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<p class="noindent">A nonlinear free-vibration analysis of an Euler–Bernoulli beam with an edge crack and a cohesive zone at the crack tip, represented by bending and shear springs, is presented. Restricting attention to bending nonlinearities, we suppose the beam is loaded statically in bending into the nonlinear region and small amplitude vibrations are then superposed. A two term perturbation expansion is used where the small parameter depends on the ratio of the first and second derivatives of the nonlinear moment-slope relations computed in Part I. The zeroth order term is the linear free-vibration solution (constant spring stiffness equal to the first derivative of the moment-slope relation). Each mode generates a second harmonic (first-order term) whose magnitude depends on the linear spring stiffness and on the small perturbation parameter. Key features of the zeroth and first-order solutions are studied as functions of the moment-slope relations computed in Part I, and the possibility of cohesive property characterization is discussed. </p>

journal of mechanics of materials and structures

Nonlinear vibration of an edge-cracked beam with a cohesive zone, II: Perturbation analysis of Euler-Bernoulli beam vibration using a nonlinear spring for damage representation