Critically Resolved Shear Stress

The extent of slip in a single crystal depends on the magnitude of the shearing stress produced by external loads, the geometry of the crystal structure, and the orientation of the active slip planes with respect to the shearing stresses. Slip begins when the shearing stress
on the slip plane in the slip direction reaches a threshold value called the critical resolved shear stress. This value is really the single crystal equivalent of the yield stress of an ordinary stress-strain curve. The value of the critical resolved shear stress depends chiefly on composition and temperature.

The fact that different tensile loads are required to produce slip in single crystals of different orientation can be rationalized by a critical resolved shear stress. To calculate the critical resolved shear stress from a single crystal tested in tension, it is necessary to know, from X-ray diffraction, the orientation with respect to the tensile axis of the plane on which slip first appears and the slip direction. Consider a cylindrical single crystal with cross-sectional area A. The angle between the normal to the slip plane and the tensile axis is ɸ, and the angle which the slip direction makes with the tensile axis is λ. The area of the slip plane inclined at the angle ɸ will be A /cos ɸ, and the component of the axial load acting in the slip plane in the slip direction is P cos λ. Therefore, the critical resolved shear stress is given by

Τ_R = P cosλ cosɸ /A

According to the law of the critical resolved shear stress, also known as Schmid's law, the limited number of slip systems allows large differences in orientation between the slip plane and the tensile axis. The magnitude of the critical resolved shear stress of a crystal is determined by the interaction of its population of dislocations with each other and with defects such as vacancies, interstitials, and impurity atoms. This stress is greater than the stress required to move a single dislocation, but it is appreciably lower than the stress required to produce slip in a perfect lattice. On the basis of this reasoning, the critical resolved shear stress should decrease as the density of defects decreases, provided that the total number of imperfections is not zero. When the last dislocation is eliminated, the critical resolved shear stress should rise abruptly to the high value predicted for the shear strength of a perfect crystal. Experimental evidence for the effect of decreasing defect density is shown by the fact that the critical resolved shear stress of soft metals can be reduced to less than one-third by increasing the purity. At the other extreme, micron-diameter single-crystal filaments, or whiskers, can be grown essentially dislocation-free. Tensile tests on these filaments have given strengths which are approximately equal to the calculated strength of a perfect crystal.