A-plates

In the case of a plate column the performance of a real plate is related to the performance of a theoretical one by the plate efficiency. In the case of a packed column the height equivalent to a theoretical plate HETP) gives a measure of the contacting efficiency of the packing. 

The source is brought to a. positive poteptial (I/) of several kilovolts and the ions are extracted by a plate at ground potential. They acquire kinetic energy and thus velocity according to their mass and charge. They enter a magnetic field whose direction is perpendicular to their trajectory. Under the effect of the field, Bg, the trajectory is curved by Lorentz forces that produce a centripetal acceleration perpendicular to both the field and the velocity. 

Small quantities of solids may be spread upon unglazed porcelain plates. The chief disadvantage of this method is the comparatively high cost of the porous plates, since they cannot be conveniently cleaned nor can the same area be used for different substances. However, a plate may be broken and used for small amounts of material. 

Now, in principle, the angle of contact between a liquid and a solid surface can have a value anywhere between 0° and 180°, the actual value depending on the particular system. In practice 6 is very difficult to determine with accuracy even for a macroscopic system such as a liquid droplet resting on a plate, and for a liquid present in a pore having dimensions in the mesopore range is virtually impossible of direct measurement. In applications of the Kelvin equation, therefore, it is almost invariably assumed, mainly on grounds of simplicity, that 0 = 0 (cos 6 = 1). In view of the arbitrary nature of this assumption it is not surprising that the subject has attracted attention from theoreticians. 

By selecting either a large positive or negative voltage on a plate with a slit in it held above the surface, the desorbed negative or positive ions can be accelerated away from the surface and into a mass analyzer. 

We call a plate the shallow shell when k =k2 = 0. This implies that the plate mid-surface coincides with the plane z = 0, and the plate is limited by the two parallel planes z = h, z = —h and a boundary contour. Let us redenote the horizontal and vertical displacements of the plate mid-surface by u = ui, u = U2, w. In this case, the plate horizontal and vertical displacements are not coupled. Indeed, it follows from (1.18), (1.19), that U = (ui,U2) is described by the following equilibrium equations ... 

Utilizing the constitutive law (1.5), the other model of a plate under the creep condition follows ... 

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... 

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. 

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 model of the plate considered in this section actually corresponds to a shallow shell having zeroth curvatures. The gradient of the punch surface is assumed to be rather small, so that the nonpenetration condition imposed in the domain is the same as in the usual case for a plate. Meanwhile, the restriction imposed on the crack faces contains three components of the displacement vector. 

Let a plate occupy a bounded domain fl c with smooth boundary F. Inside fl there is a graph Fc of a sufficiently smooth function. The graph Fc corresponds to the crack in the plate (see Section 1.1.7). A unit vector n = being normal to Fc defines the surfaces of the crack. 

L flc) be some given functions of the external forces. The equilibrium problem for a plate with a crack is formulated as the following variational inequality ... 

Thus, we obtain the nonpenetration condition of the cut faces, the same as for a plate with a crack. Later on, for simplicity we consider the case e = 1, i.e. 

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. 

In this section we deal with the simplified nonpenetration condition of the crack faces considered in the previous section. We formulate the model of a plate with a crack accounting for only horizontal displacements and construct approximate equations using penalty and iterative methods. The convergence of these solutions is proved and its application to the onedimensional problem is discussed. Analytical solutions for the model of a bar with a cut are obtained. The results of this section can be found in (Kovtunenko, 1996c, 1996d). 

We consider a boundary value problem for equations describing an equilibrium of a plate being under the creep law (1.31)-(1.32). The plate is assumed to have a vertical crack. As before, the main peculiarity of the problem is determined by the presence of an inequality imposed on a solution which represents a mutual nonpenetration condition of the crack faces... 

In the domain we consider the following equations describing quasistatic deformation of a plate ... 

The equilibrium problem for a plate is formulated as some variational inequality. In this case equations (3.92)-(3.94) hold, generally speaking, only in the distribution sense. Alongside (3.95), other boundary conditions hold on the boundary F the form of these conditions is clarified in Section 3.3.3. To derive them, we require the existence of a smooth solution to the variational inequality in question. On the other hand, if we assume that a solution to (3.92)-(3.94) is sufficiently smooth, then the variational inequality is a consequence of equations (3.92)-(3.94) and the initial and boundary conditions. All these questions are discussed in Section 3.3.3. In Section 3.3.2 we prove an existence theorem for a solution to the variational equation and in Section 3.3.4 we establish some enhanced regularity properties for the solution near F. ... 

We can now give an exact statement of the equilibrium problem for a plate. Suppose that / G L Q ). An element (0, x) G 17 is said to be a solution to the equilibrium problem for a thermoelastic plate with a crack if it satisfies the variational inequality... 

It is noteworthy that the original equilibrium problem for a plate with a crack can be stated twofold. On the one hand, it may be formulated as variational inequality (3.98). In this case all the above-derived boundary conditions are formal consequences of such a statement under the supposition of sufficient smoothness of a solution. On the other hand, the problem may be formulated as equations (3.92)-(3.94) given initial and boundary conditions (3.95)-(3.97) and (3.118)-(3.122). Furthermore, if we assume that a solution is sufficiently smooth then from (3.92)-(3.97) and (3.118)-(3.122) we can derive variational inequality (3.98). 

We consider the model of a plate with a crack describing the plate vertical displacements with a given friction between the crack surfaces. The results of this section are published in (Kovtunenko, 1998). 

We analyse the behaviour of solutions for a plate having a crack provided that the crack length tends to zero. The nonpenetration conditions are assumed to hold at the crack faces. 

Here [ ] is a jump of a function at the crack faces, v is the unit normal vector to the crack shape, and 2h is the thickness of the shell. A similar extreme crack shape problem for a plate was considered in Section 2.4. 

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