Mechanical Buckling of Circular Orthotropic Bilayer Nanoplate Embedded in an Elastic Matrix under Radial Compressive Loading

Subject Areas : micro and nano mechanics

M.  Ahmadpour ¹ (Department of Mechanical Engineering, Mashhad branch, Islamic Azad University, Mashhad, Iran)
M.E.  Golmakani ² (Department of Mechanical Engineering, Mashhad branch, Islamic Azad University, Mashhad, Iran)
M.N.  Sadraee Far ³ (Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran)

Received: 2020-02-29 Accepted : 2020-04-19 Published : 2021-03-01

Keywords: Mechanical buckling, Orthotropic Nanoplate, DQM, Nonlocal Mindlin Theory,

Abstract :

This article investigates the buckling behavior of orthotropic annular/circular bilayer graphene sheet embedded in Winkler–Pasternak elastic medium under mechanical loading. Using the nonlocal elasticity theory, the bilayer graphene sheet is modeled as a nonlocal orthotropic plate which contains small scale effect and van der Waals interaction forces. Differential Quadrature Method (DQM) is employed to solve the governing equations for various combinations of simply supported or clamped boundary conditions. The results show that small scale parameter does not have any effect on critical buckling load of cases without elastic medium in simply supported boundary condition. Also, increase of vdW coefficient leads to increase of critical buckling load smoothly then it has no impact on critical buckling load after a certain value.

References:

[1]     Sakhaee-Pour, A., Ahmadian, M. T., and Vafai, A., Applications of Single-Layered Graphene Sheets as Mass Sensors and Atomistic Dust Detectors, Solid State Commun, Vol. 145, 2008, pp. 168–172.

[2]     Wang, J., Li, Z., Fan, G., Pan, H., Chen, Z., and Zhang, D., Reinforcement with Graphene Nano-Sheets in Aluminum Matrix Composites, Scr. Mater, Vol. 66, 2012, pp. 594–597.

[3]     Drexler, K. E., Nanosystems: Molecular Machinery, Manufacturing and Computation, John Wiley & Sons Inc., 1992.

[4]     Craighead, H. G., Nano Electro Mechanical Systems, Science, Vol. 290, 2000, pp. 1532–1535.

[5]     Li, X., Bhushan, B., Takashima, K., Baek, C. W., and Kim, Y. K., Mechanical Characterization of Micro/Nanoscale Structures for MEMS/NEMS Applications Using Nanoindentation Techniques, Ultramicroscopy, Vol. 97, 2003, pp. 481–494.

[6]     Pouresmaeeli, S., Fazelzadeh, S. A., and Ghavanloo, E., Exact Solution for Nonlocal Vibration of Double-Orthotropic Nanoplates Embedded in Elastic Medium, Composites Part B, Vol. 43, No. 8, 2012, pp. 3384–3390.

[7]     Farajpour, A., ArabSolghar, A. R., and Shahidi, A. R., Postbuckling Analysis of Multilayered Graphene Sheets Under Non-Uniform Biaxial Compression, Physica E, Vol. 47, 2013, pp.197–206.

[8]     Zhou, S. J., Li, Z. Q., Length Scales in The Static, Dynamic Torsion of a Circular Cylindrical Micro-Bar, J. Shanghai Univ. Technol., Vol. 31, 2001, pp. 401–407.

[9]     Fleck, N. A., Hutchinson, J. W., Strain Gradient Plasticity, Adv. Appl. Mech., Vol. 33, 1997, pp. 296–358.

[10]  Akgoz, B., Civalek, O., A Size-Dependent Shear Deformation Beam Model Based on the Strain Gradient Elasticity Theory, Internat. J. Engrg. Sci., Vol. 70, 2013, pp. 1–14.

[11]  Malikan, M., Nguyen, V. B., Buckling Analysis of Piezo-Magnetoelectric Nanoplates in Hygrothermal Environment Based On a Novel One Variable Plate Theory Combining with Higher-Order Nonlocal Strain Gradient Theory, Physica E: Low-Dimensional System and Nanostructures, Vol. 102, 2018, pp. 8-28.

[12]  Akgoz B., Civalek O., Free vibration analysis of axially functionally graded tapered Bernoulli–Euler microbeams based on the modified couple stress theory, Compos. Struct., Vol. 98, 2013, pp. 314–322.

[13]  Yang F., Chong A.C.M., Lam D.C.C., Tong P., Couple stress based strain gradient theory for elasticity, Int. J. Solids Struct., Vol. 39, 2002, pp. 2731–2743.

[14]  Malikan, M., Elctro-Mechanical Shear Buckling of Piezoelectric Nanoplate Using Modified Couple Stress Theory Based On Simplified First Order Shear Deformation Theory, Aplied Mathematical Modeling, Vol. 48, 2017, pp. 196-207.

[15]  Eringen, A. C., Edelen, D. G. B., On Nonlocal Elasticity, Internat. J. Engrg. Sci., Vol. 10, 1972, pp. 233–248.

[16]  Eringen, A. C., Nonlocal Continuum Field Theories, Springer, New York, 2002.

[17]  Eringen, A. C., Nonlocal Continuum Mechanics Based on Distributions, Internat. J. Engrg. Sci., Vol. 44, 2006, pp. 141–147.

[18]  Malikan, M., Dimitri, R., and Tornabene, F., Transient Response of Oscillated Carbon Nanotubes with an Internal and External Damping, Composites Part B: Engineering, Vol. 158, 2019, pp. 198-205.

[19]  Reddy, J. N., Nonlocal Theories for Bending, Buckling and Vibration of Beams, Internat. J. Engrg. Sci., Vol. 45, 2007, pp. 288–307.

[20]  Reddy, J. N., Pang, S. D., Nonlocal Continuum Theories of Beams for The Analysis of Carbon Nanotubes, J. Appl. Phys., Vol. 103, 2008, 023511.

[21]  Murmu, T., Pradhan, S. C., Buckling Analysis of a Single-Walled Carbon Nanotube Embedded in an Elastic Medium Based On Nonlocal Elasticity and Timoshenko Beam Theory and Using DQM, Physica E, Vol. 41, 2009, pp. 1232–1239.

[22]  Murmu, T., Pradhan, S. C., Thermo-Mechanical Vibration of a Single-Walled Carbon Nanotube Embedded in an Elastic Medium Based On Nonlocal Elasticity Theory, Comput. Mater. Sci., Vol. 46, 2009, pp. 854–859.

[23]  Wang, C. M., Duan, W. H., Free Vibration of Nanorings/Arches Based On Nonlocal Elasticity, J. Appl. Phys., Vol. 104, 2008, 014303.

[24]  Demir, C., Civalek, O., Torsional and Longitudinal Frequency and Wave Response of Microtubules Based On the Nonlocal Continuum and Nonlocal Discrete Models, Appl. Math. Model, Vol. 37, No. 22, 2013, pp. 9355–9367.

[25]  Civalek, O., Demir, C., and Akgoz, B., Static Analysis of Single Walled Carbon Nanotubes (SWCNT) Based on Eringen’s Nonlocal Elasticity Theory, Int. J. Eng. Appl. Sci., Vol. 2, 2009, pp. 47–56.

[26]  Zenkour, A. M., Nonlocal Transient Thermal Analysis of a Single-Layered Graphene Sheet Embedded in Viscoelastic Medium, Physica E, Vol. 97, 2016, pp. 87–97.

[27]  Shen, H., Zhang, C. L., Torsional Buckling and Postbuckling of Double-Walled Carbon Nanotubes by Nonlocal Shear Deformable Shell Model, Compos. Struct., Vol. 92, 2010, pp. 1073–1084.

[28]  Poot, M., van der Zant, H. S. J., Nanomechanical Properties of Few-Layer Graphene Membranes, Appl. Phys. Lett., Vol. 92, 2008, 063111.

[29]  Pradhan, S. C., Phadikar, J. K., Small Scale Effect On Vibration of Embedded Multilayered Graphene Sheets Based On Nonlocal Continuum Models, Phys. Lett. A, Vol. 373, 2009, pp. 1062–1069.

[30]  Pradhan, S. C., Buckling of Single Layer Graphene Sheet Based On Nonlocal Elasticity and Higher Order Shear Deformation Theory, Phys. Lett. A, Vol. 373, No. 45, 2009, pp. 4182–4188.

[31]  Reddy, C. D., Rajendran, S., and Liew, K. M., Equilibrium Configuration and Continuum Elastic Properties of Finite Sized Graphene, Nanotechnology, Vol. 17, 2006, pp. 864–870.

[32]  Shokrieh, M. M., Rafiee, R., Prediction of Young’s Modulus of Graphene Sheets and Carbon Nanotubes Using Nanoscale Continuum Mechanics Approach, Mater. Des., Vol. 31, 2010, pp. 790–795.

[33]  Sakhaee-Pour, A., Elastic Buckling of Single-Layered Graphene Sheet, Comput. Mater. Sci., Vol. 45, No. 2, 2009, pp. 266–270.

[34]  Ansari, R., Rajabiehfard, R., and Arash, B., Nonlocal Finite Element Model for Vibrations of Embedded Multi Layered Graphene Sheets, Comput. Mater. Sci., Vol. 49, 2010, pp.831–838.

[35]  Jomehzadeh, E., Saidi, A. R., A Study On Large Amplitude Vibration of Multilayered Graphene Sheets, Comput. Mater. Sci., Vol. 50, 2011, pp. 1043–1051.

[36]  Aksencer, T., Aydogdu, M., Levy Type Solution Method for Vibration and Buckling of Nanoplates Using Nonlocal Elasticity Theory, Physica E, Vol. 43, 2011, pp. 954–959.

[37]  Ghasemi, A., Dardel, M., Ghasemi, M. H., and Barzegari, M. M., Analytical Analysis of Buckling and Postbuckling of Fluid Conveying Multi-Walled Carbon Nanotubes, Appl. Math. Model, Vol. 37, 2013, pp. 4972–4992.

[38]  Farajpour, A, Danesh, M, and Mohammadi, M. Buckling Analysis of Variable Thickness Nanoplates Using Nonlocal Continuum Mechanics, Physica E., Vol. 44, 2011, pp. 719–727.

[39]  Pradhan, S. C., Murmu, T., Thermo-Mechanical Vibration of fgm Sandwich Beam Under Variable Elastic Foundations Using Differential Quadrature Method, J. Sound Vib., Vol. 321, 2009, pp. 342–362.

[40]  Pradhan, S. C., Kumar, A., Vibration Analysis of Orthotropic Graphene Sheets Embedded in Pasternak Elastic Medium Using Nonlocal Elasticity Theory and Differential Quadrature Method, Comput. Mater. Sci., Vol. 50, 2010, pp. 239–245.

[41]  Civalek, O., Akgoz, B., Vibration Analysis of Micro-Scaled Sector Shaped Graphene Surrounded by an Elastic Matrix, Comput. Mater. Sci., Vol. 77, 2013, pp. 295–303.

[42]  Tsiatas, G. C., Yiotis, A. J., Size Effect On the Static, Dynamic and Buckling Analysis of Orthotropic Kirchhoff-Type Skew Micro-Plates Based On a Modified Couple Stress Theory: Comparison with The Nonlocal Elasticity Theory, Acta Mech., Vol. 226, 2015, pp. 1267–1281.

[43]  Civalek, O., Elastic Buckling Behavior of Skew Shaped Single-Layer Graphene Sheets, Thin Solid Films, Vol. 550, 2013, pp. 450–458.

[44]  Zenkour, A. M., Sobhy, M., Nonlocal Elasticity Theory for Thermal Buckling of Nanoplates Lying On Winkler–Pasternak Elastic Substrate Medium, Physica E, Vol. 53, 2013, pp. 251–259.

[45]  Ansari, R., Sahmani, S., Prediction of Biaxial Buckling Behavior of Single-Layered Graphene Sheets Based On Nonlocal Plate Models and Molecular Dynamics Simulations, Applied Mathematical Modelling, Vol. 37, 2013, pp. 7338–7351.

[46]  Farajpour, A., Dehghany, M., and Shahidi, A. R., Surface and Nonlocal Effects On the Axisymmetric Buckling of Circular Graphene Sheets in Thermal Environment, Composites: Part B, Vol. 50, 2013, pp. 333–343.

[47]  Radebe, I. S., Adali, S., Buckling and Sensitivity Analysis of Nonlocal Orthotropic Nanoplates with Uncertain Material Properties, Composites: Part B, Vol. 56, 2014, pp. 840–846.

[48]  Ansari, R., Shahabodini, A., Shojai, M. F., Mohammadi, V., and Gholami, R., On the Bending and Buckling Behaviors of Mindlin Nanoplates Considering Surface Energies, Physica E, Vol. 57, 2014, pp. 126–137.

[49]  Anjomshoa, A., Shahidi, A. R., Hassani, B., and Jomehzadeh, E., Finite Element Buckling Analysis of Multi-Layered Graphene Sheets on Elastic Substrate Based on Nonlocal Elasticity Theory, Applied Mathematical Modelling, Vol. 38, No. 24, 2014, pp. 5934–5955.

[50]  Mohammadi, M., Farajpou,r A., Moradi, A., and Ghayour, M., Shear Buckling of Orthotropic Rectangular Graphene Sheet Embedded in an Elastic Medium in Thermal Environment, Composites Part B, Vol. 56, 2014, pp. 629–637.

[51]  Radic, N., Jeremic, D., Trifkovic, S., and Milutinovic, M., Buckling Analysis of Double-Orthotropic Nanoplates Embedded in Pasternak Elastic Medium Using Nonlocal Elasticity Theory, Composites: Part B, Vol. 61, 2014, pp. 162–171.

[52]  Sarrami-Foroushani, S., Azhari, M., On the use of Bubble Complex Finite Strip Method in the Nonlocal Buckling and Vibration Analysis of Single-Layered Graphene Sheets, International Journal of Mechanical Sciences, Vol. 85, 2014, pp. 168-178.

[53]  Sarrami-Foroushani, S., Azhari, M., Nonlocal Vibration and Buckling Analysis of Single and Multi-Layered Graphene Sheets Using Finite Strip Method Including Van Der Waals Effects, Physica E: Low-dimensional Systems and Nanostructures, Vol. 57, 2014, pp. 83-95.

[54]  Murmu, T., Sienz, J., Adhikari, S., and Arnold, C., Nonlocal Buckling of Double-Nanoplate-Systems Under Biaxial Compression, Composites: Part B, Vol. 44, 2013, pp. 84–94.

[55]  Farajpour, A., Shahidi, A. R., Mohammadi, M., and Mahzoon, M., Buckling of Orthotropic Micro/Nanoscale Plates Under Linearly Varying in-Plane Load Via Nonlocal Continuum Mechanics, Composite Structures, Vol. 94, 2012, pp. 1605–1615.

[56]  Farajpour, A., Mohammadi, M., Shahidi, A. R., and Mahzoon, M., Axisymmetric Buckling of the Circular Graphene Sheets with the Nonlocal Continuum Plate Model, Physica E, Vol. 43, 2011, pp. 1820–1825.