Plastic Zone and Effect of Constraint

In Chapters 2 and 3, the restrictions in the use of linear elastic fracture mechanics (LEFM) were discussed in terms of the dimensions of the crack and the body (specimen, component, or structure) relative to the size of the crack-tip plastic zone. Simple estimates of the plastic zone sizes were given in Section 3.6. A more detailed examination of the role of constraint (plane strain versus plane stress) and the variations in plastic zone size from the surface to the interior of a body would help in understanding fracture behavior and the design of practical specimens for measurements of fracture toughness. Note that the plastic zone size in actual materials 

For mode 1 loading, the stresses near the tip of a crack in a plate are given from Eqn. (3.29) in polar-cylindrical coordinates, with the z-axis along the crack front, the x-axis in the direction of crack prolongation, and the y-axis perpendicular to the crack plane. 

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 

The size and shape of the plastic zones are illustrated in Fig. 4.1 the plane strain zone is estimated for a Poisson ratio v of 0.3. Note that the actual zone sizes would be larger to reflect redistribution of stresses associated with plastic deformation near the crack tip. The shape of the plastic zone would also change to reflect the workhardening behavior of the material. The actual sizes and shapes would need to be determined experimentally for each class of materials. 

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. 

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