to the surface and the latter in the plane of the surface. The distribution of height is measured from a reference plane say, mean plane.
The characterization of a surface may be one dimensional 1-D or two dimensional 2-D depending upon the machining and finishing process. For 1-D case,
height z varies with one of the coordinates, whereas in other coordinate there exists a lay where variation of z is comparatively small. But for surfaces made by conventional
manufacturing processes, when 1-D characterization is not proper, 2-D surface roughnesses description is required. Atomic Force Microscope AFM and 3D surface
profilometer are used to improve the resolution and accuracy of the roughness measurement. Typical 1-D and 2-D representation of nominally flat surfaces polished
are shown in Figure 2.2. a and 2.2. b. Scale used for the height is much larger than that for the wavelength because the height of the asperities from the mean plane is very
less compared to their wavelengths [16].
Figure 2.2: Typical representation of a surface: a one-dimensional b two-
dimensional [16].
2.3 Contact Problem of Smooth Surfaces
The study of the contact behavior of deformable bodies has divided into two approaches. The first, regard the bodies are smooth and can be adequately described by
their nominal geometry, the second admits that all surfaces are comprised of multitude of peaks and valleys which have multiple asperities which is regarded as a rough
surface. Hertz [1] was first to analyze the elastic contact between two non-conforming spheres. He gave the analytical solution for the normal contact between two curved
bodies for contact pressure and subsurface stress field. In Hertz analysis, following assumptions were made.
a Radii of curvature of the contacting bodies are large compared with radius of
circle of contact. b
Contact is frictionless. c
Surfaces are continuous and non-conforming. d
Strain is small. e
Each solid can be considered as half space. Based on these assumptions, the stress fields generated by an indenter contacting
an elastic solid can be analyzed [17]. For the case of elastic contact with a spherical indenter of radius R, the radius of the circle of contact a between the indenter and the
specimen surface increases with the load. The contact pressure distribution p proposed by Hertz is given by
⁄
The total contact load P can be obtained from the above pressure distribution as, ∫
Where maximum pressure = 32 , with
denoting the mean pressure. The contact circle radius a is given by,
is the effective modulus of elasticity and is the effective radius of curvature
defined as,
, ,
are the Young’s modulus, Poisson’s ratio and radius of curvature of the indenter material and
, ,
are the corresponding parameters of the specimen. The depth of indentation h is related to indenter radius by,
When two nominally flat surfaces are brought into contact under load, contact occurs only at discrete spots. The real area of contact is the summation of all individual
spots. Real area is only fraction of the nominal contact area. The real area of contact between two solid surfaces has a profound influence on friction and wear of machine
parts, thermal and electrical conduction, contact stresses and joint stiffness. Greenwood and Williamson [7] explains how Gaussian distribution can be applied to surface
parameters in practice and the reasons for regular occurrence of such a distribution governing each individual effect. For a rough surface in contact with a rigid plane, GW
model which assumes constant tip radius for the asperities gives good order of magnitude estimates the number of contacts, real area of contact and nominal pressure
at any given separation. Some other important models are available in literature like Greenwood and Tripp model GT Model [18], Whitehouse and Archard Model WA
Model [19], Onions and Archard Model OA Model [20], Nayak Model [21], The elliptic Model [22], Hisakado Model [23], Francis Model [24], McCool model [25]. In
most of these models, the asperities are assumed to be either spherical, paraboloidal or elliptic paraboloidal in shape and the corresponding Hertzian solution for single contact
is used in the analysis.
2.4 Contact Problem of Rough Surface
