Partnerzy

Streszczenie:
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.

Streszczenie:
Structural schemes of applicative interests in Engineering Science frequently encounter the intricate phenomenon of discontinuity. The present study intends to address the discontinuity in the flexure of elastic nanobeam by adopting an abstract variational scheme. The mixture unified gradient theory of elasticity is invoked to realize the size-effects at the ultra-small scale. The consistent form of the interface conditions, stemming from the established stationary variational principle, is meticulously set forth. The boundary-value problem of equilibrium is properly closed and the analytical solution of the transverse displacement field of the elastic nanobeam is addressed. As an alternative approach, the eigenfunction expansion method is also utilized to scrutinize the efficacy of the presented variational formulation in tackling the flexure of elastic nanobeams with discontinuity. The flexural characteristic of mixture unified gradient beams with diverse kinematic constraints is numerically illustrated and thoroughly discussed. The anticipated nanoscopic features of the characteristic length-scale parameters are confirmed. The demonstrated numerical results can advantageously serve as a benchmark for the analysis and design of pioneering ultra-sensitive nano-sensors. The established variationally consistent size-dependent framework paves the way ahead in nanomechanics and inspires further research contributing to fracture mechanics of ultra-small scale elastic beams.