Crack fields in fully developed plasticity

The HRR crack-tip solution of Section 12.3.3 for strain-hardening material in mode I represents asymptotic behavior considering momentary flow fields of cracks remaining perfectly sharp. In these solutions the Mises equivalent strain e is concentrated at the crack tip proportionally to (1/r) Such strain concentration is clearly incompatible with the crack remaining sharp. Instead, 

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. 

We start with the fully plastic, non-hardening plane-strain flow field of a deeply double-edge-notched plate of a rigid plastic material where the two symmetrically placed notches simulate two opposing sharp cracks shown in Fig. 12.12. The material is considered to have a tensile yield strength (To (= T) or a yield strength in shear of k = (To/ /3- 

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). 

as the inclined arrows show. Moreover, since the flank AO is traction-free, (72 = 0 and (7i = 2k, and since de3 = 0 by virtue of the plane-strain consideration, along the line AO a-i = a + ai) / = k, where all stresses and strains are principal values. Along the entire ABO domain the deformation is by simple shear and is associated with a mean normal stress oi a = +(T2 + o 3)/3 = k. 

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