Plastic Zone Correction Factor

The engineering community prefers to work with stresses rather than energy. As such, stress intensity factors K (in units of (F/is now widely used in engineering, whereas the use of strain energy release rate G (in units of (FL/L )) is limited to some scientihc areas. 

6 Plastic Zone Correction Factor and Crack-Opening Displacement 

Before closing this chapter, two plasticity-related parameters need to be introduced. The hrst parameter relates to the presence of plastic deformation at the crack tip in technologically important material (i.e., the plastic zone correction factor), and an estimation of its size. The second one relates to the extent of opening of the crack at its tip in the presence of plastic deformation, which is then used as an alternate parameter for characterizing the crack-driving force. 

An estimate of the plastic zone correction factor was made by Irwin (see [9]). He postulated that stresses ahead of the crack tip, away from a small plastic zone, can be approximated by those given by the solutions of linear elasticity, provided that an effective crack length Ug is used, where, 

In Eqn. (3.48), a is the physical (actual) crack length, and is the plastic zone correction factor. The parameter ry, for generalized plane stress, is estimated by setting the normal stress cryy(r, 0) directly in front of the crack equal to the uniaxial tensile yield strength ays of the material. From Eqn. (3.29), one obtains  

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The plastic zone correction factor, for generalized plane stress, is, therefore, given by Eqn. (3.49) below. 

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.. 

Equations (4.3) and (4.4) are written to expressly reflect Irwin s plastic zone correction factor ry (see Eqn. (3.49)). By normalizing with respect to ry, the yield loci, or... 

Figure 4.3. Schematic diagram showing the typical variation in fracture toughness with material thickness (B), or thickness relative to Irwin s plastic zone correction factor (B/ry).

In Eqn. (4.7), 2a is the effective crack diameter. Formally 2a represents the diameter of the physical crack and the associated plastic zone correction factor, namely. 

The question now is how much nonlinearity is considered to be not excessive. Formally, the nonlinearity should not exceed that which would correspond to an increase in the initial (or physical) crack diameter (2ao) by the plane strain plastic zone correction factor i.e., by Iriy (see Eqn. (3.49)). Physically, it is acceptance of the fact that a plastically deformed zone would develop at the crack tip, and its presence is equivalent to a change in the effective crack length at the onset of fracture from Itto to Itto -I- 2riy i.e.,... 

The question of how large a pop-in indication should be required can only be answered empirically. Ideally, the crack advance at pop-in should include sufficient material to be representative of the fracture property of the entire specimen. At this juncture, the consensus is that the increment of growth should be at least equal to the formally dehned plane strain plastic zone correction factor (see Eqns. (4.14) to (4.16)). This limit may be related to the load-displacement record in a manner similar to that discussed previously. 

Fig. 5. (i) Graphical determination of the size of the plastic zone, (ii) Evolution of the toughness, K, with a geometry factor a/W before correction (Kj x) and after correction (Ktn) with the size of the plastic zone. Material iPP/EPR-1 tested at 0.001 m/s and room temperature. 

Lawn, Marshall, and others have emphasized the fact that the residual stress term was not just a correction factor but played the important part in determining the final size of the radial crack system. However, like Evans and Charles, they developed the model of the plastic deformed zone acting to force open the penny-shaped median crack just beneath it, so that the basic equations to be used were still... 

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