Superposition of grain boundary and solid solution strengthening in copper-germanium alloys

Abstract

The Hall-Petch equation relates the yield strength {sigma}{sub y} of polycrystals to their grain size d: {sigma}{sub y} = {sigma}{sub o} + k d{sup {minus}1/2}, k is often referred to as the Hall-Petch slope. In a plot {sigma}{sub y} versus d{sup {minus}1/2}, the ordinate intercept equals {sigma}{sub y}. Apparently two contributions to {sigma}{sub y} are distinguished: (i) {sigma}{sub o} represents those mechanisms which govern the critical resolved shear stress (CRSS) {tau}{sub c} of single crystals of the same material; (ii) kd{sup {minus}1/2} stands for the hardening effects of the grain boundaries: they limit the slip lengths of dislocations. There are many instances where it is possible to break down {sigma}{sub y} into the two terms {sigma}{sub o} and kd{sup {minus}1/2}, but there are also many systems in which k varies with the intrinsic properties of the material, e.g., with its solute content or with its dispersion of second phase particles. Copper-germanium alloys have been evaluated as to strengthening by grain boundaries and solid solution.