Two previously undeformed specimens of the same metal are to be plastically deformed by reducing their cross-sectional areas. One has a circular cross section, and the other is rectangular; during deformation the circular cross section is to remain circular, and the rectangular is to remain as such. Their original and deformed dimensions are as follows: $$\begin{array}{lcc} \hline & \begin{array}{c} \text {Circular} \\ \text {(diameter, } \mathbf{m m} \text { ) } \end{array} & \begin{array}{c} \text {Rectangular} \\ (\mathbf{m m}) \end{array} \\ \hline \text { Original dimensions } & 18.0 & 20 \times 50 \\ \text { Deformed dimensions } & 15.9 & 13.7 \times 55.1 \\ \hline \end{array}$$ Which of these specimens will be the hardest after plastic deformation, and why?

Cross-sectional area reduction is a fundamental concept in materials science and engineering, involving the compression of a material, which alters its geometry and can subsequently change its properties. It plays a critical role in many manufacturing processes such as forging, rolling, and drawing.

During these processes, a material's original cross-sectional area is decreased, which often results in increased hardness and strength due to the phenomenon called work hardening or strain hardening. As the material undergoes plastic deformation—permanent changing of shape in response to applied stress—the movement of dislocations within the material's crystal structure becomes more difficult. This is because as the shape is altered, there becomes less room for the dislocations to move, making the material harder and stronger.

In the example given, we analyzed two different geometries: circular and rectangular. The initial and final cross-sectional areas are determined using corresponding geometric formulas. For the circular specimen, the area is calculated with the formula \( A = \pi \left(\frac{diameter}{2}\right)^2 \) and for the rectangular specimen, it’s simply the width times the height. By quantifying the area reduction, we gain insight into the intensity of the deformation process and can anticipate the material's increase in hardness.

In deformation mechanics, we must distinguish between elastic and plastic deformation. Elastic deformation is the reversible bending or stretching of bonds in a material, where it returns to its original shape after the force is removed. On the other hand, plastic deformation happens when the applied stress exceeds a material's yield strength, resulting in a permanent change in the shape.

The degree of plastic deformation can be measured as a percentage change from the original to deformed dimensions of the material. This is especially relevant when comparing the hardness of various materials or the same material under different levels of deformation. In the problem presented, we determined the percentage of deformation for both specimens using the formula \( Percentage\,of\,deformation = \frac{(Area_{original} - Area_{deformed})}{Area_{original}} \times 100\% \). Higher deformation percentages generally lead to greater hardness as a consequence of the increase in dislocation density and hindrance to dislocation movement, which is the underlying principle behind strain hardening.

Materials science is an interdisciplinary field that focuses on understanding and manipulating the properties of materials to create new applications or enhance existing ones. It combines elements of physics, chemistry, and engineering to explore how the composition, structure, and processing of materials affect their properties and performance.

In the context of plastics deformation and hardness, materials science provides the theoretical foundation for predicting and explaining how changes in a material's microstructure during deformation influence its mechanical properties. For example, introducing defects like dislocations through deformation can dramatically change a material's yield strength, hardness, and ductility. Materials scientists study these processes to develop materials that balance these properties effectively, for applications ranging from aerospace components to biomedical devices. By understanding how to manipulate the cross-sectional area and the resulting plastic deformation, materials scientists aim to tailor the hardness and strength of materials to meet specific design criteria.

When discussing engineering materials properties, we refer to a wide range of characteristics that determine how a material will respond under various conditions. These properties fall into different categories such as mechanical, thermal, electrical, and chemical. Mechanical properties, which include hardness, ductility, tensile strength, and toughness, are of paramount importance in selecting materials for structural applications.

Hardness, a measure of a material's resistance to localized plastic deformation (e.g., indentation or scratching), is particularly relevant when materials are subjected to friction and wear. As seen in the problem at hand, by calculating the hardness through the percentage of cross-sectional area reduction, we indirectly estimate how the material properties have been altered due to deformation.

To optimize engineering materials for different uses, it’s essential to understand how processes such as deformation affect their mechanical properties. With plastics deformation leading to an increase in hardness and strength, engineers are equipped to choose or process materials that will bear the stresses and strains they will face in service life while maintaining functionality and safety.