# Probability Density Function Models for Float Glass under Mechanical Loading with Varying Parameters

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Glass and Properties

Characteristic | Symbol and Unit | Value |
---|---|---|

Density | ρ [kg/m³] | 2500 [18] |

Glass transition temperature | T_{g} [°C] | 540–575 [17,19] |

Coefficient of thermal expansion | α_{T} [10^{−6} °C^{−1}] | 8.9–9 [17,18] |

(T < T_{g}) | ||

Young’s modulus | E [GPa] | 70–73 [17,18] |

Poisson’s ratio | ν [−] | 0.23 [17] |

Thermal conductivity | λ [W/mK] | 1 [18] |

Nominal tensile resistance | σ_{t} [MPa] | 45 [18] |

#### 2.2. Strength Prediction Model

#### 2.2.1. Procedure

#### 2.2.2. Parameters

#### 2.2.3. Case Study

#### 2.2.4. SPM Applied on Case Study

^{2}.

#### 2.2.5. Probability Density Distribution

#### 2.2.6. Convergence Study

#### 2.3. Model Selection and Distribution Fitting—AIC Calculation

## 3. Results and Discussion

#### 3.1. Scale Factor

#### 3.2. Width/Loading Span

#### 3.3. Loading Span/Support Span

#### 3.4. Flaw Density

^{2}. [22] This shows the importance of a parameter study on the flaw density. Table 7 displays the models used for this parameter study, all PDFs and all CDFs are visualized together in Figure 21, and Figure 22 presents the trends of the normalized AIC values, the mean values and the standard deviations.

^{2}has the best fit with a lognormal distribution. For flaw densities ranging from 1.0 to 2.5 flaw/cm

^{2}, the best fit changes to a gamma distribution. Furthermore, flaw densities higher than and equal to 3.0 flaws/cm

^{2}have a best fit with a normal distribution. The AIC values in Figure 22a also illustrate the Gumbel fit worsening when considering larger flaw densities.

#### 3.5. Flaw Shape

#### 3.6. Maximum Flaw Depth

#### 3.7. Fracture Toughness

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Representation of an elliptical surface flaw [13].

**Figure 3.**Flaw orientation distribution for (

**a**) top and (

**b**) bottom surfaces (with a flaw density of 1 flaw/cm

^{2}for visualization).

**Figure 4.**Flaw depth distribution for (

**a**) top and (

**b**) bottom surfaces (with a flaw density of 1 flaw/cm

^{2}for visualization).

**Figure 5.**The ${\sigma}_{x}$ stress states of the top and bottom surfaces for a total load of (

**a**) 1000 N and (

**b**) 1800 N.

**Figure 6.**Stress intensity factor for bottom surface (with a flaw density of 1 flaw/cm

^{2}for visualization).

**Figure 16.**(

**a**) AIC values and (

**b**) the mean and st.dev. of the normal stress at failure for a varying scale factor.

**Figure 18.**(

**a**) AIC values and (

**b**) the mean and st.dev. of the normal stress at failure for a varying ratio of the width to the loading span.

**Figure 20.**(

**a**) AIC values and (

**b**) the mean and st.dev. of the normal stress at failure for a varying ratio of the loading span to the support span.

**Figure 22.**(

**a**) AIC values and (

**b**) the mean and st.dev. of the normal stress at failure for a varying flaw density.

**Figure 24.**(

**a**) AIC values and (

**b**) the mean and st.dev. of the normal stress at failure for a varying flaw shape.

**Figure 26.**(

**a**) AIC values and (

**b**) the mean and st.dev. of the normal stress at failure for a varying maximum flaw depth.

**Figure 28.**(

**a**) AIC values and (

**b**) the mean and st.dev. of the normal stress at failure for varying fracture toughness.

Number of Simulations | Mean Value [MPa] | St.dev. [MPa] | COV [−] |
---|---|---|---|

50 | 101.43 | 15.14 | 0.149 |

100 | 96.84 | 13.11 | 0.135 |

500 | 98.84 | 13.97 | 0.141 |

1000 | 99.32 | 14.09 | 0.142 |

5000 | 98.96 | 13.84 | 0.140 |

10,000 | 98.86 | 13.86 | 0.140 |

50,000 | 98.91 | 13.88 | 0.140 |

100,000 | 98.93 | 13.83 | 0.140 |

500,000 | 98.96 | 13.84 | 0.140 |

$${\mathit{P}}_{\mathit{q}}$$
| Number of Simulations |
---|---|

10^{−1} | 460 |

10^{−2} | 5 052 |

10^{−3} | 50 970 |

10^{−4} | 510 154 |

10^{−5} | 5 101 990 |

Scale Factor [−] | Mean Value [MPa] | St.dev. [MPa] | COV [−] | Best Fit |
---|---|---|---|---|

0.25 | 148.10 | 42.11 | 0.284 | Gumbel |

0.5 | 117.62 | 21.94 | 0.187 | Gamma |

1 ^{1} | 99.28 | 13.71 | 0.138 | Gamma |

2 | 87.14 | 9.64 | 0.111 | Normal |

4 | 75.59 | 6.31 | 0.083 | Normal |

^{1}This scale factor is used in the Case Study of Section 2.2.3 and serves as a reference.

width/L_{l} [−] | Mean Value [MPa] | St.dev. [MPa] | COV [−] | Best Fit |
---|---|---|---|---|

0.1 | 118.35 | 22.63 | 0.191 | Lognormal |

0.2 | 107.94 | 17.37 | 0.161 | Gamma |

0.3 | 103.07 | 15.46 | 0.150 | Gamma |

0.4 | 99.89 | 14.09 | 0.141 | Gamma |

0.43 ^{1} | 99.25 | 13.82 | 0.139 | Gamma |

0.5 | 97.85 | 13.31 | 0.136 | Gamma |

0.6 | 95.90 | 12.44 | 0.130 | Normal |

0.7 | 94.16 | 12.06 | 0.128 | Gamma |

0.8 | 93.06 | 11.61 | 0.125 | Normal |

0.9 | 92.21 | 11.25 | 0.122 | Normal |

1.0 | 91.26 | 11.03 | 0.121 | Normal |

^{1}This ratio of the width to the loading span is used in the Case Study of Section 2.2.3 and serves as a reference.

L_{l}/L_{s} [−] | Mean Value [MPa] | St.dev. [MPa] | COV [−] | Best Fit |
---|---|---|---|---|

0.1 | 106.86 | 18.21 | 0.170 | Gamma |

0.2 | 104.37 | 16.61 | 0.159 | Gamma |

0.3 | 102.55 | 15.27 | 0.149 | Gamma |

0.4 | 100.55 | 14.28 | 0.142 | Gamma |

0.5 ^{1} | 99.25 | 13.82 | 0.139 | Gamma |

0.6 | 97.76 | 13.18 | 0.135 | Normal |

0.7 | 96.79 | 12.69 | 0.131 | Normal |

0.8 | 95.50 | 12.43 | 0.130 | Gamma |

0.9 | 94.95 | 12.09 | 0.127 | Normal |

^{1}This ratio of the loading span to the support span is used in the Case Study of Section 2.2.3 and serves as a reference.

ρ_{flaw} [flaws/cm^{2}] | Mean Value [MPa] | St.dev. [MPa] | COV [−] | Best Fit |
---|---|---|---|---|

0.5 | 111.09 | 20.79 | 0.187 | Lognormal |

1.0 | 104.18 | 16.48 | 0.158 | Gamma |

1.5 | 101.29 | 14.61 | 0.144 | Gamma |

2.0 ^{1} | 99.25 | 13.82 | 0.139 | Gamma |

2.5 | 97.74 | 12.97 | 0.133 | Gamma |

3.0 | 96.89 | 12.52 | 0.129 | Normal |

3.5 | 95.73 | 12.10 | 0.126 | Normal |

4.0 | 94.96 | 11.72 | 0.123 | Normal |

^{1}This flaw density is used in the Case Study of Section 2.2.3 and serves as a reference.

a/c [−] | Mean Value [MPa] | St.dev. [MPa] | COV [−] | Best Fit |
---|---|---|---|---|

0.4 | 109.18 | 15.10 | 0.138 | Gamma |

0.6 | 100.40 | 14.16 | 0.141 | Normal |

0.8 | 98.20 | 13.48 | 0.137 | Gamma |

1.0 ^{1} | 98.83 | 13.94 | 0.141 | Gamma |

1.2 | 101.06 | 14.09 | 0.139 | Gamma |

1.4 | 104.39 | 14.44 | 0.138 | Gamma |

1.6 | 107.77 | 15.23 | 0.141 | Normal |

^{1}This flaw shape is used in the Case Study of Section 2.2.3 and serves as a reference.

a_{max} [mm] | Mean Value [MPa] | St.dev. [MPa] | COV [−] | Best Fit |
---|---|---|---|---|

0.06 | 127.72 | 17.78 | 0.139 | Gamma |

0.08 | 111.07 | 15.62 | 0.141 | Gamma |

0.10 ^{1} | 99.44 | 13.92 | 0.140 | Gamma |

0.12 | 90.62 | 12.40 | 0.137 | Gamma |

0.14 | 83.65 | 11.65 | 0.139 | Normal |

0.16 | 78.33 | 11.05 | 0.141 | Gamma |

^{1}This maximum flaw depth is used in the Case Study of Section 2.2.3 and serves as a reference.

K_{IC} [MPa$\sqrt{\mathit{m}}$] | Mean Value [MPa] | St.dev. [MPa] | COV [−] | Best Fit |
---|---|---|---|---|

0.65 | 86.09 | 11.90 | 0.138 | Gamma |

0.70 | 92.35 | 12.76 | 0.138 | Gamma |

0.75 ^{1} | 99.25 | 13.82 | 0.139 | Gamma |

0.80 | 105.62 | 14.71 | 0.139 | Gamma |

0.85 | 112.46 | 15.67 | 0.139 | Normal |

0.90 | 118.96 | 16.37 | 0.138 | Gamma |

^{1}This fracture toughness is used in the Case Study of Section 2.2.3 and serves as a reference.

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**MDPI and ACS Style**

Symoens, E.; Van Coile, R.; Jovanović, B.; Belis, J.
Probability Density Function Models for Float Glass under Mechanical Loading with Varying Parameters. *Materials* **2023**, *16*, 2067.
https://doi.org/10.3390/ma16052067

**AMA Style**

Symoens E, Van Coile R, Jovanović B, Belis J.
Probability Density Function Models for Float Glass under Mechanical Loading with Varying Parameters. *Materials*. 2023; 16(5):2067.
https://doi.org/10.3390/ma16052067

**Chicago/Turabian Style**

Symoens, Evelien, Ruben Van Coile, Balša Jovanović, and Jan Belis.
2023. "Probability Density Function Models for Float Glass under Mechanical Loading with Varying Parameters" *Materials* 16, no. 5: 2067.
https://doi.org/10.3390/ma16052067