Elastic shear bond

Elastic shear bond is a stress in the interface when its bond strength is not exceeded it is supposed that the elastic shear bond is uniformly distributed along the fibre. 

All these tests lead to the apparently justified conclusions that before is reached the preceding linear part of the force-deflection curve corresponds to elastic shear bond, and that the debonding processes start just at the maximum pull-out force Such a simple image was completed by observations published already by Pinchin and Tabor (1975) who have shown that the debonding crack appears along the pulled fibre at a certain distance of a few micrometers from its surface. These observations in the fibre-matrix interface after the pull-out test corroborated well with the measurements of micro-hardness of the interface, which have shown that the weakest layer was situated at a distance from the fibre surface (cf. Chapter 7). 

Once the elastic shear strength has been exceeded (at the point of entry of the fibre into the matrix) debonding will occur. Assuming that the debonding is limited to the zone in which the elastic shear stress exceeds the adhesional shear bond strength, then the load transfer process will be made up of frictional slip at the debonded end and elastic shear transfer in the rest of the fibre. This implies that when the elastic shear bond strength is exceeded, catastrophic failure will not necessarily occur. This state of combined stress-transfer mechanisms was treated analytically by Lawrence [8], and was later reviewed and extended by others... 

The glide planes on which dislocations move in the diamond and zincblende crystals are the octahedral (111) planes. The covalent bonds lie perpendicular to these planes. Therefore, the elastic shear stiffnesses of the covalent bonds... 

Figure 5.6 Correlation of octahedral shear stiffnesses with bond moduli for Group IV crystals. The octahedral stiffnesses measure the elastic shear resistances of the covalent bonds across the (111) planes.

Table 8 presents a survey of the basic elastic constants of a series of polymer fibres and the relation with the various kinds of interchain bonds. As shown by this table, the interchain forces not only determine the elastic shear modulus gy but also the creep rate of the fibre. 

Therefore, the rate at which chemical bonds break increases with elastic shear stressing of the material. The rupture of chemical bonds, hence fracture of material, leads to its fragmentation into particles. This reduces the average particle size in powder as fractured particles multiply into even smaller particles. Equation (1.24) points to the importance of elastic shear strains in mechanical activation of chemical bonds for particle size refinement and production of nanoparticles. 

The elastic shear constant, ci - C 2, as obtained in the Bond Orbital Approximation of Eq. (8-14), as obtained by Sokcl (1976) and including the bonding-antibonding matrix elements from perturbation theory, and experimental values all are in units of 10 erg/enr ... 

Suppose that a thin film is bonded to one surface of a substrate of uniform thickness hs- It will be assumed that the substrate has the shape of a circular disk of radius R, although the principal results of this section are independent of the actual shape of the outer boundary of the substrate. A cylindrical r, 0, z—coordinate system is introduced with its origin at the center of the substrate midplane and with its z—axis perpendicular to the faces of the substrate the midplane is then at z = 0 and the film is bonded to the face at z = hs/2. The substrate is thin so that hs R, and the film is very thin in comparison to the substrate. The film has an incompatible elastic mismatch strain with respect to the substrate this strain might be due to thermal expansion effects, epitaxial mismatch, phase transformation, chemical reaction, moisture absorption or other physical effect. Whatever the origin of the strain, the goal here is to estimate the curvature of the substrate, within the range of elastic response, induced by the stress associated with this incompatible strain. For the time being, the mismatch strain is assumed to be an isotropic extension or compression in the plane of the interface, and the substrate is taken to be an isotropic elastic solid with elastic modulus Es and Poisson ratio Vs the subscript s is used to denote properties of the substrate material. The elastic shear modulus /Xg is related to the elastic modulus and Poisson ratio by /ig = Es/ 1 + t s). 

Beyond this point, the stress distribution in the bonded zones is governed by elastic considerations, and can be calculated by equations such as Eqs 3.8 and 3.9, assuming a fibre with a length of I - b) and a pull-out load of P (Eq. 3.12). Thus, the interfacial elastic shear stress at the end of the debonded zone is ... 

If points A and C on curves such as those shown in Figure 3.14 can be determined, one can then calculate the elastic interfacial shear bond strength (tgu) and the interfacial frictional shear bond strength (tfij) using the equations developed by Laws ... 

An insight into the performance of real bonded structures may only be obtained from knowledge of the engineering characteristics of the adhesive concerned and their interaction with a mathematical model of the structure itself The appropriate characteristics — shear modulus, elastic shear stress limit, asymptotic shear stress — are probably best obtained from the thick adherend shear test . Typical figures for two contrasting types are given in Table 7.3. 

There are two well-accepted models for stress transfer. In the Cox model [94] the composite is considered as a pair of concentric cylinders (Fig. 19). The central cylinder represents the fiber and the outer region as the matrix. The ratio of diameters r/R) is adjusted to the required Vf. Both fiber and matrix are assumed to be elastic and the cylindrical bond between them is considered to be perfect. It is also assumed that there is no stress transfer across the ends of the fiber. If the fiber is much stiffer than the matrix, an axial load applied to the system will tend to induce more strain in the matrix than in the fiber and leads to the development of shear stresses along the cylindrical interface. Cox used the following expression for the tensile stress in the fiber (cT/ ) and shear stress at the interface (t) ... 

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