A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks

1968/06/01 by J. R. Rice · 8,316 citations
Engineering · Mathematics · #Composite material #Compression (physics) #Crack tip opening displacement #Fatigue and fracture mechanics #Finite element method #Fracture mechanics #Geometry #Integral equation #Materials science #Mathematical analysis #Mathematics #Mechanical stress and fatigue analysis #Mechanics #Metal Forming Simulation Techniques #Physics #Plane stress #Plasticity #Singular integral #Singularity #Strain hardening exponent #Stress intensity factor #Structural engineering #Tension (geology)

paper · doi:10.1115/1.3601206

published in Journal of Applied Mechanics 35(2), 379-386 (ASM International)

openalex publication_date 1968/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

A line integral is exhibited which has the same value for all paths surrounding the tip of a notch in the two-dimensional strain field of an elastic or deformation-type elastic-plastic material. Appropriate integration path choices serve both to relate the integral to the near tip deformations and, in many cases, to permit its direct evaluation. This averaged measure of the near tip field leads to approximate solutions for several strain-concentration problems. Contained perfectly plastic deformation near a crack tip is analyzed for the plane-strain case with the aid of the slip-line theory. Near tip stresses are shown to be significantly elevated by hydrostatic tension, and a strain singularity results varying inversely with distance from the tip in centered fan regions above and below the tip. Approximate estimates are given for the strain intensity, plastic zone size, and crack tip opening displacement, and the important role of large geometry changes in crack blunting is noted. Another application leads to a general solution for crack tip separations in the Barenblatt-Dugdale crack model. A proof follows on the equivalence of the Griffith energy balance and cohesive force theories of elastic brittle fracture, and hardening behavior is included in a model for plane-stress yielding. A final application leads to approximate estimates of strain concentrations at smooth-ended notch tips in elastic and elastic-plastic materials.

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