Design optimization of nanostructures is a new challenging research area. The modelling of multilayer graphene sheets has a similar character as the modelling of composite laminates. However, the traditional laminate plate theories are revised in order to incorporate nonlocal elasticity. The main aim of the current study is to point out the crotchet features arising in the design optimization of graphene laminates based on the theoretical analysis performed and numerical results obtained. The study is focused on the improvement of the mechanical performance of graphene and nanostructures, particularly vibration properties of multilayer graphene laminates.
1. Elishakoff, I., Dujat, K., Muscolino, G., Bucas, S., Narsuki, T., Wand, C. M., et al. Carbon Nanotubes and Nanosensors: Vibration, Buckling and Ballistic Impact. ISTE/Wiley, Portland, 2012.
2. Eringen, A. C. and Edelen, D. G. B. On nonlocal elasticity. Int. J. Eng. Sci., 1972, 10, 233–248.
https://doi.org/10.1016/0020-7225(72)90039-0
https://doi.org/10.1016/0020-7225(72)90070-5
3. Eringen, A. C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys., 1983, 54, 4703–4710.
https://doi.org/10.1063/1.332803
4. Aydogdu, M. A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Physica E, 2009, 41, 1651–1655.
https://doi.org/10.1016/j.physe.2009.05.014
5. Aydogdu, M. and Elishakoff, I. On the vibration of nanorods restrained by a linear spring in-span. Mech. Res. Comm., 2014, 57, 90–96.
https://doi.org/10.1016/j.mechrescom.2014.03.003
6. Murmu, T. and Pradhan, S. C. Small-scale effect on the free in-plane vibration of nanoplates by nonlocal continuum model. Physica E, 2009, 41, 1628–1633.
https://doi.org/10.1016/j.physe.2009.05.013
7. Emam, S. A. A general nonlocal nonlinear model for buckling of nanobeams. Appl. Math. Model., 2013, 37, 6929–6939.
https://doi.org/10.1016/j.apm.2013.01.043
8. Mohammadi, M., Ghayour, M., and Farajpour, A. Free transverse vibration analysis of circular and annular graphene sheets with various boundary conditions using the nonlocal continuum plate model. Compos. Part B-Eng., 2013, 45, 32–42.
https://doi.org/10.1016/j.compositesb.2012.09.011
9. Ansari, R., Sahmani, S., and Arash, B. Nonlocal plate model for free vibrations of single-layered graphene sheets. Phys. Lett. A, 2010, 375, 53– 62.
https://doi.org/10.1016/j.physleta.2010.10.028
10. Ansari, R. and Sahmani, S. Surface stress effects on the free vibration behavior of nanoplates. Int. J. Eng. Sci., 2011, 49, 1204–1215.
https://doi.org/10.1016/j.ijengsci.2011.06.005
11. Malekzadeh, P. and Shojaee, M. Free vibration of nanoplates based on a nonlocal two-variable refined plate theory. Compos. Struct., 2013, 95, 443–452.
https://doi.org/10.1016/j.compstruct.2012.07.006
12. Hosseini-Hashemi, S., Zare, M., and Nazemnezhad, R. An exact analytical approach for free vibration of Mindlin rectangular nano-plates via nonlocal elasticity. Compos. Struct., 2013, 100, 290–299.
https://doi.org/10.1016/j.compstruct.2012.11.035
13. Wang, Y., Li, F., Jing, X., and Wang, Y. Nonlinear vibration analysis of double-layered nanoplates with different boundary conditions. Phys. Lett. A, 2015, 379, 1532–1537.
https://doi.org/10.1016/j.physleta.2015.04.002
14. Jomehzadeh, E. and Saidi, A. R. A study on large amplitude vibration of multilayered sheets. Comput. Mater. Sci., 2011, 50, 1043–1051.
https://doi.org/10.1016/j.commatsci.2010.10.045
15. Natarajan, S., Chakraborty, S., Thangavel, M., Bordas, S., and Rabczuk, T. Size-dependent free flexural vibration behavior of functionally graded nanoplates. Comput. Mater. Sci., 2012, 65, 74–80.
https://doi.org/10.1016/j.commatsci.2012.06.031
16. Liu, J. C., Zhang, Y. Q., and Fan, L. F. Nonlocal vibration and biaxial buckling of double-viscoelastic-FGM-nanoplate system with viscoelastic Pasternak medium in between. Compos. Part B-Eng., 2017, 118, 125–134.
https://doi.org/10.1016/j.physleta.2017.01.056
17. Zhang, L. W., Zhang, Y., and Liew, K. M. Modeling of nonlinear vibration of graphene sheets using a meshfree method based on nonlocal elasticity theory. Appl. Math. Model., 2017, 49, 691–704.
https://doi.org/10.1016/j.apm.2017.02.053
18. Pradhan, S. C. and Murmu, T. Small scale effect on the buckling analysis of single-layered graphene sheet embedded in an elastic medium based on nonlocal plate theory. Physica E, 2010, 42, 1293–1301.
https://doi.org/10.1016/j.physe.2009.10.053
19. Pradhan, S. C. and Phadikar, J. K. Small scale effect on vibration of embedded multilayered graphene sheets based on nonlocal continuum models. Phys. Lett. A, 2009, 373, 1062–1069.
https://doi.org/10.1016/j.physleta.2009.01.030
20. Nazemnezhada, R., Zarea, M., and Hosseini-Hashemi, S. Sandwich plate model of multilayer graphene sheets for considering interlayer shear effect in vibration analysis via molecular dynamics simulations. Appl. Math. Model., 2017, 47, 459–472.
https://doi.org/10.1016/j.apm.2017.03.033
21. Lu, L., Ru, C. Q., and Guo, X. M. Vibration of a multilayer graphene sheet under layerwise tension forces. Int. J. Mech. Sci., 2017, 121, 157–163.
https://doi.org/10.1016/j.ijmecsci.2017.01.007
22. Behera, L. and Chakraverty, S. Application of Differential Quadrature method in free vibration analysis of nanobeams based on various nonlocal theories. Comput. Math. Appl., 2015, 69, 1444–1462.
https://doi.org/10.1016/j.camwa.2015.04.010
23. Malekzadeh, P., Setoodeh, A. R., and Beni, A. A. Small scale effect on the free vibration of orthotropic arbitrary straight sided quadrilateral nanoplates. Compos. Struct., 2011, 93, 1631–1639.
https://doi.org/10.1016/j.compstruct.2011.01.008
https://doi.org/10.1016/j.compstruct.2011.02.013
24. Ravari, M. R. K. and Shahidi, A. R. Axisymmetric buckling of the circular annular nanoplates using finite difference method. Meccanica, 2013, 48(1), 135–144.
https://doi.org/10.1007/s11012-012-9589-3
25. Chakraverty, S. and Behera, L. Free vibration of rectangular nanoplates using Rayleigh–Ritz method. Physica E, 2014, 56, 357–363.
https://doi.org/10.1016/j.physe.2013.08.014
26. Eltaher, M. A., Emam, S. A., and Mahmoud, F. F. Static and stability analysis of nonlocal functionally graded nanobeams. Compos. Struct., 2013, 96, 82–88.
https://doi.org/10.1016/j.compstruct.2012.09.030
27. Reddy, J. N., El-Borgi, S., and Romanoff, J. Non-linear analysis of functionally graded microbeams using Eringen’s non-local differential model. Int. J. Nonlin. Mech., 2014, 67, 308–318.
https://doi.org/10.1016/j.ijnonlinmec.2014.09.014
28. Kirs, M., Mikola, M., Haavajõe, A., Õunapuu, E., Shvartsman, B., and Majak, J. Haar wavelet method for vibration analysis of nanobeams. WWFAA, 2016, 2, 20–28.
https://doi.org/10.1515/wwfaa-2016-0003
29. Majak, J., Shvartsman, B., Kirs, M., Pohlak, M., and Herranen, H. Convergence theorem for the Haar wavelet based discretization method. Compos. Struct., 2015, 126, 227–232.
https://doi.org/10.1016/j.compstruct.2015.02.050
30. Majak, J., Shvartsman, B., Karjust, K., Mikola, M., Haavajõe, A., and Pohlak, M. On the accuracy of the Haar wavelet discretization method. Compos. Part B-Eng., 2015, 80, 321–327.
https://doi.org/10.1016/j.compositesb.2015.06.008
31. Aruniit, A., Kers, J., Goljandin, D., Saarna, M., Tall, K., Majak, J., and Herranen, H. Particulate filled composite plastic materials from recycled glass fibre reinforced plastics. Mater. Sci. – Medzg., 2011, 17(3), 276−281.
32. Aruniit, A., Kers, J., Majak, J., Krumme, A., and Tall, K. Influence of hollow glass microspheres on the mechanical and physical properties and cost of particle reinforced polymer composites. Proc. Estonian Acad. Sci., 2012, 61, 160–165.
https://doi.org/10.3176/proc.2012.3.03
33. Lellep, J. and Majak, J. On optimal orientation of nonlinear elastic orthotropic materials. Struct. Optim., 1997, 14, 116–120.
https://doi.org/10.1007/BF01812513
34. Majak, J. and Hannus, S. Orientational design of anisotropic materials using the Hill and Tsai-Wu strength criteria. Mech. Compos. Mater., 2003, 39(6), 509–520.
https://doi.org/10.1023/B:MOCM.0000010623.38596.3e
35. Reddy, J. N. Mechanics of Composite Plates: Theory and Analysis. Chemical Rubber Company, Boca Raton, FL, 1997.