Flexural Frequency Analysis of Damaged Beams Using Mixture Unified Gradient Elasticity Theory

The free transverse vibration of miniaturized beams with multiple edge cracks is investigated using the mixture unified gradient elasticity theory. The model captures both possible stiffening and softening size-dependent behaviors at small scales. The problem is addressed using the Bernoulli-Euler beam theory, with the domain partitioned into distinct sections at the locations of cracked cross-sections. To account for the discontinuities in bending slope and deflection, rotational and translational springs are introduced at these cracked cross-sections. The time-dependent variational functional associated with the mixture unified gradient elasticity theory 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 model predictions demonstrate excellent agreement with experimental data from the literature for large-scale beams. Furthermore, as the crack length tends to zero, the results converge with those of crack-free mixture unified gradient elastic beams reported in prior studies. The model is applied to examine the effects of gradient characteristic parameters, crack length and location, and boundary conditions on the frequencies. Novel findings and discussions presented here hold significance for the design, condition monitoring, and maintenance of miniaturized structures.