Stress-strain relations indentation

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. 

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... 

The question of whether microhardness is a property related to the elastic modulus E or the yield stress T is a problem which has been commented on by Bowman Bevis (1977). These authors found an experimental relationship between microhardness and modulus and/or yield stress for injection-moulded semicrystalline plastics. According to the classical theory of plasticity the expected microindentation hardness value for a Vickers indenter is approximately equal to three times the yield stress (Tabor s relation). This assumption is only valid for an ideally plastic solid showing sufficiently large deformation with no elastic strains. PE, as we have seen, can be considered to be a two-phase material. Therefore, one might anticipate a certain variation of the H/ T 3 ratio depending on the proportion of the compliant to the stiff phase. 

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. 

In Fig. 5.6 the indentation pressure is plotted versus a/r. The Hertzian relation between the indentation stress, po, and the indentation strain, a/r, is linear (see, for example, [2]) and given by ... 

At room temperature (below glass transition), non-Newtonian flow has attracted much attention in the past two decades thanks to instrumented indentation tools that allow for strain rate-controlled experiments (Appendix H). Hardness H is related to yield stress y, the dependence of which on strain rate may be written in the form (Han and Tomozawa, 1990 Keulen, 1993)... 

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