A statistical theory of flow stress and work-hardening

Abstract The consequences of a random distribution of point-like obstacles to the movement of dislocations are examined. Firstly, it is proven that a critical applied stress exists; it is determined by a critical value of the fraction of passable obstacles such that the average dislocation segment can keep moving. There is thus no limit to the expandability of a convex dislocation loop; however, the moving dislocation leaves debris behind in the form of concave loops around areas surrounded entirely by impassable obstacles. The rate of this primary dislocation storage with strain, if one considers monopoles only, is very low and again depends on the critical fraction of passable obstacles only, not on any properties of the distribution as long as it is non-singular. The absence of large-scale regularity in the obstacle structure alone (whatever the nature of the obstacles may be) can thus explain “stage II” hardening which is characterized by a value of about 1/300 of the shear modulus, independent of all conceivable parameters within wide limits. The structure of the work-hardened crystal derived in this semi-phenomenological way consists of small concave loops piled up around each other and presumably arranged in strands perpendicular to the slip plane. It offers itself for appealing mechanistic interpretations of more complicated phenomena such as latent hardening, dynamic recovery, and some metallographic observations.