Singular fields

The conclusion of this subsection is that Eq. (62) for singular fields takes the form... 

The level curves of < >0 must be orthogonal, in each point, to the level curves of 0o, since we know that electromagnetic knots are singular fields (E B = 0). This condition can be written as... 

In our experiments using an optical tweezers set-up, only slightly modified compared to the conventional system described above, we have shown that using phase singular fields we can not only trap absorbing particles but also set them into rotation. This demonstrates the transfer of the angular momentum from the light beam to the particles. 

Fig. 12.8 (a) A mode I crack with a cohesive zone of length c over which the opening tractions at the two tips of the crack are limited to <7c. (b) The shape of a mode I crack with a cohesive zone of length c under a traction of compared with the shape of a mode I crack with a singular field (v is the half crack-flank displacement shown in (a) (from Williams (1984) courtesy of Wiley). 

Shih, F. C. (1983) Tables of Hutchinson-Rice-Rosengren Singular Field Quantities, MRL Report, MRLE-147, Providence, RI Brown University. 

Wang et al. (52) have given a careful analysis of the crack tip stress field in DCB adhesive fracture specimens by a finite element method. The main results are the following 1) the singular field is localized in the... 

ME, but of less mathematical rigour, is the singularity method (SM), which reconstructs the potential field by a superposition of singularity fields from point charges (Phillips 1995). Additionally, it is possible to approximate compact, isometric aggregates as porous spheres (PS), for which analytical expressions of the double layer structure are available (Ohshima 2008). Some of the mentioned methods assume a certain shape of the aggregates or primary particles, some of them are based on the Unearised PBE (BEM, ME, PS). Only a few have already been employed for particle clusters (CoeUio et al. 1996 Kwon et al. 1998 Schiefil et al. 2012). 

Any solution to the linearised PBE can be expressed as a linear combination of singularity fields (Eq. (4.117)). The problem is, thus, reduced to (i) properly locating the singularities (inside the particles) and to (ii) adjusting the singularity strengths q in such a way that the boundary condition (3.28) is fulfilled. That is verified at a number of control points on the surface of the primary particles. 

Due to the simple analytical expression of the singularity field, the gradient in Eq. (4.121) can be easily deduced ... 

Similarly, the crack edge singular fields for mode II loading are given by... 

It was noted in Section 2.3.2 that most of the current interfacial fracture mechanics methodologies describe steady-state crack propagation, but not the initiation of interfacial cracks. A recent approach to the prediction of initiation is based on the calculation of the singular stress field at the free edge of a bimaterial system loaded on the top layer [59,60]. Because the crack is assumed not to exist initially in this analysis, a very different singular field is predicted, and the results can be used to predict initiation of cracks in residually stressed coatings. Because the predictions of this theory sometimes contradict the predictions of the Suo and Hutchinson approach, we shall briefly review it as a final note. 

Three methods of mode partition may be used for fracture of adhesive joints (Femlund and Spelt 1991) the local method based on the stress singular field (Hutchinson and Suo 1992), the global method based on the beam theory (Williams 1988), and the method using the adhesive edge stresses based on the adhesive-beam model (Fernlund and Spelt 1991). 

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