Reissner stationary variational principle for nonlocal strain gradient theory of elasticity

Shaat, 2016, On a second-order rotation gradient theory for linear elastic continua, Int. J. Eng. Sci., 100, 74, 10.1016/j.ijengsci.2015.11.009 Aifantis, 1992, On the role of gradients in the localization of deformation and fracture, Int. J. Eng. Sci., 30, 1279, 10.1016/0020-7225(92)90141-3 Aifantis, 2003, Update on a class of gradient theories, Mech. Mater., 35, 259, 10.1016/S0167-6636(02)00278-8 Aifantis, 2011, On the gradient approach–Relation to Eringen's nonlocal theory, Int. J. Eng. Sci., 49, 1367, 10.1016/j.ijengsci.2011.03.016 Aifantis, 2014, On non-singular GRADELA crack fields, Theor. Appl. Mech. Lett., 4, 10.1063/2.1405105 Aifantis, 2016, Internal length gradient (ILG) material mechanics across scales and disciplines, Adv. Appl. Mech., 49, 1, 10.1016/bs.aams.2016.08.001 Aifantis, 2005, The role of interfaces in enhancing the yield strength of composites and polycrystals, J. Mech. Phys. Solid., 53, 1047, 10.1016/j.jmps.2004.12.003 Altan, 1996, Longitudinal vibrations of a beam: a gradient elasticity approach, Mech. Res. Commun., 23, 35, 10.1016/0093-6413(95)00074-7 Apuzzo, 2017, A closed-form model for torsion of nanobeams with an enhanced nonlocal formulation, Composites Part B, 108, 315, 10.1016/j.compositesb.2016.09.012 Apuzzo, 2017, Free vibrations of Bernoulli-Euler nano-beams by the stress-driven nonlocal integral model, Composites Part B, 123, 105, 10.1016/j.compositesb.2017.03.057 Askes, 2009, Gradient elasticity and flexural wave dispersion in carbon nanotubes, Phys. Rev. B, 80, 10.1103/PhysRevB.80.195412 Askes, 2011, 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, 1962, 10.1016/j.ijsolstr.2011.03.006 Askes, 2010, Review and critique of the stress gradient elasticity theories of Eringen and Aifantis, 203 Askes, 2002, A classification of higher-order strain-gradient models–linear analysis, Arch. Appl. Mech., 2, 171, 10.1007/s00419-002-0202-4 Attia, 2016, Modeling and analysis of nanobeams based on nonlocal-couple stress elasticity and surface energy theories, Int. J. Mech. Sci., 105, 126, 10.1016/j.ijmecsci.2015.11.002 Barretta, 2015, Variational formulations for functionally graded nonlocal Bernoulli–Euler nanobeams, Compos. Struct., 129, 80, 10.1016/j.compstruct.2015.03.033 Barretta, 2015, A gradient Eringen model for functionally graded nanorods, Compos. Struct., 131, 1124, 10.1016/j.compstruct.2015.06.077 Barretta, 2016, Functionally graded Timoshenko nanobeams: a novel nonlocal gradient formulation, Composites Part B, 100, 208, 10.1016/j.compositesb.2016.05.052 Barretta, 2017, Longitudinal vibrations of nano-rods by stress-driven integral elasticity, Mech. Adv. Mater. Struct. Barretta, 2017, Application of gradient elasticity to armchair carbon nanotubes: size effects and constitutive parameters assessment, Eur. J. Mech. Solid., 65, 1, 10.1016/j.euromechsol.2017.03.002 Blevins, 2016 Canadija, 2016, On functionally graded Timoshenko nonisothermal nanobeams, Compos. Struct., 135, 286, 10.1016/j.compstruct.2015.09.030 Carrera, 2001, Developments, ideas, and evaluations based upon Reissner's mixed variational theorem in the modeling of multilayered plates and shells, Appl. Mech. Rev., 54, 301, 10.1115/1.1385512 Carrera, 2005, Bending of composites and sandwich plates subject to localized lateral loadings: a comparison of various theories, Compos. Struct., 68, 185, 10.1016/j.compstruct.2004.03.013 Carrera, 2005, A unified formulation to assess theories of multilayered plates for various bending problems, Compos. Struct., 69, 271, 10.1016/j.compstruct.2004.07.003 Carrera, 2011 Challamel, 2013, Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams, Compos. Struct., 105, 351, 10.1016/j.compstruct.2013.05.026 Challamel, 2008, The small length scale effect for a non-local cantilever beam: a paradox solved, Nanotechnol, 19, 10.1088/0957-4484/19/34/345703 Demasi, 2009, Mixed plate theories based on the generalized unified formulation, I: governing equations, Compos. Struct., 87, 1, 10.1016/j.compstruct.2008.07.013 Dym, 2013 Elishakoff, 2012 Emam, 2009, Postbuckling and free vibrations of composite beams, Compos. Struct., 88, 636, 10.1016/j.compstruct.2008.06.006 Eringen, 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys., 54, 4703, 10.1063/1.332803 Eringen, 2002 Faghidian, 2017, Unified formulations of the shear coefficients in Timoshenko beam theory, J. Eng. Mech. Trans. ASCE, 143 Faghidian, 2018, On non-linear flexure of beams based on non-local elasticity theory, Int. J. Eng. Sci., 124, 49, 10.1016/j.ijengsci.2017.12.002 Forest, 2012, Stress gradient continuum theory, Mech. Res. Commun., 40, 16, 10.1016/j.mechrescom.2011.12.002 Guo, 2016, Torsional vibration of carbon nanotube with axial velocity and velocity gradient effect, Int. J. Mech. Sci., 119, 88, 10.1016/j.ijmecsci.2016.09.036 Gurtin, 2010 Güven, 2014, A generalized nonlocal elasticity solution for the propagation of longitudinal stress waves in bars, Eur. J. Mech. Solid., 45, 75, 10.1016/j.euromechsol.2013.11.014 Hadjesfandiari, 2011, Couple stress theory for solids, Int. J. Solid Struct., 48, 2496, 10.1016/j.ijsolstr.2011.05.002 Haghani, 2017, Linear and nonlinear flexural analysis of higher-order shear deformation laminated plates with circular delamination, Acta Mech. Kargarnovin, 2010, 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, 199, 10.1007/s00707-009-0253-5 Kargarnovin, 2010, 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, 1080, 10.1016/j.cnsns.2009.05.030 Khodabakhshi, 2015, A unified integro-differential nonlocal model, Int. J. Eng. Sci., 95, 60, 10.1016/j.ijengsci.2015.06.006 Li, 2015, Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory, Int. J. Eng. Sci., 97, 84, 10.1016/j.ijengsci.2015.08.013 Li, 2016, Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material, Int. J. Eng. Sci., 107, 77, 10.1016/j.ijengsci.2016.07.011 Li, 2017, Post-buckling analysis of functionally graded nanobeams incorporating nonlocal stress and microstructure-dependent strain gradient effects, Int. J. Mech. Sci., 120, 159, 10.1016/j.ijmecsci.2016.11.025 Li, 2015, Flexural wave propagation in small-scaled functionally graded beams via a nonlocal strain gradient theory, Compos. Struct., 133, 1079, 10.1016/j.compstruct.2015.08.014 Li, 2016, Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory, Int. J. Mech. Sci., 115–116, 135, 10.1016/j.ijmecsci.2016.06.011 Li, 2016, Free vibration analysis of nonlocal strain gradient beams made of functionally graded material, Int. J. Eng. Sci., 102, 77, 10.1016/j.ijengsci.2016.02.010 Li, 2016, Wave propagation in viscoelastic single-walled carbon nanotubes with surface effect under magnetic field based on nonlocal strain gradient theory, Physica E, 75, 118, 10.1016/j.physe.2015.09.028 Li, 2017, Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory, Compos. Struct., 165, 250, 10.1016/j.compstruct.2017.01.032 Li, 2017, Nonlocal vibrations and stabilities in parametric resonance of axially moving viscoelastic piezoelectric nanoplate subjected to thermo-electro-mechanical forces, Composites Part B, 116, 153, 10.1016/j.compositesb.2017.01.071 Li, 2018, Size-dependent nonlinear vibration of beam-type porous materials with an initial geometrical curvature, Compos. Struct., 184, 1177, 10.1016/j.compstruct.2017.10.052 Liao, 2004 Liao, 2012 Liao, 2013 Lim, 2015, A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation, J. Mech. Phys. Solid., 78, 298, 10.1016/j.jmps.2015.02.001 Lopatin, 2016, An analytical expression for fundamental frequency of the composite lattice cylindrical shell with clamped edges, Compos. Struct., 141, 232, 10.1016/j.compstruct.2016.01.053 Lu, 2017, Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory, Int. J. Eng. Sci., 116, 12, 10.1016/j.ijengsci.2017.03.006 Lu, 2017, A unified nonlocal strain gradient model for nanobeams and the importance of higher order terms, Int. J. Eng. Sci., 119, 265, 10.1016/j.ijengsci.2017.06.024 Marotti de Sciarra, 2014, A new nonlocal bending model for Euler-Bernoulli nanobeams, Mech. Res. Commun., 62, 25, 10.1016/j.mechrescom.2014.08.004 Mindlin, 1965, Second gradient of strain and surface-tension in linear elasticity, Int. J. Solid Struct., 1, 417, 10.1016/0020-7683(65)90006-5 Mindlin, 1968, On first strain gradient theories in linear elasticity, Int. J. Solid Struct., 4, 109, 10.1016/0020-7683(68)90036-X Papargyri-Beskou, 2003, Dynamic analysis of gradient elastic flexural beams, Struct. Eng. Mech., 15, 705, 10.12989/sem.2003.15.6.705 Polizzotto, 2001, Nonlocal elasticity and related variational principles, Int. J. Solid Struct., 38, 7359, 10.1016/S0020-7683(01)00039-7 Polizzotto, 2003, Gradient elasticity and nonstandard boundary conditions, Int. J. Solid Struct., 40, 7399, 10.1016/j.ijsolstr.2003.06.001 Polizzotto, 2003, Unified thermodynamic framework for nonlocal/gradient continuum theories, Eur. J. Mech. Solid., 22, 651, 10.1016/S0997-7538(03)00075-5 Polizzotto, 2012, A gradient elasticity theory for second-grade materials and higher order inertia, Int. J. Solid Struct., 49, 2121, 10.1016/j.ijsolstr.2012.04.019 Polizzotto, 2013, A second strain gradient elasticity theory with second velocity gradient inertia–Part I: constitutive equations and quasi-static behavior, Int. J. Solid Struct., 50, 3749, 10.1016/j.ijsolstr.2013.06.024 Polizzotto, 2013, A second strain gradient elasticity theory with second velocity gradient inertia–Part II: dynamic behavior, Int. J. Solid Struct., 50, 3766, 10.1016/j.ijsolstr.2013.07.026 Polizzotto, 2014, Stress gradient versus strain gradient constitutive models within elasticity, Int. J. Solid Struct., 51, 1809, 10.1016/j.ijsolstr.2014.01.021 Polizzotto, 2015, A unifying variational framework for stress gradient and strain gradient elasticity theories, Eur. J. Mech. Solid., 49, 430, 10.1016/j.euromechsol.2014.08.013 Polizzotto, 2016, Variational formulations and extra boundary conditions within stress gradient elasticity theory with extensions to beam and plate models, Int. J. Solid Struct., 80, 405, 10.1016/j.ijsolstr.2015.09.015 Polizzotto, 2017, A hierarchy of simplified constitutive models within isotropic strain gradient elasticity, Eur. J. Mech. Solid., 61, 92, 10.1016/j.euromechsol.2016.09.006 Raees, 2017, Homotopy shear band solutions in gradient plasticity, Z. Naturforsch., 72, 477, 10.1515/zna-2016-0475 Rao, 2007 Reddy, 2017 Reissner, 1985, Reflections on the theory of elastic plates, Appl. Mech. Rev., 38, 1453, 10.1115/1.3143699 Reissner, 1986, On a mixed variational theorem and on a shear deformable plate theory, Int. J. Numer. Meth. Eng., 23, 193, 10.1002/nme.1620230203 Romano, 2016, 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, 10.1016/j.ijengsci.2016.09.009 Romano, 2017, Stress-driven versus strain-driven nonlocal integral model for elastic nano-beams, Composites Part B, 114, 184, 10.1016/j.compositesb.2017.01.008 Romano, 2017, Nonlocal elasticity in nanobeams: the stress-driven integral model, Int. J. Eng. Sci., 115, 14, 10.1016/j.ijengsci.2017.03.002 Romano, 2016, Micromorphic continua: non-redundant formulations, Continuum Mech. Therm., 28, 1659, 10.1007/s00161-016-0502-5 Romano, 2017, Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams, Int. J. Mech. Sci., 121, 151, 10.1016/j.ijmecsci.2016.10.036 Romano, 2017, On nonlocal integral models for elastic nano-beams, Int. J. Mech. Sci., 131–132, 490, 10.1016/j.ijmecsci.2017.07.013 Salehipour, 2015, Modified nonlocal elasticity theory for functionally graded materials, Int. J. Eng. Sci., 90, 44, 10.1016/j.ijengsci.2015.01.005 Shen, 2016, Torsion of a functionally graded material, Int. J. Eng. Sci., 119, 14, 10.1016/j.ijengsci.2016.09.003 Song, 2010, Effects of initial axial stress on waves propagating in carbon nanotubes using a generalized nonlocal model, Comput. Mater. Sci., 49, 518, 10.1016/j.commatsci.2010.05.043 Thai, 2017, A review of continuum mechanics models for size-dependent analysis of beams and plates, Compos. Struct., 177, 196, 10.1016/j.compstruct.2017.06.040 Tornabene, 2014, Static analysis of doubly-curved anisotropic shells and panels using CUF approach, differential geometry and differential quadrature method, Compos. Struct., 107, 675, 10.1016/j.compstruct.2013.08.038 Wu, 1992, Cohesive elasticity and surface phenomena, Q. Appl. Math., 50, 73, 10.1090/qam/1146625 Wu, 2011, 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, 10.1016/j.compstruct.2010.11.015 Wu, 2015, 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, 10.1016/j.compstruct.2014.11.073 Wu, 2011, RMVT-based meshless collocation and element-free Galerkin methods for the quasi-3D analysis of multilayered composite and FGM plates, Compos. Struct., 93, 923, 10.1016/j.compstruct.2010.07.001 Yang, 2002, Couple stress based strain gradient theory for elasticity, Int. J. Solid Struct., 39, 2731, 10.1016/S0020-7683(02)00152-X Yang, 2017, Nonlinear dynamic characteristics of FGCNTs reinforced microbeam with piezoelectric layer based on unifying stress-strain gradient framework, Composites Part B, 111, 372, 10.1016/j.compositesb.2016.11.058 Zhang, 2010, Bending, buckling, and vibration of micro/nanobeams by hybrid nonlocal beam model, J. Eng. Mech., 136, 562, 10.1061/(ASCE)EM.1943-7889.0000107 Zhu, 2017, Twisting statics of functionally graded nanotubes using Eringen's nonlocal integral model, Compos. Struct., 178, 87, 10.1016/j.compstruct.2017.06.067 Zhu, 2017, Longitudinal and torsional vibrations of size-dependent rods via nonlocal integral elasticity, Int. J. Mech. Sci., 2017, 639, 10.1016/j.ijmecsci.2017.09.030 Zhu, 2017, On longitudinal dynamics of nanorods, Int. J. Eng. Sci., 120, 129, 10.1016/j.ijengsci.2017.08.003 Zhu, 2017, Closed form solution for a nonlocal strain gradient rod in tension, Int. J. Eng. Sci., 119, 16, 10.1016/j.ijengsci.2017.06.019