Contact problems

But the film thickness doesn t change, so how can the coercive voltage increase And the same capacitor can show its true coercive voltage of e.g. 2 Volts after a few measurements. So it appears to be a contact problem, though a pure platinum electrode is contacted to a pure Ptlr alloy coated cantilever, so an excellent contact should be expected. To investigate the reason of this poor contact behavior, afm conductivity scans were performed on a pure Pt coated 

17 Electrical Characterization of Ferroelectric Properties in the Sub-Micrometer Scale 

In the sequel, we consider concrete boundary conditions for the above models to formulate boundary value problems. Also, restrictions of the inequality type imposed upon the solutions are introduced. We begin with the nonpenetration conditions in contact problems (see Kravchuk, 1997 Khludnev, Sokolowski, 1997 Duvaut, Lions, 1972). 

The model describing interaction between two bodies, one of which is a deformed solid and the other is a rigid one, we call a contact problem. After the deformation, the rigid body (called also punch or obstacle) remains invariable, and the solid must not penetrate into the punch. Meanwhile, it is assumed that the contact area (i.e. the set where the boundary of the deformed solid coincides with the obstacle surface) is unknown a priori. This condition is physically acceptable and is called a nonpenetration condition. We intend to give a mathematical description of nonpenetration conditions to diversified models of solids for contact and crack problems. Indeed, as one will see, the nonpenetration of crack surfaces is similar to contact problems. In this subsection, the contact problems for two-dimensional problems characterizing constraints imposed inside a domain are considered. 

Let a punch shape be described by the equation z = ip(x), and xi,X2,z be the Descartes coordinate system, x = xi,X2). We assume that the mid-surface of a plate occupies the domain fl of the plane = 0 in its non-deformable state. Then the nonpenetration condition for the plate vertical displacements w is expressed by the inequalities 

Conversely, let a contact occur at the point x. This means w x) = i x) and p x) 0. Consequently, in this case 

The meaning of the relation (1.37) is the following. The punch pressure p = — / is equal to zero if a contact is absent. If the punch pressure 

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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. 

We would like to stress at this point that the derivation of (1.36) and (1.38)-(1.39) is connected with the simulation of contact problems and therefore contains some assumptions of a mechanical character. This remark is concerned with the sign of the function p in the problem (1.36) and with the direction of the vector pi,P2,p) in the problem (1.38), (1.39). Note that the classical approach to contact problems is characterized by a given contact set (Galin, 1980 Kikuchi, Oden, 1988 Grigolyuk, Tolkachev, 1980). In contact problems considered in the book, the contact set is unknown, and we obtain the so called free boundary problems. Other free boundary problems can be found in (Hoffmann, Sprekels, 1990 Elliot, Ock-endon, 1982 Antontsev et ah, 1990 Kinderlehrer et ah, 1979 Antontsev et ah, 1992 Plotnikov, 1995). 

We start with contact problems for plates. The contact problems with nonpenetration conditions can be viewed as a specific type of crack problem. On the other hand, the analysis of solution properties when the contact occurs is useful in the sequel. 

The viscoelastic contact problem for a plate with the constitutive law (see Section 1.1.4)... 

Let C be a bounded domain with smooth boundary T, <3 = x (0, T). Our object is to study a contact problem for a plate under creep conditions (see Khludneva, 1990b). The formulation of the problem is as follows. In the domain Q, it is required to find functions w, Mij, i,j = 1,2, satisfying the relations... 

We proceed with an investigation of the contact problem for a plate under creep conditions. We know that for every fixed / G L Q) there exists a unique solution w,M satisfying (2.35)-(2.37). Let G L Q) be a given element and F c (Q) be a closed convex and bounded set. We introduce the cost functional... 

We continue the investigation of the contact problem for a plate under creep conditions. In this section the case of both normal and tangential displacements of the plate is considered. 

The results on contact problems for plates without cracks can be found in (Caffarelli, Friedman, 1979 Caffarelli et al., 1982). Properties of solutions to elliptic problems with thin obstacles were analysed in (Frehse, 1975 Schild, 1984 Necas, 1975 Kovtunenko, 1994a). Problems with boundary conditions of equality type at the crack faces are investigated in (Friedman, Lin, 1996). 

We consider a problem similar to the one considered in Section 2.8. The nonpenetration condition between crack faces is taken in simplified form. Our aim is to obtain some qualitative properties of solutions for a contact problem for a plate having a crack. 

Theorem 5.10. Let all the assumption of Theorem 5.9 he fulfilled. Then, from the solutions v, w, n, of the elastoplastic contact problem... 

In this section we analyse the contact problem for a curvilinear Timoshenko rod. The plastic yield condition will depend just on the moments m. We shall prove that the solution of the problem satisfies all original boundary conditions, i.e., in contrast to the preceding section, we prove existence of the solution to the original boundary value problem. 

The contact problem for a rod under creep conditions is considered in this section. Our goal is to prove an existence theorem. We use the notations of the preceding sections. For convenience, introduce the notations... 

Demkowicz L., Oden J.T. (1982) On some existence results in contact problems with nonlocal friction. Nonlinear Anal. Theory, Meth. and Appl. 6 (10), 1075-1093. 

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

Haslinger J., Neittaanmaki P., Tiihonen T. (1986) Shape optimization in contact problems based on penalization of the state inequality. Apl. Mat. 31 (1), 54-77. 

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). 

Khludnev A.M. (1996a) Contact problem for a plate having a crack of minimal opening. Control and Cybernetics 25 (3), 605-620. 

Khludneva E. Yu. (1990a) Optimal control of external forces in contact problems for a viscoelastic plate. In Algebra and Math. Analysis. Novosibirsk, 8-14 (in Russian). 

Kovtunenko V.A. (1996a) Numerical solution of a contact problem for the Timoshenko bar model. Izvestiya Rus. Acad. Sci. Mechanics of Solid 5, 79-84 (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. 

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] ... 

Via an ad hoc extension of the viscoelastic Hertzian contact problem, Falsafi et al. [38] incorporated linear viscoelastic effects into the JKR formalism by replacing the elastic modulus with a viscoelastic memory function accounting for time and deformation, K t) ... 

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]... 

The major differences between polymer and liquid electrolytes result from the physical stiffness of the PE. PEs are either hard-to-soft solids, or a combination of solid and molten in phases equilibrium. As a result, wetting and contact problems are to be expected at the Li/PE interface. In addition, the replacement of the native oxide layer covering the lithium, under the... 

Similar contact problems are expected at the PE/carbon interface, where the wetting of the carbon by the PE is of crucial importance [128, 129]. 

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