Stress-strain relations for crystals containing plastic deformation.

Let P be one of the possible slip planes in a crystal, and let X be one of the possible slip directions in the plane P. Also let S be the magnitude of the shearing stress on the plane P resolved in the direction X, and So the critical value of S, i.e., the least value of S sufficient to permit slip or plastic deformation under the conditions specified in the following italicized statement; if such slip is realized, it will be considered to occur over the plane P and in the direction X, as is commonly assumed. The ratio S/So will, of course, depend on the plane P as well as the selection of the direction X, and the values of the various ratios S/So will not be constant throughout the crystal except in the case of a uniform stress field. Denoting by max S/SO the greatest value of the possible ratios S/So at points of the crystal, we shall now make the following assumption. Slip or plastic deformation will occur in a region R of the crystal if max S/Se> 1 (1)