Flexural frequency analysis of damaged beams using mixture unified gradient elasticity theory

Abstract

The flexural vibration of miniaturized homogeneous isotropic beams with multiple cracks is investigated using the mixture unified gradient elasticity theory. The model captures both possible stiffening and softening size-dependence at small scales. The problem is addressed using the Bernoulli-Euler beam theory, with the domain partitioned into distinct sections at cracked cross-sections. Cracks are assumed to be non-propagating, sufficiently spaced to avoid interaction, and open during vibration. The elastic spring model is employed to capture the effect of cracks on the dynamic characteristics. The time-dependent variational functional is rigorously established to derive variationally consistent and extra non-standard boundary and continuity conditions. Natural frequencies are obtained by solving the eigenvalue problem resulting from the imposition of boundary and continuity conditions. The predictions demonstrate excellent agreement with experimental, molecular dynamics, and analytical data from the literature for both large- and small-scale beams. The model is applied to examine the effects of gradient characteristic parameters, crack length and location, and boundary conditions on the frequencies. The practical application of the model to the inverse problem, where the location and length of a crack are unknown a priori, is numerically analyzed. The results indicate that the size effect significantly influences the inverse problem solution.