Determination Rule for alpha, beta Directions and phi in Teaching of Slip-Line Theory

In the teaching of plastic mechanics and applications of slip-line theory using conventional methods, multivalued results are usually caused by the uncertain direction of the slip line and dip angles. Determination rules for the alpha and beta directions and phi values are proposed to improve slip-line theory according to the particle flow law under the effect of principal stress, and slip lines and dip angles suitable for a typical stress boundary problem are described. The alpha and beta slip lines should simultaneously point to or away from the intersection, and the synthetic direction of the slip lines should point to the first principal stress sigma(1) or away from the direction of the third principal stress sigma(3). When the Hencky stress equation of the alpha line is applied, two points on the alpha line should maintain the same direction, and the absolute value of the phi difference should be less than or equal to pi. Moreover, the alpha line of two points should simultaneously point to the inner and outer normal direction of the beta line when the Hencky stress equation of the beta line is used. The average stress and critical load of plastic deformation in the plane lath V-notch tension are solved using slip-line theory. Both the calculated critical stress and the load maintain uniformity using different slip lines and dip angles, and the proposed determination rule reliably avoids multivalued solutions. This is important for students and researchers in correctly understanding and applying slip-line theory.