Bending of small-scale Timoshenko beams based on the integral/differential nonlocal-micropolar elasticity theory: a finite element approach
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摘要
A novel size-dependent model is developed herein to study the bending behavior of beam-type micro/nano-structures considering combined effects of nonlocality and micro-rotational degrees of freedom. To accomplish this aim, the micropolar theory is combined with the nonlocal elasticity. To consider the nonlocality, both integral(original)and differential formulations of Eringen's nonlocal theory are considered. The beams are considered to be Timoshenko-type, and the governing equations are derived in the variational form through Hamilton's principle. The relations are written in an appropriate matrix-vector representation that can be readily utilized in numerical approaches. A finite element(FE) approach is also proposed for the solution procedure. Parametric studies are conducted to show the simultaneous nonlocal and micropolar effects on the bending response of small-scale beams under different boundary conditions.
A novel size-dependent model is developed herein to study the bending behavior of beam-type micro/nano-structures considering combined effects of nonlocality and micro-rotational degrees of freedom. To accomplish this aim, the micropolar theory is combined with the nonlocal elasticity. To consider the nonlocality, both integral(original)and differential formulations of Eringen's nonlocal theory are considered. The beams are considered to be Timoshenko-type, and the governing equations are derived in the variational form through Hamilton's principle. The relations are written in an appropriate matrix-vector representation that can be readily utilized in numerical approaches. A finite element(FE) approach is also proposed for the solution procedure. Parametric studies are conducted to show the simultaneous nonlocal and micropolar effects on the bending response of small-scale beams under different boundary conditions.
A novel size-dependent model is developed herein to study the bending behavior of beam-type micro/nano-structures considering combined effects of nonlocality and micro-rotational degrees of freedom. To accomplish this aim, the micropolar theory is combined with the nonlocal elasticity. To consider the nonlocality, both integral(original)and differential formulations of Eringen's nonlocal theory are considered. The beams are considered to be Timoshenko-type, and the governing equations are derived in the variational form through Hamilton's principle. The relations are written in an appropriate matrix-vector representation that can be readily utilized in numerical approaches. A finite element(FE) approach is also proposed for the solution procedure. Parametric studies are conducted to show the simultaneous nonlocal and micropolar effects on the bending response of small-scale beams under different boundary conditions.
引文
[1]WANG,K.F.,WANG,B.L.,and KITAMURA,T.A review on the application of modified continuum models in modeling and simulation of nanostructures.Acta Mechanica Sininica,32,83-100(2016)
[2]ELTAHER,M.A.,KHATER,M.E.,and EMAM,S.A.A review on nonlocal elastic models for bending,buckling,vibrations,and wave propagation of nanoscale beams.Applied Mathematical Modelling,40,4109-4128(2016)
[3]ANSARI,R.,NOROUZZADEH,A.,SHAKOURI,A.H.,BAZDID-VAHDATI,M.,and ROUHI,H.Finite element analysis of vibrating micro-beams and-plates using a three-dimensional micropolar element.Thin-Walled Structures,124,489-500(2018)
[4]LI,C.A nonlocal analytical approach for torsion of cylindrical nanostructures and the existence of higher-order stress and geometric boundaries.Composite Structures,118,607-621(2014)
[5]S?MS?K,M.and REDDY,J.N.A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory.Composite Structures,101,47-58(2013)
[6]ANSARI,R.and NOROUZZADEH,A.Nonlocal and surface effects on the buckling behavior of functionally graded nanoplates:an isogeometric analysis.Physica E,84,84-97(2016)
[7]MA,H.M.,GAO,X.L.,and REDDY,J.N.A microstructure-dependent Timoshenko beam model based on a modified couple stress theory.Journal of Mechanics and Physics of Solids,56(12),3379-3391(2008)
[8]FATEMI,J.,VAN KEULEN,F.,and ONCK,P.Generalized continuum theories:application to stress analysis in bone.Meccanica,37,385-396(2002)
[9]IVANOVA,E.A.,KRIVTSOV,A.M.,MOROZOV,N.F.,and FIRSOVA,A.D.Description of crystal packing of particles with torque interaction.Mechanics of Solids,38,76-88(2003)
[10]FOREST,S.and SAB,K.Cosserat overall modeling of heterogeneous materials.Mechanics Research Communications,25,449-454(1998)
[11]YANG,J.F.C.and LAKES,R.S.Experimental study of micropolar and couple stress elasticity in compact bone in bending.Journal of Biomechanics,15,91-98(1982)
[12]ERINGEN,A.C.Microcontinuum Field Theory,I.Foundations and Solids,Springer,Berlin(1999)
[13]NEFF,P.and FOREST,S.A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure:modelling,existence of minimizers,identification of moduli and computational results.Journal of Elasticity,87(2-3),239-276(2007)
[14]ISBUGA,V.and REGUEIRO,R.A.Three-dimensional finite element analysis of finite deformation micromorphic linear isotropic elasticity.International Journal of Engineering Science,49(12),1326-1336(2011)
[15]ANSARI,R.,BAZDID-VAHDATI,M.,SHAKOURI,A.,NOROUZZADEH,A.,and ROUHI,H.Micromorphic first-order shear deformable plate element.Meccanica,51(8),1797-1809(2016)
[16]ANSARI,R.,BAZDID-VAHDATI,M.,SHAKOURI,A.H.,NOROUZZADEH,A.,and ROUHI,H.Micromorphic prism element.Mathematics and Mechanics of Solids,22(6),1438-1461(2017)
[17]MINDLIN,R.D.Micro-structure in linear elasticity.Archive for Rational Mechanics and Analysis,16(1),51-78(1964)
[18]MINDLIN,R.D.Second gradient of strain and surface-tension in linear elasticity.International Journal of Solids and Structures,1(4),417-438(1965)
[19]AIFANTIS,E.C.Fracture Scaling,Springer,Dordrecht,299-314(1999)
[20]YANG,F.A.C.M.,CHONG,A.C.M.,LAM,D.C.C.,and TONG,P.Couple stress based strain gradient theory for elasticity.International Journal of Solids and Structures,39(10),2731-2743(2002)
[21]LAM,D.C.,YANG,F.,CHONG,A.C.M.,WANG,J.,and TONG,P.Experiments and theory in strain gradient elasticity.Journal of Mechanics and Physics of Solids,51(8),1477-1508(2003)
[22]ROQUE,C.M.C.,FIDALGO,D.S.,FERREIRA,A.J.M.,and REDDY,J.N.A study of a microstructure-dependent composite laminated Timoshenko beam using a modified couple stress theory and a meshless method.Composite Structures,96,532-537(2013)
[23]REDDY,J.N.and KIM,J.A nonlinear modified couple stress-based third-order theory of functionally graded plates.Composite Structures,94(3),1128-1143(2012)
[24]ANSARI,R.,NOROUZZADEH,A.,GHOLAMI,R.,SHOJAEI,M.F.,and HOSSEINZADEH,M.Size-dependent nonlinear vibration and instability of embedded fluid-conveying SWBNNTs in thermal environment.Physica E,61,148-157(2014)
[25]REDDY,J.N.Microstructure-dependent couple stress theories of functionally graded beams.Journal of Mechanics and Physics of Solids,59(11),2382-2399(2011)
[26]ANSARI,R.,GHOLAMI,R.,NOROUZZADEH,A.,and SAHMANI,S.Size-dependent vibration and instability of fluid-conveying functionally graded microshells based on the modified couple stress theory.Microfluidics Nanofluidics,19(3),509-522(2015)
[27]ANSARI,R.,GHOLAMI,R.,and NOROUZZADEH,A.Size-dependent thermo-mechanical vibration and instability of conveying fluid functionally graded nanoshells based on Mindlin’s strain gradient theory.Thin-Walled Structures,105,172-184(2016)
[28]KIM,J.,˙ZUR,K.K.,and REDDY,J.N.Bending,free vibration,and buckling of modified couples stress-based functionally graded porous micro-plates.Composite Structures,209,879-888(2019)
[29]COSSERAT,E.and COSSERAT,F.Th′eorie des Corps D′eformables,Herman,Paris(1909)
[30]ERINGEN,A.C.and SUHUBI,E.Nonlinear theory of simple micro-elastic solids-I.International Journal of Engineering Science,2,189-203(1964)
[31]SUHUBI,E.and ERINGEN,A.C.Nonlinear theory of micro-elastic solids-II.International Journal of Engineering Science,2,389-404(1964)
[32]ERINGEN,A.C.Linear Theory of Micropolar Elasticity,Springer,New York(1965)
[33]ERINGEN,A.C.Linear theory of micropolar viscoelasticity.International Journal of Engineering Science,5,191-204(1967)
[34]ERINGEN,A.C.Theory of micropolar plates.Zeitschrift f¨ur Angewandte Mathematik und Physik(ZAMP),18,12-30(1967)
[35]MINDLIN,R.D.Stress functions for a Cosserat continuum.International Journal of Solids and Structures,1,265-271(1965)
[36]ERINGEN,A.C.and KAFADAR,C.B.Continuum Physics,Vol.IV,Academic Press,New York,1-73(1976)
[37]LU,Z.J.Micropolar continuum mechanics is more profound than classical continuum mechanics.Applied Mathematics and Mechanics(English Edition),8(10),939-946(1987)https://doi.org/10.1007/BF02454256
[38]LUKASZEWICZ,G.Micropolar Fluids:Theory and Applications,Birkh¨auser,Boston(1999)
[39]DYSZLEWICZ,J.Micropolar Theory of Elasticity,Springer,New York(2004)
[40]IESAN,D.Classical and Generalized Models of Elastic Rods,CRC Press,Boca Raton(2009)
[41]PIETRASZKIEWICZ,W.and EREMEYEV,V.A.On natural strain measures of the non-linear micropolar continuum.International Journal of Solids and Structures,46(3-4),774-787(2009)
[42]ALTENBACH,H.and EREMEYEV,V.A.Strain rate tensors and constitutive equations of inelastic micropolar materials.International Journal of Plasticity,63,3-17(2014)
[43]ANSARI,R.,SHAKOURI,A.H.,BAZDID-VAHDATI,M.,NOROUZZADEH,A.,and ROUHI,H.A nonclassical finite element approach for the nonlinear analysis of micropolar plates.Journal of Computational Nonlinear Dynamics,12(1),011019(2017)
[44]KR¨ONER,E.Elasticity theory of materials with long range cohesive forces.International Journal of Solids and Structures,3,731-742(1967)
[45]KRUMHANSL,J.Some Considerations of the Relation between Solid State Physics and Generalized Continuum Mechanics,IUTAM symposia,Springer,Berlin/Heidelberg,298-311(1968)
[46]KUNIN,I.A.The Theory of Elastic Media with Microstructure and the Theory of Dislocations,IUTAM symposia,Springer,Berlin/Heidelberg,321-329(1968)
[47]ERINGEN,A.C.Nonlocal polar elastic continua.International Journal of Engineering Science,10,1-16(1972)
[48]ERINGEN,A.C.and EDELEN,D.G.B.On nonlocal elasticity.International Journal of Engineering Science,10,233-248(1972)
[49]ERINGEN,A.C.On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves.Journal of Applied Physics,54,4703-4710(1983)
[50]ZHAO,H.S.,ZHANG,Y.,and LIE,S.T.Frequency equations of nonlocal elastic micro/nanobeams with the consideration of the surface effects.Applied Mathematics and Mechanics(English Edition),39(8),1089-1102(2018)https://doi.org/10.1007/s10483-018-2358-6
[51]EBRAHIMI,F.and BARATI,M.R.Dynamic modeling of preloaded size-dependent nanocrystalline nano-structures.Applied Mathematics and Mechanics(English Edition),38(12),1753-1772(2017)https://doi.org/10.1007/s10483-017-2291-8
[52]GUVEN,U.General investigation for longitudinal wave propagation under magnetic field effect via nonlocal elasticity.Applied Mathematics and Mechanics(English Edition),36(10),1305-1318(2015)https://doi.org/10.1007/s10483-015-1985-9
[53]LI,C.Torsional vibration of carbon nanotubes:comparison of two nonlocal models and a semicontinuum model.International Journal of Mechanical Science,82,25-31(2014)
[54]LI,C.,LI,S.,YAO,L.,and ZHU,Z.Nonlocal theoretical approaches and atomistic simulations for longitudinal free vibration of nanorods/nanotubes and verification of different nonlocal models.Applied Mathematical Modelling,39(15),4570-4585(2015)
[55]ANSARI,R.,GHOLAMI,R.,SAHMANI,S.,NOROUZZADEH,A.,and BAZDID-VAHDATI,M.Dynamic stability analysis of embedded multi-walled carbon nanotubes in thermal environment.Acta Mechanica Solida Sinica,28(6),659-667(2015)
[56]SHEN,J.P.and LI,C.A semi-continuum-based bending analysis for extreme-thin micro/nanobeams and new proposal for nonlocal differential constitution.Composite Structures,172,210-220(2017)
[57]NOROUZZADEH,A.and ANSARI,R.Isogeometric vibration analysis of functionally graded nanoplates with the consideration of nonlocal and surface effects.Thin-Walled Structures,127,354-372(2018)
[58]CHALLAMEL,N.and WANG,C.M.The small length scale effect for a non-local cantilever beam:a paradox solved.Nanotechnology,19(34),345703(2008)
[59]NOROUZZADEH,A.,ANSARI,R.,and ROUHI,H.Pre-buckling responses of Timoshenko nanobeams based on the integral and differential models of nonlocal elasticity:an isogeometric approach.Applied Physics A,123,330(2017)
[60]FARAJI-OSKOUIE,M.,ANSARI,R.,and ROUHI,H.Bending of Euler-Bernoulli nanobeams based on the strain-driven and stress-driven nonlocal integral models:a numerical approach.Acta Mechanica Sinica,34(5),871-882(2018)
[61]MAHMOUDPOUR,E.,HOSSEINI-HASHEMI,S.H.,and FAGHIDIAN,S.A.Nonlinear vibration analysis of FG nano-beams resting on elastic foundation in thermal environment using stress-driven nonlocal integral model.Applied Mathematical Modelling,57,302-315(2018)
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[64]NOROUZZADEH,A.and ANSARI,R.Finite element analysis of nano-scale Timoshenko beams using the integral model of nonlocal elasticity.Physica E,88,194-200(2017)
[65]FERN′ANDEZ-S′AEZ,J.,ZAERA,R.,LOYA,J.A.,and REDDY,J.N.Bending of Euler-Bernoulli beams using Eringen’s integral formulation:a paradox resolved.International Journal of Engineering Science,99,107-116(2016)
[66]ROMANO,G.,BARRETA,R.,and DIACO,M.On nonlocal integral models for elastic nanobeams.International Journal of Mechanical Science,131,490-499(2017)
[67]MAHMOUD,F.F.On the nonexistence of a feasible solution in the context of the differential form of Eringen’s constitutive model:a proposed iterative model based on a residual nonlocality formulation.International Journal of Applied Mechanics,9(7),1750094(2017)
[68]ANSARI,R.,TORABI,J.,and NOROUZZADEH,A.Bending analysis of embedded nanoplates based on the integral formulation of Eringen’s nonlocal theory using the finite element method.Physica B,534,90-97(2018)
[69]NOROUZZADEH,A.,ANSARI,R.,and ROUHI,H.Isogeometric analysis of Mindlin nanoplates based on the integral formulation of nonlocal elasticity.Multidiscipline Modeling in Materials and Structures,14(5),810-827(2018)
[70]LAKES,R.Experimental microelasticity of two porous solids.International Journal of Solids and Structures,22,55-63(1986)
[71]LI,L.and HU,Y.Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material.International Journal of Engineering Science,107,77-97(2016)
[72]XU,X.J.,WANG,X.C.,ZHENG,M.L.,and MA,Z.Bending and buckling of nonlocal strain gradient elastic beams.Composite Structures,160,366-377(2017)
[73]NOROUZZADEH,A.,ANSARI,R.,and ROUHI,H.Nonlinear wave propagation analysis in Timoshenko nano-beams considering nonlocal and strain gradient effects.Meccanica,53(13),3415-3435(2018)
[74]NOROUZZADEH,A.,ANSARI,R.,and ROUHI,H.Nonlinear bending analysis of nanobeams based on the nonlocal strain gradient model using an isogeometric finite element approach.Iranian Journal of Science and Technology,Transactions of Civil Engineering,1-15(2018)https://doi.org/10.1007/s40996-018-0184-2
[2]ELTAHER,M.A.,KHATER,M.E.,and EMAM,S.A.A review on nonlocal elastic models for bending,buckling,vibrations,and wave propagation of nanoscale beams.Applied Mathematical Modelling,40,4109-4128(2016)
[3]ANSARI,R.,NOROUZZADEH,A.,SHAKOURI,A.H.,BAZDID-VAHDATI,M.,and ROUHI,H.Finite element analysis of vibrating micro-beams and-plates using a three-dimensional micropolar element.Thin-Walled Structures,124,489-500(2018)
[4]LI,C.A nonlocal analytical approach for torsion of cylindrical nanostructures and the existence of higher-order stress and geometric boundaries.Composite Structures,118,607-621(2014)
[5]S?MS?K,M.and REDDY,J.N.A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory.Composite Structures,101,47-58(2013)
[6]ANSARI,R.and NOROUZZADEH,A.Nonlocal and surface effects on the buckling behavior of functionally graded nanoplates:an isogeometric analysis.Physica E,84,84-97(2016)
[7]MA,H.M.,GAO,X.L.,and REDDY,J.N.A microstructure-dependent Timoshenko beam model based on a modified couple stress theory.Journal of Mechanics and Physics of Solids,56(12),3379-3391(2008)
[8]FATEMI,J.,VAN KEULEN,F.,and ONCK,P.Generalized continuum theories:application to stress analysis in bone.Meccanica,37,385-396(2002)
[9]IVANOVA,E.A.,KRIVTSOV,A.M.,MOROZOV,N.F.,and FIRSOVA,A.D.Description of crystal packing of particles with torque interaction.Mechanics of Solids,38,76-88(2003)
[10]FOREST,S.and SAB,K.Cosserat overall modeling of heterogeneous materials.Mechanics Research Communications,25,449-454(1998)
[11]YANG,J.F.C.and LAKES,R.S.Experimental study of micropolar and couple stress elasticity in compact bone in bending.Journal of Biomechanics,15,91-98(1982)
[12]ERINGEN,A.C.Microcontinuum Field Theory,I.Foundations and Solids,Springer,Berlin(1999)
[13]NEFF,P.and FOREST,S.A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure:modelling,existence of minimizers,identification of moduli and computational results.Journal of Elasticity,87(2-3),239-276(2007)
[14]ISBUGA,V.and REGUEIRO,R.A.Three-dimensional finite element analysis of finite deformation micromorphic linear isotropic elasticity.International Journal of Engineering Science,49(12),1326-1336(2011)
[15]ANSARI,R.,BAZDID-VAHDATI,M.,SHAKOURI,A.,NOROUZZADEH,A.,and ROUHI,H.Micromorphic first-order shear deformable plate element.Meccanica,51(8),1797-1809(2016)
[16]ANSARI,R.,BAZDID-VAHDATI,M.,SHAKOURI,A.H.,NOROUZZADEH,A.,and ROUHI,H.Micromorphic prism element.Mathematics and Mechanics of Solids,22(6),1438-1461(2017)
[17]MINDLIN,R.D.Micro-structure in linear elasticity.Archive for Rational Mechanics and Analysis,16(1),51-78(1964)
[18]MINDLIN,R.D.Second gradient of strain and surface-tension in linear elasticity.International Journal of Solids and Structures,1(4),417-438(1965)
[19]AIFANTIS,E.C.Fracture Scaling,Springer,Dordrecht,299-314(1999)
[20]YANG,F.A.C.M.,CHONG,A.C.M.,LAM,D.C.C.,and TONG,P.Couple stress based strain gradient theory for elasticity.International Journal of Solids and Structures,39(10),2731-2743(2002)
[21]LAM,D.C.,YANG,F.,CHONG,A.C.M.,WANG,J.,and TONG,P.Experiments and theory in strain gradient elasticity.Journal of Mechanics and Physics of Solids,51(8),1477-1508(2003)
[22]ROQUE,C.M.C.,FIDALGO,D.S.,FERREIRA,A.J.M.,and REDDY,J.N.A study of a microstructure-dependent composite laminated Timoshenko beam using a modified couple stress theory and a meshless method.Composite Structures,96,532-537(2013)
[23]REDDY,J.N.and KIM,J.A nonlinear modified couple stress-based third-order theory of functionally graded plates.Composite Structures,94(3),1128-1143(2012)
[24]ANSARI,R.,NOROUZZADEH,A.,GHOLAMI,R.,SHOJAEI,M.F.,and HOSSEINZADEH,M.Size-dependent nonlinear vibration and instability of embedded fluid-conveying SWBNNTs in thermal environment.Physica E,61,148-157(2014)
[25]REDDY,J.N.Microstructure-dependent couple stress theories of functionally graded beams.Journal of Mechanics and Physics of Solids,59(11),2382-2399(2011)
[26]ANSARI,R.,GHOLAMI,R.,NOROUZZADEH,A.,and SAHMANI,S.Size-dependent vibration and instability of fluid-conveying functionally graded microshells based on the modified couple stress theory.Microfluidics Nanofluidics,19(3),509-522(2015)
[27]ANSARI,R.,GHOLAMI,R.,and NOROUZZADEH,A.Size-dependent thermo-mechanical vibration and instability of conveying fluid functionally graded nanoshells based on Mindlin’s strain gradient theory.Thin-Walled Structures,105,172-184(2016)
[28]KIM,J.,˙ZUR,K.K.,and REDDY,J.N.Bending,free vibration,and buckling of modified couples stress-based functionally graded porous micro-plates.Composite Structures,209,879-888(2019)
[29]COSSERAT,E.and COSSERAT,F.Th′eorie des Corps D′eformables,Herman,Paris(1909)
[30]ERINGEN,A.C.and SUHUBI,E.Nonlinear theory of simple micro-elastic solids-I.International Journal of Engineering Science,2,189-203(1964)
[31]SUHUBI,E.and ERINGEN,A.C.Nonlinear theory of micro-elastic solids-II.International Journal of Engineering Science,2,389-404(1964)
[32]ERINGEN,A.C.Linear Theory of Micropolar Elasticity,Springer,New York(1965)
[33]ERINGEN,A.C.Linear theory of micropolar viscoelasticity.International Journal of Engineering Science,5,191-204(1967)
[34]ERINGEN,A.C.Theory of micropolar plates.Zeitschrift f¨ur Angewandte Mathematik und Physik(ZAMP),18,12-30(1967)
[35]MINDLIN,R.D.Stress functions for a Cosserat continuum.International Journal of Solids and Structures,1,265-271(1965)
[36]ERINGEN,A.C.and KAFADAR,C.B.Continuum Physics,Vol.IV,Academic Press,New York,1-73(1976)
[37]LU,Z.J.Micropolar continuum mechanics is more profound than classical continuum mechanics.Applied Mathematics and Mechanics(English Edition),8(10),939-946(1987)https://doi.org/10.1007/BF02454256
[38]LUKASZEWICZ,G.Micropolar Fluids:Theory and Applications,Birkh¨auser,Boston(1999)
[39]DYSZLEWICZ,J.Micropolar Theory of Elasticity,Springer,New York(2004)
[40]IESAN,D.Classical and Generalized Models of Elastic Rods,CRC Press,Boca Raton(2009)
[41]PIETRASZKIEWICZ,W.and EREMEYEV,V.A.On natural strain measures of the non-linear micropolar continuum.International Journal of Solids and Structures,46(3-4),774-787(2009)
[42]ALTENBACH,H.and EREMEYEV,V.A.Strain rate tensors and constitutive equations of inelastic micropolar materials.International Journal of Plasticity,63,3-17(2014)
[43]ANSARI,R.,SHAKOURI,A.H.,BAZDID-VAHDATI,M.,NOROUZZADEH,A.,and ROUHI,H.A nonclassical finite element approach for the nonlinear analysis of micropolar plates.Journal of Computational Nonlinear Dynamics,12(1),011019(2017)
[44]KR¨ONER,E.Elasticity theory of materials with long range cohesive forces.International Journal of Solids and Structures,3,731-742(1967)
[45]KRUMHANSL,J.Some Considerations of the Relation between Solid State Physics and Generalized Continuum Mechanics,IUTAM symposia,Springer,Berlin/Heidelberg,298-311(1968)
[46]KUNIN,I.A.The Theory of Elastic Media with Microstructure and the Theory of Dislocations,IUTAM symposia,Springer,Berlin/Heidelberg,321-329(1968)
[47]ERINGEN,A.C.Nonlocal polar elastic continua.International Journal of Engineering Science,10,1-16(1972)
[48]ERINGEN,A.C.and EDELEN,D.G.B.On nonlocal elasticity.International Journal of Engineering Science,10,233-248(1972)
[49]ERINGEN,A.C.On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves.Journal of Applied Physics,54,4703-4710(1983)
[50]ZHAO,H.S.,ZHANG,Y.,and LIE,S.T.Frequency equations of nonlocal elastic micro/nanobeams with the consideration of the surface effects.Applied Mathematics and Mechanics(English Edition),39(8),1089-1102(2018)https://doi.org/10.1007/s10483-018-2358-6
[51]EBRAHIMI,F.and BARATI,M.R.Dynamic modeling of preloaded size-dependent nanocrystalline nano-structures.Applied Mathematics and Mechanics(English Edition),38(12),1753-1772(2017)https://doi.org/10.1007/s10483-017-2291-8
[52]GUVEN,U.General investigation for longitudinal wave propagation under magnetic field effect via nonlocal elasticity.Applied Mathematics and Mechanics(English Edition),36(10),1305-1318(2015)https://doi.org/10.1007/s10483-015-1985-9
[53]LI,C.Torsional vibration of carbon nanotubes:comparison of two nonlocal models and a semicontinuum model.International Journal of Mechanical Science,82,25-31(2014)
[54]LI,C.,LI,S.,YAO,L.,and ZHU,Z.Nonlocal theoretical approaches and atomistic simulations for longitudinal free vibration of nanorods/nanotubes and verification of different nonlocal models.Applied Mathematical Modelling,39(15),4570-4585(2015)
[55]ANSARI,R.,GHOLAMI,R.,SAHMANI,S.,NOROUZZADEH,A.,and BAZDID-VAHDATI,M.Dynamic stability analysis of embedded multi-walled carbon nanotubes in thermal environment.Acta Mechanica Solida Sinica,28(6),659-667(2015)
[56]SHEN,J.P.and LI,C.A semi-continuum-based bending analysis for extreme-thin micro/nanobeams and new proposal for nonlocal differential constitution.Composite Structures,172,210-220(2017)
[57]NOROUZZADEH,A.and ANSARI,R.Isogeometric vibration analysis of functionally graded nanoplates with the consideration of nonlocal and surface effects.Thin-Walled Structures,127,354-372(2018)
[58]CHALLAMEL,N.and WANG,C.M.The small length scale effect for a non-local cantilever beam:a paradox solved.Nanotechnology,19(34),345703(2008)
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