A mixed variational framework for higher-order unified gradient elasticity
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.
Section snippets
Higher-order unified gradient elasticity
Variational frameworks can be introduced based on energy considerations or an inverse process to derive the variational forms from the known governing equations. Stationary principles, also known as mixed variational principles, are of particular importance in the classical mechanics of structures. The stationary functional is generally constructed from the equations of an elastic continuum by treating all the variables to be independent of each other; further details of mixed variational
Torsion mechanics of higher-order elastic bars
Implementation of the conceived higher-order unified gradient elasticity in structural schemes of interest to the Engineering Science is evinced via studying the mechanics of torsion. A homogenous straight bar under torsion of length, elastic shear modulus G and annular cross-section with polar moment of area about the annulus center J is considered. The bar is referred to orthogonal axes with abscissa x being coincident with the centroid axis and r denotes the position vector of a
Size-dependent shear modulus
To demonstrate the efficacy of the conceived higher-order unified gradient elasticity in capturing the peculiar response of materials with nano-structural features, the size-dependent shear modulus of SWCNTs is examined. It is well-established that SWCNTs can be effectively treated as a uniform homogenous thin-shell bar. A fully-fixed bar, i.e. classically fixed at both ends, subjected to a linearly distributed torsional couple is accordingly considered. It is also worthy of notice,
Concluding remarks
In the present study, a mixed variational principle associated with the higher-order unified gradient elasticity is proposed in the intrinsic form and employed to study the mechanics of torsion of nano-bars. The differential and boundary conditions of dynamic equilibrium are consistently derived according to the higher-order unified gradient elasticity theory in conjunction with the higher-order constitutive relations. The conceived stationary variational formulation can realize the
Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
References (75)
On the gradient approach–relation to Eringen's nonlocal theory
International Journal of Engineering Science
(2011)- et al.
Stress gradient, strain gradient and inertia gradient beam theories for the simulation of flexural wave dispersion in carbon nanotubes
Composites Part B: Engineering
(2018) - et al.
Gradient elasticity and dispersive wave propagation: Model motivation and length scale identification procedures in concrete and composite laminates
International Journal of Solids and Structures
(2019) - et al.
Hearing distributed mass in nanobeam resonators
International Journal of Solids and Structures
(2020) - et al.
Saint-Venant torsion of cylindrical orthotropic elliptical cross section
Mechanics Research Communications
(2019) On non-linear flexure of beams based on non-local elasticity theory
International Journal of Engineering Science
(2018)- et al.
A phenomenological theory for strain gradient effects in plasticity
Journal of the Mechanics and Physics of Solids
(1993) - et al.
Stress gradient continuum theory
Mechanics Research Communications
(2012) - et al.
Analytical solutions for bending, vibration, and buckling of FGM plates using a couple stress-based third-order theory
Composite Structures
(2013) - et al.
Bending, free vibration, and buckling of modified couples stress-based functionally graded porous micro-plates
Composite Structures
(2019)
Dislocations in second strain gradient elasticity
International Journal of Solids and Structures
A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation
Journal of the Mechanics and Physics of Solids
A microstructure-dependent Timoshenko beam model based on a modified couple stress theory
Journal of the Mechanics and Physics of Solids
Second gradient of strain and surface-tension in linear elasticity
International Journal of Solids and Structures
On dual-complementary variational principles in mathematical physics
International Journal of Engineering Science
Gradient elasticity and nonstandard boundary conditions
International Journal of Solids and Structures
A unifying variational framework for stress gradient and strain gradient elasticity theories
European Journal of Mechanics A/Solids
Nonlocal theories for bending, buckling and vibration of beams
International Journal of Engineering Science
Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates
International Journal of Engineering Science
Microstructure-dependent couple stress theories of functionally graded beams
Journal of the Mechanics and Physics of Solids
A nonlinear modified couple stress-based third-order theory of functionally graded plates
Composite Structures
Nonlinear finite element analysis of functionally graded circular plates with modified couple stress theory
European Journal of Mechanics-A/Solids
Experimental validation of the modified couple stress Timoshenko beam theory for web-core sandwich panels
Composite Structures
Torsion of a functionally graded material
International Journal of Engineering Science
A model for a constrained, finitely deforming, elastic solid with rotation gradient dependent strain energy, and its specialization to von Karman plates and beams
Journal of Physics and Mechanics of Solids
Modeling of anisotropic elastic properties of multi-walled zigzag carbon nanotubes
International Journal of Engineering Science
Practical bayesian inference: A primer for physical scientists
Aifantis versus Lam strain gradient models of Bishop elastic rods
Acta Mechanica
Nonlocal strain gradient torsion of elastic beams: Variational formulation and constitutive boundary conditions
Archive of Applied Mechanics
On the mechanical analysis of microcrystalline cellulose sheets
International Journal of Engineering Science
Generalization of Eringen's result for random response of a beam on elastic foundation
European Journal of Mechanics A/Solids
On nonlinear dilatational strain gradient elasticity
Continuum Mechanics and Thermodynamics
Flexoelectricity and apparent piezoelectricity of a pantographic micro-bar
International Journal of Engineering Science
Nonlocal continuum field theories
Inverse determination of the regularized residual stress and eigenstrain fields due to surface peening
Journal of Strain Analysis for Engineering Design
Analytical inverse solution of eigenstrains and residual fields in autofrettaged thick-walled tubes
Journal of Pressure Vessel Technology
Analytical approach for inverse reconstruction of eigenstrains and residual stresses in autofrettaged spherical pressure vessels
Journal of Pressure Vessel Technology
Cited by (61)
Elastostatics of nonuniform miniaturized beams: Explicit solutions through a nonlocal transfer matrix formulation
2024, International Journal of Engineering ScienceWave propagation in periodic nano structures through second strain gradient elasticity
2023, International Journal of Mechanical SciencesFormulation of non-local space-fractional plate model and validation for composite micro-plates
2023, International Journal of Engineering ScienceExploring the effect of stress-strain behavior of cemented tailings backfill on the stability of inclined sill mat: A numerical study
2023, Alexandria Engineering Journal