Axially symmetric contact problem for half spaces with geometrically perturbed surface

1999 ◽  
Vol 35 (6) ◽  
pp. 777-782 ◽  
Author(s):  
B. E. Monastyrs'kyi
Keyword(s):  
Contact Problem ◽  
Axially Symmetric ◽  
Axially Symmetric Contact

A Numerical Solution for an Axially Symmetric Contact Problem

1967 ◽  
Vol 34 (2) ◽  
pp. 283-286 ◽  
Author(s):  
Yih-O Tu
Keyword(s):  
Stress Distribution ◽  
Contact Problem ◽  
Contact Stress ◽  
Plate Thickness ◽  
Contact Radius ◽  
Axially Symmetric ◽  
Pressure Distributions ◽  
Linear Algebraic Equations ◽  
Axially Symmetric Contact ◽  
Contact Stress Distribution

A numerical scheme for the axially symmetric contact problem of a plate pressed between two identical spheres is given. The axially symmetric contact stress distribution is represented by a finite set of pressure distributions linearly varying with the radius between values defined in a set of concentric circles. The normal displacements of the bodies in contact resulting from these pressure distributions are matched at every radius of the discrete set of radii of these circles. The integral equation for the unkown contact stress distribution is thus approximated by a set of linear algebraic equations whose solution yields the unknown pressure values of the approximate distribution. The contact radius, relative approach, and the maximum contact stress are then computed numerically from this solution and are presented in terms of the total load, the radius of the sphere, and the plate thickness.

Rough Contacts Modeled by Nonlinear Coatings

2007 ◽  
Author(s):  
I. I. Kudish
Keyword(s):  
Contact Problem ◽  
Numerical Solutions ◽  
Asymptotic Methods ◽  
Experimental Studies ◽  
Nonlinear Function ◽  
Normal Displacement ◽  
Experimental Fact ◽  
Axially Symmetric ◽  
Elastic Solids ◽  
Local Pressure

A number of experimental studies [1–3] revealed that the normal displacement in a contact of rough surfaces due to asperities presence is a nonlinear function of local pressure and it can be approximated by a power function of pressure. Originally, a linear mathematical model accounting for surface roughness of elastic solids in contact was introduced by I. Shtaerman [4]. He assumed that the effect of asperities present in a contact of elastic solids can be essentially replaced by the presence of a thin coating simulated by an additional normal displacement of solids’ surfaces proportional to a local pressure. Later, a similar but nonlinear problem formulation that accounted for the above mentioned experimental fact was proposed by L. Galin. In a series of papers this problem was studied by numerical and asymptotic methods [5–9]. The present paper has a dual purpose: to analyze the problem analytically and to provide some asymptotic and numerical solutions. The results presented below provide an overview of the results obtained on the topic and published by the author earlier in the journals hardly accessible to the international tribological community (such as Russian and mathematical journals) and, therefore, mostly unknown by tribologists. A number of recent publications on contacts of rough elastic solids supports the view that these results are still of value to the specialists involved in nanotribology. The existence and uniqueness of a solution of a contact problem for elastic bodies with rough (coated) surfaces is established based on the variational inequalities approach. Four different equivalent formulations of the problem including three variational ones were considered. A comparative analysis of solutions of the contact problem for different values of initial parameters (such as the indenter shape, parameters characterizing roughness, elastic parameters of the substrate material) is done with the help of calculus of variations and the Zaremba-Giraud principle of maximum for harmonic functions [10,11]. The results include the relations between the pressure and displacement distributions for rough and smooth solids as well as the relationships for solutions of the problems for rough solids with fixed and free contact boundaries. For plane and axially symmetric cases some asymptotic and numerical solutions are presented.

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