Hall-Petch relation

As was the case for metals, poly crystalline ceramics show a dependence of mechanical properties on grain size, as given earlier by the Hall-Petch relation in Eq. (5.22), which in this case applies to fracture strength, ay ... 

As a reminder of the way interfaces, as key microstructural elements, can impact material response, we note one of the celebrated relations between a material s microstructure and its physical response, namely, the Hall-Petch relation which holds that the yield strength of a material scales as... 

VHN. Hardness through the weld region varied between approximately 240 and 280 VHN, with an average hardness of approximately 260 VHN. A slight increase in hardness in the stir zone may be inferred from the results. The increased hardness in the stir zone may have stemmed from the locally refined grain size through the Hall-Petch relation. 

The Hall-Petch relation (Eq. 14.8) indicates the effect of grain size, d, on the stress required to make the dislocation move in a polycrystalline sample. The origin of the relation is that the stress to operate a Frank-Read source increases as the size of the source decreases. If the grain size decreases, then the maximum size of the Frank-Read source also decreases. The result is the famous d relationship. 

The Hall-Petch relation relates the flow stress to the grain size ... 

Similar results are obtained as shown in Fig. 20 for high-purity Cu wires (99.999%) of various diameters. The variation of their strength with annealing results from the polycrystalline strengthening which, as Fig. 21 shows, follows the well known l/Vd dependence of the Hall-Petch relation d = grain size). 

Fig. 2.17 a Bending strength as a function of average grain size b shows agreement between the experimental data and the Hall-Petch relation [9]. With kind permission of Mr. Rothman for the... 

C.S. Pande, B.B. Rath, M.A. Imam, Effect of annealing twins on Hall-Petch relation in polycrystalline materials. Mater. Sci. Eng. A-Struct Mater. Prop. MicrostrucL Process. 367(1-2), 171-175 (2004)... 

G.J. Fan, H. Choo, P.K. Liaw, E.J. Lavernia, A model for the inverse Hall-Petch relation of nanocrystalline materials. Mater. Sci. Eng. 409(1-2), 243-248 (2005). Structural Materials Properties Microstructure and Processing... 

H.W. Song, S.R. Guo, Z.Q. Hu, A coherent polycrystal model for the inverse Hall-Petch relation in nanocrystalline materials. Nanostract. Mater. 11(2), 203-210 (1999)... 

V. Bata, E.V. Pereloma, An alternative physical explanation of the Hall-Petch relation. Acta Mater. 52(3), 657-665 (2004)... 

Equation 6.25 has the form of the Hall-Petch relation. Here, d represents the grain size. It says that the fracture stress should increase as the grain size decreases. In the case of briffle materials, if the initial flaw size is limifed by the grain size, the relation between the fracture strength and grain size is given by the Orowan relation. This relation is expressed by Equation 6.27. 

Another factor affecting the strength is the grain size. The Hall-Petch relation holds well in this case. The strength is inversely proportional to the square root of the grain size. 

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