Tabors relation

The question whether hardness is a property related to modulus (E) or yield stress (Y) is a problem which has been commented before by Bowman and Bevis 13). These authors found an experimental relationship between microhardness and modu-lus/yield-stress for injection-moulded semicrystalline plastics. According to the clasical theory of plasticity the expected indentation hardness value for a Vickers indenter is approximaterly equal to three times the yield stress (Tabor s relation). This assump-... 

Fig. 16. Correlation between microhardness at 0.1 min and true yield stress of PE M = 170000 (O) Mw = 2x 106 (A). The solid line is drawn according to Tabor s relation 28 )...

This analysis is consistent with the conclusion of Gerk (1977) that the behavior that determines hardness is deformation-hardening not the yield stress. He was one of the first authors to point this out. For other types of materials, it is the maximum stress that the material can bear after deformation (plastic, or that associated with phase transitions in eluding twinning). Hardness is not directly related to the elastic limit, although there is an indirect connection with the offset plastic deformation of metals as demonstrated by Tabor (1951). 

In textbooks, plastic deformation is often described as a two-dimensional process. However, it is intrinsically three-dimensional, and cannot be adequately described in terms of two-dimensions. Hardness indentation is a case in point. For many years this process was described in terms of two-dimensional slip-line fields (Tabor, 1951). This approach, developed by Hill (1950) and others, indicated that the hardness number should be about three times the yield stress. Various shortcomings of this theory were discussed by Shaw (1973). He showed that the experimental flow pattern under a spherical indenter bears little resemblance to the prediction of slip-line theory. He attributes this discrepancy to the neglect of elastic strains in slip-line theory. However, the cause of the discrepancy has a different source as will be discussed here. Slip-lines arise from deformation-softening which is related to the principal mechanism of dislocation multiplication a three-dimensional process. The plastic zone determined by Shaw, and his colleagues is determined by strain-hardening. This is a good example of the confusion that results from inadequate understanding of the physics of a process such as plasticity. 

As we discussed in Section II in relation to (2.41), a survival amplitude has a semiclassical behavior that is directly related to the periodic orbits by the Gutzwiller or the Berry-Tabor trace formulas, in contrast to the quasi-classical quantities (2.42) or (3.3). Therefore, we may expect the function (3.7) to present peaks on the intermediate time scale that are related to the classical periodic orbits. For such peaks to be located at the periodic orbits periods, we have to assume that die level density is well approximated as a sum over periodic orbits whose periods Tp = 3eSp and amplitudes vary slowly over the energy window [ - e, E + e]. A further assumption is that the energy window contains a sufficient number of energy levels. At short times, the semiclassical theory allows us to obtain... 

The experimental vibrogram shows an important recurrence around 160 fs, which may be assigned to the edge periodic orbit (3,2°, -)n0rmai- Recently, the vibrogram analysis has been carried out by Michaille et al. [113] on the basis of another model proposed by Joyeux [118] as well as on an ab initio potential fitted to the experimental data of Pique [119]. Essentially the same classical periodic orbits appear in the different models at low energies. In the same context, let us add that Joyeux has recently applied the Berry-Tabor trace formula to a IF Fermi-resonance Hamiltonian model of CS2 [120] and carried out a classical analysis of several related resonance Hamiltonians [121]. 

Urban and suburban air concentrations have been found to range from less than 0.005 to 1.5 mg/m3 (Tabor and Warren 1958). No distinct pattern related to industrialization appeared in the results reported on 754 samples from 18 cities and four suburban areas in the United States. For example, in Houston, Texas and its suburbs, 76% of the samples contained barium at levels ranging from 0.005 to 1.5 mg/m, whereas in Fort Worth, Texas, 66% of the samples had values below 0.005 mg/m (Tabor and Warren 1958). 

Interactions between crossed cylinders of mica in air, uncoated or coated with fatty acid monolayers, are described in J. N. Israelachvili and D. Tabor, "The measurement of van der Waals dispersion forces in the range 1.5 to 130 nm," Proc. R. Soc. London Ser. A, 331, 19-38 (1972). An excellent review of this and related work is given in J. N. Israelachvili and D. Tabor, Van der Waals Forces Theory and Experiment, Vol. 7 of Progress in Surface and Membrane Science Series (Academic Press, New York and London, 1973). Later reconciliation of theory and experiment required taking note of cylinder radius L. R. White, J. N. Israelachvili, and B. W. Ninham, "Dispersion interaction of crossed mica cylinders A reanalysis of the Israelachvili-Tabor experiments," J. Chem. Soc. Faraday Trans. 1, 72, 2526-36 (1976). 

Tabor, H., 1954, Metabolic studies on histidine, histamine and related imidazoles, Pharmacol. Rev., 6 299iB43. 

Another motivation for measurement of the microhardness of materials is the correlation of microhardness with other mechanical properties. For example, the microhardness value for a pyramid indenter producing plastic flow is approximately three times the yield stress, i.e. // 3T (Tabor, 1951). This is the basic relation between indentation microhardness and bulk properties. It is, however, only applicable to an ideally plastic solid showing no elastic strains. The correlation between H and Y is given in Fig. 1.1 for linear polyethylene (PE) and poly(ethylene terephthalate) (PET) samples with different morphologies. The lower hardness values of 30-45 MPa obtained for melt-crystallized PE materials fall below the /// T cu 3 value, which may be related to a lower stiff-compliant ratio for these lamellar structures (BaM Calleja, 1985b). PE annealed at ca 130 °C... 

According to Tabor, the ratio between the indentation pressure Pm and Y for a Vickers diamond pyramid is about 3.3 H = 0.921 Pm) (Tabor, 1979). This relation applies for materials which behave as ideally plastic solids, but fails for materials in which the elastic strains are non-negligible (March, 1964 Hirst Howse, 1969). 

For the yield stress in compression, deviations from Tabor s relation giving values of 2Yc are found. This is presumably due to the elastic strain of the indented material. A detailed analysis of the H/Yc ratio on the basis of mechanical models of elastoplastic indentation reveals that H/Yc linearly increases with ln[(tan/3Ec)/Yc]. Compression-moulded (chain-folded) PE samples, which present the lowest crystallinity of all the samples investigated, also show the lowest H/Yc ratio as a consequence of the comparatively large elastic strains. 

Tabor H. and Tabor C.W. (1964) Spermidine, spermine and related amines. Pharmacol. Rev. 16, 245. 

Figure 8-3. Relation of the tangential traction coefficient to the area ratio A /A. 0 Clean platinum, t Lubricated platinum. Data computed by Tabor [5] from results by Courtney-Pratt and Eisner [10].

The adhesive transfer of organic plastics has some special features of it own. Makinson and Tabor [24] observed that polytetrafluoroethylene sliding on glass left transferred material on the counter surface in the form of lumps, ribbons, sheets or very thin films, depending on the rubbing conditions. Pooley and Tabor [25], who studied the transfer process more intensively, also reported the behavior of other polymers such as fluorocarbon copolymers, polyethylene, polypropylene, polystyrene, polymethylmethacrylate and polyvinyl chloride. Descriptions of transfer in relation to wear were reported for PTFE by Tanaka tt ai. [20] and for polyethylene by Miller a.1. [21]... 

The previous sections present many process types and variations, and a question naturally arises as to how one can choose the most economical process embodiment for a particular separation. The purpose of this section is to present a method by which a reasonable answer can be given to this question. Economics of the processes are chiefly a function of capital (investment-related) costs and energy costs, and to a lesser extent tabor and other costs. Since capital and energy costs can vary from site to site, hard-and-fast criteria cannot be given, but the following procedure will seldom mislead. 

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