A mixed variational framework for higher-order unified gradient elasticity

Highlights

• Stationary variational framework for higher-order unified gradient elasticity theory.

• Higher-order boundary value problem and gradient constitutive laws in intrinsic form.

• Elimination of recognized restrictions typical of nonlocal gradient elasticity model.

• Examination of well-posed generalized elasticity theory in nano-mechanics of torsion.

• Determination of size-dependent shear modulus of SWCNTs with dissimilar chirality.

Abstract

The higher-order unified gradient elasticity theory is conceived in a mixed variational framework based on suitable functional space of kinetic test fields. The intrinsic form of the differential and boundary conditions of equilibrium along with the constitutive laws is consistently established. Various forms of the gradient elasticity theory, in the sense of stress or strain gradient models, can be retrieved as particular cases of the introduced generalized elasticity theory. The proposed stationary variational principle can effectively realize the nanoscopic structural effects while being exempt of restrictions typical of the nonlocal gradient elasticity model. The well-posed generalized gradient elasticity theory is invoked to study the mechanics of torsion and the torsional behavior of elastic nano-bars is analytically examined. The closed-form analytical formulae of the size-dependent shear modulus of nano-sized bar is determined and efficiently applied to reconstruct the shear modulus of SWCNTs with dissimilar chirality in comparison with the numerical simulation data. A practical approach to calibrate the characteristic lengths associated with the higher-order unified gradient elasticity theory is introduced. Numerical results associated with the torsion of higher-order unified gradient elastic bars are demonstrated and compared with the counterpart size-dependent elasticity theories. The conceived generalized gradient elasticity theory can beneficially characterize the nanoscopic response of advanced nano-materials.

Introduction

The current technological challenges of nano-engineering and developing new manufacturing methods have led to building of new devices with reduced size and wide application. Nanostructures such as carbon nanotubes, graphene sheets and carbon-based heterostructures have shown great potential to be used as mechanical components in macro-, micro-, and nano-scale systems (Almagableh, Omari & Sevostianov, 2019; Fang, Chen, Zhang, Xia & Weng, 2021; Pelliciari & Tarantino, 2021). Nano-structured materials are also extensively employed to enhance the physical properties of advanced composites (Kushch & Sevostianov, 2021; Mazloum, Kováčik, Zagrai & Sevostianov, 2020) as pioneering homogenization techniques are recently developed to identify their equivalent physical properties (Sevostianov & Kachanov, 2021; Vilchevskaya, Kushch, Kachanov & Sevostianov, 2021).

In these ultra-small systems, understanding of the mechanical characteristics is crucial in order to improve the efficiency and functionality of these complex systems. Three approaches are commonly adopted for investigating the mechanical responses of structures at nano-scales: experimental techniques, molecular dynamics (MD) simulations, and size-dependent continuum modeling. The second and third approaches are efficient theoretical tools to explain the underlying reasons behind experimentally observed responses, and to extract the mechanical characteristics where performing reliable experiments is still impossible and/or highly expensive. The main postulate governing the strain gradient theory is the assumption of material response dependence on both the classical strain and the strain gradients of various orders (Fleck & Hutchinson, 1993; Mindlin, 1965). Different forms of the strain gradient theories are recognized within the literature to be capable of describing the stiffening material behavior; as put into evidence in flexure mechanics of nano-scale bars (Barretta, Faghidian & Marotti de Sciarra, 2019; Fu, Zhou & Qi, 2020), flexoelectric nano-beams (Dilena, Fedele Dell'Oste, Fernández-Sáez, Morassi & Zaera, 2020; Li, Wang & Yang, 2021), flexoelectricity and piezoelectricity of a micro-bar (Eremeyev, Ganghoffer, Konopińska-Zmysłowska & Uglov, 2020), higher-order shear deformable micro-beams (Ma, Gao & Reddy, 2008, 2010; Reddy & Arbind, 2012) and micro-plates (Ma, Gao & Reddy, 2011; Srinivasa & Reddy, 2013), functionally graded micro-beams (Reddy, 2011; Romanoff & Reddy, 2014) and micro-plates (Kim & Reddy, 2012, 2013, 2015), microcrystalline cellulose sheets (Dastjerdi, Naeijian, Akgöz & Civalek, 2021), functionally graded porous micro-plates (Kim, Żur & Reddy, 2019; Reddy, Romanoff & Loya, 2016), and wave dispersion analysis (Mikhasev, Botogova & Eremeyev, 2021). To capture the softening behavior of materials with nano-structural feature, either the nonlocal elasticity theory or the stress gradient theory is exploited in nano-mechanics; see e.g. representative studies on nanobeams (Elishakoff, Ajenjo & Livshits, 2020; Reddy, 2007) and nano-plates (Hache, Challamel & Elishakoff, 2019, 2019a; Reddy, 2010b). While the nonlocal elasticity comprised the size-dependent effects of the long-range interatomic forces (Eringen, 2002), the stress gradient theory is introduced as a counterpart of the strain gradient model wherein the material response depends on both the classical stress and the stress gradients of various orders (Forest & Sab, 2012).

Nano-structured materials, nevertheless, may reveal either of stiffening or softening structural responses contingent on the material particular state conditions (Pisano, Fuschi & Polizzotto, 2021). To characterize a wider spectrum of nanoscopic phenomena in material response, diverse forms of gradient elasticity models are combined with the nonlocal elasticity, such as nonlocal strain gradient theory (Aifantis, 2011; Lim, Zhang & Reddy, 2015; Polizzotto, 2015), nonlocal modified gradient theory (Faghidian, 2021b; 2021a), nonlocal surface elasticity (Li, Lin & Ng, 2020; Zhu & Li, 2019), and strain gradient nonlocal Biot poromechanics (Tong, Ding, Yan, Xu & Lei, 2020). Among the aforementioned size-dependent elasticity models, the nonlocal strain gradient theory has drawn much attention in the literature notwithstanding the controversial issue of non-standard boundary conditions (Zaera, Serrano & Fernández-Sáez, 2020, 2020a). By way of illustration, see its recent implementation in nano-structural analysis of beams (Malikan, Uglov & Eremeyev, 2020a), plates (Farajpour, Howard & Robertson, 2020), and shells (Malikan, Krasheninnikov & Eremeyev, 2020b).

The size-dependent elasticity theories, incorporating higher-order nanoscopic effects, can more effectively capture the peculiar structural characteristics at nano-scale and, thus, can represent a more precise interpretation of size-effect phenomena particularly at the presence of topographical defects such as edges and corners. The classical elasticity theory is well-known to result in inadmissible singularities in the vicinity of topographical defects; on the contrary, introducing higher-order size-dependent theories allows a continuum to sustain boundary conditions on edges and corners, see recent advances as addresses in (Eremeyev, Cazzani and dell'Isola (2021), Lurie, Kalamkarov, Solyaev and Volkov, (2021) and Yang, Timofeev, Emek Abali, Li and Müller, (2021)). A general variational framework, based on suitable functional space of kinetic test fields, is conceived to establish the higher-order unified gradient elasticity theory in the absence of topographical defects on the continuum boundary. The intrinsic form of the mixed variational principle associated with the higher-order unified gradient elasticity is introduced in Section 2. The differential and boundary conditions of dynamic equilibrium consistent with the higher-order unified gradient elasticity theory in conjunction with the higher-order constitutive laws are consistently determined. The proposed generalized gradient elasticity theory is equipped with consistent numbers of gradient length-scale parameters contrary to similar size-dependent elasticity models suffering from imbalanced numbers of characteristic lengths (Faghidian, 2021b, 2021a). As the stiffening structural response is realized via the strain gradient model, the softening behavior is captured through the stress gradient model rather than the nonlocal elasticity theory. Introducing the nonlocal integral convolutions and prescribing the sophisticated equivalence theorems between the integral and differential formulations are, therefore, superfluous (Faghidian, 2020b, 2020a, 2020c). The proposed stationary variational formulation can efficiently introduce the higher-order gradient effects while being exempt of the restrictions associated with the nonlocal elasticity model in view of limited available nonlocal kernels mainly dedicated to one-dimensional structural analysis. Evidence of well-posedness of the proposed higher-order unified gradient elasticity framework is demonstrated in Section 3 wherein the elastostatic torsional behavior of structural schemes of interest to the Engineering Science community is rigorously examined. Section 4 is devoted to study the size-dependent shear modulus of single-walled carbon nanotubes (SWCNTs) with dissimilar chirality via introducing the closed-form analytical formulae. The analytically determined size-dependent shear modulus and ensuing results are subsequently compared with the pertinent numerical simulation data. A practical approach to calibrate the characteristic lengths associated with the higher-order unified gradient elasticity theory is introduced and advantageously exploited. Section 4 is furthermore enriched by making comparison of the nano-material response of SWCNTs detected based on the higher-order unified gradient elasticity with the corresponding results achieved in accordance with the higher-order nonlocal gradient theory along with the first-order unified gradient theory. Concluding remarks are summarized in Section 5.