Elastic solutions contact

The physical processes that occur during indentation are schematically illustrated in Fig. 31. As the indenter is driven into the material, both elastic and plastic deformation occurs, which results in the formation of a hardness impression conforming to the shape of the indenter to some contact depth, h. During indenter withdrawal, only the elastic portion of the displacement is recovered, which facilitates the use of elastic solutions in modeling the contact process. 

PAPER Vl(iii) Solutions for isoviscous line contacts using a closed form elasticity solution... 

Hartnett,M. "A General Solution for Elastic Bodies Contact Problem," in Solid Contact and Lubrication, edited by Cheng and Kerr, A.S.M.E. publlcaation number AMD-39, pp.51-66. 1980. 

This displacement estimate must also be obtained from the elasticity solution for the half space problem it is exactly twice the normal displacement of the surface at the center of the contact area relative to the normal displacement at a remote point on the half space surface in the limit as r/a —> oo. 

Consider the case where the contact area C(t) is contracting, or at least nonexpanding, for all t. This is the region on which displacement, or rather its derivative, is specified. It follows that v x, t) is known in C(t), so that the general solutions of the Hilbert problem (3.3.2) discussed in Sect. 3.3 are in fact final solutions of the problem. These are identical in form to the corresponding elastic solutions but where v x,t) takes the place of the displacement derivative. This is a special case of the Extended Correspondence Principle discussed in Sect. 2.6. 

One would be in an ad-hoc fashion to assume that, because of the tendency toward interpenetration, near the crack tips the crack surfaces would come in smooth contact and form a cusp, and the resulting contact region would consist of a single uninterrupted zone rather than the sum of a series of discrete zones as implied by the oscillatory nature of the elastic solution (see Comninou [ll], Atkinson [l2]). Another way is to assume that near the crack tip the linear theory is not valid and to use a large deformation nonlinear theory. An asymptotic analysis using such a theory was provided by Knowles and Sternberg [l3] for the plane stress interface crack problem in two bonded dissimilar incompressible Neo-Hookean materials which shows no oscillatory behavior for stresses or... 

The magnitude and distributions of stress in the knee are different from the hip. In the hip, the spherical contacting surfaces are highly conforming, and the effective (von Mises) stress levels are below yield, and, thus, below the onset of irrecoverable plastic deformation. Consequently, for hip components, UHMWPE can reasonably be considered to behave as an elastic material at the continuum level. Elasticity solutions have been developed to calculate... 

A possible solution to overcome this difficulty consists in obtaining the pressure field in mixed film lubrication through the conjugation of full film and dry contact (elastic) solutions. 

Our aim is to analyze the solution properties of the variational inequality describing the equilibrium state of the elastic plate. The plate is assumed to have a vertical crack and, simultaneously, to contact with a rigid punch. 

Properties of solutions in contact problems for elastic plates and shells having cracks. 

The new approach to crack theory used in the book is intriguing in that it fails to lead to physical contradictions. Given a classical approach to the description of cracks in elastic bodies, the boundary conditions on crack faces are known to be considered as equations. In a number of specific cases there is no difflculty in finding solutions of such problems leading to physical contradictions. It is precisely these crack faces for such solutions that penetrate each other. Boundary conditions analysed in the book are given in the form of inequalities, and they are properly nonpenetration conditions of crack faces. The above implies that similar problems may be considered from the contact mechanics standpoint. 

We have to stress that the analysed problems prove to be free boundary problems. Mathematically, the existence of free boundaries for the models concerned, as a rule, is due to the available inequality restrictions imposed on a solution. As to all contact problems, this is a nonpenetration condition of two bodies. The given condition is of a geometric nature and should be met for any constitutive law. The second class of restrictions is defined by the constitutive law and has a physical nature. Such restrictions are typical for elastoplastic models. Some problems of the elasticity theory discussed in the book have generally allowable variational formulation... 

Viscoelastic contact problems have drawn the attention of researchers for some time [2,3,104,105]. The mathematical peculiarity of these problems is their time-dependent boundaries. This has limited the ability to quantify the boundary value contact problems by the tools used in elasticity. The normal displacement (u) and pressure (p) fields in the contact region for non-adhesive contact of viscoelastic materials are obtained by a self-consistent solution to the governing singular integral equation given by [106] ... 

Hertz [27] solved the problem of the contact between two elastic elliptical bodies by modeling each body as an infinite half plane which is loaded over a contact area that is small in comparison to the body itself. The requirement of small areas of contact further allowed Hertz to use a parabola to represent the shape of the profile of the ellipses. In essence. Hertz modeled the interaction of elliptical asperities in contact. Fundamental in his solution is the assumption that, when two elliptical objects are compressed against one another, the shape of the deformed mating surface lies between the shape of the two undeformed surfaces but more closely resembles the shape of the surface with the higher elastic modulus. This means the deformed shape after two spheres are pressed against one another is a spherical shape. 

Boussinesq and Cerruti made use of potential theory for the solution of contact problems at the surface of an elastic half space. One of the most important results is the solution to the displacement associated with a concentrated normal point load P applied to the surface of an elastic half space. As presented in Johnson [49]... 

With good experimental technique and careful analysis, the hardness and elastic modulus of many materials can be measured using these methods with accuracies of better than 10 % [59]. There are, however, some materials in which the methodology signihcantly overestimates H and E, spe-cihcally, materials in which a large amount of pile-up forms around the hardness impression. The reason for the overestimation is that Eqs (22) and (24) are derived from a purely elastic contact solution, which accounts for sink-in only [65]. 

This chapter describes a DML model proposed by the authors, based on the expectation that the Reynolds equation at the ultra-thin film limit would yield the same solutions as those from the elastic contact analysis. A unified equation system is therefore applied to the entire domain, which gives rise to a stable and robust numerical procedure, capable of predicting the tribological performance of the system through the entire process of transition from full-film to boundary lubrication. 

T. W. Shield and D. M. Bogy, Some Axisymmetric Problems for Layered Elastic Media Part I — Multiple Region Contact Solutions for Simply-Connected Indenters, Thansac-tions of the ASME, vol. 56, pp. 798-806, Dec 1989. 

Solution The maximum contact radius and contact time for an elastic sphere collision can be estimated from Eqs. (2.133) and (2.136), respectively. For collisions among... 

Foam films of different size, shape and spatial orientation are obtained at the approach of individual bubbles or the surfaces of a biconcave drop, or at bubble contact with the solution/air interface, or at withdrawing a frame from a solution, etc. Individual foam bubbles are usually used in the study of foam properties. They prove to be most useful in many cases, for example, in the determination of foam film elasticity, the estimation of gas diffusion from the bubble through the film, the detection of the rupture of the foam bubble films [e.g. 1], Beginning with the remarkable bubbles of Boys [2] and reaching to present day studies, single foam bubbles have since long attracted a considerable interest (see, for instance, the monograph of Dukhin, Kretzschmar and Miller [3]). 

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