Lateral constraints

In order to stably levitate an object, the net force on it must be zero, and the forces on the body, if it is perturbed, must act to return it to its original position. The object must be at a local potential minimum that is, the second derivatives with respect to all spatial coordinates of the potential must be positive. This may seem, at first sight, to be trivial to arrange. However, any system whose potential is a solution to Laplace s equation is automatically unstable A statement in words of Laplace s equation is that the sum of the second partial derivatives of the potential is zero, and so not all can be simultaneously positive. This has long been known for electrostatic potentials, having been stated by Earnshaw(n) Millikan s scheme for suspending charged particles is thus only neutrally stable, since the fields within a Millikan capacitor provide no lateral constraint. 

The Lucas jigs use a ball contact to make and break the electrical circuit which is very effective but, as they have no lateral constraint, are prone to tilting and, hence, can only be used with rings having a large diameter to height ratio. 

These IF statements are really a form of discrete decision making embedded within the model. One possible approach to remove the difficulties it caused is to move the discrete decisions to the outside of the model and the continuous variable optimizer. For example, the friction factor equation can be selected to be the laminar one irrespective of the Reynolds number that is computed later. Constraints can be added to forbid movement outside the laminar region or to forbid movement too far outside the laminar region. If the solution to the well-behaved continuous variable optimization problem (it is solved with few iterations) is on such a constraint boundary, tests can be made to see if crossing the constraint boundary can improve the objective function. If so, the boundary is crossed—i.e., a new value is given to the discrete decision, etc. 

If the craze layer extends with complete lateral constraint, the strain in the craze is related to the change in its density. From a relationship between density and refractive index, an equation between strain in the craze and its refractive index can be derived. Although it is usual to start with the Lorenz-Lorentz equation, this may not be the correct relationship for a material having the structure of the craze (9). For the present purposes a linear relationship is assumed. The error introduced is at most 10% and only a few percent for the stretched craze with a high void content. 

There are a wide variety of static shear strength tests as shown in Figure 6.19. These various tests differ in the manner of lateral constraint, shearing plane, and whether they are undrained/drained. 

In the case of thin film sintering the lateral constraint imposed by a rigid substrate allows a shrinkage only in the direction (z-direction) perpendicular to the film. The imposed constraint induces an in-plane tensile stress and the stress accelerates the shrinkage in the z-direction. From the lateral constraint Ex = Sy = 0, cr = 0, and cTx = cry = cr, and the expressions of the uniaxial strain rates in Eq. (5.33), the densification rate in the z-direction is expressed as... 

P(6) Bearing failure (Figure 5.11 (b)) occurs in the material immediately adjacent to the contact area of fastener and laminate and is caused primarily by compressive stresses acting on the hole surface. It is likely to occur when the ratio of d/w is low or when the ratio of by-pass load to bearing load is low. Bearing failure is strongly affected by lateral constraint (clamping force), since lateral constraint prevents delamination of plies. 

P(9) In pin-loaded joints there is neither any clamping provided by tensile forces in the fasteners nor any lateral constraint provided in the vicinity of the fastener holes. 

P(10) Finger-tight joints are those with some clamping and lateral constraint in the vicinity of bolt holes provided by lightly torqued fasteners. 

P(11) Torqued fasteners are those with substantial clamping and lateral constraint provided by fasteners tightened to a pre-set torque. 

Fig. 6.10. Modelling the lateral constraints on the chain motion imposed by the entanglements by a tube . The average over the rapid wriggling motion within the tube defines the primitive path (continuous dark line)...

In this section we discuss the influence of lateral constraints, of the form used to define the tube diameter in sections II, III and IV, on polymer dynamics. Here we offer an alternative derivation of the reptation theory... 

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