# An Investigation into the Densification-Affected Deformation and Fracture in Fused Silica under Contact Sliding

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## Abstract

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## 1. Introduction

## 2. Scratching Tests

## 3. Finite Element Modeling

_{b}is the hydrostatic yield stress for pure compression. The relationship between hydrostatic pressure p and the densification $\zeta $ is modeled by:

_{0}(GPa) is the hydrostatic pressure under which a densification of ${\zeta}_{\mathrm{max}}$/2 is produced. The parameters of the modified elliptical model used in this study are taken from the ref. [28].

_{e}is used in a parallelepiped with a dimension of a × a × (l + 2W), and graded FE mesh is used in the residual region. For conical scratching, the semi-included angle α of the conical indenter is set as 70.3°, to ensure that the projected area-to-indentation depth function is the same as the commonly used Vickers and Berkovich indenters. A = 2.79 h

_{max}is the nominal contact radius for 70.3° conical scratching.

_{max}, l, and v, respectively. The Coulomb friction model is used to model the adhesion friction behavior between the indenter and the sample. The coefficient of adhesion friction f was determined to be 0.04, by comparing FEA and scratching tests.

_{e}. The appropriate parameters result in stable and convergent normal and tangential loads, and apparent coefficient of friction in the sliding stage. h

_{max}is assumed to be 1 μm. Results show that a cross-section dimension of 5a × 5a, a sliding length of 10 h

_{max}, and a mesh size of 1/8 h

_{max}are appropriate for the simulations.

## 4. Verification of Finite Element Models

#### 4.1. Experimental Verification of Elastic Recovery

_{e}for scratching reflects the extent of elastic deformation relative to the whole deformation. In addition, f

_{e}can be conveniently measured by AFM. Therefore, f

_{e}is used to verify the finite element model in this study.

_{f}(see Figure 5) slightly decreases with the distance d to the unloading position of indenter tip. The profile shown in Figure 5 is obtained by averaging five equally spaced cross-section profiles of the middle part of the impression. The residual scratch depth h

_{f}after elastic recovery, determined from Figure 5, is 668 nm.

_{2}to t

_{1}in the stage Ⓑ from the uncorrected displacement from t

_{3}to t

_{4}. Similarly, the corrected normal displacement in the postmortem profiling stage Ⓕ was calculated by subtracting the uncorrected displacement from t

_{1}to t

_{2}from the uncorrected displacement from t

_{5}to t

_{6}. The evolution of the corrected normal displacement with time is shown in Figure 6. The maximum scratching depth and residual depth are 1063 nm and 479 nm, respectively. It is worth noting that this value of residual depth is smaller than that measure by AFM (i.e., h

_{f}= 668 nm). This is possibly because the indenter did not strictly follow the scratching path in stage Ⓕ, due to the movement of the sample or the motion error of the indentation test in the lateral direction. By contrast, the AFM probe accurately detects the lowest positions of the residual scratching profiles for two reasons. First, the tip radius of the AFM probe is much smaller than that of the Berkovich indenter. Second, the AFM probe is scanning across the impression. Using the AFM-measured h

_{f}, the elastic recovery ratio is calculated to be f

_{e}= 1−h

_{f}/h

_{amx}= 37.2%. Ba analyzing the FEA-predicted profiles of the scratching impression at the fully-loaded and fully-unloaded states shown in Figure 7, the value of f

_{e}predicted by FEA is 37%, which is very close to the experimental value.

#### 4.2. Theoretical Verification of Hardness Ratio

_{T}and dF

_{N}, have the following relationship:

_{pl}and A

_{pv}are the laterally and vertically projected contact areas, respectively; p(h,β) is the contact pressure at the point (h,β) on the indenter surface; and β and h are the phase and the height measured from the indenter tip, respectively. According to the definition of hardness, the ploughing hardness can be expressed by:

_{H}= H

_{T}/H

_{s}is the ratio of tangential hardness and scratching hardness; and ${\mu}_{0}$ is the friction coefficient induced by ploughing. As ${k}_{H}^{p}\approx 1$, we can conclude from Equation (8) that k

_{H}> 1 when friction exists, i.e., H

_{T}> H

_{s}. This is consistent with the scratching tests [34]. It is worth noting that the above analysis considers the non-uniform distribution of the contact pressure. This is more accurate than the widely adopted assumption that the contact pressure is uniformly distributed [35].

## 5. Deformation and Fracture in Fused Silica under Scratching

#### 5.1. Scratching Hardness

_{i}(the hardness at the end of stage ①) for edge-leading Berkovich scratching is slightly increased with the rise in friction. By contrast, the scratching hardness (the hardness in the right red box) is nearly independent of friction.

_{s}for Berkovich indenter is independent of f, H

_{s}for conical indenter is linearly decreased with f, as demonstrated in Figure 11. The scratching hardness induced by ploughing, i.e., ${H}_{s}^{p}$, remains nearly unchanged when f increases from 0 to 0.2.

#### 5.2. Plastic Deformation

_{max}> 10), and a cylindroid at the rear of the indenter (x/h

_{max}≤ 10).

_{r}predicted by FEA equals 2.24 h

_{max}. Figure 15a shows that m is bigger than a, while n is smaller than a. After unloading, m remains nearly unchanged, but n in the cross-section close to point B increases, due to the significant elastic recovery. m is significantly larger than n in the steady sliding stage. Therefore, the prediction accuracy of the existent sliding stress field models may be greatly improved if the spherical/cylindrical yield region is replaced by an ellipsoid/cylindroid. Figure 15b shows that the depth of the elastic–plastic boundary center in the yz-cross-section behind the indenter decreases rapidly when the indenter moves far away from it. ξ for scratching is much bigger than that for indentation (close to ξ at l

_{B}= 9.5 h

_{max}) at both the fully loaded and the fully unloaded states. This indicates that the Ahn and Wang models should be refined to allow for the embedding of the center of the plastic zone.

#### 5.3. Stress Fields and Cracking Behavior

#### 5.3.1. In the Sliding Stage

_{x}and σ

_{y}are identified in the regions below the indenter tip just outside the elastic–plastic boundary, i.e., the regions R

_{1}and R

_{2}shown in Figure 16, respectively. They are the driving forces of median cracks. As the maximal σ

_{y}is higher than the maximal σ

_{x}, the median crack along the sliding direction tends to initiate prior to that along the lateral direction. The maximal value of σ

_{z}locates at the far rear of the indenter (region R

_{3}), which is the driving force of lateral cracks. In the sliding stage, the driving force of median cracks is higher than that of lateral cracks.

#### 5.3.2. At the Fully-Unloaded State

_{y}on the sample surface at the front of the indenter (region R

_{4}), which is the driving force of radial cracks along the sliding direction, i.e., the radial crack 1 in Figure 18a. σ

_{y}at the bottom of the yield region remains nearly unchanged after unloading. Therefore, median cracks along the sliding direction remain open in the unloading stage if they initiate in the sliding stage. The maximal σ

_{z}increases from 0.087H to 0.137H during the unloading process. This indicates that the lateral crack emerges more easily in the unloading stage compared with the sliding stage.

#### 5.3.3. Maximum Principal Stress

_{1}/H shown in Figure 19 indicate that the median crack along the sliding direction tends to be the first crack to appear during the sliding stage. It initiates below the yield region at the rear of the indenter, i.e., in the region R

_{2}. Once initiated in the xz-cross-section, the median crack propagates along the sliding direction during scratching. As the value of σ

_{1}in region R

_{3}is smaller than that in region R

_{2}, the initiation load of median crack under indentation is higher than that under scratching. This is consistent with experimental observations. Though median crack is absent during indentation tests under the normal load of 40 N [39], it is observed during scratching tests under the normal load of 600 mN, as shown in Figure 18b. During the unloading stage, radial cracks may initiate from the sample surface at the front of the indenter, i.e., region R

_{4}, as shown in Figure 18a. As σ

_{y}in region R

_{4}is small before unloading, and significantly increases during the unloading process, radial cracks tend to emerge in the unloading stage. If both radial and median cracks form, they coalesce to form a big median–radial crack that penetrates through the entire yield region, as verified by the experiments (see Figure 18).

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Schematic diagram of edge-forward Berkovich scratching, (

**b**) the applied normal force and lateral displacement for the normal load of 200 mN, and (

**c**) the resulting normal displacement and lateral force.

**Figure 4.**The AFM-measured three-dimensional topography of the scratching impression induced by an edge-forward Berkovich indenter under the normal load of 200 mN.

**Figure 6.**The evolution of the corrected normal displacement with time for scratching with an edge-forward Berkovich indenter, under the normal load of 200 mN.

**Figure 7.**The FEA-simulated profiles of the scratching impression at the fully loaded and fully unloaded states for an edge-forward Berkovich indenter.

**Figure 8.**The variation of hardness ratio ${k}_{H}^{p}$ with the half-included angle of the conical indenters.

**Figure 9.**The variation of hardness with time at various values of adhesion friction coefficient f for edge-leading Berkovich scratching.

**Figure 10.**The evolutions of (

**a**) contact area and (

**b**) normal force with time at various friction coefficients for edge-leading Berkovich scratching.

**Figure 11.**The hardness during scratching as a function of adhesion friction coefficient f for scratching with a 70.3° conical indenter. The value pairs of (H

_{s}, f) is fitted to obtain the dash line.

**Figure 12.**Contours of densification at the fully unloaded state induced by scratching with a 70.3° conical indenter in (

**a**) the top surface and (

**b**) xz-cross-section.

**Figure 13.**The elastic–plastic boundaries defined by a von Mises equivalent plastic strain of 10

^{−2}in the yz-cross-section at the fully unloaded state for 70.3° conical scratching. The simulated boundaries are fitted by (

**a**) a circular arc and (

**b**) an elliptical arc.

**Figure 14.**The contours of von Mises equivalent plastic strain in the xz-cross-section (

**a**) at the fully loaded and (

**b**) fully unloaded states.

**Figure 15.**The fitted (

**a**) dimension and (

**b**) depth of the elastic–plastic boundary normalized by h

_{max}. The indenter moves from the point C to the point B in the scratching stage (see Figure 2).

**Figure 16.**The contours of (

**a**) σ

_{x}/H, (

**b**) σ

_{y}/H, and (

**c**) σ

_{z}/H at the end of the sliding stage ② in the xz-cross-section, where a

_{r}= 2.24 h

_{max}is the real contact radius evaluated by FEA, and H is the hardness measured by indentation tests. The indenter moves along the positive direction of the x-axis.

**Figure 17.**The contours of (

**a**) σ

_{y}/H and (

**b**) σ

_{z}/H at the fully unloaded state in the xz-cross-section predicted by FEA. The indenter moves along the positive direction of x-axis.

**Figure 18.**The SEM images of the impression at the (

**a**) end, (

**b**) middle, and (

**c**) start of the edge-forward Berkovich scratching, under the normal load of 600 mN.

**Figure 19.**The contours of σ

_{1}/H at the (

**a**) fully loaded and (

**b**) fully unloaded states in the xz-cross-section. The thick dash line is the boundary between σ

_{1}≈ σ

_{y}and σ

_{1}≈ σ*, where σ* is the in-plane principal stress.

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**MDPI and ACS Style**

Li, C.; Ma, Y.; Sun, L.; Zhang, L.; Wu, C.; Ding, J.; Duan, D.; Wang, X.; Chang, Z.
An Investigation into the Densification-Affected Deformation and Fracture in Fused Silica under Contact Sliding. *Micromachines* **2022**, *13*, 1106.
https://doi.org/10.3390/mi13071106

**AMA Style**

Li C, Ma Y, Sun L, Zhang L, Wu C, Ding J, Duan D, Wang X, Chang Z.
An Investigation into the Densification-Affected Deformation and Fracture in Fused Silica under Contact Sliding. *Micromachines*. 2022; 13(7):1106.
https://doi.org/10.3390/mi13071106

**Chicago/Turabian Style**

Li, Changsheng, Yushan Ma, Lin Sun, Liangchi Zhang, Chuhan Wu, Jianjun Ding, Duanzhi Duan, Xuepeng Wang, and Zhandong Chang.
2022. "An Investigation into the Densification-Affected Deformation and Fracture in Fused Silica under Contact Sliding" *Micromachines* 13, no. 7: 1106.
https://doi.org/10.3390/mi13071106