# Modeling Deformation and Fracture of Boron-Based Ceramics with Nonuniform Grain and Phase Boundaries and Thermal-Residual Stress

## Abstract

**:**

_{4}C) and boron carbide-titanium diboride (B

_{4}C-TiB

_{2}), the latter a dual-phase composite. Recent advancements in processing technology enable the production of these materials via spark-plasma sintering (SPS) at nearly full theoretical density. Numerical simulations invoking biaxial loading (e.g., pure shear) demonstrate how properties and mechanisms at the scale of the microstructure influence overall strength and ductility. In agreement with experimental inferences, simulations show that plasticity is more prevalent in the TiB

_{2}phase of the composite and reduces the tendency for transgranular fracture. The composite demonstrates greater overall strength and ductility than monolithic B

_{4}C in both simulations and experiments. Toughening of the more brittle B

_{4}C phase from residual stress, in addition to crack mitigation from the stronger and more ductile TiB

_{2}phase are deemed advantageous attributes of the composite.

## 1. Introduction

## 2. Phase Field Mechanics

#### 2.1. Coordinates and Order Parameters

#### 2.2. Kinematics

#### 2.3. Balance Laws and Thermodynamics

#### 2.4. Twinning, Plastic Shear, and Dilatation

#### 2.5. Isotropic Elastic Formulation

#### 2.6. Initial Stress

## 3. Material Properties and Polycrystalline Microstructures

#### 3.1. Boron Carbide Phase

#### 3.2. Titanium Diboride Phase

Parameter (Units) | B${}_{4}$C | TiB${}_{2}$ | Boundary | Description (Refs.) |
---|---|---|---|---|

${\rho}_{0}$ (g/cm${}^{3}$) | 2.52 | 4.52 | - | mass density ${}^{1}$ [34,58,59] |

${\mathsf{B}}_{0}$ (GPa) | 205 | 240 | 211 | initial bulk modulus [14,59,60] |

${\mathsf{G}}_{0}$ (GPa) | 187 | 255 | 200 | initial shear modulus [14,59,60] |

${\beta}_{0}^{\prime}$ (-) | 8.40 | 3.04 | 7.16 | bulk stiffening in (38) [14,61,62] |

${{\rm Y}}_{0}$ (J/m${}^{2}$) | 3.27 | 4.14 | 3.47 | nominal fracture energy [34,48,63] |

$\widehat{\alpha}$ | 10 | 100 | - | cleavage anisotropy [27,34,48,52] |

$\Gamma $ (J/m${}^{2}$) | 0.54 | 0.12 | - | twin boundary or SF energy [12,34,48] |

${\gamma}_{0}$ | 0.31 | 0.015 | - | max twin shear or plastic slip [12,34,48] |

$\widehat{A}$ (MPa) | 188 | 11.4 | - | phase or dislocation energy [34,48,64] |

${l}_{\xi}={l}_{\eta}$ ($\mathsf{\mu}$m) | 0.1 | 0.1 | 0.1 | regularization length [34,48,65] |

${\tilde{\sigma}}_{0}$ (GPa) | −0.496 | 1.660 | - | nominal residual stress in (46) [4,6,34] |

${K}_{R}$ (MPa·m${}^{1/2}$) | 1.54 | - | 1.25 | residual toughening in (47) [4] |

#### 3.3. Grain and Phase Boundaries

#### 3.4. Residual Stresses

## 4. Numerical Methods

#### 4.1. Geometric Rendering and Investigated Parameters

- Composition: B${}_{4}$C-23 vol. % TiB${}_{2}$ versus pure B${}_{4}$C;
- Residual stress: nonzero ${\tilde{\sigma}}_{0}$ enabled via (46) or suppressed (${\tilde{\sigma}}_{0}$ = 0);
- Twinning and slip: $\eta >0$ enabled or suppressed ($\eta =0$);
- Grain morphology: effects examined via different loading directions;
- Lattice orientation: randomized to activate different cleavage, habit, and slip planes.

#### 4.2. Boundary Conditions and Homogenization

## 5. Model Results

- Plasticity, when it occurs, is much more prevalent in the TiB${}_{2}$ phase (basal slip) than the B${}_{4}$C phase (twinning, shear bands).
- Plasticity reduces the tendency for transgranular fracture, especially in TiB${}_{2}$ grains of the composite.
- Average peak pressure $\overline{p}$ is always slightly compressive, but average pressure is negligible compared to effective deviatoric stress $\overline{\Sigma}$, as expected for equi-biaxial loading.
- Thermal-residual stress enhances overall strength and ductility, primarily via toughening of the B${}_{4}$C phase initially under residual compression.
- Heterogeneous grain and phase boundary energies from the Weibull strength statistics lead to more cracks and lower overall strength, in general, than constant boundary energies, which correspond to fewer very weak links in the microstructure.
- The composite, in which intergranular fractures dominate (with transgranular fractures arising sometimes, but less often) demonstrates greater overall strength and ductility than pure B${}_{4}$C, in which transgranular fractures dominate.
- Peak effective stress for the B${}_{4}$C-TiB${}_{2}$ composite, averaged over Sims 1, 2, and 3 from Table 2, is 1.58 GPa. Peak effective stress for pure B${}_{4}$C, averaged over Sims 13, 14, and 15, is 1.20 GPa. The ratio of composite-to-monolithic material effective strength is 1.58/1.20 = 1.32.
- When plasticity is suppressed in constitutive models of both materials (Sims 4, 5, and 6 vs. 16, 17, and 18), the ratio of composite-to-monolithic material effective strength is 1.17.

#### Comparison with Experiments and Prior Modeling

- Elastic modulus increase of approximately 20%;
- Static flexure strength increase of approximately 20%;
- Dynamic flexure strength increase of approximately 30%;
- Static fracture toughness increase on the order of 100%;
- Increased dislocation mechanisms;
- Increased tendency for intergranular over transgranular fracture;
- Vickers hardness decrease of approximately 10%;
- Mass density increase of approximately 20%.

## 6. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Boron carbide-titanium diboride initial rendering for phase field simulations: (

**a**) FE mesh of phases (yellow = TiB${}_{2}$, white = B${}_{4}$C, dark = binder); (

**b**) equilibrated initial stress component ${P}_{22}$ prior to biaxial tension–compression loading in the ${X}_{1}{X}_{2}$ plane.

**Figure 2.**Simulation 2, $\overline{\u03f5}=8.4\times {10}^{-3}$ (solid rendering): (

**a**) normal stress component ${P}_{22}$; (

**b**) fracture order parameter $\xi $; (

**c**) slip/twinning order parameter $\eta $.

**Figure 3.**Fracture order parameter $\xi $ (transparent rendering): (

**a**) Sim 1, $\overline{\u03f5}=7.8\times {10}^{-3}$; (

**b**) Sim 4, $\overline{\u03f5}=7.2\times {10}^{-3}$; (

**c**) Sim 7, $\overline{\u03f5}=6.0\times {10}^{-3}$; (

**d**) Sim 10, $\overline{\u03f5}=8.4\times {10}^{-3}$; (

**e**) Sim 14, $\overline{\u03f5}=8.4\times {10}^{-3}$; (

**f**) Sim 17, $\overline{\u03f5}=8.4\times {10}^{-3}$.

**Figure 4.**Effective average stress $\overline{\Sigma}$ versus applied biaxial strain $\overline{\u03f5}$: (

**a**) Sims 1, 2, and 3 vs. Sims 4, 5, and 6; (

**b**) Sims 1, 2, and 3 vs. Sims 7, 8, and 9; (

**c**) Sims 1, 2, and 3 vs. Sims 10, 11, and 12; (

**d**) Sims 1, 2, and 3 vs. Sims 13, 14, and 15.

**Figure 5.**Average values for simulations with B${}_{4}$C-TiB${}_{2}$ (with and without residual stress) and pure B${}_{4}$C: (

**a**) maximum effective stress $\overline{\Sigma}$; (

**b**) ductility measured by applied strain $\overline{\u03f5}$ at peak stress; (

**c**) plasticity measured by averaged slip/twinning order parameter $\overline{\eta}$ at peak stress.

**Table 2.**Phase field simulations: biaxial loading, different microstructures, and physics. Right four columns: peak average von Mises stress and average pressure and order parameters at peak stress.

Sim. | Material | Lattice | Bound. | Slip/Twin | Res. Stress | Weibull | $\overline{\Sigma}$ | $\overline{\mathit{p}}$ | $\overline{\mathit{\eta}}$ | $\overline{\mathit{\xi}}$ |
---|---|---|---|---|---|---|---|---|---|---|

Cond. | $\mathit{\eta}>0$ | $|{\tilde{\mathit{\sigma}}}_{0}|>0$ | ${{\rm Y}}^{\left(2\right)}$ | (GPa) | (GPa) | |||||

1 | B${}_{4}$C-TiB${}_{2}$ | 1 | ${\overline{\u03f5}}_{11}-{\overline{\u03f5}}_{22}$ | Y | Y | Y | 1.6349 | 0.006347 | 0.1451 | 0.1196 |

2 | B${}_{4}$C-TiB${}_{2}$ | 2 | ${\overline{\u03f5}}_{22}-{\overline{\u03f5}}_{33}$ | Y | Y | Y | 1.4603 | 0.006776 | 0.1390 | 0.1261 |

3 | B${}_{4}$C-TiB${}_{2}$ | 3 | ${\overline{\u03f5}}_{33}-{\overline{\u03f5}}_{11}$ | Y | Y | Y | 1.6419 | 0.007327 | 0.1420 | 0.1359 |

4 | B${}_{4}$C-TiB${}_{2}$ | 1 | ${\overline{\u03f5}}_{11}-{\overline{\u03f5}}_{22}$ | N | Y | Y | 1.1341 | 0.005450 | 0.0000 | 0.1584 |

5 | B${}_{4}$C-TiB${}_{2}$ | 2 | ${\overline{\u03f5}}_{22}-{\overline{\u03f5}}_{33}$ | N | Y | Y | 1.1302 | 0.003949 | 0.0000 | 0.1232 |

6 | B${}_{4}$C-TiB${}_{2}$ | 3 | ${\overline{\u03f5}}_{33}-{\overline{\u03f5}}_{11}$ | N | Y | Y | 1.2422 | 0.004801 | 0.0000 | 0.1480 |

7 | B${}_{4}$C-TiB${}_{2}$ | 1 | ${\overline{\u03f5}}_{11}-{\overline{\u03f5}}_{22}$ | Y | N | Y | 1.1776 | 0.005507 | 0.0758 | 0.1128 |

8 | B${}_{4}$C-TiB${}_{2}$ | 2 | ${\overline{\u03f5}}_{22}-{\overline{\u03f5}}_{33}$ | Y | N | Y | 1.0666 | 0.005629 | 0.0766 | 0.1064 |

9 | B${}_{4}$C-TiB${}_{2}$ | 3 | ${\overline{\u03f5}}_{33}-{\overline{\u03f5}}_{11}$ | Y | N | Y | 1.1479 | 0.007116 | 0.0784 | 0.1122 |

10 | B${}_{4}$C-TiB${}_{2}$ | 1 | ${\overline{\u03f5}}_{11}-{\overline{\u03f5}}_{22}$ | Y | Y | N | 1.8663 | 0.009083 | 0.1782 | 0.1478 |

11 | B${}_{4}$C-TiB${}_{2}$ | 2 | ${\overline{\u03f5}}_{22}-{\overline{\u03f5}}_{33}$ | Y | Y | N | 1.7105 | 0.008339 | 0.1618 | 0.1323 |

12 | B${}_{4}$C-TiB${}_{2}$ | 3 | ${\overline{\u03f5}}_{33}-{\overline{\u03f5}}_{11}$ | Y | Y | N | 1.8670 | 0.010320 | 0.1813 | 0.1662 |

13 | B${}_{4}$C | 1 | ${\overline{\u03f5}}_{11}-{\overline{\u03f5}}_{22}$ | Y | N | N | 1.2460 | 0.008489 | 0.0707 | 0.1485 |

14 | B${}_{4}$C | 2 | ${\overline{\u03f5}}_{22}-{\overline{\u03f5}}_{33}$ | Y | N | N | 1.1580 | 0.007554 | 0.0749 | 0.1392 |

15 | B${}_{4}$C | 3 | ${\overline{\u03f5}}_{33}-{\overline{\u03f5}}_{11}$ | Y | N | N | 1.1903 | 0.006258 | 0.0612 | 0.1217 |

16 | B${}_{4}$C | 1 | ${\overline{\u03f5}}_{11}-{\overline{\u03f5}}_{22}$ | N | N | N | 1.0212 | 0.006541 | 0.0000 | 0.2383 |

17 | B${}_{4}$C | 2 | ${\overline{\u03f5}}_{22}-{\overline{\u03f5}}_{33}$ | N | N | N | 1.0240 | 0.005984 | 0.0000 | 0.2374 |

18 | B${}_{4}$C | 3 | ${\overline{\u03f5}}_{33}-{\overline{\u03f5}}_{11}$ | N | N | N | 1.0203 | 0.006182 | 0.0000 | 0.2383 |

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Clayton, J.D.
Modeling Deformation and Fracture of Boron-Based Ceramics with Nonuniform Grain and Phase Boundaries and Thermal-Residual Stress. *Solids* **2022**, *3*, 643-664.
https://doi.org/10.3390/solids3040040

**AMA Style**

Clayton JD.
Modeling Deformation and Fracture of Boron-Based Ceramics with Nonuniform Grain and Phase Boundaries and Thermal-Residual Stress. *Solids*. 2022; 3(4):643-664.
https://doi.org/10.3390/solids3040040

**Chicago/Turabian Style**

Clayton, John D.
2022. "Modeling Deformation and Fracture of Boron-Based Ceramics with Nonuniform Grain and Phase Boundaries and Thermal-Residual Stress" *Solids* 3, no. 4: 643-664.
https://doi.org/10.3390/solids3040040