Surface deformations

Knoop developed an accepted method of measuring abrasive hardness using a diamond indenter of pyramidal shape and forcing it into the material to be evaluated with a fixed, often 100-g, load. The depth of penetration is then determined from the length and width of the indentation produced. Unlike WoodeU s method, Knoop values are static and primarily measure resistance to plastic flow and surface deformation. Variables such as load, temperature, and environment, which affect determination of hardness by the Knoop procedure, have been examined in detail (9). 

Case Hardening by Surface Deformation. When a metaUic material is plastically deformed at sufficiently low temperature, eg, room temperature for most metals and alloys, it becomes harder. Thus one method to produce a hard case on a metallic component is to plastically deform the surface region. This can be accomplished by a number of methods, such as by forcing a hardened rounded point onto the surface as it is moved. A common method is to impinge upon the surface fine hard particles such as hardened steel spheres (shot) at high velocity. This process is called shot... 

The surface may gain a very high (eg, 1000 Vickers) hardness from this process. Surface deformation also produces a desired high compressive residual stress. Figure 9 illustrates the improvement in fatigue properties of a carburized surface that has been peened (18). 

Second, deformation twins were observed in metal grains at the damaged surfaces. Deformation twinning cannot result from corrosion but is the consequence of shock loading of the metal, precisely the effects of microjets of water impacting on the metal surface. 

Fig. 26.7. Skidding on a rough road surface deforms the tyre material elastically.

Dutrowski [5] in 1969, and Johnson and coworkers [6] in 1971, independently, observed that relatively small particles, when in contact with each other or with a flat surface, deform, and these deformations are larger than those predicted by the Hertz theory. Johnson and coworkers [6] recognized that the excess deformation was due to the interfacial attractive forces, and modified the original Hertz theory to account for these interfacial forces. This led to the development of a new theory of contact mechanics, widely referred to as the JKR theory. Over the past two decades or so, the contact mechanics principles and the JKR theory have been employed extensively to study the adhesion and friction behavior of a variety of materials. 

The effect of surface deformation in the Helmholtz layer should also be involved in Eq. (35). In consideration of specific adsorption of anions, such effects can be expressed by the potential gradient Lj y, z, as follows,... 

Substituting Eqs. (35) and (36) into Eq. (34), the electrochemical potential fluctuation of dissolved metal ions at OHP is deduced. Then, disregarding the fluctuation of the chemical potential due to surface deformation, the local equilibrium of reaction is expressed as fi% = 0. With the approximation cm x, y, 0, if cm(x, y, (a, tf, we can thus derive the following equation,... 

By integration of the loeal slopes, we have reconstructed the micro-mirror surface. An example is shown in Fig.4, along the line indicated by an arrow on the slope map. The surface deformations do not exceed 1 nm along the studied profile. Although surface shapes vary from mirror to mirror, deformations in the nanometer range demonstrate the remarkable quality of this device. 

During the past three decades, many efforts have been made to calculate the surface deformation more efficiently and with less computer storage. A new method, known as the Multi-level Multi-integration (MLMI), which was orders of magnitude faster than the conventional methods, was developed by Lubrecht and loannides [33,34]. The method has been proven to be very efficient in saving CPU times though it costs a complicated procedure in programming. 

Noticing the fact that the formula for determining surface deformation takes the form of convolution, the fast Fourier transform (FFT) technique has been applied in recent years to the calculations of deformation [35,36]. The FFT-based approach would give exact results if the convolution functions, i.e., pressure and surface topography take periodic form. For the concentrated contact problems, however. 

Consider a distributed pressure acting on an elastic halfspace, and let the pressure distribution and the normal surface deformation be denoted hy p(x) and v x) for line contacts, or hypix, y) and v(x, y) for point contacts, respectively. According to the theory of contact mechanics [18], the normal surface deformation v(x) or v x,y) caused by a distributed pressure may be written in the forms of... 

In numerical analysis, both functions of normal surface deformation and pressure distribution have to be discretized in a space domain over U grid points for a line load, or grid points for two-dimensional distributed load. As an example, the deformation for line loading can be rewritten in discrete form as follows ... 

Having the influence coefficients obtained, the normal surface deformations can be obtained from the multisummation as described in Eq (27). The computation may be implemented using different numerical approaches, including direct summation (DS), MLMI, and DC-FFT based methods, which will be briefly described in this section. 

To use the DFT properly for evaluating normal surface deformation, the linear convolution appearing in Eq (27) has to be transformed to the circular convolution. This requires a pretreatment for the influence coefficient Kj and pressure pj so that the convolution theorem for circle convolution can be applied. The pretreatment can be performed in two steps ... 

In summary, when the DC-FFT algorithm is applied, the calculation of normal surface deformation within a region [xq, Xe] can be implemented in the following procedure ... 

Comparisons of the accuracy and efficiency for three numerical procedures, the direct summation, DC-FFT-based method and MLMI, are made in this section. The three methods were applied to calculating normal surface deformations at different levels of grids, under the load of a uniform pressure on a rectangle area 2a X 2fo, or a Hertzian pressure on a circle area in radius a. The calculations were performed on the same personal computer, the computational domain was set as -1.5a=Sx 1.5a and -1.5a=Sy 1.5a, and covered... 

Concerning the numerical accuracy, the closed form solutions of normal surface deformation have been compared to the numerical results calculated through the three methods of DS, DC-FFT, and MLMI. The influence coefficients used in the numerical analyses were obtained from three different schemes Green function, piecewise constant function, and bilinear interpolation. The relative errors, as defined in Eq (39), are given in Table 2 while Fig. 4 provides an illustration of the data. 

Chang, L. "An Efficient and Accurate Formulation of the Surface Deformation Matrix in Elastohydrodynamic Point Contacts, ASME 7. Tribol,Vol.in, 1989, pp. 642-647. 

Onuma T. and Ohkawa S. Detection of surface deformation related to with C02 injection by DInSAR at In Salah, Algeria. 2009 Energy Procedia 1 2177-2184. 

Surface curvature, color coding for, 10 340 Surface defect densities, 9 731 Surface deformation, case hardening by, 16 207-208 Surface diffusion... 

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