Two previously undeformed cylindrical specimens of an alloy are to be strain hardened by reducing their cross-sectional areas (while maintaining their circular cross sections). For one specimen, the initial and deformed radii are 15 and \(12 \mathrm{~mm}\), respectively. The second specimen, with an initial radius of \(11 \mathrm{~mm}\), must have the same deformed hardness as the first specimen; compute the second specimen's radius after deformation.

Understanding true strain is vital for studying the behavior of materials under deformation. True strain, represented as \( \epsilon_{true} \), measures how much a material has been deformed from its original shape. It's calculated using the natural logarithm of the ratio of the final dimension to the original dimension. In the case of cylindrical objects, like the metal specimens in our exercise, we focus on their cross-sectional area changes. The formula for true strain in terms of radii is:

\( \epsilon_{true} = \frac{r_{0}^2}{r^2} \)

where \( r_{0} \) is the initial radius and \( r \) is the deformed radius. It is crucial to note that true strain provides an accurate measure regardless of the amount of deformation, making it an essential concept in material science and engineering.

When metals undergo deformation, there's more than just a change in shape; their internal structure and properties can change as well. This process, known as strain hardening or work hardening, occurs when metal is subjected to stresses beyond its elastic limit. As a result, the metal becomes stronger and harder as its crystal lattice is distorted. One must consider the extent of deformation to predict the resulting hardness or strength of a metal, and this is where true strain becomes a critical parameter.

By reducing the cross-sectional area while maintaining the circular cross section, as in our exercise, we apply uniform stress that leads to a uniform distribution of strain throughout the metal specimen.

The mechanical properties of materials describe their behavior under various loads and stresses. These properties encompass hardness, ductility, toughness, and strength, among others. They are influenced by factors such as the material's composition, microstructure, and the way it has been processed.

Strain hardening is a technique used to enhance some of these properties—specifically, the strength and hardness of metals. By understanding the deformation process, engineers can design metals that are suited for specific applications, ensuring they perform reliably when in use. This relationship between deformation and mechanical properties is what makes our exercise so relevant to material science and engineering applications.

The stress-strain relationship is pivotal in understanding material behavior under applied forces. Stress is the force applied to a material over a certain area, and strain is the deformation that results from applying stress. The stress-strain curve which plots stress against strain for a material provides a visual representation of this relationship and includes key information such as the yield strength, ultimate strength, and fracture point.

This relationship directly correlates to our exercise where the deformation of cylindrical metal specimens is considered. By comparing true strains, we ensure that both specimens endure the same degree of work hardening, thereby achieving similar mechanical properties. This concept is foundational for predicting how a material's dimensions affect its capacity to withstand forces without permanently deforming or breaking.