# 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

## References

- Ko, Y.; Tsurumi, T.; Fukunaga, O.; Yano, T. High pressure sintering of diamond-SiC composite. J. Mater. Sci.
**2001**, 36, 469–475. [Google Scholar] [CrossRef] - Herrmann, M.; Matthey, B.; Hohn, S.; Kinski, I.; Rafaja, D.; Michaelis, A. Diamond-ceramics composites–new materials for a wide range of challenging applications. J. Eur. Ceram. Soc.
**2012**, 32, 1915–1923. [Google Scholar] [CrossRef] - Clayton, J.; Guziewski, M.; Ligda, J.; Leavy, R.; Knap, J. A multi-scale approach for phase field modeling of ultra-hard ceramic composites. Materials
**2021**, 14, 1408. [Google Scholar] [CrossRef] [PubMed] - Taya, M.; Hayashi, S.; Kobayashi, A.; Yoon, H. Toughening of a particulate-reinforced ceramic-matrix composite by thermal residual stress. J. Am. Ceram. Soc.
**1990**, 73, 1382–1391. [Google Scholar] [CrossRef] - Sigl, L. Microcrack toughening in brittle materials containing weak and strong interfaces. Acta Mater.
**1996**, 44, 3599–3609. [Google Scholar] [CrossRef] - Rubink, W.; Ageh, V.; Lide, H.; Ley, N.; Young, M.; Casem, D.; Faierson, E.; Scharf, T. Spark plasma sintering of B
_{4}C and B_{4}C-TiB_{2}composites: Deformation and failure mechanisms under quasistatic and dynamic loading. J. Eur. Ceram. Soc.**2021**, 41, 3321–3332. [Google Scholar] [CrossRef] - Gao, Y.; Tang, T.; Yi, C.; Zhang, W.; Li, D.; Xie, W.; Huang, W.; Ye, N. Study of static and dynamic behavior of TiB
_{2}–B_{4}C composite. Mater. Des.**2016**, 92, 814–822. [Google Scholar] [CrossRef] - Pittari, J., III; Subhash, G.; Zheng, J.; Halls, V.; Jannotti, P. The rate-dependent fracture toughness of silicon carbide-and boron carbide-based ceramics. J. Eur. Ceram. Soc.
**2015**, 35, 4411–4422. [Google Scholar] [CrossRef] - Wenbo, H.; Jiaxing, G.; Jihong, Z.; Jiliang, Y. Microstructure and properties of B
_{4}C-ZrB_{2}ceramic composites. J. Eng. Innov. Technol.**2013**, 3, 163–166. [Google Scholar] - Sadowski, T.; Marsavina, L. Multiscale modelling of two-phase ceramic matrix composites. Comput. Mater. Sci.
**2011**, 50, 1336–1346. [Google Scholar] [CrossRef] - Ortiz, M.; Suresh, S. Statistical properties of residual stresses and intergranular fracture in ceramic materials. J. Appl. Mech.
**1993**, 60, 77–84. [Google Scholar] [CrossRef] - Vanderwalker, D.; Croft, W. Dislocations in shock-loaded titanium diboride. J. Mater. Res.
**1988**, 3, 761–763. [Google Scholar] [CrossRef] - Vanderwalker, D. Fracture in titanium diboride. Phys. Status Solidi A
**1989**, 111, 119–126. [Google Scholar] [CrossRef] - Clayton, J. Nonlinear thermodynamic phase field theory with application to fracture and dynamic inelastic phenomena in ceramic polycrystals. J. Mech. Phys. Solids
**2021**, 157, 104633. [Google Scholar] [CrossRef] - Clayton, J.; Leavy, R.; Knap, J. Phase field mechanics of residually stressed ceramic composites. Philos. Mag.
**2022**, 102, 1891–1944. [Google Scholar] [CrossRef] - Bryant, E.; Sun, W. A mixed-mode phase field fracture model in anisotropic rocks with consistent kinematics. Comput. Methods Appl. Mech. Eng.
**2018**, 342, 561–584. [Google Scholar] [CrossRef] - Na, S.; Sun, W. Computational thermomechanics of crystalline rock, Part I: A combined multi-phase field/crystal plasticity approach for single crystal simulations. Comput. Methods Appl. Mech. Eng.
**2018**, 338, 657–691. [Google Scholar] [CrossRef] - Nguyen, T.T.; Réthoré, J.; Yvonnet, J.; Baietto, M.C. Multi-phase field modeling of anisotropic crack propagation for polycrystalline materials. Comput. Mech.
**2017**, 60, 289–314. [Google Scholar] [CrossRef][Green Version] - Shahba, A.; Ghosh, S. Coupled phase field finite element model for crack propagation in elastic polycrystalline microstructures. Int. J. Fract.
**2019**, 219, 31–64. [Google Scholar] [CrossRef] - Del Piero, G.; Lancioni, G.; March, R. A variational model for fracture mechanics: Numerical experiments. J. Mech. Phys. Solids
**2007**, 55, 2513–2537. [Google Scholar] [CrossRef] - Agrawal, V.; Dayal, K. Dependence of equilibrium Griffith surface energy on crack speed in phase field models for fracture coupled to elastodynamics. Int. J. Fract.
**2017**, 207, 243–249. [Google Scholar] [CrossRef] - Amirian, B.; Jafarzadeh, H.; Abali, B.; Reali, A.; Hogan, J. Thermodynamically-consistent derivation and computation of twinning and fracture in brittle materials by means of phase field approaches in the finite element method. Int. J. Solids Struct.
**2022**, 252, 111789. [Google Scholar] [CrossRef] - Needleman, A. A continuum model for void nucleation by inclusion debonding. J. Appl. Mech.
**1987**, 54, 525–531. [Google Scholar] [CrossRef] - Clayton, J. Dynamic plasticity and fracture in high density polycrystals: Constitutive modeling and numerical simulation. J. Mech. Phys. Solids
**2005**, 53, 261–301. [Google Scholar] [CrossRef] - Foulk, J.; Vogler, T. A grain-scale study of spall in brittle materials. Int. J. Fract.
**2010**, 163, 225–242. [Google Scholar] [CrossRef] - Clayton, J.; Knap, J. A phase field model of deformation twinning: Nonlinear theory and numerical simulations. Phys. D
**2011**, 240, 841–858. [Google Scholar] [CrossRef][Green Version] - Clayton, J.; Knap, J. A geometrically nonlinear phase field theory of brittle fracture. Int. J. Fract.
**2014**, 189, 139–148. [Google Scholar] [CrossRef] - Clayton, J.; Williams, C. Modelling the anomalous shock response of titanium diboride. Proc. R. Soc. Lond. A
**2022**, 478, 20220253. [Google Scholar] [CrossRef] - Clayton, J. Finsler differential geometry in continuum mechanics: Fundamental concepts, history, and renewed application to ferromagnetic solids. Math. Mech. Solids
**2022**, 27, 910–949. [Google Scholar] [CrossRef] - Johnson, K. Contact Mechanics; Cambridge University Press: Cambridge, UK, 1985. [Google Scholar]
- Chen, X.; Hutchinson, J.; Evans, A. The mechanics of indentation induced lateral cracking. J. Am. Ceram. Soc.
**2005**, 88, 1233–1238. [Google Scholar] [CrossRef] - Quinn, G.; Bradt, R. On the Vickers indentation fracture toughness test. J. Am. Ceram. Soc.
**2007**, 90, 673–680. [Google Scholar] [CrossRef] - Kraft, R.; Molinari, J. A statistical investigation of the effects of grain boundary properties on transgranular fracture. Acta Mater.
**2008**, 56, 4739–4749. [Google Scholar] [CrossRef] - Clayton, J.; Rubink, W.; Ageh, V.; Choudhuri, D.; Chen, R.; Du, J.; Scharf, T. Deformation and failure mechanics of boron carbide-titanium diboride composites at multiple scales. JOM
**2019**, 71, 2567–2575. [Google Scholar] [CrossRef] - Zavattieri, P.; Raghuram, P.; Espinosa, H. A computational model of ceramic microstructures subjected to multi-axial dynamic loading. J. Mech. Phys. Solids
**2001**, 49, 27–68. [Google Scholar] [CrossRef] - Wereszczak, A.; Kirkland, T.; Strong, K.; Jadaan, O.; Thompson, G. Size scaling of tensile failure stress in boron carbide. Adv. Appl. Ceram.
**2010**, 109, 487–492. [Google Scholar] [CrossRef] - Giannakopoulos, A.; Larsson, P.L.; Vestergaard, R. Analysis of Vickers indentation. Int. J. Solids Struct.
**1994**, 31, 2679–2708. [Google Scholar] [CrossRef] - Zeng, K.; Giannakopoulos, A.; Rowcliffe, D. Vickers indentations in glass II. Comparison of finite element analysis and experiments. Acta Metall. Et Mater.
**1995**, 43, 1945–1954. [Google Scholar] [CrossRef] - Gurtin, M. Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Phys. D
**1996**, 92, 178–192. [Google Scholar] [CrossRef] - Clayton, J.; Knap, J. Phase field modeling of coupled fracture and twinning in single crystals and polycrystals. Comput. Methods Appl. Mech. Eng.
**2016**, 312, 447–467. [Google Scholar] [CrossRef] - Allen, S.; Cahn, J. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall.
**1979**, 27, 1085–1095. [Google Scholar] [CrossRef] - Clayton, J. Nonlinear Mechanics of Crystals; Springer: Dordrecht, The Netherlands, 2011. [Google Scholar]
- Brace, W.; Paulding, B.; Scholz, C. Dilatancy in the fracture of crystalline rocks. J. Geophys. Res.
**1966**, 71, 3939–3953. [Google Scholar] [CrossRef] - Curran, D.; Seaman, L.; Cooper, T.; Shockey, D. Micromechanical model for comminution and granular flow of brittle material under high strain rate application to penetration of ceramic targets. Int. J. Impact Eng.
**1993**, 13, 53–83. [Google Scholar] [CrossRef] - Clayton, J.; Knap, J. Phase field modeling of directional fracture in anisotropic polycrystals. Comput. Mater. Sci.
**2015**, 98, 158–169. [Google Scholar] [CrossRef] - Borden, M.; Verhoosel, C.; Scott, M.; Hughes, T.; Landis, C. A phase field description of dynamic brittle fracture. Comput. Methods Appl. Mech. Eng.
**2012**, 217, 77–95. [Google Scholar] [CrossRef] - Amor, H.; Marigo, J.J.; Maurini, C. Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments. J. Mech. Phys. Solids
**2009**, 57, 1209–1229. [Google Scholar] [CrossRef] - Clayton, J.; Leavy, R.; Knap, J. Phase field modeling of heterogeneous microcrystalline ceramics. Int. J. Solids Struct.
**2019**, 166, 183–196. [Google Scholar] [CrossRef] - An, Q.; Goddard, W.A.; Cheng, T. Atomistic explanation of shear-induced amorphous band formation in boron carbide. Phys. Rev. Lett.
**2014**, 113, 095501. [Google Scholar] [CrossRef][Green Version] - An, Q.; Goddard, W. Atomistic origin of brittle failure of boron carbide from large-scale reactive dynamics simulations: Suggestions toward improved ductility. Phys. Rev. Lett.
**2015**, 115, 105051. [Google Scholar] [CrossRef][Green Version] - Clayton, J.; Knap, J. Continuum modeling of twinning, amorphization, and fracture: Theory and numerical simulations. Contin. Mech. Thermodyn.
**2018**, 30, 421–455. [Google Scholar] [CrossRef] - Clayton, J.; Zorn, J.; Leavy, R.; Guziewski, M.; Knap, J. Phase field modeling of diamond-silicon carbide ceramic composites with tertiary grain boundary phases. Int. J. Fract.
**2022**, 237, 101–138. [Google Scholar] [CrossRef] - Patil, S.; Heider, Y.; Padilla, C.; Cruz-Chú, E.; Markert, B. A comparative molecular dynamics-phase field modeling approach to brittle fracture. Comput. Methods Appl. Mech. Eng.
**2016**, 312, 117–129. [Google Scholar] [CrossRef] - Hu, S.; Henager, C.; Chen, L.Q. Simulations of stress-induced twinning and de-twinning: A phase field model. Acta Mater.
**2010**, 58, 6554–6564. [Google Scholar] [CrossRef] - Jafarzadeh, H.; Levitas, V.; Farrahi, G.; Javanbakht, M. Phase field approach for nanoscale interactions between crack propagation and phase transformation. Nanoscale
**2019**, 11, 22243–22247. [Google Scholar] [CrossRef] [PubMed] - Li, W.; Hahn, E.; Branicio, P.; Yao, X.; Germann, T.; Feng, B.; Zhang, X. Defect reversibility regulates dynamic tensile strength in silicon carbide at high strain rates. Scr. Mater.
**2022**, 213, 114593. [Google Scholar] [CrossRef] - Clayton, J.; Knap, J. Phase field modeling of twinning in indentation of transparent single crystals. Model. Simul. Mater. Sci. Eng.
**2011**, 19, 085005. [Google Scholar] [CrossRef] - Thevenot, F. A review on boron carbide. Key Eng. Mater.
**1991**, 56, 59–88. [Google Scholar] [CrossRef] - Munro, R. Material properties of titanium diboride. J. Res. Natl. Inst. Stand. Technol.
**2000**, 105, 709–720. [Google Scholar] [CrossRef] - Swab, J.; Meredith, C.; Casem, D.; Gamble, W. Static and dynamic compression strength of hot-pressed boron carbide using a dumbbell-shaped specimen. J. Mater. Sci.
**2017**, 52, 10073–10084. [Google Scholar] [CrossRef] - Dandekar, D.; Benfanti, D. Strength of titanium diboride under shock wave loading. J. Appl. Phys.
**1993**, 73, 673–679. [Google Scholar] [CrossRef] - Dodd, S.; Saunders, G.; James, B. Temperature and pressure dependences of the elastic properties of ceramic boron carbide (B
_{4}C). J. Mater. Sci.**2002**, 37, 2731–2736. [Google Scholar] [CrossRef] - Beaudet, T.; Smith, J.; Adams, J. Surface energy and relaxation in boron carbide (10$\underset{1}{\xaf}$1) from first principles. Solid State Commun.
**2015**, 219, 43–47. [Google Scholar] [CrossRef] - Fanchini, G.; McCauley, J.; Chhowalla, M. Behavior of disordered boron carbide under stress. Phys. Rev. Lett.
**2006**, 97, 035502. [Google Scholar] [CrossRef] [PubMed][Green Version] - Clayton, J. Computational modeling of dual-phase ceramics with Finsler-geometric phase field mechanics. Comput. Model. Eng. Sci. CMES
**2019**, 120, 333–350. [Google Scholar] [CrossRef][Green Version] - Miehe, C.; Schaenzel, L.M.; Ulmer, H. Phase field modeling of fracture in multi-physics problems. Part I. Balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids. Comput. Methods Appl. Mech. Eng.
**2015**, 294, 449–485. [Google Scholar] [CrossRef] - Clayton, J.; Knap, J. Geometric micromechanical modeling of structure changes, fracture, and grain boundary layers in polycrystals. J. Micromech. Mol. Phys.
**2018**, 3, 1840001. [Google Scholar] [CrossRef] - Hwang, C.; Du, J.; Yang, Q.; Celik, A.; Christian, K.; An, Q.; Schaefer, M.; Xie, K.; LaSalvia, J.; Hemker, K.; et al. Addressing amorphization and transgranular fracture of B
_{4}C through Si doping and TiB_{2}microparticle reinforcing. J. Am. Ceram. Soc.**2022**, 105, 2959–2977. [Google Scholar] [CrossRef] - Yang, Q.; Celik, A.; Du, J.; LaSalvia, J.; Hwang, C.; Haber, R. Advancing the mechanical properties of Si/B co-doped boron carbide through TiB
_{2}reinforcement. Mater. Lett.**2020**, 266, 127480. [Google Scholar] [CrossRef] - Hu, G.; Chen, C.; Ramesh, K.; McCauley, J. Mechanisms of dynamic deformation and dynamic failure in aluminum nitride. Acta Mater.
**2012**, 60, 3480–3490. [Google Scholar] [CrossRef] - Shen, Y.; Li, G.; An, Q. Enhanced fracture toughness of boron carbide from microalloying and nanotwinning. Scr. Mater.
**2019**, 162, 306–310. [Google Scholar] [CrossRef] - Ye, K.; Wang, Z. Twins enhanced mechanical properties of boron carbide. Ceram. Int.
**2022**, 48, 14499–14506. [Google Scholar] [CrossRef] - Chen, M.; McCauley, J.; Hemker, K. Shock-induced localized amorphization in boron carbide. Science
**2003**, 299, 1563–1566. [Google Scholar] [CrossRef] [PubMed]

**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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**MDPI and ACS Style**

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