Plasticity, A Mechanics of the Plastic State of Matter

Flows of dilatant fluids

“…(3.8) and the condition that Ui be continuous on r determines Ui uniquely: Eq. (2.12), (3.1), (3.3) and the assumed continuity of stress imply that (3.4) and (3.5) hold with n replaced by a'n where a' = 1 + a/(2n) satisfies the equation Equations (3.6) through (3.9) are identical to the system of equations which occur in the study of the torsion of a cylinder with simply connected cross-section which is made of an elastic, perfectly plastic solid satisfying the von Mises yield condition [2].…”

“…( [1][2][3][4][5] In generalising the mechanical behavior described by (1.1) and (1.2) we will assume that with every dilatant fluid there can be associated a continuous function <p(epq) of the components of the velocity strain tensor satisfying *(0) < 0; (1.6) only flows for which^( eP0) < 0, (1.7) are admissible for the model. The function <p will be called the "dilatancy function"; it may, or may not, be analytic in the components e"q .…”

“…2, for a dilatant fluid in simple shear can be alternatively interpreted as the relation between axial strain (s) and axial stress (e) for an elastic, perfectly plastic solid in simple tension [2], In this section we will develop a model of a dilatant fluid, for a general state of stress, in much the same way as the elastic, perfectly plastic model of a plastic solid is developed for a general state of strain [2].…”

“…The mathematical difficulties and the analogy are fully discussed in Ref. [2], The only problem that has been solved directly is that which corresponds to the flow in a circular pipe. The most rewarding technique used to date in the torsion problem is the semiinverse method of Sokolovsky [5].…”

Spacetime Interpretation of Torsion in Prismatic Bodies

“…According to Nádai's sandpile analogy [2,6,7] one may regard a Prandtl potential satisfying the Eikonal equation (11) as the height function φ of a sand-heap or sand-pile of constant angle of repose α = tan −1 k located in same Euclidean space E 3 . The strategy for solving the for φ adopted by Prandtl is to erect over the base B, a sand-pile, that is the solution φ Eikonal of the Eikonal equation (11).…”