A three-point bending test is performed on a spinel \(\left(\mathrm{MgAl}_{2} \mathrm{O}_{4}\right)\) specimen having a rectangular cross section of height \(d=3.8 \mathrm{~mm}(0.15 \mathrm{in}\).) and width \(b=9 \mathrm{~mm}(0.35 \mathrm{in}\).); the distance between support points is \(25 \mathrm{~mm}(1.0 \mathrm{in}\).). (a) Compute the flexural strength if the load at fracture is \(350 \mathrm{~N}\left(80 \mathrm{lb}_{\mathrm{f}}\right)\). (b) The point of maximum deflection \(\Delta y\) occurs at the center of the specimen and is described by $$ \Delta y=\frac{F L^{3}}{48 E I} $$ where \(E\) is the modulus of elasticity and \(I\) is the cross-sectional moment of inertia. Compute \(\Delta y\) at a load of \(310 \mathrm{~N}\left(70 \mathrm{lb}_{\mathrm{}}\right)\).

When materials undergo a three-point bending test, an important characteristic measured is their flexural strength. This is the ability of a material to resist deformation under load, and in the context of our exercise, it refers to the stress the spinel specimen endures at the point of fracturing. The flexural strength, denoted as \(\sigma_f\), is calculated using the formula \[ \sigma_f = \frac{3FL}{2bd^2} \].

Applying this equation involves using the applied force at fracture (\(F\)), the distance between the supports (\(L\)), the width (\(b\)), and the thickness (\(d\)) of the specimen, ultimately yielding the material's flexural strength in megapascals (MPa). It's crucial to ensure units are consistent and according to standard measurement units when plugging in the values to obtain an accurate result.

Maximum deflection is a term used to describe the greatest displacement experienced by a material when a load is applied. During the bending test, it occurs at the center of the specimen. To calculate this, the formula used is \[ \Delta y = \frac{FL^3}{48EI} \], where \(\Delta y\) is the deflection, \(F\) is the applied load, \(L\) is the span length between supports, \(E\) is the modulus of elasticity, and \(I\) is the moment of inertia.

Understanding how a material deflects under a load is essential in the design and evaluation of structural components. Proper calculation assures that the material will perform as expected without failing under the worst conditions it may encounter in real-world applications.

Moment of inertia is a geometric property of the cross-sectional area of a beam and is crucial when assessing the beam's ability to resist bending. It is denoted by \(I\) and for a rectangular cross-section, it is calculated with \[ I = \frac{bd^3}{12} \]. This computation factors in the width \(b\) and the depth \(d\) of the specimen, to the third power, indicating how the distribution of material affects the specimen's rigidity.

High moment of inertia means the material is more resistant to bending, which is essential knowledge for engineers who need to ensure that structural elements can withstand imposed loads without excessive bending.

The modulus of elasticity, symbolized as \(E\), is a measure of a material's stiffness or rigidity and forms a core aspect of the deflection calculation in bending tests. It quantifies the relationship between the stress (force per unit area) and strain (proportional deformation) in a material. While it is often provided as a material constant, it can be determined experimentally under certain conditions.

The higher the modulus of elasticity, the less a material will deform when a load is applied, hence a stiffer material. This is key to understanding how a material will behave under load - knowledge that can prevent material failures in practical applications like bridge construction, the manufacturing of machine parts, or the design of any structure where the flexibility or rigidity of the materials used is crucial to the structure's safety and functionality.