References
- E. Carrera and V.V. Zozulya, Carrera unified formulation (CUF) for the micropolar plates and shells. I. Higher order theory, Mech. Adv. Mater. Struct., pp. 1–23, 2020. DOI: https://doi.org/10.1080/15376494.2020.1793241
- E. Carrera, and V.V. Zozulya, Carrera unified formulation (CUF) for the micropolar beams: Analytical solutions, Mech. Adv. Mater. Struct., pp. 1–25, 2019. DOI: https://doi.org/10.1080/15376494.2019.1578013
- M. Asghari, M.H. Kahrobaiyan, M. Rahaeifard, and M.T. Ahmadian, Investigation of the size effects in Timoshenko beams based on the couple stress theory, Arch. Appl. Mech., vol. 81, no. 7, pp. 863–874, 2011. DOI: https://doi.org/10.1007/s00419-010-0452-5.
- Y.M. Ghugal and R.P. Shimpi, A review of refined shear deformation theories for isotropic and anisotropic laminated beams, J. Reinf. Plast. Compos., vol. 20, no. 3, pp. 255–272, 2001. DOI: https://doi.org/10.1177/073168401772678283.
- I. Elishakoff, J. Kaplunov, and E. Nolde, Celebrating the centenary of Timoshenko’s study of effects of shear deformation and rotary inertia, Applied Mechanics Reviews., vol. 67, no. 6, pp. 802–811 2015. DOI: https://doi.org/10.1115/1.4031965.
- A. Banerjee, Non-dimensional analysis of the elastic beam having periodic linear spring mass resonators, Meccanica., vol. 55, no. 5, pp. 1181–1111, 2020. DOI: https://doi.org/10.1007/s11012-020-01151-z.
- A. Labuschagne, N.J. van Rensburg, and A.J. Van der Merwe, Comparison of linear beam theories, Math. Comput. Modell., vol. 49, no. 1/2, pp. 20–30, 2009. DOI: https://doi.org/10.1016/j.mcm.2008.06.006.
- S.P. Timoshenko, On the additional deflection due to shearing, Glas. Hrvat. Prirodosl. Drus., Zagreb., vol. 33, no. Part 1, pp. 50–52, 1921. Nr. 1),
- M.B. Rubin, Cosserat Theories: shells, Rods and Points, (Vol. 79). Springer Science & Business Media, Berlin/Heidelberg, Germany, 2013.
- A. Czekanski and V.V. Zozulya, Vibration analysis of nonlocal beams using higher-order theory and comparison with classical models, Mech. Adv. Mater. Struct., pp. 1–17, 2019. DOI: https://doi.org/10.1080/15376494.2019.1665761.
- B. Wu, A. Pagani, W.Q. Chen, and E. Carrera, Geometrically nonlinear refined shell theories by Carrera unified formulation, Mech. Adv. Mater. Struct., pp. 1–21, 2019. DOI: https://doi.org/10.1080/15376494.2019.1702237
- V.V. Zozulya, Couple stress theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models, Curved Layered Struct., vol. 4, no. 1, pp. 119–133, 2017. DOI: https://doi.org/10.1515/cls-2017-0009.
- Z. Xue, Y. Huang, and M. Li, Particle size effect in metallic materials: a study by the theory of mechanism-based strain gradient plasticity, Acta Mater. , vol. 50, no. 1, pp. 149–160, 2002. DOI: https://doi.org/10.1016/S1359-6454(01)00325-1.
- Y.P. Cao and J. Lu, Size-dependent sharp indentation-I: a closed-form expression of the indentation loading curve, J. Mech. Phys. Solids., vol. 53, no. 1, pp. 33–48, 2005. DOI: https://doi.org/10.1016/j.jmps.2004.06.005.
- Z.H. Sun, X.X. Wang, A.K. Soh, H.A. Wu, and Y. Wang, Bending of nanoscale structures: Inconsistency between atomistic simulation and strain gradient elasticity solution, Comput. Mater. Sci., vol. 40, no. 1, pp. 108–113, 2007. DOI: https://doi.org/10.1016/j.commatsci.2006.11.015.
- A.T. Karttunen, J. Romanoff, and J.N. Reddy, Exact microstructure-dependent Timoshenko beam element, Int. J. Mech. Sci., vol. 111, pp. 35–42, 2016.
- M. Sobhy and A.M. Zenkour, The modified couple stress model for bending of normal deformable viscoelastic nanobeams resting on visco-Pasternak foundations, Mech. Adv. Mater. Struct., vol. 27, no. 7, pp. 525–538, 2020. DOI: https://doi.org/10.1080/15376494.2018.1482579.
- W. Chen and J. Si, A model of composite laminated beam based on the global–local theory and new modified couple-stress theory, Compos. Struct., vol. 103, pp. 99–107, 2013. DOI: https://doi.org/10.1016/j.compstruct.2013.03.021.
- F. Ebrahimi and M.R. Barati, Vibration analysis of parabolic shear-deformable piezoelectrically actuated nanoscale beams incorporating thermal effects, Mech. Adv. Mater. Struct., vol. 25, no. 11, pp. 917–929, 2018a. DOI: https://doi.org/10.1080/15376494.2017.1323141.
- F. Ebrahimi and M.R. Barati, Longitudinal varying elastic foundation effects on vibration behavior of axially graded nanobeams via nonlocal strain gradient elasticity theory, Mech. Adv. Mater. Struct., vol. 25, no. 11, pp. 953–963, 2018b. DOI: https://doi.org/10.1080/15376494.2017.1329467.
- W. Chen and Y. Wang, A model of composite laminated Reddy plate of the global-local theory based on new modified couple-stress theory, Mech. Adv. Mater. Struct., vol. 23, no. 6, pp. 636–651, 2016. DOI: https://doi.org/10.1080/15376494.2015.1028691.
- E. Ventsel, T. Krauthammer, and E. J. A. M. R. Carrera, Thin plates and shells: theory, analysis, and applications, Appl. Mech. Rev., vol. 55, no. 4, pp. B72–B73, 2002.
- R.A. Toupin, 1964. Theories of Elasticity with Couple-Stress. BM Watson Research Center Yorktown Heights, New York.
- E. Cosserat and F. Cosserat, 1909. Théorie des corps déformables. A. Hermann et fils, Paris.
- A.T. Karttunen, J.N. Reddy, and J. Romanoff, Micropolar modeling approach for periodic sandwich beams, Compos. Struct., vol. 185, pp. 656–664, 2018. DOI: https://doi.org/10.1016/j.compstruct.2017.11.064.
- S. Ramezani, R. Naghdabadi, and S. Sohrabpour, Analysis of micropolar elastic beams, Eur J Mech – A/Solids, vol. 28, no. 2, pp. 202–208, 2009. DOI: https://doi.org/10.1016/j.euromechsol.2008.06.006.
- A.K. Noor, and M.P. Nemeth, Micropolar beam models for lattice grids with rigid joints, Comp Methods Appl Mech Eng., vol. 21, no. 2, pp. 249–263, 1980. DOI: https://doi.org/10.1016/0045-7825(80)90034-1.
- V.V. Zozulya, Higher order theory of micropolar plates and shells, Z Angew. Math. Mech., vol. 98, no. 6, pp. 886–918, 2018. DOI: https://doi.org/10.1002/zamm.201700317.
- R.D. Mindlin, Second gradient of strain and surface-tension in linear elasticity, Int. J. Solids Struct., vol. 1, no. 4, pp. 417–438, 1965. DOI: https://doi.org/10.1016/0020-7683(65)90006-5.
- W. Nowacki, Theory of Micropolar Elasticity. Department for Mechanics of Deformable Bodies, (No. 25). Springer, Berlin, 1972.
- A.C. Eringen, Mechanics of micromorphic continua. In Mechanics of Generalized Continua, E. Kroner (Ed.), IUTAM Symposium, Freudenstadt, pp. 18–35, 1968.
- R.D. Mindlin, and H.F. Tiersten, 1962. Effects of Couple-Stresses in Linear Elasticity (No. CU-TR-48). Columbia University, New York.
- W. Nowacki, The linear theory of micropolar elasticity. In Micropolar Elasticity Springer, Vienna, pp. 1–43, 1974.
- A.C. Eringen, Theory of micropolar elasticity. In Microcontinuum Field Theories, Springer, New York, NY, pp. 101–248, 1999.
- A.C. Eringen, Microcontinuum field theories: II. Fluent Media, (Vol. 2). Springer Science & Business Media, Berlin/Heidelberg, Germany, 2001.
- A.C. Eringen, Microcontinuum field theories: I. Foundations and Solids. Springer Science & Business Media, Berlin/Heidelberg, Germany, 2012.
- R. Kumar and P. Ailawalia, Deformation in micropolar cubic crystal due to various sources, Int. J. Solids Struct., vol. 42, no. 23, pp. 5931–5944, 2005. DOI: https://doi.org/10.1016/j.ijsolstr.2005.01.022.
- A. Gharahi and P. Schiavone, Uniqueness of solution for plane deformations of a micropolar elastic solid with surface effects, Continuum Mech. Thermodyn., vol. 32, no. 1, pp. 9–22, 2020. DOI: https://doi.org/10.1007/s00161-019-00779-x.
- R.D. Mindlin, Influence of couple-stresses on stress concentrations, Exp. Mech., vol. 3, no. 1, pp. 1–7, 1963. DOI: https://doi.org/10.1007/BF02327219.
- A.R. Khoei, S. Yadegari, and S.O.R. Biabanaki, 3D finite element modeling of shear band localization via the micro-polar Cosserat continuum theory, Comput. Mater. Sci., vol. 49, no. 4, pp. 720–733, 2010. DOI: https://doi.org/10.1016/j.commatsci.2010.06.015.
- J.R. Banerjee, Dynamic stiffness formulation and free vibration analysis of centrifugally stiffened Timoshenko beams, J. Sound Vib., vol. 247, no. 1, pp. 97–115, 2001. DOI: https://doi.org/10.1006/jsvi.2001.3716.
- J.M. Dion, and C. Commault, Feedback decoupling of structured systems, IEEE Trans. Automat. Contr., vol. 38, no. 7, pp. 1132–1135, 1993. DOI: https://doi.org/10.1109/9.231471.
- C.E. Augarde, and A.J. Deeks, The use of Timoshenko’s exact solution for a cantilever beam in adaptive analysis, Finite Elem. Anal. Des., vol. 44, no. 9/10, pp. 595–601, 2008. DOI: https://doi.org/10.1016/j.finel.2008.01.010.
- R.E. Hoffman, and T. Ariman, 1968. The application of Micropolar mechanics to composites (No. Themis-UND-68-3). Notre Dame Univ Ind Coll of Engineering.
- V.V. Vasiliev, V.A. Barynin, and A.F. Rasin, Anisogrid lattice structures-survey of development and application, Compos. Struct., vol. 54, no. 2/3, pp. 361–370, 2001. DOI: https://doi.org/10.1016/S0263-8223(01)00111-8.
- J.N. Reddy, Microstructure-dependent couple stress theories of functionally graded beams, J. Mech. Phys. Solids., vol. 59, no. 11, pp. 2382–2399, 2011. DOI: https://doi.org/10.1016/j.jmps.2011.06.008.
- S.K. Park and X.L. Gao, Micromechanical modeling of honeycomb structures based on a modified couple stress theory, Mech. Adv. Mater. Struct., vol. 15, no. 8, pp. 574–593, 2008. DOI: https://doi.org/10.1080/15376490802470499.
- D.C. Lam, F. Yang, A.C.M. Chong, J. Wang, and P. Tong, Experiments and theory in strain gradient elasticity, J. Mech. Phys. Solids., vol. 51, no. 8, pp. 1477–1508, 2003. DOI: https://doi.org/10.1016/S0022-5096(03)00053-X.
- J.N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. CRC Press, Boca Raton, Florida, United States, 2003.
- R. De Borst, and L.J. Sluys, Localisation in a Cosserat continuum under static and dynamic loading conditions, Comp. Methods Appl. Mech. Eng., vol. 90, no. 1–3, pp. 805–827, 1991. DOI: https://doi.org/10.1016/0045-7825(91)90185-9.
- R. De Borst, É. Simulation of strain localization: a reappraisal of the Cosserat continuum, Eng. Comput., vol. 8, no. 4, pp. 317–332, 1991. DOI: https://doi.org/10.1108/eb023842.
- T.H. Tran, V. Monchiet, and G. Bonnet, A micromechanics-based approach for the derivation of constitutive elastic coefficients of strain-gradient media, Int. J. Solids Struct., vol. 49, no. 5, pp. 783–792, 2012. DOI: https://doi.org/10.1016/j.ijsolstr.2011.11.017.
- A. Tordesillas, J.F. Peters, and B.S. Gardiner, Shear band evolution and accumulated microstructural development in Cosserat media, Int. J. Numer. Anal. Meth. Geomech., vol. 28, no. 10, pp. 981–1010, 2004. DOI: https://doi.org/10.1002/nag.343.
- M. Asghari, M. Rahaeifard, M.H. Kahrobaiyan, and M.T. Ahmadian, The modified couple stress functionally graded Timoshenko beam formulation, Mater. Design, vol. 32, no. 3, pp. 1435–1443, 2011.
- A.R. Hadjesfandiari, A. Hajesfandiari, H. Zhang, and G.F. Dargush, 2017. Size-dependent couple stress Timoshenko beam theory. arXiv preprint arXiv:1712.08527.
- A.R. Khoei, and K. Karimi, An enriched-FEM model for simulation of localization phenomenon in Cosserat continuum theory, Comput. Mater. Sci., vol. 44, no. 2, pp. 733–749, 2008. DOI: https://doi.org/10.1016/j.commatsci.2008.05.019.
- ZdeneˇkP. Bazˇant, and Gilles Pijaudier-Cabot, Nonlocal continuum damage, localization instab convergence, Journal of Applied Mechanics, Transactions of the ASME, vol. 55, no. 2, pp. 287–293, 1988. DOI: https://doi.org/10.1115/1.3173674.
- A. Needleman, Material rate dependence and mesh sensitivity in localization problems, Comp. Methods Appl. Mech. Eng., vol. 67, no. 1, pp. 69–85, 1988. DOI: https://doi.org/10.1016/0045-7825(88)90069-2.
- P.V. O’neil, Advanced Engineering Mathematics. Cengage Learning, Boston, Massachusetts, United States, 2011.
- K.T. Chau, Theory of Differential Equations in Engineering and Mechanics. CRC Press, Boca Raton, Florida, United States, 2017.
- J.R. Hutchinson, Shear coefficients for Timoshenko beam theory, J. Appl. Mech., vol. 68, no. 1, pp. 87–92, 2001. DOI: https://doi.org/10.1115/1.1349417.
- A. Mukherjee, and G. Agnivo, 2010. Determination of Natural Frequency of Euler’s Beams Using Analytical and Finite Element Method. Department of Mechanical Engineering.
- D. Singh, and S.K. Tomar, Longitudinal waves at a micropolar fluid/solid interface, Int. J. Solids Struct., vol. 45, no. 1, pp. 225–244, 2008. DOI: https://doi.org/10.1016/j.ijsolstr.2007.07.015.
- H. Reda, Y. Rahali, J.F. Ganghoffer, and H. Lakiss, Wave propagation in 3D viscoelastic auxetic and textile materials by homogenized continuum micropolar models, Compos. Struct. , vol. 141, pp. 328–345, 2016. DOI: https://doi.org/10.1016/j.compstruct.2016.01.071.
- B. Karami, D. Shahsavari, and M. Janghorban, Wave propagation analysis in functionally graded (FG) nanoplates under in-plane magnetic field based on nonlocal strain gradient theory and four variable refined plate theory, Mech. Adv. Mater. Struct., vol. 25, no. 12, pp. 1047–1057, 2018. DOI: https://doi.org/10.1080/15376494.2017.1323143.
- D.J. Colquitt, I.S. Jones, N.V. Movchan, and A.B. Movchan, Dispersion and localization of elastic waves in materials with microstructure, Proc. R Soc. A., vol. 467, no. 2134, pp. 2874–2895, 2011. DOI: https://doi.org/10.1098/rspa.2011.0126.
- S. Papargyri-Beskou, D. Polyzos, and D.E. Beskos, Wave dispersion in gradient elastic solids and structures: a unified treatment, Int. J. Solids Struct., vol. 46, no. 21, pp. 3751–3759, 2009. DOI: https://doi.org/10.1016/j.ijsolstr.2009.05.002.
- M. Lawson, Light Steel Framing and Modular Construction, Steel Technology International, STERLING PUBLICATIONS LTD, India, pp. 104–110, 2001.
- E. Carrera, and V.V. Zozulya, Carrera unified formulation (CUF) for the micropolar plates and shells. II. Complete linear expansion case, Mech. Adv. Mater. Struct., pp. 1–20, 2020. DOI: https://doi.org/10.1080/15376494.2020.1793242