Asymptotic Analysis Of Near Tip Stress And Deformation Fields For Cracks In Porous Elastic Plastic Materials PDF Download

The elastic--plastic stress and deformation fields at the tip of a crack which grows in an ideally plastic solid under plane strain, shows small scale yielding conditions. Results of an asymptotic analysis suggests the existence of a crack tip stress state similar to that of the classical Prandtl field, but containing a zone of elastic unloading between the centered fan region and the trailing constant stress plastic region. The near tip expression for the rate of opening displacement delta at distance r from the growing tip is found to have the form delta-. = .cap alpha. J-.?sub o/ + .beta.(sigma/sub o//E) a-. ln(R/r) but the presence of the elastic wedge causes .beta. to have the revised value of 5.08 (for Poisson ratio .nu. = 0.3); also, (a = crack length, sigma/sub o/ = yield strength, E = elastic modulus, and J denotes the far-field value), and (1-.nu.2)K2/E for the small scale yielding conditions considered. The parameters .cap alpha. and R cannot be determined from the asymptotic analysis, but comparisons with finite element solutions suggest that, for small amounts of growth, .cap alpha. is approximately the same for stationary and growing cracks, and R scales approximately with the size of the plastic zone, being about 15% to 30% larger. For large scale yielding, a similar form applies with possible variations in .cap alpha. and .beta., in cases which maintain triaxial constraint at the crack tip. In the fully yielded case R is expected to be proportional to the dimension of the uncracked ligament. Model crack growth criterion requiring a critical delta at some fixed r from the tip, is re-examined in light of the more accurate solution. Results suggest that the J versus .delta.a relation describing growth is dependent on the extent of yielding. It is suggested that this dependency might be small for highly ductile materials, provided that a similar triaxial constraint is maintained in all cases.