Estimate the theoretical fracture strength of a brittle material if it is known that fracture occurs by the propagation of an elliptically shaped surface crack of length \(0.5 \mathrm{mm}(0.02 \mathrm{in.})\) and having a tip radius of curvature of \(5 \times\) \(10^{-3} \mathrm{mm}\left(2 \times 10^{-4} \mathrm{in.}\right),\) when a stress of \(1035 \mathrm{MPa}(150,000 \mathrm{psi})\) is applied.

The Griffith equation plays a fundamental role in understanding fracture mechanics, particularly in brittle materials. Originally proposed by A.A. Griffith in 1921, it describes the condition of fracture in terms of the energy balance during crack propagation. According to Griffith, a crack will grow if the decrease in elastic energy is greater than the increase in surface energy. In simple terms, when an object with a crack is stressed, it may reach a point where growing the crack further costs less energy than keeping it the same length, leading to the material's failure.

Mathematically expressed, the Griffith equation is \[ \sigma_f = \sqrt{\frac{2E\gamma}{\pi a}} \.\]Here, \(\sigma_f\) is the theoretical fracture strength, \(E\) is the Young's modulus of the material, \(\gamma\) is the surface energy per unit area, and \(a\) is the length of a surface crack. This equation implies that the smaller the crack size or the greater the material's toughness (resistance to crack growth), the higher the theoretical fracture strength will be.

To properly apply this principle, one would calculate \(E\) and \(\gamma\) experimentally or derive them from literature values. However, in our exercise improvement advice, we will use the given stress as an approximation of the fracture strength, simplifying the process for educational purposes.

Brittle materials, such as glass, ceramics, and some polymers, are known to fracture with minimal plastic deformation. Fracture in these materials often occurs suddenly and with catastrophic consequences because they are unable to absorb much energy through deformation before breaking. Unlike ductile fracture, brittle fracture doesn't involve a significant amount of stretching or necking. Instead, it immediately follows the elastic deformation stage and appears as a cleavage or transgranular breakage.

When a brittle material contains flaws or cracks, especially those of a critical size, the applied stress becomes concentrated at the crack tips. This phenomenon is known as stress concentration and can drastically reduce the material's strength. Even cracks that are microscopic in size can lead to failure at stress levels much lower than the material's inherent theoretical strength. Therefore, understanding and predicting the behavior of cracks in brittle materials are essential for ensuring the structural integrity and safety of components made from such materials.

The stress intensity factor, denoted as \(K\), is a fundamental quantity in fracture mechanics that characterizes the intensity of the stress field near the tip of a crack. It is an essential parameter for predicting the onset of crack growth in brittle materials. The stress intensity factor depends on the applied load, the size and shape of the crack, and the geometry of the component.

For a given material, there is a critical stress intensity factor, known as \(K_{IC}\), or the fracture toughness, which is a measure of the material's resistance to crack growth. If the stress intensity factor exceeds \(K_{IC}\), a crack will propagate, leading to failure. The equation \[ K = \sigma \sqrt{\pi a} \.\]expresses the relationship between the stress intensity factor \(K\) and the applied stress \(\sigma\), where \(a\) is the crack length. When dealing with real-world applications, engineers must carefully assess the stress intensity factor to ensure the integrity of materials and prevent unexpected fracturing.

In the context of the exercise, the value of \(K\) is not calculated directly, but the idea of balancing the applied stress with the stress at the crack tip gives us the same end result of estimating the theoretical fracture strength.