Elastic contact

A number of nitroso compounds, A-nitrosamines among them, are potent carcinogens. The most common carcinogenic nitrosamines, found mainly in protein food, are A-nitroso-dimethylamine (NDMA), A-nitroso-diethylamine (NDEA), A-nitroso-pyrrolidine (N-Pyr), and A-nitroso-piperidine (N-Pip). These compounds supposedly increase the risk of colon, rectum, stomach, pancreas, and bladder cancers. Nitrosamines are most prevalent in cured meats, but have also been detected in smoked fish, soy protein foods dried by direct flame, and food-contact elastic nettings. Dietary surveys indicated weekly mean intakes of these compounds amounting to about 3 pg per person (Anon., 1988 Cassens, 1995). In addition, the precursors of nitrosamines, especially nitrate, are abundant in some leafy and root vegetables (Table 14.1). 

Soy protein foods dried by direct flame Some alcoholic beverages Food-contact elastic nettings Rubber baby bottle nipples Cosmetics... 

While the surface energy of the solid favors an extended contact, elasticity strenuously opposes it. The JKR formula expresses the fact that the radius I results from a compromise between these two forms of energy. It reads... 

The two contacting elastic bodies, (A) and (B), shown In Figure 1(a) were represented by an equivalent ellipsoid near a plane, as shown In Figure 1(b). If the principal radii of curvature of the undeformed solids are (r, r. ) and (tBx Bv ... 

The stress value for making surface damage is related to the Vickers hardness value. It can be divided into three levels 1. contacting elastically, 2. contact area is partly damaged, 3. contact area is fully damaged and indent is made. 

Fogden, A. and White, L. R., Contact elasticity in the presence of capillary condensation. The nonadhesive Hertz problem, JCIS, 138, 414-430 (1990). 

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. 

Contacting was performed by mechanically pressing platinum contacts elastically on the film. By means of small screws, the pressure of the contacts could be adjusted. For the Montgomery method, this adjustment could be done for each of the contacts separately. Different sample holders were used for different sample sizes. For most of the conductivity experiments presented here, we used samples of 19 X 2 mm2 and 14 x 6 mm in size for the standard four-probe and Montgomery techniques, respectively. 

The second factor contains the influence of the contact region with its different properties related to the matrix material this influence is expressed by the ratio of the two moduli. The term also contains the pressure dependence of the velocity controlled by a strong pressure dependence of the contact elasticity (modulus Me). 

Thus, the contact modulus equals the solid material modulus divided by 2.8 and the pressure dependence follows with the exponent m = 0.33 = 1 /3, exactly Hertz s contact elasticity for a spherical contact. 

Pressure Calculation of pressure effect upon velocity controlled by parameters of the contact elasticity. 

Substances in this category include Krypton, sodium chloride, and diamond, as examples, and it is not surprising that differences in detail as to frictional behavior do occur. The softer solids tend to obey Amontons law with /i values in the normal range of 0.5-1.0, provided they are not too near their melting points. Ionic crystals, such as sodium chloride, tend to show irreversible surface damage, in the form of cracks, owing to their brittleness, but still tend to obey Amontons law. This suggests that the area of contact is mainly determined by plastic flow rather than by elastic deformation. 

A number of friction studies have been carried out on organic polymers in recent years. Coefficients of friction are for the most part in the normal range, with values about as expected from Eq. XII-5. The detailed results show some serious complications, however. First, n is very dependent on load, as illustrated in Fig. XlI-5, for a copolymer of hexafluoroethylene and hexafluoropropylene [31], and evidently the area of contact is determined more by elastic than by plastic deformation. The difference between static and kinetic coefficients of friction was attributed to transfer of an oriented film of polymer to the steel rider during sliding and to low adhesion between this film and the polymer surface. Tetrafluoroethylene (Telfon) has a low coefficient of friction, around 0.1, and in a detailed study, this lower coefficient and other differences were attributed to the rather smooth molecular profile of the Teflon molecule [32]. 

In AFM, the relative approach of sample and tip is nonnally stopped after contact is reached. Flowever, the instrument may also be used as a nanoindenter, measuring the penetration deptli of the tip as it is pressed into the surface of the material under test. Infomiation such as the elastic modulus at a given point on the surface may be obtained in tliis way [114], altliough producing enough points to synthesize an elastic modulus image is very time consuming. 

Carpick et al [M] used AFM, with a Pt-coated tip on a mica substrate in ultraliigh vacuum, to show that if the defonnation of the substrate and the tip-substrate adhesion are taken into account (the so-called JKR model [175] of elastic adliesive contact), then the frictional force is indeed proportional to the contact area between tip and sample. Flowever, under these smgle-asperity conditions, Amontons law does not hold, since the statistical effect of more asperities coming into play no longer occurs, and the contact area is not simply proportional to the applied load. 

Johnson K L, Kendall K and Roberts A D 1971 Surface energy and the contact of elastic solids Proc. R. Soc. A 324 301... 

Modelling of the tme contact area between surfaces requires consideration of the defonnation that occurs at the peaks of asperities as they come into contact with mating surfaces. Purely elastic contact between two solids was first described by H Hertz [7], The Hertzian contact area (A ) between a sphere of radius r and a flat surface compressed under nonnal force N is given by... 

Detennining the contact area between two rough surfaces is much more difficult than the sphere-on-flat problem and depends upon the moriDhology of the surfaces [9]. One can show, for instance, that for certain distributions of asperity heights the contact can be completely elastic. However, for realistic moriDhologies and macroscopic nonnal forces, the contact region includes areas of both plastic and elastic contact with plastic contact dominating. 

Submitting the main topic, we deal with models of solids with cracks. These models of mechanics and geophysics describe the stationary and quasi-stationary deformation of elastic and inelastic solid bodies having cracks and cuts. The corresponding mathematical models are reduced to boundary value problems for domains with singular boundaries. We shall use, if it is possible, a variational formulation of the problems to apply methods of convex analysis. It is of importance to note the significance of restrictions stated a priori at the crack surfaces. We assume that nonpenetration conditions of inequality type at the crack surfaces are fulfilled, which improves the accuracy of these models for contact problems. We also include the modelling of problems with friction between the crack surfaces. 

Thus, the relations (1.36) or (1.37) describe the interaction between a plate and a punch. To derive the contact model for an elastic plate, one needs to use the constitutive law (1.25). Contact problems for inelastic plates are derived by the utilizing of corresponding inelastic constitutive laws given in Section 1.1.4. 

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. 

Galin L.A. (1980) Contact problems in elasticity and viscoelasticity. Nauka, Moscow (in Russian). 

Khludnev A.M. (1983) A contact problem of a linear elastic body and a rigid punch (variational approach). Appls. Maths. Mechs. 47 (6), 999-1005 (in Russian). 

Progress in modelling and analysis of the crack problem in solids as well as contact problems for elastic and elastoplastic plates and shells gives rise to new attempts in using modern approaches to boundary value problems. The novel viewpoint of traditional treatment to many such problems, like the crack theory, enlarges the range of questions which can be clarified by mathematical tools. 

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... 

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