Deformation zone modelling

Step 3. The set of fracture properties G(t) are related to the interfaee structure H(t) through suitable deformation mechanisms deduced from the micromechanics of fracture. This is the most difficult part of the problem but the analysis of the fracture process in situ can lead to valuable information on the microscopic deformation mechanisms. SEM, optical and XPS analysis of the fractured interface usually determine the mode of fracture (cohesive, adhesive or mixed) and details of the fracture micromechanics. However, considerable modeling may be required with entanglement and chain fracture mechanisms to realize useful solutions since most of the important events occur within the deformation zone before new fracture surfaces are created. We then obtain a solution to the problem. 

When relating interface structure to strength, the literature is replete with analyses, which are based on the nail solution [1,58], as shown in Fig. 10. This model is excellent when applied to very weak interfaces (Gic 1 J/m ) where most of the fracture events in the interface occur on a well-defined 2D plane. However, the nail solution is not applicable to strong interfaces (Gic 100-1000 J/m ), where the fracture events occur in a 3D deformation zone, at the crack tip. In Fig. 10, two beams are bonded by E nails per unit area of penetration length L. The fracture energy G c, to pull the beams apart at velocity V is determined by... 

In the traditional Dugdale model [56], a = Oy and the familiar result is obtained, Gic = cTySc- In the EPZ model, cr exceeds critical crack opening displacement <5c is proportional to the maximum stresses cr in the deformation zone... 

Fig. 5.1. The electrostatic configurations of the Neilson-Benedick three-zone model describe a piezoelectric solid subject to elastic-inelastic shock deformation which divides the crystal into three distinct zones. Zone 1, ahead of the elastic wave, is unstressed. Zone 2 is elastically stressed at the Hugoniot elastic limit. Zone 3 is isotropically pressurized to the input pressure value (after Graham [74G01]).

The deformation zones were calculated for the polymers of Table 5.1 and Table 6.1 according to the Dugdale-Barenblatt-model. Yield stress ay from tensile tests was used instead of the cohesive stress ctc since a reasonable agreement of ay and ctc... 

Fig. 7.3. Deformation zone as calculated from the Dugdale-Barenblatt-model (Eq. 7.2). In order to magnify the displacements, E/cry = 7 was assumed for the diagram, whereas, in reality the ratio is about 38 (Tables 5.1 and 6.1)...

Figure 8 also shows values of f that have been calculated by two other methods. In the first, Jaswal (19) has used lattice-vibration eigen-frequencies and eigenvectors which have been calculated in the first Brillouin zone using the deformation-dipole model for the lattice. This... 

Various models 1-2,42 43) have been proposed to describe the extent and shape of the localised plastic deformation zone at the crack tip. From these models one may define a parameter known as the crack opening displacement, 5, (see Fig. 16) and the value of 5,c for the onset of crack growth is given by... 

The existence of a wedge-shaped cavitated or fibrillar deformation zone or craze, ahead of the crack-tip in mode I crack opening, has led to widespread use of models based on a planar cohesive zone in the crack plane [39, 40, 41, 42]. The applicability of such models to time-dependent failure in PE is the focus of considerable attention at present [43, 44, 45, 46, 47]. However, given the parallels with glassy polymers, a recent static model for craze breakdown developed for these latter, but which may to some extent be generalised to polyolefins [19, 48, 49], will first be introduced. This helps establish important links between microscopic quantities and macroscopic fracture, to be referred to later. 

Fig. 1.3 Comparison of elastic Hertzian contact (left) and adhesive JKR contact (right), (a) Hertzian contact Dashed line (sphere) shape of contacting spherical lens prior to pressing to the flat surface by force L. Hertzian contact profile shown by solid line, with radius under external load L aH (b) JKR contact Schematic of adhesion force (adhesive zone model, forces schematically indicated by vectors) further deforming a spherical lens from Hertzian contact (solid line) to JKR contact (dotted line) with radius aJKR. Reproduced from [7] with permission copyright Springer Verlag...

Abagyan, R., Batalov, S., Cardozo, T., Totrov, M., Webber, J., and Zhou, Y. (1997) Homology modeling with internal coordinate mechanics deformation zone mapping and improvements of models via conformational search. Proteins Suppl 1, 29-37. 

The modelling of regional and local deformation zones in three dimensions at Aspo was conducted on the basis of their ductile history. A ductile precursor has been documented for all modelled zones but one (and may be valid for that one as well). The assumption is that ductile high strain zones of the scale considered (half a metre or more) are normally persistent over distances of more than a few tens of metres and are commonly the location of later reactivations. Ductile deformation was therefore normally a pre-requisite for interpolating structures with a distinct strike and dip over distances larger than a few tens of metres and also to extrapolate structures beyond the point of observation. Four deformation zones penetrate the central tunnel spiral at Aspb in the new model. 

A theory of fracture was developed for such entangled polymers that was based on the vector percolation model of Kantor and Webman. The percolation model is used to describe connectivity between the chains and to relate the interfacial structure to the breakdown process of the deformation zone at the crack tip. Vector percolation involves the transmission of forces (vectors) through a two- or three-dimensional lattice where a certain fraction of the bonds are missing or broken. Thus, we can examine how the stiffness or strength of a lattice changes with bond fracture or disentanglement. 

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