Elasticity Versus Hardness

Theoretically, the dimensionality for elasticity, stress and hardness is identical in Pascal unit (energy density) but at different states. The ideal form represents the intrinsic property change without experimental artifacts being involved. However, the softening and the slope transition in the IHPR plastic deformation arises from the extrinsic competition between activation of and resistance to glide dislocations, which is absent in the elastic deformation in particular using the non-contact measurement such as SWA techniques and Raman measurements. 

A measurement of the size dependence of the hardness, shear stress, and elastic modulus of copper nanoparticles, as shown in Fig. 28.8a, confirmed this expectation. The shear stress and elastic modulus of Cu reduce monotonically with the solid size but the hardness shows the strong IHPR. Therefore, the extrinsic factors become dominant in the plastic deformation of nanocrystals, which triggers the HPR and IHPR being actually a response to the contacting detection. However, as compared in Fig. 28.8b and c, the hardness and Young s modulus of Ni films are linearly interdependent. This observation indicates that extrinsic factors dominating the IHPR of nanograins contribute little to the nanoindentation test of film materials. 

By definition, the Young s modulus is for the regime of elastic deformation, while its inverse, or the extensibility/compressibility, covers both elastic and plastic types of deformation. However, either the elastic or the plastic process is 

However, for plastic deformation, the competition between dislocation activation and dislocation resistance becomes dominant, which presents a difference in the plastic deformation from the elastic response in terms of the IHPR features. In the nanoindentation test, errors may arise because of the shapes and sizes of the tips. The stress-strain profiles of a nanosolid are not symmetrical when comparing the situation under tension to that under compression [112]. The flow stress is dependent of strain rate, loading mode and duration, and material compactness, as well as size distribution. These factors may influence measured data that are seen 

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Figure 2.7 Plastic flow stresses of from Brinell spherical indentation-hardnesses versus elastic shear moduli. Nominally pure fee metals at 200K (Gilman, 1960).

The structures of the prototype borides, carbides, and nitrides yield high values for the valence electron densities of these compounds. This accounts for their high elastic stiffnesses, and hardnesses. As a first approximation, they may be considered to be metals with extra valence electrons (from the metalloids) that increase their average valence electron densities. The evidence for this is that their bulk modili fall on the same correlation line (B versus VED) as the simple metals. This correlation line is given in Gilman (2003). 

Fig. 5 Variations of the interaction energy between particles versus the center-to-center distance, (a) Hertzian potentials for particles with bulk elasticity the dashed line represents the usual Hertz potential (1), and the solid line the generalized Hertzian potential (2). (b) Potentials for emulsions with surface elasticity the dashed line denotes the approximate solution for small compression ratios (3), and the solid line the general solution (4). (c) Ultrasoft potentials for star polymers (where a 1.3Rg, with Rq being the radius of gyration of the star, following [123]) (5) dashed line f = 256 solid line f = 128 dashed and dotted line f = 64. (d) Hard-sphere potential...

The theoretical curve defined by the reciprocal of Eq. (4-14) is calculated as follows. is obtained from a plot of In/oo versus 1/T the high-pressure rate constant 4 0 is measured directly or obtained from extrapolation of the plot of l//exp versus 1/P the effective number of oscillators n/2 is obtained by locating the pressure at which begins to fall off it is assumed that at this pressure the rate of activation is equal to the first-order rate of reaction, that is, ac W = > exp 9 relation which will yield a value of n/2 after insertion of the experimental value of and a reasonable value for the elastic-hard-sphere diameter d. 

FIGURE 4.25 Force versus displacement curves on PNiPAAm gel in pure water at 10 and 35°C and on mica in pure water at room temperature. The same z-piezo displacement results in a smaller cantilever deflection on the soft gel surface in comparison with the hard mica sample because of elastic indentation. Source Matzelle et al. [55]. Reproduced with permission of American Chemical Society. 

Fig. 50. Hardness versus elastic modulus for several fee metals, essentially covalent crystals, and ionic materials (after Gilman and Chin ).

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