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Reissner stationary variational principle for nonlocal strain gradient theory of elasticity. (English) Zbl 1406.74102

Summary: The general form of Reissner stationary variational principle is established in the framework of the nonlocal strain gradient theory of elasticity. Including two size-dependent characteristic parameters, the nonlocal strain gradient elasticity theory can demonstrate the significance of the strain gradient as well as the nonlocal elastic stress field. Based on the Reissner functional, the governing differential and boundary conditions of dynamic equilibrium and differential constitutive equations of the classical and first-order nonlocal stress tensor are derived in the most general form. Additionally, the boundary congruence conditions are formulated and discussed for the nonlocal strain gradient theory. To exhibit the application value of Reissner variational principle, it is employed to examine the nonlinear vibrations of size-dependent Bernoulli-Euler and Timoshenko beams. In the case of immovable boundary conditions, employing the weighted residual Galerkin method, the homotopy analysis method is also utilized to determine the closed form analytical solutions of the geometrically nonlinear vibration equations. Consequently, the analytical expressions for the nonlinear natural frequencies of Bernoulli-Euler and Timoshenko nonlocal strain gradient beams are derived.

MSC:

74B20 Nonlinear elasticity
74K10 Rods (beams, columns, shafts, arches, rings, etc.)

Software:

MUL2; BVPh
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[1] Shaat, M.; Abdelkefi, A., On a second-order rotation gradient theory for linear elastic continua, Int. J. Eng. Sci., 100, 74-98, (2016) · Zbl 1406.74539
[2] Aifantis, E. C., On the role of gradients in the localization of deformation and fracture, Int. J. Eng. Sci., 30, 10, 1279-1299, (1992) · Zbl 0769.73058
[3] Aifantis, E. C., Update on a class of gradient theories, Mech. Mater., 35, 3-6, 259-280, (2003)
[4] Aifantis, E. C., On the gradient approach-relation to Eringen’s nonlocal theory, Int. J. Eng. Sci., 49, 12, 1367-1377, (2011) · Zbl 06923523
[5] Aifantis, E. C., On non-singular GRADELA crack fields, Theor. Appl. Mech. Lett., 4, 5, (2014), 051005-1:7
[6] Aifantis, E. C., Internal length gradient (ILG) material mechanics across scales and disciplines, Adv. Appl. Mech., 49, 1-110, (2016)
[7] Aifantis, K. E.; Willis, J. R., The role of interfaces in enhancing the yield strength of composites and polycrystals, J. Mech. Phys. Solid., 53, 5, 1047-1070, (2005) · Zbl 1120.74316
[8] Altan, B. S.; Evensen, H. A.; Aifantis, E. C., Longitudinal vibrations of a beam: a gradient elasticity approach, Mech. Res. Commun., 23, 1, 35-40, (1996) · Zbl 0843.73048
[9] Apuzzo, A.; Barretta, R.; Canadija, M.; Feo, L.; Luciano, R.; Marotti de Sciarra, F., A closed-form model for torsion of nanobeams with an enhanced nonlocal formulation, Composites Part B, 108, 315-324, (2017)
[10] Apuzzo, A.; Barretta, R.; Luciano, R.; Marotti de Sciarra, F.; Penna, R., Free vibrations of Bernoulli-Euler nano-beams by the stress-driven nonlocal integral model, Composites Part B, 123, 105-111, (2017)
[11] Askes, H.; Aifantis, E. C., Gradient elasticity and flexural wave dispersion in carbon nanotubes, Phys. Rev. B, 80, 19, (2009)
[12] Askes, H.; Aifantis, E. C., Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results, Int. J. Solid Struct., 48, 13, 1962-1990, (2011)
[13] Askes, H.; Gitman, I., Review and critique of the stress gradient elasticity theories of eringen and aifantis, (Maugin, G.; Metrikine, A., Mechanics of Generalized Continua, (2010), Springer New York), 203-210 · Zbl 1396.74032
[14] Askes, H.; Suiker, A. S.J.; Sluys, L. J., A classification of higher-order strain-gradient models-linear analysis, Arch. Appl. Mech., 2, 2-3, 171-188, (2002) · Zbl 1065.74004
[15] Attia, M. A.; Mahmoud, F. F., Modeling and analysis of nanobeams based on nonlocal-couple stress elasticity and surface energy theories, Int. J. Mech. Sci., 105, 126-134, (2016)
[16] Barretta, R.; Feo, L.; Luciano, R.; Marotti de Sciarra, F., Variational formulations for functionally graded nonlocal Bernoulli-Euler nanobeams, Compos. Struct., 129, 80-89, (2015)
[17] Barretta, R.; Feo, L.; Luciano, R.; Marotti de Sciarra, F., A gradient eringen model for functionally graded nanorods, Compos. Struct., 131, 1124-1131, (2015)
[18] Barretta, R.; Feo, L.; Luciano, R.; Marotti de Sciarra, F.; Penna, R., Functionally graded Timoshenko nanobeams: a novel nonlocal gradient formulation, Composites Part B, 100, 208-219, (2016)
[19] Barretta, R.; Faghidian, S. A.; Luciano, R., Longitudinal vibrations of nano-rods by stress-driven integral elasticity, Mech. Adv. Mater. Struct., (2017), Accepted for publication
[20] Barretta, R.; Brcic, M.; Canadija, M.; Luciano, R.; Marotti de Sciarra, F., Application of gradient elasticity to armchair carbon nanotubes: size effects and constitutive parameters assessment, Eur. J. Mech. Solid., 65, 1-13, (2017) · Zbl 1406.74373
[21] Blevins, R. D., Formulas for dynamics, acoustics and vibration, (2016), John Wiley & Sons United Kingdom
[22] Canadija, M.; Barretta, R.; Marotti de Sciarra, F., On functionally graded Timoshenko nonisothermal nanobeams, Compos. Struct., 135, 286-296, (2016)
[23] Carrera, E., Developments, ideas, and evaluations based upon Reissner’s mixed variational theorem in the modeling of multilayered plates and shells, Appl. Mech. Rev., 54, 4, 301-329, (2001)
[24] Carrera, E.; Ciuffreda, A., Bending of composites and sandwich plates subject to localized lateral loadings: a comparison of various theories, Compos. Struct., 68, 2, 185-202, (2005)
[25] Carrera, E.; Ciuffreda, A., A unified formulation to assess theories of multilayered plates for various bending problems, Compos. Struct., 69, 3, 271-293, (2005)
[26] Carrera, E.; Brischetto, S.; Nali, P., Plates and shells for smart structures: classical and advanced theories for modeling and analysis, (2011), John Wiley & Sons United Kingdom · Zbl 1241.74001
[27] Challamel, N., Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams, Compos. Struct., 105, 351-368, (2013)
[28] Challamel, N.; Wang, C. M., The small length scale effect for a non-local cantilever beam: a paradox solved, Nanotechnol, 19, 34, (2008), 1-7
[29] Demasi, L., Mixed plate theories based on the generalized unified formulation, I: governing equations, Compos. Struct., 87, 1, 1-11, (2009)
[30] Dym, C. L.; Shames, I. H., Solid mechanics a variational approach, (2013), Springer New York, (Augmented Edition) · Zbl 1279.74001
[31] Elishakoff, I.; Pentaras, D.; Dujat, K.; Versaci, C.; Muscolino, G.; Storch, J.; Bucas, S.; Challamel, N.; Natsuki, T.; Zhang, Y. Y.; Wang, C. M.; Ghyselinck, G., Carbon nanotubes and nanosensors: vibrations, buckling and ballistic impact, (2012), Wiley-ISTE London
[32] Emam, S. A.; Nayfeh, A. H., Postbuckling and free vibrations of composite beams, Compos. Struct., 88, 4, 636-642, (2009)
[33] Eringen, A. C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys., 54, 9, 4703-4710, (1983)
[34] Eringen, A. C., Nonlocal continuum field theories, (2002), Springer US · Zbl 1023.74003
[35] Faghidian, S. A., Unified formulations of the shear coefficients in Timoshenko beam theory, J. Eng. Mech. Trans. ASCE, 143, 9, (2017), 06017013-1:8
[36] Faghidian, S. A., On non-linear flexure of beams based on non-local elasticity theory, Int. J. Eng. Sci., 124, 49-63, (2018) · Zbl 1423.74471
[37] Forest, S.; Sab, K., Stress gradient continuum theory, Mech. Res. Commun., 40, 16-25, (2012)
[38] Guo, S.; He, Y.; Liu, D.; Lei, J.; Shen, L.; Li, Z., Torsional vibration of carbon nanotube with axial velocity and velocity gradient effect, Int. J. Mech. Sci., 119, 88-96, (2016)
[39] Gurtin, M. E.; Fried, E.; Anand, L., The mechanics and thermodynamics of continua, (2010), Cambridge University Press Cambridge
[40] Güven, U., A generalized nonlocal elasticity solution for the propagation of longitudinal stress waves in bars, Eur. J. Mech. Solid., 45, 75-79, (2014) · Zbl 1406.74352
[41] Hadjesfandiari, A. R.; Dargush, G. F., Couple stress theory for solids, Int. J. Solid Struct., 48, 18, 2496-2510, (2011)
[42] Haghani, A.; Mondali, M.; Faghidian, S. A., Linear and nonlinear flexural analysis of higher-order shear deformation laminated plates with circular delamination, Acta Mech., (2017), accepted for publication · Zbl 1390.74021
[43] Kargarnovin, M. H.; Jafari-Talookolaei, R. A., Application of the homotopy method for the analytic approach of the nonlinear free vibration analysis of the simple end beams using four engineering theories, Acta Mech., 212, 3-4, 199-213, (2010) · Zbl 1397.74089
[44] Kargarnovin, M.; Faghidian, S. A.; Farjami, Y.; Farrahi, G. H., Application of homotopy-Padé technique in limit analysis of circular plates under arbitrary rotational symmetric loading using von-Mises yield criterion, Commun. Nonlinear Sci. Numer. Simulat., 15, 4, 1080-1091, (2010) · Zbl 1221.74052
[45] Khodabakhshi, P.; Reddy, J. N., A unified integro-differential nonlocal model, Int. J. Eng. Sci., 95, 60-75, (2015) · Zbl 1423.74133
[46] Li, L.; Hu, Y., Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory, Int. J. Eng. Sci., 97, 84-94, (2015) · Zbl 1423.74495
[47] Li, L.; Hu, Y., Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material, Int. J. Eng. Sci., 107, 77-97, (2016) · Zbl 1423.74496
[48] Li, L.; Hu, Y., Post-buckling analysis of functionally graded nanobeams incorporating nonlocal stress and microstructure-dependent strain gradient effects, Int. J. Mech. Sci., 120, 159-170, (2017)
[49] Li, L.; Hu, Y.; Ling, L., Flexural wave propagation in small-scaled functionally graded beams via a nonlocal strain gradient theory, Compos. Struct., 133, 1079-1092, (2015)
[50] Li, L.; Hu, Y.; Li, X., Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory, Int. J. Mech. Sci., 115-116, 135-144, (2016)
[51] Li, L.; Li, X.; Hu, Y., Free vibration analysis of nonlocal strain gradient beams made of functionally graded material, Int. J. Eng. Sci., 102, 77-92, (2016) · Zbl 1423.74399
[52] Li, L.; Hu, Y.; Ling, L., Wave propagation in viscoelastic single-walled carbon nanotubes with surface effect under magnetic field based on nonlocal strain gradient theory, Physica E, 75, 118-124, (2016)
[53] Li, X.; Li, L.; Hu, Y.; Ding, Z.; Deng, W., Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory, Compos. Struct., 165, 250-265, (2017)
[54] Li, C.; Liu, J. J.; Cheng, M.; Fan, X. L., Nonlocal vibrations and stabilities in parametric resonance of axially moving viscoelastic piezoelectric nanoplate subjected to thermo-electro-mechanical forces, Composites Part B, 116, 153-169, (2017)
[55] Li, L.; Tang, H.; Hu, Y., Size-dependent nonlinear vibration of beam-type porous materials with an initial geometrical curvature, Compos. Struct., 184, 1177-1188, (2018)
[56] Liao, S., Beyond perturbation: introduction to homotopy analysis method, (2004), Chapman and Hall/CRC press New York · Zbl 1051.76001
[57] Liao, S., Homotopy analysis method in nonlinear differential equations, (2012), Springer New York · Zbl 1253.35001
[58] Liao, S., Advances in the homotopy analysis method, (2013), World Scientific New Jersey
[59] Lim, C. W.; Zhang, G.; Reddy, J. N., A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation, J. Mech. Phys. Solid., 78, 298-313, (2015) · Zbl 1349.74016
[60] Lopatin, A. V.; Morozov, E. V.; Shatov, A. V., An analytical expression for fundamental frequency of the composite lattice cylindrical shell with clamped edges, Compos. Struct., 141, 232-239, (2016)
[61] Lu, L.; Guo, X.; Zhao, J., Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory, Int. J. Eng. Sci., 116, 12-24, (2017) · Zbl 1423.74499
[62] Lu, L.; Guo, X.; Zhao, J., A unified nonlocal strain gradient model for nanobeams and the importance of higher order terms, Int. J. Eng. Sci., 119, 265-277, (2017) · Zbl 06985749
[63] Marotti de Sciarra, F.; Barretta, R., A new nonlocal bending model for Euler-Bernoulli nanobeams, Mech. Res. Commun., 62, 1, 25-30, (2014)
[64] Mindlin, R. D., Second gradient of strain and surface-tension in linear elasticity, Int. J. Solid Struct., 1, 4, 417-438, (1965)
[65] Mindlin, R. D.; Eshel, N. N., On first strain gradient theories in linear elasticity, Int. J. Solid Struct., 4, 1, 109-124, (1968) · Zbl 0166.20601
[66] Papargyri-Beskou, S.; Polyzos, D.; Beskos, D. E., Dynamic analysis of gradient elastic flexural beams, Struct. Eng. Mech., 15, 6, 705-716, (2003) · Zbl 1022.74010
[67] Polizzotto, C., Nonlocal elasticity and related variational principles, Int. J. Solid Struct., 38, 42-43, 7359-7380, (2001) · Zbl 1014.74003
[68] Polizzotto, C., Gradient elasticity and nonstandard boundary conditions, Int. J. Solid Struct., 40, 26, 7399-7423, (2003) · Zbl 1063.74015
[69] Polizzotto, C., Unified thermodynamic framework for nonlocal/gradient continuum theories, Eur. J. Mech. Solid., 22, 5, 651-668, (2003) · Zbl 1032.74505
[70] Polizzotto, C., A gradient elasticity theory for second-grade materials and higher order inertia, Int. J. Solid Struct., 49, 15-16, 2121-2137, (2012)
[71] Polizzotto, C., A second strain gradient elasticity theory with second velocity gradient inertia-part I: constitutive equations and quasi-static behavior, Int. J. Solid Struct., 50, 24, 3749-3765, (2013)
[72] Polizzotto, C., A second strain gradient elasticity theory with second velocity gradient inertia-part II: dynamic behavior, Int. J. Solid Struct., 50, 24, 3766-3777, (2013)
[73] Polizzotto, C., Stress gradient versus strain gradient constitutive models within elasticity, Int. J. Solid Struct., 51, 9, 1809-1818, (2014)
[74] Polizzotto, C., A unifying variational framework for stress gradient and strain gradient elasticity theories, Eur. J. Mech. Solid., 49, 430-440, (2015) · Zbl 1406.74091
[75] Polizzotto, C., Variational formulations and extra boundary conditions within stress gradient elasticity theory with extensions to beam and plate models, Int. J. Solid Struct., 80, 405-419, (2016)
[76] Polizzotto, C., A hierarchy of simplified constitutive models within isotropic strain gradient elasticity, Eur. J. Mech. Solid., 61, 92-109, (2017) · Zbl 1406.74090
[77] Raees, A.; Xu, H.; Aifantis, E. C., Homotopy shear band solutions in gradient plasticity, Z. Naturforsch., 72, 5, 477-486, (2017)
[78] Rao, S. S., Vibration of continuous systems, (2007), John Wiley & Sons New Jersey
[79] Reddy, J. N., Energy principles and variational methods in applied mechanics, (2017), Wiley New Jersey
[80] Reissner, E., Reflections on the theory of elastic plates, Appl. Mech. Rev., 38, 11, 1453-1464, (1985)
[81] Reissner, E., On a mixed variational theorem and on a shear deformable plate theory, Int. J. Numer. Meth. Eng., 23, 2, 193-198, (1986) · Zbl 0579.73018
[82] Romano, G.; Barretta, R., Comment on the paper “exact solution of Eringen’s nonlocal integral model for bending of Euler-Bernoulli and Timoshenko beams” by meral tuna and mesut kirca, Int. J. Eng. Sci., 109, 240-242, (2016) · Zbl 1423.74511
[83] Romano, G.; Barretta, R., Stress-driven versus strain-driven nonlocal integral model for elastic nano-beams, Composites Part B, 114, 184-188, (2017)
[84] Romano, G.; Barretta, R., Nonlocal elasticity in nanobeams: the stress-driven integral model, Int. J. Eng. Sci., 115, 14-27, (2017) · Zbl 1423.74512
[85] Romano, G.; Barretta, R.; Diaco, M., Micromorphic continua: non-redundant formulations, Continuum Mech. Therm., 28, 6, 1659-1670, (2016) · Zbl 1365.74135
[86] Romano, G.; Barretta, R.; Diacom, M.; Marotti de Sciarra, F., Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams, Int. J. Mech. Sci., 121, 151-156, (2017)
[87] Romano, G.; Barretta, R.; Diaco, M., On nonlocal integral models for elastic nano-beams, Int. J. Mech. Sci., 131-132, 490-499, (2017)
[88] Salehipour, H.; Shahidi, A. R.; Nahvi, H., Modified nonlocal elasticity theory for functionally graded materials, Int. J. Eng. Sci., 90, 44-57, (2015) · Zbl 1423.74138
[89] Shen, Y.; Chen, Y.; Li, L., Torsion of a functionally graded material, Int. J. Eng. Sci., 119, 14-28, (2016) · Zbl 1423.74410
[90] Song, J.; Shen, J.; Li, X. F., Effects of initial axial stress on waves propagating in carbon nanotubes using a generalized nonlocal model, Comput. Mater. Sci., 49, 3, 518-523, (2010)
[91] Thai, H.-T.; Vo, T. P.; Nguyen, T.-K.; Kim, S.-E., A review of continuum mechanics models for size-dependent analysis of beams and plates, Compos. Struct., 177, 196-219, (2017)
[92] Tornabene, F.; Fantuzzi, N.; Viola, E.; Carrera, E., Static analysis of doubly-curved anisotropic shells and panels using CUF approach, differential geometry and differential quadrature method, Compos. Struct., 107, 675-697, (2014)
[93] Wu, C. H., Cohesive elasticity and surface phenomena, Q. Appl. Math., 50, 1, 73-103, (1992) · Zbl 0817.73056
[94] Wu, C. P.; Chiu, K. H., RMVT-based meshless collocation and element-free Galerkin methods for the quasi-3D free vibration analysis of multilayered composite and FGM plates, Compos. Struct., 93, 5, (2011), 1433-1148
[95] Wu, C. P.; Lai, W. W., Reissner’s mixed variational theorem-based nonlocal Timoshenko beam theory for a single-walled carbon nanotube embedded in an elastic medium and with various boundary conditions, Compos. Struct., 122, 390-404, (2015)
[96] Wu, C. P.; Chiu, K. H.; Wang, Y. M., RMVT-based meshless collocation and element-free Galerkin methods for the quasi-3D analysis of multilayered composite and FGM plates, Compos. Struct., 93, 2, 923-943, (2011)
[97] Yang, F.; Chong, A. C.M.; Lam, D. C.C.; Tong, P., Couple stress based strain gradient theory for elasticity, Int. J. Solid Struct., 39, 10, 2731-2743, (2002) · Zbl 1037.74006
[98] Yang, W. D.; Fang, C. Q.; Wang, X., Nonlinear dynamic characteristics of FGCNTs reinforced microbeam with piezoelectric layer based on unifying stress-strain gradient framework, Composites Part B, 111, 372-386, (2017)
[99] Zhang, Y. Y.; Wang, C. M.; Challamel, N., Bending, buckling, and vibration of micro/nanobeams by hybrid nonlocal beam model, J. Eng. Mech., 136, 5, 562-574, (2010)
[100] Zhu, X.; Li, L., Twisting statics of functionally graded nanotubes using Eringen’s nonlocal integral model, Compos. Struct., 178, 87-96, (2017)
[101] Zhu, X.; Li, L., Longitudinal and torsional vibrations of size-dependent rods via nonlocal integral elasticity, Int. J. Mech. Sci., 2017, 133, 639-650, (2017)
[102] Zhu, X.; Li, L., On longitudinal dynamics of nanorods, Int. J. Eng. Sci., 120, 129-145, (2017) · Zbl 06985786
[103] Zhu, X.; Li, L., Closed form solution for a nonlocal strain gradient rod in tension, Int. J. Eng. Sci., 119, 16-28, (2017) · Zbl 1423.74334
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