# Computational Determination of Macroscopic Mechanical and Thermal Material Properties for Different Morphological Variants of Cast Iron

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## Abstract

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## 1. Introduction

## 2. Creating Simulation Domains for Cast Iron with Lamellar Graphite

#### 2.1. Modelling the Morphology of Cast Iron with Lamellar Graphite

#### 2.2. The Volume Elements

## 3. Model Formulation

#### 3.1. Mechanical and Thermal Model for Pearlite

#### 3.2. Mechanical and Thermal Model for the Graphite Lamellae

## 4. Mechanical Simulations

#### 4.1. Key Points of the Numerics and the Simulation Setup

#### 4.2. Analysis of the Mechanical Model

#### 4.3. Mechanical Simulation Study

#### 4.3.1. Results of the Mechanical Simulation Study

#### 4.3.2. Mechanical Parameters

## 5. Thermal Simulations

## 6. Correlation of Thermal and Mechnical Properties

## 7. Summary and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic representation of an idealised graphite lamella, in the form of a flat corrugated ellipsoid. The ellipsoid is shown in the particle’s own $\tilde{x},\tilde{y},\tilde{z}$ coordinate system and the axes of the hexagonal crystal lattice are given.

**Figure 2.**“Micrograph” of a synthetically generated GJL-150 material, with a graphite flake volume fraction of $10\mathrm{v}.-\%$.

**Figure 3.**Micrograph of a specimen taken from a truck brake disc, made of a GG-15 material [25].

**Figure 4.**Spatial images of the graphite morphology of statistical volume elements of different graphite characteristics. The simulation domains have a spatial extent of $1.25\text{}\mathrm{m}\mathrm{m}\times 1.25\text{}\mathrm{m}\mathrm{m}\times 1.25\text{}\mathrm{m}\mathrm{m}$, for a spatial discretisation using voxels of size $5\text{}\mathsf{\mu}\mathrm{m}\times 5\text{}\mathsf{\mu}\mathrm{m}\times 5\text{}\mathsf{\mu}\mathrm{m}$, and include a graphite flake volume fraction of $10\text{}\mathrm{v}.-\%$.

**Figure 5.**Temperature-dependent flow curves of the pearlitic matrix. The Young’s moduli and the yield strengths decrease at higher temperatures. According to Bonora and Ruggiero [1], all flow curves are self-similar to the material behaviour at room temperature.

**Figure 6.**Temperature-dependent thermal conductivities. Isotropic thermal conductivity of the pearlitic matrix, ${\lambda}^{\mathrm{m}}$, according to Helsing and Grimvall [34]. Components of the transversely isotropic thermal conductivity of the graphite lamellae ${\lambda}_{\phantom{\rule{4pt}{0ex}}\mathrm{texthexa}}^{\mathrm{g}}$ and ${\lambda}_{\mathrm{hex}-\mathrm{c}}^{\mathrm{g}}$, according to Helsing and Grimvall [34].

**Figure 8.**Degradation function of the graphite stiffness. Threshold value of the volume change, ${\epsilon}_{\mathrm{vol}}^{\mathrm{crit}}$; minimum value of the degradation function, ${d}_{\mathrm{min}}$.

**Figure 9.**Influence of the graphite class on anisotropy, with a constant domain size and graphite volume fraction. The macroscopic stress–strain characteristics averaged over three variants of a graphite class are shown. The corresponding standard deviations of the various graphite morphologies are given as error bars.

**Figure 10.**Influence of the minimum value of the degradation function on the macroscopic stress–strain behaviour. Simulations at room temperature, using GJL-150 SVEs with a graphite flake volume fraction of 10 $\mathrm{v}.-\%$. For a constant threshold of volume change, ${\epsilon}_{\mathrm{vol}}^{\mathrm{crit}}=0.001$, the minimum value of the degradation function varies for the values ${d}_{\mathrm{min}}=\{0.05,0.06,0.07\}$.

**Figure 11.**Influence of the threshold value of the volume change on the macroscopic stress–strain behaviour. Simulations at room temperature, using GJL-150 SVEs with a graphite flake fraction of 10 $\mathrm{v}.-\%$. For a constant minimum value of the degradation function of ${d}_{\mathrm{min}}=0.05$, the threshold of the volume change is varied for the values ${\epsilon}_{\mathrm{vol}}^{\mathrm{crit}}=\{0.00075,0.001,0.002\}$.

**Figure 12.**Stress–strain diagram of an IA2 SVE: one-dimensional and moment-free for tensile and compressive load. For some strain states, the contour plots of the degradation function of the graphite lamellae, $degrad$, are given as an inlay. The pearlitic matrix is shown in grey. The tension–compression stress–strain asymmetry of grey cast iron can clearly be seen.

**Figure 13.**Contour plots for tensile and compressive load of a statistical volume element of the class IA2, with a graphite flake volume fraction of 12 $\mathrm{v}.-\%$. The von Mises equivalent stress, ${\sigma}_{\mathrm{v}.\mathrm{M}.}$, and the accumulated plastic strain, ${\epsilon}_{\mathrm{akk}}$, are shown, at a macroscopic strain of the same magnitude in the x direction of $\epsilon =0.0035$.

**Figure 14.**Contour plots for tensile loading of different statistical volume elements with a graphite flake volume fraction of 10 $\mathrm{v}.-\%$. The von Mises equivalent stress, ${\sigma}_{\mathrm{v}.\mathrm{M}.}$, and the accumulated plastic strain, ${\epsilon}_{\mathrm{akk}}$, are shown, at a macroscopic tensile strain of $\epsilon =0.0035$ in the x direction.

**Figure 15.**Exemplary Kadi4Mat workflow for conducting the simulation study and persisting the results in the Kadi4Mat repository.

**Figure 16.**Stress–strain curves for tensile and compressive load of all graphite classes with 12 $\mathrm{v}.-\%$ of graphite precipitations and for $\theta =20\text{}{}^{\circ}\mathrm{C}$. For a clearer representation, the stress–strain curves of the compressive simulations are given by positive values.

**Figure 17.**Stress–strain curves for tensile and compressive load of all graphite classes with 12 $\mathrm{v}.-\%$ of graphite precipitations and for $\theta =750\text{}{}^{\circ}\mathrm{C}$. For a clearer representation, the stress–strain curves of the compressive simulations are given by positive values.

**Figure 18.**Stress–strain diagrams of a GJL-150 graphite morphology under tensile load, with 12 $\mathrm{v}.-\%$ of graphite precipitations and variable temperature.

**Figure 19.**Stress–strain diagrams of a GJL-150 graphite morphology under compressive load, with 12 $\mathrm{v}.-\%$ of graphite precipitations and variable temperature. The stress–strain curves are given by positive values.

**Figure 20.**Stress–strain diagrams of GJL-150 graphite morphologies under tensile and compressive load, with a variable graphite volume fraction at room temperature. For a clearer representation, the stress–strain curves of the compressive simulations are given by positive values.

**Figure 21.**Schematic evaluation of the secant module and the yield strength. The secant moduli are evaluated at strains of $0.1$% and $0.2$%. The straight lines for evaluating the yield strengths, ${\mathrm{R}}_{\mathrm{p}}$, run parallel to the Young’s moduli.

**Figure 22.**Secant moduli of all graphite classes with graphite flake volume fractions of 10, 11 and 12 $\mathrm{v}.-\%$, evaluated for the temperatures 20, 150, 300, 450, 600 and 750 ${}^{\circ}\mathrm{C}$, at a tensile strain of $0.1$ %.

**Figure 23.**Secant moduli of all graphite classes with graphite flake volume fractions of 10, 11 and 12 $\mathrm{v}.-\%$, evaluated for the temperatures 20, 150, 300, 450, 600 and 750 ${}^{\circ}\mathrm{C}$, at a tensile strain of $0.2$%.

**Figure 24.**Poisson’s ratios of all graphite classes with graphite flake volume fractions of 10, 11 and 12 $\mathrm{v}.-\%$, evaluated for the temperatures 20, 150, 300, 450, 600 and 750 ${}^{\circ}\mathrm{C}$, at a tensile strain of $0.1$%.

**Figure 25.**Poisson’s ratios of all graphite classes with graphite flake volume fractions of 10, 11 and 12 $\mathrm{v}.-\%$, evaluated for the temperatures 20, 150, 300, 450, 600 and 750 ${}^{\circ}\mathrm{C}$, at a tensile strain of $0.2$%.

**Figure 26.**Yield strengths of all graphite classes with graphite flake volume fractions of 10, 11 and 12 $\mathrm{v}.-\%$, evaluated for the temperatures 20, 150, 300, 450, 600 and 750 ${}^{\circ}\mathrm{C}$, at a tensile strain of $\epsilon =0.1\%$.

**Figure 27.**Yield strengths of all graphite classes with graphite flake volume fractions of 10, 11 and 12 $\mathrm{v}.-\%$, evaluated for the temperatures 20, 150, 300, 450, 600 and 750 ${}^{\circ}\mathrm{C}$, at a tensile strain of $0.2$%.

**Figure 28.**Secant moduli, Poisson’s ratios and yield strengths of all graphite classes with a graphite flake volume fraction of 12 $\mathrm{v}.-\%$, evaluated for temperatures 20, 150, 300, 450, 600 and 750 ${}^{\circ}\mathrm{C}$, at a compressive strain of $0.1$%.

**Figure 29.**Secant moduli, Poisson’s ratios and yield strengths of all graphite classes with a graphite flake volume fraction of 12 $\mathrm{v}.-\%$, evaluated for the temperatures 20, 150, 300, 450, 600 and 750 ${}^{\circ}\mathrm{C}$, at a compressive strain of $0.2$%.

**Figure 30.**Thermal conductivities of all graphite classes with graphite flake volume fractions of 10, 11 and 12 $\mathrm{v}.-\%$, for the temperatures 20, 150, 300, 450, 600 and 750 ${}^{\circ}\mathrm{C}$.

**Figure 31.**Secand moduli and yield strengths vs. thermal conductivities of all graphite classes with graphite flake volume fractions of 10, 11 and 12 $\mathrm{v}.-\%$ at room temperature. The properties capture tensile load for strains of $\epsilon =0.1\%$ and $\epsilon =0.2\%$.

**Figure 32.**Secand moduli and yield strengths vs. thermal conductivities of all graphite classes with graphite flake volume fractions of 10, 11 and 12 $\mathrm{v}.-\%$ at a temperature of $\theta =300\text{}{}^{\circ}\mathrm{C}$. The properties capture tensile load for strains of $\epsilon =0.1\%$ and $\epsilon =0.2\%$.

Guide Number | Value | Unit |
---|---|---|

IA2 | $0.5$ to $<1.0$ | $\mathrm{m}\mathrm{m}$ |

IA3 | $0.25$ to $<0.5$ | $\mathrm{m}\mathrm{m}$ |

IA4 | $0.12$ to $<0.25$ | $\mathrm{m}\mathrm{m}$ |

IA5 | $0.06$ to $<0.12$ | $\mathrm{m}\mathrm{m}$ |

**Table 2.**Temperature-dependent mechanical material parameters of the pearlitic matrix. Temperature $\theta $; reduction factor of the Young’s modulus, ${k}_{\mathrm{E}}$; Young’s modulus ${E}^{\mathrm{m}}$; reduction factor of the yield strength, ${k}_{\mathrm{y}}$; yield strength ${\sigma}_{\mathrm{y},0}^{\mathrm{m}}$; saturation stress ${\sigma}_{\infty}^{\mathrm{m}}$; saturation rate ${\eta}^{\mathrm{m}}$.

$\mathit{\theta}\phantom{\rule{4pt}{0ex}}{[}^{\circ}\mathbf{C}]$ | ${\mathit{k}}_{\mathbf{E}}$ | ${\mathit{E}}^{\mathbf{m}}\phantom{\rule{4pt}{0ex}}\left[\mathbf{GPa}\right]$ | ${\mathit{k}}_{\mathbf{y}}$ | ${\mathit{\sigma}}_{\mathbf{y},0}^{\mathbf{m}}\phantom{\rule{4pt}{0ex}}\left[\mathbf{MPa}\right]$ | ${\mathit{\sigma}}_{\mathit{\infty}}^{\mathbf{m}}\phantom{\rule{4pt}{0ex}}\left[\mathbf{MPa}\right]$ | ${\mathit{\eta}}^{\mathbf{m}}\phantom{\rule{4pt}{0ex}}\left[\mathbf{GPa}\right]$ |
---|---|---|---|---|---|---|

20 | 1.0 | 210.0 | 1.0 | 230.0 | 772.0 | 19.0 |

150 | 0.95 | 199.5 | 1.0 | 230.0 | 772.0 | 19.0 |

300 | 0.8 | 168.0 | 1.0 | 230.0 | 772.0 | 19.0 |

450 | 0.65 | 136.5 | 0.89 | 204.7 | 687.1 | 19.0 |

600 | 0.31 | 65.1 | 0.47 | 108.1 | 362.8 | 19.0 |

750 | 0.11 | 23.1 | 0.17 | 39.1 | 131.2 | 19.0 |

Pascal et al. [41] | Bosak et al. [40] | Blakslee et al. [39] | |
---|---|---|---|

${C}_{11}^{\mathrm{g}}$ in MPa | 1126.0 | 1109.0 | 1060.0 |

${C}_{12}^{\mathrm{g}}$ in MPa | 200.0 | 139.0 | 180.0 |

${C}_{13}^{\mathrm{g}}$ in MPa | 39.5 | 0.0 | 15.0 |

${C}_{33}^{\mathrm{g}}$ in MPa | 40.7 | 38.7 | 36.5 |

${C}_{44}^{\mathrm{g}}$ in MPa | 4.51 | 5.0 | 0.18 |

${C}_{66}^{\mathrm{g}}$ in MPa | 485.0 |

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**MDPI and ACS Style**

Herrmann, C.; Schmid, S.; Schneider, D.; Selzer, M.; Nestler, B.
Computational Determination of Macroscopic Mechanical and Thermal Material Properties for Different Morphological Variants of Cast Iron. *Metals* **2021**, *11*, 1588.
https://doi.org/10.3390/met11101588

**AMA Style**

Herrmann C, Schmid S, Schneider D, Selzer M, Nestler B.
Computational Determination of Macroscopic Mechanical and Thermal Material Properties for Different Morphological Variants of Cast Iron. *Metals*. 2021; 11(10):1588.
https://doi.org/10.3390/met11101588

**Chicago/Turabian Style**

Herrmann, Christoph, Stefan Schmid, Daniel Schneider, Michael Selzer, and Britta Nestler.
2021. "Computational Determination of Macroscopic Mechanical and Thermal Material Properties for Different Morphological Variants of Cast Iron" *Metals* 11, no. 10: 1588.
https://doi.org/10.3390/met11101588