assume that the distance from the center in the longitudinal direction of the core material 3 to the left end in the longitudinal direction of the core material 3 is a, and the distance from the center in the longitudinal direction of the core material 3 to the right end in the longitudinal direction of the core material 3 is b. In the following description, assume that the core material 3 is thicker than the skin materials 4 and 5, and the skin materials 4 and 5 have the same thickness. Assume that for the core material 3, both the skin materials 4 and 5 have a small high-temperature deformation resistance at the hot rolling temperature, and the high-temperature deformation resistance of the skin material 4 is smaller than that of the skin material 5.

[0021] In this embodiment, since the high-temperature deformation resistance of the core material 3 is greater than that of the skin materials 4 and 5, in the bonding process which is the initial stage of hot rolling, the elongation amounts of the skin materials 4 and 5 are larger than the elongation amount of the core material 3. Also, when the strength of the core material 3 is higher than that of the skin materials 4 and 5, in the bonding process, since rolling is performed in an unbonded state, the core material 3 hardly deforms. Therefore, in order to prevent the peeling at the ends accompanying the elongation of the skin materials 4 and 5 during the subsequent hot rolling, as shown in Fig. 1, at the stage of the clad material 6, the skin materials 4 and 5 are made shorter than the core material 3 by a predetermined length in advance. In Fig. 1, the hot rolling rolls 1 and 2 are arranged near the center in the longitudinal direction of the clad material 6, and the clad material 6 is hot rolled while being fed leftward along the longitudinal direction while narrowing the interval between the hot rolling rolls 1 and 2 from the state shown in Fig. 1, or while being fed rightward and performing hot rolling. For this reason, in the core material 3 of Fig. 1, a portion where the skin materials 4 and 5 do not exist, indicated by the length a', is provided at the left end, and a portion where the skin materials 4 and 5 do not exist, indicated by the length b', is provided at the right end.

[0022] When the clad material 6 is fed leftward along the longitudinal direction with respect to the hot rolling rolls 1 and 2 and the right half in the longitudinal direction of the clad material 6 is hot rolled, next, the clad material 6 is fed rightward in the longitudinal direction and the left half in the longitudinal direction of the clad material 6 is hot rolled. When the clad material 6 is fed rightward along the longitudinal direction with respect to the hot rolling rolls 1 and 2 and the left half in the longitudinal direction of the clad material 6 is hot rolled, next, the clad material 6 is fed leftward in the longitudinal direction and the right half in the longitudinal direction of the clad material 6 is hot rolled.

[0023] Through the hot rolling in the first pass described above, a joining process is performed over the entire length of the clad material 6 to closely adhere the skin materials 4 and 5 to the core material 3. After that, hot rolling is performed for the required number of passes to obtain an aluminum clad material 10 in which the skin materials 8 and 9 are integrated with the core material 7 as shown in FIG. 3. In the aluminum clad material 10 in which the skin materials 8 and 9 are integrated with the core material 7, when the target clad ratio is the thickness (A) of the skin material 8: the thickness (B) of the core material 7: the thickness (C) of the skin material 9, for example, it can be 20%, 70%, 10%, etc. The target clad ratio can also be, for example, 30%, 60%, 10%, etc. When starting the hot rolling in the first pass from the starting pattern 1 shown in FIG. 1, it is preferable to have the relationship of a:b≒a’:b’.

[0024] When performing hot rolling according to the starting pattern shown in FIG. 1, numerical analysis is applied to predict how much the skin materials 4 and 5 will elongate in the length direction as described below, and a predicted value is calculated. First, as a first prerequisite, it is assumed that the length in the rolling direction of the core material and the skin material before being subjected to the joining process is determined so that the length in the rolling direction of the skin material 8 or 9 after the joining process is substantially equal to the length in the rolling direction of the core material 7 after the joining process. Here, “substantially equal” indicates the difference within the range where no peeling of the skin material occurs in the hot rolling following the joining process, the amount of trimming of the skin material after hot rolling is reduced, and an appropriate clad ratio can be obtained. The difference between the length in the rolling direction of the skin material after the joining process and the length in the rolling direction of the core material after the joining process is preferably 100 mm or less for preventing peeling of the skin material and obtaining an appropriate clad ratio. Regarding the above-mentioned difference, more preferably, it is 50 mm or less, and the amount of trimming of the skin material after hot rolling can be further reduced. It is obvious that the closer the difference is to 0 mm, the more preferable it is. The reason for setting the difference described above to 100 mm or less is that when manufacturing an aluminum clad material in an actual hot rolling facility, if the skin materials 8 and 9 extend with respect to the core material 7 and the skin materials 8 and 9 protrude from the end face of the core material 7, as long as the protrusion length is 100 mm or less, there is no risk of dropping or the like occurring in the conveying path. Also, considering the case where the skin materials 8 and 9 are shorter than the core material 7 after hot rolling, in that case, it is premised that the difference, assuming that the skin materials 8 and 9 are shorter than the length of the core material 7, is 100 mm or less. Considering both the case where the skin materials 8 and 9 extend with respect to the core material 7 and the skin materials 8 and 9 protrude from the end face of the core material 7, and the case where the skin materials 8 and 9 are shorter than the core material 7 and the difference is 100 mm or less, being substantially equal means that the difference between the rolling direction length of the skin material after the joining process and the rolling direction length of the core material after the joining process is within ±100 mm.

[0025] In this embodiment, for the predicted elongation amount value of the skin material obtained by numerical analysis, the actual elongation amount of the skin material by actual hot rolling using the core material and the skin material is obtained, and a correction coefficient α is obtained so that the predicted elongation amount value obtained by numerical analysis can be approximated to the actually measured elongation amount obtained by actual hot rolling. Next, when manufacturing the aluminum clad material 10, the corrected predicted lengths of the skin materials 8 and 9 obtained by multiplying the predicted elongation amount value by the correction coefficient α are obtained. When the corrected predicted lengths of the skin materials 8 and 9 are obtained, it is possible to grasp how much elongation each of them will have by hot rolling. By obtaining the predicted length of the skin material 8 and taking the difference from the initial length of the core material 7, the predicted elongation amount of the skin material 8 with respect to the core material 7 can be known. Details of the calculation method of the correction coefficient α will be described later.

[0026] Therefore, as shown in FIG. 1, at the stage before hot rolling, it is preferable to use the skin material 4 that is shorter in length than the core material 3 by the amount of elongation that will occur in the subsequent joining process. In the case of FIG. 1, a'+b' corresponds to the predicted elongation amount. Also, if the skin materials 4 and 5 are made of the same aluminum alloy and have the same thickness, the skin material 5 may have the same length as the skin material 4. However, when different aluminum alloys are used or the thicknesses are different, the predicted length of the skin material 5 with respect to the core material 3 can be obtained by the same method as described above, and the length of the skin material 5 before hot rolling can be determined.

[0027] In this embodiment, as an example, when calculating the predicted elongation value in the joining process using numerical analysis (in the case of analytical hot rolling), the initial total plate thickness is set in the range of 450 to 650 mm, the clad ratio of the skin materials 8 and 9 with respect to the core material 7 is set in the range of 5 to 30%, and the length of the initial core material can be set in the range of 3000 to 5000 mm. It is preferable that the initial lengths of the skin materials 8 and 9 and the core material 7 are aligned. Also, as an example of the analysis conditions, the reduction amount is set to 20 mm, and the actual rolling and the analysis conditions are made to match.

[0028] The method of numerical analysis is not particularly limited, but a general-purpose non-linear structural analysis software using the finite element method is suitable because it can be easily calculated for various materials to be rolled. In this embodiment, as an example of numerical analysis, an elastoplastic finite element method software for structural analysis (LS-DYNA, Ver R10.2.0, manufactured by Livermore Software Technology Corporation (LSTC)) can be used. Note that the method of numerical analysis used here may be an elementary analysis such as the slab method. Also, when using the elastoplastic finite element method for structural analysis, since it is necessary to consider plastic deformation for the core material 3 and the skin materials 4 and 5, a rigid-plastic body or an elastoplastic body can be used. Since the elastic deformation amount in the elongation of the skin material is minute and can be ignored, it is more preferable to use a rigid-plastic body from the viewpoint of calculation time. In this embodiment or the examples described later, an elastoplastic body was used. The hot rolling rolls 1 and 2 are assumed to be rigid bodies that do not deform at all, and as the hot deformation resistance curves of the core material 3 and the skin materials 4 and 5, the hot deformation resistance curves obtained by hot compression tests at 480 °C and a strain rate of 1 / s for each material of the core material 3 and the skin materials 4 and 5 are applied. In addition, the element type in the elastoplastic finite element method software for structural analysis is not particularly limited as long as it is a so-called solid element that can consider triaxial stresses. In plate rolling, since the amount of strain in the plate width direction is negligibly small compared to the amounts of strain in the length direction and the plate thickness direction, it is also possible to use a so-called plane strain element that does not consider the strain in the plate width direction. In this embodiment, among the solid elements implemented in LS-DYNA described above, a fully integrated S / R solid element is used, and the plate width directions of the core material 3 and the skin materials 4 and 5 are displacement-restrained to assume a plane strain state in which the deformation in the plate width direction is ignored.

[0029] Next, it is assumed that the core material 3 and the skin materials 4 and 5 do not shift relative to each other in the initial biting region where the hot rolling rolls 1 and 2 first bite the core material 3 and the skin materials 4 and 5. For the other regions of the core material 3 and the skin materials 4 and 5 excluding the initial biting region, the contact analysis between the core material 3 and the skin materials 4 and 5 is calculated by the penalty method assuming Coulomb friction, and the friction coefficient used in the penalty method can be obtained by a friction behavior evaluation test at the hot rolling temperature. Note that the actual rolling method assumed in the present invention is to perform the first-pass rolling from a position in the middle of the clad material (slab) 6 as shown in FIGS. 1 and 2, and to perform rolling with the skin materials 4 and 5 shorter than the core material 3. On the other hand, in the skin material elongation prediction analysis method of the embodiment, for the sake of simplifying the prediction analysis, a bonding condition of performing rolling in one pass over the entire length of the clad material (slab) 6 from one side to the opposite side is adopted. At this time, the end on the side that first bites is set as the initial biting region, and the numerical analysis is performed after simplification by aligning the initial lengths of the skin materials 4 and 5 and the core material 3.

[0030] In this embodiment, the contact condition at the interface between the core material 3 and the skin materials 4 and 5 assumes Coulomb friction, but it is not limited thereto, and definitions such as shear friction can also be used. In Coulomb friction, a value of about 0.2 can be used as the Coulomb friction coefficient, and in shear friction, a value of about 0.9 can be used as the shear friction coefficient. Although the predicted elongation value of the skin material changes depending on the value of the friction coefficient, since the correction coefficient α is finally used to correct so that the actual elongation amount of the skin material in actual rolling and the elongation prediction result are well approximated, here, it is not necessary to exactly match the friction coefficient with the actual phenomenon. In this embodiment, as the contact condition in which the Coulomb friction coefficient can be set, CONTACT_AUTOMATIC_SURFACE_TO_SURFACE implemented in the above-mentioned LS-DYNA was selected and used.

[0031] Next, a correction coefficient α is obtained so that the predicted elongation value obtained by the non-linear structural analysis software can approximate the actually measured elongation value obtained by the hot rolling test. Based on the actual results of the elongation amounts of the core material 3 and the skin materials 4 and 5 obtained by the hot rolling test and the predicted elongation values, the correction coefficient α that minimizes the sum of squared residuals represented by the following formula (1) so that they approximate each other can be obtained by an optimization calculation.

[0032]

Equation

[0033] However, in formula (1), i indicates the sample number, xi indicates the predicted value of the corresponding sample number, and yi indicates the actual value of the corresponding sample number. Note that in formula (1), the term (xi×α) means correction by first-order integration, but the form of the correction formula is not limited to the correction by the above-mentioned first-order integration. For example, second-order integration or shifting using an intercept can also be considered. For example, regarding the above-mentioned formula for the sum of squared residuals, a quadratic polynomial of the following formula (2) can be used instead of formula (1). For the above-mentioned formula for the sum of squared residuals, other polynomials, exponential formulas, logarithmic formulas, etc. can be appropriately used.

[0034]

Number

[0035] However, in formula (2), i represents the sample number, xi represents the predicted value of the corresponding sample number, yi represents the actual value of the corresponding sample number, and a, b, and c represent correction coefficients.

[0036] Figure 4 shows a graph with the vertical axis showing the actual value of the elongation of the skin material when hot-rolling the clad material using an actual hot-rolling apparatus, and the horizontal axis showing the predicted value of the elongation calculated by the numerical analysis described above so as to correspond to the case where the actual value shown on the vertical axis is obtained. The solid line drawn diagonally in Figure 4 is a straight line where the value on the vertical axis and the value on the horizontal axis are 1:1 (that is, the predicted value and the actual value coincide), and the purpose is to correct the value deviated from the solid line in Figure 4. When graphing the relationship shown in Figure 4, the actual value of the elongation when manufacturing the clad material by hot-rolling using an actual hot-rolling apparatus and the predicted value of the elongation obtained by the above-mentioned non-linear structure analysis software deviate. In the example of Figure 4, the predicted value of the elongation by the numerical analysis before correction is larger than the actual value of the elongation, and the predicted value before correction overestimates the elongation amount of the skin material compared to the actual phenomenon. Therefore, the correction coefficient α that minimizes the above-mentioned sum of squared residuals is obtained so that the values of both approximate each other.

[0037] Using Figure 5, the method for determining the correction coefficient α will be described more specifically. For a plurality of clad materials A to G..., the actual value y of the elongation of the skin material in the bonding rolling is measured. For these clad materials, the elongation amount is predicted by numerical analysis to obtain the predicted value x of the elongation amount. Here, since the predicted value x of the elongation amount has not been corrected so as to approximate the actually measured value of the elongation amount, as exemplified in Figure 4, the elongation amount of the skin material cannot be accurately predicted, so it is necessary to determine the correction coefficient α. Therefore, assuming the correction coefficient α is 1, a value x×α obtained by multiplying the prediction value x by the correction coefficient α is calculated for the clad materials A to G.... Here, since α is assumed to be 1, the value of x×α coincides with the elongation prediction value x. Next, the square of the difference between the actual elongation value y and the predicted value x×α multiplied by the correction coefficient α is calculated for the clad materials A to G.... The sum of the values of {y-(x×α)} calculated for the clad materials A to G.... is calculated. 2 The sum of these values is calculated. Since the sum of squared residuals calculated in this way increases or decreases with respect to the value of α, if α is varied and the sum of squared residuals is calculated, the α that minimizes the sum of squared residuals can be determined. The predicted value x×α of the corrected skin material elongation multiplied by the correction coefficient α obtained by this method closely approximates the actual skin material elongation y.

[0038] Based on the above numerical analysis, if the predicted elongation amounts (mm) of the core material 3, the skin materials 4 and 5 are calculated, then according to the calculation results of the predicted elongation amounts, the lengths of the skin materials 4 and 5 are set shorter in advance so as to cancel the calculation results. For example, assuming that the core material 3 hardly elongates and the length of the core material is 3500 mm, if it is predicted that the skin material 4 elongates about 350 mm and the skin material 5 elongates 380 mm, then the length of the core material is set to 3500 mm, the skin material 4 is set to 3150 mm, and the skin material 5 is set to 3120 mm. Note that, based on the calculation result of the predicted elongation amount, the lengths of the skin materials 4 and 5 can be set shorter in advance so that the core material 7, the skin material 8, and the skin material 9 all have the same length after hot rolling. However, in this embodiment, cases where the skin materials 8 and 9 are slightly too long or too short with respect to the core material 7 after hot rolling are also allowed. In the hot rolling following the joining process, within the range where the skin material does not fall off, and in the range where the amount of trimming of the skin material after hot rolling is reduced and an appropriate clad ratio can be obtained, the difference between the length in the rolling direction of the skin material after the joining process and the length in the rolling direction of the core material after the joining process is preferably 100 mm or less in order to prevent the skin material from falling off and obtain an appropriate clad ratio. Therefore, as a rough guideline in this embodiment, if the difference in the length in the rolling direction between the skin materials 8 and 9 and the length in the rolling direction of the core material 7 after the joining process is 100 mm or less, it is determined that a good joining result is obtained. More preferably, the difference is 50 mm or less, and the amount of trimming of the skin material after hot rolling can be further reduced. However, it is obvious that the closer the difference is to 0 mm, the more preferable it is. If the skin materials 8 and 9 are shorter than the core material 7 by more than 100 mm after hot rolling, a large amount of wasted unclad portions will occur, resulting in a decrease in yield. If the skin materials 8 and 9 are longer than the core material 7 by more than 100 mm after hot rolling, the protruding portions of the skin materials 8 and 9 may fall off onto the conveying path, which may hinder productivity.

[0039] As described above, the elongation amounts of the skin materials 4 and 5 are calculated and predicted by numerical analysis. As a result of the elongation of the skin materials 4 and 5, the lengths of the core material 3 and the skin materials 4 and 5 are approximately aligned after the first-pass hot rolling, or the rolled skin materials 8 and 9 are slightly shorter or slightly longer than the core material 7. After adjusting the lengths of the skin materials 4 and 5, the first-pass hot rolling is performed. By this first-pass hot rolling, the skin materials 4 and 5 can be adhered to the core material 3. In the first-pass hot rolling, even if there are some errors, since the skin material is shortened in advance according to the prediction by the above numerical analysis, the difference in length between the core material 3 and the skin materials 4 and 5 after hot rolling can be suppressed shorter than before. Therefore, even if the skin materials 4 and 5 protrude from the longitudinal ends of the core material 3, the amount of protrusion can be significantly reduced compared to the prior art, resulting in prevention of the detachment of the skin material ends. Further, even if the skin materials 4 and 5 become shorter than the core material 3 after hot rolling, the amount of shortening can be reduced, so that the amount of the core material 3 that is cut off and wasted after hot rolling can be reduced.

[0040] By subjecting the rolled material after the first-pass hot rolling to hot rolling for the required number of passes from the second pass onward, a clad material 10 having the structure shown in FIG. 3 with the target thickness can be obtained. If the skin materials 4 and 5 are brought into close contact with the core material 3 by the first-pass hot rolling, then in the hot rolling from the second pass onward, the skin materials 4 and 5 are hot rolled with almost no sliding (displacement) at the interface with the core material in their longitudinal directions, and a clad material with the target clad ratio can be obtained.

[0041] When the hot deformation resistance of the core material 3 is greater than that of the skin materials 4 and 5, it can be assumed that the core material 3 hardly elongates in the joining process. When comparing the skin materials 4 and 5, there are cases where the hot deformation resistance of the skin material 5 is greater than and less than that of the skin material 4. When comparing the skin materials 4 and 5 and the hot deformation resistance of the skin material 5 is greater than that of the skin material 4, the elongation amounts of the skin materials 4 and 5 change depending on the magnitude of that value. In the above numerical analysis, the predicted elongation amount value changes according to the degree to which the hot deformation resistance of the skin material 5 is greater than that of the skin material 4.

[0042] Note that since the clad ratios in the clad materials to be manufactured vary, the individual elongation amounts of the skin materials 4 and 5 will differ depending on what values the clad ratios of the skin materials 4 and 5 are respectively. In the above numerical analysis, this relationship can also be considered to accurately predict the elongation amount.

[0043] By starting the hot rolling process from the start pattern 1 shown in FIG. 1, rolling can be performed while avoiding a large displacement that occurs when the skin materials 4 and 5 are bitten in at the stage where the work rolls 1 and 2 bite in the skin materials 4 and 5 to start hot rolling. In addition, when controlling hot rolling by the load applied to the clad material 6 from the work rolls 1 and 2, rolling control can be performed with an accurate load. Therefore, as a final product, as shown in FIG. 3, an aluminum clad material 10 in a state where the skin materials 8 and 9 are surely adhered to the core material 7 can be obtained. In the hot rolling following the joining process, the protrusion amounts of the skin materials 8 and 9 from the longitudinal end portions of the core material 7 are reduced, and the clad material 10 having the target clad ratio can be manufactured while preventing the skin material from peeling off when the skin materials 8 and 9 protrude from the core material 7.

[0044] FIG. 2 shows another example (starting pattern 2) of the material arrangement employed in the first rolling pass when implementing the method for manufacturing a clad material according to the present invention. A clad material 6 including a core material 3 and skin materials 4 and 5 is disposed between the work rolls 1 and 2 that are vertically spaced apart. In this example of the material arrangement, it is assumed that the high-temperature deformation resistance (MPa) of the core material 3 is greater than the high-temperature deformation resistance (MPa) of the skin materials 4 and 5, and it is assumed that the high-temperature deformation resistance (MPa) of the skin material 4 is smaller than the high-temperature deformation resistance (MPa) of the skin material 5. Based on these assumptions, the various conditions such as when forming a clad material by rolling are the same as those in the previous example.

[0045] The starting pattern 2 shown in FIG. 2 indicates that the work rolls 1 and 2 are disposed on the length end side of the clad material 6 (the longitudinal end side of the core material 3). In the case shown in FIG. 2, assume that the distance from the left end in the longitudinal direction of the core material 3 to the position where the work rolls 1 and 2 are disposed is a, and the distance from the right end in the longitudinal direction of the core material 3 to the position where the work rolls 1 and 2 are disposed is b. In the following description, the thickness relationship between the core material 3 and the skin materials 4 and 5 and the relationship between the high-temperature deformation resistances of the core material 3 and the skin materials 4 and 5 are also equivalent to those in the previous example.

[0046] In order to prevent the peeling off at the ends due to the elongation of the skin materials 4 and 5 during hot rolling, as shown in Fig. 2, at the stage of the clad material 6, the skin materials 4 and 5 are made shorter than the core material 3 in advance. In Fig. 2, the work rolls 1 and 2 are arranged on the left end side in the length direction of the clad material 6, and hot rolling is performed while feeding the clad material 6 leftward along the length direction while narrowing the interval between the work rolls 1 and 2 from the state shown in Fig. 2. For this reason, in the core material 3 of Fig. 2, a portion where the skin materials 4 and 5 do not exist, indicated by the length a', is provided at the left end portion, and a portion where the skin materials 4 and 5 do not exist, indicated by the length b', is provided at the right end portion.

[0047] When the clad material 6 is fed leftward along the length direction with respect to the work rolls 1 and 2 and most of the right side in the length direction of the clad material 6 is hot rolled, next, the clad material 6 is fed rightward in the length direction, and the left end side in the length direction of the clad material 6 is hot rolled. By performing hot rolling over the entire length of the core material 3 by the above first-pass hot rolling, the skin materials 4 and 5 are brought into close contact with the core material 3, and after performing hot rolling the required number of times, a clad material 10 with the target thickness in which the skin materials 8 and 9 are integrated with respect to the core material 7 can be obtained as shown in Fig. 3.

[0048] Even in the case of the start pattern 2 shown in Fig. 2, rolling can be performed while avoiding a large deviation that occurs when the work rolls 1 and 2 bite into the skin materials 4 and 5 at the start of hot rolling. For this reason, when controlling hot rolling by applying a load from the work rolls 1 and 2 to the clad material 6, accurate rolling control can be achieved while avoiding deviation.

[0049] Even in the case of the start pattern 2 shown in Fig. 2, similar to the case where hot rolling is started from the start pattern 1 shown in Fig. 1 above, hot rolling with a reduced elongation amount of the skin materials 4 and 5 can be performed, and while preventing the peeling off of the ends of the skin materials 4 and 5, a clad material 10 with the target clad ratio can be obtained.

[0050] In the method for manufacturing the aluminum clad material according to the embodiment described above, on the premise that the initial total plate thickness is set in the range of 450 to 650 mm, the clad ratio of the skin material is set in the range of 5 to 30%, and the length of the initial core material is set in the range of 3000 to 5000 mm, as the non-linear structural analysis software, elastoplastic finite element method software for structural analysis can be used. Also, in the elastoplastic finite element method software for structural analysis, it is assumed that the core material and the skin material are elastoplastic bodies and the rolling rolls are rigid bodies that do not deform at all. As the hot deformation resistance curves of the core material and the skin material, the hot deformation resistance curves obtained by hot compression tests at 480 °C for each material of the core material and the skin material can be applied. Also, as the element type, a fully integrated S / R solid element can be used, and six or more integration points in the thickness direction of the core material and the skin material can be set, and it can be assumed that the width direction is displacement-restrained and the plane strain state is the restraint condition. However, when the skin material is clad on both sides of the core material, the clad ratio of one skin material = the thickness of one skin material / (the thickness of one skin material + the thickness of the core material + the thickness of the other skin material). Note that the element type is not limited to the fully integrated S / R solid element described above, and shell elements or the like may be used, and the numerical analysis is not limited to the elastoplastic finite element method for structural analysis, and prediction may be performed using elementary solution methods or slab methods.

[0051] Also, in the calculation of the predicted elongation amount using the numerical analysis described above, it is assumed that the core material and the skin material in the initial biting region where the rolling roll first bites the core material and the skin material do not shift relative to each other in the rolling direction, and in the other regions of the core material and the skin material excluding the initial biting region, the contact force between the core material and the skin material is calculated by the penalty method, and it is preferable to obtain the friction coefficient used in the penalty method by a friction behavior evaluation test at the hot rolling temperature.

Example

[0052] As shown in Table 1 below, for each of the clad materials No. A to E shown in Table 1 having a three-layer structure of a skin material (brazing material) A made of an Al-Si alloy, a core material B made of an Al-Zn alloy, and a skin material (sacrificial material layer) C made of an Al-Mn alloy, the following analysis was performed.

[0053] Assume that hot rolling is started from the starting pattern 1 shown in Fig. 1, and assume that the skin material A, the core material B, and the skin material C have the hot deformation resistance (MPa) shown in Table 1 below and the initial longitudinal dimensions (mm) shown in Table 1 below. When the target clad ratio (%) of the skin material A, the core material B, and the skin material C shown in Table 1 is used, the elongation prediction results (mm) based on the finite element method described in the embodiment are shown in Table 1 below. The total thickness of the clad material including the skin material A, the core material B, and the skin material C was set to 600 mm, and the reduction amount by the work roll was set to 20 mm. The prediction based on the finite element method is based on the calculation method of the finite element method based on the parameter settings detailed in the embodiment above.

[0054] In addition, the calculation results of Sample No. B applying the core material D made of Al-Zn alloy to Table 1 and the elongation prediction results of Sample No. C with different initial lengths for Sample No. A are shown. The elongation prediction results of Sample No. D with a three-layer structure using the skin material (brazing material) E made of Al-Si alloy, the skin material (sacrificial material) F made of Al-Mn alloy, and the core material B made of Al-Zn alloy in Table 1 are shown. The elongation prediction results of Sample No. E with a three-layer structure using the skin material A, the core material B, and the skin material C in Table 1, and the target clad ratio thereof set to 30%:60%:10% are shown.

[0055]

Table 1

[0056] As shown in Table 1, the case where hot rolling is carried out with the initial longitudinal dimension set to a predetermined value of skin material A:core material B:skin material C = 3500 mm:3500 mm:3500 mm in Sample No. A will be described. As shown in Table 1, regarding the sample of No. A, when calculating the elongation prediction value by applying the finite element method described in the embodiment, it was possible to calculate that the skin material A: core material B: skin material C = 225 mm: 0 mm: 240 mm. From this calculation result, it is estimated that for the core material B, the skin materials A and C have a large elongation of about 225 to 240 mm, and it is predicted that the skin materials A and C protrude about 125 to 140 mm from the end of the core material B.

[0057] Therefore, in the present invention, for the purpose of absorbing 225 mm and 240 mm of the elongation prediction results shown in Table 1 calculated based on the above-mentioned finite element method, the initial longitudinal dimensions were set as skin material A: core material B: skin material C = 3275 mm: 3500 mm: 3260 mm as shown in the sample of No. A in Table 2 below. Also, the initial clad ratio is skin material A: core material B: skin material C = 21.2%: 68.0%: 10.8%, and the overall thickness is set to 600 mm. The method for adjusting the initial clad ratio is to set the clad ratio so that when the predicted elongation occurs, the clad ratio does not exceed ±1.5% of the target. Also, since the volume does not change according to the law of conservation of mass, the decrease amount of the clad ratio is calculated from the elongation amount. At this time, it is assumed that there is no variation in the clad ratio and it is uniform. For the sample of No. A, hot rolling was performed under the condition of a reduction amount of 20 mm by the first-pass hot rolling. The results are shown in Table 2 below.

[0058]

Table 2

[0059] As shown in Table 2, the actual elongation amounts were skin material A: core material B: skin material C = 215 mm: 0 mm: 255 mm. As a result, the lengths after the first-pass hot rolling were skin material A: core material B: skin material C = 3490 mm, 3500 mm, 3515 mm, respectively, and in all cases, the skin material A was 10 mm shorter than the core material B, and the skin material C was 15 mm longer than the core material B. This means that even if the end of the skin material C protrudes from the end of the core material B, the protrusion amount is as small as 15 mm, and the skin material A is 10 mm shorter, so no peeling of the skin material occurred on the end side of the core material. In addition, the final clad ratio of the product was skin material A: core material B: skin material C = 19.3%, 71.6%, 9.2%, which was within the range of ±1.5% with respect to the target clad ratio, and a good clad ratio was obtained.

[0060] In contrast, as shown in Sample A' in Table 3 below, when hot rolling in the first pass was performed under the conditions of skin material A: core material B: skin material C = 3400 mm, 3500 mm, 3400 mm, the actual elongation amounts were 230 and 240 mm, the protrusion amounts were 130 - 140 mm, skin material peeling occurred, and the clad ratio also exceeded ±1.5% with respect to the target clad ratio. Note that the final clad ratio (%) of the product shown in each table means the clad ratio of the product after performing rolling by the normal hot rolling method after the first pass of hot rolling so that the final total plate thickness after hot rolling is in the range of 5 - 30 mm.

[0061]

Table 3

[0062]

Table 4

[0063] Table 4 and Figure 6 show the values when the correction coefficient α is obtained based on the predicted value x, actual value y, predicted value x×α, and {actual value y - (predicted value x×α)} from a plurality of samples as shown in Figure 5 when analyzing Samples No. A - E. In this example, the correction coefficient α is applied to all of Samples A - E. 2 The values of are shown. In this embodiment, the correction coefficient α is applied to all of Samples A - E. The relationship between the predicted value and the actual value shown in Table 4 can be displayed in a graph with the vertical axis (actual value) and the horizontal axis (predicted value, predicted value×α) as shown in Figure 6. The actual value indicates the value when the sample is rolled under the condition of a reduction amount of 20 mm by the actual hot rolling apparatus. The correction coefficient α obtained from the relationship shown in Table 4 and Figure 6 is 0.6299995, but it is judged to be approximately 0.63. As shown as an example in Table 5 below, when the analysis result of Sample No. A is 355, multiplying by 0.63 gives an elongation prediction result of 225 mm, and when the analysis result of Sample No. A is 380, multiplying by 0.63 gives an elongation prediction result of 240 mm.

[0064]

Table 5

[0065] Next, as shown in the sample of No. B in Table 1, predictions were made when a sample with a high-temperature deformation resistance of 30 MPa was applied as the core material D. The skin materials A and C are the same as those of the sample of No. A. As shown in the sample of No. B, when the initial longitudinal dimensions are skin material A: core material D: skin material C = 3500 mm, 3500 mm, 3500 mm, the predicted elongation amounts of the skin materials A and C are 225 mm and 240 mm shown in Table 1. Therefore, it is expected that the skin materials A and C will protrude a considerable amount from the end of the core material D, similar to the case of the sample of No. A. Therefore, as shown in the sample of No. B in Table 2, when the initial longitudinal dimensions are set to skin material A: core material D: skin material C = 3275 mm: 3500 mm: 3260 mm, the actual elongation amounts are skin material A: core material D: skin material C = 225 mm: 0 mm: 235 mm. In the case of the sample of No. B in Table 2, as shown in Table 2, no skin peeling occurred, and the final clad rate of the product was also good.

[0066] On the other hand, when hot rolling in the first pass is performed under the conditions of skin material A: core material D: skin material C = 3400 mm, 3500 mm, 3400 mm as shown in the sample of No. B' in Table 3, the actual elongation amounts of the skin materials A and C are 210 mm and 275 mm, and the protrusion amounts become as large as 110 mm and 175 mm, resulting in skin peeling, and the clad rate also exceeded ±1.5% of the target clad rate, causing variations.

[0067] Next, as shown in the sample No. C in Table 1, although the high-temperature deformation resistance is equivalent to that of the sample No. A in Table 1, prediction by the finite element method described in the embodiment was performed for the sample with the initial length of skin material A: core material B: skin material C set to 5000 mm: 5000 mm: 5000 mm. As shown in the sample No. C in Table 1, since the predicted elongation amounts of skin material A and skin material C are as large as 330 mm and 355 mm, skin materials A and C will protrude by a considerable amount from the end of core material B, similar to the case of the sample No. A. Therefore, as shown in the sample No. C in Table 2, when the initial longitudinal dimensions are set to skin material A: core material B: skin material C = 4670 mm: 5000 mm: 4645 mm, the actual elongation amounts are skin material A: core material B: skin material C = 335 mm: 0 mm: 355 mm. In the case of the sample No. C in Table 2, as shown in Table 2, no skin peeling occurred, and the final clad rate of the product was also good.

[0068] On the other hand, as shown in the sample C' in Table 3, when hot rolling in the first pass is performed under the conditions of skin material A: core material B: skin material C = 4800 mm, 5000 mm, 4800 mm, the actual elongation amounts of skin material A and skin material C are 385 mm and 430 mm, respectively, and the protrusion amounts are as large as 185 mm and 230 mm. Skin peeling occurred, and the clad rate exceeded the target clad rate by ±1.5%, resulting in variations.

[0069] Next, as shown in the sample No. D in Table 1, skin materials E and F were provided for the sample No. A in Table 1, and a sample with a large difference in high-temperature deformation resistance was applied. Based on the conditions of the same initial length and the same target clad rate as the sample No. A, prediction of the elongation amount by the finite element method described in the embodiment was performed. As shown in the sample No. D, the predicted elongation amount is 340 mm in skin material E, and skin material E will protrude by a considerable amount from the end of core material B. Therefore, as shown in the sample D in Table 2, when the initial longitudinal dimensions are set to skin material E: core material B: skin material F = 3160 mm: 3500 mm: 3495 mm, the actual elongation amounts are skin material E: core material B: skin material F = 360 mm: 0 mm: 5 mm. In the case of the sample No. D, as shown in Table 2, no skin peeling occurred, and the final clad rate of the product was also good.

[0070] In contrast, as shown in Sample D’ of Table 3, when applying Skin Material E and Skin Material F, which have a large difference in high-temperature deformation resistance with respect to Skin Material A and Skin Material C, to the Sample No. A in Table 1 and performing the first-pass hot rolling under the conditions of the same initial length and the same target clad ratio as the Sample No. A, the actual elongation of Skin Material D’ becomes as large as 410 mm, skin peeling occurs, and the clad ratio also exceeds ±1.5% with respect to the target clad ratio, resulting in variations.

[0071] Next, as shown in the Sample No. E in Table 1, when applying a sample in which the target clad ratio with respect to Skin Material A and Skin Material C is changed to 30:60:10 for the Sample No. A in Table 1, and performing the elongation prediction based on the aforementioned finite element method on the basis of the same initial length and the same high-temperature deformation resistance as the Sample No. A. As shown in the Sample No. E, the predicted elongation values are 145 mm for Skin Material A and 260 mm for Skin Material C. With respect to the Core Material B, it is predicted that Skin Material A will protrude by 45 mm and Skin Material C will protrude by 160 mm. Therefore, as shown in the Sample No. E in Table 2, when setting the initial longitudinal dimensions to Skin Material A:Core Material B:Skin Material C = 3355 mm:3500 mm:3240 mm, the actual elongation amounts are Skin Material A:Core Material B:Skin Material C = 125 mm:0 mm:275 mm. In the case of the Sample No. E, as shown in Table 2, no skin peeling occurred, and the final clad ratio of the product was also good.

[0072] In contrast, as shown in Sample E’ of Table 3, when applying a sample in which the target clad ratio with respect to Skin Material A and Skin Material C is changed to 30:60:10 for the Sample No. A in Table 1 and performing the first-pass hot rolling under the condition of the same initial length as Sample A in Table 1, the actual elongation of Skin Material A is 140 mm and the actual elongation of Skin Material C is 300 mm, skin peeling occurs, and the clad ratio also exceeds ±1.5% with respect to the target clad ratio, resulting in variations.

[0073] As described above based on Tables 1 to 4, in accordance with the present invention, considering the high-temperature deformation resistance of the material, the cladding rate, the length of the cladding material, etc., based on the elongation prediction results predicted by numerical analysis such as the finite element method, the initial lengths of the core material and the skin materials on both sides thereof are adjusted. As a result, it has become clear that good elongation prediction can be performed and skin peeling can be prevented in any case where skin materials with different high-temperature deformation resistances are provided on both sides of the core material, where the cladding rate of the skin material with respect to the core material is changed, or where the length of the cladding material is changed. Also, in accordance with the present invention, by using numerical analysis such as the finite element method to adjust the strength of the skin materials provided on both sides of the core material, adjusting the initial lengths of the core material and the skin materials on both sides thereof, adjusting the skin material strength difference between both sides of the skin material, and adjusting the cladding rate of the skin material with respect to the core material, it has been found that a good cladding rate can be obtained in the final product in any case.

Explanation of Reference Signs

[0074] 1, 2... work rolls, 3... core material, 4, 5... skin materials, 6... cladding material, 7... core material, 8, 9... skin materials, 10... clad material.

Claims

1. A method for manufacturing an aluminum clad material in which two or more aluminum or aluminum alloy materials are joined in a joining step using hot rolling rolls, and the joined body obtained by this joining is subsequently rolled using the hot rolling rolls to join at least a core material and a skin material, comprising: predicting the difference in elongation amounts in the rolling direction between the core material and the skin material generated in the joining step by numerical analysis for predicting the elongation of unknown actual hot rolling, assuming that the length of the core material in the rolling direction does not change, and joining by preliminarily determining the length of the skin material so that the length of the skin material in the rolling direction after the joining step is substantially equal to the length of the core material in the rolling direction with reference to the length of the core material in the rolling direction; obtaining a correction coefficient that can be approximated using the elongation prediction value obtained by the same method as the numerical analysis for the known elongation amount of actual hot rolling; When manufacturing the aluminum clad material, determining the length of the skin material in the rolling direction before being subjected to the joining step based on the elongation prediction value of the skin material obtained by the same method as the numerical analysis based on the correction coefficient. A method for manufacturing an aluminum clad material, characterized by this.

2. When obtaining a correction coefficient α so that the elongation prediction value obtained by the numerical analysis can be approximated to the actually measured elongation amount obtained by the actual hot rolling, the following formula (1) is used so that the relationship between the actual results of the elongation amount of the core material and the skin material by the actual hot rolling and the elongation prediction value approximates each other. The method for manufacturing an aluminum clad material according to claim 1, characterized in that the correction coefficient α that minimizes the sum of squared residuals is obtained by optimization calculation. 【Number 1】 However, in formula (1), i indicates the sample number, xi indicates the predicted value of the corresponding sample number, and yi indicates the actual result value of the corresponding sample number.

3. When obtaining correction coefficients a, b, c so that the elongation prediction value obtained by the numerical analysis can be approximated to the actually measured elongation amount obtained by the actual hot rolling, the following formula (2) is used so that the relationship between the actual results of the elongation amount of the core material and the skin material by the actual hot rolling and the elongation prediction value approximates each other. The method for manufacturing an aluminum clad material according to claim 1, characterized in that the correction coefficients a, b, c that minimize the sum of squared residuals are obtained by optimization calculation. 【Number 2】 However, in formula (2), i represents the sample number, xi represents the predicted value of the corresponding sample number, yi represents the actual value of the corresponding sample number, and a, b, and c represent correction coefficients.

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