Indenter geometries

Indentation has been used for over 100 years to determine hardness of materials [8J. For a given indenter geometry (e.g. spherical or pyramidal), hardness is determined by the ratio of the applied load to the projected area of contact, which was determined optically after indentation. For low loads and contacts with small dimensionality (e.g. when indenting thin films or composites), a new way to determine the contact size was needed. Depth-sensing nanoindentation [2] was developed to eliminate the need to visualize the indents, and resulted in the added capability of measuring properties like elastic modulus and creep. 

Equation (24) is originally derived for a conical indenter. Pharr et al. showed that Eq (24) holds equally well to any indenter, which can be described as a body of revolution of a smooth function [67]. Equation (24) also works well for many important indenter geometries, which cannot be described as bodies of revolution. 

Since the strain arising from an ideal sharp indenter cannot be wholly elastic (as is the case with a blunt indenter), a number of new specific features of failure of a particular material may arise, especially in the early stages of crack formation under the influence of surface penetration at low loads. It is reasonable to suppose at the same time that as the crack region extends widely below the contact zone, the influence of indenter geometry should become significant. 

For any indenter geometry it has been found that the relationship between stiffness S and elastic modulus Kmaybe defined as follows [133] ... 

From a more fundamental point of view, the selection of different inden-ter geometries and loading conditions offer the possibility of exploring the viscoelastic/viscoplastic response and brittle failure mechanisms over a wide range of strain and strain rates. The relationship between imposed contact strain and indenter geometry has been quite well established for normal indentation. In the case of a conical or pyramidal indenter, the mean contact strain is usually considered to depend on the contact slope, 0 (Fig. 2a). For metals, Tabor [32] has established that the mean strain is about 0.2 tanG, i.e. independent of the indentation depth. A similar relationship seems to hold for polymers although there is some indication that the proportionality could be lower than 0.2 for viscoelastic materials [33,34], In the case of a sphere, an... 

Figure 2.4 schematically depicts the indentation geometry for a Vickers indenter penetrating a cylindrical surface with a radius r. In the case of an ideal plastic deformation (i.e. when elastic stresses are absent) after load removal, the square pyramidal indenter leaves a rhombic indentation with one of its diagonals parallel to the filament axis. Let 2BC be the measured indentation length, f x, normal to the filament axis and 2DE the indentation length, which would arise on a flat surface for the same penetration depth. For an isotropic material, = C. However, as a result of the existing curvature, i > (anisometric indentation). From Fig. 2.4, DE = BC + BD tan a/2) and since tan(a/2) —1/2 one has = li + IBD.

Kopsch, H.. Zum Verhalten von polymermodifizierten Bitumen bei tiefen Temptcraturcn (On the condition of pol modified bitumen at low temperatures). Bitumen, I. 10 13 (1990). Brisco, B., Sebastian, K and Adams, M., The effect of indenter geometry on the elastic response to indentation, J. Phys. D. Applied Pliy.sics, 27. 1156 1162 (1994). 

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... 

The repeated measurement of the size of indents, and the interpretation of indent geometry for the purposes of calculation, may be tedious, and operator bias is almost unavoidable. The edge of the impression is not always well defined, and misleading edge effects may be associated with anisotropic plasticity or plastic recovery. Faceted and elongated grains, or other microstructural features, together with the limitations of contrast and resolution in the optical microscope, complicate the interpretation, while the shape of the indent may differ in different materials so-called pin-cushion or barreled indents, associated with different constitutive relations and frictional shear on the faces of the indentor in contact with the plastic zone [3]. Mismeasurement of indent size is a major source of scatter in the experimental data and the relative errors in the results of different operators. 

Parameter Indenter geometry/ material Test force/ indentation depth Definition Special requirements/aspects... 

Knowing the indenter geometry, the average pressure Pttp on the indenter surface during the scratching is simply given as... 

With these complications in mind, research in this area has blossomed rapidly. Two main foci of research in this area are on (1) how external conditions (such as levels of loadings, the use of different indenters, and scratch rate) and (2) intrinsic materials properties (such as modulus and crystallinity) affect the tribological behaviors of the polymers. Apart from examining the scratch resistance of polymers, a closely related quantity which is of interest would be changes in coefficient of friction. Studies relating mechanical properties (3-5,9,36,71,75,76), deformation patterns (18,33,63,71,77-81), fabrication process (3,5,35,72,77,82-86) with respect to experimental parameters, snch as temperature (18), loading effect (24,71,72,87-96), indenter geometry (21,33,75,82,95,97), and scratch velocity (21,56,57,59,64,65,96,98) have been carried ont. In addition, scratch maps for different polymers have been produced (32,33), and various scratch resistance properties estimated (33,37,56,58,59,99). 

For depth-sensing nanoindentation, a controlled, variable force is applied to a sample by the indenter and the resulting displacement of the indenter is measured. The resulting load vs. displacement data, together with the indenter geometry, can be analyzed to obtain hardness and elastic modulus using well established mechanical models (14). The simultaneous measurement of load and displacement also allows study of creep (time dependent strain response due to a step change in stress) (15,16). 

A typical load-displacement curve is shown in Fig. 2. The loading portion of the curve results from both plastic and elastic deformation response of the contact, while the unloading portion of the curve is related to the elastic recovery of the contact. If the indenter geometry and materials properties are known, the modulus can be obtained by fitting the unloading curve to determine the contact stiffness at maximum load (i4, 17). In this case,... 

It should be noted at this stage that variations in micro- and low-load hardness observed as functions of indenter geometry and crystal orientation are in fact only reflections of different distributions of shear stress within the bulk of the crystal. In this respect, too, the variations in chemical bond type found in ceramics must be important since ionic bonding proves less of a barrier to plastic flow than the strictly directed covalent bonds. 

In these equations is a function of mechanical properties of the material and the indenter geometry, E is Young s modulus for the material, V is its Poisson s ratio, and 6 is the indenter cone angle. 

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