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Schmid's law (also Schmid factor, $$m$$) describes the slip plane and the slip direction of a stressed material, which can resolve the most shear stress. The factor is named after Erich Schmid who coauthored a book with Walter Boas introducing the concept in 1935.

Derivation


Schmid's Law states that the resolved shear stress ($$\tau_{RSS}$$), the shear component of an applied tensile or compressive stress along a slip plane not perpendicular or parallel to the stress axis, is equal to the stress applied to the material (σ) multiplied by the cosine of the angle with the vector between the glide plane normal and the applied stress direction ($$\phi$$) and the cosine of the angle between the glide direction and the applied stress direction ($$\lambda$$).

Consider a single crystal material under an applied tensile force $$F$$. The cross-sectional area of the crystal is $$A_0$$, and the cross-sectional area of an arbitrary slip plane in the material is $$A_s$$. These areas are related by $$A_s=\frac{A_0}{\cos\phi}$$. Moreover, the resolved force along the slip direction is given by $$F_s=F\cos\lambda$$. Recall also that $$\sigma=\frac{F}{A_0}$$. The resolved shear stress is then


 * $$ \tau_{RSS} = \frac{F_s}{A_s} = \frac{F}{A_0}\cos\phi\cos\lambda $$
 * $$ \tau_{RSS} = m \sigma $$,

where $$ m =\cos\phi\cos\lambda $$ is known as the Schmid factor.

Each slip system will have a different value of $$\tau_{RSS}$$, but the one with the maximum value of $$m$$ will be the one in which plastic deformation begins. This critical value is known as the critical resolved shear stress and is a constant material parameter.

It is worth noting that the above derivation is for a single crystal material. In the case of a polycrystalline material, an average Schmid factor is employed due to the random orientation of grains.

Limitations
First noticed by Taylor, body-centered cubic transition metals show notable deviations from Schmid's law, due to what is overarchingly termed "anomolous" slip. More specifically, the non-planar character of screw dislocation cores in these materials gives rise to twinning/anti-twinning (TAT) and tension/compression (TC) asymmetries of the yield and flow stresses, which allow for more complicated slip mechanisms. Thus, Schmid's law does not apply well to these BCC materials because it assumes {110} slip. Attempts have been made to modify Schmid's law to account for the fact that slip only occurs in these directions on average.