Novel local/nonlocal formulation of the stress-driven model through closed form solution for higher vibrations modes

Highlights

• New two-phases stress-based closed form solution for free vibrations is proposed.

• Local/nonlocal model is defined by a convex combination through mixture parameter.

• Normalised natural frequencies for four study cases are determined.

• Effectiveness is discussed through a comparison with literature results.

• Normalised frequency for higher vibrations modes is computed for each study case.

Abstract

In the present work, an innovative two-phases local/nonlocal constitutive mixture model based on the stress-driven model for free vibrations problems is presented. The above two-phases model has been formulated by defining a convex combination of local/nonlocal phases through a mixture parameter and the closed form solution for free vibrations problems is given. The proposed local/nonlocal model has been applied to compute the natural frequencies of four nanobeams study cases. The results are presented in terms of normalised natural frequency as function of both nonlocal and mixture parameters. First, the influence of the two parameters is studied. The effectiveness of the present model is discussed by comparing the results with those obtained by applying the Gradient Elasticity theory. Finally, the normalised natural frequencies for the second, third and fourth modes of vibrations are presented and discussed for each study case.

Introduction

In the last decades, the interest in nanomaterials for engineering applications, such as electronic, biotechnology and nanostructure, is remarkably increased. The success of nanomaterials may be mainly due to their versatility and peculiar mechanical, electrical and thermal properties resulting from the nanoscale dimensions (i.e. 1–100 nm). According to their shape, they may be classified as nanorods, nanobeams, nanoribons, nanoplates and nanosheets.

Among the multitude of nanostructures, Carbon NanoTubes (CNTs), after their discovery in 1991 by Iijima [1], have gained the attention of the Scientific Community for their interesting applications [2], [3] and their peculiarity to be easily theoretically modelled as nanobeams [4]. Although a great research work has been done by researchers in the last years [4], CNTs mechanical and dynamic behaviour needs to be further investigated and properly predicted. In order to achieve this purpose, two ways may be followed: experimental characterisation and theoretical modelling.

Since performing experimental tests at nanoscale may be quite expensive and time consuming, the second way is often preferred and several theoretical models have been developed to find a reliable and low-cost tool to describe CNTs mechanical behaviours [4].

Generally, the theoretical models may be mainly subdivided into atomistic models [4], continuum based models (classical and non-classical continuum models) [5], [6], [7], [8], [9], [10], [11] and hybrid atomistic-continuum models, developed by combining the atomistic and continuum theories to study multiscale problems and overcome the limitations of the two above theories [4].

The first studies on CNTs were conducted by applying classical continuum models together with various beam theories, like Euler–Bernoulli and Timoshenko theories [5], [6]. However, structures at the nanoscale have a discrete nature, due to the atoms lattice and interactions between atoms, therefore continuum mechanics may have some limitations in describing this type of structures. Furthermore, as observed through experimental investigations and atomistic simulations, forces that are completely irrelevant at the macroscale, such as long range inter-atomic and inter-molecular cohesive ones, may change the physical properties at nanoscale influencing significantly both the static and the dynamic responses of the nanostructures [4], [12]. As a consequence, size effects are present and neglecting the small scale effects in nanostructures may cause completely incorrect simulations and, hence, give an improper design.

In such a context, several attempts to capture these small scale effects in nanostructures, by accounting for the nonlocal nature of the phenomenon, have been done and various non-classical continuum theories have been developed, such as strain gradient theory [13], [14], [15], [16], [17], [18]; couple stress theory [19], [20], [21]; micropolar theory [12] and nonlocal elasticity theory [7], [8], [9], [10].

By focusing the attention on the dynamic behaviour of CNTs, the nonlocal elasticity theory has been widely applied in vibration problems of nanotubes. According to such a theory, originally proposed by Eringen [7], [22], [23], the nonlocal continuum mechanics differs from the classical continuum mechanics since the stress at a point is assumed to be influenced by the strain at all points of the whole body [24]. More precisely, Eringen observed that the classical continuum theories failed to predict accurate results if the ratio between an external characteristic length (such as, for example, the crack length) and the internal characteristic length (related to the lattice structure) approaches the unit. To account for that small scale effect according to the Eringen theory, the stress field is obtained through an integral convolution (strain-driven integral convolution) between the elastic strain field and a suitable averaging kernel [23]. Since the solution of the original strain-driven integral problem was difficult mathematically, the same Author, in 1983, proposed a more convenient differential approach, named Eringen Differential Formulation (EDF) [24], [25].

Although the original strain-driven integral formulation is still assumed valid for some problems and several nonlocal theories have been developed by starting from the first Eringen’s works [26], [27], [28], [29], some Authors [25], [30], [31] have observed that the constitutive equations based on the EDF are in contrast with the equilibrium conditions [25]. In particular, it was observed that the nonlocal solution based on EDF coincided with the local one in the case of a cantilever beam under end-point loading [25], [30].

Different strategies have been proposed to solve these paradoxes. To overcome ill-posedness of strain-driven nonlocal elastic problems, a mixture of local/nonlocal elasticity was adopted by different Authors [32], [33], [34] on the basis of the original proposal by Eringen [35], [36]. Accordingly, Eringen’s two-phase local/nonlocal model, as a well-posed model, became one of the most reliable forms of Eringen’s nonlocal theory capable of accurately modeling the nanostructures with different boundary conditions. In the last years, Eringen’s two-phase local/nonlocal model has been used for analyzing the behaviour of different nanostructures, such as micro/nanorods [32], different loading conditions, such as bending [33], and vibrations [34].

Another way to overcome the above-mentioned drawback was proposed by Romano and Barretta through the stress-driven nonlocal model [37], [38]. Such a model is based on the same philosophy of original Eringen’s theory, but by writing the integral convolution as a function of the stress field instead of the strain one. Such a model has been applied to study nanobeams subjected to bending [37], [38], [39], axial load [40], torsion [41] buckling [42], [43], [44], [45] and free vibrations [8], [46], [47]. Moreover, the model has been recently extended to 2D nanostructures [48]. An attempt to use the stress-driven model together with local/nonlocal mixture formulation has been addressed in Ref. [49] for a free bar under uniform tension and a cantilever beam under uniform bending.

In the present work, the stress-driven nonlocal model is, for the first time, employed in conjunction with local/nonlocal mixture formulation for free vibrations problems by means the closed form solution presented in Ref. [46]. The main novelties of the present work are both the implementation of the mixture parameter in the close form solution and the application of the proposed formulations to determine the frequencies of higher vibrations modes. In particular, the effect of both nonlocal and mixture parameters on the nanobeam natural frequency is investigated focusing on the higher vibrations modes. The paper is, then, structured as in the following. In Section 2, the equations governing the free vibrations problem of a Euler-Bernoulli nanobeam expressed by means of the local/nonlocal model are presented. The boundary conditions are presented in Section 3 together with the four study cases. The obtained results in terms of normalized natural frequency, also for higher vibrations modes, are reported and discussed in Section 4 and some data are compared with literature results. Finally, the conclusions are summarised in Section 5.