# 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

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**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 |

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**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