Cylindrical Inclusions

The purpose of this chapter is to extend the ideas in Chapter 16 to the situation where one material occurs as an inclusion in the other. In Chapters 13 through 15, most of the discussion centered on conditions that varied along one direction, x, but not along orthogonal directions. At several points, a cylindrical tube with fixed radius was imagined, but this was only a handy visualization of the condition where all velocities are zero in planes normal to x. The use of the equations in Chapters 13 through 16 is to describe conditions close to a planar interface of large extent. If the ratio (distance from interface)/(breadth of a planar portion of interface) is small, behavior is as if the interface were infinitely extensive, and it is to this condition that the equations apply. As a step toward understanding behavior around an inclusion, we now consider a long cylinder of one material embedded in an unlimited extent of a second material. The axis of the cylinder is taken as the y-direction and we continue to assume that everything is uniform in this direction: all properties and all behaviors are uniform along y and all velocities along y are zero. But in xz planes we now see a circular cross-section as in Figure 18.1 instead of just a planar boundary. As regards stress state, let this be uniform throughout the host material except insofar as the inclusion causes variation; let the remote stress state have principal compressive stresses σxx and σzz with σzz larger. To start, we make the same assumptions as in Chapter 13, namely that the two materials are uniform and of the same chemical composition, differing only in viscosity; and let the material of the inclusion be stiffer. Because of σzz being larger, the entire assembly will at any moment be shortening along z and elongating along x, and if the cross-section is circular at the moment we inspect it, it will be elliptical at all later times. An impression of how deformation proceeds is given in Figure 18.2, which shows how a grid would look at a later time if it had been a square grid at the moment when the inclusion was circular.