Problem 8 - An Introduction to Materials Engineering and Science for Chemical and Materials Engineers

Question

A glass plate contains an atomic-scale surface crack. Assume that the crack tip radius is approximately equal to the diameter of an \(\mathrm{O}^{2-}\) ion. Given that the crack is \(1 \mu \mathrm{m}\) long and the theoretical strength of the defect-free glass is \(7.0 \mathrm{GPa}\), calculate the breaking strength of the glass plate.

Step-by-Step Solution

Answer

The breaking strength of the glass plate is approximately 1.169 MPa.

Step 1: Understanding the Problem

We need to calculate the breaking strength of a glass plate that contains a surface crack. The crack's radius of curvature is approximately the diameter of an \( \mathrm{O}^{2-} \) ion. The theoretical strength of defect-free glass is given as \( 7.0 \mathrm{GPa} \), and the crack is \( 1 \mu m \) long.

Step 2: Determine Crack Tip Radius

The radius of curvature at the tip of the crack is approximately the diameter of an \( \mathrm{O}^{2-} \) ion, typically about \( 0.14 \mathrm{nm} \) (or \( 1.4 \times 10^{-10} \mathrm{m} \)).

Step 3: Using the Griffith Criterion Formula

Griffith's criterion for fracture in brittle materials is given by: \[ \sigma_f = \sqrt{\frac{2\gamma E}{\pi a}} \] where \( \sigma_f \) is the fracture strength, \( \gamma \) is the surface energy per unit area, \( E \) is Young's modulus, and \( a \) is the half-length of the crack. Use an approximation since exact values for \( \gamma \) and \( E \) are often not given, simplifying it by comparing the theoretical strength to that of defect-free glass.

Step 4: Estimate Fracture Strength Using Surface Energy Approximation

We approximate using the given theoretical strength of defect-free glass (7.0 GPa) and considering the crack parameters. The modified formula becomes: \[ \sigma_f \approx \sigma_{th} \left(\frac{r}{a}\right)^{1/2} \] where \( \sigma_{th} = 7.0 \mathrm{GPa} \), \( r \approx 1.4 \times 10^{-10} \mathrm{m} \), and \( a = 0.5 \times 1 \mu m = 0.5 \times 10^{-6} \mathrm{m} \).

Step 5: Calculate Breaking Strength

Substitute the values into the formula: \[ \sigma_f \approx 7.0 \mathrm{GPa} \times \left(\frac{1.4 \times 10^{-10}}{0.5 \times 10^{-6}}\right)^{1/2} \] Calculate the expression: \( \sigma_f \approx 7.0 \mathrm{GPa} \times \left(2.8 \times 10^{-4}\right)^{1/2} \).

Step 6: Perform Final Calculation

Calculate the square root: \( \left(2.8 \times 10^{-4}\right)^{1/2} \approx 1.67 \times 10^{-2} \). Thus, \( \sigma_f \approx 7.0 \mathrm{GPa} \times 1.67 \times 10^{-2} = 1.169 \mathrm{MPa} \).

Step 7: Conclude with the Breaking Strength

The calculated breaking strength of the glass with the surface crack is approximately \( 1.169 \mathrm{MPa} \).

Key Concepts

Fracture Strength Calculation

Calculating the fracture strength is crucial for understanding how much stress a material can withstand before failing. In this specific case, we are looking at a glass plate with a surface crack. The fracture strength is determined using the well-known Griffith's criterion, which is tailored for brittle materials like glass. According to Griffith's theory, the presence of a crack, even at the atomic scale, significantly reduces the strength of a material compared to its theoretical strength when defect-free.

The fracture strength, denoted as \( \sigma_f \), can be estimated using the formula:

• \( \sigma_f = \sigma_{th} \left(\frac{r}{a}\right)^{1/2} \)

Where \( \sigma_{th} \) is the theoretical strength of the material, \( r \) is the radius of curvature at the crack tip, and \( a \) is the half-length of the crack. This formula helps us calculate how defects reduce the load a brittle material can tolerate.

Brittle Materials

Brittle materials, such as glass and ceramics, are known for their inability to deform plastically. Instead, they fracture suddenly when their strength is exceeded. This characteristic is primarily due to their atomic structure, which doesn't allow for dislocation movement like more ductile materials (e.g., metals) do.

• Brittle fracture is usually quite sudden and occurs without much warning.


• Such materials often fail by crack propagation, making the study of cracks crucial for predicting failure.

Understanding the behavior of brittle materials under stress helps engineers prevent abrupt fractures, ensuring safety and reliability in applications that involve these materials.

Atomic-Scale Surface Crack

Even a small crack at the atomic level can severely impact the strength of a material. These microscopic imperfections serve as stress concentrators—points within a material that experience a much greater stress level than the rest of the object.

The radius of the crack, often on the order of nanometers, affects the amount of stress concentration, influencing the material's effective breaking strength. The Griffith's criterion is particularly useful in quantifying the effect of these atomic-scale surface cracks on a material's fracture strength.

• Even though these cracks are atomic-scaled, their impact on material strength is large.


• Engineers must take care to minimize these imperfections to maximize the material's load-carrying capacity.

Young's Modulus

Young's modulus, represented usually as \( E \), is a fundamental material property that measures the stiffness of a solid material. It describes the relationship between stress (force per unit area) and strain (deformation) in a linear elastic material.

• Young's modulus is a measure of a material's elasticity, an important characteristic for calculating fracture strength.


• A higher Young's modulus indicates that a material is stiffer, which means it deforms less under stress.

Griffith's criterion includes Young's modulus in its formula because the resistance to crack propagation is partly dependent on the material's stiffness. This relation shows the importance of understanding both the elastic and fracture properties of materials when designing and evaluating their structural integrity.