The mechanical properties of cobalt may be improved by incorporating fine particles of tungsten carbide (WC). Given that the moduli of elasticity of these materials are, respectively, \(200 \mathrm{GPa}\) \(\left(30 \times 10^{6} \mathrm{psi}\right)\) and \(700 \mathrm{GPa}\left(102 \times 10^{6} \mathrm{psi}\right)\), plot the modulus of elasticity versus the volume percent of WC in Co from 0 to 100 vol \(\%\), using both upperand lower- bound expressions.

The term 'mechanical properties' relates to how materials respond to external forces, such as tension, compression, and shear. In material science, these properties often dictate the applicability of materials in various engineering contexts. One such property is the modulus of elasticity, which is a measure of a material's stiffness. It essentially reflects the material's ability to deform elastically when a stress is applied. Different materials have distinct moduli of elasticity. For example, cobalt (Co) is used for its toughness and strength, while tungsten carbide (WC), known for its extreme hardness, shows a much greater modulus of elasticity.

Understanding these properties helps engineers design components that can withstand specific loads without deforming permanently. When cobalt is enhanced with tungsten carbide, we aim to improve its mechanical properties, particularly the modulus of elasticity, which can be visualized through a plotted graph against the volume percent of WC.

The volume percent (vol%) is a measure used in material science to describe the concentration of one substance within another in terms of volume. It's analogous to saying how much space one material takes up in a mixture relative to the total space available. In our exercise, the volume percent pertains to how much tungsten carbide is present in the cobalt matrix. This is significant because the mechanical properties of composite materials, like the Co-WC mixture, change as the proportion of their constituents change. A volume percent of 0 means pure cobalt, while 100 vol% signifies pure tungsten carbide. Analyzing properties at different volume percents aids in producing a composite material with optimized performance. The relationship between volume percent and mechanical properties is often nonlinear and must be investigated for precise material design.

In the context of materials, upper and lower bound expressions are mathematical models used to estimate the extent of a property that a composite material could possess. They provide theoretical limits between which the actual property value is likely to fall. The upper bound is calculated using the rule of mixtures which assumes that the strain is uniform across both materials. In contrast, the lower bound is based on the inverse rule of mixtures, assuming stress is the same throughout the materials in the composite.

The upper bound typically assumes perfect load transfer between the two materials, leading to optimistic estimations, while the lower bound assumes imperfect load transfer, yielding conservative estimations. When plotting these bounds against volume percent, the actual modulus of elasticity for the composite will exist within the area enclosed by the upper and lower bound curves, helping engineers to understand the potential variability in material properties and design safety margins appropriately.

Material science is the study and application of the properties of matter in various fields of science and engineering. It plays a critical role in developing new materials and enhancing the functionalities of existing ones. It integrates knowledge from physics and chemistry to explain why materials behave the way they do and how they can be altered to serve specific applications. In the example of improving cobalt's mechanical properties with tungsten carbide, material science principles are applied to find the right combination that yields the right balance of stiffness and strength for use in hard-wearing applications, such as cutting tools and aerospace components.

Material scientists utilize sophisticated models, such as the upper and lower bound expressions mentioned previously, to predict how changes in composition, such as volume percent variations, affect overall mechanical properties. The ultimate goal is to engineer materials with predictable properties tailored to meet the demands of modern technology and innovation.