# Accessing Colony Boundary Strengthening of Fully Lamellar TiAl Alloys via Micromechanical Modeling

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## Abstract

**:**

## 1. Introduction

#### 1.1. Microstructure and Micromechanics of Fully Lamellar TiAl

#### 1.1.1. Lattice Structures and Orientation Relation

#### 1.1.2. Influence of Microstructural Boundaries on Strength

- ${K}_{C}$ is a function of ${\lambda}_{L}$ and ${\lambda}_{D}$, since strengths of slip/twinning systems in adjacent colonies are determined by lamella and domain boundary strengthening and
- experimentally determined ${K}_{C}$ values are only valid for the given combination of ${\lambda}_{L}$ and ${\lambda}_{D}$, rendering identification of the functional relation ${K}_{C}=f({\lambda}_{L},{\lambda}_{D})$ unreasonably labor-intensive.

#### 1.2. Scope of Present Paper

## 2. Modeling

#### 2.1. Kinematics and Stress Measures

#### 2.2. Thermomechanics and Temperature Evolution

#### 2.3. Crystal Plasticity

#### 2.3.1. Flow and Twinning Rule

#### Slip

#### Twinning

#### 2.3.2. Defect Density Evolution

#### Dislocation Density Evolution

#### Twin Evolution

#### 2.3.3. Critical Resolved Shear Stresses

#### Initial Slip/Twinning System Strength

#### Evolution of Slip System Strength

#### Evolution of Twinning System Strength

#### 2.4. Helmholtz Free Energy

#### 2.5. RVE Generation and Discretization

#### 2.5.1. RVE of a Polysynthetically Twinned Crystal

#### 2.5.2. RVE of a Polycolony Microstructure

## 3. Parameter Identification

#### 3.1. Constitutive Assumptions

#### 3.1.1. Morphological Classification

- longitudinal ($\mathit{s}\parallel $ lamellar plane; $\mathit{n}\perp $ lamellar plane),
- mixed ($\mathit{s}\parallel $ lamellar plane; $\mathit{n}$ lamellar plane) or
- transversal ($\mathit{s}\nparallel $ lamellar plane; $\mathit{n}$ lamellar plane).

#### 3.1.2. Hall–Petch Strengthening by Evolving Twins

#### 3.1.3. Modeling Super Slip

#### Initial Critical Resolved Shear Stress

#### Taylor Hardening

#### 3.1.4. Recovery

#### 3.2. Model Parameters

#### 3.2.1. Onset of Yield

#### 3.2.2. Dislocation Accumulation and Hardening Interaction

#### 3.3. Results

## 4. Determining the Hall–Petch Coefficient for Colony Boundary Strengthening

#### 4.1. Calculation Scheme

#### 4.2. Simulation Results

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Schematic illustration of fully lamellar microstructure with magnification of lamellae. ${\lambda}_{L}$: lamella thickness; ${\lambda}_{D}$: domain size; ${\lambda}_{C}$: colony size. (SEM micrograph: courtesy of Michael Oehring, Helmholtz-Zentrum Geesthacht)

**Figure 2.**Crystallographic lattices of constituent phases with most relevant deformation mechanisms. left: hexagonal D0${}_{19}$ lattice of ${\alpha}_{2}$ phase (Ti${}_{3}$Al); right: tetragonal L1${}_{0}$ lattice of $\gamma $ phase (TiAl).

**Figure 3.**Illustration of dislocation pile-up stress at a colony boundary possibly activating slip/twinning in the adjacent colony. For ${\lambda}_{C}^{II}={\lambda}_{C}^{I}$ but ${\lambda}_{L}^{II}<{\lambda}_{L}^{I}$ and ${\lambda}_{D}^{II}<{\lambda}_{D}^{I}$, the strength of respective systems in the right image will be higher than in the left one, requiring a higher pile-up stress to be activated and thus making colony boundary strengthening a function of ${\lambda}_{L}$ and ${\lambda}_{D}$.

**Figure 4.**Qualitative illustration of hardening laws, i.e., Equations (24), (25), (27) and (28). (

**a**) strengthening of slip systems $\alpha $ and twinning systems $\beta $ with total dislocation density ${\rho}^{\mathrm{dis}}$; influence of interaction coefficient ${C}_{\beta \alpha}$; (

**b**) strengthening of slip systems $\alpha $ and twinning systems $\beta $ with volume fraction of non-coplanar twins; influence of hardening coefficient ${h}_{\alpha \beta}$ resp. ${h}_{\beta {\beta}^{\prime}}$.

**Figure 5.**Representative volume element of polysynthetically twinned crystal. $\phi $: angle between uniaxial load and lamella plane; ${\gamma}_{M/T}^{I-III}$: six orientation variants of $\gamma $ phase (three matrix and three twin orientations).

**Figure 6.**Representative volume element of polycolony fully lamellar microstructure consisting of 36 lamellar colonies. Separate depiction of the ${\alpha}_{2}$ phase and the orientation variants of the $\gamma $ phase shows their distribution within the colonies.

**Figure 7.**Comparison of possible evolutions of longitudinal and transversal twins in $\gamma $ lamellae.

**Figure 8.**Orientation dependent yield and hardening behavior of polysynthetically twinned crystals. Experimental results taken from [9].

**Figure 9.**Interpolation scheme for determining ${K}_{C}({\lambda}_{L}^{i},{\lambda}_{D}^{i})$ by combination of simulation and experimental results. Applying the constitutive model of a polysynthetically twinned crystal to a polycolony RVE, yields ${\sigma}_{0}^{\mathrm{sim}}({\lambda}_{L}^{i},{\lambda}_{D}^{i},{\lambda}_{C}=\infty )$ for a given combination of ${\lambda}_{L}^{i}$ and ${\lambda}_{D}^{i}$. With corresponding experimental results, the relation ${K}_{C}({\lambda}_{L}^{i},{\lambda}_{D}^{i})=\frac{{\sigma}_{Y}^{\mathrm{exp}}({\lambda}_{L}^{i},{\lambda}_{D}^{i},{\lambda}_{C}^{i})-{\sigma}_{0}^{\mathrm{sim}}({\lambda}_{L}^{i},{\lambda}_{D}^{i})}{{\lambda}_{C}^{{i}^{-0.5}}}$ is evaluated. Repeating this for different combinations of ${\lambda}_{L}$ and ${\lambda}_{D}$ reveals ${K}_{C}=f({\lambda}_{L},{\lambda}_{D})$.

**Figure 10.**Colony boundary Hall–Petch coefficient ${K}_{C}$ plotted over ${\lambda}_{L}$. Full symbols: determined via the scheme illustrated in Figure 9; solid line: interpolation of calculated values via ${K}_{C}\left({\lambda}_{L}\right)={K}_{C,0}+{K}_{C,{\lambda}_{L}}\frac{1}{\sqrt{{\lambda}_{L}}}.$; open symbols: experimentally determined in [12,13,14,16]. ${K}_{C}$ value from [13] was determined from experiments with the indicated range of lamella thicknesses ${\lambda}_{L}$ (

**b**,

**c**): comparison of experimentally determined ${\sigma}_{Y}^{\mathrm{exp}}({\lambda}_{L},{\lambda}_{D},{\lambda}_{C})$ [10,13,14]and simulated yield stresses ${\sigma}_{Y}^{\mathrm{exp}}({\lambda}_{L},{\lambda}_{D},{\lambda}_{C})$. In simulations, colony boundary strengthening was incorporated by introducing the interpolation from Figure 10a to Equations (21) and (22). Despite the scattering of KC values, the simulated yield stresses reproduce the experimental results very well.

**Table 1.**Slip and twinning systems in $\gamma $ and ${\alpha}_{2}$ phase with morphological classification according to [6].

$\mathit{\gamma}$ Phase | ||||

System | Mechanism | Classification | Index | |

$\left(111\right)\left[1\overline{1}0\right]$ | ordinary slip | longitudinal | 1 | |

$\left(111\right)\left[01\overline{1}\right]$ | super slip | longitudinal | 2 | |

$\left(111\right)\left[10\overline{1}\right]$ | super slip | longitudinal | 3 | |

$\left(\overline{1}\overline{1}1\right)\left[1\overline{1}0\right]$ | ordinary slip | mixed | 4 | |

$\left(1\overline{1}\overline{1}\right)\left[01\overline{1}\right]$ | super slip | mixed | 5 | |

$\left(1\overline{1}1\right)\left[10\overline{1}\right]$ | super slip | mixed | 6 | |

$\left(1\overline{1}1\right)\left[110\right]$ | ordinary slip | transversal | 7 | |

$\left(\overline{1}11\right)\left[110\right]$ | ordinary slip | transversal | 8 | |

$\left(11\overline{1}\right)\left[0\overline{1}\overline{1}\right]$ | super slip | transversal | 9 | |

$\left(11\overline{1}\right)\left[\overline{1}0\overline{1}\right]$ | super slip | transversal | 10 | |

$\left(1\overline{1}1\right)\left[0\overline{1}\overline{1}\right]$ | super slip | transversal | 11 | |

$\left(1\overline{1}\overline{1}\right)\left[\overline{1}0\overline{1}\right]$ | super slip | transversal | 12 | |

$\left(111\right)\left[11\overline{2}\right]$ | twinning | longitudinal | 1 | |

$\left(\overline{1}11\right)\left[\overline{1}1\overline{2}\right]$ | twinning | transversal | 2 | |

$\left(1\overline{1}1\right)\left[1\overline{1}\overline{2}\right]$ | twinning | transversal | 3 | |

$\left(11\overline{1}\right)\left[112\right]$ | twinning | transversal | 4 | |

${\mathit{\alpha}}_{\mathbf{2}}$ Phase | ||||

System | Mechanism | Classification | Index | |

$\langle 11\overline{2}0\rangle \left(0001\right)$ | basal slip | longitudinal | 1–3 | |

$\langle 11\overline{2}0\rangle \left\{1\overline{1}00\right\}$ | prismatic slip | mixed | 4–6 | |

$\langle \overline{1}\overline{1}26\rangle \left\{11\overline{2}1\right\}$ | pyramidal slip | transversal | 7–12 |

Phase | Symbol | Value | Annotation | Ref. | |
---|---|---|---|---|---|

material parameters | $\gamma $ | E | $173.59\phantom{\rule{0.166667em}{0ex}}\left[\mathrm{GPa}\right]-0.0342[T-{T}_{0}]\phantom{\rule{0.166667em}{0ex}}\left[\frac{\mathrm{GPa}}{{}^{\circ}\mathrm{C}}\right]$ | ${T}_{0}=25<T<935$ | [62] |

$\nu $ | $0.234+6.7\times {10}^{-6}[T-{T}_{0}]\phantom{\rule{0.166667em}{0ex}}\left[\frac{1}{{}^{\circ}\mathrm{C}}\right]$ | ${T}_{0}=25<T<847$ | [62] | ||

$\frac{c}{a}$ | $1.00356+7.2\times {10}^{-6}[T-{T}_{0}]\phantom{\rule{0.166667em}{0ex}}\left[\frac{1}{{}^{\circ}\mathrm{C}}\right]$ | ${T}_{0}=20<T<1450$ | [63] | ||

${\gamma}_{T}$ | $\frac{1}{\sqrt{2}}[-]$ | [5] | |||

${\alpha}_{2}$ | E | $147.05\phantom{\rule{0.166667em}{0ex}}\left[\mathrm{GPa}\right]-0.0525[T-{T}_{0}]\phantom{\rule{0.166667em}{0ex}}\left[\frac{\mathrm{GPa}}{{}^{\circ}\mathrm{C}}\right]$ | ${T}_{0}=25<T<954$ | [62] | |

$\nu $ | $0.295-5.9\times {10}^{-5}[T-{T}_{0}]\phantom{\rule{0.166667em}{0ex}}\left[\frac{1}{{}^{\circ}\mathrm{C}}\right]$ | ${T}_{0}=25<T<954$ | [62] | ||

$\frac{c}{a}$ | $0.804\approx const.$ | ${T}_{0}=20<T<1450$ | [63] | ||

$\gamma $/${\alpha}_{2}$ | ${\rho}_{0}$ | $4.219\phantom{\rule{0.166667em}{0ex}}\left[\frac{\mathrm{g}}{{\mathrm{cm}}^{3}}\right]-1.579\times {10}^{-4}[T-{T}_{0}]\phantom{\rule{0.166667em}{0ex}}\left[\frac{\mathrm{g}}{{\mathrm{cm}}^{3}{\phantom{\rule{4pt}{0ex}}}^{\circ}\mathrm{C}}\right]$ | ${T}_{0}=25<T<1150$ | [64] | |

${c}_{p}$ | $0.6207\phantom{\rule{0.166667em}{0ex}}\left[\frac{\mathrm{J}}{\mathrm{g}{\phantom{\rule{0.166667em}{0ex}}}^{\circ}\mathrm{C}}\right]+1.5897\times {10}^{-4}[T-{T}_{0}]\phantom{\rule{0.166667em}{0ex}}\left[\frac{\mathrm{J}}{\mathrm{g}{{[}^{\circ}\mathrm{C}]}^{2}}\right]$ | ${T}_{0}=20<T<900$ | [65] | ||

$\kappa $ | $15.35\phantom{\rule{0.166667em}{0ex}}\left[\frac{\mathrm{W}}{\mathrm{m}{\phantom{\rule{0.166667em}{0ex}}}^{\circ}\mathrm{C}}\right]+1.364\times {10}^{-2}[T-{T}_{0}]\phantom{\rule{0.166667em}{0ex}}\left[\frac{\mathrm{W}}{\mathrm{m}{{[}^{\circ}\mathrm{C}]}^{2}}\right]$ | ${T}_{0}=100<T<900$ | [65] | ||

${\alpha}_{t}$ | $8.936\times {10}^{-6}\phantom{\rule{0.166667em}{0ex}}\left[\frac{1}{{}^{\circ}\mathrm{C}}\right]+3.4\times {10}^{-9}[T-{T}_{0}]\phantom{\rule{0.166667em}{0ex}}\left[\frac{1}{{{[}^{\circ}\mathrm{C}]}^{2}}\right]$ | ${T}_{0}=100<T<900$ | [65] | ||

model parameters | $\gamma $ | ${k}_{D}$ | ${k}_{D}\left(0{\phantom{\rule{0.166667em}{0ex}}}^{\circ}\mathrm{C}\right)+{k}_{D}\left(T\right)$ | [18] | |

${k}_{D}\left(0{\phantom{\rule{0.166667em}{0ex}}}^{\circ}\mathrm{C}\right)$ | 0.125 [MPa$\sqrt{\mathrm{m}}$] | [18] | |||

${k}_{D}\left(T\right)$ | $\mathrm{sin}\left(0.00395\left[\frac{1}{{}^{\circ}\mathrm{C}}\right]T\right)\left[2.41\times {10}^{-6}\phantom{\rule{0.166667em}{0ex}}\frac{\left[\mathrm{Pa}\sqrt{\mathrm{m}}\right]}{{{[}^{\circ}\mathrm{C}]}^{3.61}}\phantom{\rule{0.166667em}{0ex}}{T}^{3.61}\right]$ | [18] | |||

${k}_{L}$ | ${k}_{L}\left(0{\phantom{\rule{0.166667em}{0ex}}}^{\circ}\mathrm{C}\right)+{k}_{L}\left(T\right)$ | [18] | |||

${k}_{L}\left(0{\phantom{\rule{0.166667em}{0ex}}}^{\circ}\mathrm{C}\right)$ | 0.125 [MPa$\sqrt{\mathrm{m}}$] | [18] | |||

${k}_{L}\left(T\right)$ | $\mathrm{sin}\left(0.00462\left[\frac{1}{{}^{\circ}\mathrm{C}}\right]T\right)\left[2.64\phantom{\rule{0.166667em}{0ex}}\frac{\left[\mathrm{Pa}\sqrt{\mathrm{m}}\right]}{{{[}^{\circ}\mathrm{C}]}^{1.54}}\phantom{\rule{0.166667em}{0ex}}{T}^{1.54}\right]$ | [18] | |||

$\gamma $/${\alpha}_{2}$ | ${\nu}_{0}$ | $0.001\phantom{\rule{0.166667em}{0ex}}\left[\frac{1}{s}\right]$ | [18] | ||

n | $50\phantom{\rule{0.166667em}{0ex}}[-]$ | [18] |

**Table 3.**Identified model parameters; ls: longitudinal slip, ms: mixed slip, ts: transversal slip, lt: longitudinal twinning, tt: transversal twinning.

Phase | Symb | Value | Unit | System Index (cf. Table 1) | Annotation | |
---|---|---|---|---|---|---|

dislocation accumulation | $\gamma $ | ${A}_{\alpha ,0}$ | $1\times {10}^{9}$ | [$\frac{1}{{\mathrm{mm}}^{2}}$] | $\alpha =\phantom{\rule{0.166667em}{0ex}}$1–3 | ls |

${A}_{\alpha ,0}$ | $2\times {10}^{9}$ | [$\frac{1}{{\mathrm{mm}}^{2}}$] | $\alpha =\phantom{\rule{0.166667em}{0ex}}$4–6 | ms | ||

${A}_{\alpha ,0}$ | $2\times {10}^{9}$ | [$\frac{1}{{\mathrm{mm}}^{2}}$] | $\alpha =\phantom{\rule{0.166667em}{0ex}}$7–12 | ts | ||

${\rho}_{\alpha ,\mathrm{sat}}^{\mathrm{dis}}$ | $1\times {10}^{8}$ | [$\frac{1}{{\mathrm{mm}}^{2}}$] | $\alpha =\phantom{\rule{0.166667em}{0ex}}$1–12 | all slip systems | ||

${p}_{\alpha}$ | 0.05 | [–] | for $\alpha =\phantom{\rule{0.166667em}{0ex}}$1–12 | all slip systems | ||

${\alpha}_{2}$ | ${A}_{\alpha ,0}$ | $2\times {10}^{9}$ | [$\frac{1}{{\mathrm{mm}}^{2}}$] | $\alpha =\phantom{\rule{0.166667em}{0ex}}$4–6 | ms | |

${A}_{\alpha ,0}$ | $2\times {10}^{9}$ | [$\frac{1}{{\mathrm{mm}}^{2}}$] | $\alpha =\phantom{\rule{0.166667em}{0ex}}$7–12 | ts | ||

${\rho}_{\alpha ,\mathrm{sat}}^{\mathrm{dis}}$ | $1\times {10}^{8}$ | [$\frac{1}{{\mathrm{mm}}^{2}}$] | $\alpha =\phantom{\rule{0.166667em}{0ex}}$4–12 | ms and ts | ||

${p}_{\alpha}$ | 0.05 | [–] | $\alpha =\phantom{\rule{0.166667em}{0ex}}$4–12 | ms and ts | ||

hardening parameters | $\gamma $ | ${h}_{\alpha \beta}$ | 0 | [MPa] | $\alpha =\phantom{\rule{0.166667em}{0ex}}$1–3 and $\beta =\phantom{\rule{0.166667em}{0ex}}$1 | ls by lt |

${h}_{\alpha \beta}$ | 100 | [MPa] | $\alpha =\phantom{\rule{0.166667em}{0ex}}$4–12 and $\beta =\phantom{\rule{0.166667em}{0ex}}$1 | ms and ts by lt | ||

${h}_{\alpha \beta}$ | 1500 | [MPa] | $\alpha =\phantom{\rule{0.166667em}{0ex}}$1–6 and $\beta =\phantom{\rule{0.166667em}{0ex}}$2–4 | ls and ms by tt | ||

${h}_{\alpha \beta}$ | 300 | [MPa] | $\alpha =\phantom{\rule{0.166667em}{0ex}}$7–12 and $\beta =\phantom{\rule{0.166667em}{0ex}}$2–4 | ts by tt | ||

${h}_{\beta {\beta}^{{}^{\prime}}}$ | 0 | [MPa] | $\beta =\phantom{\rule{0.166667em}{0ex}}$1 and ${\beta}^{{}^{\prime}}=\phantom{\rule{0.166667em}{0ex}}$1 | lt by lt | ||

${h}_{\beta {\beta}^{{}^{\prime}}}$ | 1500 | [MPa] | $\beta =\phantom{\rule{0.166667em}{0ex}}$1 and ${\beta}^{{}^{\prime}}=\phantom{\rule{0.166667em}{0ex}}$2–4 | lt by tt | ||

${h}_{\beta {\beta}^{{}^{\prime}}}$ | 300 | [MPa] | $\beta =\phantom{\rule{0.166667em}{0ex}}$2–4 and ${\beta}^{{}^{\prime}}=\phantom{\rule{0.166667em}{0ex}}$2–4; $\beta \ne {\beta}^{{}^{\prime}}$ | tt by tt | ||

${h}_{\beta {\beta}^{{}^{\prime}}}$ | 100 | [MPa] | $\beta =\phantom{\rule{0.166667em}{0ex}}$2–4 and ${\beta}^{{}^{\prime}}=\phantom{\rule{0.166667em}{0ex}}$1 | tt by lt | ||

${C}_{\beta \alpha}$ | 900 | [–] | $\beta =\phantom{\rule{0.166667em}{0ex}}$1 and $\alpha =\phantom{\rule{0.166667em}{0ex}}$1–12 | lt by all slip systems | ||

${C}_{\beta \alpha}$ | 150 | [–] | $\beta =\phantom{\rule{0.166667em}{0ex}}$2–4 and $\alpha =\phantom{\rule{0.166667em}{0ex}}$1–12 | tt by all slip systems |

**Table 4.**Microstructural data from literature experiments with fully lamellar alloys. *: not reported. If no ${\alpha}_{2}$ was reported in corresponding reference, it was set to 10 Vol. %; the domain sizes ${\lambda}_{D}$ are assumed to be $50\phantom{\rule{0.166667em}{0ex}}{\lambda}_{L}$

Composition | ${\mathit{\alpha}}_{2}$ Content [Vol. %] | ${\mathit{\lambda}}_{\mathit{C}}$ [$\mathsf{\mu}$m] | ${\mathit{\lambda}}_{\mathit{D}}$ [$\mathsf{\mu}$m] | ${\mathit{\lambda}}_{\mathit{L}}$ [$\mathsf{\mu}$m] | Ref. |
---|---|---|---|---|---|

Ti-45.3Al-2.1Cr-2Nb | 20 | 75 | 4.4 * | 0.088 | [13] |

Ti-45.3Al-2.1Cr-2Nb | 29 | 78 | 2.9 * | 0.058 | [13] |

Ti-45.3Al-2.1Cr-2Nb | 32 | 56 | 1.75 * | 0.035 | [13] |

Ti-45.5Al-2Cr-1.5Nb-1V | 10 * | 260 | 8 * | 0.16 | [14] |

Ti-45.5Al-2Cr-1.5Nb-1V | 10 * | 390 | 8 * | 0.16 | [14] |

Ti-45.5Al-2Cr-1.5Nb-1V | 10 * | 690 | 8 * | 0.16 | [14] |

Ti-45.5Al-2Cr-1.5Nb-1V | 10 * | 920 | 8.5 * | 0.17 | [14] |

Ti-45.5Al-2Cr-1.5Nb-1V | 10 * | 370 | 0.75 * | 0.015 | [14] |

Ti-45.5Al-2Cr-1.5Nb-1V | 10 * | 360 | 4.75 * | 0.095 | [14] |

Ti-45.5Al-2Cr-1.5Nb-1V | 10 * | 380 | 25 * | 0.5 | [14] |

Ti-47Al-2Cr-2Nb | 10 * | 65 | 5 * | 0.1 | [10] |

Ti-47Al-2Cr-2Nb | 10 * | 62 | 19.5 * | 0.39 | [10] |

Ti-47Al-2Cr-2Nb-0.15B | 10 * | 33 | 22 * | 0.44 | [10] |

Ti-47Al-2Cr-1.8Nb-0.2W-0.15B | 10 * | 31 | 15 * | 0.3 | [10] |

Ti-47Al-2Cr-1.8Nb-0.2W-0.15B | 10 * | 25 | 7.05 * | 0.141 | [10] |

Ti-46Al-2Cr-1.8Nb-0.2W-0.15B | 10 * | 26 | 5.25 * | 0.105 | [10] |

Ti-47Al-2Cr-1Nb-1Ta | 10 * | 60 | 4.3 * | 0.086 | [10] |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Schnabel, J.E.; Bargmann, S. Accessing Colony Boundary Strengthening of Fully Lamellar TiAl Alloys via Micromechanical Modeling. *Materials* **2017**, *10*, 896.
https://doi.org/10.3390/ma10080896

**AMA Style**

Schnabel JE, Bargmann S. Accessing Colony Boundary Strengthening of Fully Lamellar TiAl Alloys via Micromechanical Modeling. *Materials*. 2017; 10(8):896.
https://doi.org/10.3390/ma10080896

**Chicago/Turabian Style**

Schnabel, Jan Eike, and Swantje Bargmann. 2017. "Accessing Colony Boundary Strengthening of Fully Lamellar TiAl Alloys via Micromechanical Modeling" *Materials* 10, no. 8: 896.
https://doi.org/10.3390/ma10080896