Vector Projection
The component of an applied force acting along a specific slip direction on a specific slip plane governs the onset of plastic flow. Calculating resolved shear stress requires knowledge of the orientation of the crystal relative to the loading axis. This value dictates whether a particular slip system will activate.
Geometric Calculation
Trigonometric relationships between the tensile axis, the slip plane normal and the slip direction define the magnitude of the force. The formula for resolved shear stress multiplies the nominal stress by the cosine of the two relevant angles. This projection accounts for the anisotropy of single crystals.
Yield Initiation
Plasticity begins only when the force on a slip system exceeds a critical threshold. A high resolved shear stress on a primary system leads to immediate deformation, while systems oriented poorly remain inactive. This selective activation produces the complex surface textures seen in deformed metals.
Interference Effect
Grain boundaries and internal defects alter the local force distribution compared to the macroscopic average. While the global resolved shear stress provides a baseline, local variations can lead to premature failure at stress concentrators. Accurate modeling requires accounting for these microstructural inhomogeneities.
This effect is especially pronounced in materials with large grains where the orientation of a single neighbor can significantly restrict the available slip systems. Localized stress concentrations often exceed the average value by a factor of three or more depending on the geometry of the defect.
