Experimentally, it has been observed for single crystals of a number of metals that the critical resolved shear stress \(\tau_{\text {crsss }}\) is a function of the dislocation density \(\rho_{D}\) as $$ \tau_{\text {crss }}=\tau_{0}+A \sqrt{\rho_{D}} $$ where \(\tau_{0}\) and \(A\) are constants. For copper, the critical resolved shear stress is \(2.10 \mathrm{MPa}\) (305 psi) at a dislocation density of \(10^{5} \mathrm{~mm}^{-2}\). If it is known that the value of \(A\) for copper is \(6.35\) \(\times 10^{-3} \mathrm{MPa} \cdot \mathrm{mm}(0.92 \mathrm{psi} \cdot \mathrm{mm})\), compute \(\tau_{\text {crss }}\) at a dislocation density of \(10^{7} \mathrm{~mm}^{-2}\).

Understanding the concept of dislocation density is pivotal when studying the mechanical properties of crystalline materials. In simple terms, dislocation density, denoted as \( \rho_{D} \) in materials science, refers to the number of dislocations within a given area of the crystal lattice. Dislocations are essentially defects or irregularities within the crystal structure that play a significant role in determining the strength and plasticity of the material.

The measure of dislocation density gives scientists and engineers an idea of how crowded the lattice is with dislocations. It's commonly expressed in terms of the number of dislocations per square millimeter (\(\text{mm}^{-2}\)). High dislocation densities mean there are many obstacles to dislocation movement, leading to an increase in strength according to the Hall-Petch relationship and this sort of interplay is foundational to the concept of work hardening in metals.

In practice, materials with low dislocation densities can deform more easily because the dislocations can move with less interference. Conversely, materials with high dislocation densities are stronger but less ductile, as the dislocations 'tangle' and impede each other's movement, thus requiring more force to induce deformation. This direct relationship between dislocation density and mechanical properties is key to understanding how materials can be engineered for specific applications.

The term single crystals of metals refers to a piece of metal in which the crystal lattice is continuous and unbroken to the edges of the sample, with no grain boundaries. This uniformity means that the properties of single-crystal metals are highly anisotropic — they vary based on the direction of measurement.

Single-crystal structures are particularly important in materials science because they can be ideal for studying the fundamental properties of metals, free of the complexities introduced by grain boundaries in polycrystalline samples.

In the context of shear stress and dislocation density, single crystals often exhibit clearer and more predictable relationships because their behavior isn't complicated by interactions at grain boundaries. Understanding how single crystals deform under stress is critical for applications such as turbine blades in jet engines where uniformity in mechanical properties can lead to improvements in performance and safety. Single-crystal metals can also serve as the benchmark for comparing the properties of more commonly used polycrystalline metals.

The shear stress calculation is fundamental to understanding how materials respond to applied forces. Shear stress, represented by the Greek letter \( \tau \) in equations, refers to the component of stress coplanar with a material cross-section. It arises from forces that cause layers of the material to slide past each other.

In the context of the given exercise, the critical resolved shear stress (\( \tau_{\text{crss}} \)) is the shear stress resolved along a particular slip direction and plane, necessary to move dislocations and cause slipping or plastic deformation in the crystal. The equation \( \tau_{\text{crss }}=\tau_{0}+A \sqrt{\rho_{D}} \) indicates that this stress has a base value (\( \tau_{0} \)) and increases with the square root of the dislocation density, modified by a constant \( A \).

When calculating the critical resolved shear stress for a different dislocation density using the given constants, engineers and material scientists can predict the required force to cause slip in a material. This is crucial for designing applications where precise mechanical properties are required and for selecting appropriate materials for specific stress conditions.