Surface traction

User-Related Properties. The most important element in the player s contact with the surface is traction. Shoe traction for light-duty consumer purposes need address only provision of reasonable footing. The frictional characteristics are obviously of much greater importance in surfaces designed for athletic use. For specialized surfaces such as a track, shoe traction is especially critical. With grass-like surfaces, traction is significantly affected by pile density and height, and other aspects of fabric constmction. 

At the instant of contact between a sphere and a flat specimen there is no strain in the specimen, but the sphere then becomes flattened by the surface tractions which creates forces of reaction which produce strain in the specimen as well as the sphere. The strain consists of both hydrostatic compression and shear. The maximum shear strain is at a point along the axis of contact, lying a distance equal to about half of the radius of the area of contact (both solids having the same elastic properties with Poisson s ratio = 1/3). When this maximum shear strain reaches a critical value, plastic flow begins, or twinning occurs, or a phase transformation begins. Note that the critical value may be very small (e.g., in pure simple metals it is zero) or it may be quite large (e.g., in diamond). 

It has long been known that the spatial variation in surface tension at a li-quid/vapor surface results in added tangential stresses at the surface this results in a surface traction that acts on the adjoining fluid, giving rise to the... 

We now consider some models of polymer structure and ascertain their usefulness as representative volume elements. The Takayanagi48) series and parallel models are widely used as descriptive devices for viscoelastic behaviour but it is not correct to use them as RVE s for the following reasons. First, they assume homogeneous stress and displacement throughout each phase. Second, they are one-dimensional only, which means that the modulus derived from them depends upon the directions of the surface tractions. If we want to make up models such as the Takayanagi ones in three dimensions then we shall have a composite brick wall with two or more elements in each of which the stress is non-uniform. 

A conscious choice of such elements can be made but in general the equilibrium distribution of stress cannot be found except for particular geometries. The assumptions of uniform strain throughout the assembly or of uniform stress were respectively made by Voigt and by Reuss. Returning to the structures actually perceivable in polymers one may consider the spherulite in a semi crystalline polymer as being unsuitable as a RVE because the boundary is not included. However, an assembly of spherulites would be acceptable, since it would contain sufficient to make it entirely typical of the bulk and because such an assembly would have moduli independent of the surface tractions and displacements. The linear size of such a representative volume element of spherulites would be perhaps several hundred microns. 

Where a melt-crystallized polymer has been processed by drawing, rolling or other means to produce an aligned structure in which lamellae as well as polymer chains have discernible order, a pseudocrystalline unit cell is present. Provided that this unit cell contains elements of the crystals as well as the boundaries between crystals and that it is entirely typical of the material as a whole then it could be considered as a RVE within the meaning defined above. The lamella crystal itself sometimes considered as embedded in an amorphous matrix would not seem to be an acceptable RVE for reasons similar to those advanced against the Takayanagi model, namely that its modulus is dependent upon the surface tractions. The boundaries between lamella crystals in the matrix must be included in an acceptable RVE. 

We need to consider the size of the representative volume element (RYE) for polymers bearing in mind the requirements of its definition. These were 1) the RVE should be entirely typical of the material on average 2) it must contain sufficient of the inclusions (phases) for the overall moduli to be independent of the surface tractions and displacements provided these are macroscopically uniform. 

Boundary element methods can be used for particulate flows where direct1 formulations can be used. The surface tractions on the solids are integrated to compute the hydrodynamic force and torque on those particles, which for suspended particles must be zero. 

The direct boundary integral formulation was used to simulate suspended spheres in simple shear flow. The viscosity was then calculated by integration of the surface tractions on the moving wall. Figure 10.28 shows a typical mesh for the domain and spheres for these simulations in this mesh, the box has dimensions of 1 x 1 x 1 (Length units)3 and 40 spheres of radius of 0.05 length units. 

Here ma is the bulk solid-fluid interaction force, T.s the partial Cauchy stress in the solid, p/ the hydrostatic pressure in the perfect fluid, IIS the second-order stress in the solid, ha the density of partial body forces, ta the partial surface tractions, ts the traction corresponding to the second-order stress tensor in the solid and dvs/dn the directional derivative of v.s. along the outward unit normal n to the boundary cXl of C. 

In this paper we limit our attention to external actions for which bs 0 and by 0, i.e. only to external surface tractions. In order to find the partial tractions we assume the existence of a potential function such that the working, li/ex of external surface tractions is given by... 

We analyze, within a linearized second gradient theory, the static infinitesimal deformations of an annular porous cylinder filled with an inviscid fluid and with the inner and the outer surfaces subjected to uniform external pressures pj xt and iff1 respectively. We assume that surface tractions on the inner and the outer surfaces of the cylinder, in the reference configuration, equal -po and postulate that... 

Surface tractions or contact forces produce a stress field in the fluid element characterized by a stress tensor T. Its negative is interpreted as the diffusive flux of momentum, and x x (—T) is the diffusive flux of angular momentum or torque distribution. If stresses and torques are presumed to be in local equilibrium, the tensor T is easily shown to be symmetric. 

For SDE, with a constant sliding stress, t0, the sliding distance,/, in the absence of fiber failure, is related to the crack surface tractions, ab, by14,31... 

Fig. 6 Schematic representation of the cohesive surface traction-opening law (1) no crazing, (2) craze widening with (2a) hardening-like response or (2b) softening-like response depending on the prescribed opening rate, and (3) craze breakdown at An = A r...

In order to satisfy the stress-free boundary condition, coupled compressional and shear waves propagate together in a SAW such that surface traction forces are zero (i.e., T y = 0, where y is normal to the device surface). The generalized surface acoustic wave, propagating in the z-direction, has a displacement profile u(y) that varies with depth y into the crystal as... 

Surface tractions pull fluid in wherever a is locally higher... 

If the loading applied to the cracked body is distributed as stresses over the boundary instead of being a concentrated k)ad, then the various work terms must be evaluated as integrals taken over the boundary. Thus, if we consider a body as shown in Fig. 3 with normal and shear surface tractions o and Os giving dis-... 

We now examine the question of how in continuum mechanics the forces due to material external to the region are communicated to it. Note that we will adopt the notation dQ to characterize the boundary of the region Q. In simplest terms, forces are transmitted to a continuum either by the presence of body forces or via surface tractions . Body forces are those such as that due to gravity which... 

The Cauchy stress principle arises through consideration of the equilibrium of body forces and surface tractions in the special case of the infinitesimal tetrahedral volume shown in fig. 2.7. Three faces of the tetrahedron are perpendicular to the Cartesian axes while the fourth face is characterized by a normal n. The idea is to insist on the equilibrium of this elementary volume, which results in the observation that the traction vector on an arbitrary plane with normal n (such as is shown in the figure) can be determined once the traction vectors on the Cartesian planes are known. In particular, it is found that = crn, where a is known as the stress tensor, and carries the information about the traction vectors associated with the Cartesian planes. The simple outcome of this argument is the claim that... 

In order to point out the essential difference between deformations induced by body forces and surface tractions we recall Ericksen s theorem [88]. The theorem states that homogeneous deformations are the only deformations that can be achieved by the application of surface tractions alone, considering a homogeneous and isotropic material characterized by an arbitrary strain-energy function. In other words, we cannot induce diverse non-homogeneous deformations with surface tractions. In contrast to surface tractions, application of fields that act as body forces leads to non-homogeneous deformations without any additional constraints for the material. 

Apply sufficient surface traction to restore the elements to their original shape (Fig. 13.1r). 

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