Half-space problems

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... 

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. 

The corresponding relative displacement of the centers of the spheres , based on the solutions of the relevant elastic half-space problem, is... 

The Green s functions will be attributed a dependence upon co to indicate that the moduli have been replaced by the complex moduli. We will omit subscripts, in this and the next two sections, which amounts to developing a one-dimensional rather than a three-dimensional theory. This results in a considerable tidying of the equations and the loss of generality is irrelevant in the present context, because only the one-dimensional theory is required, in any case, in later chapters. This is because attention is focussed on crack and half-space problems of such a nature that only the normal displacement and pressure are relevant to the solution of the problem. 

It is to be expected that three-dimensional boundary value problems will present greater difficulties than plane problems. In particular, with the far wider choice of boundary regions on which to specify displacement and stress, one rapidly meets problems that are unsolvable - at least analytically. This is true even for elastic materials. In fact, the contact problem with an elliptical contact area is the most general problem that allows an explicit analytic solution - for elastic materials [Galin (1961), Lur e (1964)], in the case of half-space problems. This corresponds to an ellipsoidal indentor, according to classical Hertz theory. The theory can be extended to cover contact between two gently curved bodies. The solution is valid only for quasi-static conditions. 

Secondly, the linearized inverse problem is, as well as known, ill-posed because it involves the solution of a Fredholm integral equation of the first kind. The solution must be regularized to yield a stable and physically plausible solution. In this apphcation, the classical smoothness constraint on the solution [8], does not allow to recover the discontinuities of the original object function. In our case, we have considered notches at the smface of the half-space conductive media. So, notche shapes involve abrupt contours. This strong local correlation between pixels in each layer of the half conductive media suggests to represent the contrast function (the object function) by a piecewise continuous function. According to previous works that we have aheady presented [14], we 2584... 

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]... 

As an example of the use of Bessel functions in potential theory we shall consider the problem of determining a function ip[g, z) lor the half-space n Si 0, z 0 satisfying the differential equation... 

Another example of half-space diffusion problem is as follows. A thin-crystal wafer initially contains some " Ar (e.g., due to decay of " K). When the crystal is heated up (metamorphism), " Ar will diffuse out and the surface concentration of " Ar is zero. If the temperature is constant and, hence, the diffusivity is constant, the problem is also a half-space diffusion problem. 

A third example is as follows. Initially a crystal has a uniform 8 0. Then the crystal is in contact with a fluid with a higher 8 0. Ignore the dissolution of the crystal in the fluid (e.g., the fluid is already saturated with the crystal). Then would diffuse into the crystal. Because fluid is a large reservoir and mass transport in the fluid is rapid, 8 0 at the crystal surface would be maintained constant. Hence, this is again a half-space diffusion problem with uniform initial concentration and constant surface concentration. The evolution of 8 0 with time is shown in Figure l-8b. 

For a diffusion couple, the definition of Amid requires some thinking because the mid-concentration of the whole diffusion couple is right at the interface, which does not move with time. This is because for a diffusion couple every side is diffusing to the other side. On the other hand, if a diffusion couple is viewed as two half-space diffusion problems with the interface concentration viewed as the fixed surface concentration, then. Amid equals 0.95387(Df), the same as the half-space diffusion problem. 

Another experimental method to investigate diffusion is the so-called half-space method, in which the sample (e.g., rhyolitic glass with normal oxygen isotopes) is initially uniform with concentration C, but one surface (or all surfaces, as explained below) is brought into contact with a large reservoir (e.g., water vapor in which oxygen is all 0). The surface concentration of the sample is fixed to be constant, referred to as Cq. The duration is short so that some distance away from the surface, the concentration is unaffected by diffusion. Define the surface to be X = 0 and the sample to be at x > 0. This diffusion problem is the so-called halfspace or semi-infinite diffusion problem. 

A typical feature of the statement of the problem is the vacuum in the left half-space, x < 0, at the initial time. Gas particles located initially at the surface (at x = 0) fly out into the vacuum after the action of the external pressure is completed. 

PROBLEM L2.5 Derive approximation (L2.145) by expansion of Eq. (L2.144) and by differentiation of — [AHam/12jrZ2] for the interaction of half-spaces. 

Diffusion time (diffusion time constant) — This parameter appears in numerous problems of - diffusion, diffusion-migration, or convective diffusion (- diffusion, subentry -> convective diffusion) of an electroactive species inside solution or a solid phase and means a characteristic time interval for the process to approach an equilibrium or a steady state after a perturbation, e.g., a stepwise change of the electrode potential. For onedimensional transport across a uniform layer of thickness L the diffusion time constant, iq, is of the order of L2/D (D, -> diffusion coefficient of the rate-determining species). For spherical diffusion (inside a spherical volume or in the solution to the surface of a spherical electrode) r spherical diffusion). The same expression is valid for hemispherical diffusion in a half-space (occupied by a solution or another conducting medium) to the surface of a disk electrode, R being the disk radius (-> diffusion, subentry -> hemispherical diffusion). For the relaxation of the concentration profile after an electrical perturbation (e.g., a potential step) Tj = L /D LD being - diffusion layer thickness in steady-state conditions. All these expressions can be derived from the qualitative estimate of the thickness of the nonstationary layer... 

The problem is specified as the determination of the state of stress and the deformation produced in viscoelastic half-space (z < 0) by a circular punch of radius a whose force is P (Fig. 16.4). As is well known (Ref. 15, p. 25), the displacement of a half-space caused by forces applied to its free surface with the condition of null deformation at infinite distance is given by... 

SC Hunter. The Hertz problem for a rigid spherical indentor and a viscoelastic half-space. J Mech Phys Solids 8 219, 1960. 

The Kirchhoff approximation is based on expressing the scattered field p (r, oj) at some point r in the upper half-space, using its values at the reflecting boundary (see Figure 14-2). To solve this problem we apply the Kirchhoff integral formula (13.197) to the scattered field in the upper half-space ... 

We turn now to the other side of the colloidal particle interaction problem idealised to the case of two half spaces separated by salt water [3-10, 22-24]. Typically such particles will contain ionisable groups at their surfaces, so that the surfaces are charged. Imagine that the water, as before, retains its bulk properties up to the surface of the half spaces. The charged surfaces create an inhomogeneous profile of cationic and anionic density. For an isolated surface at the simplest level of approximation and schematically only, this distribution follows from the equation ... 

Eqn (2.92) is the culmination of our efforts to compute the displacements due to an arbitrary distribution of body forces. Although this result will be of paramount importance in coming chapters, it is also important to acknowledge its limitations. First, we have assumed that the medium of interest is isotropic. Further refinements are necessary to recast this result in a form that is appropriate for anisotropic elastic solids. A detailed accounting of the anisotropic results is spelled out in Bacon et al. (1979). The second key limitation of our result is the fact that it was founded upon the assumption that the body of interest is infinite in extent. On the other hand, there are a variety of problems in which we will be interested in the presence of defects near surfaces and for which the half-space Green function will be needed. Yet another problem with our analysis is the assumption that the elastic constants... 

Fig. 11.15. Conformal transformation from wedge + dislocation to the problem of a half-space containing dislocation (adapted from Ohr et al. (1985)).

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