Use of Published Experimental Results to Validate Approaches to Gray and Ductile Iron Mechanical Properties Prediction

Abstract

Cast iron is one of the most widely used materials in the world. Numerous experiments have been conducted in the last 50 years to develop specific microstructure and properties of cast iron. Chemical composition and processing variables have been studied to obtain optimum properties. Efforts have been made to use experimental variables (composition, cooling rate, liquid treatment) and nondestructive tests to predict cast iron’s mechanical properties. Most of these relationships are limited to the small data set conducted by a single research project. Researchers at the University of Alabama at Birmingham have collected published cast iron experimental results from the last 40 years to validate the usability of such relationships with a large set of experimental results. In this study, about 2000 cast iron sample data were collected from literature. Statistical analysis was done to compare predicted mechanical properties and experimental measurements. Mean percentage error and mean absolute error are summarized for the whole dataset and for applicable ranges. This summary provides suggestion for which model would be applicable for specific cast iron.

Introduction

Mechanical property can be defined as a property that involves a relationship between stress and strain or a reaction to an applied force.1 The mechanical properties determine the range of usefulness and establish the service life that can be expected. Mechanical properties are also used to classify and identify a material. Strength, ductility, hardness, impact resistance, and fracture toughness are the most common mechanical properties considered. These properties of cast iron are related to the microstructure, which is dependent on process variables such as chemical compositions (carbon equivalent, alloying elements), cooling conditions (section size, pouring temperature, and mold material2), liquid treatment (spherodizing, inoculation, holding time) and heat treatment.3,4, ^– 5 Due to the wide variation of properties resulting from the above-mentioned variables, it is difficult to propose any rigorous expression which would predict the magnitude of a property.6 Noticeable statistical data scatter is reported even for identical sand-cast gray iron parts from three different foundries.7 It is desirable to have the capability to predict properties in cast iron to permit conservation of alloys while meeting the desired mechanical property requirements and anticipate process design needed to produce specific properties in new or unfamiliar parts. Researchers over the years have applied various methods such as statistical analyses6 ^, 8 ^, 9 (regression, separation of variables) of experimental data,10 numerical analysis,11 ^, 12 artificial neural network (ANN),13,14,15, ^– 16 nondestructive tests17 ^, 18 to study these variables to correlate and predict microstructure and mechanical properties. In general, these efforts can be classified into two broad approaches—(a) correlation of process variables to microstructure and mechanical properties and (b) correlation of mechanical properties.

Statistical analysis is performed generally by collection and organization of experimental data (chemical composition, section thickness) from industrial and/or research laboratory cast iron heats. The data were interpreted, correlated, and presented as a mathematical equation. The predictive analyses are performed separately for ductile and gray iron by the researchers. One of the earliest such work was done by McElwee and Schneidewind6 to correlate between properties (tensile and compressive strength, Brinell hardness number (HB), modulus of rupture) of gray iron. McElwee and Schneidewind also proposed the tensile strength (TS) prediction model as a function of composition and section thickness. Modification was published on the previously mentioned relationship between gray iron chemistry and tensile strength by Bates8 and then Shturmakov and Loper.19 Such multiple regression equation models for ductile iron have been published by researchers.9 ^, 20 ^, 21 Graphical representations (contour plots) were published showing combined effects of 11 elements (C, Si, Mg, Ce, Sn, Pb, Ti, Bi, Sb, Cu and Cr) on ductile iron properties.22

Numerical analysis creates, analyzes, and implements algorithms for solving the problems of microstructure evolution numerically. Numerical simulations of different processes have seen an extensive and increasing effort due to the advent of computation capability in recent times. Commercial software packages are available to calculate cooling curves, flow rate, and solidification processes. The output of such microstructure evolution is used to predict final microstructure, properties, and quality of final castings.12 The uses of such software packages are still limited because of the expense and proprietary calculation methodology involved. In literature, researchers have applied thermal analysis and phase growth kinetics within numerical analysis to predict microstructure.23 The output of these numerical models is used to calculate mechanical properties.11 ^, 21 Mampaey10 proposed a two-step TS prediction method for any location of a casting. In the first step, TS is calculated based on calculated solidification time. In the second step, influence of the cooling rate during the eutectoid transformation is used to adjust TS calculated in the first step.

Artificial neural network (ANN) is described as mathematical models which solve by means of learning rather than by specific programming based on well-defined rules. ANN has been utilized by researchers to investigate the possibility of predicting mechanical properties. Input variables (composition, inoculation temperature, holding time, casting modulus, etc.) and the output parameter (mechanical properties) are used for training the ANN. A multilayer network with nonlinear perceptrons (functions that can decide whether an input belongs to one class or another) was applied to predict output based on the training. The principle drawback of this method is that the resultant output is specific to training dataset. Also, there is no applicable mathematical relation available in the literature to test the applicability of such ANN method to a different set of experimental data set.

Nondestructive evaluation (NDE) techniques such as ultrasonic velocity,24 Brinell hardness (HB) measurements,17 density measurements,18 and eddy current25 have been applied to identify different types of cast iron microstructures and correlated with its properties. Cast iron is essentially a mixture of graphite in a matrix of ferrite, pearlite, or carbides. Since graphite has a much lower density compared to other phases, it affects the alloy density directly. Li et al. used this assumption and density measurements of ranges of gray and ductile irons to evaluate cast iron microstructures.18 Researchers have used ultrasonic and resonance techniques and investigated correlations between acoustic response and certain mechanical properties.26,27,28,29,30, ^– 31 HB measurements have been correlated by Basaj and Dorn17 with tensile and yield strength and percent elongation.

The mechanical property prediction methods mentioned above compares very well (R ² value of at least 0.8) with experimental results compared in respective publications. However, the prediction methodologies in general have been established by using a limited set of laboratory experiments or data from few foundries. Therefore, the question arises: Are these relationships universally applicable to all cast iron foundries? This article will concentrate on validating those relationships with a diverse set of data from sources of different time and production set up.

Methodology

Data Collection

Experimental results of about 2000 heats on cast iron have been collected from literature and internal pours at The University of Alabama at Birmingham (UAB)32 and the Southern Research Institute (SRI).33 Cast irons were classified in three categories namely gray (GI), compacted (CGI), and ductile (DI). These categories were specified by the authors of the publications reviewed. The authors of this paper did not make any judgment on determining cast iron classification.

The American Foundry Society (AFS) Transactions have become a rich source of such experimental data over the years.34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89, ^– 110 Figure 1 shows the distribution of number of AFS Transactions papers and heats reported from 1971 to 2014 from when data was collected. Chemical composition, experimental variables, microstructure analysis, and physical, mechanical, thermal and acoustic properties were tabulated using a spreadsheet. Heat-treated cast iron results were not included in this study. The prediction models available in the literature applicable to only as-cast samples have been considered for this study. Details on the methodology followed in collecting, storing, and categorizing the data has been published by the authors previously.111 It was also shown that the experimental results collected conform to generally accepted values of chemical composition and mechanical properties. The exceptions can be attributed to experimental variability and measurement techniques.

Figure 1

Timeline of data collected from AFS transactions.

Available standards generally list minimum acceptable values of tensile strength, yield strength, and percent elongation for each grade of cast iron. A range of HB values expected to have similar strength, and elongation minima as equivalent standard grades are also available. The collected database for this paper is compared with the Society of Automotive Engineers (SAE) standards for GI112 in Figure 2 and the American Society for Testing and Materials (ASTM) standards for CGI and DI113 ^, 114 in Figures 3 and 4. These figures illustrate the reliability and breadth of the collected data. They also show results which does not conform to any available standard grades. Such results come from research publications dedicated to identify the effect of different casting condition variables over the years. The range of chemical composition and section size is tabulated in Table 1. To summarize, it can be said that the database not only covers industrially accepted values but also includes highs and lows of mechanical properties due to different casting variables.

Figure 2

Collected gray iron experimental measurements plotted with SAE standards.

Figure 3

Collected compacted graphite iron experimental measurements plotted with ASTM standards.

Figure 4

Collected ductile iron experimental measurements plotted with ASTM standards.

Table 1 Range of Chemical Composition and Section Size (Sample Thickness or Bar Diameter) in the Database Used for Analysis in this Article

Analysis

A literature review was made to find mechanical property and microstructure prediction models. Mathematical models with a proposed mathematical equation to predict properties were considered for study in this article.

Mathematical equations from different methods/models have been used to calculate mechanical properties by using experimental values from the database compiled by the authors of this paper.

Models11 ^, 21 with variables (such as cooling rate of the casting at 900 °C) which are generally not measured are computed using commercially available software.

When using a reference model, data from that article was not included for the analysis. The residual (the difference between the observed value of the dependent variable and the predicted value) was calculated for each data point to find outliers. Outliers are values that fall outside of an overall trend in the data. Sometimes they are caused by error. Other times outliers indicate the presence of a previously unknown phenomenon.

A boxplot, sometimes called a box and whisker plot, was plotted for each set of residuals from the individual model. An example is shown in Figure 5. A boxplot splits the data set into quartiles. The body of the boxplot consists of a “box” (hence, the name), which goes from the first quartile (Q1) to the third quartile (Q3). Within the box, a horizontal line is drawn at the Q2, the median of the data set. The lower and upper limits of acceptable values were calculated using the following equations to identify outliers.

Figure 5

Schematic of box and whisker plot.

• Lower limit: Q1 − 1.5*IQ

• Upper limit: Q3 + 1.5*IQ

• where IQ = interquartile range = Q3 − Q1

Two vertical lines, called whiskers, extend from the top and bottom of the box. The top whisker goes from Q3 to the upper limit of nonoutlier in the data set, and the bottom whisker goes from Q1 to the lower limit of nonoutlier.

Outliers were then checked for errors from documentation mistakes, unit conversion and not reported input values which would give a false prediction. If no error was detected, the outliers were not discarded. An average value of the residual which is called mean absolute error for each model is reported in Tables 4 and 5. Finally, percentage error was calculated for each data point using Eqn. 1. Mean absolute error (MAE) and mean percentage error of all the analyzed models are summarized in Tables 4 and 5.

$$ {\text{MAE}} = \left| {{\text{calculated}} - {\text{measured}}} \right| $$

(1)

$$ \% {\text{Error}} = \frac{{\left| {{\text{calculated}} - {\text{measured}}} \right|}}{{\left| {\text{measured}} \right|}} \times 100\% $$

(2)

If the authors of the specific model specify the applicability range, a similar study as stated above was done within the specified range.

Results

This section summarizes description of test conditions of the sample set for different model, comparative results and statistical variation. Comparisons between experimental and predicted properties by various authors are done in this section. The comparative results within specified applicability range are shown by a solid black circle. The results are shown for gray and ductile iron and arranged chronologically.

Gray Iron Property Prediction

Prediction of gray iron mechanical properties is a complex proposition. Researchers have been utilizing different process variables to propose models which capture the scattering of properties accurately. Most of the models use chemical composition and section size or cooling rate, but there are exceptions too. Jura et al.115 published a mathematical equation correlating Brinell hardness number (HB) of unalloyed gray cast iron with several parameters of cooling curves. Goettsch et al.23 developed a gray iron microstructure prediction model base on growth kinetics of gray cast iron. Hua-Qin et al.116 published a tensile strength prediction model based on inoculants containing Rare Earth, Al, Ca, Cu, Cr, Mn, and Si. In this section, gray iron properties prediction models with mathematical equation with commonly measured and reported variables are studied.

The first gray iron model studied in this article was published in 1950 by McElwee and Schneidewind.6 They developed a tensile strength (TS) prediction method for selection of alloy additions necessary to meet requirements of critical casting sections. The method was expressed as the following equation:

$$ {\text{TS}}\;\left( {\text{ksi}} \right) = 10 \left( {b - 2 \left( {\% {\text{CE}}} \right)} \right)\left( {f_{1} } \right)\left( {f_{2} } \right)\left( {f_{3} } \right) \ldots \left( {f_{n} } \right) $$

(3)

where b is a constant depending on section size; C.E. is carbon equivalent; and the fs are multiplying factors for the alloys. The factors can be read off from the plot of alloy factor as a function of percent alloy in the article (Figure 6). The predicted TS is compared with experimental measurements in Figure 7. The mean percentage error was calculated to be ±11.5% for the complete database.

Figure 6

Multiplication factors for McElwee tensile strength prediction model (reproduced from Reference 6).

Figure 7

Comparison of values of tensile strength measured versus predicted by McElwee and Schneidewind using alloy addition factors.

They also suggested a rapid approximation of tensile strength (in psi) based on a given section size, D (in inches) and carbon equivalent. For unalloyed gray cast iron, the equation is expressed as:

$$ {\text{TS}}\;\left( {\text{ksi}} \right) = 10\left( {11.68 - 2\left( {\% {\text{CE}}} \right)} \right) - 2.3\log D $$

(4)

The simplified equation was used to compare the model prediction with the experimental tensile strength measurement collected in the database. The comparison is shown in Figure 8. Surprisingly, this model had the least mean percent error (±11%) among all the models compared.

Figure 8

Comparison of values of tensile strength measured versus predicted by McElwee and Schneidewind.

Bates8 at Southern Research Institute (SRI) developed multiple regression equations based on experiments on gray iron. Eleven heats of cast iron were produced to determine the effect of individual ally elements and section size on mechanical properties. The iron was poured at a temperature of 2500–2525 °F. Four 0.875-in-diameter “A” bars, six 1.2-in-diameter “B” bars, and a 2-in-diameter “C” bar were poured from each heat. The following correlation equation between strength and yield strength to the chemical compositions and sample diameter was proposed after statistical analysis.

$$ {\text{TS}}\;\left( {\text{ksi}} \right) = b_{0} + b_{1} \left( {\% {\text{C}}} \right) + b_{2} \left( {\% {\text{Si}}} \right) + b_{3} \left( {\% {\text{Mn}} - 1.7(\% {\text{S}}} \right) + b_{4} \left( {\% {\text{Cr}}} \right) + b_{5} \left( {\% {\text{Ni}}} \right) + b_{6} \left( {\% {\text{Cu}}} \right) + b_{7} \left( {\% {\text{Mo}}} \right) + \frac{{b_{8} }}{{{\text{dia}}\left( {\text{in}} \right)}} + b_{9} \left( {\% {\text{Si}}} \right)^{2} + b_{10} \left( {\% {\text{Cu}}} \right)^{2} + b_{11} \left( {\% {\text{Mo}}} \right)^{2} $$

(5)

The values of coefficients (b’s) for tensile and yield strength and limiting ranges of chemical composition and sample diameter for the use of this equation are reported in Tables 2 and 3, respectively.

Table 2 List of Coefficients for Bates Gray Iron Prediction Model (Reproduced from Reference 8)

Table 3 Applicable Condition for Bates Regression Equation (Reproduced from Reference 8)

The comparison between Bates model and experimental measurements is shown in Figure 9. The percentage error was calculated to be ±24% for the complete dataset and also within the specified range.

Figure 9

Comparison of values of tensile strength measured versus predicted by Bates et al.

Yang et al.117 expressed experimental measurements of HB (Eqn. 6) and percent pearlite content (Eqn. 7) relative to various alloying elements and cooling rate. A total of 16 experimental heats were melted in an induction furnace, and five sample cylinders with varying diameters and standardized tensile test bar were cast. Cooling curve data was recorded, and first- and second-order polynomial approximation of each was constructed. Cooling rate at 900 °C and composition were correlated with hardness and percent pearlite. The equations obtained are shown below.

$$ {\text{HB}} = 106.7 + 111 \left( {\% {\text{Cr}}} \right) + 508\left( {\% {\text{Cr}}} \right)^{2} + 150.8\left( {\% {\text{V}}} \right) - 96\left( {\% {\text{V}}} \right)^{2} - 93.7\left( {\% {\text{Mo}}} \right) + 167.36\left( {\% {\text{Mo}}} \right)^{2} + 20.7\left( {\% {\text{Cu}}} \right) - 10.6\left( {\% {\text{Ni}}} \right) + 74.1 \left( {v900} \right) - 15.3\left( {v900} \right)^{2} $$

(6)

$$ \% {\text{Pe}} = 63.3 + 90.8 \left( {\% {\text{Cr}}} \right) - 93.1\left( {\% {\text{Cr}}} \right)^{2} + 60.9\left( {\% {\text{V}}} \right) - 126.4\left( {\% {\text{V}}} \right)^{2} - 174.9 \left( {\% {\text{Mo}}} \right) + 199.6 \left( {\% {\text{Mo}}} \right)^{2} + 25.7\left( {\% {\text{Cu}}} \right) + 9.4\left( {\% {\text{Ni}}} \right) + 25.2\left( {v900} \right) - 6.8\left( {v900} \right)^{2} $$

(7)

$$ v900 = {\text{cooling}}\; {\text{rate}}\;{\text{at}}\; 900\;^\circ {\text{C}} = 0.4283\left( {{\text{modulus}},\;{\text{cm}}} \right)^{2} - 2.0444 $$

(8)

A linear regression was done between tensile strength and hardness.

$$ {\text{TS}} \left( {\text{ksi}} \right) = \frac{{ - 3.2997 + 1.4335{\text{HB}}}}{6.89};\quad \left[ {R^{2} = 0.94} \right] $$

(9)

The comparative results between prediction and measurement of hardness are shown in Figure 10 and of tensile strength is shown in Figure 11. Cooling rate at 900 °C for different section sizes were calculated using a regression curve fitting the equation obtained from a commercially available solidification software package. The details are described in the “Appendix”.

Figure 10

Comparison of values of hardness measured versus predicted by Yang et al.

Figure 11

Comparison of values of tensile strength measured versus predicted by Yang et al.

Figure 12

Comparison of values of tensile strength measured versus predicted by modified model of McElwee by Bates et al.

Bates118 work modified the tensile strength model of McElwee. The effects of alloying elements were adjusted, and the following equations were reported. The calculated tensile strength results are compared in Figure 12.

$$ {\text{TS}} = {\text{A}} \times {\text{B}} $$

(10)

where

$$ A = 101.1193 - 20.3283\left( {\left( {\% {\text{C}}} \right) + \left( {\% {\text{Si}}} \right)/4 + \left( {\% {\text{P}}} \right)/2} \right) + 4.3887/\left( {{\text{cast}}\; {\text{bar}}\; {\text{radius}}} \right) $$

$$ B = 1.000 + 0.1371\left( {\% {\text{Si}}} \right) - 0.0021\left( {\left( {\% {\text{Mn}}} \right) - 1.7\left( {\% {\text{S}}} \right)} \right) - 0.3132\left( {\% {\text{S}}} \right) + 0.3562\left( {\% {\text{Cr}}} \right) + 0.0282 \left( {\% {\text{Ni}}} \right) + 0.1107\left( {\% {\text{Cu}}} \right) + 0.6297\left( {\% {\text{Mo}}} \right) - 5.2985\left( {\% {\text{Ti}}} \right) - 0.2305\left( {\% {\text{Sn}}} \right) $$

Shturmakov and Loper19 used mechanical property and chemical composition data from a commercial gray iron (ASTM classes 30B, 35B and 40B) foundry to evaluate by means of multiple regression, using the least squares method. Samples were melted in a cupola furnace and tapped into a channel holding furnace and then transferred to a pouring ladle. A total of 3117 test ASTM B bars were cast to be evaluated, and from the analysis the following predictive equations for hardness and tensile strength were obtained.

$$ \begin{aligned} {\text{HB}} & = 470 - 77.6\left( {\% {\text{C}}} \right) - 15.8\left( {\% {\text{Si}}} \right) + 52.7\left( {\% {\text{Mn}}} \right) + 65.2\left( {\% {\text{S}}} \right) \\ & \quad + 69.3\left( {\% {\text{P}}} \right) + 45.8\left( {\% {\text{Cr}}} \right) - 5.2\left( {\% {\text{Ni}}} \right) + 28.8\left( {\% {\text{Cu}}} \right) + 102\left( {\% {\text{Al}}} \right) \\ & \quad - 971\left( {\% {\text{Ti}}} \right) - 109\left( {\% {\text{V}}} \right) + 71.6\left( {\% {\text{Sn}}} \right) + 101\left( \% \right){\text{Mo}} \\ \end{aligned} $$

(11)

$$ {\text{TS}} = 157.176 - 32.031\left( {\% {\text{C}}} \right) - 4.388\left( {\% {\text{Si}}} \right) - 1.427 \left( {\% {\text{Mn}}} \right) + 16.653\left( {\% {\text{S}}} \right) - 3.524 \left( {\% {\text{P}}} \right) + 9.675\left( {\% {\text{Cr}}} \right) - 3.768 \left( {\% {\text{Ni}}} \right) + 2.419\left( {\% {\text{Cu}}} \right) + 69.209 \left( {\% {\text{Al}}} \right) - 220.366\left( {\% {\text{Ti}}} \right) - 30.383\left( {\% {\text{V}}} \right) - 9.520\left( {\% {\text{Sn}}} \right) + 13.619\left( {\% {\text{Mo}}} \right) $$

(12)

The equations were used for calculating hardness and tensile strength for the database collected by the authors of this article. The calculated results are compared in Figure 13 for tensile strength and in Figure 14 for hardness. The range of hardness and strength cast iron samples used to formulate the equations by Shturmakov and Loper is applied, and the comparative results are shown with solid black circle in Figures 13 and 14.

Figure 13

Comparison of values of tensile strength measured versus predicted by Shturmakov and Loper.

Figure 14

Comparison of values of hardness measured versus predicted by Shturmakov and Loper.

The mean percent error analysis shows an error of ±20% for hardness and ±24% for tensile strength. Application of the applicable range improved the prediction results by approximately ±2%.

The results of the statistical analysis to compute mean percent error and mean absolute error are summarized in Table 4. The results are also listed for available application range specified with the respective model.

Table 4 Summary of the Statistical Analysis of Comparisons Between Prediction Models and Experimental Measurements for Gray Iron

Ductile Iron Property Prediction

The application of ductile iron (DI) is increasing more and more due to its excellent properties and castability. Microstructural factors such as nodularity of graphite, nodule count, matrix microstructure, and presence of nonmetallic inclusions determine properties of DI castings. Microstructure is dependent on composition, inoculation treatment, and cooling rate. A large number of researchers have studied possible ways of predicting mechanical properties based on the above-mentioned variables. In this section, the published models are introduced and then a comparative analysis is done with the database collected by the authors of this article.

Yu et al.20 studied the effects of alloying elements (Mo, Ni, Cu) on HB of ductile iron. In total, 19 DI heats were produced to cast cylindrical bars of five different diameters. The hardness measurements were taken from the center of the bar. A ductile iron HB prediction model was proposed based on regression analysis of chemical composition and section size (D) in inches. A similar equation to predict percent ferrite (f _α) present in the casting was also published. It was reported that HB is related exponentially with percent pearlite content (f _Pe). The published equations to calculate percent ferrite and hardness are shown below.

$$ f_{\alpha } = 15.6 - 11.9\left( {\% {\text{Mo}}} \right) - 13.6\left( {\% {\text{Cu}}} \right) + 19 \left( D \right) - 8.1\left( {\% {\text{Ni}}} \right)\left( D \right) - 15.9\left( {\% {\text{Cu}}} \right)\left( D \right) $$

(13)

$$ {\text{HB}} = 237 + 151\left( {\% {\text{Mo}}} \right) + 68.4\left( {\% {\text{Ni}}} \right) + 89\left( {\% {\text{Cu}}} \right) - 21.3\left( D \right) - 51.7\left( D \right)\left( {\% {\text{Mo}}} \right) - 20.4\left( D \right)\left( {\% {\text{Ni}}} \right) $$

(14)

$$ {\text{HB}} = \exp \left( {5.01 + f_{\text{Pe}} } \right) $$

(15)

The experimental measurements were reported to be scattered for castings with over 90% pearlite. Once the matrix is fully pearlitic, or nearly so, hardness is affected by solid solution strengthening and pearlite fineness. Yu et al. prediction model was applied to calculate hardness and then compared in Figure 15. Hardness prediction was also done for castings with pearlite content <90% and is shown with solid black circles in figure. Equation 14 was not included in the figure because the agreement was poor.

Figure 15

Comparison of values of hardness measured versus predicted by Yu et al. (Eqn 14).

Venugopalan and Alagarsamy9 investigated the effects of chemical composition and microstructure on the mechanical properties of ductile iron. Their study consisted of 15 different experimental ductile iron castings. Base iron was melted in a commercial cupola and then treated with Mg–Fe–Si alloy containing 0.5% cerium by the sandwich method in a tundish ladle before pouring keel blocks (1 × 1.5 × 8 in legs) and Y blocks (with 1 × 2 in legs) in green sand molds. Chemical composition (C, Si, Mn, P, Cu, Ni, and Mo) of the irons were varied to create ductile iron with varying ferrite in the matrix microstructure. Linear multiple regression equations, one including phosphorus content as an independent variable (Eqn. 16) and one without (Eqn. 17), were generated from the results to the predicted percent of ferrite from the experimental measurements.

$$ f_{\alpha } = 66\left( {\% {\text{Si}}} \right) + 721\left( {\% {\text{P}}} \right) + 226\left( {\% {\text{Mo}}} \right) - 29\left( {\% {\text{Mn}}} \right) - 100\left( {\% {\text{Cu}}} \right) - 16.5\left( {\% {\text{Ni}}} \right) - 234\left( {\% {\text{Mo}}} \right)\left( {\% {\text{Ni}}} \right) - 113\left( {\% {\text{Cu}}} \right)\left( {\% {\text{Mo}}} \right) - 86 $$

(16)

$$ f_{\alpha } = 69\left( {\% {\text{Si}}} \right) + 198\left( {\% {\text{Mo}}} \right) - 35\left( {\% {\text{Mn}}} \right) - 109\left( {\% {\text{Cu}}} \right) - 22\left( {\% {\text{Ni}}} \right) - 202\left( {\% {\text{Ni}}} \right)\left( {\% {\text{Mo}}} \right) - 74\left( {\% {\text{Cu}}} \right)\left( {\% {\text{Mo}}} \right) - 73 $$

(17)

Vickers indenter was used to measure micro hardness of the matrix phases with a 100-gf load. Knoop indenter was used when the ferrite ring thickness was too small. Ferrite and pearlite microhardness measurements were then fitted in multiple regression equations with the composition. The simple rule of mixture was utilized to get the composite matrix micro hardness (CMMH). Finally, CMMH was used in linear regression expressions to get tensile strength, yield strength, and elongation prediction equations shown below.

$$ {\text{Ferrite}}\; {\text{hardness}} \left( {\text{HF}} \right) = 64 + 44\left( {\% {\text{Si}}} \right) + 9\left( {\% {\text{Mn}}} \right) + 114\left( {\% {\text{P}}} \right) + 10\left( {\% {\text{Cu}}} \right) + 7\left( {\% {\text{Ni}}} \right) + 22\left( {\% {\text{Mo}}} \right) $$

(18)

$$ {\text{Pearlite}}\;{\text{hardness}} \left( {\text{HP}} \right) = 249 + 26\left( {\% {\text{Si}}} \right) + 12\left( {\% {\text{Mn}}} \right) + 234\left( {\% {\text{P}}} \right) + 16\left( {\% {\text{Cu}}} \right) + 17.5\left( {\% {\text{Ni}}} \right) + 26\left( {\% {\text{Mo}}} \right) $$

(19)

$$ {\text{Composite}}\;{\text{matrix}}\;{\text{micro}}\;{\text{hardness}} \left( {\text{CMMH}} \right) = \left( {\left( {\text{HF}} \right) \times \left( {\% f_{\alpha } } \right) + \left( {\text{HP}} \right) \times \left( {\% {\text{Pe}}} \right)} \right)/100 $$

(20)

$$ {\text{TS}} \left( {\text{ksi}} \right) = 0.10 + 0.36\left( {\text{CMMH}} \right) $$

(21)

$$ {\text{YS}} \left( {\text{ksi}} \right) = 12 + 0.18\left( {\text{CMMH}} \right) $$

(22)

$$ \% {\text{EL}} = 37.85 - 0.093\left( {\text{CMMH}} \right) $$

(23)

These equations (which include P as an independent variable) were used to calculate hardness and tensile strength for published experimental compositions of ductile iron. The calculated results were then compared with reported values. The comparisons for Brinell hardness are shown in Figure 16 and for tensile strength are shown in Figure 17. Statistical analyses of the comparisons are shown in Table 5.

Figure 16

Comparison of values of hardness measured versus predicted by Venugopalan and Alagarsamy.

Figure 17

Comparison of values of tensile strength measured versus predicted by Venugopalan and Alagarsamy.

Table 5 Summary of the Statistical Analysis of Comparisons Between Prediction Models and Experimental Measurements for Ductile Iron

In 1993, an equation to calculate Brinell hardness for low alloyed ductile iron from ferrite fraction (f _α) and silicon content was published by Svensson et al.119 The relationship was developed by using solidification and solid-state transformation modeling. The model takes into account the impacts of silicon addition such as solution hardening effect and increased driving force for ferrite precipitation. These are equations are reported to be valid in the range of 1.7–4.9% silicon.

$$ {\text{HB}} = \left( {{\text{HB}}_{\alpha }^{{\left( {\% {\text{Si}}} \right)}} } \right)(f_{\alpha } ) + {\text{HB}}_{\text{Pe}}^{{\left( {\% {\text{Si}}} \right)}} \left( {1 - f_{\alpha } } \right) $$

(24)

$$ {\text{HB}}_{\alpha }^{{\left( {\% {\text{Si}}} \right)}} = 54 + 37\left( {\% {\text{Si}}} \right) $$

(25)

$$ {\text{HB}}_{\text{Pe}}^{{\left( {\% {\text{Si}}} \right)}} = 167 + 31\left( {\% {\text{Si}}} \right) $$

(26)

Svensson model was used to calculate hardness for experimental heats with necessary variables available. The comparison between measured and predicted values is shown in Figure 18. Mean percent error reduced from ±8.9 to ±7.1 when applied within the specified silicon range and unalloyed DI castings. It can be seen from Figure 18 that, around above HB 250, the model fails to predict as soon as the effect of alloying element comes into play, which is expected for this simple model.

Figure 18

Comparison of values of hardness measured versus predicted by Svensson et al.

Guo et al.21 proposed a mechanical property prediction model by taking microstructural evolution into account and utilizing microstructural features such as fractions of graphite, ferrite, and pearlite, and nodularity of graphite. Regression analysis of the above-mentioned variables was done to achieve the following equations to formulate hardness, tensile strength, and elongation.

$$ {\text{HB}} = 100f_{\text{Gr}} + {\text{HB}}_{\alpha } f_{\alpha } + {\text{HB}}_{\text{Pe}} f_{\text{Pe}} $$

(27)

where HB_α and HB_Pe are hardness of ferrite and pearlite, respectively, f _Gr, f _α, f _Pe are the fraction of graphite, ferrite, and pearlite, respectively.

$$ {\text{HB}}_{\text{Pe}} = 223 + 50\left( {\% {\text{Mn}} + \% {\text{Cu}} + \% {\text{Cr}} + \% {\text{Mo}}} \right) + 10\left( {\% {\text{Ni}}} \right) + 20\left( {\frac{{{\text{d}}T}}{{{\text{d}}t}} - 0.5} \right) $$

(28)

where, ${\text{d}}T / {\text{d}}t = {\text{cooling}}\;{\text{rate}}\;{\text{at}}\; 850\;^\circ {\text{C}}.$

$$ {\text{TS}} \left( {\text{MPa}} \right) = \left( {1 - f_{\text{Gr}}^{n} } \right)\left( {482.2f_{\alpha } + 991.5f_{\text{Pe}} } \right) $$

(29)

$$ \% {\text{EL}} = \left( {1 - f_{\text{Gr}}^{n} } \right)\left( {26.2f_{\alpha } + 5.61f_{\text{Pe}} } \right) $$

(30)

where n is an indication of nodularity. The influence of nodularity was not studied by Guo et al. For their study, n was taken as unity since all samples analyzed by them had good nodularity.

To validate the hardness model, two test castings—one ASTM 65-45-12 and one ASTM 100-70-03 grade having six different diameter cylindrical bars—were poured in resin-bonded sand molds. The tensile strength and elongation model were validated using commercial DI data. Mean percent error was calculated to be ±4.13, ±4.06, and ±9.14, respectively, for hardness, tensile strength, and elongation.

Guo et al. used mechanical properties model to calculate hardness and tensile strength for published experimental results with documented microstructural and compositional values necessary. Cooling rate dT/dt at 850 °C was not available in the database collected by the authors of this article. A commercially available solidification model was utilized to generate cooling rate data. The procedure followed is described in detail in “Appendix.” The comparison for hardness is shown in Figure 19 and for tensile strength is shown in Figure 20. Guo et al. model provides tensile strength in MPa. Conversion was done to ksi to accommodate comparison with other models. Statistical analysis showed mean percent error was ±6.5 and ±14.9 for hardness and tensile strength, respectively. The hardness model was found to have the best fit among the models compared in this study.

Figure 19

Comparison of values of hardness measured versus predicted by Guo et al.

Figure 20

Comparison of values of tensile strength measured versus predicted by Guo et al.

Basaj and Dorn17 studied HB and tensile properties from two commercial foundries. Both foundries produce cupola and electric melt process iron. Keel block test bars were produced for hardness and tensile property measurements. Hardness values were obtained using the optical scan measurement of the indentation at the cross section of the tensile bar shank and midway between the center and surface. Statistical regression analysis was done to obtain equations to predict tensile properties and percent pearlite from hardness measurements. Two separate set of equations denoted by A and B were obtained to have a better fit for individual foundries. R ² values in excess of 0.8 were obtained for all the cases. The equations published are shown here.

$$ {\text{A}}1 :\;{\text{TS}}\left( {\text{psi}} \right) = 504\left( {\text{HB}} \right) - 13574;\quad \left[ {R^{2} = 0.981} \right] $$

(31)

$$ {\text{A}}2 :\;{\text{YS}}\left( {\text{psi}} \right) = 223 \left( {\text{HB}} \right) + 8179;\quad \left[ {R^{2} = 0.92} \right] $$

(32)

$$ {\text{A}}3 :\;\log \left( {\text{HB}} \right) = 2.44796 + 0.210607 \log \left( {\% {\text{EL}}} \right) - 0.321105 \left( {\log {\text{EL}}} \right)2 ;\quad \left[ {R^{2} = 0.924} \right] $$

(33)

$$ {\text{A}}8 :\; f_{\text{Pe}} = 0.7351\left( {\text{HB}} \right) - 105.3;\quad \left[ {R^{2} = 0.89} \right] $$

(34)

$$ {\text{B}}1{\text{a:}}\;{\text{TS}} = 0.4519\left( {\text{HB}} \right) - 4.726;\quad \left[ {R^{2} = 0.93} \right] $$

(35)

$$ {\text{B}}2 :\; {\text{YS}}\left( {\text{ksi}} \right) = 0.2254\left( {\text{HB}} \right) + 8.2557;\quad \left[ {R^{2} = 0.87} \right] $$

(36)

$$ {\text{B}}3 :\;\% {\text{EL}} = 0.0003\left( {\text{HB}} \right)^{2} - 0.2401\left( {\text{HB}} \right) + 48.569;\quad \left[ {R^{2} = 0.86} \right] $$

(37)

$$ {\text{B}}5 :\;\% {\text{Pearlite}} = - 0.000084973\left( {\text{HB}} \right)^{3} + 0.055\left( {\text{HB}} \right)^{2} - 11.009\left( {\text{HB}} \right) + 715.2;\quad \left[ {R^{2} = 0.83} \right] $$

(38)

For comparison in this study, equations from foundry A were used. Figure 21 shows the tensile strength and yield strength comparison between experimental measurements and prediction from Basaj et al. models. The strength models were found to have the best fit (least error) within the models studied in this study with mean error ±11%.

Figure 21

Comparison of values of tensile and yield strength measured versus predicted by Basaj et al.

The results of the statistical analysis to compute mean percent error and mean absolute error are summarized in Table 5. The results are also listed for available application range specified with the respective model.

Discussion

Comparison Between Models

All the models analyzed above uses positive or negative influence coefficient related to input variables such as chemical composition and section size. Comparison between models is possible by calculating property using same chemical composition. If one of the element composition is varied, the trend or influence of that element on different models can be observed. To obtain such comparative results, copper (Cu) composition was chosen as the varying element. Copper has a distinctive effect on transforming matrix structure from ferritic to pearlitic. And as the percent pearlite increases, mechanical properties such as tensile strength and hardness increases too. Figure 22 shows the effect of increase in percentage pealite in matrix on tensile strength.

Figure 22

Ductile iron tensile strength as a function of percentage pearlite.

The effect of copper addition on mechanical properties and microstructure are well understood. All the models include copper as an input variable. In this study copper composition was varied between zero to one percent. Average composition of ASTM class 40 and ASTM grade 80-65-6 was used, respectively, for gray and ductile iron to calculate tensile strength. The section thickness/bar diameter was kept constant at 1 inch for all the calculations. The compositions used for calculations are given in Table 6.

Table 6 Iron Composition Used for Comparison Between Property Prediction Models

Figure 23 shows the calculated tensile strengths as lines as a function of copper composition. The black solid dots represent the measured tensile strength. The composition of the experimental results IS not limited to class 40. All the models, except Bates, show an increase in strength with copper addition. The Bates model predicts a maximum strength and then reduction as more copper is added.

Figure 23

Tensile strength as a function of percentage copper for gray iron.

Similar study can be done for other elements. Carbon has a negative effect on strength and is shown in Figure 24. Yang model does not have carbon as input variable and that is why it appears as a flat line on the figure. Carbon composition was varied between 3 and 3.5% and all the other elements were kept constant as stated in Table 6. Copper composition was kept fixed at 0.3%.

Figure 24

Tensile strength as a function of percentage carbon for gray iron.

In case of ductile iron, average composition of ASTM grade 80-65-6 was used to calculate strength with varying copper percentage. The results are shown in Figure 25. Venugopalan and Guo model is shown to overlap. For Basaj model, hardness was used from Venugopalan model. Although all the models predict a linear increase in tensile strength with increase in copper, the experimental measurements shows increase up to 0.5% Cu, and then tensile strength reaches a plateau. Figure 26 shows effect of copper addition on amount of pearlite present in matrix microstructure. Percent pearlite reaches 90–100% with 0.5% copper addition.

Figure 25

Tensile strength as a function of percentage copper for ductile iron.

Figure 26

Percent pearlite in microstructure as a function of Copper addition.

The database can be used to conduct similar studies for other variables to improve current property prediction models. Decreasing percentage error would enable use of such models not only to a certain set of samples but to a broader spectrum of cast iron production facilities.

Conclusion

This article studies the capability of published models to predict mechanical properties of cast iron (GI and DI). The prediction results are compared with experimental measurements collected from the literature. The comparison shows promising results with percent error ±25 or less. Property prediction as a function of a measured property had the least error. Industrial cast iron production facilities can conduct a similar study to find an applicable model. Based on the study the following conclusions can be inferred:

• The deviation between measured and predicted values of mechanical properties could be due to testing procedure difference and/or due to lack of variables such as raw materials, melting procedure, pouring temperature, superheat, inoculation amount and method, and magnesium treatment in the models.

• Limiting the statistical analysis within applicable range did not improve percent error in most cases.

• Property prediction models for ductile iron have less percent error compared to gray iron. Improvement in prediction of ductile iron property could be due to consideration of microstructural evolution in ductile iron models.

• The authors could not find any model for compacted graphite iron, which provides simple mathematical expression to be readily useable. There are future research opportunities to come up with such models.

• The database compiled and used by the authors for this article could be used for validation of relationships correlating cast iron composition, process variables and properties.

• Availability of such database of experimental results from diverse sources enables researchers to validate novel ideas without running rigorous experiments.

• It is of utmost importance to establish a standard procedure to document and publish experimental results in a standard tabular form to enable such future studies and improve current property prediction models.

Appendix

A sample casting, with 0.5-, 1-, and 2-inch-diameter bar (Figure 27), was used to acquire cooling rate values. Solidification only simulation was done to get the cooling rate (dT/dt) at the center of each bar at 850 and 900 °C, respectively, for casting grade GJS 400 and GJL 250. Regression analysis was done to obtain equation to calculate the cooling rate for other thicknesses within 0.5 inch to 2 inch. The regression analysis for ductile and gray iron is shown, respectively, in Figures 28 and 29.

Figure 27

Schematic of the casting design used for cooling rate calculation.

Figure 28

Cooling rate calculation for ductile iron (GJS 400) at 850 °C.

Figure 29

Cooling rate calculation for gray iron (GJL 250) at 900 °C.