# Phase-Field Modeling of Fused Silica Cone-Crack Vickers Indentation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{IT}), the indentation modulus (E

_{IT}), the plane strain modulus (E*), the indentation creep (C

_{IT}), the indentation relaxation (R

_{IT}), and the elasto-plastic behavior (work, W), can be determined in one measurement [1]. Based on the measurement results, it is possible to construct stress–strain diagrams, which are of great importance for materials and coatings for which the conventional static tensile test is not applicable. In addition to the above-mentioned mechanical properties, by applying higher loads, it is possible to estimate the indentation fracture toughness of the material based on the cracks at the tips of the Vickers indent.

- To the authors’ knowledge, the present paper represents the first 3D phase-field model of the Vickers indentation fracture to this date.
- An asymmetric model, where the energy decomposition is applied for both the crack driving force and the stress field, is used.
- Contact analysis with friction is applied.

## 2. Materials and Methods

#### 2.1. Material and Experimental Setup

^{3}) produced by Anton Paar, TriTec SA, (Corcelles-Cormondrèche, Switzerland) according to EN ISO 14577-1:2016, at room temperature using a certified Vickers diamond pyramidal indenter. The measurements were performed using different loads. The load of 50 mN was selected to determine the plane strain modulus, E*, and the indentation modulus, E

_{IT}, in accordance with recommendations from the calibration certificate not to perform indentations over 100 mN using the Berkovich or the Vickers indenter. Higher loads could cause cracks that may influence the results. Based on the measurement, it was concluded that the measured values of E* are in excellent correlation with certified values. The certified and the measured E*, the certified Poisson’s ratio, ν, and the fracture toughness value, K

_{IC}, from the literature, used for further phase-field modeling, are shown in Table 1.

_{IC}, could not be directly measured by the instrumented indentation test, but it could be calculated by measuring the total length of cracks emanating from the corners of indentations at loads higher then critical force, Pc, at which the first cracks occur. The model used for the calculation of K

_{IC}is dependent on the type of crack, median, radial, half-penny, cone, or lateral, using the Palmquist method [15] or the method proposed by Anstis et al. [16]. As mentioned in the Introduction Section, during indentation, the load was increased to determine the critical force, to pinpoint the start of crack formation, and possibly, to calculate K

_{IC}.

#### 2.2. Numerical Modeling

#### 2.2.1. Phase-Field Formulation

^{ext}represents the vector of external forces. The history field, H(t), in Equation (4) is employed instead of ${\psi}_{e}$ to prevent the crack “healing”.

#### 2.2.2. Energy Split

#### 2.2.3. Indentation Modeling

^{®}processor, with a 3.80 GHz clock speed and 128 GB RAM memory. Additionally, ABAQUS was accelerated with a NVidia

^{®}RTX™ GPU unit. The average duration of the simulations was around 10 days.

## 3. Results and Discussion

## 4. Conclusions

_{2}, and can be assumed to be homogeneous. The use of graphic accelerated servers will enhance the computational speeds in future investigations, especially in combination with efficient adaptive remeshing and faster equation solvers. The use of a ductile formulation instead of a brittle one will possibly describe the real indentation curve even better. Additionally, the use of ductile formulation with different energy decompositions could possibly describe the radial (also median and half-penny) cracks, which appear as the second-dominant crack pattern in the fused silica indentation. This could explain why different crack patterns appear on the same specimen at the same indentation force.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- EN ISO 14577-1:2015; Metallic Materials–Instrumented Indentation Test for Hardness and Materials Parameters–Part 1: Test Method. ISO: Geneva, Switzerland, 2015.
- Bruns, S.; Johanns, K.E.; Rehman, H.U.R.; Pharr, G.M.; Durst, K. Constitutive modeling of indentation cracking in fused silica. J. Am. Ceram. Soc.
**2017**, 100, 1928–1940. [Google Scholar] [CrossRef] - Lee, J.H.; Gao, Y.F.; Johanns, K.E.; Pharr, G.M. Cohesive interface simulations of indentation cracking as a fracture toughness measurement method for brittle materials. Acta Mater.
**2012**, 60, 5448–5467. [Google Scholar] [CrossRef] - Bruns, S. The Indentation Densification and Cracking Behavior of Fused Silica. 2020. Available online: https://www.semanticscholar.org/paper/The-Indentation-Densification-and-Cracking-Behavior-Bruns/4b0a4ca825c17178367a0c2db4fc1b252d2b088c (accessed on 30 May 2022).
- Hagan, J.T. Cone cracks around Vickers indentations in fused silica glass. J. Mater. Sci.
**1979**, 14, 462–466. [Google Scholar] [CrossRef] - Michel, M.D.; Serbena, F.C.; Lepienski, C.M. Effect of temperature on hardness and indentation cracking of fused silica. J. Non. Cryst. Solids
**2006**, 352, 3550–3555. [Google Scholar] [CrossRef] - Kocer, C.; Collins, R.E. Angle of Hertzian Cone Cracks. J. Am. Ceram. Soc.
**1998**, 81, 1736–1742. [Google Scholar] [CrossRef] - Strobl, M.; Seelig, T. Phase field modeling of Hertzian indentation fracture. J. Mech. Phys. Solids
**2020**, 143, 104026. [Google Scholar] [CrossRef] - Lawn, B.R.; Evans, A.G. A model for crack initiation in elastic/plastic indentation fields. J. Mater. Sci.
**1977**, 12, 2195–2199. [Google Scholar] [CrossRef] - Rickhey, F.; Marimuthu, K.P.; Lee, H. Investigation on Indentation Cracking-Based Approaches for Residual Stress Evaluation. Materials
**2017**, 10, 404. [Google Scholar] [CrossRef] [PubMed][Green Version] - Cao, Y.; Kazembeyki, M.; Tang, L.; Krishnan, N.M.A.; Smedskjaer, M.M.; Hoover, C.G.; Bauchy, M. Modeling the nanoindentation response of silicate glasses by peridynamic simulations. J. Am. Ceram. Soc.
**2021**, 104, 3531–3544. [Google Scholar] [CrossRef] - Steinke, C.; Kaliske, M. A phase-field crack model based on directional stress decomposition. Comput. Mech.
**2019**, 63, 1019–1046. [Google Scholar] [CrossRef] - Kindrachuk, V.M.; Klunker, A. Phase Field Modeling of Hertzian Cone Cracks Under Spherical Indentation. Strength Mater.
**2020**, 52, 967–974. [Google Scholar] [CrossRef] - Wu, J.-Y.; Huang, Y.; Nguyen, V.P.; Mandal, T.K. Crack nucleation and propagation in the phase-field cohesive zone model with application to Hertzian indentation fracture. Int. J. Solids Struct.
**2022**, 241, 111462. [Google Scholar] [CrossRef] - Roebuck, B.; Bennett, E.; Lay, L.; Morrell, R. Palmqvist Toughness for Hard and Brittle Materials Measurement Good Practice Guide No. 9; National Physical Laboratory: Teddington, UK, 2008; p. 48. [Google Scholar]
- Anstis, G.R.; Chantikul, P.; Lawn, B.R.; Marshall, D.B. A Critical Evaluation of Indentation Techniques for Measuring Fracture Toughness: I, Direct Crack Measurements. J. Am. Ceram. Soc.
**1981**, 64, 533–538. [Google Scholar] [CrossRef] - Giannakopoulos, A.E.; Suresh, S. Determiantion of elastoplastic properties by instrumented sharp indentation. Scr. Mater.
**1999**, 40, 1191–1198. [Google Scholar] [CrossRef][Green Version] - Dao, M.; Chollacoop, N.; Van Vliet, K.J.; Venkatesh, A.; Suresh, S. Computational modeling of the forward and reverse problems in instrumented sharp indentation. Acta Mater.
**2001**, 49, 3899–3918. [Google Scholar] [CrossRef][Green Version] - Seleš, K. Abaqus Code for a Residual Control Staggered solution Scheme for the Phase-Field Modeling of Brittle Fracture. Eng. Fract. Mech.
**2019**, 205, 370–386. [Google Scholar] [CrossRef] - Miehe, C.; Welschinger, F.; Hofacker, M. Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations. Int. J. Numer. Methods Eng.
**2010**, 83, 1273–1311. [Google Scholar] [CrossRef] - Seleš, K.; Aldakheel, F.; Tonkovic, Z.; Sorić, J.; Wriggers, P. A General Phase-Field Model for Fatigue Failure in Brittle and Ductile Solids. Comput. Mech.
**2021**, 67, 1431–1452. [Google Scholar] - Seleš, K. Numerical Phase-Field Modeling of Damage in Heterogeneous Materials. Ph.D. Thesis, University of Zagreb, Zagreb, Croatia, 2020. [Google Scholar]
- Ambrosio, L.; Tortorelli, V.M. On The Approximation of Free Discontinuity Problems. Boll. Della Unione Mat. Ital.
**1992**, 6, 105–123. [Google Scholar] - Alessi, R.; Ambati, M.; Gerasimov, T.; Vidoli, S.; De Lorenzis, L. Comparison of Phase-Field Models of Fracture Coupled with Plasticity. In Advances in Computational Plasticity; Oñate, E., Peric, D., de Souza Neto, E., Chiumenti, M., Eds.; Springer: Cham, Switzerland, 2018. [Google Scholar] [CrossRef]
- De Lorenzis, L.; Gerasimov, T. Numerical implementation of phase-field models of brittle fracture. In Modeling in Engineering Using Innovative Numerical Methods for Solids and Fluids; De Lorenzis, L., Ed.; Springer: Berlin/Heidelberg, Germany, 2020; pp. 75–101. [Google Scholar] [CrossRef]
- Freddi, F.; Royer-Carfagni, G. Regularized variational theories of fracture: A unified approach. J. Mech. Phys. Solids
**2010**, 58, 1154–1174. [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] - Strobl, M.; Seelig, T. On constitutive assumptions in phase field approaches to brittle fracture. Procedia Struct. Integr.
**2016**, 2, 3705–3712. [Google Scholar] [CrossRef][Green Version] - Fan, M.; Jin, Y.; Wick, T. A quasi-monolithic phase-field description for mixed-mode fracture using predictor–corrector mesh adaptivity. Eng. Comput.
**2021**, 1–25. [Google Scholar] [CrossRef] - Yu-Sheng, L.; Michael, J.; Borden, K.; Ravi-Chandar, C.M.L. A phase-field model for fatigue crack growth. J. Mech. Phys. Solids
**2019**, 132. ISSN 0022-5096. [Google Scholar] - He, Q.-C.; Shao, Q. Closed-Form Coordinate-Free Decompositions of the Two-Dimensional Strain and Stress for Modeling Tension–Compression Dissymmetry. J. Appl. Mech.
**2019**, 86, 031007. [Google Scholar] [CrossRef] - Nguyen, T.-T.; Yvonnet, J.; Waldmann, D.; He, Q.-C. Implementation of a new strain split to model unilateral contact within the phase field method. Int. J. Numer. Methods Eng.
**2020**, 121, 4717–4733. [Google Scholar] [CrossRef] - Wu, J.-Y.; Nguyen, V.P. A length scale insensitive phase-field damage model for brittle fracture. J. Mech. Phys. Solids
**2018**, 119, 20–42. [Google Scholar] [CrossRef] - Abaqus 6.14-1, Dassault Systems Simulia Corp., Providence, RI, USA. 2014. Available online: http://130.149.89.49:2080/v6.14/pdf_books/ANALYSIS_4.pdf (accessed on 30 May 2022).
- Seleš, K.; Tomić, Z.; Tonković, Z. Microcrack propagation under monotonic and cyclic loading conditions using generalised phase-field formulation. Eng. Fract. Mech.
**2021**, 255, 107973. [Google Scholar] [CrossRef] - Bruns, S.; Petho, L.; Minnert, C.; Michler, J.; Durst, K. Fracture toughness determination of fused silica by cube corner indentation cracking and pillar splitting. Mater. Des.
**2020**, 186, 108311. [Google Scholar] [CrossRef] - Li, C.; Ding, J.; Zhang, L.; Wu, C.; Sun, L.; Lin, Q.; Liu, Y.; Jiang, Z. Densification effects on the fracture in fused silica under Vickers indentation. Ceram. Int.
**2021**, 48, 9330–9341. [Google Scholar] [CrossRef]

**Figure 1.**Indentation curve comparison between numerical brittle phase-field formulation and experimental measurement. Area under the experimental indentation curve corresponds to elastic (${W}_{e}$) and plastic (${W}_{p}$) indentation work.

**Figure 2.**Residual indents and different fracture patterns after Vickers indentation on fused silica glass, (

**a**) indent showing only primary cone crack, (

**b**) indent with secondary cone crack and radial cracks at the edge of the indent.

**Figure 4.**Numerical model of Vickers indentation: (

**a**) full model, (

**b**) quarter symmetrical model with symmetry boundary conditions, and (

**c**) idealized crack.

**Figure 6.**Comparison of the crack growth for different values of the friction coefficient, µ. All deformed models are under the same indenter penetration, i.e., $h=3\mathsf{\mu}\mathrm{m}$.

**Figure 7.**Positive maximum principal stress (in MPa) on specimen surface as a cause of cone-crack initiation, (

**a**) stress distribution (in MPa) in front plane, (

**b**) stress distribution in isometric view (without the indenter), showing ring-like distribution outside the contact region, and (

**c**) isometric view (without the indenter) of the initiation ring.

**Figure 8.**Stable crack growth with inclination angle at different penetration depths: (

**a**) $u=1.2\mathsf{\mu}\mathrm{m}$, (

**b**) $u=1.65\mathsf{\mu}\mathrm{m}$, (

**c**) $u=2.025\mathsf{\mu}\mathrm{m}$, (

**d**) $u=2.4\mathsf{\mu}\mathrm{m}$, (

**e**) $u=2.8\mathsf{\mu}\mathrm{m}$, (

**f**) $u=3.375\mathsf{\mu}\mathrm{m}$, front view for different penetration depths, and (

**g**) isometric view without indenter at $u=3.375\mathsf{\mu}\mathrm{m}$ (max. loading).

Type of Value | Plane Strain Modulus, E* (MPa) | Poisson’s Ratio, $\mathit{\nu}$(-) |
---|---|---|

Certified value | 75,100 ± 300 | 0.16 |

Measured value | 74,990 | - |

SD | 1360 | - |

Total Energy Functional | |

$\Psi ={\Psi}^{b}+{\Psi}^{s}={{\displaystyle \int}}_{\Omega /\Gamma}{\psi}_{e}\left(\epsilon \right)d\Omega +{{\displaystyle \int}}_{\Gamma}{G}_{c}d\Gamma $ | (1) |

Elastic deformation energy density | |

${\psi}_{e}=\frac{1}{2}\lambda t{r}^{2}\left(\epsilon \right)+\mu tr\left({\epsilon}^{2}\right)$ | (2) |

Regularized energy functional | |

$\Psi \left(u,\varphi \right)={{\displaystyle \int}}_{\Omega}g\left(\varphi \right){\psi}_{e}\left(\epsilon \left(u\right)\right)d\Omega +{{\displaystyle \int}}_{\Omega}\frac{{G}_{c}}{2}\left[l{\left(\nabla \varphi \right)}^{2}+\frac{1}{l}{\varphi}^{2}\right]d\Omega $ | (3) |

Governing equations through the principle of virtual work | |

${{\displaystyle \int}}_{\Omega}{\left(1-\varphi \right)}^{2}\frac{\partial {\psi}_{e}\left(\epsilon \right)}{\partial \epsilon}\delta \epsilon d\Omega ={F}^{\mathrm{ext}}\delta u$ ${{\displaystyle \int}}_{\Omega}\left\{{G}_{c}l\Delta \varphi \Delta \left(\delta \varphi \right)+\frac{{G}_{c}}{l}\varphi \delta \varphi \right\}d\Omega ={{\displaystyle \int}}_{\Omega}2\left(1-\varphi \right)H\left(t\right)\delta \varphi d\Omega $ $H\left(t\right):={\mathrm{max}}_{\tau =\left[0,t\right]}{\psi}_{e}\left(\tau \right)$ | (4) |

Degradation function | |

$g\left(\varphi \right)={\left(1-\varphi \right)}^{2}$ | (5) |

Crack density function (AT2—Ambrosio-Tortorelli [24,25]) | |

$\gamma \left(\varphi ,\nabla \varphi \right)=\frac{1}{2}\left[\frac{1}{l}{\varphi}^{2}+l{\left|\nabla \varphi \right|}^{2}\right]$ | (6) |

Length scale parameter | |

$l=\frac{27}{256}\frac{{G}_{C}E}{{\left({\sigma}^{max}\right)}^{2}}$ | (7) |

**Table 3.**Spectral split energy decomposition main equations, In the table, the well-known formula of the model firstly proposed in [30] are given for completeness.

if ${\epsilon}_{1}>0$ | |

${\psi}_{0}^{+}=\frac{Ev}{2\left(1+v\right)\left(1-2v\right)}{\left({\epsilon}_{1}+{\epsilon}_{2}+{\epsilon}_{3}\right)}^{2}+\frac{E}{2\left(1+v\right)}\left({\epsilon}_{1}^{2}+{\epsilon}_{2}^{2}+{\epsilon}_{3}^{3}\right),{\psi}_{0}^{-}=0$ | (11) |

else if ${\epsilon}_{2}+v{\epsilon}_{1}>0$ | |

${\psi}_{0}^{+}=\frac{Ev}{2\left(1+v\right)\left(1-2v\right)}{\left({\epsilon}_{3}+{\epsilon}_{2}+2v{\epsilon}_{1}\right)}^{2}+\frac{E}{2\left(1+v\right)}\left({\left({\epsilon}_{3}+v{\epsilon}_{1}\right)}^{2}+{\left({\epsilon}_{2}+v{\epsilon}_{1}\right)}^{2}\right),{\psi}_{0}^{-}=\frac{E}{2}{\epsilon}_{1}^{2}$ | (12) |

else if $\left(1-v\right){\epsilon}_{3}+v\left({\epsilon}_{1}+{\epsilon}_{2}\right)>0$ | |

${\psi}_{0}^{+}=\frac{E}{2\left(1-{v}^{2}\right)\left(1-2v\right)}{\left(\left(1-v\right){\epsilon}_{3}+v{\epsilon}_{2}+v{\epsilon}_{1}\right)}^{2},{\psi}_{0}^{-}=\frac{E}{2\left(1-{v}^{2}\right)}\left({\epsilon}_{1}^{2}+{\epsilon}_{2}^{2}+2v{\epsilon}_{1}{\epsilon}_{2}\right)$ | (13) |

else | |

${\psi}_{0}^{-}=\frac{Ev}{2\left(1+v\right)\left(1-2v\right)}{\left({\epsilon}_{1}+{\epsilon}_{2}+{\epsilon}_{3}\right)}^{2}+\frac{E}{2\left(1+v\right)}\left({\epsilon}_{1}^{2}+{\epsilon}_{2}^{2}+{\epsilon}_{3}^{3}\right)$ | (14) |

Modulus of Elasticity, E, MPa | Poisson’s Ratio, ν | Fracture Toughness/Energy Release Rate, ${\mathit{G}}_{\mathit{C}},\mathbf{N}/\mathbf{mm}$ | Tensile Strength, ${\mathit{\sigma}}^{\mathit{m}\mathit{a}\mathit{x}},\mathbf{MPa}$ |
---|---|---|---|

75,000 | 0.16 | 0.006 | 4000 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tomić, Z.; Jukić, K.; Jarak, T.; Fabijanić, T.A.; Tonković, Z. Phase-Field Modeling of Fused Silica Cone-Crack Vickers Indentation. *Nanomaterials* **2022**, *12*, 2356.
https://doi.org/10.3390/nano12142356

**AMA Style**

Tomić Z, Jukić K, Jarak T, Fabijanić TA, Tonković Z. Phase-Field Modeling of Fused Silica Cone-Crack Vickers Indentation. *Nanomaterials*. 2022; 12(14):2356.
https://doi.org/10.3390/nano12142356

**Chicago/Turabian Style**

Tomić, Zoran, Krešimir Jukić, Tomislav Jarak, Tamara Aleksandrov Fabijanić, and Zdenko Tonković. 2022. "Phase-Field Modeling of Fused Silica Cone-Crack Vickers Indentation" *Nanomaterials* 12, no. 14: 2356.
https://doi.org/10.3390/nano12142356