Particle on or near a Plane Surface

The geometry of the scattering problem is shown in Fig. 2.14. An axisymmetric particle is situated in the neighborhood of a plane surface so that its axis of symmetry is normal to the plane surface. The 2-axis of the particle coordinate system Oxyz is directed along the axis of symmetry and the origin O is situated at the distance zg below the plane surface. The incident radiation is a linearly polarized vector plane wave propagating in the ambient medium (the medium below the surface S) 

The incident wave strikes the particle either directly or after interacting with the surface. The direct and the reflected incident fields are expanded in terms of regular vector spherical wave functions 

The scattered field is expanded in terms of radiating vector spherical wave functions 

Inserting (2.198) into (2.197), we derive a series representation for the interacting field in terms of regular vector spherical wave functions 

In the null-field method, the scattered field coefficients are related to the expansion coefficients of the fields striking the particle by the transition matrix T. For an axisymmetric particle, the equations become uncoupled, permitting a separate solution for each azimuthal mode. Thus, for a fixed azimuthal mode m, we tnmcate the expansions given by (2.194)-(2.196) and (2.199), and derive the following matrix equation  

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A homogeneous, dielectric or perfectly conducting, axisymmetric particle on or near a plane surface (TPARTSUB.f90)... 

In this section, we present scattering results for an axis3nnmetric particle situated on or near a plane surface. For this purpose we use the TPARTSUB routine and a computer program based on the discrete sources method [59,60]. 

A more efficient approach is to base the boundary-integral formulation on a fundamental solution (or more accurately a Green s function) that incorporates the relevant boundary conditions at one or more of the surfaces. In the case of a particle or drop moving near an infinite plane wall, this means finding a solution for a point force that exactly satisfies the no-slip and kinematic boundary conditions at the wall. If we were to consider the motion of a particle or drop in a tube, it would be useful to have the solution for a point force satisfying the same conditions on the tube walls. 

ABS and HIPS. The yield stress vs. W/t curves of ABS and HIPS are very similar. They are somewhat surprising because the yield stresses reach their respective maximum values near the W/t (or W/b) where plane strain predominates. This behavior is not predicted by either the von Mises-type or the Tresca-type yield criteria. This also appears to be typical of grafted-rubber reinforced polymer systems. A plausible explanation is that the rubber particles have created stress concentrations and constraints in such a way that even in very narrow specimens plane strain (or some stress state approaching it) already exists around these particles. Consequently, when plane strain is imposed on the specimen as a whole, these local stress state are not significantly affected. This may account for the similarity in the appearance of fracture surface electron micrographs (Figures 13a, 13b, 14a, and 14b), but the yield stress variation is still unexplained. 

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