Let's solve the problem step by step: --- ## **Step 1: Estimate the inherent flaw size for each glass type** The **ultimate tensile stress** (\(\sigma_{ult}\)) is related to the **fracture toughness** (\(K_{IC}\)) and the inherent flaw size (\(a\)) by the following equation (for a surface flaw): \[ \sigma_{ult} = \frac{K_{IC}}{Y \sqrt{\pi a}} \] where - \(\sigma_{ult}\) = ultimate tensile strength (MPa) - \(K_{IC}\) = fracture toughness (\(MPa\sqrt{m}\)) - \(Y\) = geometry factor (for surface cracks, typically \(Y \approx 1\)) - \(a\) = flaw size (m) **Rearrange to solve for \(a\):** \[ a = \left( \frac{K_{IC}}{Y \sigma_{ult} \sqrt{\pi}} \right)^2 \] Assume \(Y = 1\). --- #### **Calculate for Each Glass Type** #### **1. Gorilla Glass 6 (Jul 2018):** - \(\sigma_{ult} = 65\) MPa - \(K_{IC} = 0.75\) MPa\(\sqrt{m}\) \[ a = \left( \frac{0.75}{1 \times 65 \times \sqrt{\pi}} \right)^2 \] \[ \sqrt{\pi} \approx 1.772 \] \[ a = \left( \frac{0.75}{65 \times 1.772} \right)^2 = \left( \frac{0.75}{115.18} \right)^2 = (0.00651)^2 \approx 4.24 \times 10^{-5} \text{ m} = 42.4 \ \mu m \] --- #### **2. Gorilla Glass Victus 2 (Nov 2022):** - \(\sigma_{ult} = 70\) MPa - \(K_{IC} = 0.82\) MPa\(\sqrt{m}\) \[ a = \left( \frac{0.82}{70 \times 1.772} \right)^2 = \left( \frac{0.82}{124.04} \right)^2 = (0.00661)^2 \approx 4.37 \times 10^{-5} \text{ m} = 43.7 \ \mu m \] --- #### **3. Gorilla Glass Armor (Jan 2024):** - \(\sigma_{ult} = 72\) MPa - \(K_{IC} = 0.86\) MPa\(\sqrt{m}\) \[ a = \left( \frac{0.86}{72 \times 1.772} \right)^2 = \left( \frac{0.86}{127.58} \right)^2 = (0.00674)^2 \approx 4.54 \times 10^{-5} \text{ m} = 45.4 \ \mu m \] --- #### **4. Gorilla Glass Armor (Jun 2024):** - \(\sigma_{ult} = 78\) MPa - \(K_{IC} = 0.78\) MPa\(\sqrt{m}\) \[ a = \left( \frac{0.78}{78 \times 1.772} \right)^2 = \left( \frac{0.78}{138.22} \right)^2 = (0.00564)^2 \approx 3.18 \times 10^{-5} \text{ m} = 31.8 \ \mu m \] --- ### **Summary Table of Inherent Flaw Sizes** | Version | Inherent Flaw Size (\(\mu m\)) | |---------------------------|-------------------------------| | Gorilla Glass 6 (2018) | 42.4 | | Gorilla Glass Victus 2 | 43.7 | | Gorilla Glass Armor (Jan) | 45.4 | | Gorilla Glass Armor (Jun) | 31.8 | --- ## **Step 2: Calculate Strength as a Function of Scratch Depth (\(a\))** Using the same equation, for **scratch depths** \(a\) ranging between **10 and 100 \(\mu m\)**: \[ \sigma = \frac{K_{IC}}{Y \sqrt{\pi a}} \] Where \(a\) is the **scratch depth** (meters). Let's express \(a\) in meters: \(a =\) 10, 20, ..., 100 \(\mu m = 10^{-6}\) m. You can write a general form: \[ \sigma(a) = \frac{K_{IC}}{\sqrt{\pi a}} \] Let's do one sample calculation for Gorilla Glass 6 at \(a = 10 \mu m = 1 \times 10^{-5}\) m: \[ \sigma = \frac{0.75}{\sqrt{3.1416 \times 1 \times 10^{-5}}} \] \[ = \frac{0.75}{\sqrt{3.1416 \times 10^{-5}}} \] \[ = \frac{0.75}{\sqrt{3.1416} \times 10^{-2.5}} \] \[ \sqrt{3.1416} \approx 1.772} \] \[ = \frac{0.75}{1.772 \times 0.00316} \] \[ = \frac{0.75}{0.00560} \] \[ \sigma \approx 134 \text{ MPa} \] You can repeat this calculation for all glasses and all \(a\) values (make a table or plot). --- ## **Step 3: Plot Strength vs. Scratch Depth** **On the X-axis:** Scratch depth (\(\mu m\)), **On the Y-axis:** Calculated strength (\(MPa\)). **One curve for each glass.** --- ## **Step 4: Scratches Near a Circular Cut-out** When flaws are near a cut-out (e.g., camera), the geometry factor \(Y\) increases due to stress concentration. Assume \(Y = 1.5\). The formula becomes: \[ \sigma = \frac{K_{IC}}{1.5\sqrt{\pi a}} \] This will reduce the calculated strength by a factor of 1.5 at each depth. --- ## **Final Answers Summary** ### **1. Inherent Flaw Sizes (\(\mu m\)):** - Gorilla Glass 6: **42.4** - Gorilla Glass Victus 2: **43.7** - Gorilla Glass Armor (Jan 2024): **45.4** - Gorilla Glass Armor (Jun 2024): **31.8** ### **2. Strength vs. Scratch Depth Equation:** - \(\sigma(a) = \frac{K_{IC}}{\sqrt{\pi a}}\) (for normal surface) - \(\sigma(a) = \frac{K_{IC}}{1.5\sqrt{\pi a}}\) (near camera cut-out) ### **3. Plots:** - Plot 1: Four curves, \(\sigma\) vs. \(a\) (10–100 \(\mu m\)), for each glass. - Plot 2: Same, but with strength values divided by 1.5 (for the cut-out condition). --- #### **If you need the full set of values and a sample plot, let me know!**