Point force

The most celebrated textual embodiment of the science of energy was Thomson and Tait s Treatise on Natural Philosophy (1867). Originally intending to treat all branches of natural philosophy, Thomson and Tait in fact produced only the first volume of the Treatise. Taking statics to be derivative from dynamics, they reinterpreted Newton s third law (action-reaction) as conservation of energy, with action viewed as rate of working. Fundamental to the new energy physics was the move to make extremum (maximum or minimum) conditions, rather than point forces, the theoretical foundation of dynamics. The tendency of an entire system to move from one place to another in the most economical way would determine the forces and motions of the various parts of the system. Variational principles (especially least action) thus played a central role in the new dynamics. 

Fig. 8. Velocity field caused by a point force in a pure solvent and in...

Here V represents the local volume of a computational cell and Va the volume of particle a. The 5-function ensures that the drag force acts as a point force at the (central) position of this particle. In Eq. (42), [ > is the momentum transfer coefficient, which will be discussed in more detail in Section III.D. The gas phase density p is calculated from the ideal gas law ... 

In the point force approximation technique (see Section Ic), Burgers (BIO) suggested a polynomial approximation for the distributed line force along the axis of a body of large aspect ratio ... 

A compressive force in terms of a 2 point or a 12 point force breaks open the capsule by mechanical means ... 

The total resisting force F, is found by multiplying the point force fr by the circumference. 

Here u (x - xo) represents the velocity field generated by a point force in the -direction at a point x0 and qk is the corresponding pressure. 

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]... 

Note that all the stresses in Eq. (2.27) become singular at the origin where the point force is applied. To avoid this singularity, consider a small spherical cavity whose center is located at the origin of the coordinates as shown in Fig. 2.4. The coordinates are arranged such that the force is in the z-direction and is applied at the origin of the coordinates. Thus, the summation of the surface forces from the stresses in the direction of the z-axis balances the point force inside the solid medium. 

Since Fz is equal to the applied point force, the constant A is obtained from the preceding equation as... 

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. 

Figure 2.7. A point force on the boundary of a semiinfinite medium (a) r — z coordinates (b) R — coordinates.

Consider the case where a point force is acting on the boundary of a semiinfinite body, as shown in Fig. 2.7. Let the boundary be in the plane z = 0. Then, the shear stress on the boundary would be given by Eq. (2.27) as... 

The balance between the point force and the integration of Tz over the hemispherical surface yields... 

As a result of applying a point force on the boundary, displacements in the semiinfinite solid are produced. These displacements can be calculated from Hooke s law and the displacement-strain relationships. The displacement in the r-direction, /r, is given by Eq. (2.20) and Eq. (2.23) as... 

The displacements given in Eq. (2.55) for a point force on a semiinfinite solid indicate that the displacements would be infinite when r approaches zero at the origin where the point force is applied. This singularity may be avoided by using a load distribution over a finite area to replace the point force. 

The influence of this pressure distribution on the displacements in a neighboring area on the same boundary surface can be analyzed as shown in Fig. 2.8. Let M be an arbitrary point outside the circular area (Fig. 2.8(a)) or inside the circular area (Fig. 2.8(b)) with a distance r from the center of the circular area. A small element within the loaded zone is chosen as s d/5 ds, where s is the distance between this element and M. From the solution of lz for a point force on a semiinfinite solid, the increment of the vertical displacement at location M under the influence of this element force would be... 

A more common situation, however, is that before there is any significant frictional heating, an axial position is reached where the barrel is heated to well above the melting point, forcing the creation of a him of melt. In either case, this marks the end of that portion of the process in the extruder called the solids conveying zone, where only solids are present and the only elementary step that occurs is handling of solids. 

The hierarchy of equations thereby obtained can be closed by truncating the system at some arbitrary level of approximation. The results eventually obtained by various authors depend on the implicit or explicit hypotheses made in effecting this closure—a clearly unsatisfactory state of affairs. Most contributions in this context aim at calculating the permeability (or, equivalently, the drag) of a porous medium composed of a random array of spheres. The earliest contribution here is due to Brinkman (1947), who empirically added a Darcy term to the Stokes equation in an attempt to represent the hydrodynamic effects of the porous medium. The so-called Brinkman equation thereby obtained was used to calculate the drag exerted on one sphere of the array, as if it were embedded in the porous medium continuum. Tam (1969) considered the same problem, treating the particles as point forces he further assumed, in essence, that the RHS of Eq. (5.2a) was proportional to the average velocity and hence was of the explicit form... 

Fig. 23.3. (a) Schematic depiction of the creation of a lipid nanotube by application of a point force, allowing interconnection of vesicles, (b) Mechanical stresses and physical properties of planar phospholipid assemblies... 

In the longer term, picoindentation instruments are likely to be widely used to extend the technique to a still smaller scale, with the help of techniques developed for atomic force microscopy. Already, plastic deformation at depths of a few atomic layers, as well as the effect of surface forces, have been quantified by means of depth-load measurements, using a point force microscope, i.e. an AFM operated in static (non-scanning) mode (Burnham Colton, 1989). 

The waterproofing body underneath the mat may be a plastic membrane, a thin aluminium sheet, or a bituminous/ concrete bed. These mats may be loosely placed or may be glued at certain points on the sub-layer. When placed on the concrete or on the asphalt, point forces may be exerted by the underlying aggregate. In addition, during construction activities, falling of reinforcements and other objects may induce local damage in the mat and in the... 

The presence of a (planar) body located at the origin is equivalent, in the far field, to a point force located at the origin (eg. see [75]). The flow u = (ux, uy, uz) satisfies the linearised momentum equation,... 

Vorticity is advected downstream and diffuses, primarily, cross-stream. The presence of a point force creates a dipole source of vorticity (last term in (7.2)) - corresponding to a source and sink or equal strength, close together. An exact solution to (7.2) is... 

Within the context of the elastic Green function, the reciprocal theorem serves as a jumping off point for the construction of fundamental solutions to a number of different problems. For example, we will first show how the reciprocal theorem may be used to construct the solution for an arbitrary dislocation loop via consideration of a distribution of point forces. Later, the fundamental dislocation solution will be bootstrapped to construct solutions associated with the problem of a cracked solid. 

The basic idea is to exploit the fact that the displacements associated with the point force are already known, and correspond to the elastic Green function. Further, the displacements associated with the dislocation are prescribed on the slip plane. We can rewrite the surface integral that spans the slipped region as... 

In this case f )(+) refers to the traction associated with the dislocation just above the slip plane while t (—) refers to the value of the traction just below the slip plane. However, since the displacement fields (u ) associated with the point force are continuous across the slip plane, Similarly, the tractions... 

Thus far, we have treated only one side of eqn (8.27). Our next task is to probe the significance of the term involving the body force. Through the artifice of choosing the fields associated with a point force as our complementary fields, the dislocation solutions can be obtained directly. In particular, we note that if... 

Now, we have expressed the general streamfunction, (7-149), and the disturbance flow contribution in (7-150) and (7-151), in terms of spherical coordinates. However, we have not yet specified a body shape. Thus the linear decrease of the disturbance flow with distance from the body must clearly represent a property of creeping-flows that has nothing to do with specific coordinate systems. Indeed, this is the case, and the velocity field (7-151) plays a very special and fundamental role in creeping-flow theory. It is commonly known as the Stokeslet velocity field and represents the motion induced in a fluid at Re = 0 by a point force at the origin (expressed here in spherical coordinates).17 We shall see later that the Stokeslet solution plays an important role in many aspects of creeping-flow theory. 

---