Mechanics of a nanocrystalline coating and grain-size dependence of its plastic strength

Abstract

From the mechanics perspective the most critical properties of a coating material are likely its yield strength, hardness, and wear resistance. These properties are intimately related to each other as a material with high yield strength usually also possesses superior hardness and strong wear resistance. Our objective in this study is to seek for the strongest material state in terms of its plastic strength, and the critical grain size at which this strength can be attained. Based on the cross-sectional morphology of a nanocrystalline coating we first conceive a composite plate model consisting of the columnar grains and the grain boundary affiliated region (broadly called the GB zone), and present a plane-stress theory to investigate its in-plane behavior. We then make use of a linear comparison composite and a field fluctuation approach to explore the competition between the grain interior and the plastically softer GB zone as the grain size decreases from the coarse grain to the nanometer range. Based on the developed model the anisotropic stress–strain relations of a ZrN coating are illustrated as a function of grain size in both in-plane and out-of-plane directions. It is demonstrated that, as the grain size decreases, the variation of plastic strength in the Hall–Petch plot undergoes a transition from a positive to a negative slope, exhibiting the Hall–Petch and the inverse Hall–Petch effects. The strength-differential effect between tension and compression is also observed. The critical grain sizes at which the maximum strength develops are found to be lower along the out-of-plane direction than along the in-plane ones, and also lower under compression than under tension. Based on Tabor’s law, our calculated critical grain size for the hardness was about 16 nm, while a recent indentation test indicated a range of 14.2–19 nm. The existence of Hall–Petch, inverse Hall–Petch, and strength-differential effects, and of an optimal grain size, is thus theoretically proven for the first time for nanocrystalline coatings.