Free vibration of an edge-cracked beam with a Dugdale–Barenblatt cohesive zone

Abstract

This paper presents a theoretical investigation into the effect of crack-plane plasticity on the vibration of a transversely edge-cracked beam. The crack-plane is presumed to be loaded statically in either bending or shear such that Dugdale–Barenblatt type cohesive zones are activated in the crack-plane ahead of the crack tip. Then the beam undergoes small amplitude vibrations about this state. Euler–Bernoulli beam theory is used for the vibration analysis in which an infinitesimally thin beam element surrounding the crack-plane is replaced with a bending and a shear line-spring. The line-springs by nature account for the discontinuities in slope and deflection, respectively, across the crack-plane. Previous studies of the elastic problem (no cohesive zones) calculate the compliances of the line-springs from tabulated linear elastic fracture mechanics solutions and the compliances depend only on the flexural rigidity of the beam and on the crack length to beam depth ratio. Here the compliance of the line-spring for the mode loaded statically in the nonlinear range is calculated using the Boundary Element Method (BEM) in an iterative nonlinear analysis and the compliance depends nonlinearly on the load across the crack plane. This nonlinear load–deformation relation is linearized about the applied static load level, and the resulting compliance used in the vibration analysis. Among other results for mode I cohesive zones, there is a strong reduction of the fundamental frequency as both load and crack length increase and as yield strength decreases. The results show that there is some potential for designing a non-destructive material characterization technique which would use the changes in frequency to infer the properties of material behavior laws in the cohesive zone.