Continuum surface stress

The second fairly modern group of methods introduces (of numerical reasons) a 3D continuous surface force (CSF) or a 3D continuum surface stress (CSS) acting locally within the whole transition region constituting a meso-scale interface. Notice that since we are primarily interested in the interfacial forces, the latter group of techniques were used approximating the surface effects without actually reconstructing the interface. 

For this reason the surface tension force can be interpreted as a correction to the momentum stress tensor, i.e., it represents a continuum surface stress (CSS) tensor. 

Fig. 40. Co-induced surface stress on Cu(lOO). Deposition proceeds beyond 15 ML. The slope corresponds to a tensile film stress of 2.5 GPa. The lattice mismatch of 2 % is expected to induce a tensile film stress of 2.9 GPa as calculated ftom 3 order continuum elasticity. Data from [99Gut] (1 ML 1.53xl0 atoms / cm ).

When a fluid flows past a solid surface, the velocity of the fluid in contact with the wall is zero, as must be the case if the fluid is to be treated as a continuum. If the velocity at the solid boundary were not zero, the velocity gradient there would be infinite and by Newton s law of viscosity, equation 1.44, the shear stress would have to be infinite. If a turbulent stream of fluid flows past an isolated surface, such as an aircraft wing in a large wind tunnel, the velocity of the fluid is zero at the surface but rises with increasing distance from the surface and eventually approaches the velocity of the bulk of the stream. It is found that almost all the change in velocity occurs in a very thin layer of fluid adjacent to the solid surface ... 

When considering boundary conditions, a useful dimensionless hydrodynamic number is the Knudsen number, Kn = X/L, the ratio of the mean free path length to the characteristic dimension of the flow. In the case of a small Knudsen number, continuum mechanics will apply, and the no-slip boundary condition assumption is valid. In this formulation of classical fluid dynamics, the fluid velocity vanishes at the wall, so fluid particles directly adjacent to the wall are stationary, with respect to the wall. This also ensures that there is a continuity of stress across the boundary (i.e., the stress at the lower surface—the wall—is equal to the stress in the surface-adjacent liquid). Although this is an approximation, it is valid in many cases, and greatly simplifies the solution of the equations of motion. Additionally, it eliminates the need to include an extra parameter, which must be determined on a theoretical or experimental basis. 

Lu et al. [7] extended the mass-spring model of the interface to include a dashpot, modeling the interface as viscoelastic, as shown in Fig. 3. The continuous boundary conditions for displacement and shear stress were replaced by the equations of motion of contacting molecules. The interaction forces between the contacting molecules are modeled as a viscoelastic fluid, which results in a complex shear modulus for the interface, G = G + mG", where G is the storage modulus and G" is the loss modulus. G is a continuum molecular interaction between liquid and surface particles, representing the force between particles for a unit shear displacement. The authors also determined a relationship for the slip parameter Eq. (18) in terms of bulk and molecular parameters [7, 43] ... 

A third model for feature-scale polish was proposed by Runnels [40], and focuses on stresses created by flowing slurry on feature surfaces under continuum mechanics. The model incorporates fracture mechanics and chemistry through empirical means. The geometry of a typical structure under study is shown in Fig. 8. 

When the experimentalist set an ambitious objective to evaluate micromechanical properties quantitatively, he will predictably encounter a few fundamental problems. At first, the continuum description which is usually used in contact mechanics might be not applicable for contact areas as small as 1 -10 nm [116,117]. Secondly, since most of the polymers demonstrate a combination of elastic and viscous behaviour, an appropriate model is required to derive the contact area and the stress field upon indentation a viscoelastic and adhesive sample [116,120]. In this case, the duration of the contact and the scanning rate are not unimportant parameters. Moreover, bending of the cantilever results in a complicated motion of the tip including compression, shear and friction effects [131,132]. Third, plastic or inelastic deformation has to be taken into account in data interpretation. Concerning experimental conditions, the most important is to perform a set of calibrations procedures which includes the (x,y,z) calibration of the piezoelectric transducers, the determination of the spring constants of the cantilever, and the evaluation of the tip shape. The experimentalist has to eliminate surface contamination s and be certain about the chemical composition of the tip and the sample. 

The dilational rheology behavior of polymer monolayers is a very interesting aspect. If a polymer film is viewed as a macroscopy continuum medium, several types of motion are possible [96], As it has been explained by Monroy et al. [59], it is possible to distinguish two main types capillary (or out of plane) and dilational (or in plane) [59,60,97], The first one is a shear deformation, while for the second one there are both a compression - dilatation motion and a shear motion. Since dissipative effects do exist within the film, each of the motions consists of elastic and viscous components. The elastic constant for the capillary motion is the surface tension y, while for the second it is the dilatation elasticity e. The latter modulus depends upon the stress applied to the monolayer. For a uniaxial stress (as it is the case for capillary waves or for compression in a single barrier Langmuir trough) the dilatational modulus is the sum of the compression and shear moduli [98]... 

The exact laws, based on continuum analysis of the fibers and the matrix, would be very complicated. The analysis would involve equilibrium of stresses around, and in, the fibers and compatibility of matrix deformation with the fiber strains. Furthermore, end and edge effects near the free surfaces of the composite material would introduce complications. However, a simplified model can be developed for the interior of the composite material based on the notion that the fibers and the matrix interact only by having to experience the same longitudinal strain. Otherwise, the phases behave as two uniaxially stressed materials. McLean5 introduced such a model for materials with elastic fibers and he notes that McDanels et al.6 developed the model for the case where both the fibrous phase and the matrix phase are creeping. In both cases, the longitudinal parameters are the same, namely... 

We have been able to interpret this experimental data by means of a very simple continuum model. Assuming the cluster to be plastic with a uniform internal stress, the relationship between this internal stress and the radius of curvature of the cluster s free surface is given by the Young equation... 

For an arbitrary volume x of the multicomponent continuum, the total rate of change of linear momentum in the 7th coordinate direction must equal the sum of the following (1) the surface integral of the stress vector Y,k where afj equals the component in the direction Xj of the stress vector acting on that face of an elemental parallelepiped of species K which has an... 

Generally, for any dimension therefore, if a crack of length I already exists in an infinite elastic continuum, subject to uniform tensile stress a perpendicular to the length of the crack, then for the onset of brittle fracture, Griffith equates (the differentials of) the elastic energy E with the surface energy E ... 

The introduction of the concepts of stress and deformation at a point has been a fundamental concept in the development of the mechanics of continuum media. From a physical point of view, only the displacement is a real quantity, while stress imphes an idealized situation that is not directly measurable the value of a stress can only be inferred from its effects. The effects of the force at a point P depend on the orientation of the element surface SA comprising the point, which in turn is characterized by a vector rij (j =1,2,3) normal to the surface at P, as shown in Figure 4.1. The stress vector at the point P can be written as... 

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