## Q O

Figure 6.7. An edge dislocation in a two-dimensional rectangular lattice.

Figure 6.7. An edge dislocation in a two-dimensional rectangular lattice.

along the dislocation are weaker. One method of increasing the stress at which the brittle-to-ductile transition occurs is to impede die movement of the dislocation« by introducing tiny particles of another material into the lattice. This process is used to harden steel, where particles of iron carbide are precipitated into the steel. The iron carbide particles block die movement of die dislocations.

### 6.1.3. Mechanical Properties

The intrinsic elastic modulus of a nanostructured material is essentially the same as that of the bulk material having micrometer-sized grains until the grain size becomes very small, less than 5nm. As we saw in Chapter 5, Young's modulus is die factor relating stress and strain. It is the slope of the stress-strain curve in the linear region. The larger the value of Young's modulus, the less elastic the material. Figure 6.8 is a plot of the ratio of Young's modulus E in nanograined iron, to its value in conventional grain-sized iron E0, as a function of grain size. We see from the figure that below ~20nm, Young's modulus begins to decrease from its value in conventional grain-sized materials.

The yield strength ay of a conventional grain-sized material is related to the grain size by the Hall-Petch equation ay = a„ +Kd~(l,2) (6.1)

GRAIN SIZE (nm)

Figure 6.8. Plot of the ratio of Young's modulus E in nanogram iron to its value Eq in conventional granular iron as a function of grain size.

GRAIN SIZE (nm)

Figure 6.8. Plot of the ratio of Young's modulus E in nanogram iron to its value Eq in conventional granular iron as a function of grain size.

where a0 is die fiictional stress opposing dislocation movement, K is a constant, and is the grain size in micrometers. Hardness can also be described by a similar equation. Figure 6.9 plots die measured yield strength of Fe-Co alloys as a function of <r(1/2), showing the linear behavior predicted by Eq. (6.1). Assuming that the equation is valid for nanosized grains, a bulk material having a 50-run grain size would have a yield strength of 4.14 GPa. The reason for the increase in yield strength with smaller grain size is that materials having smaller grains have more grain boundaries, blocking dislocation movement Deviations from die HalT-Petch behavior have been observed for materials made of particles less than 20 nm in size. The deviations involve no dependence on particle size (zero slope) to decreases in yield strength with particle size (negative slope). It is believed that conventional dislocation-based deformation is not possible in bulk nanostructured materials with sizes less than 30 nm because mobile dislocations are unlikely to occur. Examination of small-grained bulk nanomaterials by transmission electron microscopy during deformation does not show any evidence for mobile dislocations.

Most bulk nanostructured materials are quite brittle and display reduced ductility under tension, typically having elongations of a few percent for grain sizes less than 30 nm. For example, conventional coarse-grained annealed polycrystalline copper is very ductile, having elongations of up to 60%. Measurements in samples with grain sizes less than 30 nm yield elongations no more than 5%. Most of these measurements have been performed on consolidated particulate samples, which have large residual stress, and flaws due to imperfect particle bonding, which restricts disloca x

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