Indentation Strain Field

When the polymeric material is compressed the local deformation beneath the indenter will consist of a complex combination of effects. The specific mechanism prevailing will depend on the strain field depth round the indenter and on the morphology of the polymer. According to the various mechanisms of the plastic deformation for semicrystalline polymers 40 the following effects may be anticipated ... 

Cook suggests that water entry under the load imposed by the indenter is similar to water entry under the load imposed by the abrasive particle. As the abrasive particle moves across the surface, a strain field develops in the glass surface due to the load and velocity of the particle, with compressive strain in front of the particle and tensile strain behind the particle. In front of the particle, hydrostatic pressure leads to water entry into the oxide as the abrasive pushes water into the surface. However, diffusion of water into the oxide is inhibited by the compressive strain occurring in front of the particle. The diffusion coefficient of water decreases exponentially with compressive strain and increases exponentially with tensile strain. The difference between the traveling indenter (i.e., the abrasive particle) and the static indenter (i.e., the Knoop indenter) is the tensile strain that occurs in back of the traveling indenter leads to accelerated diffusion of water into the oxide. Thus, one of the functions of the abrasive particle is to pump water into the oxide surface. Water enters the oxide under the influence of the hydrostatic pressure in front of the particle and diffuses further into die oxide in back of the particle. The depth to which water diffuses into the oxide is a function of abrasive particle... 

Schall et al. indented colloidal crystals using a needle with an almost hemispherical tip of diameter 40 dm, inducing a strain field in which tine maximum shear strain lies well below the contact surface. The tip diameter, particle radius and crystal thickness in their experiments were chosen to be similar to parameters in typical metallic nano-indentation experiments. The authors discussed their observations using a model that addresses the role played by thermal fluctuations in the nucleation and growth of dislocations. 

From the above example it can be seen that a complex system needing careful analysis is present in each case, but the underlying fact is that the type of dislocation and their interactions are intimately concerned with the stress-strain field imposed by the geometry of the indenter. The implication of this is that hardness anisotropy is an obvious manifestation of dislocation interactions and indenter facet geometry. Simplified interpretations of this have been sought, of which the Brookes Resolved Shear Stress model, given in Section 3.6.1, is an important development. 

Another useful parameter in characterizing the stiffness of a gel network is hardness. Hardness is measured by indenting a probe into the gel at a specified velocity while measuring the force required for the indentation The force required to indent the gel to a certain depth is the hardness. While the measured hardness does depend strongly on the modulus of the gel, it also depends on many other measurement parameters such as the size and shape of the gel sample, probe size, speed, and indentation depth. Additionally, the applied strain field is very non-uniform. The strain and strain rate near the probe can be high, but because the gel is incompressible, the entire volume of the gel experiences deformation due to the displacement of gel by the probe. Thus hardness is at best a relative measure of gel material properties. However, since hardness is used by some common gel manufacturers to specify their materials, it is important to understand it in the context of other rheological characterization methods. 

A similar procedure can be applied to determine the hardness of the films. In fact, the combined contributions of the coating and the substrate are measured. Notably, at equal indentation depth, the influence of the substrate on the measured hardness is less than on the indentation modulus. Indeed, the plastic strain field is much less extended spatially than the elastic strain field (Figure H.9). From Figure H.9, one further observes that the 10% rule of thumb (in other words, film response is determined with relative penetration depth less than 0.1) has to be used very carefiilly. 

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. 

In summary, it is clear that there are substantial effects that vary systematically with the wavelength of the multilayer due both to internal stresses and the microstructure of the coatings. It has also been seen that deformation can occur not just by dislocation flow, as the initial analyses have assumed, but by mechanisms such as lattice rotations and shear along column boundaries. In addition, the use of indentation complicates the deformation field, so that the assumption that equal strains in both layers are required need not be correct. These effects all influence the hardness but have not so far been included in analyses. 

It can be easily shown that compressive uniaxial or biaxial stress results in the upshift of the Raman band to higher wavenumbers, whereas tensile stress decreases the Raman frequency in these cases [47]. However, in the complex stress field under the indenter, further complicated by the volumetric changes during possible phase transformations and the breakdown of constitutive equations due to macro- and microcracking, determination of the strain tensor components becomes a challenging task and the simplifying analytical models already discussed here need to be used. 

Figure 12.31 The slip-line field for a deep symmetrical notch (a) is identical to that for the frictionless punch indenting a plate under conditions of plane strain (b). (Reproduced with permission from Cottrell, The Mechanical Properties of Matter, Wiley, New York, 1964)...

These cracks are much shallower and are formed on unloading at the boundary between elastic, nonpermanently deformed material and the plastically deformed material close to the indenter the interface between these two regions is the source of a stress field because material that has been plastically strained has a different stress-strain behavior than the normal material. 

Field and Swain have studied elasto-plastic response on a number of brittle materials, including silicate glass, silicon single crystal and single crystalline sapphire, by the indentation with small micron sized spherical tipped indenters. The analysis of indentation stress-strain curve was performed from the indentation on... 

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