Slip line field

If slip-line fields do not control plastic indentation, what does The answer is not the beginning of the plastic deformation, but the end of it. The end means after deformation hardening has occurred. That is, it is not the initial yield stress, Y0, that controls indentation, but the limiting yield stress, Y. This is... 

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. 

Careful examination of a notched region, after a slight impact load has started the yielding process, reveals a pattern of shear bands (Fig. 9.6a) in some glassy plastics. The plastic strain occurs inhomogeneously. The shear strain is about 1 within the shear bands, and zero between them. The overall pattern is remarkably similar to a particular slip line field pattern (Fig. 9.6b). 

Slip line fields are used to analyse metal plasticity under plane strain conditions (see Appendix C). The slip line field consists of two families of logarithmic spirals, with equations in polar coordinates r, d... 

Analysis of yielding at a notch (a) shear band patterns seen in a thin section cut from a polycarbonate specimen (b) slip line field pattern for yielding (c) Mohr circle diagram for the states of stress at points A and B in (b) (d) stress components on the surface of the prism marked out by neighbouring a and j3 slip lines. 

Several researchers tried to replace the single-shear plane model by a shear zone model. Lee and Shaffer (1951) provided a slip-line solution by applying the theory of plasticity. In the slip-line model, the metal is assumed to flow along the line of maximum shear lines. The slip-line field solution cannot be applied easily to three-dimensional as well as strain-hardening cases. Sidjanin and Kovac (1997) applied the concept of fracture mechanics in chip formation process. Atkins (2003) demonstrated that the work for creation of new surfaces in metal cutting is significant. He also points out that Shaw (1954) has shown this work to be insignificant. However, when this work is included based on the modem ductile fracture mechanics, even the Merchant analysis provides reasonable results. 

The construction of a slip-line field is a difficult task. 

Five nonlinear equations are established to solve for the five slip-line field angles, Oj, 62 j/, rjj, and rj2 -... 

Abebe M, Appl FC (1981) A slip line field for negative rake angle cutting. In Proceedings of the North American manufacturing research conference SME, p 341... 

The actual behavior of the blunting crack requires for its analysis numerical approaches that we consider below. However, here we try first to capture the essential features of the flow pattern from the ideally plastic, non-hardening material solutions using slip-line-field approaches of plasticity theory. 

When the central ligament of thickness 2b becomes just fully plastic, a symmetrical Prandtl slip-line field is established. This moves material by plastic shear from the crack flanks into the central ligament (McClintock 1969). 

Fig. 12.12 The mode I rigid-plastic, non-hardening slip-line field solution in a deeply double-edge-notched (DEN) thick plate with a remaining central ligament of extent 2b (from...

In the BOC fan the slip lines undergo a smooth rotation from BO to CO by an angle Af = ujl, which, according to slip-line-field theory, results in a monotonic change with of the mean normal stress by... 

Thus, the slip-line-field solutions show that for this deeply double-edge-notched plate, considered as two opposing deep cracks, there is a concentration of mean... 

The accompanying distributions of principal stresses ui, 0-2,and 0-3 and the mean normal stress a in the crack plane at y = 0 can be determined readily from the slip-line field and are, for 0 < x < x, with x given by (12.47),... 

The solutions are the HRR field solution, the slip-line field solution, and the FEM solution. The comparison is presented in Fig. 12.15, which shows the o /oo distribution of the FEM solution along y = 0, in the x direction, normalized with the current crack-opening displacement... 

Fig. 12.15 A comparison of three plastic-field solutions at the mode I crack tip the HRR solution the slip-line field solution, and the numerical FEM solution of McMeeking (1977).

Figure 12.31 The slip-line field for a deep symmetrical notch (a) is identical to that for the frictionless punch indenting a plate under conditions of plane strain (b). (Reproduced with permission from Cottrell, The Mechanical Properties of Matter, Wiley, New York, 1964)...

Based on the slip-line field theory [e.g., see Hill (1950)], Adachi and Yoshioka (1973) also extended the analysis of Ansley and Smith (1967) for spheres to include the creeping cross-flow over cylinders and obtained the following approximation expression for X ... 

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