Analytical solution approach for nonlinear buckling and postbuckling analysis of cylindrical nanoshells based on surface elasticity theory

doi:10.1007/s10483-016-2100-9

Abstract

Abstract:

The size-dependent nonlinear buckling and postbuckling characteristics of circular cylindrical nanoshells subjected to the axial compressive load are investigated with an analytical approach. The surface energy effects are taken into account according to the surface elasticity theory of Gurtin and Murdoch. The developed geometrically nonlinear shell model is based on the classical Donnell shell theory and the von Kármán's hypothesis. With the numerical results, the effect of the surface stress on the nonlinear buckling and postbuckling behaviors of nanoshells made of Si and Al is studied. Moreover, the influence of the surface residual tension and the radius-to-thickness ratio is illustrated. The results indicate that the surface stress has an important effect on prebuckling and postbuckling characteristics of nanoshells with small sizes.

Key words: cylindrical nanoshell, postbuckling, analytical method, surface stress, nonlinear buckling

2010 MSC Number:

R. ANSARI, T. POURASHRAF, R. GHOLAMI, H. ROUHI. Analytical solution approach for nonlinear buckling and postbuckling analysis of cylindrical nanoshells based on surface elasticity theory. Applied Mathematics and Mechanics (English Edition), 2016, 37(7): 903-918.

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References

[1] Eringen, A. C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54(9), 4703-4710(1983)
[2] Eringen, A. C. Nonlocal Continuum Field Theories, Springer, New York (2002)
[3] Mindlin, R. D. and Tiersten, H. F. Effects of couple-stresses in linear elasticity. Archive for Rational Mechanics and Analysis, 11(1), 415-448(1962)
[4] Koiter, W. T. Couple stresses in the theory of elasticity. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen (B), 67(1), 17-44(1964)
[5] Yang, F., 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)
[6] Mindlin, R. D. Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis, 6(1), 51-78(1964)
[7] Mindlin, R. D. Second gradient of strain and surface tension in linear elasticity. International Journal of Solids and Structures, 1(4), 417-438(1965)
[8] Lam, D. C. C., Yang, F., Chong, A. C. M., Wang, J., and Tong, P. Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids, 51(8), 1477-1508(2003)
[9] Eringen, A. C. and Suhubi, E. Nonlinear theory of simple micro-elastic solids-I. International Journal of Engineering Science, 2(2), 189-203(1964)
[10] Suhubi, E. and Eringen, A. C. Nonlinear theory of micro-elastic solids-Ⅱ. International Journal of Engineering Science, 2(4), 389-404(1964)
[11] Ansari, R., Arash, B., and Rouhi, H. Vibration characteristics of embedded multi-layered graphene sheets with different boundary conditions via nonlocal elasticity. Composite Structures, 93(9), 2419-2429(2011)
[12] Peddieson, J., Buchanan, G. R., and McNitt, R. P. Application of nonlocal continuum models to nanotechnology. International Journal of Engineering Science, 41(3-5), 305-312(2003)
[13] Sudak, L. J. Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. Journal of Applied Physics, 94(11), 7281-7287(2003)
[14] Rouhi, H. and Ansari, R. Nonlocal analytical flugge shell model for axial buckling of double-walled carbon nanotubes with different end conditions. Nano, 7, 1250018(2012)
[15] Ansari, R. and Rouhi, H. Analytical treatment of the free vibration of single-walled carbon nanotubes based on the nonlocal flugge shell theory. Journal of Engineering Materials and Technology, 134(1), 011008(2012)
[16] Chang, T. P. Thermal-mechanical vibration and instability of a fluid-conveying single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory. Applied Mathematical Modelling, 36(5), 1964-1973(2012)
[17] 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 Sinica, 32(1), 83-100(2016)
[18] Ansari, R., Gholami, R., and Rouhi, H. Size-dependent nonlinear forced vibration analysis of magneto-electro-thermo-elastic Timoshenko nanobeams based upon the nonlocal elasticity theory. Composite Structures, 126, 216-226(2015)
[19] Ansari, R., Rouhi, H., and Sahmani, S. Calibration of the analytical nonlocal shell model for vibrations of double-walled carbon nanotubes with arbitrary boundary conditions using molecular dynamics. International Journal of Mechanical Sciences, 53(9), 786-792(2011)
[20] Ansari, R. and Rouhi, H. Explicit analytical expressions for the critical buckling stresses in a monolayer graphene sheet based on nonlocal elasticity. Solid State Communications, 152(2), 56-59(2012)
[21] Gibbs, J. W. The Scientific Papers of J. Willard Gibbs, Vol. 1, Longmans-Green, London (1906)
[22] Cammarata, R. C. Surface and interface stress effects on interfacial and nanostructured materials. Materials Science and Engineering, A237(2), 180-184(1997)
[23] Gurtin, M. E. and Murdoch, A. I. A continuum theory of elastic material surfaces. Archive for Rational Mechanics and Analysis, 57(4), 291-323(1975)
[24] Gurtin, M. E. and Murdoch, A. I. Surface stress in solids. International Journal of Solids and Structures, 14(6), 431-440(1978)
[25] Arefi, M. Surface effect and non-local elasticity in wave propagation of functionally graded piezoelectric nano-rod excited to applied voltage. Applied Mathematicsand Mechanics (English Edition), 37(3), 289-302(2016) DOI 10.1007/s10483-016-2039-6
[26] Ansari, R. and Sahmani, S. Bending behavior and buckling of nanobeams including surface stress effects corresponding to different beam theories. International Journal of Engineering Science, 49(11), 1244-1255(2011)
[27] Eltaher, M. A., Mahmoud, F. F., Assie, A. E., and Meletis, E. I. Coupling effects of nonlocal and surface energy on vibration analysis of nanobeams. Applied Mathematics and Computation, 224, 760-774(2013)
[28] Ansari, R., Mohammadi, V., Faghih-Shojaei, M., Gholami, R., and Sahmani, S. Postbuckling characteristics of nanobeams based on the surface elasticity theory. Composites Part B:Engineering, 55, 240-246(2013)
[29] Hossieni-Hashemi, S. and Nazemnezhad, R. An analytical study on the nonlinear free vibration of functionally graded nanobeams incorporating surface effects. Composites Part B:Engineering, 52, 199-206(2013)
[30] Ansari, R., Mohammadi, V., Faghih-Shojaei, M., Gholami, R., and Rouhi, H. Nonlinear vibration analysis of Timoshenko nanobeams based on surface stress elasticity theory. European Journal of Mechanics-A/Solids, 45, 143-152(2014)
[31] Amirian, B., Hosseini-Ara, R., and Moosavi, H. Surface and thermal effects on vibration of embedded alumina nanobeams based on novel Timoshenko beam model. Applied Mathematics and Mechanics (English Edition), 35(7), 875-886(2014) DOI 10.1007/s10483-014-1835-9
[32] Ansari, R., Mohammadi, V., Faghih-Shojaei, M., Gholami, R., and Sahmani, S. Postbuckling analysis of Timoshenko nanobeams including surface stress effect. International Journal of Engineering Science, 75, 1-10(2014)
[33] Ansari, R., Mohammadi, V., Faghih-Shojaei, M., Gholami, R., and Sahmani, S. On the forced vibration analysis of Timoshenko nanobeams based on the surface stress elasticity theory. Composites Part B:Engineering, 60, 158-166(2014)
[34] Chen, X. and Meguid, S. A. Asymmetric bifurcation of initially curved nanobeam. Journal of Applied Mechanics, 82, 091003(2015)
[35] Ansari, R., Gholami, R., Norouzzadeh, A., and Darabi, M. A. Surface stress effect on the vibration and instability of nanoscale pipes conveying fluid based on a size-dependent Timoshenko beam model. Acta Mechanica Sinica, 31(5), 708-719(2015)
[36] Wang, K. F. and Wang, B. L. A general model for nano-cantilever switches with consideration of surface effects and nonlinear curvature. Physica E, 66, 197-208(2015)
[37] Ansari, R., Pourashraf, T., and Gholami, R. An exact solution for the nonlinear forced vibration of functionally graded nanobeams in thermal environment based on surface elasticity theory. Thin-Walled Structures, 93, 169-176(2015)
[38] Lu, Z., Xie, F., Liu, Q., and Yang, Z. Surface effects on mechanical behavior of elastic nanoporous materials under high strain. Applied Mathematics and Mechanics (English Edition), 36(7), 927-938(2015) DOI 10.1007/s10483-015-1958-9
[39] Duan, J., Li, Z., and Liu, J. Pull-in instability analyses for NEMS actuators with quartic shape approximation. Applied Mathematics and Mechanics (English Edition), 37(3), 303-314(2016) DOI 10.1007/s10483-015-2007-6
[40] He, L. H., Lim, C. W., and Wu, B. S. A continuum model for size-dependent deformation of elastic film of nano-scale thickness. International Journal of Solids and Structures, 41(3), 847-857(2004)
[41] Ansari, R. and Sahmani, S. Surface stress effects on the free vibration behavior of nanoplates. International Journal of Engineering Science, 49(11), 1204-1215(2011)
[42] Lu, P., He, L. H., Lee, H. P., and Lu, C. Thin plate theory including surface effects. International Journal of Solids and Structure, 43(16), 4631-4647(2006)
[43] Ansari, R., Gholami, R., Faghih-Shojaei, M., Mohammadi, V., and Darabi, M. A. Surface stress effect on the pull-in instability of hydrostatically and electrostatically actuated rectangular nanoplates with various edge supports. Journal of Engineering Materials and Technology, 134(4), 041013(2012)
[44] Huang, D. W. Size-dependent response of ultra-thin films with surface effects. International Journal of Solids and Structures, 45(2), 568-579(2008)
[45] Ansari, R., Gholami, R., Faghih-Shojaei, M., Mohammadi, V., and Sahmani, S. Surface stress effect on the vibrational response of circular nanoplates with various edge supports. Journal of Applied Mechanics, 80(2), 021021(2013)
[46] Narendar, S. and Gopalakrishnan, S. Study of Terahertz wave propagation properties in nanoplates with surface and small-scale effects. International Journal of Mechanical Sciences, 64(1), 221-231(2012)
[47] Ansari, R., Mohammadi, V., Faghih-Shojaei, M., Gholami, R., and Sahmani, S. Surface stress effect on the postbuckling and free vibrations of axisymmetric circular Mindlin nanoplates subject to various edge supports. Composite Structures, 112, 358-367(2014)
[48] Shaat, M., Mahmoud, F. F., Gao, X. L., and Faheem, A. F. Size-dependent bending analysis of Kirchhoff nano-plates based on a modified couple-stress theory including surface effects. International Journal of Mechanical Sciences, 79, 31-37(2014)
[49] Ansari, R., Mohammadi, V., Faghih-Shojaei, M., Gholami, R., and Darabi, M. A. A geometrically non-linear plate model including surface stress effect for the pull-in instability analysis of rectangular nanoplates under hydrostatic and electrostatic actuations. International Journal of Non-Linear Mechanics, 67, 16-26(2014)
[50] Ansari, R., Gholami, R., Faghih-Shojaei, M., Mohammadi, V., and Sahmani, S. Surface stress effect on the pull-in instability of circular nanoplates. Acta Astronautica, 102, 140-150(2014)
[51] Radebe, I. S. and Adali, S. Effect of surface stress on the buckling of nonlocal nanoplates subject to material uncertainty. Latin American Journal of Solids and Structures, 56, 840-846(2015)
[52] Ansari, R., Ashrafi, M. A., Pourashraf, T., and Sahmani, S. Vibration and buckling characteristics of functionally graded nanoplates subjected to thermal loading based on surface elasticity theory. Acta Astronautica, 109, 42-51(2015)
[53] Ansari, R., Shahabodini, A., Faghih-Shojaei, M., Mohammadi, V., and Gholami, R. On the bending and buckling behaviors of Mindlin nanoplates considering surface energies. Physica E, 57, 126-137(2014)
[54] Miller, R. E. and Shenoy, V. B. Size-dependent elastic properties of nanosized structural elements. Nanotechnology, 11, 139-147(2000)
[55] Park, H. S. Surface stress effects on the critical buckling strains of silicon nanowires. Computational Materials Science, 51(1), 396-401(2012)
[56] Chiu, M. S. and Chen, T. Bending and resonance behavior of nanowires based on Timoshenko beam theory with high-order surface stress effects. Physica E, 54, 149-156(2013)
[57] Qiao, L. and Zheng, X. Effect of surface stress on the stiffness of micro/nanocantilevers:nanowire elastic modulus measured by nano-scale tensile and vibrational techniques. Journal of Applied Physics, 113(1), 013508(2013)
[58] Rouhi, H., Ansari, R., and Darvizeh, M. Size-dependent free vibration analysis of nanoshells based on the surface stress elasticity. Applied Mathematical Modelling, 40(4), 3128-3140(2016)
[59] Rouhi, H., Ansari, R., and Darvizeh, M. Analytical treatment of the nonlinear free vibration of cylindrical nanoshells based on a first-order shear deformable continuum model including surface influences. Acta Mechanica, 227(2016) DOI 10.1007/s00707-016-1595-4
[60] Zabow, G., Dodd, S. J., Moreland, J., and Koretsky, A. P. Fabrication of uniform cylindrical nanoshells and their use as spectrally tunable MRI contrast agents. Nanotechnology, 20, 385301(2009)
[61] Weingarten, V. I. and Seide, P. Elastic stability of thin-walled cylindrical and conical shells under combined external pressure and axial compression. AIAA Journal, 3, 913-920(1965)
[62] Hutchinson, J. W. Imperfection sensitivity of externally pressurized spherical shells. Journal of Applied Mechanics, 34(1), 49-55(1967)
[63] Kirshnamoorthy, G. Buckling of thin cylinders under combined external pressure and axial compression. Journal of Aircraft, 11, 65-68(1974)
[64] Bisagni, C. Numerical analysis and experimental correlation of composite shell buckling and postbuckling. Composites Part B:Engineering, 31(8), 655-667(2000)
[65] Teng, J. G. and Hong, T. Postbuckling analysis of elastic shells of revolution considering mode switching and interaction. International Journal of Solids and Structures, 43(3-4), 551-568(2006)
[66] Amabili, M. Nonlinear Vibrations and Stability of Shells and Plates, Cambridge University Press, Cambridge (2008)
[67] Donnell, L. H. Beam, Plates and Shells, McGraw-Hill, New York (1976)
[68] Vol'mir, A. S. Stability of Elastic Systems, No. FTD-MT-64-335, Foreign Technology Division, Wright-Patterson AFB, Ohio (1965)
[69] Ogata, S., Li, J., and Yip, S. Ideal pure shear strength of aluminum and copper. Science, 298, 807-811(2002)
[70] Zhu, R., Pan, E., Chung, P. W., Cai, X., Liew, K. M., and Buldum, A. Atomistic calculation of elastic moduli in strained silicon. Semiconductor Science and Technology, 21, 906-911(2006)
[71] Prinz, V. Y. and Golod, S. V. Elastic silicon-film-based nanoshells:formation, properties, and applications. Journal of Applied Mechanics and Technical Physics, 47(6), 867-878(2006)
[72] Huang, H. and Han, Q. Nonlinear buckling and postbuckling of heated functionally graded cylindrical shells under combined axial compression and radial pressure. International Journal of Non-Linear Mechanics, 44(2), 209-218(2009)