Thermoelastic analysis of rotating functionally graded micro/nanodisks of variable thickness

In the present manuscript, the second strain gradient (SSG) is utilized to investigate the primary resonance of a nonlinear Euler-Bernoulli nanobeam is analyzed in this paper for the first time. To that end, the second strain gradient theory, a higher-order continuum theory capable of taking the size effects into account, is utilized and the governing equation of the motion for an Euler-Bernoulli nanobeam is derived with sixteen higher-order material constants. Then by implementing the Galerkin’s method,the Duffing equation for the vibration of a hinged-hinged nanobeam is obtained and its primary resonance is studied utilizing the method of multiple scales. The size effects and impact of various system parameters on the amplitude of the response are then investigated for three different materials and the results are compared to thatof the first strain gradient and classical theories. The results of this manuscript clearly shows that the nonlinear vibration of a second strain gradient nanobeam is size-dependent and although the difference between the results obtained by the second strain gradient theory and the first strain gradient theory is negligible for thicker beams, as the thickness decreases, the difference becomes more prominent. Also, the effects of nonlinearity on the forced vibration nonlinear response of an SSG beam are investigated and some observations are reported.

Sound radiation analysis with an enriched Timoshenko beam model based on second strain gradient theory

Modeling deformation of auxetic and non-auxetic polymer gels

The usual continuum theories are inadequate in predicting the mechanical behavior of solids in the presence of small defects and stress concentrators; it is well known that such continuum methods are unable to detect the change of the size of the inhomogeneities and defects. For these reasons various augmented continuum theories and strain gradient theories have been proposed in the literature. The major difficulty in implication of these theories lies in the lack of information about the additional material constants which appear in such theories. For fcc metals, for the calculation of the associated characteristic lengths which arise in first strain gradient theory, an atomistic approach based on the Sutton–Chen interatomic potential function is proposed. For the validity of the computed characteristic lengths, the phenomenon of the size effect pertinent to a nano-sized circular void within an fcc (111) plane is examined via both first strain gradient theory and lattice statics. Comparison of the results explains the physical ramifications of the characteristic lengths in improving the usual continuum results. Moreover, by reconsideration of the Kelvin problem it is shown that a commonly employed variant of the first strain gradient theory is only valid for a few fcc metals.

We recently proposed a strain gradient theory to account for the size dependence of plastic deformation at micron and submicron length scales. The strain gradient theory includes the effects of both rotation gradient and stretch gradient such that the rotation gradient influences the material character through the interaction between the Cauchy stresses and the couple stresses; the stretch gradient measures explicitly enter the constitutive relations through the instantaneous tangent modulus. Indentation tests at scales on the order of one micron have shown that measured hardness increases significantly with decreasing indent size. In the present paper, the strain gradient theory is used to model materials undergoing small-scale indentations. A strong effect of including strain gradients in the constitutive description is found with hardness increasing by a factor of two or more over the relevant range behavior. Comparisons with the experimental data for polycrystalline copper and single crystal copper indeed show an approximately linear dependence of the square of the hardness, H 2, on the inverse of the indentation depth, 1/h, I.e., H-2 proportional to 1/h, which provides an important self-consistent check of the strain gradient theory proposed by the authors earlier.

The governing equations of motion, together with the associated boundary conditions, are derived for the second strain gradient Timoshenko micro- and nano-beams. The second strain gradient theory is a highly powerful nonclassical continuum theory, capable of capturing the size effects in micro- and nano-scale structures. In case studies, the static and free-vibration behaviors of a hinged–hinged beam are investigated utilizing the presented second strain gradient theory-based Timoshenko beam model. The obtained results are compared with those of the available models in the literature, which are based on the (first) strain gradient theory, the modified couple stress theory, and the classical continuum theory.

Large-amplitude coupled scale-dependent behaviour of geometrically imperfect NSGT nanotubes

Implicit iterative finite element scheme for a strain gradient crystal plasticity model based on self-energy of geometrically necessary dislocations

Recent experiments have shown that the microscale material behavior is very different from that of bulk materials, and displays strong size effects when the characteristic length associated with the deformation is on the order of microns. Conventional continuum theories, however, can not predict this size dependence because they do not have an intrinsic length in their constitutive models. A new continuum theory, namely the strain gradient theory, has been proposed to investigate the deformation of solids at the microscale. For materials undergoing plastic deformation, the basis of strain gradient theory is the dislocation theory in materials science, and strain gradient plasticity has agreed remarkably well with experiments. For elastic materials with microstructures, it has also been established that the material behavior can be represented by an elastic strain gradient theory. A general approach to investigate fracture of materials with strain gradient effects is established. Both the near-tip asymptotic fields and the elastic full-field solutions are obtained in closed form. Due to stain gradient effects, stresses ahead of a crack tip are significantly higher than those in the classical K field. The plastic zone size surronunding a crack tip is estimated by elastic near-tip fields, as well as by the Dugdale model. It is established that the plastic zone is, in general, much more round and larger than that estimated from the classical K field.

Coupled longitudinal-transverse-rotational free vibration of post-buckled functionally graded first-order shear deformable micro- and nano-beams based on the Mindlin's strain gradient theory

Nonlinear vibration and stability analysis of the curved microtube conveying fluid as a model of the micro coriolis flowmeters based on strain gradient theory

In this paper, the flexural free vibrations of three dimensional micro beams are investigated based on strain gradient theory. The most general form of the strain gradient theory which contains five higher-order material constants has been applied to the micro beam to take the small-scale effects into account. Having considered the Euler-Bernoulli beam model, governing equations of motion are written by utilizing the Hamilton’s principle. Then, the state-space solution technique is used to find some solutions for natural frequencies of the beam under various boundary conditions. The numerical results show that the resonant frequencies are significantly dependent on the length scale parameter of the micro beam. The less the non-dimensional length scale is, the more deviation appears between results obtained for natural frequencies of micro shaft by strain gradient theory and classical continuum theory. Moreover, except for a micro shaft which is simply supported at both ends, the extra type of boundary conditions emerges from using strain gradient theory significantly affects the results.

In this paper, for the first time based on the nonlocal strain gradient theory the effect of size dependency in torsional vibration of bi-direction functionally graded (FG) nonlinear nano-cone is study. The material properties were assumed to vary according to the arbitrary function in radial and axial directions. The Navier equation and boundary conditions of the size-dependent bidirectional FG nonlinear nano-cone were derived by Hamilton's principle. These equations were solved by employing the generalized differential quadrature method (GDQM). The presented model can turn into the classical model if the material length scale parameters are taken to be zero. The effects of some parameters, such as inhomogeneity constant, cross-sectional area parameter and small-scale parameters, were studied. As an essential result of this study can be stated that an FG nano-cone model based on the nonlocal elasticity theory behaves softer and based on the strain gradient theory behaves harder.

The second strain gradient functionally graded beam formulation

Non-linear behaviors of carbon nanotubes under electrostatic actuation based on strain gradient theory