A Finite-Element Study of the Asymptotic Near-Tip Fields for Mode I Plane-Strain Cracks Growing Stably in Elastic-Ideally Plastic Solids

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Abstract
The asymptotic near-tip stress and deformation fields for a Mode I plane-strain crack growing quasi-statically in an elastic-ideally plastic solid under small-scale yielding conditions are analyzed numerically by the finite-element method. The asymptotic analyses of Rice and Sorensen; Rice, Drugan, and Sham; and Drugan, Rice, and Sham predict that the crack opening rate ˙δ at a small distance r from the growing crack tip under contained yielding conditions is given by ˙δ=˙Jσ0+βσ0E˙aℓnRrasr→0 where a is the crack length, σ0 the tensile yield strength, E Young's modulus, J the farfield value of the J-integral evaluated on contours in the elastic region, β = 5.462 (Drugan et al) for Poisson ratio ν = 0.3, and the parameters α and R are undetermined by the asymptotic analysis. The finite-element solutions generated are very detailed, with the maximum extent of the plastic zone being about 100 times the smallest element size. This high resolution in the finite-element solution enables correlation of the finite-element results with the asymptotic crack opening rate to be made and the expressions for the parameters α and R to be determined. Various crack growth histories are simulated by relaxing the nodal force at the crack tip and simultaneously increasing the external applied load in order to investigate the possible dependence of α and R on the crack growth history. The maximum amount of crack growth simulated is about 20 percent of the maximum extent of the plastic zone size. The numerical results reveal that a well-defined elastic unloading sector develops and moves with the advancing crack tip and its location coincides well that predicted by the asymptotic analysis. Estimation from the numerical results gives β = 5.46, which is in very good agreement with the asymptotic result. The parameter α is found to be independent of the crack growth histories examined and has a value of 0.58. An attempt to relate R to the quantity EJ/σ02 is made. However, it is found that the parameter s, defined by s ≡ R/(EJ/σ02), varies in a range 0.113 to 0.133 for the crack growth histories simulated. Thus, the precise definition for R remains unclear.

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