Infinite solid

FIGURE 26.18 Theoretical temperature rise in the contact area of a pad sliding over a semi-infinite solid for different depths from the surface. Width 2b 2 mm, speed 3 m/s, pressure 2 Mp, p—l, heat conductivity 0.15 W/m/K, heat diffusivity 10 " m /s. 

From cluster to infinite solid a quantum study of the electronic properties of M0O3 A. Rahmouni and C. Barbier 427... 

Only a finite difference numerical solution can give exact results for conduction. However, often the following approximation can serve as a suitable estimation. For the unsteady case, assuming a semi-infinite solid under a constant heat flux, the exact solution for the rate of heat conduction is... 

The semiconductor support can be treated in a similar manner (Bose and Foo 1974), since the Greenians G2 and g2 of the infinite and semi-infinite solids, respectively, are also linked by the Dyson equation, i.e.,... 

Approach to the Simulation of Elastic, Semi-Infinite Solids. 

The diffusion coefficients for Rb, Cs and Sr in obsidian can be calculated from the aqueous rate data in Table 1 as well as from the XPS depth profiles. A simple single-component diffusion model (9j characterizes onedimensional transport into a semi-infinite solid where the diffusion coefficient (cm2-s 1) is defined by ... 

The solution to this partial differential equation depends upon geometry, which imposes certain boundary conditions. Look np the solution to this equation for a semi-infinite solid in which the surface concentration is held constant, and the diffusion coefficient is assumed to be constant. The solution should contain the error function. Report the following the bonndary conditions, the resulting equation, and a table of the error function. 

In an infinite solid this set of critical points obeys a number of theorems, the chief being the Euler equation (eqn (14.1)) ... 

Fig. 4. Rate of melting of semi-infinite solid for various values of mi (L4). Reproduced by permission of Quarterly of Applied Mathematics.

Citron (C4) generalizes Landau s derivation of the steady-state melting rate of a semi-infinite solid with instantaneous removal of the melt to temperature-dependent thermal conductivity and specific heat, expressible in the form... 

Concentrated Force at a Point in an Infinite Solid Medium... 

Consider the case of a concentrated force applied to a point in an infinite solid medium. To find the relationships between the point force and the resulting stresses, Love s stress function may be selected, from sets of solutions of Eq. (2.25), as [Timoshenko and Goodier, 1970]... 

Figure 2.4. Surface stresses on a spherical cavity in an infinite solid medium.

In this section, the case of a semiinfinite solid with a concentrated force acting on the boundary is introduced. This case was originally solved by Boussinesq (1885). It should be noted that the only difference between this case and the case of a point force in an infinite solid medium is the boundary conditions. Shear stresses vanish on the boundary of the semiinfinite solid. In the following, the concept of a center of compression is introduced. The stress field in a semiinfinite solid with a boundary force can be obtained by superimposing the stress fields from a point force and a series of centers of compression. A center of compression is defined as the combination of three perpendicular pair forces. 

Consider a case of two equal but opposite forces (in the z-direction) positioned a small distance 5 apart in an infinite solid body, as shown in Fig. 2.5. At any point M, the stresses... 

Assume that the centers of pressure are uniformly distributed along the z-axis from z = 0 to z = —oo. The stresses in an infinite solid medium can be obtained by integration from... 

As pointed out in the previous section, melting can often be modeled in terms of simple geometries. Here we analyze the transient conduction problem in a semi-infinite solid. We compare the solutions of this problem, assuming first (a) constant thermophysical properties, then (b) variable thermophysical properties and finally, and (c) a phase transition with constant thermophysical properties in each phase. These solutions, though useful by themselves, also help demonstrate the profound effect of the material properties on the mathematical complexities of the solution. 

Example 5.2 Semi-infinite Solid with Constant Thermophysical Properties and a Step Change in Surface Temperature Exact Solution The semi-infinite solid in Fig. E5.2 is initially at constant temperature Tq. At time t — 0 the surface temperature is raised to T. This is a one-dimensional transient heat-conduction problem. The governing parabolic differential equation... 

Fig. E5.2 Temperature profiles in a semi-infinite solid with a step change in temperature at the boundary.

Example 5.3 The Semi-infinite Solid with Variable Thermophysical Properties and a Step Change in Surface Temperature Approximate Analytical Solution We have stated before that the thermophysical properties (k, p, Cp) of polymers are generally temperature dependent. Hence, the governing differential equation (Eq. 5.3-1) is nonlinear. Unfortunately, few analytical solutions for nonlinear heat conduction exist (5) therefore, numerical solutions (finite difference and finite element) are frequently applied. There are, however, a number of useful approximate analytical methods available, including the integral method reported by Goodman (6). We present the results of Goodman s approximate treatment for the problem posed in Example 5.2, for comparison purposes. 

Example 5.4 Melting of a Semi-infinite Solid with Constant Thermophysical Properties and a Step Change in Surface Temperature The Stefan-Neumann Problem The previous example investigated the heat conduction problem in a semi-infinite solid with constant and variable thermophysical properties. The present Example analyzes the same conduction problem with a change in phase. 

Fig. E5.4 Melting in a semi-infinite solid. X/(t) is the thickness of the molten layer at time t, Xs(t) is the distance of the interface from the location of external surface at time t — 0. The temperature profile in the solid is expressed in coordinate xs, which is stationary, whereas the temperature profile in the melt is expressed in coordinate xh which has its original outer surface of melt, hence, it slowly moves with time if ps pt...

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