Flexural wave propagation in small-scaled functionally graded beams via a nonlocal strain gradient theory

Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory

“…Furthermore, the stiffness softening effects or the stiffness enhancement effects can be observed from the nonlocal strain gradient models, depending on the values of the nonlocal parameter and the material length scale parameter [36,39,[42][43][44]. The nonlocal strain gradient theory has been employed to investigate the size-dependent mechanical behaviors of linear and nonlinear beams [39,[42][43][44][45] and plates [46].…”

“…Under such case, ∇=∂/∂x. Thus, the general constitutive relation can be then simplified to[34,[37][38][39][40][41]45] 1 − (ea) 2 ∇ 2 t xx = 1 − l 2 ∇ 2 Eε xx(5) …”

Bifurcation analysis of the rotating axially compressed nano‐rod with imperfections

“…[2] Micro-and nano-electromechanical systems (MEMS and NEMS) also require the use of nonlocal stress-strain constitutive equations in modelling their behavior, see [10,24,26,27,37]. Functionally graded materials (FGM) display non-local properties as well and thus being modeled using different variants of the Eringen constitutive equations, as in [11,12,19,35,36] for buckling problems and in [16,18,31,32,34] for free vibrations problem. Unlike the atomic modelling of small-scale structures, continuum approach requires less computational effort and, moreover, in many cases of buckling and free vibrations of such structures as well as for the wave propagation analytical solutions are also available.…”

Wave Propagation Characteristics in Functionally Graded Double-Beams

“…When the stress-strain relations are substituted into the force and moment definitions: (6) the following constitutive equations are obtained as follows: (7) where: A ij , B ij and D ij denote the extensional, coupling and bending stiffnesses respectively. The extensional, coupling and bending stiffnesses are defined in the following way: (8) The governing equations of the beam can be obtained variationally by use of Hamilton's principle as follows: (9) where: the subscript ",tt" denotes the derivation with respect to time. ρ i is the density function of the beam and defined in the following form: (10) By neglecting the rotary inertias, the equations of motion of a double-beam system with an elastic layer, can be defined as (Filiz and Aydogdu [10]),…”

“…They used two continuous resilient layers such as upper and lower Euler-Bernoulli beams in the model to obtain the displacements that are used to calculate the cut on frequencies and critical velocities of the track. Li et al [9] developed an analytical model of smallscaled functionally graded beams for the flexural wave propagation analysis and investigated the dispersion relation between the phase velocity and wave number. Shamalta and Metrikine [11] investigated the steady-state dynamic response of an embedded railway track by use of a model composing of two Euler-Bernoulli beams, a Kirchoff plate, two continuous visco-elastic elements that connect the beams and plate and 2-D elastic foundation.…”