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In order to study effects of constraint on fracture toughness, it is reasonable to study the region of large strains close to the crack tip within which the microscopic separations that lead to fracture often take place. The first step in this direction was taken in 1950 by Hill, who postulated that close to a circular notch tip the principal stress directions would be radial and circumferential, so that the plastic slip lines (maximum shear stress trajectories) would be logarithmic spirals. The resulting equation for stress normal to the notch symmetry plane, neglecting strain hardening, was identical to that for the circumferential stress near the bore of an ideally plastic thick-walled hollow cylinder under external radial tension, because the relevant geometries are identical. Hill's hypothesis was extended algebraically by Merkle to include strain hardening with a generalized-plane-strain small-strain hollow cylinder analogy, and numerically in a more general way geometrically for plane strain and large strains by Rice and Johnson. Large strain finite element analyses have shown that a wedge-shaped zone ahead of a blunting crack tip deforms like a hollow cylinder. This paper extends the generalized plane strain hollow cylinder analogy to large strains. The strain equations are derived by analyzing the constant volume deformation of a differential cylindrical element. The circumferential strain is singular at the tip of an initially sharp crack. With the strain distribution determined, the stresses are obtained by integrating the equation of radial equilibrium. An approximation is developed for the first increment of radial stress near the strain singularity. Calculations show that the in-plane stresses are only slightly sensitive to transverse plastic strain.
fracture mechanics, crack-tip blunting, transverse strain, constraint effects, large strains, fatigue (materials)
Research specialist, Oak Ridge National Laboratory, Oak Ridge, TN