Consider a metal single crystal oriented such that the normal to the slip plane and the slip direction are at angles of \(60^{\circ}\) and \(35^{\circ},\) respectively, with the tensile axis. If the critical resolved shear stress is \(6.2 \mathrm{MPa}(900 \mathrm{psi}),\) will an applied stress of \(12 \mathrm{MPa}(1750 \mathrm{psi})\) cause the single crystal to yield? If not, what stress will be necessary?

Schmid's Law plays a pivotal role in understanding the plastic deformation of materials, particularly in single crystals. Fundamentally, it relates to how external stress translates into shear stress on specific planes and directions within a crystalline material. To put it into perspective, imagine the internal structure of a crystal being like a stack of playing cards. When force is applied, it may not cause immediate slipping or shifting unless it is at the right angle—the same principle applies to crystals at the atomic level.

Mathematically, Schmid's Law is expressed as \(\tau = \sigma \times \cos\alpha \times \cos\beta\), where \(\tau\) is the resolved shear stress on the slip system, \(\sigma\) is the externally applied stress, \(\alpha\) is the angle between the action of the applied stress and the normal to the slip plane, and \(\beta\) is the angle between the applied stress and the slip direction. \(\alpha\) and \(\beta\) are critical because they dictate the component of the applied stress that effectively works to cause dislocation motion—the fundamental process behind plastic deformation in crystals.

In a simple view, Schmid's Law helps us to pinpoint the 'easy' paths for deformation in a crystal and predict how stress will be 'resolved' into shear along these paths.

Yielding, in metallurgical terms, refers to the onset of plastic deformation — the point at which a material transitions from elastic behavior to permanent, irreversible deformation. For single crystals, yielding doesn't occur uniformly as it might in polycrystalline materials; it's highly dependent on the crystallographic orientation of the slip systems.

Single crystals yield when the resolved shear stress on the most favorably oriented slip system exceeds the critical resolved shear stress (CRSS). CRSS is a material property indicating the stress level required to initiate slip — the movement of dislocations — on a slip plane in a crystal. It is the sheer stress value needed to overcome the atomic bonds and cause a dislocation to move and multiply, leading to plastic deformation. Depending on the purity and temperature, the CRSS can vary significantly.

Understanding the yielding of single crystals is crucial for applications involving single-crystal materials, like turbine blades in jet engines, where the orientation can be engineered to maximize strength and durability under high-stress conditions.

Calculating the resolved shear stress effectively gauges whether a single crystal will yield under a given stress regime. It is the shear stress component that contributes to dislocation motion along a slip plane and slip direction. Calculation involves knowing not just the overall stress applied to the material, but also the angles \(\alpha\) and \(\beta\) specific to the crystal's orientation.

Following Schmid's Law, we use the cosines of these angles in our formula \(\tau = \sigma \times \cos\alpha \times \cos\beta\) to resolve the actual stress acting along the slip system. By plugging in the values from a real-world problem, as seen in the textbook exercise, one computes the resolved shear stress and compares it against the critical resolved shear stress (CRSS) of the material. If the calculated value is less than CRSS, as in the provided example, the crystal does not yield. On the other hand, if it equals or exceeds CRSS, yielding will occur, and the material will deform plastically.

In contexts like materials engineering and physics, these calculations are not just academic exercises but essential for designing and predicting the behavior of materials under various stress conditions.