Indentation stress distribution

For a one-material case, analytic solutions exist for both the deformation profile of the elastic material as well as the pressure and stress distributions for the indenter (approximating wafer features). Consider a single-layer pad that is thick relative to the vertical deformation and has a deformation force applied over a circular region of radius a. The deformation is given by a set of two equations that represent deformations within and outside the circular radius over which the force is applied. The deformation at any radius r less than a is given by [59] ... 

Figure 2.1 illustrates the stress distribution on an amorphous PET sample at an indentation depth, h = 2 mi (Rikards et al, 1998). It can be seen that the depth of the plastic zone shown is here about five times the penetration distance of the Vickers indenter. 

Figure 2.1. Stress distribution, in MPa, for amorphous PET at an indentation depth h = 2 um (c is the depth of the plastic zone and a is one-half the projected length of the indentation diagonal). Stresses larger than 78 MPa are elastic. (After Rikards et ah, 1998.)...

Crack propagation by the Vickers indentation was observed to prove the presence of residual stresses. When the surface of SiC layer was indented, the crack propagation was serious near disk center. This means that tensile stress expanded the cracks on SiC surface. Figure 4 shows the crack propagation on the section of the ceramic part. It was found that the cracks parallel to interface were longer than those perpendicular to interface. This means that tensile stress was large in the perpendicular direction to expand the cracks. These results are well consistent with the expectation from the calculation of stress distributions. 

These results can be interpreted in terms of the subsurface stress distribution beneath the indenter [30]. For instance, the vertical extent of the stress field be-... 

Stress distribution in indentation is largely affected by the indenter tip geometry, which is a vital factor in determining the boundary conditions for the field. The major types of indenter tips shown schematically in Figure 3 may be separated into two groups, viz. point-force (pyramidal and conical) and spherical indenters. Correspondingly, Boussinesq and Hertzian stress fields will describe point-force and spherical indentation in the case of purely elastic loading (Fig. 4). To account for possible elastic compliance of the indenter, a reduced elastic modulus Er is... 

Solutions for simplified cases in which it is assumed that plane sections of the coating remain plane after indentation-induced compression, can be found in the book by Johnson [75]. Johnson s approach was subsequently extended by Jaffar [76]. An early work [24] assumed frictionless contact and used elastic continuum mechanics to estimate the stress distribution in layered materials. In this analysis, force and displacement continuity at the interfaces was satisfied exactly, while the surface displacement condition (profile of the indenter), was only approximated. These authors analyzed indentation stresses created by circular flat-ended and parabolic indenters numerically on both single and multilayer systems, with good results. 

A similar, but further simplified approach is that of Matthewson [34], The assumptions are that the indenter is rigid, and that the coating is thin enough so that the stresses in the coating may be adequately described by the average of each stress through the thickness. This leads to an over determined problem in which not all of the continuity conditions can be met, but can be satisfied approximately. The approach, however, matches experimental observations quite well, and can be used to obtain estimates of the stress distribution due to contact of axisynunetric profiles. 

Because of its thickness, the specimen bends upon application of the indentation force, resulting in radial compression at the top and tension at the bottom. The resulting stress distribution is calculated from the elastic analysis of the theory of plates according to ... 

The relations for contact area, indentation, and vertical stress distribution were found... 

This property may indicate the rubber like nature of the matrix the indenter stress is probably distributed to other parts of the shell and stored as elastic energy which is released when the indenter is raised. ... 

The values and spatial distribution of the residual stresses around indentations can be assessed by means of the Raman imaging technique. In general, mechanical strain may shift the frequencies of the Raman modes and lift their degeneracy. 

Outside the contact area, the tensile stresses fall off as r where r is the radial distance from the center of the contact area. Inside the contact area, the stresses are compressive and form a hemispherical distribution. It is also found that the region below the indenter is compressive to a depth 2a, in contrast to the tensile stress field found for a point indentation. At larger distances, the stress field (necessarily) converges to that for the point-force problem. From Eq. (8.90), it is clear, even for elastic contacts, that the load eflection behavior is non-linear. This is a reflection of the growth in the contact area as the load increases. If the contact area can be measured, it is sometimes useful to rearrange Eq. (8.88) into the (linear) form... 

Figure 10. Distribution of normalized maximum principal stress in the vicinity of indenter tip.

---