Introduction

Cast iron alloys are the most produced casting material in the world, primarily due to their excellent castability, recyclability, in addition to well-balanced thermal and mechanical properties.1 In the international classification system, the different cast iron grades are designated by their ultimate tensile strength, highlighting the importance of this property. Other mechanical properties such as the fatigue limit can be determined by empirical relationships based on the ultimate tensile strength (UTS).2 The mechanical properties, including the UTS, are often predicted from the microstructure, and it is utilized to simulate the local properties of the components.3,4 It is believed that the UTS of cast irons is influenced by factors such as chemical composition, graphite shape and size, primary dendrites morphology, pearlite/ferrite fraction, pearlite interlamellar spacing and ferrite grain size.5 The size, amount and shape of graphite significantly impact the UTS, and for this reason, the maximum graphite length has been treated as the maximum crack length in the Griffith fracture equation for UTS prediction in fully pearlitic lamellar graphite iron.6,7,8,9 Bates proposed a UTS model based on the Griffith criterion.6 However, as Bates wrote in his paper, “If the equation is applicable to cast iron,” it might be possible to develop the model based on the Griffith criterion, but unfortunately, it is not proven if it is applicable or not in the above-mentioned paper. The derivation of the Griffith criterion assumes that all elastic strain energy is utilized for the creation of the new surface (corresponding to an increase in surface energy), a premise that doesn’t hold true for metallic (ductile) materials due to other energy dissipation mechanisms, such as the motion of dislocations. When plastic deformation occurs, Young’s modulus, \(E\), in the equation becomes irrelevant, and the surface energy, \(\gamma \), takes an unrealistic value. Consequently, the equation loses its physical meaning, even though the relationship \(\sigma \propto 1/\sqrt{a}\) might still be valid (where \(\sigma \) is the stress at fracture and \(a\) is the length of flaw). Irwin’s modified Griffith criterion, \(\sigma >\sqrt{\frac{2\Gamma E}{\pi a}}\), is applicable to ductile materials (where \(\Gamma \) is the effective surface energy and \(E\) is the Young’s modulus). However, in practice, the term \(\Gamma E\) in the numerator cannot be determined from microstructure information or experiments, reducing it to a mere fitting parameter. In such cases, an empirical equation with a polynomial form might suffice.

Numerous empirical models have been developed for UTS prediction in different cast iron types.10,11 These models are polynomial equations that consider various microstructural features, microconstituent fractions, alloying elements, casting thicknesses, cooling rates, etc. However, these models typically apply only within the specified ranges studied in specific works.12,13,14,15,16,17,18 Moreover, in most cases, the focus is solely on one characteristic, such as the size of graphite or nodularity, without considering variations in other properties, like changes in the amount of pearlite in the matrix. Consequently, it remains uncertain whether the obtained results genuinely stem from the single characteristic under consideration or if other factors play a role. For instance, it is recognized by some researchers that the UTS of cast irons increases as nodularity increases to approximately 75–80%, after which the UTS value plateaus with further increases in nodularity. However, it is unlikely that the parameters other than nodularity remain consistent across various types of cast irons (lamellar graphite iron/LGI, compacted graphite iron/CGI and spheroidal graphite iron/SGI), especially when it is investigated for typical microstructures for each type of cast iron. Consequently, the influence of the other factors, such as the difference in pearlite amount, is ignored. Thus, in some cases, it cannot be concluded that it is solely the influence of nodularity.

Not just for UTS, but in general, understanding phenomena solely through statistics or empirical equations is not possible. Even if a wide range of data is used for the analysis,19 the approach by the statistics or empirical equations only allows us to judge the influence of each factor, whether a specific factor has a positive or negative effect without understanding the physical meaning, and sometimes it shows unphysical results. Creating a model that has physical meaning becomes essential for a comprehensive understanding of the phenomena. In the present study, the influence of different factors will be discussed by creating a model to describe the UTS of the cast irons.

Cast iron can be considered as a composite material with graphite embedded in a pearlitic and/or ferritic metallic matrix. In this study, a UTS model was developed based on the rule of mixture. Unlike other properties, the rule of mixtures for strength is not simple as discussed in the modeling section of this paper and is seldom used, often in a highly simplified form.20,21

In this study a more detailed explanation of the UTS for lamellar graphite iron (LGI), compacted graphite iron (CGI), aka. vermicular graphite iron, and spheroidal graphite cast iron (SGI), aka. ductile iron will be provided. However, the influence of graphite shape on stress concentration, which is often discussed, will not be discussed in this paper. This is because the concept of stress concentration primarily applies to the elastic region. In cases where failures are not linked to significant plastic deformation, stress concentration becomes crucial (e.g., impact or fatigue strength). However, failures at room temperature typically occur after significant plastic deformation, minimizing the role of stress concentration.

In the present work, the key factor lies in the continuity of the matrix, as highlighted by Bates who mentioned, “The available data indicate that cast iron can be modeled as a steel with dispersed graphite flakes disrupting the continuity of the matrix,” although the specifics are not extensively addressed in his paper. The aim of this study is to describe the influence of different factors, such as eutectic cell size, pearlite amount and nodularity, on the UTS by considering the physical meanings. The influence of chemical composition on UTS is not discussed in the model. The parameters, such as the UTS of a specific phase used in the model, might be influenced by chemical composition, but the current focus of this work is on showing how one can describe the UTS with a physical meaning.

Experimental

Table 1 shows the chemical composition of the base alloy employed in the experiment. In this study, the base alloy for SGI was subjected to remelting and held under isothermal conditions. The details of the experimental method are described elsewhere,22 and so here only a brief summary of the method is given.

Table 1 Chemical Composition of the Base Alloy (mass%)
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The base alloy is cast into two furan sand molds, each containing 25 open cylindrical cavities of 50 mm in diameter and 300 mm in height. These molds are produced using a sand printing technique and are specifically designed to facilitate a bottom-to-top filling of the cavities. The two molds are poured from the same ladle in a sequential operation to minimize the fading of nodularization treatment and the decarburization of the alloy. As shown in Table 1, the magnesium content in the base alloy is relatively high, and the magnesium content was regulated by varying the holding time at 1450 °C during the remelting cycle, resulting in samples with diverse nodularities, ranging from 0 to 85%. This regulation technique was employed in some researches.22,23 It is not mentioned in these papers, but it is believed that the magnesium content decreases with increasing holding time due to the evaporation of magnesium. The cooling rate during solidification remained consistent across all samples.

The tensile strength was measured using a Kammrath and Weiss testing module, following ASTM E8 standard, employing flat miniaturized tensile test bars with dimensions of 12 mm gauge length and 2 mm thickness. Microstructural analysis was conducted for the location in the vicinity of the fracture surface. Forty micrographs, each with a magnification of 10x, were captured from the polished surface of each sample to quantify nodularity and the area fraction of graphite. The evaluation was made using an Olympus GX71F optical microscope with Olympus Stream Desktop 2.3 image analysis software. The samples underwent etching in a picric acid-based solution to reveal the eutectic cells. The etching reagent contains picric acid, NaOH, KOH, and distilled water in a ratio of 1:1:4:5, respectively. Eutectic cell size was determined based on the average size of the 5–7 largest eutectic cells located in the area near the fracture surface. The percentage area of ferrite and pearlite was measured from nital-etched microstructural samples (4 ml HNO3 and 100 ml alcohol) using the open-source image analysis software ImageJ.

Modelling

Basic Description

In the present study, a model was developed based on the rule of mixtures. Firstly, let’s consider Component 1, which is more ductile and has higher strength than Component 2. Then, the stress-strain curves will become as shown in Figure 1. In the figure, \({\sigma }_{\text{UTS},1}\) is the stress at which the matrix phase fractures (i.e., UTS of Component 1), \({\sigma }_{\text{UTS},2}\) is the stress at which Component 2 fractures (i.e., UTS of Component 2), \({\varepsilon }_{2}\) is the strain at which Component 2 fractures and \({\sigma }_{1}^{*}\) is the stress acting on Component 1 at \({\varepsilon }_{2}\).

Figure 1
figure 1

Schematic of the stress-strain curves for two components of a composite material.

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In this scenario, when the fraction of Component 1 is relatively low, the stress at which Component 2 fractures ( \({\sigma }_{1}^{*}{f}_{1}+{\sigma }_{\text{UTS},2}\left(1-{f}_{1}\right)\), shown as linear segment (1) in red in Figure 2) becomes higher than the stress that Component 1 can withstand (\({\sigma }_{\text{UTS},1}{f}_{1}\), shown as dotted linear segment (2) in Figure 2). Consequently, the composite will fracture when Component 2 fails, i.e., at \({\sigma }_{1}^{*}{f}_{1}+{\sigma }_{\text{UTS},2}\left(1-{f}_{1}\right)\) where \({f}_{1}\) is the fraction of Component 1.

Figure 2
figure 2

Schematic illustration of ultimate tensile strength against the fraction of Component 1, \({f}_{1}\).

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On the other hand, when the fraction of Component 1 is relatively high, the stress along linear segment (2) in blue becomes higher than the stress shown by the dotted linear segment (1). In this case, even if Component 2 fractures, Component 1 will bear the load, and the composite will fracture when the stress reaches \({\sigma }_{\text{UTS},1}{f}_{1}\).

A challenge in providing a quantitative description of the rule of mixture for strength lies in estimating \({\varepsilon }_{2}\), i.e., \({\sigma }_{1}^{*}\). Therefore, the actual behaviour (linear segments (1) and (2)) of the UTS is approximated by linear segment (3), i.e., Eqn. 1.

$${\sigma }_{\text{UTS}}={f}_{1}{\sigma }_{\text{UTS},1}+\left(1-{f}_{1}\right){\sigma }_{\text{UTS},2}$$
(1)

where \({f}_{1}\) is the fraction of Component 1, \({\sigma }_{\text{UTS},1}\) is the UTS of Component 1, and \({\sigma }_{\text{UTS},2}\) is the UTS of Component 2.

The deviation from the actual UTS behaviour is addressed by introducing an interaction parameter as detailed in the next section. As can be seen in Figure 2, this simplification results in an overestimation of UTS. This kind of overestimation could potentially be adjusted by introducing an interaction parameter as detailed in the next section.

Modelling for Cast Irons

The LGI microstructure includes eutectic cells containing flaky graphite (0% nodularity). The SGI microstructure includes nodular graphite (relatively high nodularity). The microstructure of CGI bears the morphology of both cast iron types, LGI and SGI (intermediate nodularity), which is implemented in the developed UTS model. Thus, we may apply the rule of mixture for the LGI (weak and brittle phase) and SGI (strong and ductile phase with 100% nodularity). In this context, the strength of the mixture can be expressed as a function of the fraction of SGI part. Components 1 and 2 mentioned in the previous section will be replaced by SGI and LGI, respectively. Then, by referring Figure 2, the UTS of the mixture can be described as follows:

$${\sigma }_{\text{UTS},\text{mix}}={f}_{\text{SGI}}{\sigma }_{\text{UTS},\text{SGI}}+{f}_{\text{LGI}}{\sigma }_{\text{UTS},\text{LGI}}$$
(2)

where \({\sigma }_{\text{UTS},\text{mix}}\) is the UTS of the mixture, \({f}_{\text{SGI}}\) is the fraction of SGI, \({\sigma }_{\text{UTS},\text{SGI}}\) is the UTS of SGI, \({f}_{\text{LGI}}\) is the fraction of LGI and \({\sigma }_{\text{UTS},\text{LGI}}\) is the UTS of LGI.

The nodularity (%), \(N\), will be expressed through the fraction of SGI, i.e., \(N=100{f}_{\text{SGI}}\).

Regarding the UTS of LGI (\({\sigma }_{\text{UTS},\text{LGI}}\)), in the present model, it is assumed that the eutectic cell region is fully ferritic and the cell is surrounded by the matrix with a certain amount of pearlite. The strength of the graphite phase, and the strength of the graphite and matrix interface are assumed to be zero, essentially treating the graphite phase as pore defect. Under these assumptions, \({\sigma }_{\text{UTS},\text{LGI}}\) can be expressed by Eqn. 3 by applying the rule of mixture to the “LGI part” which consists of eutectic cell and the surrounding pearlite phase.

$${\sigma }_{\text{UTS},\text{LGI}}={\sigma }_{\text{UTS},\alpha }\left(1-{f}_{\text{G}}\right){f}_{\text{eut}}+{\sigma }_{\text{UTS},\text{p}}{f}_{\text{p},\text{LGI}}$$
(3)

where \({\sigma }_{\text{UTS},\alpha }\) is the UTS of ferrite, \({f}_{\text{G}}\) is the fraction of graphite, \({f}_{\text{eut}}\) is the fraction of eutectic cell in the LGI part, \({\sigma }_{\text{UTS},\text{p}}\) is the UTS of pearlite and \({f}_{\text{p},\text{LGI}}\) is the fraction of pearlite in the LGI part.

The UTS of LGI is also influenced by the eutectic cell size. As shown in Figure 3, when the crack propagates through the eutectic cells, when the eutectic cell size is smaller, there are more discontinuities of the graphite phase, acting as resistance against crack propagation. This type of resistance, characterized by the discontinuity of the graphite phase, might be referred as “matrix bridges.”24 Based on these considerations, it is fair to say that the UTS of the LGI part decreases with increasing the eutectic cell size. In a different perspective, if the eutectic cell is large, the size of the graphite within the eutectic cell is also large, which corresponds to a large defect. In the present model, it is assumed that the UTS of the LGI part linearly decreases with increasing the eutectic cell size. Note that the UTS of the pearlite part will not be influenced by the eutectic cell size. Then, the UTS of the LGI part, \({\sigma }_{\text{UTS},\text{LGI}}\), can be expressed by Eqn. 4.

$${\sigma }_{\text{UTS},\text{LGI}}=\left(1-a{D}_{\text{eut}}\right){\sigma }_{\text{UTS},\alpha }\left(1-{f}_{\text{G}}\right){f}_{\text{eut}}+{\sigma }_{\text{UTS},\text{p}}{f}_{\text{p},\text{LGI}}$$
(4)

where \({D}_{\text{eut}}\) is the diameter of the eutectic cell, \(a\) is the coefficient to describe the degree of influence of the eutectic cell diameter.

Figure 3
figure 3

Schematic illustration of the eutectic cell and crack propagation mechanism.

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Regarding the UTS of SGI (\({\sigma }_{\text{UTS},\text{SGI}}\)), the graphite phase is treated as a defect, and its strength is assumed to be negligible, similar to the approach taken for the LGI case. Then,

$${\sigma }_{\text{UTS},\text{SGI}}={\sigma }_{\text{UTS},\text{mat}}\left(1-{f}_{\text{G}}\right)$$
(5)

where \({\sigma }_{\text{UTS},\text{mat}}\) is the UTS of matrix.

The UTS of the matrix (\({\sigma }_{\text{UTS},\text{mat}}\)) can be obtained using the rule of mixture for the ferrite and pearlite phases in the matrix as follows.

$${\sigma }_{\text{UTS},\text{mat}}={f}_{\text{p}}{\sigma }_{\text{UTS},\text{p}}+\left(1-{f}_{\text{p}}\right){\sigma }_{\text{UTS},\alpha }$$
(6)

where \({f}_{\text{p}}\) is the fraction of pearlite in the matrix, \({\sigma }_{\text{UTS},\alpha }\) is the UTS of ferrite and \({\sigma }_{\text{UTS},\text{p}}\) is the UTS of pearlite. Note that \({\sigma }_{\text{UTS},\alpha }\) and \({\sigma }_{\text{UTS},\text{p}}\) depend on the composition, grain size, etc., and thus, for the accurate prediction of UTS value (\({\sigma }_{\text{UTS},\text{mix}}\)), these values must be determined by experiments or any other methods including modelling.

Hence, Eqn. 2 can be rewritten as follows.

$${\sigma }_{\text{UTS},\text{mix}}=\left\{\left(1-a{D}_{\text{eut}}\right){\sigma }_{\text{UTS},\alpha }\left(1-{f}_{\text{G}}\right){f}_{\text{eut}}+{\sigma }_{\text{UTS},\text{p}}{f}_{\text{p},\text{LGI}}\right\}{f}_{\text{LGI}}+\left(\left(1-{f}_{\text{p},\text{SGI}}\right){\sigma }_{\text{UTS},\alpha }+{f}_{\text{p},\text{SGI}}{\sigma }_{\text{UTS},\text{p}}\right)\left(1-{f}_{\text{G}}\right){f}_{\text{SGI}}$$
(7)

Equation 7 corresponds to the line for \({\sigma }_{\text{UTS},\text{LGI}}\times {f}_{\text{LGI}}+{\sigma }_{\text{UTS},\text{SGI}}\times {f}_{\text{SGI}}\) in Figure 4, i.e., the ideal mixture case.

Figure 4
figure 4

Schematic illustration of Eqns. (7) and (8).

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In the current model, it is assumed that the matrix part takes the stress, and the degree of continuity of the matrix is considered a crucial factor determining the UTS, and it is fair to say that the continuity of the matrix is increased with increasing of the frction of SGI (\({f}_{\text{SGI}}\)). Given that the degree of continuity of the matrix in the case of mixture, including CGI, differs from a simple mixture of LGI and SGI (i.e., CGI is not a simple mixture of LGI and SGI), deviations from the ideal mixture case are addressed by introducing an interaction parameter. Consequently, an interaction term, \(\Omega \cdot {f}_{\text{CGI}} \cdot {f}_{\text{SGI}}\) is introduced to Eqn. 7.

Thus, Eqn. 7 is modified as follows:

$${\sigma }_{\text{UTS},\text{mix}}=\left\{\left(1-a{D}_{\text{eut}}\right){\sigma }_{\text{UTS},\alpha }\left(1-{f}_{\text{G}}\right){f}_{\text{eut}}+{\sigma }_{\text{UTS},\text{p}}{f}_{\text{p},\text{LGI}}\right\}{f}_{\text{LGI}}+\left(\left(1-{f}_{\text{p},\text{SGI}}\right){\sigma }_{\text{UTS},\alpha }+{f}_{\text{p},\text{SGI}}{\sigma }_{\text{UTS},\text{p}}\right)\left(1-{f}_{\text{G}}\right){f}_{\text{SGI}}+\Omega \cdot {f}_{\text{LGI}} \cdot {f}_{\text{SGI}}$$
(8)

where \(\Omega \) is the interaction parameter.

The interaction term is represented as the dotted curve in Figure 4, and the summation of the interaction term and the ideal mixture term results in \({\sigma }_{\text{mix}}\). The interaction parameter \(\Omega \), i.e., deviation from the ideal mixture case, may depend on the amount of graphite, but for the present study, a constant graphite amount is considered. It should be noted that the UTS of LGI (\({\sigma }_{\text{UTS},\text{LGI}}\)) depends on the eutectic cell size as shown in Eqn. 4, and the interaction parameter \(\Omega \) may depend on the \({\sigma }_{\text{UTS},\text{LGI}}\). However, for simplicity, it is assumed that the \(\Omega \) value does not depend on \({\sigma }_{\text{UTS},\text{LGI}}\). The interaction parameter, \(\Omega \), describes the geometric interaction, justifying this assumption. As can be seen in Figure 4, the interaction parameter term is symmetrical for \({f}_{\text{SGI}}\). In practice, it is probably not symmetrical, but due to the limitations of the currently available data, there is no possibility for an alternative description. Therefore, as a first approximation, the symmetrical shaped function is assumed.

Results and Discussion

In this section, some applications of Eqn. 8 will be demonstrated. The required parameters were measured through the microstructure observations. Typical micrographs for the measurements of the nodularity/fraction of graphite, eutectic cell diameter, and fraction of pearlite are shown in Figures 5, 6 and 7, respectively.

Figure 5
figure 5

A typical micrograph for the graphite fraction and nodularity measurements. Nodularity: (a) 6% (b) 10% and (c) 33%.

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Figure 6
figure 6

A typical micrograph (Color-etched micrograph) for the eutectic cell size measurements.

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Figure 7
figure 7

A typical micrograph (nital 4% etched) for the measurement of ferrite and pearlite fraction measurements. Nodularity: (a) 6% (b) 10% and (c) 33% . Pearlite/Ferrite/Graphite fractions: (a) 0.63/0.27/0.1, (b) 0.74/0.17/0.09 and (c) 0.83/0.08/0.09.

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The UTS and the necessary parameters for the model were measured for 11 samples, and the measurement results are summarized in Table 2. As described in the Experimental section, the sample thickness is relatively thin (2 mm), and one might suspect problems with it since the thickness of the sample is only several times larger than the eutectic cell size. However, Noguchi25 compared the UTS of 2 mm and 8 mm samples for LGI and SGI and concluded that there is no significant difference in the average UTS values. In addition, in the present study, the UTS values obtained from the samples with the same dimension (2 mm thick), are reported. Thus, consistency is maintained, and the structure of the suggested model is not influenced by the accuracy of the obtained UTS values.

Table 2 Summary of the Required Parameters
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In the case of the experimental results presented in Table 2, where the pearlite amount is relatively high, we can assume that the fraction of pearlite in the SGI part matrix (\({f}_{\text{p},\text{SGI}}\)) is equal to 1, indicating fully pearlitic matrix. To achieve the specified pearlite fraction in the sample (\({f}_{\text{p}}\)) as shown in Table 2, the pearlite fraction in the LGI part is increased accordingly, i.e., it is considered that the LGI part consists of eutectic cell and pearlite. The fraction of pearlite in the LGI part was calculated from the difference between the measured total pearlite amount and the amount of pearlite in the SGI part. The fraction of eutectic cell in the LGI part (\({f}_{\text{eut}}\)) is equal to \(1-{f}_{\text{p},\text{LGI}}\). \({f}_{\text{p},\text{LGI}}\) can be calculated from the total pearlite amount in a sample. The fraction of the LGI part (\({f}_{\text{LGI}}\)) is equal to \(1-{f}_{\text{SGI}}\). The fraction of the SGI part (\({f}_{\text{SGI}}\)) can be described by the nodularity as \({f}_{\text{SGI}}=N/100\).

Then, Eqn. 8 can be rewritten as follows:

$${\sigma }_{\text{UTS},\text{mix}}=\left\{\left(1-a{D}_{\text{eut}}\right){\sigma }_{\text{UTS},\alpha }\left(1-{f}_{\text{G}}\right)\left(1-{f}_{\text{p},\text{LGI}}\right)+{\sigma }_{\text{UTS},\text{p}}{f}_{\text{p},\text{LGI}}\right\}\left(1-N/100\right)+{\sigma }_{\text{UTS},\text{p}}\left(1-{f}_{\text{G}}\right)\left(N/100\right)+\Omega \cdot \left(1-N/100\right) \cdot N/100$$
(9)

The parameters in Eqn. 9 can be determined as follows. The coefficient to describe the influence of the eutectic cell size on the UTS, \(a\), will be determined by fitting. The measured values will be used for the eutectic cell diameter (\({D}_{\text{eut}}\)), the fraction of graphite (\({f}_{\text{G}}\)) and the nodularity (\(N\)). The interaction parameter \(\Omega \) will be determined by the fitting. The UTS of ferrite (\({\sigma }_{\text{UTS},\alpha }\)) and pearlite (\({\sigma }_{\text{UTS},\text{p}}\)) will also be determined by fitting.

To obtain the values of \(a\), \(\Omega \), \({\sigma }_{\text{UTS},\alpha }\) and \({\sigma }_{\text{UTS},\text{p}}\), the fitting routine was performed using Eqn. 9 and the data presented in Table 2. For the fitting, the random search method was employed and implemented using the commercially available software, MATLAB. The data for Samples 2, 5 and 10 were excluded from the fitting dataset, allowing them to be reserved for validation (Note: The exclusion of Samples 2, 5 and 10 were arbitrary, and any of the 11 samples could have been chosen). Then, the fitting results become, \(a=8.63\times {10}^{-4}\), \(\Omega =-62.2\), \({\sigma }_{\text{UTS},\alpha }=254\) and \({\sigma }_{\text{UTS},\text{p}}=632\).

Regarding the values for \({\sigma }_{\text{UTS},\alpha }\) and \({\sigma }_{\text{UTS},\text{p}}\), if we apply the rule of mixture for Samples 9–11, which have relatively high nodularity and a sufficiently high degree of continuity of the matrix, and estimate the \({\sigma }_{\text{UTS},\alpha }\) and \({\sigma }_{\text{UTS},\text{p}}\) by fitting, the obtained values are \(321\) MPa and \(760\) MPa, respectively. In addition, according to,26 \({\sigma }_{\text{UTS},\alpha }\) is \(196-392\) MPa and \({\sigma }_{\text{UTS},\text{p}}\) is \(785-883\) MPa. Considering that these values depend on the grain size, composition, etc., the obtained values (\({\sigma }_{\text{UTS},\alpha }=254\) and \({\sigma }_{\text{UTS},\text{p}}=632\)) seem reasonable. Any errors in \({\sigma }_{\text{UTS},\alpha }\) and \({\sigma }_{\text{UTS},\text{p}}\) will propagate to the fitting parameters \(a\) and \(\Omega \).

The calculated UTS and experimental values are shown in Figure 8. Considering the errors in the UTS and microstructural parameters, it can be concluded that the calculated values and experimental values reasonably agree with each other. Since the value of \(a\) is \(8.63\times {10}^{-4}\), the UTS of the eutectic cell is expected to reach zero at \({D}_{\text{eut}}=\) 1159 µm. As shown in Table 2, the eutectic cell diameter of Sample 1 exceeds this critical value. The lower UTS was observed may be attributed to uncertainties associated with the measurement of each parameter, including the UTS value. In addition, a certain variation of chemical composition is expected even though the same base alloy is used. Therefore, \({\sigma }_{\text{UTS},\alpha }\) and \({\sigma }_{\text{UTS},\text{p}}\) are expected to vary. Considering these uncertainties, the observed difference is likely within the range of experimental and measurement errors.

Figure 8
figure 8

Calculated UTS values against experimental values.

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The results of Samples 2, 5 and 10 which were excluded from the fitting and reserved for validation, are shown as triangles in Figure 8. These results also show reasonable agreements between the calculated and experimental values. The agreement between the calculated and experimental values is less critical in this context since the calculated values are obtained through fitting. The more significant aspect lies in the model’s ability to describe the influence of each factor on the UTS. In the following section, the influence of the each factor on the UTS will be discussed.

Based on the current model (Eqns. (8) and (9)), the influence of specific factors, such as eutectic cell size, nodularity and fraction of pearlite can be discussed. In this section, the analysis of some selected concrete cases will be shown.

To examine how the eutectic cell size affects the UTS, let us take the conditions for Sample 5 as an example. If the eutectic cell diameter is decreased by 100 µm (under the same values of the other parameters), the UTS will be increased by 18.0 MPa, which corresponds to 11.7% increase.

The impact of nodularity on the UTS cannot be solely assessed by taking the partial differentiation of Eqn. 9 with respect to nodularity. This is due to the fact that the pearlite amount is altered by the fraction of the SGI part, namely, by the nodularity itself. With the aim to explore the impact of nodularity, consider a fully ferritic matrix to avoid the mathematical complexity.

In this case, Eqn. 8 would be re-written as Eqn. 10:

$${\sigma }_{\text{UTS},\text{mix}}=\left(1-a{D}_{\text{eut}}\right){\sigma }_{\text{UTS},\alpha }\left(1-{f}_{\text{G}}\right)\left(1-N/100\right)+\left({\sigma }_{\text{UTS},\alpha }\right)\left(1-{f}_{\text{G}}\right)\left(N/100\right)+\Omega \cdot \left(1-N/100\right) \cdot \left(N/100\right)$$
(10)

Then, on the example of the sample with \(N=10\%\), \({D}_{\text{eut}}=500\) µm and \({f}_{\text{G}}=0.1\), the calculation results in \({\sigma }_{\text{UTS},\text{mix}}\) = 136 MPa. If the nodularity is increased to e.g., 60%, then \({\sigma }_{\text{UTS},\text{mix}}\) increases to 179 MPa. Equation (8) also reveals that instead of increasing the nodularity from 10% to 60%, an equivalent outcome (\({\sigma }_{\text{UTS},\text{mix}}\)=179 MPa) can be achieved by either incorporating 10% of pearlite in the matrix or reducing the eutectic cell diameter from 500 µm to 254 µm. For instance, increasing nodularity from 10% to 60% will have a smaller impact on hardness than increasing pearlite by 10%. Thus, if one aims to increase the UTS without increasing hardness, nodularity should be increased, whereas if both UTS and hardness are desired to be increased, the amount of pearlite should be increased. The impact of each factor depends on the microstructure conditions, and the above calculation is just an example to give a feeling of the impact of each factor. However, the present model can be utilized to derive guidelines for alloy design. For instance, the developed UTS model is expected to be used in combination with the other material property models, such as thermal conductivity, based on the microstructural parameters,27 thereby designing the alloys with desired performance.

Conclusion

A model was developed to describe the ultimate tensile strength (UTS) of cast iron with different microstructures. This model integrated the rule of mixtures for different sections of the microstructure and included an interaction term to account for the impact of mixing on the continuity of the matrix. As a result, the UTS was expressed as a function of microstructural parameters such as pearlite fraction, eutectic cell size, and nodularity. This made it possible to estimate the influence of individual microstructural parameters on the UTS. The demonstration examples showed how to attain a specific strength by adjusting each microstructural parameter, thereby revealing the impact of each parameter. It contributes to the model being applied as a “tool” for the cast iron design to achieve the desired mechanical properties.