# Lath Martensite Microstructure Modeling: A High-Resolution Crystal Plasticity Simulation Study

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## Abstract

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## 1. Introduction

_{s}. Carbon, present in solid solution in austenite, remains in solid solution in the new martensitic phase [1], which usually distorts the crystal lattice. The kinetics of the transformation and the morphology of the martensite are driven by the minimization of the strain energy in the presence of constraints from the neighboring microstructure which gives rise to elastic and plastic deformation [2].

_{$\gamma $}$\left|\right|${0 1 1}

_{${\alpha}^{\prime}$}and <1 0 1>

_{$\gamma $}$\left|\right|$<1 1 1>

_{${\alpha}^{\prime}$}(Kurdjumov–Sachs, KS [7]), {1 1 1}

_{$\gamma $}$\left|\right|${0 1 1}

_{${\alpha}^{\prime}$}and <1 1 2>

_{$\gamma $}$\left|\right|$<0 1 1>

_{${\alpha}^{\prime}$}(Nishiyama-Wassermann, NS [8,9]), and {1 1 1}

_{$\gamma $}$\left|\right|${1 0 1}

_{${\alpha}^{\prime}$}and <5 12 17>

_{$\gamma $}$\left|\right|$<7 17 17>

_{${\alpha}^{\prime}$}(Greninger–Troiano, GT [10]). These models are still widely used, even though theoretical considerations predict deviations from orientation relationships in terms of low-indexed planes and directions [11]. Experiments by Morito et al. [12,13] in Fe-C alloys with different carbon contents, in two Mn-containing steels, and in a maraging steel confirmed this: in all these alloys they found an orientation relationship that can be described as a near KS orientation relationship that deviates towards NW orientation relationship and is very close to GT orientation relationship. In view of the small angular differences between the different orientation relationships and the fact that all of them are just an approximation to the theoretically expected values, the features of lath martensite will be discussed in the following under the assumption of a KS orientation relationship. The 24 variants (V1–V24) defined by the KS model are given in Appendix A.

## 2. Generating Lath Martensitic Microstructures

- Packet generation: The austenitic grain (Figure 3a) is subdivided by two flat boundaries into three packets with approximately the same volume. Since no rules are established on how the packets geometrically partition the prior austenite grain, the boundaries are modelled to be perpendicular to each other. The resulting T-shaped grain boundary network is randomly rotated in space (Figure 3b).
- Subblock generation: For each packet, a different habit plane is selected that is parallel to a {111} plane of the austenitic grain. The packets are then subdivided into subblocks of thickness, ${t}_{\mathrm{subblock}}$, parallel to the habit plane (Figure 3c). According to Morito et al. [12], subblocks in low-carbon steels appear in pairs of crystallographic orientations. For example, the 6 variants of the ${\left(111\right)}_{\gamma}$ habit plane occur in the following pairs: V1–V4, V2–V5, and V3–V6. This variant selection is considered when assigning the crystallographic orientations. The order of the variants within a pair and the arrangement of the pairs is random, where for the former a direct repetition is disallowed.
- Lath generation: A Voronoi tessellation is performed in each subblock where each seed corresponds to one individual lath. The volume of the lath, ${V}_{\mathrm{lath}}$, is inversely proportional to the number of seeds. By distorting the resulting structure of equiaxed grains, laths with an average shape of length (${l}_{\mathrm{lath}}$) > width (${w}_{\mathrm{lath}}$) > thickness (${t}_{\mathrm{lath}}$) are achieved. The longest direction, ${l}_{\mathrm{lath}}$, of the laths is aligned parallel to one of the <110> directions in the respective {111} plane, the shortest direction, ${t}_{\mathrm{lath}}$, is aligned normal to the plane. Each lath gets a crystallographic orientation assigned that deviates slightly from the nominal orientation according to the KS model (Figure 3d). More precisely, a random misorientation axis is chosen and the misorientation angle scatters randomly by a value between 0 and ${\theta}_{\mathrm{max}}$.

- The thickness of the subblocks in the direction normal to the habit plane, ${t}_{\mathrm{subblock}}$. It is measured in units of length (UL) which corresponds to the side length of a voxel.
- The average volume of the lath, ${V}_{\mathrm{lath}}$, controlled via the number density of seeds used in the Voronoi tessellation. It is measured in units of volume (UV) which corresponds to the volume of a voxel, i.e., UL
^{3}. - The average aspect ratio of the lath’s dimensions, ${l}_{\mathrm{lath}}\ge {w}_{\mathrm{lath}}\ge {t}_{\mathrm{lath}}$, controlled via the respective stretch factor.
- The maximum misorientation angle of the individual lath with respect to the nominal KS orientation, ${\theta}_{\mathrm{max}}$. It is measured in degrees (${}^{\circ}$).

- The rotation of the packet geometry.
- Sequence of variants within a subblock.
- Sequence of pairs within a block.
- Misorientaton distribution of the laths within the same subblock.

## 3. Modeling Framework

#### 3.1. Numerical Solution Strategy

#### 3.2. Constitutive Model

#### 3.3. Constitutive Parameters

## 4. Simulation Setup

#### 4.1. Simulations Based on Experimental Microstructures

- Experimental microstructure: This is a direct 2D takeover of the measured crystallographic orientation of each of the 1143 × 1143 = 1,306,449 material points after cleaning out the retained austenite (Figure 4a). It is the same model that was used for the parameter adjustment (Section 3.3).
- 3D RVEs: A regular grid of 256 × 256 × 256 = 16,777,216 material points with a $0.5$ $\mathsf{\mu}$$\mathrm{m}$ resolution that contains 86 equiaxed austenitic grains serves as the starting point. The values of the parameters used to create the martensitic structure from this microstructure are: ${t}_{\mathrm{subblock}}=15$ UL, ${V}_{\mathrm{lath}}=1400$ UV, ${l}_{\mathrm{lath}}:{w}_{\mathrm{lath}}:{t}_{\mathrm{lath}}=$ 9:3:1, ${\theta}_{max}=3{}^{\circ}$. A total of ten 3D RVEs are created using different random seeds.
- 2D RVEs: These models are created by selecting a slice from a 3D model that contains 90 austenitic grains in a 600 × 600 × 50 = 18,000,000 grid. The 3D RVE used for slicing was created using the same parameters as for the 3D models. A total of three 2D RVEs with 600 × 600 = 360,000 points are used, choosing different slices from the same 3D model.

#### 4.2. 3D Simulations with Systematically Varied Microstructural Features

- Lath volume: The value is set to ${V}_{\mathrm{lath}}=$ 320, 1400, and 4600 UV. Since subblocks are entirely filled with laths, a decrease in lath volume directly results in more lath per subblock and vice versa.
- Lath aspect ratio: Different lath shapes are created modifying ${l}_{\mathrm{lath}}$, ${w}_{\mathrm{lath}}$, and ${t}_{\mathrm{lath}}$. Rectangular cuboid-shaped laths are created with aspect ratio 9:3:1. Plate-shaped laths are created with aspect ratio 8:8:1. Rod-shaped laths are created with aspect ratio 5:1:1. Cube-shaped laths are created with aspect ratio 1:1:1.
- Scatter: The misorientation angle is chosen as ${\theta}_{\mathrm{max}}=$ 0, 3, and 5${}^{\circ}$. ${\theta}_{\mathrm{max}}=0{}^{\circ}$ represents a 3D RVE made only of subblocks since all laths in a subblock will have the same orientation. Limiting ${\theta}_{\mathrm{max}}<5{}^{\circ}$ is based on experimental evidence showing that the misorientation angle of a laths within a subblock does not exceed 5 ${}^{\circ}$.
- Subblock thickness: Subblock thickness is set to ${t}_{\mathrm{subblock}}=$ 8, 15, and 20 UL.

## 5. Results & Discussion

#### 5.1. Simulations Based on Experimental Microstructures

#### 5.1.1. Average Stress–Strain Response

#### 5.1.2. Correlation of Stress and Strain Fields

#### 5.1.3. Micromechanics of 2D and 3D Models

#### 5.2. 3D Simulations with Systematically Varied Microstructural Features

## 6. Summary and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Variant | Plane | Direction | Variant | Plane | Direction |
---|---|---|---|---|---|

1 | (111)${}_{\gamma}$ $\left|\right|$(011)${}_{{\alpha}^{\prime}}$ | [1 0 1]${}_{\gamma}$ $\left|\right|$ [1 1 1]${}_{{\alpha}^{\prime}}$ | 13 | (111)${}_{\gamma}$ $\left|\right|$(011)${}_{{\alpha}^{\prime}}$ | [0 1 1]${}_{\gamma}$ $\left|\right|$ [1 1 1]${}_{{\alpha}^{\prime}}$ |

2 | [1 0 1]${}_{\gamma}$ $\left|\right|$ [1 1 1]${}_{{\alpha}^{\prime}}$ | 14 | [0 1 1]${}_{\gamma}$ $\left|\right|$ [1 1 1]${}_{{\alpha}^{\prime}}$ | ||

3 | [0 1 1]${}_{\gamma}$ $\left|\right|$ [1 1 1]${}_{{\alpha}^{\prime}}$ | 15 | [1 0 1]${}_{\gamma}$ $\left|\right|$ [1 1 1]${}_{{\alpha}^{\prime}}$ | ||

4 | [0 1 1]${}_{\gamma}$ $\left|\right|$ [1 1 1]${}_{{\alpha}^{\prime}}$ | 16 | [1 0 1]${}_{\gamma}$ $\left|\right|$ [1 1 1]${}_{{\alpha}^{\prime}}$ | ||

5 | [1 1 0]${}_{\gamma}$ $\left|\right|$ [1 1 1]${}_{{\alpha}^{\prime}}$ | 17 | [1 1 0]${}_{\gamma}$ $\left|\right|$ [1 1 1]${}_{{\alpha}^{\prime}}$ | ||

6 | [1 1 0]${}_{\gamma}$ $\left|\right|$ [1 1 1]${}_{{\alpha}^{\prime}}$ | 18 | [1 1 0]${}_{\gamma}$ $\left|\right|$ [1 1 1]${}_{{\alpha}^{\prime}}$ | ||

7 | (111)${}_{\gamma}$ $\left|\right|$(011)${}_{{\alpha}^{\prime}}$ | [1 0 1]${}_{\gamma}$ $\left|\right|$ [1 1 1]${}_{{\alpha}^{\prime}}$ | 19 | (111)${}_{\gamma}$ $\left|\right|$(011)${}_{{\alpha}^{\prime}}$ | [1 1 0]${}_{\gamma}$ $\left|\right|$ [1 1 1]${}_{{\alpha}^{\prime}}$ |

8 | [1 0 1]${}_{\gamma}$ $\left|\right|$ [1 1 1]${}_{{\alpha}^{\prime}}$ | 20 | [1 1 0]${}_{\gamma}$ $\left|\right|$ [1 1 1]${}_{{\alpha}^{\prime}}$ | ||

9 | [1 1 0]${}_{\gamma}$ $\left|\right|$ [1 1 1]${}_{{\alpha}^{\prime}}$ | 21 | [0 1 1]${}_{\gamma}$ $\left|\right|$ [1 1 1]${}_{{\alpha}^{\prime}}$ | ||

10 | [1 1 0]${}_{\gamma}$ $\left|\right|$ [1 1 1]${}_{{\alpha}^{\prime}}$ | 22 | [0 1 1]${}_{\gamma}$ $\left|\right|$ [1 1 1]${}_{{\alpha}^{\prime}}$ | ||

11 | [0 1 1]${}_{\gamma}$ $\left|\right|$ [1 1 1]${}_{{\alpha}^{\prime}}$ | 23 | [1 0 1]${}_{\gamma}$ $\left|\right|$ [1 1 1]${}_{{\alpha}^{\prime}}$ | ||

12 | [0 1 1]${}_{\gamma}$ $\left|\right|$ [1 1 1]${}_{{\alpha}^{\prime}}$ | 24 | [1 0 1]${}_{\gamma}$ $\left|\right|$ [1 1 1]${}_{{\alpha}^{\prime}}$ |

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**Figure 1.**(

**a**) Map of the Inverse Pole Figure (IPF) along the normal/out-of-plane direction of a Fe-0.13C-5.1Ni (wt%) model alloy. The heat treatment consisted of austenization at 900${}^{\circ}$ for 5 min and subsequently quenching in water to obtain a fully martensitic microstructure [17]. (

**b**) Schematic of the hierarchical microstructure of lath martensite.

**Figure 2.**(

**a**) Inverse Pole Figure (IPF) map of a 18Ni-8Co-5Mo maraging steel and (

**b**) 100 pole figure, showing the crystal-orientation. Blocks are made of 2 subblocks with a low-angle misorientation of about 10 degrees (V1–V4, V2–V5 and V3–V6). Color legend is given in Figure 1a. Adapted with permission from ref. [13]. Copyright 2006 Elsevier.

**Figure 3.**Creation of a martensitic microstructure from an austenitic microstructure: (

**a**) Austenitic microstructure with highlighted grain. (

**b**) Division of the grain in 3 packets. (

**d**) Creation of the subblocks. (

**e**) Creation of the laths by means of Voronoi tessellation. Legend for Inverse Pole Figure (IPF) in (

**a**,

**c**,

**d**) is given in Figure 1a.

**Figure 4.**(

**a**) Inverse Pole Figure (IPF) map of the experimental microstructure after cleaning. (

**b**) Experimental and simulated stress–strain and hardening rate curves.

**Figure 5.**Inverse Pole Figure (IPF) maps, color legend is given in Figure 2a: (

**a**) Raw experimental microstructure (includes retained austenite). (

**b**) Prior austenitic microstructure. Black color means no indexing was possible. (

**c**) Synthetic austenitic microstructure in 3D. (

**d**) Martensitic microstructure. Black lines in (

**c**,

**d**) represents austenite grain boundaries.

**Figure 7.**Heatmaps showing stress versus strain distributions in Rolling Direction (RD). (

**a**) Measured microstructure. (

**b**) 2D Representative Volume Element (RVE). (

**c**) 3D RVE.

**Figure 8.**Statistical evaluation: (

**a**) Strain and (

**b**) stress distributions. (

**c**) Number of points with values above a certain threshold. The threshold values are selected to obtain around 1% frequency for the data of the experimental microstructure.

**Figure 9.**Spatially resolved results for the experimental microstructure of size 400 μm

^{2}(

**top**), a 2D Representative Volume Element (RVE) of size 300 μm

^{2}(

**middle**) and a 3D RVE of size 128 μm

^{3}(

**bottom**).

**Figure 10.**Spatially resolved results of a 3D RVE in a section normal to the rolling direction: Inverse Pole Figure (IPF), equivalent strain, and equivalent stress.

**Figure 11.**Inverse Pole Figure (IPF) (for color legend see Figure 2a) and equivalent strain of a part of the generated microstructure corresponding to 3D RVEs with ${\theta}_{\mathrm{max}}=0{}^{\circ}$ (

**left**) and ${\theta}_{\mathrm{max}}=5{}^{\circ}$ (

**right**). Two packets belonging to a prior austenite grain are distinguishable.

**Figure 12.**Distributions of strain (

**a**) and stress (

**b**) in the rolling direction for different subblock sizes.

**Figure 13.**Inverse Pole Figure (IPF) and equivalent strain and stress distribution for an RVE with ${t}_{\mathrm{subblock}}=20$ (

**top**) and ${t}_{\mathrm{subblock}}=8$ (

**bottom**).

**Figure 14.**Strain (

**a**) and stress (

**b**) distributions. Number of points with values above a certain threshold (

**c**). The threshold values are selected to obtain around 1–2% frequency.

**Table 1.**Nominal chemical composition of the martensitic steel in wt%. Conventional quenching was used to produce lath martensite.

C | Si | Mn | P | S | Cu | Al | Nb | Mo | Ni | Cr |
---|---|---|---|---|---|---|---|---|---|---|

≤0.25 | ≤0.70 | ≤1.60 | ≤0.025 | ≤0.010 | ≤0.30 | ≤0.03 | ≤0.05 | ≤0.50 | ≤0.80 | ≤1.50 |

Property | Symbol | Value | Unit |
---|---|---|---|

Elastic constant | ${C}_{11}$ | 417.4 | GPa |

Elastic constant | ${C}_{12}$ | 242.4 | GPa |

Elastic constant | ${C}_{44}$ | 211.1 | GPa |

Initial resistance | ${g}_{0}$ | 160.0 | MPa |

Saturation resistance | ${g}_{\infty}$ | 555.0 | MPa |

Initial hardening rate | ${h}_{0}$ | 90.0 | GPa |

Reference shear rate | ${\dot{\gamma}}_{0}$ | 10^{−3} | s^{−1} |

Stress exponent | n | 20 | |

Strain hardening exponent | a | 2.0 |

$\mathit{r}\left(\right)open="("\; close=")">{\mathit{\epsilon}}_{11}-{\mathit{\epsilon}}_{\mathbf{vM}}$ | $\mathit{r}\left(\right)open="("\; close=")">{\mathit{\sigma}}_{11}-{\mathit{\sigma}}_{\mathbf{vM}}$ | $\mathit{r}\left(\right)open="("\; close=")">{\mathit{\epsilon}}_{11}-{\mathit{\sigma}}_{11}$ | $\mathit{r}\left(\right)open="("\; close=")">{\mathit{\epsilon}}_{11}-{\mathit{\sigma}}_{\mathbf{vM}}$ | |
---|---|---|---|---|

2D measured | 0.98 | 0.13 | −0.15 | 0.12 |

2D RVEs | 0.98 | 0.14 | −0.16 | 0.13 |

3D RVEs | 0.97 | 0.26 | −0.13 | 0.04 |

**Table 4.**Fraction of points above the threshold value for stress and strain. The threshold values are selected to obtain around 1–2%.

Lath Aspect Ratio (${\mathit{l}}_{\mathbf{lath}}$:${\mathit{w}}_{\mathbf{lath}}$:${\mathit{t}}_{\mathbf{lath}}$) | Lath Volume (${\mathit{V}}_{\mathbf{lath}}$/UV) | Scatter (${\mathit{\theta}}_{\mathbf{max}}$/${}^{\circ}$) | Subblock Size (${\mathit{t}}_{\mathbf{subblock}}$/UL) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

8:8:1 | 5:1:1 | 9:3:1 | 1:1:1 | 320 | 1400 | 4600 | 0 | 3 | 5 | 8 | 15 | 20 | |

$f({\epsilon}_{\mathrm{vM}}>0.105)$/% | 1.508 | 1.457 | 1.498 | 1.456 | 1.470 | 1.498 | 1.533 | 1.510 | 1.498 | 1.475 | 1.248 | 1.498 | 1.599 |

$f({\sigma}_{\mathrm{vM}}>2.2\mathrm{GPa})$/% | 1.179 | 1.271 | 1.246 | 1.121 | 1.213 | 1.246 | 1.225 | 1.225 | 1.246 | 1.261 | 1.234 | 1.246 | 1.481 |

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**MDPI and ACS Style**

Gallardo-Basile, F.-J.; Naunheim, Y.; Roters, F.; Diehl, M.
Lath Martensite Microstructure Modeling: A High-Resolution Crystal Plasticity Simulation Study. *Materials* **2021**, *14*, 691.
https://doi.org/10.3390/ma14030691

**AMA Style**

Gallardo-Basile F-J, Naunheim Y, Roters F, Diehl M.
Lath Martensite Microstructure Modeling: A High-Resolution Crystal Plasticity Simulation Study. *Materials*. 2021; 14(3):691.
https://doi.org/10.3390/ma14030691

**Chicago/Turabian Style**

Gallardo-Basile, Francisco-José, Yannick Naunheim, Franz Roters, and Martin Diehl.
2021. "Lath Martensite Microstructure Modeling: A High-Resolution Crystal Plasticity Simulation Study" *Materials* 14, no. 3: 691.
https://doi.org/10.3390/ma14030691