Stress surface

In Chapter III, surface free energy and surface stress were treated as equivalent, and both were discussed in terms of the energy to form unit additional surface. It is now desirable to consider an independent, more mechanical definition of surface stress. If a surface is cut by a plane normal to it, then, in order that the atoms on either side of the cut remain in equilibrium, it will be necessary to apply some external force to them. The total such force per unit length is the surface stress, and half the sum of the two surface stresses along mutually perpendicular cuts is equal to the surface tension. (Similarly, one-third of the sum of the three principal stresses in the body of a liquid is equal to its hydrostatic pressure.) In the case of a liquid or isotropic solid the two surface stresses are equal, but for a nonisotropic solid or crystal, this will not be true. In such a case the partial surface stresses or stretching tensions may be denoted as Ti and T2-... 

Shuttleworth [26] (see also Ref. 27) gives a relation between surface free energy and stretching tension as follows. For an anisotropic solid, if the area is increased in two directions by dAi and dA2, as illustrated in Fig. VII-1, then the total increase in free energy is given by the reversible work against the surface stresses, that is. 

Calculated Surface Energies and Surface Stresses (ergs/cm )... 

Surface Stresses and Edge Energies. Some surface tension values, that is, values of the surface stress t, are included in Table VII-2. These are obtained by applying Eq. Vll-5 to the appropriate lattice sums. The calculation is very sensitive to the form of the lattice potential. Earlier calculations have given widely different results, including negative r [43, 51, 52]. 

Actual crystal planes tend to be incomplete and imperfect in many ways. Nonequilibrium surface stresses may be relieved by surface imperfections such as overgrowths, incomplete planes, steps, and dislocations (see below) as illustrated in Fig. VII-5 [98, 99]. The distribution of such features depends on the past history of the material, including the presence of adsorbing impurities [100]. Finally, for sufficiently small crystals (1-10 nm in dimension), quantum-mechanical effects may alter various physical (e.g., optical) properties [101]. 

An indirect estimate of surface tension may be obtained from the change in lattice parameters of small crystals such as magnesium oxide and sodium chloride owing to surface tensional compression [121] however, these may represent nonequilibrium surface stress rather than surface tension [68]. Surface stresses may produce wrinkling in harder materials [122]. 

Bikerman [182] criticized the derivation of Eq. X-18 out of concern for die ignored vertical component of On soft surfaces a circular ridge is raised at the periphery of a drop (see Ref. 67) on harder solids there is no visible effect, but the stress is there. It has been suggested that the contact angle is determined by the balance of surface stresses rather than one of surface free energies, the two not necessarily being the same for a... 

Characterization. The proper characterization of coUoids depends on the purposes for which the information is sought because the total description would be an enormous task (27). The foUowiag physical traits are among those to be considered size, shape, and morphology of the primary particles surface area number and size distribution of pores degree of crystallinity and polycrystaUinity defect concentration nature of internal and surface stresses and state of agglomeration (27). Chemical and phase composition are needed for complete characterization, including data on the purity of the bulk phase and the nature and quaHty of adsorbed surface films or impurities. 

Figure 12.2 shows that the stress in the beam is zero along the neutral axis at its centre, and is a maximum at the surface, at the mid-point of the beam (because the bending moment is biggest there). The maximum surface stress is given by... 

Because of this, the data listed in Table 15.7 for ceramic materials differ in emphasis from those listed for metals. In particular, the Table shows the modulus of rupture (the maximum surface stress when a beam breaks in bending) and the thermal shoek resist-anee (the ability of the solid to withstand sudden changes in temperature). These, rather than the yield strength, tend to be the critical properties in any design exercise. 

The main causes of failure in gear couplings are wear or surface fatigue caused by lack of lubricant, incorrect lubrication, or excessive surface stresses. Component fracture caused by overload or fatigue is generally of secondary importance. 

A meehanical analysis of the near surface stresses in a wedge test as a function of surface geometry shows that tan a is equal to the ratio of the shear stress to the peel stresses (Fig. 4). 

It is possible to determine the x-component of the momentum equation by setting the rate of change of x-momentum of the fluid particle equal to the total force in the x-direction on the element due to surface stresses plus the rate of increase of x-momentum due to sources, which gives ... 

The effects of the surface stresses are accounted for explicitly, and source terms Sj y, and include contributions due to the body process only. For example, the body force due to gravity would be modeled by S x = 0, S y = 0, and S z = -pg-... 

The surface stress free (SSF) approach for removal of the Cu layer and planarization without polishing is critical for manufacturing a new generation of IC wafer composed of soft low-k materials [54]. 

Lubrecht, A. A. and loaimides, E. A., Fast Solution of the Dry Contact Problem and Associated Surface Stress Field, Using Multilevel Techniques, ASME7. Tribal, Vol. 113, 1991, PP-128-133. 

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