# The Role of Microstructure on the Tensile Plastic Behaviour of Ductile Iron GJS 400 Produced through Different Cooling Rates—Part II: Tensile Modelling

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Material

#### 2.2. Tensile Tests and Microstructure Plasticity Model

_{0}= 12.5 mm and a length of l

_{0}= 50 mm, complying with the standard ASTM E8-8M with a strain rate of 10

^{−4}s

^{−1}. The true stress–true plastic strains (σ vs. ε

_{p}) were used, where σ = S∙(1 + e) and ε

_{p}= ε – σ/E = ln(1 + e) – σ/E (with S and e as the engineering stress and elongation, respectively) and E is the experimental Young modulus.

_{o}+ σ

_{G}(ε

_{P}) = σ

_{o}+ Mα

_{o}Gbρ

^{1/2},

_{o}is the initial stress because of the solid solution or precipitation strengthening, and σ

_{G}is the component of stress depending on the increase of the dislocation density ρ because of strain ε

_{P}. M is the Taylor factor (3.01 in BCC materials), α

_{o}the dislocation-dislocation interaction strength (0.5) [22], G the elastic shear modulus for ferrite (64 GPa), and b the Burgers vector length of ferrite (0.248 nm). At strain ε

_{P}= 0, σ

_{G}was assumed to be nil because of the negligible initial dislocation density of GJS 400 in the cast conditions. The total dislocation-density ρ increases because of straining, according to the mechanistic evolution equation by Kocks-Mecking-Estrin [22,23,24,25,26,27,28]:

_{o}is the dynamic recovery term that describes the softening of materials during straining because of dislocation annihilation and low energy dislocation structure formation. Λ, D, and λ are the microstructure characteristic lengths; Λ is the dislocations mean free path related to the dislocation cells in ferrite with Λ = β/ρ

^{1/2}and a β constant [22], D is the ferritic grain size or pearlitic island size, and λ is the interlamellar spacing in pearlite.

_{G}= Mα

_{o}Gbρ

^{1/2}and Λ = β/ρ

^{1/2}, and considering that in ferrite the grain boundaries and dislocation cells are the obstacles to dislocation motion (for ferrite Equation (2)), results in

_{o}(= 1.538 × 10

^{5}MPa) and K

_{1}(= 7.565 × 10

^{6}MPa

^{2}∙μm) are constants depending on the BCC ferritic crystal, while β and 1/ε

_{c,F}are outputs from fitting. The detailed calculations to obtain Equation (3) from Equations (1) and (2) are reported in Appendix A.

_{Ferrite}, and the pearlitic colony size, D

_{Pearlite}. Thus, pearlite Equation (2) results in

_{c,P}is the output from the fitting, if λ is known. The detailed calculations to obtain Equation (4) from Equations (1) and (2) are reported in Appendix A. The equation σ

_{G}vs. ε

_{P}, resulting from integrating Equation (4), is an exponential decay equation with a saturation stress σ

_{S,P}that is the maximum stress achieved asymptotically at the condition dσ

_{G}/dε

_{P}= 0, while ε

_{c,P}is the critical strain that defines the rate at which σ

_{S,P}is achieved. However, if an average characteristic λ cannot be measured (like in the present investigation of GJS 400 because of the complexity of pearlitic microstructures [17]), Equation (4) can be fitted to the experimental data considering the quantity (K

_{1}/λ) as a further output from the fitting. Then, from (K

_{1}/λ) and ε

_{c,P}, the saturation stress σ

_{S,P}= [(K

_{1}/λ)∙ε

_{c,P}] can be found to test the physical meaning of the fitting results.

_{P}) in GJS 400 produces a mixture rule:

_{P}) = (1 − X

_{Pearlite})·σ

_{Ferrite}(ε

_{P}) + X

_{Pearlite}·σ

_{Pearlite}(ε

_{P})

_{Pearlite}is the pearlite volume fraction, and (1 – X

_{Pearlite}) is the ferrite volume fraction. The rule of mixture that has been usually used for all two phase materials [29] has also been successfully used in Dual Phase (DP) steels [30,31,32] whose microstructures consist of soft ferrite and hard martensite. Equation (5) was used successfully in DP steels for hardness, Yield Stress (YS), and Ultimate Tensile Stress (R

_{m}). In terms of mechanical constituents, DP’s microstructure has similarities with the investigated GJS 400, consisting of soft ferrite and hard pearlite (and graphite nodules), so Equation (5) was used for the present investigation.

## 3. Results

#### 3.1. Model Calibration

_{o}= 243.1 ± 6.2 MPa (see Equation (1)) was found. Indeed, the plastic flow curves of GJS 400 from different moulds did not change during the early stages of deformations but was significant at high strains, which could be rationalized by the findings that ferrite was the dominant softer constituent that deformed first at yielding, while the smaller volume fractions (<4%) of harder pearlite contributed significantly later at higher strains. Thus, σ

_{o}= 243.1 MPa was used to model all the flow curves of GJS 400 from the other moulds. The strain hardening data of the tensile flow curves of GJS 400 from the Lynchburg mould with an average ferrite grain size of 37.3 ± 3.2 μm were fitted, yielding the following average values for the equation parameters: 1/ε

_{c,F}= 6.36 ± 0.25 and β = 119.1 ± 8.7 MPa (see Equation (3)). These values are consistent with the literature, where β has been reported to be between 100 and 200 [22], proving the physical meaning and, in turn, the validity of the model.

_{Pearlite}(ε

_{P}) vs. ε

_{P}, Equation (5) was fitted to the GJS 400 Y 25 mm tensile data considering the quantity (K

_{1}/λ) and the parameter 1/ε

_{c,P}as outputs. In the GJS 400 produced with Y 25 mm, the average volume fraction of the pearlite was 3.8% ± 0.4%, and the average grain size was 39.2 ± 2.3 μm (see Table 2). The fitting resulted in a pearlite flow curve with an average saturation stress of σ

_{S,P}= 1094.4 ± 106.0 MPa and an average critical strain parameter of 1/ε

_{c,P}= 22.1 ± 3.3. In Figure 2a,b, the fitting results are reported for a typical flow curve of GJS 400 from the Y 25 mm mould sample, detailing the contributions from the ferrite and pearlite. The fit was excellent at high stresses, while at low strains there was some mismatch. Though σ

_{S,P}was consistent with the results reported in the literature for Isothermed Ductile Irons 1000 with a pearlite volume fraction higher than 80% [27], the 1/ε

_{c,P}for pearlite was quite low, considering that it should have been just slightly lower than 40. In other words, the pearlite contribution to the flow curve in Figure 2b should have increased faster while keeping the same saturation stress, σ

_{Pearlite,V}. The reasons for this result are not evident and need further investigation.

#### 3.2. Model Prediction

## 4. Discussion

#### 4.1. Considerations of the Minimum Requirements of Data Statistics Complying with the Standards ASTM E2567-16a and ASTM E112-13

#### 4.2. Microstructure Parameters Relevant to Describing the Plastic Behaviour of GJS 400

_{m}) built with the average microstructure parameters for the four different moulds in Table 2 are reported in Figure 5, while the R

_{m}values, the elongations at R

_{m}, e

_{n}(n after necking), and the yield stress, YS, are reported in Table 3. In fact, the comparison of the model flow curves in engineering stress–strain affords an extended evaluation of the model results, since all flow curves strained beyond necking correspond to the end of uniform elongation and the occurrence of localised deformation. Since the final rupture e

_{R}could be affected by local defects in the necking, e

_{R}prediction was beyond the aims of the present investigation.

_{m}= 440.5 MPa) and the least ductile (elongation at R

_{m}= 15.8%) (in Figure 5) because of the combination of its small ferritic grain size (39.2 ± 2.3 μm) and high pearlite volume fraction (3.8 ± 0.4%), while in the Lynchburg mould, even if the ferritic microstructure was the finest (37.3 ± 3.2 μm) because it had the highest solidification rate, the absence of pearlite produced the softest microstructure with the lowest R

_{m}(424.4 MPa) and the most ductile microstructure with the largest elongation at R

_{m}(16.6%). The significant increases of ferritic grain size in the Y 50 mm (48.6 ± 4.7 μm) and Y 75 mm (47.7 ± 7.0 μm) samples, which should have significantly weakened the GJS 400 microstructures, were, indeed, compensated by the significant presence of pearlite (4.0% ± 1.6% in Y 50 mm and 3.0% ± 0.5% in the Y 75 mm mould). Thus, the engineering flow curves in Figure 6c for the Lynchburg and Y 50 mm and 75 mm moulds were finally comparable. It is noteworthy that the model also correctly described the elongations at R

_{m}, since in the comparable flow curves (the Lynchburg, Y 50 mm, and Y 75 mm moulds), the microstructures with the higher pearlite volume fractions presented shorter elongations to R

_{m}(15.9% in Y 50 mm and 16.0% in the Y 75 mm mould), which is consistent with the fact that the microstructure constituents that strengthen materials reduced their ductility.

#### 4.3. Considerations of Other Microstructural Parameters

_{c1}when the eutectoid transformation starts, should have effect on mechanical properties. In the present GJS 400 investigation [17], the cooling rates at Ac1 were 2.40 °C/min in the Y 75 mm mould, 3.54 °C/min in the Y 50 mm mould, and 5.47 °C/min in Y 25 mm. In the GJS 400 samples produced with different moulds [17], the pearlite was irregular, and its shape was rarely lamellar. It depended instead on grain orientation in agreement with [7,35], so it was not possible to measure any characteristic interlamellar spacing. However, the results reported in Figure 4, where a single pearlite flow curve was valid for all samples, and the fact that the pearlite volume fraction was the only significant parameter, suggest that the pearlite’s characteristic widths likely did not change significantly in the range of the investigated cooling rates through A

_{c1}. However, the pearlite volume fractions reported in Table 2 varied slightly from 0% to 4% in the different moulds, and this could be another possible reason why a single pearlite flow curve (i.e., a single pearlite characteristic width) could be used successfully.

## 5. Conclusions

- This model described very well the experimental flow curves at high strains, while at low strains, minor mismatching was present. This mismatching was ascribed to the graphite-matrix decohesion;
- The plastic behaviour of the GJS 400 with different microstructures depended mainly on the ferritic grain size and pearlitic volume fraction, while the other microstructure parameters were not needed to rationalize the GJS 400’s plastic behaviour;
- The correlation between the mechanical constituents (ferrite and pearlite), physical parameters, and microstructure was validated, so the use of dislocation-related-dislocation density constitutive equations (like the Voce and Estrin equations) for different DI grades reported in previous investigations was also validated;
- The results proved that the data gathered while complying with the minimum requirements of the standards’ statistics were not enough to produce accurate microstructural data.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Mechanistic Equation of Strain Hardening and Physical Parameters

_{o}= dynamic recovery term;

- Λ = dislocation mean free path related to the dislocation cells in ferrite;
- D = ferritic grain size or pearlitic island size;
- λ = interlamellar spacing in pearlite.

^{1/2}, with a β constant of the magnitude between 100 and 200 [22], Equation (A1) becomes

_{o}the dislocation–dislocation interaction strength (0.5) [22], G the elastic shear modulus for ferrite (64 GPa), and b the Burgers vector length of the ferrite (0.248 nm), which results in

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**Figure 1.**Scanning electron microscopy (SEM) micrograph with secondary electron imaging of GJS 400 produced with the Y 75 mm mould after etching with Nital 2%: nodular graphite (black) in the ferritic matrix with bright grain boundaries and pearlitic islands (light grey).

**Figure 2.**(

**a**) Example of fitting Equation (5) with a typical tensile flow curve of GJS 400 from the Y 25 mm mould sample; (

**b**) the same fitting in (

**a**) at a different scale to highlight the contributions from the pearlite.

**Figure 3.**Examples of modelling with Equations (3)–(5) the tensile flow curve of the GJS 400 from different moulds: (

**a**) and (

**b**) from Y 50 mm; (

**c**) and (

**d**) from Y 75 mm.

**Figure 4.**Modelling the tensile flow curves of the GJS 400 reported in Figure 3 using the average microstructure values in Table 2: (

**a**) and (

**b**) Y 50 mm with the average ferrite grains size = 48.6 μm, and average pearlite volume fraction = 4.0%; (

**c**) and (

**d**) Y 75 mm with average ferrite grains size = 47.7 μm, and average pearlite volume fraction = 3.0%.

**Figure 5.**Engineering stress–strain flow curves (up to ultimate tensile stress R

_{m}) built with the average microstructure parameters for the four different moulds reported in Table 2.

**Figure 6.**Engineering stress–strain curves (up to an ultimate tensile stress of R

_{m}): (

**a**) Lynchburg and Y 25 mm moulds and (

**b**) ferrite contribution to flow curves only; (

**c**) Lynchburg, Y 50 mm and Y 75 mm moulds; (

**d**) ferrite contribution to flow curves only.

C | Si | Mg | Mn | Cu | Ni | Cr | P | S | Fe |
---|---|---|---|---|---|---|---|---|---|

3.63 | 2.45 | 0.046 | 0.129 | 0.133 | 0.0168 | 0.023 | 0.038 | 0.0043 | Bal. |

**Table 2.**Average parameters of the GJS 400 microstructures produced with four different moulds. The errors associated with the measurements are the standard deviations.

Mould | Nodule Count (mm^{−2}) | Nodule Size (μm) | Nodularity (%) | Pearlite Volume Fraction (%) | Ferrite Grain Size (μm) |
---|---|---|---|---|---|

Lynchburg 25 mm | 261 ± 15 | 24.3 ± 0.6 | 89.8 ± 3.0 | - | 37.3 ± 3.2 |

Y 25 mm | 242 ± 11 | 24.9 ± 0.5 | 91.2 ± 1.6 | 3.8 ± 0.4 | 39.2 ± 2.3 |

Y 50 mm | 116 ± 14 | 31.5 ± 1.0 | 87.1 ± 1.4 | 4.0 ± 1.6 | 48.6 ± 4.7 |

Y 75 mm | 105 ± 9 | 34.5 ± 0.5 | 83.2 ± 4.6 | 3.0 ± 0.5 | 47.5 ± 7.2 |

**Table 3.**Comparison between the predicted (engineering flow curves reported in Figure 5) and experimental (

^{exp}) average tensile properties, ultimate tensile strength R

_{m}, yield stress YS, and elongation at necking e

_{n}.

Mould | R_{m} (MPa) | R_{m}^{exp} (MPa) | e_{n} (%) | e_{n}^{exp} (%) | YS (MPa) | YS^{exp} (MPa) |
---|---|---|---|---|---|---|

Lynchburg 25 mm | 424.4 | 424.3 | 16.6 | 16.7 | 277.2 | 288.3 |

Y 25 mm | 440.5 | 440.7 | 15.8 | 16.0 | 277.9 | 294.2 |

Y 50 mm | 428.4 | 429.8 | 15.9 | 16.2 | 278.7 | 288.8 |

Y 75 mm | 424.5 | 426.5 | 16.0 | 16.0 | 277. | 287.7 |

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**MDPI and ACS Style**

Angella, G.; Donnini, R.; Ripamonti, D.; Górny, M.; Zanardi, F.
The Role of Microstructure on the Tensile Plastic Behaviour of Ductile Iron GJS 400 Produced through Different Cooling Rates—Part II: Tensile Modelling. *Metals* **2019**, *9*, 1019.
https://doi.org/10.3390/met9091019

**AMA Style**

Angella G, Donnini R, Ripamonti D, Górny M, Zanardi F.
The Role of Microstructure on the Tensile Plastic Behaviour of Ductile Iron GJS 400 Produced through Different Cooling Rates—Part II: Tensile Modelling. *Metals*. 2019; 9(9):1019.
https://doi.org/10.3390/met9091019

**Chicago/Turabian Style**

Angella, Giuliano, Riccardo Donnini, Dario Ripamonti, Marcin Górny, and Franco Zanardi.
2019. "The Role of Microstructure on the Tensile Plastic Behaviour of Ductile Iron GJS 400 Produced through Different Cooling Rates—Part II: Tensile Modelling" *Metals* 9, no. 9: 1019.
https://doi.org/10.3390/met9091019