Abstract
The unified higher-order theory of two-phase nonlocal gradient elasticity is conceived via consistently introducing the higher-order two-phase nonlocality to the higher-order gradient theory of elasticity. The unified integro-differential constitutive law is established in an abstract variational framework equipped with ad hoc functional space of test fields. Equivalence between the higher-order integral convolutions of the constitutive law and the nonlocal gradient differential formulation is confirmed by prescribing the non-classical boundary conditions. The strain-driven and stress-driven nonlocal approaches are exploited to simulate the long-range interactions at nano-scale. A range of generalized continuum models are restored under special ad hoc assumptions. The established unified higher-order elasticity theory is invoked to analytically examine the wave dispersion phenomenon. In contrast to the first-order size-dependent elasticity model, the higher-order two-phase nonlocal gradient theory can efficiently capture the wave dispersion characteristics observed in experimental measurements. The precise description of nano-scale wave phenomena noticeably underlines the importance of applying the proposed higher-order size-dependent elasticity theory. A viable approach to tackle peculiar dynamic phenomena at nano-scale is introduced.
Similar content being viewed by others
References
Lee S, Pugno NM, Ryu S (2019) Atomistic simulation study on the crack growth stability of graphene under uniaxial tension and indentation. Meccanica 54:1915–1926. https://doi.org/10.1007/s11012-019-01027-x
Almagableh A, Omari MA, Sevostianov I (2019) Modeling of anisotropic elastic properties of multi-walled zigzag carbon nanotubes. Int J Eng Sci 144:103127. https://doi.org/10.1016/j.ijengsci.2019.103127
He Z, Wang G, Pindera M-J (2019) Multiscale homogenization and localization of materials with hierarchical porous microstructures. Compos Struct 222:110905. https://doi.org/10.1016/j.compstruct.2019.110905
Marami G, Adib Nazari S, Faghidian SA, Vakili-Tahami F, Etemadi S (2016) Improving the mechanical behavior of the adhesively bonded joints using RGO additive. Int J Adhes Adhes 70:277–286. https://doi.org/10.1016/j.ijadhadh.2016.07.014
Bianchi G, Radi E (2020) Analytical estimates of the pull-in voltage for carbon nanotubes considering tip-charge concentration and intermolecular forces. Meccanica 55:193–209. https://doi.org/10.1007/s11012-019-01119-8
Rezaei M, Khadem SE, Friswell MI (2020) Energy harvesting from the secondary resonances of a nonlinear piezoelectric beam under hard harmonic excitation. Meccanica 55:1463–1479. https://doi.org/10.1007/s11012-020-01187-1
Ghavanloo E, Rafii-Tabar H, Fazelzadeh SA (2019) Computational continuum mechanics of nanoscopic structures: nonlocal elasticity approaches. Springer, Berlin. https://doi.org/10.1007/978-3-030-11650-7
Rafii-Tabar H, Ghavanloo E, Fazelzadeh SA (2016) Nonlocal continuum-based modeling of mechanical characteristics of nanoscopic structures. Phys Rep 638:1–97. https://doi.org/10.1016/j.physrep.2016.05.003
Elishakoff I, Pentaras D, Dujat K, Versaci C, Muscolino G, Storch J, Bucas S, Challamel N, Natsuki T, Zhang YY, Wang CM, Ghyselinck G (2012) Carbon Nanotubes and Nano Sensors: Vibrations, Buckling, and Ballistic Impact. ISTE-Wiley, London
Aifantis EC (2010) A personal view on current generalized theories of elasticity and plastic flow. In: Maugin GA, Metrikine AV (eds) Mechanics of generalized continua: one hundred years after the cosserats. Springer, New York, pp 191–202
Aifantis EC (1992) On the role of gradients in the localization of deformation and fracture. Int J Eng Sci 30:1279–1299. https://doi.org/10.1016/0020-7225(92)90141-3
Altan SB, Aifantis EC (1992) On the structure of the mode III crack-tip in gradient elasticity. Scr Metall Mater 26:319–324. https://doi.org/10.1016/0956-716X(92)90194-J
Askes H, Aifantis EC (2006) Gradient elasticity theories in statics and dynamics-a unification of approaches. Int J Fract 139:297–304. https://doi.org/10.1007/s10704-006-8375-4
Askes H, Aifantis EC (2011) Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results. Int J Solids Struct 48:1962–1990. https://doi.org/10.1016/j.ijsolstr.2011.03.006
Polizzotto C (2012) A gradient elasticity theory for second-grade materials and higher order inertia. Int J Solids Struct 49:2121–2137. https://doi.org/10.1016/j.ijsolstr.2012.04.019
Polizzotto C (2003) Gradient elasticity and nonstandard boundary conditions. Int J Solids Struct 40:7399–7423. https://doi.org/10.1016/j.ijsolstr.2003.06.001
Lazar M, Maugin GA, Aifantis EC (2006) Dislocations in second strain gradient elasticity. Int J Solids Struct 43:1787–1817. https://doi.org/10.1016/j.ijsolstr.2005.07.005
Askes H, Aifantis EC (2009) Gradient elasticity and flexural wave dispersion in carbon nanotubes. Phys Rev B 80:195412. https://doi.org/10.1103/PhysRevB.80.195412
Polizzotto C (2014) Stress gradient versus strain gradient constitutive models within elasticity. Int J Solids Struct 51:1809–1818. https://doi.org/10.1016/j.ijsolstr.2014.01.021
Fuschi P, Pisano AA, Polizzotto C (2019) Size effects of small-scale beams in bending addressed with a strain-difference based nonlocal elasticity theory. Int J Mech Sci 151:661–671. https://doi.org/10.1016/j.ijmecsci.2018.12.024
Li L, Lin R, Ng TY (2020) Contribution of nonlocality to surface elasticity. Int J Eng Sci 152:103311. https://doi.org/10.1016/j.ijengsci.2020.103311
Babu B, Patel BP (2020) An improved quadrilateral finite element for nonlinear second-order strain gradient elastic Kirchhoff plates. Meccanica 55:139–159. https://doi.org/10.1007/s11012-019-01087-z
Liu C, Yu J, Xu W, Zhang X, Zhang B (2020) Theoretical study of elastic wave propagation through a functionally graded micro-structured plate base on the modified couple-stress theory. Meccanica 55:1153–1167. https://doi.org/10.1007/s11012-020-01156-8
Pisano AA, Fuschi P, Polizzotto C (2020) A strain-difference based nonlocal elasticity theory for small-scale shear-deformable beams with parametric warping. Int J Multiscale Comput Eng 18(1):83–102. https://doi.org/10.1615/IntJMultCompEng.2019030885
Eringen A (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54:4703–4710. https://doi.org/10.1063/1.332803
Lazar M, Maugin GA, Aifantis EC (2006) On a theory of nonlocal elasticity of bi-Helmholtz type and some applications. Int J Solids Struct 43:1404–1421. https://doi.org/10.1016/j.ijsolstr.2005.04.027
Jena SK, Chakraverty S, Malikan M, Mohammad-Sedighi H (2020) Implementation of Hermite-Ritz method and Navier’s technique for vibration of functionally graded porous nanobeam embedded in Winkler-Pasternak elastic foundation using bi-Helmholtz nonlocal elasticity. J Mech Mater Struct 15:405–434. https://doi.org/10.2140/jomms.2020.15.405
Faghidian SA (2020) Higher-order nonlocal gradient elasticity: a consistent variational theory. Int J Eng Sci 154:103337. https://doi.org/10.1016/j.ijengsci.2020.103337
Faghidian SA (2020) Two-phase local/nonlocal gradient mechanics of elastic torsion. Math Methods Appl Sci. https://doi.org/10.1002/mma.6877
Faghidian SA (2020) Higher-order mixture nonlocal gradient theory of wave propagation. Math Methods Appl Sci. https://doi.org/10.1002/mma.6885
Barretta R, Fazelzadeh SA, Feo L, Ghavanloo E, Luciano R (2018) Nonlocal inflected nano-beams: a stress-driven approach of bi-Helmholtz type. Compos Struct 200:239–245. https://doi.org/10.1016/j.compstruct.2018.04.072
Romano G, Diaco M (2020) On formulation of nonlocal elasticity problems. Meccanica. https://doi.org/10.1007/s11012-020-01183-5
Eroglu U (2020) Perturbation approach to Eringen’s local/non-local constitutive equation with applications to 1-D structures. Meccanica 55:1119–1134. https://doi.org/10.1007/s11012-020-01145-x
Tuna M, Kirca M, Trovalusci P (2019) Deformation of atomic models and their equivalent continuum counterparts using Eringen’s two-phase local/nonlocal model. Mech Res Commun 97:26–32. https://doi.org/10.1016/j.mechrescom.2019.04.004
Apuzzo A, Barretta R, Fabbrocino F, Faghidian SA, Luciano R, Marotti de Sciarra F (2019) Axial and torsional free vibrations of elastic nano-beams by stress-driven two-phase elasticity. J Appl Comput Mech 5:402–413. https://doi.org/10.22055/jacm.2018.26552.1338
Zhu X, Li L (2017) Longitudinal and torsional vibrations of size-dependent rods via nonlocal integral elasticity. Int J Mech Sci 133:639–650. https://doi.org/10.1016/j.ijmecsci.2017.09.030
Zhu X, Li L (2017) On longitudinal dynamics of nanorods. Int J Eng Sci 120:129–145. https://doi.org/10.1016/j.ijengsci.2017.08.003
Zhu X, Li L (2017) Closed form solution for a nonlocal strain gradient rod in tension. Int J Eng Sci 119:16–28. https://doi.org/10.1016/j.ijengsci.2017.06.019
Jena SK, Chakraverty S, Malikan M, Tornabene F (2020) Effects of surface energy and surface residual stresses on vibro-thermal analysis of chiral, zigzag, and armchair types of SWCNTs using refined beam theory. Mech Based Des Struct Mach. https://doi.org/10.1080/15397734.2020.1754239
Jena SK, Chakraverty S, Malikan M (2020) Vibration and buckling characteristics of nonlocal beam placed in a magnetic field embedded in Winkler-Pasternak elastic foundation using a new refined beam theory: an analytical approach. Eur Phys J Plus 135:164. https://doi.org/10.1140/epjp/s13360-020-00176-3
Tuna M, Leonetti L, Trovalusci P, Kirca M (2020) ‘Explicit’ and ‘implicit’ non-local continuous descriptions for a plate with circular inclusion in tension. Meccanica 55:927–944. https://doi.org/10.1007/s11012-019-01091-3
Maneshi MA, Ghavanloo E, Fazelzadeh SA (2020) Well-posed nonlocal elasticity model for finite domains and its application to the mechanical behavior of nanorods. Acta Mech. https://doi.org/10.1007/s00707-020-02749-w
Ghavanloo E, Rafii-Tabar H, Fazelzadeh SA (2019) New insights on nonlocal spherical shell model and its application to free vibration of spherical fullerene molecules. Int J Mech Sci 161–162:105046. https://doi.org/10.1016/j.ijmecsci.2019.105046
Hache F, Challamel N, Elishakoff I (2019) Asymptotic derivation of nonlocal beam models from two-dimensional nonlocal elasticity. Math Mech Solids 24:2425–2443. https://doi.org/10.1177/1081286518756947
Hache F, Challamel N, Elishakoff I (2019) Asymptotic derivation of nonlocal plate models from three-dimensional stress gradient elasticity. Continuum Mech Thermodyn 31:47–70. https://doi.org/10.1007/s00161-018-0622-1
Ghavanloo E, Fazelzadeh SA (2019) Wave propagation in one-dimensional infinite acoustic metamaterials with long-range interactions. Acta Mech 230:4453–4461. https://doi.org/10.1007/s00707-019-02514-8
Numanoğlu HM, Civalek Ö (2019) On the torsional vibration of nanorods surrounded by elastic matrix via nonlocal FEM. Int J Mech Sci 161–162:105076. https://doi.org/10.1016/j.ijmecsci.2019.105076
Pisano AA, Fuschi P (2018) Structural symmetry and boundary conditions for nonlocal symmetrical problems. Meccanica 53:629–638. https://doi.org/10.1007/s11012-017-0684-3
Shaat M (2018) Correction of local elasticity for nonlocal residuals: application to Euler-Bernoulli beams. Meccanica 53:3015–3035. https://doi.org/10.1007/s11012-018-0855-x
Challamel N, Aydogdu M, Elishakoff I (2018) Statics and dynamics of nanorods embedded in an elastic medium: nonlocal elasticity and lattice formulations. Eur J Mech A Solids 67:254–271. https://doi.org/10.1016/j.euromechsol.2017.09.009
Hache F, Challamel N, Elishakoff I (2018) Lattice and continualized models for the buckling study of nonlocal rectangular thick plates including shear effects. Int J Mech Sci 145:221–230. https://doi.org/10.1016/j.ijmecsci.2018.04.058
Civalek Ö, Demir Ç (2016) A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method. Appl Math Comput 289:335–352. https://doi.org/10.1016/j.amc.2016.05.034
Aifantis EC (2003) Update on a class of gradient theories. Mech Mater 35:259–280. https://doi.org/10.1016/S0167-6636(02)00278-8
Aifantis EC (2011) On the gradient approach–Relation to Eringen’s nonlocal theory. Int J Eng Sci 49:1367–1377. https://doi.org/10.1016/j.ijengsci.2011.03.016
Aifantis EC (2014) On non-singular GRADELA crack fields. Theor Appl Mech Lett 4:051005. https://doi.org/10.1063/2.1405105
Lim CW, Zhang G, Reddy JN (2015) A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J Mech Phys Solids 78:298–313. https://doi.org/10.1016/j.jmps.2015.02.001
Faghidian SA (2018) Reissner stationary variational principle for nonlocal strain gradient theory of elasticity. Eur J Mech A Solids 70:115–126. https://doi.org/10.1016/j.euromechsol.2018.02.009
Faghidian SA (2018) On non-linear flexure of beams based on non-local elasticity theory. Int J Eng Sci 124:49–63. https://doi.org/10.1016/j.ijengsci.2017.12.002
Faghidian SA (2018) Integro-differential nonlocal theory of elasticity. Int J Eng Sci 129:96–110. https://doi.org/10.1016/j.ijengsci.2018.04.007
Xu X-J, Wang X-C, Zheng M-L, Ma Z (2017) Bending and buckling of nonlocal strain gradient elastic beams. Compos Struct 160:366–377. https://doi.org/10.1016/j.compstruct.2016.10.038
Xu X-J, Zheng M (2019) Analytical solutions for buckling of size-dependent Timoshenko beams. Appl Math Mech-Engl Ed 40:953–976. https://doi.org/10.1007/s10483-019-2494-8
Zaera R, Serrano Ó, Fernández-Sáez J (2020) Non-standard and constitutive boundary conditions in nonlocal strain gradient elasticity. Meccanica 55:469–479. https://doi.org/10.1007/s11012-019-01122-z
Zaera R, Serrano Ó, Fernández-Sáez J (2019) On the consistency of the nonlocal strain gradient elasticity. Int J Eng Sci 138:65–81. https://doi.org/10.1016/j.ijengsci.2019.02.004
Torabi J, Ansari R, Zabihi A, Hosseini K (2020) Dynamic and pull-in instability analyses of functionally graded nanoplates via nonlocal strain gradient theory. Mech Based Des Struct Mach. https://doi.org/10.1080/15397734.2020.1721298
Barretta R, Faghidian SA, Marotti de Sciarra F, Vaccaro MS (2020) Nonlocal strain gradient torsion of elastic beams: variational formulation and constitutive boundary conditions. Arch Appl Mech 90:691–706. https://doi.org/10.1007/s00419-019-01634-w
Barretta R, Faghidian SA, Marotti de Sciarra F, Penna R, Pinnola FP (2020) On torsion of nonlocal Lam strain gradient FG elastic beams. Compos Struct 233:111550. https://doi.org/10.1016/j.compstruct.2019.111550
Pinnola FP, Faghidian SA, Barretta R, Marotti de Sciarra F (2020) Variationally consistent dynamics of nonlocal gradient elastic beams. Int J Eng Sci 149:103220. https://doi.org/10.1016/j.ijengsci.2020.103220
Jena SK, Chakraverty S, Malikan M (2020) Stability analysis of nanobeams in hygrothermal environment based on a nonlocal strain gradient Timoshenko beam model under nonlinear thermal field. J Comput Des Eng. https://doi.org/10.1093/jcde/qwaa051
Jena SK, Chakraverty S, Malikan M, Sedighi HM (2020) Hygro-magnetic vibration of the single-walled carbon nanotube with nonlinear temperature distribution based on a modified beam theory and nonlocal strain gradient model. Int J Appl Mech. https://doi.org/10.1142/S1758825120500544
Zabihi A, Ansari R, Torabi J, Samadani F, Hosseini K (2019) An analytical treatment for pull-in instability of circular nanoplates based on the nonlocal strain gradient theory with clamped boundary condition. Mater Res Express 6:0950b3. https://doi.org/10.1088/2053-1591/ab31bc
Barretta R, Faghidian SA, Marotti de Sciarra F, Pinnola FP (2019) Timoshenko nonlocal strain gradient nanobeams: variational consistency, exact solutions and carbon nanotube Young moduli. Mech Adv Mater Struct. https://doi.org/10.1080/15376494.2019.1683660
Norouzzadeh A, Ansari R, Rouhi H (2018) Nonlinear wave propagation analysis in Timoshenko nano-beams considering nonlocal and strain gradient effects. Meccanica 53:3415–3435. https://doi.org/10.1007/s11012-018-0887-2
Shodja HM, Ahmadpoor F, Tehranchi A (2012) Calculation of the additional constants for fcc materials in second strain gradient elasticity: behavior of a nano-size Bernoulli-Euler beam with surface effects. J Appl Mech 79:021008. https://doi.org/10.1115/1.4005535
Ojaghnezhad F, Shodja HM (2013) A combined first principles and analytical determination of the modulus of cohesion, surface energy, and the additional constants in the second strain gradient elasticity. Int J Solids Struct 50:3967–3974. https://doi.org/10.1016/j.ijsolstr.2013.08.004
Barretta R, Faghidian SA, Marotti de Sciarra F (2019) Aifantis versus Lam strain gradient models of Bishop elastic rods. Acta Mech 230:2799–2812. https://doi.org/10.1007/s00707-019-02431-w
Barretta R, Faghidian SA, Marotti de Sciarra F (2020) A consistent variational formulation of Bishop nonlocal rods. Continuum Mech Thermodyn 32:1311–1323. https://doi.org/10.1007/s00161-019-00843-6
Yosida K (1978) Functional Analysis. Springer, New York
Polyanin A, Manzhirov A (2008) Handbook of Integral Equations. CRC Press, New York
Barretta R, Faghidian SA, Marotti de Sciarra F, Pinnola FP (2020) On nonlocal Lam strain gradient mechanics of elastic rods. Int J Multiscale Comput Eng 18:67–81. https://doi.org/10.1615/IntJMultCompEng.2019030655
Polizzotto C (2016) Variational formulations and extra boundary conditions within stress gradient elasticity theory with extensions to beam and plate models. Int J Solids Struct 80:405–419. https://doi.org/10.1016/j.ijsolstr.2015.09.015
Li L, Hu Y (2016) Wave propagation in fluid-conveying viscoelastic carbon nanotubes based on nonlocal strain gradient theory. Comput Mater Sci 112:282–288. https://doi.org/10.1016/j.commatsci.2015.10.044
De Domenico D, Askes H, Aifantis EC (2019) Gradient elasticity and dispersive wave propagation: model motivation and length scale identification procedures in concrete and composite laminates. Int J Solids Struct 158:176–190. https://doi.org/10.1016/j.ijsolstr.2018.09.007
De Domenico D, Askes H, Aifantis EC (2018) Capturing wave dispersion in heterogeneous and microstructured materials through a three-length-scale gradient elasticity formulation. J Mech Behav Mater 27:20182002. https://doi.org/10.1515/jmbm-2018-2002
De Domenico D, Askes H (2018) Nano-scale wave dispersion beyond the First Brillouin Zone simulated with inertia gradient continua. J Appl Phys 124:205107. https://doi.org/10.1063/1.5045838
Dederichs PH, Schober H, Sellmyer DJ (1981) Phonon states of elements, electron states and fermi surfaces of alloys. Springer, Berlin
Farrahi GH, Faghidian SA, Smith DJ (2010) An inverse method for reconstruction of residual stress field in welded plates. J Pressure Vessel Technol 132:061205. https://doi.org/10.1115/1.4001268
Faghidian SA (2014) A smoothed inverse eigenstrain method for reconstruction of the regularized residual fields. Int J Solids Struct 51:4427–4434. https://doi.org/10.1016/j.ijsolstr.2014.09.012
Faghidian SA (2015) Inverse determination of the regularized residual stress and eigenstrain fields due to surface peening. J Strain Anal Eng Des 50:84–91. https://doi.org/10.1177/0309324714558326
Caprio MA (2005) LevelScheme: a level scheme drawing and scientific figure preparation system for Mathematica. Comput Phys Commun 171:107–118. https://doi.org/10.1016/j.cpc.2005.04.010
Acknowledgements
This study is dedicated to Elias C. Aifantis in recognition of his contributions on the gradient elasticity.
Funding
This research did not receive any specific grant-based funding.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Faghidian, S.A., Ghavanloo, E. Unified higher-order theory of two-phase nonlocal gradient elasticity. Meccanica 56, 607–627 (2021). https://doi.org/10.1007/s11012-020-01292-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11012-020-01292-1