Deformation zone

Fig. 3. Schematic of a sphere in contact with a flat surface, (a) The deformation when surfaces are in contact. The radius of the deformed zone is a, and the separation profile is given by D versus r. The central displacement, S, is shown as the distance between the center of the deformed zone and the tip of the undeformed sphere, represented by the bold line. S characterizes the displacement of the applied load, (b) When the applied load is —/ s, the pull-off force, the surfaces jump out of contact, and the undeformed shape of the surfaces is attained.

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... 

Studies of PMMA-based ionomers also demonstrate the influence of thermal treatment on deformation modes (16). For Na salts of PMMA-based ionomers of 6 and 12 mol% that were cast from DMF, only crazes were observed on straining. However, after an additional heat treatment (48 h at 160°C), which also removes any DMF solvent that is present, shear deformation zones are induced. Hence, the ionic cluster phase, which was destroyed by the polar solvent, has been restored by the heat treatment. 

As one example, in thin films of Na or K salts of PS-based ionomers cast from a nonpolar solvent, THF, shear deformation is only present when the ion content is near to or above the critical ion content of about 6 mol% and the TEM scan of Fig. 3, for a sample of 8.2 mol% demonstrates this but, for a THF-cast sample of a divalent Ca-salt of an SPS ionomer, having only an ion content of 4.1 mol%, both shear deformation zones and crazes are developed upon tensile straining in contrast to only crazing for the monovalent K-salt. This is evident from the TEM scans of Fig. 5. For the Ca-salt, one sees both an unfibrillated shear deformation zone, and, within this zone, a typical fibrillated craze. The Ca-salt also develops a much more extended rubbery plateau region than Na or K salts in storage modulus versus temperature curves and this is another indication that a stronger and more stable ionic network is present when divalent ions replace monovalent ones. Still another indication that the presence of divalent counterions can enhance mechanical properties comes from... 

The combined effects of a divalent Ca counterion and thermal treatment can be seen from studies of PMMA-based ionomers [16]. In thin films of Ca-salts of this ionomer cast from methylene chloride, and having an ion content of only 0.8 mol%, the only observed deformation was a series of long, localized crazes, similar to those seen in the PMMA homopolymer. When the ionomer samples were subject to an additional heat treatment (8 h at 100°C), the induced crazes were shorter in length and shear deformation zones were present. This behavior implies that the heat treatment enhanced the formation of ionic aggregates and increased the entanglement strand density. The deformation pattern attained is rather similar to that of Na salts having an ion content of about 6 mol% hence, substitution of divalent Ca for monovalent Na permits comparable deformation modes, including some shear, to be obtained at much lower ion contents. 

In order to supplement micro-mechanical investigations and advance knowledge of the fracture process, micro-mechanical measurements in the deformation zone are required to determine local stresses and strains. In TPs, craze zones can develop that are important microscopic features around a crack tip governing strength behavior. For certain plastics fracture is preceded by the formation of a craze zone that is a wedge shaped region spanned by oriented micro-fibrils. Methods of craze zone measurements include optical emission spectroscopy, diffraction... 

The boundary of the proposed deformation zone is shown in Fig. 7.3 according to Eq. (7.2). The displacement at the crack tip at x = 0 is called crack opening displacement 5 or critical crack opening displacement 8C if the crack is going to propagate. 

The work done in the deformation zone can be estimated assuming the deformation zone propagates together with the crack and maintains its steady state displacement profile [86]... 

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)...

Apparently, the width and the length of the deformation zone are simply proportional to the molecular mass of the network strands The more highly crosslinked a polymer, the smaller is its deformation zone. On the other hand, polymers with few crosslinks will exhibit large deformation zones ahead of the growing crack. Since deformation zones pick up energy proportional to their crack... 

Usually, the molecular strands are coiled in the glassy polymer. They become stretched when a crack arrives and starts to build up the deformation zone. Presumably, strain softened polymer molecules from the bulk material are drawn into the deformation zone. This microscopic surface drawing mechanism may be considered to be analogous to that observed in lateral craze growth or in necking of thermoplastics. Chan, Donald and Kramer [87] observed by transmission electron microscopy how polymer chains were drawn into the fibrils at the craze-matrix-interface in PS films [92]. One explanation, the hypothesis of devitrification by Gent and Thomas [89] was set forth as early as 1972. 

Of course, the network strands cannot be stretched completely. Stretching ratios of 1.4 for PC [31, 90] and of 1.3 for epoxy polymers [37] have been reported. The chain contour length of the strands is an appropriate measure for a simple estimation of the number of strands that are stretched across the deformation zone. The chain contour length of the strands is assumed to be proportional to... 

M L equals the mass of a typical polymer chain devided by the number of atoms in its backbone. Slightly more than 1000 strands (length lc) are needed to cross the deformation zone of width 8 = 2w for all the polymers listed in Table 6.1. It is one of the essential findings of this report that the size of the deformation zone is scaled according to the length of the molecular strands. 

Alle the deformation zones contain a finite and equal number of extended chains in their most highly stretched strands. This surprising conformity of the deformation zones may well be the consequence of the imposed plane-strain fracture condition which impedes lateral contraction of the material. However, no quantitative explanation has been presented as yet. A plausible explanation would be to assume that due to the hindered lateral contraction additional tensile stresses are transferred to the most extended strand with each additional chain pulled out of the matrix [112]. 

The fracture behavior can be attributed to strain softening [91] in the deformation zone [92, 93] or to stress-activated devitrification [89, 96]. The strands are comparatively free to move in the strain softened regions of the deformation zone. The van der Waals interaction between adjacent strands is greatly reduced and the clearence between molecular segments is enlarged. 

Both sets of experiments seem to support the proportionality of crack opening displacement 5C = 2w and molecular mass Mc between crosslinks as indicated by the slope 1 in the double logarithmic plot (Fig. 7.5). Even if Mc had to be adjusted due to doubts about the front factor in Eq. (4.3), the proportionality would stay unaffected. Consequently, the size of the deformation zone ahead of the crack is determined by the length of the molecular strands in the chemical network. 

The presented results and the additional information taken from various references indicate the direct relevance of the size of the network strands for the crack opening displacement and consequently for the toughness of the polymer. In polymers under load, the molecular chains at the tip of the crack break after the deformation zone ahead of the crack has grown to a critical width 5C, that is the crack opening displacement. This value 5C is proportional to the length of the molecular strands of the network and is linked in this way to the molecular structure of the polymer. However, the molecular mechanism for chain breakage in the deformation zone is not known at present. 

The lengths of the molecular chains dominate large strain behavior and crack propagation in contrast to their minimal influence at small strain levels. Consequently, thermosets are characterized by small deformation zones (Fig. 8.2) and brittle fracture. 

Figure 8.11 Effect of orientation on deformation zones observed during puncture testing, viewed from above ...

The study area is located within in the eastern tip of the Indian Mountain Deformed Zone, southeastern New Brunswick, Canada. The area is underlain by Upper Carboniferous sedimentary rock and bounded by the Smith Creek and Berry Mills faults to the north and south, respectively (St. Peter 2006). These... 

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