Half-plane problems

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. 

Derive a nondimensional system of equations that describes the fluid-flow, thermal-energy, and mass-transfer problem for the ideal rotating-disk problem in the semiinfinite half plane. 

More generally the following question may be asked. When the values of x are confined to a certain subset / of the real axis, how does this show up in the properties of G If / is the interval — a < x < a it is known that G(k) is analytic in the whole complex k-plane and of exponential type . If I is the semi-axis x 0 the function G(k) is analytic and bounded in the upper half-plane. But no complete answer to the general question is available, although it is important for several problems. 

Superposition of the first and second half-plane loading problems gives the desired description of a region under a distant traction of and a craze traction of tr, as... 

To solve this problem, we shall refer to the elastic solution given in Ref. 15 (p. 25). According to this approach, the displacement of a point x, j) of the half-plane caused by the load applied at (x, j ) is given by... 

Now we will show how the electromagnetic field migration introduced above is related to minimization of the energy flow functional. The important step in the solution of the functional minimization problem (11.13) is calculating the steepest ascent direction (or the gradient) of the functional. To solve this problem, let us perturb the conductivity distribution crj, (x, z) = at, x, z)+Sa x, z). Actually, we have to perturb the conductivity only within the inhomogeneous domain F of the lower half-plane ... 

Let us consider as an example the 2-D gravity problem, generated by some mass distributed in the domain 5 of a lower half plane. In this model the observed data set do is equal to observed gravity field We can divide the domain S into a system of smaller subdomains (blocks Si) with constant densities Pj inside each block, as shown in Figure B-3. The gravity field of each elementary block Si with the unit density can be calculated using the formula ... 

Solving this system, we find the density distribution in the domain S of the lower half plane. Figure B-4 provides an example of a gravity inverse problem solution using this technique. 

Now we will obtain asymptotic formulae for the field in the far zone (a 1). In deriving a formula we will deform the contour of integration in eq. 10.33 on the complex plane of variable m. However, such a procedure requires either the proof of absence of poles of the integrand or evaluation of their contribution to the integral value. The problem of determination of poles is extremely difficult because of the complexity of the integrand. At the same time sufficient agreement of results of calculations by asymptotic and exact formulae allows us to think that if there are poles in the upper half-plane of m, their contribution in a considered part of the spectrum is sufficiently small. Let us present integral in eq. 10.33 in the form ... 

For the half-space problem where a gas fills the space x O bounded by a plane wall, one can define a set of dynamic quantities Qj = QjdVg U v, where v = (mg/2kT ) Vg is the dimensionless molecular velocity and T is the surface temperature. In Kuscer s analysis, the velocity distribution of molecules incident upon a surface is assumed to have the form of a slightly distorted... 

The number of scattering problems that can be solved analytically is severly limited by the inseparability of the vector wave equation in all but a very few coordinate systems. In the majority of cases various approximate methods have to be used. An excellent review of the analytic results for perfectly conducting bodies has been given by BOWMAN et al. [4.291. These include circular, elliptic, parabolic, and hyperbolic cylinders the wedge, the half plane, and other geometries. For infinite dielectric circular cylinders, see the review in KERKER [4.2]. 

Erdogan, F. and Gupta, G. D. (1971), The problem of an elastic stiffer bonded to a half plane, Journal of Applied Mechanics 38, 937-941. 

We consider the linear source q x, y) = h y) 6 x). Here (5(x) is a Dirac function and h y) is a sufficiently smooth function. Then problem (1) can be reduced to the following problem in a half-plane... 

IX. Causality. The requirement of Causality, namely that the current situation can be influenced only by past and contemporaneous events, may be shown to impose a constraint on the analytic structure of the complex moduli in the complex (JD plane, and also on combinations of the moduli multiplying Green s functions in the solution of non-inertial boundary value problems. These quantities can have no singularities in the lower half-plane. In a restricted sense, this can be shown directly for certain combinations of complex moduli using properties of the individual complex moduli. 

Our object is to apply (2.8.9) to problems involving loads on viscoelastic halfspaces. For such problems, it is desirable to re-express these equations in an alternative form [Muskhelishvili (1963)] which facilitates reduction to a Hilbert problem. Let the material occupy the upper half-plane > >0 so that (piz, t) is analytic in this region. It is convenient to extend the region of analyticity of (p(z, t) to the lower half-plane also. Then, as we shall see, it is possible to explore the discontinuities in this function across the real axis, which gives a Hilbert problem. Another approach, possibly more direct, is that of Galin, mentioned previously, which leads to problems of the Riemann-Hilbert type. These however are somewhat more difficult to deal with, from a mathematical point of view. 

The function 0(Zy t) is evaluated in the lower half-plane so that as z approaches the real axis from within the material, 0(z, t) and 0(z, 0 approach 0 x, t) and 0 Xyt), respectively, which are the limits of this complex function from above and below. We see from (3.1.3b), that, at points on the real axis where the boundary stresses are zero, 0(z, t) has no discontinuity. This is the essential reason for the choice of (3.1.1), that it gives this property. For contact problems, it means that the discontinuities in 0(Zy t) are confined to the regions of contact. 

In the case of the first boundary value problem, the stresses are given at every point on the real axis, at all times, and are zero at infinity. We let p(x, t)y t) be the applied pressure and shear on the (upper) half-plane. Then the complex stress is given by... 

We will consider the problem of a series of rigid indentors pressed into a viscoelastic half-space (y>0) and moving across it. If plane strain conditions are to hold, the indentors must be infinitely long in one direction, taken to be the z direction, and of uniform cross-section. Also the loading distribution must be uniform along each punch. We consider a typical cross-section of this configuration. All subsequent discussion refers to this cross-section of which the material occupies the half-plane y>0. 

We discuss the integral equation (3.4.2) in more detail for the problem of inden-tors in contact with a half-plane under the action of certain loads, and moving across it. In principle, the method outlined in the last section could handle any specified individual motion of the indentors. However, only the simplest will be considered, namely where the indentors are all moving in the same direction, taken to be along the negative x direction. 

This problem has also been considered in detail for an indentor moving over a layer of finite thickness, rather than a half-plane. We mention Alblas and Kuipers (1970), Margetson (1971, 1972), and Nachman and Walton (1978). Kalker (1975, 1977) reviews this topic in a systematic manner. 

The problem will be solved for the case where the viscoelastic half-plane is characterized by a discrete spectrum model (Sect. 1.6). The more general continuous spectrum model is discussed by Golden (1977). The proportionality assumption (Sect. 1.9) will be adopted for the material so that a unique Poisson s ratio exists. Therefore, from (1.6.25, 28, 29), (3.5.20) and (3.5.22), we have... 

IV. Hysteretic Friction. The hysteretic friction coefficient due to a moving inden-tor in lubricated contact with the half-plane is given by (3.8.1) and in the frictional case, by (3.8.9) or alternatively by (3.8.10) or (3.8.11). On an incompressible half-plane, it reduces to (3.8.12) which has the same form as for lubricated contact, apart from a factor (1 + / ). These expressions cannot be evaluated until the implicit equations governing the problem are solved, since the pressure is required. 

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