Stress singularities

Fig. 18. Adhesive contact of elastic spheres. pH(r) and pa(r) are the Hertz pressure and adhesive tension distributions, (a) JKR model uses a Griffith crack with a stress singularity at the edge of contact (r = a) (b) Maugis model uses a Dugdale crack with a constant tension aa in a < r < c [1111.

As indicated, an implicit assumption of the JKR theory is that there are no interactions outside the contact radius. More specifically, the energy arguments used in the development of the JKR theory do not allow specific locations of the adhesion forces to be determined except that they must be associated with the contact line where the two surfaces of the particle and substrate become joined. Adhesion-induced stresses act at the surface and not a result of action-at-a-distance interatomic forces. This results in a stress singularity at the circumference of the contact radius [41]. The validity of this assumption was first questioned by Derjaguin et al. [42], who proposed an alternative model of adhesion (commonly referred to as the DMT theory ). Needless to say, the predictions of the JKR and DMT models are vastly different, as discussed by Tabor [41]. 

The stress singularity for this and the other stress components Oy and... 

In reality, around an inclusion embedded in a matrix a rather complex situation develops, consisting of areas of imperfect bonding, permanent stresses due to shrinkage, high stress-gradients or even stress-singularities, due to the geometry of the inclusions, voids, microcracks etc. 

Wang, S.S. and Choi, 1. (1982). Boundary layer effects in composite laminates. Part I free-edge stress singularities J. Appl. Mech. 49, 541-548. 

According to LEFM the stress singularity at the tip of a (sharp) crack of length a can be conveniently described using a stress intensity factor K(a) in the crack opening mode K(a) is given by ... 

Another mathematical approach to modeling cohesive zones is to consider the crack tip fully shielded, that is Keff = 0, rather than partially shielded as in the case considered above. In this case, a cohesive zone lies in front of a traction free crack (Zone 1). This is the classical Barenblatt-Dugdale model in which the stress-intensity factor at the end of the cohesive zone is now zero that is, stress singularities are completely removed by the cohesive forces.29 The requirement of complete shielding results in a cusp-shaped cohesive zone or bridging zone profile. This approach has advantages, particularly for the elevated temperature case, in that the cohesive zone can... 

As pointed out in Section II-3, numerical modelling of viscoelastic flows leads to numerical difficulties related to the mixed character (elliptic - hypeiholic) of the constitutive equation, to the propagation of "stress singularities" and to the so-called "High Weissenberg Number" problem. 

By requiring that the stress singularity at the stationary crack tip must vanish the level of applied stress can be determined to be... 

Adhesive Strengths. The strengths of the adhesive bonds between the inclusion materials and the epoxy were determined with an experimental technique described in detail elsewhere (24). The technique involves embedding a sphere of the candidate material in the contoured-neck portion of a cylindrical epow bar. The decohesion of the sample from the epoxy is then observed as the bar is strained in tension. The radial stress at decohesion is then detennined by means of a numerical solution of the deformation problem. Because no stress singularities exist at the interface prior to debonding, as is often the case with the usual popular tests, the technique used here provides a measure of the "true adhesive strength for the bimaterial systems employed. 

FIGURE 39.2 Schematic illustration of the stress distrihution at the tip of a delamination crack the presence of a stress singularity necessitates an energy-based approach. 

Fig. 5. Contact areas (dashed lines) and contact stresses at the interface of the regular tetrahedral pyramid penetrating into the half-space. The maximum contact stresses are observed in the center of the indentation zone and decrease according to the stress isobars (solid lines), (a) Stress singularities in the purely elastic loading and (b) their reduced form when the nonlinearity of the half-space is taken into account. After Reference [33].

The matrix layer guarantees that there is no interface where three different materials meet, i.e. two yams and matrix. It has been shown by Sihn et al. [76,77] that such a contact leads to a stress singularity. 

The fracture toughness determines the critical amplitude of the stress singularity at the crack tip. For an ideal brittle continuum, provided the crack length, a, is... 

The stress distribution in the area around the hole demonstrates a stress concentration through which stress singularities can affect the membrane. These must not lead to failure, therefore part of the arising membrane forces should be taken up by friction which results from compression. The hole spacings must not be so small that these singularities on top of the other forces could lead to damage. 

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