#### Vol. 3, No. 8, 2008

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Nonlinear vibration of an edge-cracked beam with a cohesive zone, I: Nonlinear bending load-displacement relations for a linear softening cohesive law

### Prasad S. Mokashi and Daniel A. Mendelsohn

Vol. 3 (2008), No. 8, 1573–1588
##### Abstract

Part I of this paper describes the computations of the quasistatic nonlinear moment-slope relation for an edge-cracked beam element with a strictly linear softening cohesive zone ahead of the crack tip. A static plane stress linear elastic boundary element analysis is used in which the cohesive nonlinearity appears in the crack plane boundary conditions only. An iterative solution scheme is used to determine the unknown cohesive zone length, the cohesive displacement jumps, and the bending mode $J$-integral. Interpreting the moment-slope relation as a generalized load-displacement relation the bending compliance (and slope) at a given applied moment are calculated from computed $J$-integral values over a grid of applied moment and crack-length values. The dependence of the moment-slope relation on the cohesive law parameters is studied and the various computed moment-slope relations are then used in Part II to model the dynamic effect of the cohesive zone and law on the free-vibration of an edge-cracked simply-supported beam.

##### Keywords
cohesive zone, linear softening, compliance, $J$-integral, nonlinear load-displacement