Local/nonlocal mixture integral models with bi-Helmholtz kernel for free vibration of Euler-Bernoulli beams under thermal effect

Abstract

We present predictive models of the free vibration of Euler-Bernoulli beams subjected to a uniformly thermal environment using two-phase local/nonlocal mixture theory of strain- and stress-driven types. The equation of motion and standard boundary conditions are derived via Hamilton's principle and the constitutive equation is expressed in local/nonlocal mixed integral form with bi-Helmholtz kernel. The temperature effect is taken as equivalent to a thermal load taking into account the two-phase nonlocal effects. After transforming the local/nonlocal mixture integral constitutive equation to differential form with four constitutive boundary conditions, the problem is solved numerically via a generalized differential quadrature method (GDQM). We provide a series of numerical examples in which we present predictions of the size-effects on beams with various boundary edges using different local/nonlocal mixture models. The influence of the two-phase thermal load on the natural frequencies of the beam is also examined. Furthermore, it can be observed that the normalized frequency value varies consistently with the increase of the order of the vibration mode.