Indent equation

Thus for a Vickers indenter equation (5.51) becomes Kic = 0M9(HvW), which in turn is... 

The proposed contact force model can be used for the impact between two spheres, if the parameters in the model are known. The generalized parameter K may be evaluated from the radii and the material properties of the two spheres using equation (2). The remaining parameters are 5, f , 5p, which can be determined by integrating the relative indentation equations of motion twice, substitution in the contact force expression, and integrating the contact force around the hysteresis loop and equating it to the kinetic energy loss, as [15]... 

Once the calculation of effective masses of the two colliding bodies is performed right before impact (at the time of initial contact), the direct-centri impact model of the two spheres is reconstructed with effective masses m/ and and initial velocities and The relative indentation equation of motion of the two spheres during the period of contact then becomes... 

After obtaining a set of fitted velocity versus time data for a particular test specimen, we can extract the contact force and depth of indentation by mathematical operations. The differentiation of the indenter velocity gives the equation for contact force while impact ... 

Because the indentation varies with time, the modulus must be specified for a certain indentation time, eg, a 10-s modulus. The Hertz equation holds only for purely elastic materials. However, it has been appHed to viscoelastic materials, including polymers and coatings, with excellent results (249—256). Indentation hardness vs temperature curves are shown in Figure 40 (249,251). 

As we press a flat indenter into the material, shear takes place on the 45° planes of maximum shear stress shown in Fig. 11.4, at a value of shear stress equal to k. By equating the work done by the force F as the indenter sinks a distance u to the work done against k on the shear planes, we get ... 

Here M is the moment and Mp the fully-plastic moment of, for instance, a beam P/A is the indentation pressure and H the hardness of, for example, armour plating.) The left-hand side of each of these equations describes the loading conditions the right-hand side is a material property. When the left-hand side (which increases with load) equals the right-hand side (which is fixed), failure occurs. 

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. 

Molecular dynamics (MD) permits the nature of contact formation, indentation, and adhesion to be examined on the nanometer scale. These are computer experiments in which the equations of motion of each constituent particle are considered. The evolution of the system of interacting particles can thus be tracked with high spatial and temporal resolution. As computer speeds increase, so do the number of constituent particles that can be considered within realistic time frames. To enable experimental comparison, many MD simulations take the form of a tip-substrate geometry correspoudiug to scauniug probe methods of iuvestigatiug siugle-asperity coutacts (see Sectiou III.A). 

For large indentations, H = BY and for small indentations, FI 1/r. This equation is not expected to be exact because both Y and a may depend on r, but its general form matches the observations. 

Finally, in Section V, we compare the results of Bogolubov,3 Choh and Uhlenbeck, and Cohen8 with the generalized Boltzmann equation in Prigogine s formalism. The equivalence of the two methods is well known for the two-body and Cohen s three-body results 23 the demonstration of the same indentity is extended to the three-body results of Choh and Uhlenbeck and to Cohen s four-body expression. We also present the principles of the extension of this comparison for arbitrary concentration. 

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] ... 

B3, W3) while the rear is indented or dimpled. Skirt formation may also occur, as discussed in Section D. The wake angle for spherical-caps (expressed in degrees) is well represented by the empirical equation... 

The Bib and BI are defined as shown below in Equations (2) and (3), respectively. Both indices describe the bonding ability of the tablets by using the ratio of tensile strength and indentation hardness. However, Bib is determined at low-indentation speed (Hq) and BIw is determined at high speed (Hio). Hence, BI is considered more indicative of actual tableting conditions, whereas Bib relates to slower compactions speeds typical of development or roller compaction processes. 

Proceeding from equation (4.3.19), we obtain, as for the Vickers indenter (4.3.22) and (4.3.23), the following hardness formulae for commonly used indenters ... 

At a certain critical load, the crack spontaneously breaks through to the free surface (the cut-through ), usually with additional crack depth increase D, and changes into a well developed semicircle (solid-line semicircle). This transformation may also be induced in another, stable, manner, e.g., by the action of rosette type stresses around the strain region while loading the indenter before the cut-through . The model given in Fig. 6.2.8 is described by the equation... 

The equation, unlike that for a spherical indenter, contains no terms relating to the state of the sample surface (flaw population, etc.). Instead, it considers the cone angle as a variable through its influence on the penetration field. 

Usually, the contact is not smooth. It can be demostrated that the result of allowing for friction on the indenter-sample contact surface is equivalent to replacing f by y> = +arc tan p (p —coefficient of friction), which corresponds to a blunting of the indenter point. If friction is neglected, this may result in too high a value of K% in equation (6.2.9). Simultaneously the intersecting cracks may merge into others. 

More complex expressions arise if the sphere is taken to be elastic, so that the above equation should only apply to soft materials indented by very stiff materials. 

In these equations, the repulsion of the sample became interrelated with the adhesion force via the contact area. Several models have been developed to include the effect of the adhesion forces [80-83]. Johnson, Kendall, and Roberts derived the following expression for the contact radius and surface indentation ... 

There are two major sources of the deformation in contact-mode SFM the elasticity of the cantilever and the adhesion between the tip and sample surface. For purely elastic deformation, a variety of models have been developed to calculate the contact area and sample indentation. The lower limit for the contact diameter and sample indentation can be determined based on the Hertz model without taking into account the surface interactions [79]. For two bodies, i.e. a spherical tip and an elastic half-space, pressed together by an external force F the contact radius a and the indentation depth 8 are given by the following equations ... 

It should be pointed out that Equation (11) should be modified to Equation (1) when compression of a single particle between two surfaces is modelled, that is, spherical indentation is replaced by compression between two surfaces. Applications in which these models were applied to experimental data from compression testing are described later. 

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