A geometrically nonlinear beam model based on the second strain gradient theory

Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory

“…According to the Euler-Bernoulli beam theory, the formula of the nonzero component of the geometrically nonlinear strain tensor can be expressed as (Ghayesh & Farokhi 2015;Karparvarfard et al, 2015) ε xx = ∂u ∂x…”

“…Dai et al (2015) studied the nonlinear dynamics of cantilevered microbeams via modified couple stress theory. Karparvarfard, Asghari and Vatankhah (2015) derived a geometrically nonlinear beam model based on the second strain gradient theory. Ghayesh and Farokhi (2015) examined the complex dynamics of a micro-scaled nonlinear beam under a timedependent longitudinal load.…”

Free and forced vibration nonlinear analysis of a microbeam using finite strain and velocity gradients theory

“…In order to derive the equations of motion, Hamilton's principle (14) is used. Substituting the variations of strain energy (19), kinetic energy (22) and external energy (23) into Hamilton's principle, the equation of motion is achieved:…”

“…Wang et al [35] performed a similar study first for a microscale Timoshenko beam model and then for a size-dependent Reddy-Levinson beam model [36]. Kahrobaiyan et al [12] presented a nonlinear strain gradient beam formulation and analysis, while later, Karparvarfard et al [14] developed a geometrically nonlinear size-dependent beam formulation in the framework of the second strain gradient theory. Both Ghayesh et al [11] and Vatankhah et al [34] also investigated the nonlinear forced vibration of strain gradient microbeams but solved the governing equations using different analytical and numerical schemes.…”

Chaotic motion of a parametrically excited microbeam

“…Microscale continuous elements (Dehrouyeh-Semnani, 2014;Dehrouyeh-Semnani, 2015;Kahrobaiyan, Rahaeifard, Tajalli, & Ahmadian, 2012;Karparvarfard, Asghari, & Vatankhah, 2015;Mohammad-Abadi & Daneshmehr, 2014;Mohammadabadi, Daneshmehr, & Homayounfard, 2015;Sahmani, Ansari, Gholami, & Darvizeh, 2013;Tang, Ni, Wang, Luo, & Wang, 2014b), such as microbeams, are present in many microelectromechanical systems, such as in micro energy harvesters, microswitches, airbag accelerometers, vibration and shock sensors, biosensors, and microactuators, just to name a few. In some of these applications, microbeams are subject to longitudinal forces; these forces, under dynamical working conditions, vary with time.…”