Radius of contact

Actual sintering occurs by flow of mass from each sphere to the mutual point of contact, which gradually thickens. We can estimate the volume of mass, V, at the contact area. A, in terms of the foUoAving parameters r, the radius of the spheres p, the thickness of the layer buildup and x, the radius of contact of the built-up layer. If we have shrinkage, then we must also evaluate h, the amount of shrinkage, shown above as the height of interlinking layer. 

Two cases of sintering Transport of material from the spherical surfaces to the neck (top) does not contribute to densifi-cation. Transport of material from the interface between the particles to the neck (bottom) does contribute to densification. p is the neck s radius of curvature, r is the particle radius, 2h is the decrease of distance between particle centers, and x is the radius of contact. 

Therefore, for Hertzian contact, using Eq. (2.75) and Eq. (2.76), the maximum radius of contact rc, the maximum approaching distance a, and the corresponding maximum pressure can be calculated on the basis of the contact force, the elastic material properties of the spheres, and the radii of the spheres. 

Furthermore, assume that Eq. (2.74), obtained for Hertzian contact, is valid with rc being the actual radius of contact area so that Ar F ... 

Here t is measured from the moment of initial contact and, thus, can be expressed as a function of the radius of contact area rc, as shown in Fig. 2.16. The total heat exchange per impact is... 

Ke Effective thermal conductivity of rc Radius of contact area... 

Qx=8foG -8x=klat ts-8x [144]. Here, b denotes the effective radius of contact and G the reduced shear modulus of tip and sample ... 

AFM can also be used to probe local mechanical properties of thin films of food biopolymers, which are difficult to measure using traditional rheological methods. Several mechanical models have been developed to analyze the Young s modulus of food systems. One of the simplest models, the Hertz model, assumes that only the elastic deformation exists in a surface with spherical contacts, and the adhesion force can be neglected (Hugel and Seitz 2001). Equation (8.2) describes the relationship between the loading force, F and the penetration depth, d, where a is the radius of contact area, R the curvature of the tip radius, Vi and the Poisson s ratios of the two contact materials that have Young s modulus, Ei and E2. ... 

Figure 6.34 Pt particles on a surface can form hemispherical or raft-like particles, (a) The two geometrical models, with r representing the radius of contact ofthe particle and h its height (b) Height versus radius graph of the Pt particles obtained from the analysis of sections across 25 particles. All sections were performed on the same AFM image [145].

The compliance, C, of the system can also be used to relate the radius of contact during the test to a change in the displacement [10, 13] according to Eq. (3). 

This additional force means that, even in the absence of an external load, the hemisphere will be distorted and there will be a finite radius of contact a. The modified relation between the radius of contact and the applied load may be found by more detailed analysis (Johnson et al. 1971). Equation (7.2.3) is modified to... 

Fig. 11. Use of the JKR equation (Eq. 18) relating applied force, F, to radius of contact, a, to analyse results from the surface forces apparatus. Surface energy, y, of surfactant layers (DMPE), cf. (CTAB) (after Chen et al. [24]).

Fig. 11. Relation between elastic displacement 6 and radius of contact a of two spheres in an unloadingloading experiment. After an unloading from P to (path LMN), a loading from V to P leads to an instantaneous displacement first at constant a (branch NS) and then along the Hertz s curve (branch ST). This path (NST) is followed by an evolution along the curve (d)p towards the equilibrium point L. Experimental results for loading step from P = 0 and V = 4000 dynes to P = 5000 dynes are given (from ref. 2).

The situation is shown in Fig. 19. The opening of the conical pit has a radius / q. The conditions (contact angle, radius of contact) are such that the cavity is subthreshold, (A), threshold, (B) or superthreshold. ... 

In Eq. (27) the bracketed term represents the correction to the Hertz theory for solids with surface energy. Under zero load the radius of contact is given by... 

FIGURE 4. Relation between elastic displacement 8 and radius of contact a of a sphere on a plane. The equilibrium curve is 8(a). Curves (8)/ show the variation of 8 with a at fixed load, and are independent of w. (From Reference 10.)... 

The effective strain associated with a spherical tipped indenter may be written as the ratio of radius of contact circle to the radius of indenter, ajR ... 

It follows from the data presented that the area of contact has a considerable value (radius of contact of a particle tenth parts of a micron) on reducing the elasticity of the substrate ffor example, in the case of painted surfaces), the area increases. However, formula (III.41) and the data presented are valid for the particular case in which smooth spherical particles stick to a smooth surface. 

An observed penetration depth rewritten as 8=8h+(8-Sh) corresponds to the addition of a vertical displacement 5-6h at constant radius of contact ou, that is to say to the addition of the vertical movement of a flat rigid axi-symetrical punch inducmg a strong divergence of the normal stresses on the border of the contact area. 

Equation 6-27 can then be substituted into Eq. 6-26 to give the radius of contact ... 

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