Schmid's law
/ SHMIT /
Schmid's law is the bookkeeping that connects the pull you apply to a bar with the sideways shear a dislocation actually feels. Even when you pull straight along a bar, the slip plane inside sits at a slant, so only part of your force becomes the shear that drives slip. Schmid's law tells you exactly how much.
In symbols, the resolved shear stress is tau_R = sigma times cos(phi) times cos(lambda), where sigma is the applied tensile stress, phi is the angle between the pull direction and the normal (perpendicular) to the slip plane, and lambda is the angle between the pull direction and the slip direction. The product cos(phi) times cos(lambda) is called the Schmid factor. Slip begins on the system with the largest Schmid factor once tau_R reaches the critical value, so the yield stress is sigma_y = tau_CRSS / (cos phi cos lambda). The Schmid factor is largest, exactly 0.5, when both angles are 45 degrees, the orientation that yields at the lowest pull.
This is why a single crystal's strength depends on how it is oriented in the grip: pull it so the slip plane is near 45 degrees and it yields early; pull nearly along or across the slip plane (Schmid factor near zero) and it seems much stronger. In an ordinary polycrystal the thousands of randomly oriented grains average this out, but Schmid's law still governs which grains slip first.
Take tau_CRSS = 1 MPa. With both angles at 45 degrees the Schmid factor is cos45 times cos45 = 0.707 x 0.707 = 0.5, so the bar yields at sigma_y = 1 / 0.5 = 2 MPa. Tilt the crystal so phi = 30 and lambda = 60 degrees: the factor is cos30 times cos60 = 0.866 x 0.5 = 0.433, and sigma_y rises to 1 / 0.433 = about 2.3 MPa.
Same material, same CRSS; the pull needed to yield changes with orientation, exactly as the Schmid factor predicts.
The maximum Schmid factor is 0.5, so the resolved shear stress can never exceed half the applied tensile stress, a quick sanity check when you compute one.