Indent size equation

However, not all work shows the simple relationship of equation (4.1), but when examined more closely, those investigations that do not follow equation (4.1) usually involve tests where the indent size and the grain size are comparable, and grain boundaries may then be important. Since the ISE exists in correctly chosen grain-size-indent-size regimes, we can accept that hardness increases as the load decreases, which means that from equations (1.6) and (1.7) there will be more complex relationships among applied load, hardness, and grain size. 

It is the value of n in equation (4.2) that is used to express a measure of the ISE of a material. Equation (1.6) gives the relationship of Vickers hardness to load, and for there to be no dependence of hardness on indentation size, n in equation (4.2) must equal 2. When hardness changes as the half diagonal a decreases, then clearly the relationship between P and a is not simple and n will not equal 2. Deviations from n = 2.0 are then a measure of the ISE effect and for a given ceramic, variations in n will be a measure of microstructural variations, surface layers, surface chemistry, or mechanochemical effects. 

Clearly all indents made with a pyramidal indenter should have the same shape regardless of their size. Thus, since we take pressure used to make this shape to be a measure of hardness—see equations (1.6) and (1.7)—we would expect hardness to be the same and there to be no load effect. Therefore when hardness increases as the applied load decreases, as shown in Figure 1.3, it must be because the volume of material used to yield is smaller and the mechanism for yielding is dependent on a volume term which becomes more significant as the indent size decreases. The most obvious development of this idea is that the shallow near-surface volume of the deformation zone can become a significant fraction of the total affected volume when a very small load is used to make the indent. Thus, work hardened layers, surface compressed layers, ion-implanted layers, and the possibility of chemical reactions between the atmosphere and the surface can dominate the yielding mechanism to produce nonstandard hardness values. Conversely we can say that these phenomena could be studied by measuring the ISE of a ceramic. 

Naturally, following from the success of this original work, others have attempted to rationalize the approach and make it less empirical. Lawn and co-workers have provided justification for the approach and in the process found the H/E exponent in equation (5.37) by describing an approximate function relating plastic zone size to indent size. This approach leads to a dimensionless, independent constant that has to be calibrated by means of experimental data for chosen ceramics systems. In general, glass is chosen for this purpose. 

Fracture toughness is usually measured by inducing a large flaw with a diamond indenter or sawed notch, then loading the specimen to failure. Upon measuring the flaw size and load, one can use equation (c) to calculate Kic. 

The development of indentation fracture mechanics has also allowed fracture toughness to be determined using indentation cracks. Indeed, the ease with which these cracks can be introduced and the simple specimen preparation involved has popularized this approach. Moreover, it has allowed crack behavior to be studied for cracks in a size range that is close to that found in practice. There are two main approaches for determining fracture toughness from indentation cracks. In the first approach, the size of the radial cracks that emanate from the hardness impression are measured. It is recognized that the parameter x depends on the elastic properties of the indented material. It has been proposed that x=P E H), where )8 is a constant that depends only on the indenter geometry and H is hardness. Equation (8.64) can, therefore, be written as... 

Vickers indentations were made on the polished surface with loads of49,98,196,294 and 490 N to vary the crack size over a broad range. The lengths of the impression diagonals, 2o, and sizes of surface cracks, 2c, were measured with a traveling microscope immediately after the unloading. The fracture resistance, was determined from the Miyoshi s equation as follows ... 

In equation (4.4) d,- is the mean of the indent half diagonal sizes. From equation (4.3) we can write... 

Also referred to as median vent cracks, these are caused to pop-in by exceeding a critical indenter load. It is the pop-in phenomenon that is important to the development of this subject in ceramic science because the halfpenny crack has the surface trace which allows opaque materials to be analyzed by recording the radial crack size as a function of increasing load. There is, however, the implication that the surface must be prepared carefully by polishing to an optical finish in order to see the radial cracks. If necessary, samples must be annealed to remove polishing stresses. Radial cracks are the result of surface tensile stresses, (Xyy in equation (1.29). Such stresses are at a maximum at the elastic-plastic boundary. 

A recent use of equation (5.73) to determine K,c values as a function of grain size for alumina compares results to notched beam tests. The broken arms of the beams were used for the indentation-strength in bending test (ISB). Thus a good comparative study was possible. Figure 5.16 shows the close correspondence between the ISB results and the earlier results of Rice et using a double cantilever beam method. 

There will be a threshold load that will produce a cracked indent such that we can imagine the situation where this is reached and the crack is just equal in size to the diagonal of the impression 2a. Thus we can rewrite equation (5.88) using a, as the threshold indent diagonal size... 

Thus the limiting-sized indent that can be generated without producing the flawed specimen is, from equation (5.89), equal to... 

Equation (5.67) is telling us that the ceramic strength is controlled by the indentation flaw c, but two points must be remembered. First, the size of the radial c may increase due to stress corrosion to a value c before the sample is loaded in tension to breaking. And, second, we must consider the residual crack driving force, arising from the permanently deformed... 

Flaws of the size estimated from equation (5.82) are the median, radial, and lateral cracks caused by indentation, and the questions are where and why do they nucleate in a noncrystalline material Examination of the deformed zone beneath an indent or an impact shows that a series of intersecting flow lines is produced. Plastic strain is concentrated on the flow lines while the material between them is only strained elastically. Median cracks arise from the need to accommodate strains at the intersections of flow lines in a way analogous to crack nucleation from dislocations on intersecting slip planes in fully crystalline materials. 

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