A circular specimen of \(\mathrm{MgO}\) is loaded using a three-point bending mode. Compute the minimum possible radius of the specimen without fracture, given that the applied load is \(425 \mathrm{~N}\left(95.5 \mathrm{lb}_{\mathrm{f}}\right)\), the flexural strength is \(105 \mathrm{MPa}(15,000 \mathrm{psi})\), and the separation between load points is \(50 \mathrm{~mm}\) (2.0 in.).

Flexural strength, also known as bend strength or modulus of rupture, is a material property that denotes the ability of a beam or slab to resist failure in bending. It is measured by applying a load at a specific rate to a beam supported on both ends until the material fractures.

In the context of a three-point bending test, like the one in the provided exercise, the flexural strength is the maximum stress experienced by the material at the moment of fracture. This stress depends on the load applied and the dimensions and properties of the material.

It is important to not exceed the flexural strength of a material in practical applications because surpassing this limit will result in failure. In our exercise, maintaining the bending stress below the given flexural strength of the sample ensures that the material will not fracture under the applied load.

Bending stress is a measure of the internal stress caused by an external bending load. It varies along the cross-section of a structural element, being highest at the surface farthest from the neutral axis and zero at the neutral axis itself.

In the three-point bending scenario, the formula \(\sigma = \dfrac{3FL}{2\pi r^3}\) provides a way to calculate the bending stress (\(\sigma\)) in terms of the applied force (\(F\)), the distance between the supports (\(L\)), and the radius of the circular specimen (\(r\)). It's critical to consider bending stress during the design phase to ensure that the structure can support the anticipated loads without deforming or fracturing.

The solution to our exercise involved calculating the bending stress at the precise moment before material fracture to find the minimum allowable radius of the specimen that would withstand the given applied load.

Material fracture mechanics is a field of study that deals with the formation, growth, and consequence of cracks in materials. It involves the understanding of the mechanisms behind material failure under various stress conditions and the ability to predict failure using analytical and empirical methods.

One key application of fracture mechanics is the prediction of critical flaws that lead to catastrophic failure. In the exercise, the application of a three-point bending test allows us to apply knowledge of fracture mechanics to determine the minimum radius at which the MgO specimen can withstand the load without experiencing a fracture.

Materials designers use fracture mechanics principles to prevent unexpected failures by ensuring that the designs are robust enough to tolerate intended loads, and to identify flaws that could propagate and grow into critical fractures under stress. In the exercise, knowing the precise flexural strength aids in preventing fractures during practical applications where similar stress might be applied.