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The yield stress of ductile metals is observed to increase with decreasing grain size (Figure 4.14), according to the well‐known Hall–Petch equation, given by

(4.35)

where, σ_y is the yield stress, σ_o is the yield stress for the easiest slip system in a single crystal, and G is the grain size of the polycrystalline metal. Thus, reduction of grain size provides a powerful method to enhance the yield stress of metals (Chapter 6). Equation (4.35) can be derived theoretically by assuming that dislocations pile up at the grain boundaries that provide obstacles to dislocation motion. As the grain size decreases, the number of boundaries per unit volume increases, providing a larger number of obstacles.

Figure 4.14 The influence of grain size on the yield strength of a 70Cu–30Zn brass alloy.

Source: From Callister (2007) / with permission of John Wiley & Sons.

The effect of grain size on the strength of brittle ceramics is more complicated than that for ductile metals. This is because the strength of brittle materials is strongly influenced by the presence of flaws such as microcracks, particularly those at their surface (Section 4.4.2). For ceramics composed of grains of size smaller than ~10–20 μm, the flaw size is often larger than the grain size and, consequently, the strength of these ceramics should be independent of the grain size. However, data for ceramics often do show an increase in average strength with decreasing grain size below 10–20 μm. Although there is a wide scatter in the data due to various degrees of surface finish of the specimens prior to testing, the average flexural strength of alumina, the most widely studied ceramic, shows an increase from ~400 MPa to ~600 MPa with a decrease in grain size from 10 to 1 μm (Wachtman et al. 2009).