MATHEMATICAL REPRESENTAITON OF A MODIFIED
SCHAEFFLER DIAGRAM
V. Mazurovsky, M. Zinigrad, and A. Zinigrad
College of Judea and Samaria, Ariel, Israel
ABSTRACT. A new approach to predicting the structure of a weld metal based on mathematical models of technological welding processes and the nonequilibrium crystallization of the weld metal, which allows the construction of a real constitution diagram (a modified Schaeffler diagram), was presented in [1]. This paper presents a complete mathematical description of this diagram and a computer implementation of it (a program for calculating the structure of a weld metal).
Introduction
The structure of a weld metal largely determines the properties of a welded joint. This structure depends, in turn, on the chemical composition of the weld metal and the thermal cycle of the welding process. Predicting the phase-structure composition of a weld metal has been the subject of numerous papers [2-15]. These papers included graphical representations of the phase-structure composition of the metal as a function of its chemical composition, as well as computational methods for determining the percentage of the ferrite phase in a two-phase austenitic-ferritic weld metal. As was previously noted [1], each of these papers examines a definite alloying range of the weld metal. The modified Schaeffler diagram that we previously developed in [1] predicts the phase-structure composition of a weld metal with a broad alloying range, since the calculation of the chromium and nickel equivalents takes into account the consumption of alloying elements in the formation of carbides and other strengthening phases. However, it is difficult to use the diagram directly in its graphical form because it is impossible to precisely determine the percentages of the structural components without resorting to extrapolation. This problem can be solved on the basis of a mathematical description of the modified Schaeffler constitution diagram. This paper focuses on the solution of this problem, which was accomplished in several stages. In the first stage, the boundary conditions for regions of the diagram (i.e., the conditions under which the particular phase does not exist or exists in contact with other phases) were determined. The second stage involved a regression analysis of experimental data based on structural components from various groups in the weld metal. Regression equations that permit calculation of the phase-structure composition for a specific region of the diagram (for specific values of the chromium and nickel equivalents) and the conditions that determine the applicability of specific regression equations as a function of the alloying level in the alloy were obtained. We note that the values of the chromium and nickel equivalents are obtained as a result of the calculation described in [1], i.e., with consideration of the consumption of alloying elements in carbide formation and the formation of other strengthening phases. Therefore, a calculation of the phase-structure composition from these equations specifies only the composition of the matrix of the weld metal. To obtain the complete phase-structure composition, the data on the content of carbides and other strengthening phases obtained in the calculation of the chromium and nickel equivalents must also be taken into account.
Determination of the boundary conditions for the regions of the diagram
The boundary conditions for the regions of the diagram can be assigned using the mathematical equations for the dividing lines that specify the boundaries of the structural regions. These equations describe the dependence of the nickel equivalent (Nieq) on the chromium equivalent (Creq):
(1)
or, since they are linear functions,
. (2)
Thus, the position of a specific line on the diagram is specified by the coefficients a and b. Figure 1 presents the overall form of the modified Schaeffler diagram [1] with labeling of the boundary lines.
Fig 1. – Modified Schaeffler diagram taking into consideration the effect of carbide formation.
The values of the coefficients a and b for each specific boundary line are presented below (Table 1).
Table 1. Values for the coefficients a and b for the equations of the boundary lines.
b / a / Line+ 7.2 / - 2.45 /
K
+ 19 / - 0.8 / L+ 25.4 / - 0.8 / M
- 8.1 / 1.13 / N
- 4.0 / 0.35 / Q
Now, having the values for the boundary conditions, we can devise an algorithm that determines whether a point with the coordinates (Creq, Nieq) is found in each region. Any region on the diagram can be represented in the form of an irregular quadrangle (Fig. 2), which degenerates to a triangle in some cases (the F + M and ferrite regions).
Fig. 2. - Schematic representation of the ith region on the diagram.
In Fig. 2 the boundary conditions are designated as Eq1, Eq2, Eq3, and Eq4. Then, the condition that satisfies finding the point for specific values of Creq and Nieqin a specific region can be described in the following form
. (3)
In Table 2, the boundary conditions are represented by the names of the boundary lines, and the coefficients of Eq. (2) corresponding to them are listed in Table 1. If the boundaries of the regions coincide with boundaries of the diagram (i.e., lines that coincide with or are parallel to the Creq, Nieqaxes), the equations for the boundary conditions degenerate into constants.
Table 2. Boundary conditions for specific regions of the diagram
No. / Region / Eq1 / Eq2 / Eq3 / Eq41. / Ferrite + Martensite (F+M) / 0.0 / 0.0 / K / 7.2
2. / Martensite / K / N / L / 19.0
3. / Martensite + Ferrite( M+F) / 0 / Q / L / N
4. / M + A + F / L / Q / M / N
5. / Austenite + Ferrite ( A+F) / M / Q / 37.5 / N
6. / Ferrite / 0.0 / 0.0 / 10 / Q
7. / Austenite / M / N / ∞* / ∞*
8. / Austenite + Martensite ( A+M) / L / N / M / 25.4
*In the actual calculation, the infinity symbol (∞) must be replaced by a very large real number.
The algorithm for solving the problem of determining the regions where a specific point with the coordinates (Creq, Nieq) is found contains a cycle. According to the algorithm, the values of Creq and Nieq corresponding to the coordinates of the point sought (these values do not vary during the cycle) and the set of data with the number corresponding to the cycle counter (Table 2) are successively inserted into condition (3). This set of data contains the name of the region and the values of its boundary conditions. If the condition (3) is satisfied in any step of the cycle, the cycle is terminated, and the row variable with a code indicating the region sought is returned. Otherwise, the next iteration of the cycle proceeds. A block diagram of the algorithm is shown in Fig. 3.
Fig.3. - Block diagram of the algorithm for determining the region where a point with the coordinates( Creq , Nieq)is found.
Determination of the structural components within each region
As we have already noted, functional relations (regression equations) for calculating the structural components were determined by carrying out a regression analysis of the experimental data for each of the regions of the modified Schaeffler diagram. The conditions for applicability of these equations for calculating the amounts of the individual phases in multiphase regions of the diagram were also determined. These conditions and the equations for each of the eight regions are presented below. It should be noted that these conditions and equations could be refined as more experimental data are accumulated. Calculations performed using these equations with consideration of their applicability conditions exhibit close convergence of the calculated values with the experimental data.
Ferrite + Martensite (F + M) region
This region lies in the lower left-hand corner of the diagram, and the structures specified by it are characteristic of a low-alloy weld metal. If the condition
,(4)
where SmLg is the sum of the ferrite former alloying elements with the exception of Cr, holds, the structure of the region will consist of two phases, viz., perlite and bainite with the content of the bainite phase being determined by the following relation:
. (5)
The content of the perlite phase is accordingly the remainder from 100 wt.%:
.(6)
As we have already noted, to determine the complete phase-structure composition, we must take into account the previously calculated content of carbides and other strengthening phases. Then the complete phase-structure composition of the weld metal (wt. %) is
,(7)
where QHd is the total content (wt. %) of carbides and other strengthening phases and kf is a coefficient that takes into account the ratio between the phases:
. (8)
Relations (7) and (8) are characteristic of all the regions. We shall not dwell further on determining the complete phase-structure composition of the weld metal and shall focus only to the structure of the matrix.
If condition (4) does not hold, the structure of the region consists of ferrite and martensite. The content of the martensite phase (wt. %) is determined from relation (5’), and the ferrite content (wt. %) is determined from relation (6’):
,(5’)
.(6’)
Martensite region
As follows form the experimental data, the appearance of a second phase, viz., bainite, in this region is possible when the following condition is satisfied:
(9)
The bainite content can be calculated from the following relation:
(10)
The remainder is martensite, and its content (wt. %) is accordingly
(11)
If the condition (9) does not hold, the matrix consists completely of the martensite phase.
Martensite + ferrite (M + F) region
This region is located in the lower central portion of the diagram and specifies an area with a two-phase weld metal. If the ratio has a value
,(12)
the content of the ferrite phase (wt. %) is
.(13)
Otherwise, the content of the ferrite phase can be determined from the relation
.(14)
The other matrix component is martensite, and its content (wt. %) is
.(15)
Martensite + austenite + ferrite (M + A + F) region
This region is located at the center of the diagram and comprises an area with a three-phase structure. The condition and the relations for calculating thee content of the ferrite phase are similar to (12)-(14). We can determine the content of the matrensite phase (wt. %) from the following relation:
,(16)
where is a coefficient that takes into account the ratio between the phases and QZ = Nieq + 0.8Creq. Then the content (wt. %) of the austenite phase in the matrix is
.(17)
Austenite + ferrite (A + F) region
This region encompasses a broad spectrum of austenite-ferrite weld metal compositions and is extremely important for determining the content of the ferrite phase. We use the condition and relations (12)-(14) for it. The austenite content (wt. %) in the matrix is
.(18)
Ferrite region
This region is located in the lower right-hand portion of the diagram and comprises a single-phase region. Thus, if condition (3) specifies that the point with the coordinates (Creq, Nieq) is found in this region, the matrix consists of ferrite.
Austenite region
This region is located in the upper portion of the diagram and encompasses a broad spectrum of single-phase austenitic steels. A perlite component can be appear in the weld metal at certain values of the ratio between the alloying components and cooling rates as a result of the decomposition of austenite. Its content generaly does not exceed 15%. As was pointed out above, it was established on the basis of a regression analysis of the experimental data that when the condition
(19)
is satisfied, the content fo the perlite phase can be determiend from the relation
,(20)
where QP is the perlite content (wt. %) and QR = Nieq + 0.8Creq.
The remainder of the matrix consists of austenite, and its content (wt. %) is
.(21)
If the condition (20) is not satisfied, the matrix of the weld metal consists completely of austenite.
Austenite + martensite (A + M) region
This region is located in the left-hand central portion of the diagram and encompasses an extremely important and interesting spectrum of weld metal compositions, which is characteristic of many hard-facing electrodes, wires, etc. that are intended for hard-facing layers operating under the conditions of shock-abrasive loading, hydro-abrasive wear, cavitation, etc. Relation (16) gives the martensite content for these types of weld metals. Then the austenite content (wt. %) is
.(22)
Thus, for each region of the modified Schaeffler diagram we have mathematical relations that permit calculation of the content of the structural components (phases) in a weld metal with a definite chemical composition, which is characterized by the point with the coordinates (Creq, Nieq) in a specific region.
Method for calculating the phase-structure composition of a weld metal. Computer implementation of the model
It is wise to implement the foregoing mathematical description of the Schaeffler diagram in the form of a computer program for calculating the phase-structure composition of a weld metal. An extended algrorithm of such an implementation, which is essentially a method for calculation the phase-structure compoidition of a weld metal, is presented in Fig. 4.
Fig. 4. - Algorithm for calculating the phase-structure composition of a weld metal.
We have developed a complex program that includes a program for calculating the chemical composition of a weld metal [1] and a prgram for implementing the algorithm in Fig. 4, i.e., for calculating the phase-structure composition of a weld metal. They permit a quick qualitative calculation of new welding materials for solving an extensive range of problems related to the performance of welding and hard-facing operations.
Conclusion
Mathematical relations for determining the boundary conditions for the existence of the various structural regions on a modified Schaefler diagram have been obtained.
A regression analysis of experimental data has yielded regression equations that can be used to calculate the content of various phases within each region.
An algorithm has been developed for obtaining the region in which a designed weld metal of definite chemical composition, which is characterized by the values of the chromium and nickel equivalents, i.e., the point with the coordinates (Creq, Nieq), is found.
A method for determining the phase-structure composition of a weld metal has been determined.
A complex program for calculating (predicting) the chemical and phase-structure composition of a weld metal has been created.
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