# Determination of Mechanical and Fracture Properties of Silicon Single Crystal from Indentation Experiments and Finite Element Modelling

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

_{max}, corresponding damage-initiating displacement ∆

_{c}and failure displacement ∆

_{sep}. It was shown in [12] that the cohesive energy density and the peak cohesive traction play a far more important role than the shape of the cohesive traction—separation curve in predicting the final fracture behaviour. In case of the bilinear form of the cohesive traction—separation law the cohesive energy density (critical fracture energy) 2Γ can be calculated by 2$\Gamma =\frac{1}{2}{\sigma}_{\mathrm{max}}{\Delta}_{c}.$Critical review of various cohesive zone models is given in [13]. Contrary to analytical approaches, cohesive interface FE simulations exhibit a natural advantage consisting in no need to specify the crack front a priori. Namely, the crack front is found as a result of the solution of the boundary value problem. Moreover, the influence of residual stresses developing under the indent due to inelastic compressive behaviour of brittle materials [10,14,15,16,17] is more reliably captured. Some care is needed with respect to the elasticity of the cohesive interface, specifically one should avoid double-counting the elasticity—once in the cohesive law and a second time as part of the bulk behaviour. Nevertheless, the effect of this issue is negligible when cohesive surfaces are only specified along a potential single crack path such as in the case of indentation cracking, or if the stiffness of cohesive surfaces is infinite [18]. Currently, there is an increasing effort to combine cohesive zone models with extended finite element method (XFEM) to model crack growth [19,20,21]. XFEM can avoid remeshing near the crack tip as the crack grows and all other difficulties connected with it. With respect to indentation crack modelling, remeshing is not needed as the indentation cracks extend only over short distances without kinking and a zone of the potential crack formation is covered with cohesive elements. Thus, the application of XFEM does not seem to bring any other benefits in this context. To interpret the results of simulation of the growth of indentation cracks in terms of the linear fracture mechanics, the cohesive (bridging) zone must be significantly smaller than the crack. Hence, great care is needed in applying the simulation results to short crack problems under indentation tests [22,23].

_{c}(interplanar separation) leading to the loss of the crystal bearing capacity is 0.2 nm, the corresponding peak stress is of the order of theoretical strength, and the cohesive energy density 2$\Gamma \cong 5.2\mathrm{J}/{\mathrm{m}}^{2}$. Moreover, the length of the cohesive zone is very small, approximately 0.6 nm. It means that macroscopic FE simulation would require extremely fine mesh, which is often unfeasible. Nguyen and Ortiz [25] suggested a way to the macroscopic form of the cohesive law by considering the cooperative behaviour of a large number N of interatomic planes forming a cohesive layer. The thickness of the cohesive layer in FE simulations is given by the local element size D. Thus, the number of atomic planes in the cohesive layer is $N=2D/d$, where the factor 2 was added due to symmetry. Nguyen and Ortiz showed that for sufficiently large N the macroscopic critical opening displacement ∆

_{c}and the corresponding macroscopic cohesive stress ${\sigma}_{\mathrm{max}}$for the separation of a single atomic plane asymptotically scale as

_{c}and the corresponding macroscopic cohesive stress ${\sigma}_{\mathrm{max}}$ do depend on the element size, the cohesive energy density 2Γ is independent of the size element. The aim of this study is to use experimental microindentation data, FE simulations with cohesive zone modelling, and an optimisation procedure to determine the cohesive energy density of single crystals without having to check whether the size of the cohesion zone is considerably less than the crack size and thus to analyse the problems of short cracks. Obviously, such a procedure is particularly suitable for determining fracture properties of MEMS/NEMS parts or thin films using micro/nano indentation tests.

## 2. Materials and Methods

_{4}F = 12.5:87.5%).

_{Z}= 0) representing the crystal storage were used in numerical simulations. Non-elastic response of the Si crystal (denoted as SC), that tends to accommodate the contact stresses under the indenter, was modelled in terms of ideally elastoplastic material defined by Young’s modulus E

_{SC}=129.5 GPa, Poisson’s ratio ν

_{SC}= 0.278, the shear modulus G

_{SC}=79.6 GPa and the yield stress σ

_{y}

_{, SC}which is initially unknown. In this context it should be noted that the elastic-perfectly plastic material behaviour according to the von Mises yield condition accurately describes the compressive behaviour of many brittle materials [27,28]. Vickers diamond indenter (VDI) was considered as a linear isotropic body defined by Young´s modulus E

_{VDI}= 1220 GPa and Poisson´s ratio ν

_{VDI}= 0.20. Elastic properties of individual components (crystal, indenter) were chosen on the basis of available literature data. Linear isotropic behaviour was also assumed for the load cell (LC) defined by Poisson´s ratio ν

_{LC}= 0.3 and Young´s modulus E

_{LC}which takes the stiffness of the test equipment into account and is also initially unknown. Reduced Young´s modulus E

_{r}is then given by

_{r}was searched together with the yield stress σ

_{k}

_{, SC}based on load-depth curves from indentation tests, see Figure 3a. The indenter tip shape deviation from the ideal shape, see Figure 2a, was also taken into account when searching for the yield strength. Crack initiation and growth was not considered in this stage. It should be pointed out that the effect of cracks on the force-depth curve is negligible for lower loading force values. The nonlinear least-squares routine to get the best fit between the given indentation data and the optimised indentation data, produced by FE analysis, was applied to determine the aforementioned parameters. The corresponding objective functional $\mathcal{F}\left(c\right)$is given by, see [29,30,31,32,33]

_{y}

_{, SC}was determined from unloading stage of each of the force-depth curves, where linear behaviour exists (approx. to 10% decrease from maximal value of the applied force) in accordance with Oliver and Pharr method [34]. The best fit was obtained with the yield stress σ

_{y}

_{, SC}= 6.4 GPa, the value which is close to the values applied for silicon in studies [7,8,10]. Further decrease in the applied force cannot be employed for a correct fitting because the FE model does not include pop-out effect which occurs approximately at 50% decrease of the applied force. The real shape of the indenter tip, which is used in numerical simulations, was found on the basis of a calibration curve of differential hardness, which is performed before the measurement itself. It is therefore a matter of finding a match between the calculated and measured dependence of load vs indentation depth. The shape of the indenter tip and at the same time the required yield strength are calibrated here. The calibration was performed using the universal hardness HU which takes elastic and plastic deformations into account and is defined by the following relation

_{max}denotes the maximal force acting on the ideal Vickers indenter during a particular indentation test, h

_{max}denotes the corresponding maximal depth of indentation into the Si crystal, and S

_{c}is the contact area between the indenter tip and the Si crystal. The calibration was solved as an inverse problem by using incremental iteration procedure where the universal hardness and the loading force are known, and the contact area is searched. When the contact area is found the shape of indenter tip is modified and the force is incrementally increased. This procedure runs until the maximal loading (here 1000 mN) is reached. Then the calibration procedure is finished. The ideal and real indenter tip shape of Vickers indenter are shown in Figure 2b. The difference between the ideal and the real shape of indenter tip is irrelevant in terms of the force-depth dependence but essential for the development of cracks in the near vicinity of the indenter tip.

_{c}which corresponds to the maximal normal traction σ

_{max}. Numerical simulations were performed for radial cracks propagating along the (101) cleavage plane in the direction [100] and along the (011) cleavage plane in the direction [10], see Figure 2a. The complete elastic–plastic stress field during the unloading stage of the indentation is given by a superposition of the elastic contact stress field ${\mathbf{\sigma}}_{}^{m}$ and a residual stress field ${\mathbf{\sigma}}_{}^{r}$ generated due to the permanent deformation ${\mathbf{\epsilon}}_{}^{p}$under the contact. While with decreasing contact force P(t) the elastic contact stress field decreases, the residual stress field remains largely unchanged and promotes cracks extension. The boundary value problem to be solved during the unloading stage is to find the complete stress-strain field $\mathbf{\sigma}={\mathbf{\sigma}}^{\mathit{m}}+{\mathbf{\sigma}}^{\mathit{r}}$, $\mathbf{\epsilon}={\mathbf{\epsilon}}^{m}+{\mathbf{\epsilon}}^{r}$:

_{max}. The virtual crack area increment $\delta {S}_{crack}$ is given by

_{max.}

_{c}is based upon the best fit between the visible crack length on the top surface and its numerical prediction obtained by FE analysis under full unloading. The optimisation model is

_{max,j}and M = 25 is the number of performed indentation tests for each loading force. Simultaneously, the minimisation of the total energy is controlled. Observe, that as the independent cohesive material parameters also σ

_{max}and ∆

_{c}can be chosen, see Equation (9).

## 3. Results

_{max}. These data are used in the following subsection to find the cohesive energy density of the silicon crystal.

#### 3.1. Cohesive Energy Density of Silicon Crystal

_{c}. Both, the cohesive energy density 2Γ and the critical opening displacement ∆

_{c}form the output of the inverse problem. The inverse analysis starts with an initial estimate of Γ. Subsequently ∆

_{c}is sought so that the total energy Є reaches a minimum. In the next step the cohesive energy density 2Γ is adjusted to minimise the discrepancy between the measured crack length on the top surface, ${L}_{ij}^{exp}$, and its theoretical prediction ${L}_{j}^{pred}\left(\Gamma ,{\Delta}_{c}\right)$. With a new value of Γ a corrected value of ∆

_{c}is sought. This process is iteratively repeated until convergence criteria are met.

_{max}= 300 mN, 500 mN, 750 mN, and 1000 mN, and for each of 25 performed indentation tests corresponding to a particular value of P

_{max}. Subsequently, by averaging the iteratively received values of Γ, an estimate for the cohesive energy density of the analysed silicon crystal was obtained. The reliability of the used numerical model follows from the comparison of the determined values of Γ with values reported in literature. Figure 6 shows the dependency of crack length on the indentation depth and the iteratively received values of Γ for particular loading force. If a particular crack length for an appropriate indentation depth is selected in Figure 6, a corresponding value of the cohesive energy density 2Γ can be read off. For tested forces/depths, see Figure 5, we get Γ = 3.06 J/m

^{2}(crack length 7.5 μm and the indentation depth 1.18 μm), Γ = 2.81 J/m

^{2}(crack length 11.4 μm and indentation depth 1.57 μm), Γ = 2.63 J/m

^{2}(crack length 15.4 μm and indentation depth 1.97 μm) and Γ = 2.70 J/m

^{2}(crack length 18.9 μm and indentation depth 2.33 μm). It is seen that with increasing indentation depth the estimate of the cohesive energy density converges to the value 2Γ = 5.30 J/m

^{2}which agrees well with the silicon cohesive energy density values reported in literature. This convergence is due to a decrease in the measurement error with increasing indentation depth. In general, measurements at a lower indentation depth (a lower applied force) are subject to a larger error. The critical crack opening displacement is ${\Delta}_{c}=$ 13 nm. Let us however point out again that the critical opening displacement ∆

_{c}does depend on the element size D, c.f. Equation (1), which, as already mentioned, is 0.25 μm.

#### 3.2. Crack Extension during Indentation Test

_{max}= 750 mN are displayed in Figure 10. Figure 10 shows the distribution of crack opening along the crack flanks within the cohesive area which allows to identify the crack front. It is clearly seen that crack grows during the unloading stage of indentation test due to residual stress field.

#### 3.3. Mesh Density of Cohesive Zone Area

_{c}depends on the element size D. As mentioned in Introduction, Nguyen and Ortiz [25] suggested a way to the macroscopic form of the cohesive law by considering the cooperative behaviour of a large number N of interatomic planes forming a cohesive layer. This approach then shows that ∆

_{c}scales with $\sqrt{D}$. A distinctively weaker dependency on the element size D used for discretisation of the cohesive zone area, see Figure 2, was observed for the radial crack length and for the work of cohesive forces as well. Several loading forces were tested and for each the size of the cohesive zone area was adjusted, and thus the discretisation density with respect to the crack surface area. The greater the loading force and, as a result, the greater the crack surface area, the lower is the discretisation error due to the size of the element used. For that reason, the sensitivity to the size of the element (ESIZE) was performed, especially because the ideal size of the element would be at the atomic level—this would of course correspond to a different traction-separation T(Δ) dependence. Figure 11 shows a linear regression of the dependency of the crack length on the element size which was used for prediction of the crack length in case when the element size is approaching zero. These data were determined for each indentation loading force and then they were compared with experimental observation and measurement of the radial crack length, see Equation (18).

## 4. Discussion and Conclusions

^{2}the Formula (19) gives K

_{IC}= 0.86 MPa·m

^{1/2}. In contrast to the previous studies, there is no need to control the premise of the linear fracture mechanics that the cohesive zone is much shorter than the crack length. Hence, the developed approach is suitable also for short cracks for which the linear fracture mechanics premise is violated. Besides, in spite of the previous improvements of the indentation cracking formulas, they are still relatively inaccurate to predict the fracture toughness in comparison to the proposed approach based on evaluation of the cohesive energy density.

_{y}

_{, SC}was 6.4 GPa. The computed residual stress/strain field enters the analysis of the inverse problem for identification of the cohesive energy density 2Γ and the critical crack opening displacement ∆

_{c}. The inverse problem solution requires to find the best fit between the visible crack length on the top surface of the silicon crystal and its numerical prediction obtained by FE analysis under full unloading and simultaneously to ensure minimisation of the total energy. The solution results are presented in the form of a diagram which links together the cohesive energy density, the crack length, and the indentation depth, from which the cohesive energy density 2Γ can be easily read off for particular crack length and indentation depth. Nevertheless, in case of a lower indentation depth the measurements are subject to a larger error which is reflected in the estimation error of the cohesive energy density. As the indentation depth increases, the error decreases and the estimate of the cohesive energy density converges to the value 2Γ = 5.30 J/m

^{2}.

_{y}to ensure initiation of crack in a linear fracture mechanics context. Moreover, care is needed when changing ${\sigma}_{\mathrm{max}}$ because then the crack bridging zone also changes which affects the choice of cohesive element size D. In the papers [7,10] ${\sigma}_{\mathrm{max}}$ was chosen from the range <$0.5,1$> GPa, the typical value of the yield strength was ${\sigma}_{y}=5\text{}\mathrm{GPa}$, and the typical value of the Young modulus was E = 200 GPa. The fracture toughness K

_{IC}ranged from 0.7 to 1 MPa.m

^{1/2}. All these papers used the bilinear form of the cohesive traction—separation law. In this paper, the parameter ${\sigma}_{\mathrm{max}}$, and/or the critical opening displacement ∆

_{c}together with the cohesive energy density 2Γ were selected to provide best fit between the visible crack length on the top surface and its numerical prediction according to Equation (18). The optimal values were found as ${\sigma}_{\mathrm{max}}=150\text{}\mathrm{MPa}$, ∆

_{c}= 13 nm, and 2Γ = 5.30 J/m

^{2}. It is interesting to notice that the asymptotic scaling rule in Equation (1)

_{1}predicts the macroscopic critical opening displacement as ∆

_{c}$\cong $ 14.4 nm, see the text below Equation (3).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

c | optimisation variable |

d | interplanar spacing |

h_{max} | maximal depth of indentation |

f | friction coefficient |

i | position along the force-depth curve |

$\mathit{p}$ | actual tractions |

$\mathit{n}$ | unit normal to the surface |

A | size of cohesive zone |

C | interplanar modulus |

D | finite element size |

$\mathit{C}$ | stiffness tensor |

$\mathit{D}$ | compliance tensor |

$\mathcal{F}$ | objective functional |

HU | universal hardness |

N | number of interatomic planes forming a cohesive layer |

E_{LC} | Young´s modulus of test equipment load cell |

E_{r} | Reduced Young´s modulus |

E_{SC} | Young´s modulus of the Si crystal |

E_{VDI} | Young´s modulus of Vickers diamond indenter |

G_{SC} | shear modulus of the Si crystal |

${K}_{IC}$ | material fracture toughness |

${L}_{ij}^{exp}$ | measured crack length on the top surface at i-th test |

${L}_{j}^{pred}$ | predicted crack length |

P | contact force |

${P}_{}^{comp}$ | predicted loading force |

${P}_{}^{exp}$ | experimental loading force |

P_{max} | maximal force acting on the ideal Vickers indenter |

S_{c} | contact area between the indenter tip and the Si crystal |

${S}_{crack}$ | crack surface |

${S}_{u}$ | part of the boundary where displacement are prescribed |

T | cohesive traction |

${\mathbf{\epsilon}}_{}^{p}$ | permanent deformation |

δ | microscopic crack opening |

δ_{c} | critical microscopic crack opening (interplanar separation) |

$\delta \mathit{L}$ | local virtual crack extension |

$\mathbf{\upsilon}$ | local unit normal vector to the crack front |

ν_{SC} | Poisson´s ratio of the Si crystal |

ν_{VDI} | Poisson´s ratio of Vickers diamond indenter |

ν_{LC} | Poisson´s ratio of test equipment load cell |

ψ | centreline-to-face angle |

Є | total energy |

$\mathbf{\sigma}$ | complete stress-strain field |

${\mathbf{\sigma}}_{}^{m}$ | elastic contact stress field |

${\mathbf{\sigma}}_{}^{r}$ | residual stress field |

σ_{y, SC} | yield stress of the Si crystal |

σ_{max} | peak cohesive traction |

2Γ | cohesive energy density (critical fracture energy) |

∆ | macroscopic opening displacement |

∆_{c} | corresponding damage-initiating displacement |

∆_{sep} | failure displacement |

$\varphi $ | interplanar cohesive potential |

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**Figure 2.**(

**a**) FE model of Vickers indentation test on silicon crystal. (

**b**) Calibration of the tip shape based on the hardness.

**Figure 3.**Representative force–depth curves from indentation tests: (

**a**) experiment, (

**b**) numerical simulation after calibration.

**Figure 4.**Cohesive zone model (

**a**) definition and boundary conditions, (

**b**) the traction-crack opening length relationship.

**Figure 7.**Crack extension during the indentation test for maximal loading force of 300 mN: loading force (

**a**) F = 105 mN, (

**b**) F = 230 mN, (

**c**) F = 300 mN and (

**d**) F = 0 N.

**Figure 9.**Indentation with the maximal loading force P

_{max}= 300 mN: (

**a**) state at maximal load (numerical simulation), (

**b**) state after complete unloading (numerical simulation), (

**c**) state after complete unloading (experimental observation).

**Figure 10.**Distribution of the crack opening ∆ along the crack flanks for P

_{max}= 300 mN (

**a**) maximal loading, (

**b**) complete unloading.

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**MDPI and ACS Style**

Skalka, P.; Kotoul, M. Determination of Mechanical and Fracture Properties of Silicon Single Crystal from Indentation Experiments and Finite Element Modelling. *Materials* **2021**, *14*, 6864.
https://doi.org/10.3390/ma14226864

**AMA Style**

Skalka P, Kotoul M. Determination of Mechanical and Fracture Properties of Silicon Single Crystal from Indentation Experiments and Finite Element Modelling. *Materials*. 2021; 14(22):6864.
https://doi.org/10.3390/ma14226864

**Chicago/Turabian Style**

Skalka, Petr, and Michal Kotoul. 2021. "Determination of Mechanical and Fracture Properties of Silicon Single Crystal from Indentation Experiments and Finite Element Modelling" *Materials* 14, no. 22: 6864.
https://doi.org/10.3390/ma14226864