Hall-Petch relationship

Relaxation of the Chemical Bond, Springer Series in Chemical Physics 108, DOI 10.1007/978-981-4585-21-7 28, Springer Science+Business Media Singj mte 2014 

The mechanical strengthening with grain refinement in the size range of 100 nm or larger (region 1) has traditionally been rationalized with the so-called T-independent HPR that can be simplified in a dimensionless form being normalized by the bulk strength, r(oo), measured at the same temperature and under the same conditions  

The slope A or A is an adjustable parameter for fitting to measurements, which represents both the intrinsic properties and the extrinsic artifacts such as defects, the pileup of dislocations, shapes of indentation tips, strain rates, load scales, and directions in the test. The and b correspond, respectively, to the shear modulus and the Burger s vector modulus reduced by atomic size, d. The bulk modulus B is related to the shear modulus and the Poisson ratio vhy — B/[2(l -I- v)]. Using the dimensionless form of the normalized, yield strength aims to minimizing the contribution from artifacts due to processing conditions, crystal orientations, and the purity of the specimens if the measurement is conducted under the identical conditions throughout the course of the experiment. For convenience, we use both X = K and K as indicators of the dimensionless form of sizes. 

There has been a concerted global effort underway using a combination of novel processing routes, the state-of-the-art experimental measurements, and large-scale computations to develop deeper insight into the IHPR phenomena. For example, by squeezing Si nanospheres of different sizes between a diamond-tipped probe and the sapphire surface, Gerberich et al. [18] determined at room 

---

Hall-Petch relationship, 13 497 Haloacetic acids (HAAs), removal from drinking water, 17 806—807 Haloacetones, 1 163 Haloalkylating agents, 12 167 Haloalkylation, 12 166-168 Haloalkylbenzenes, intramolecular alkylation of, 12 169 IV-Halo-a-amino acids, 13 107 Haloaromatics, 13 573 Halobenzenes, as Friedel-Crafts arylating agents, 12 171 Halobismuthines, 4 28-29 Haloboration, 14 270 Halobrom (BCDMH,... 

Since the grain boundaries essentially act as barriers to slip in adjacent crystals (see Figure 5.14), it makes sense that the yield strength should depend on grain size. This is indeed the case, and the Hall-Petch relationship shows an inverse square-root dependence of yield strength on grain size, d ... 

Hardness and a ductile-to-brittle transition temperature (DBTT) have also been noted to follow a Hall-Petch relationship (Meyers, and Chalwa, 1984). Ductility increases as the grain size decreases. Decreasing grain size tends to improve fatigue resistance but increases creep rate. Electrical resistivity increases as grain size decreases, as the mean free path for electron motion decreases. 

From the Hall-Petch relationship, we know that (1/V[Pg.124]

The hardness of polycrystalline tungsten increases appreciably as the grain size is reduced. It was shown that hardness and average grain size follow a Hall-Petch relationship ... 

It is well known that dramatic strengthening can be produced by reducing the relevant length scales for deformation in metallic alloys. Three length scales are shown in Figure 5. Reduction of the grain diameter (D) produces the well-known Hall-Petch relationship, while reducing the diameter (S) of... 

The decrease of the grain size d strengthens metallic alloys (Hall-Petch relationship, third term)... 

The hardness of WC has also been measured on polycrystalline samples [8]. Figure 3 reports results from several investigators [9] which satisfy the following Hall Petch relationship ... 

Many measurements in polycrystalline Cu, Au and Al wires have shown that the yield stress Oy varies as shown in Fig. 21 with the reciprocal square root of the grain size. This relation between yield stress and grain size d, usually referred to as the Hall-Petch relationship, expresses the strengthening effect of the grain boundaries. 

The relationship in equation (4.1) has a familiar form, being the same as the Hall-Petch relationship for fracture stress or yield stress-grain size dependence. Through this connection the dominance of flow mechanisms in establishing hardness values is once again emphasized. 

Half penny crack, 156, l 9, 260 Hall-Petch relationship, 120 in MgO, 265 Hardening constant, 141 Hardening in glass, 194 Hardness absolute, 72 anisotropy, 65, 262, 293 and applied load, 7, 119-124, 251, 255, 281... 

Recalling the weU-known Hall-Petch relationship [13, 14] between yield strength Oy and grain d dimension... 

N. Ono, R. Nowak, S. Miura, Effect of deformation temperature on Hall-Petch relationship registered for polycrystalline magnesium. Mater. Lett. 58(1-2), 39-43 (2004)... 

X.J. Liu, L.W. Yang, Z.E. Zhou, P.K Chu, C.Q. Sun, Inverse Hall-Petch relationship of nanostructured Ti02 Skin-depth energy pinning versus surface preferential melting. J. Appl. Phys. 108, 073503 (2010)... 

M. Zhao, J.C. Li, Q. Jiang, Hall-Petch relationship in nanometer size range. J. Alloy. Compd. 361(1-2), 160-164 (2003)... 

Y. Giga, Y. Kimoto, Demonstration of an inverse Hall-Petch relationship in electrodeposited nanocrystalline Ni-W alloys through tensile testing. Scripta Mater. 55(2), 143-146 (2006)... 

J.A. Knapp, D.M. Follstaedt, Hall-Petch relationship in pulsed-laser deposited nickel films. 

T.G. Nieh, J.G. Wang, Hall-Petch relationship in nanocrystalline Ni and Be-B alloys. 

---