Nonlinear dynamics size-dependent geometrically nonlinear Tymoshenko beams based on a modified moment theory

Abstract

We study non-linear vibrations of the geometrically non-linear Timoshenko beams on a basis of the modified couple stress theory, and taking into account the functionally graded material (FGM) of a beam. It is assumed that the studied beams are functionally graded along their thickness. In particular, investigation of influence of the size dependent coefficient and the coefficient responsible for material non-homogeneity/grading on the beam vibrations are studied. It has been discovered that the beams modelled on a basis of the modified couple stress theory are more stiff versus the beams modelled using the classical theory of continuum. The latter statement is valid for any distribution of the material characteristics along beam thickness. It has been illustrated that both mentioned coefficients, i.e. size dependent and characterizing the material non-homogeneity have essential influence of the beams vibrations.

Well-posed two-phase nonlocal integral models for free vibration of nanobeams in context with higher-order refined shear deformation theory

“…So far, the linear free vibration problem of high-order refined shear deformation beams based on the two-phase nonlocal integral models has been effectively solved. The analysis process given can provide a reference for more complex problems, such as considering von Kármán geometric nonlinearity (Awrejcewicz et al, 2017a(Awrejcewicz et al, , 2017bAwrejcewicz and Krysko, 2020;Krysko et al, 2017aKrysko et al, , 2017bKrysko et al, , 2017cKrysko et al, , 2018.…”

“…Compared with the high computational costs requirement of Molecular Dynamics (MD) simulations, as well as the implementation difficulty of experimental tests at the nanoscale, nonclassical continuum mechanics is convenient for addressing size-dependent behaviors of nanostructures. A practical nonclassical continuum mechanics approach is to extend the classical theories by introducing some material length-scale parameters, such as couple stress theory (Awrejcewicz et al, 2017a(Awrejcewicz et al, , 2017bAwrejcewicz and Krysko, 2020;Krysko et al, 2017aKrysko et al, , 2017bKrysko et al, , 2017cKrysko et al, , 2018, strain gradient theory, and nonlocal elastic theory. Since the governing equations and boundary conditions remain in the classical forms, nonlocal elasticity is widely used to capture the size-dependent response of microstructures.…”

Dynamic instability analysis of timoshenko FG sandwich nanobeams subjected to parametric excitation

“…To bring the effect of small-scale length of the nanobeams, several methods have been introduced including Nonlocal Strain Gradient Theory (NSGT), 35,36 Nonlocal Elasticity Theory (NET), 37 and Modified Coupled Stress Theory (MCST). [38][39][40][41][42] Incorporating the strain gradient parameter and nonlocal parameter, NSGT has been employed as a reliable theory to consider nanoscale effects. Gholipour and Ghayesh 43 developed the coupled vibration of FG nanobeam under a harmonic transverse load using the NSGT and Hamilton's principle.…”

Exact and nonlocal solutions for vibration of multiply connected bar-chain system with direct and indirect neighbouring interactions

“…Actually, if the segmental length of the proposed lattice system is taken equal to the characteristic length of nano-/micro-beams (i.e., atomic bond length, granular radius, etc. ), the proposed lattice system or nonlocal model are able to model small scale beams as investigated by Krysko, et al [12] and Awrejcewicz, et al [13] using modified coupled stress theory.…”