Yield zones, plastic deformation

In rubber-toughened epoxy resin materials, the particles act in the usual way as stress concentrators which initiate shear yielding of the matrix and give rise to increases in the critical size of the deformation zone. The particles first cavitate and then dilate during further deformation of the material. There is clear experimental evidence that cavitation of the rubber occurs first and is followed by shear yielding of the epoxy matrix [84,85,102,122], this being interpreted in terms of the need to relieve local constraint in the matrix before shear yielding and plastic deformation of the matrix can occur. The necessity for particle... 

Another approach to the fracture of ductile polymers stems from the recognition that for such materials the crack tip deformation zone has two components, as shown in Figure 12.21. There is an inner zone where the fracture process occurs -which could involve a combination of shear yielding and crazing - and an outer zone where extensive yielding and plastic deformation occur. This approach was originally proposed by Broberg [70], and has been developed by Mai and Cottrell [71], Hashemi and Williams [72], Mai [73] and others. 

In metals, inelastic deformation occurs at the crack tip, yielding a plastic zone. Smith [34] has argued that the elastic stress intensity factor is adequate to describe the crack tip field condition if the inelastic zone is limited in size compared with the near crack tip field, which is then assumed to dominate the crack tip inelastic response. He suggested that the inelastic zone be 1/5 of the size of the near crack tip elastic field (a/10). This restriction is in accordance with the generally accepted limitation on the maximum size of the plastic zone allowed in a valid fracture toughness test [35,36]. For the case of crack propagation, the minimum crack size for which continuum considerations hold should be at least 50 x (r ,J. 

In textbooks, plastic deformation is often described as a two-dimensional process. However, it is intrinsically three-dimensional, and cannot be adequately described in terms of two-dimensions. Hardness indentation is a case in point. For many years this process was described in terms of two-dimensional slip-line fields (Tabor, 1951). This approach, developed by Hill (1950) and others, indicated that the hardness number should be about three times the yield stress. Various shortcomings of this theory were discussed by Shaw (1973). He showed that the experimental flow pattern under a spherical indenter bears little resemblance to the prediction of slip-line theory. He attributes this discrepancy to the neglect of elastic strains in slip-line theory. However, the cause of the discrepancy has a different source as will be discussed here. Slip-lines arise from deformation-softening which is related to the principal mechanism of dislocation multiplication a three-dimensional process. The plastic zone determined by Shaw, and his colleagues is determined by strain-hardening. This is a good example of the confusion that results from inadequate understanding of the physics of a process such as plasticity. 

The (jjj expression indicated above shows that at the crack tip (r = 0), the stress tensor components become infinite. Actually, for a material able to undergo plastic deformation, above some stress level yielding occurs and limits the stress to the corresponding value, ay. Thus, around the crack tip a zone exists in which the material is plastically deformed. Such a zone is called the plastic zone, and it is represented in Fig. 8 in the case of a crack across a plate thickness. 

Equation (3) indicates that near the crack tip the stresses can exceed the yield stress cjy of real materials. Therefore, in a certain zone around the crack tip the material undergoes plastic deformations. To account for this two basically different models of plastic zones have been developed. 

From the foregoing is clear that the material directly under the indenter consists of a zone of severe plastic deformation. It is known that macroscopic yielding of a crystalline polymer involves a local irreversible mechanism of fracture of original lamellae into smaller units (Grubb Keller, 1980). The heat generated during... 

Grinding. An important aspect is often whether such a material can be ground, and how small then are the particles obtained. It can be derived from the theory that the thickness of the zone near a crack in which plastic deformation (yielding) occurs in a homogeneous isotropic material is given by... 

Because of the constraint imposed under plane strain conditions, yielding (onset of plastic deformation) would occur at a higher stress level. A number of estimates were made with different assumed constraint and yielding criteria [9]. But, because of the approximate nature of these estimates, the plastic zone correction factor for plane strain is taken to be that given by Eqn. (3.50). 

As a result of stress redistribution due to yielding, plastic deformation is expected to extend further ahead of the crack tip than that indicated by the plastic zone correction factors. For simphcity, and to an acceptable degree of accuracy for engineering analyses, the plastic zone size is taken to be equal to twice the plastic zone correction factor, i.e.. 

The key points to be gleaned from this exercise are that the plastic zone size depends on the state of stress (or constraint) and is proportional to (Ki/oys). Its size is expected to vary through the thickness of a plate, and it would increase with increasing stress intensity factor Kj and decreasing yield strength ays. The consequence of plastic deformation on fracture behavior and fracture toughness measurements is considered briefly in the next section. 

Some fracture mechanics jargon can be confusing, because similar expressions have different meanings elsewhere in mechanics. In Appendix C, plane strain elastic deformation means that the non-zero strains (in a pipe wall) occur in one plane. In plane strain fracture, the non-zero plastic strains in the yielded zone occur in thexy plane (Fig. 9.9a), that is perpendicular to the crack tip line. The strain e z = 0, so the sides of the specimen do not move inwards, and the fracture surface appears macroscopically flat. If a crack grows through a craze, a plane strain fracture will result. Voiding in the craze allows it to open, while the strain e z in the craze remains zero, and the surrounding material remains elastic. 

Even the most brittle polymers demonstrate some localized plastic deformation - in front of the crack tip, there exists a small plastic zone where stretching of chains, chain scission, and crack propagation appear in a small volume. The size of that plastic zone is too small to manifest in macroscopic plastic yielding and the crack propagates in a brittle manner. The relative low energy absorbed by the sample on its fracture is almost entirely that dissipated inside the small plastic zone. 

The selection of the dominant deformation mechanism in the matrix depends not only on the properties of this matrix material but also on the test temperature, strain rate, as well as the size, shape, and internal morphology of the rubber particles (BucknaU 1977, 1997, 2000 Michler 2005 Michler and Balta-Calleja 2012 Michler and Starke 1996). The properties of the matrix material, defined by its chemical structure and composition, determine not rally the type of the local yield zones and plastic deformation mechanisms active but also the critical parameters for toughening. In amorphous polymers which tend to form fibrillated crazes upon deformation, the particle diameter, D, is of primary importance. Several authors postulated that in some other amorphous and semiciystalline polymers with the dominant formation of dUatational shear bands or extensive shear yielding, the other critical parameter can be the interparticle distance (ID) (the thickness of the matrix ligaments between particles) rather than the particle diameter. 

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