Bending Sector Graphene Sheet Based on the Elastic Winkler-Pstrnak with the Help of Nonlocal Elasticity Theory Using Developed Kantorovich Method

Document Type: Persian

Authors

1 MSc Student, Department of Mechanical Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran.

2 Assistant Prof., Department of Mechanical Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran

Abstract

In this study, the elastic bending of sector graphene sheet has been studied based on elasticity using Eringen Nonlocal Elasticity Theory. In order to do this, the balance equations governing the sector graphene sheet have been solved in terms of displacements with regard to nonlocal relationship of stress, shear theory of the first order, and obtained linear strains using developed Kantorovich method. In this method, the obtained partial differential equations are converted into two categories that can be solved using analytical and numerical methods. Developed Kantorovich method is a method with a high rate of convergence, in which the expected convergence is achieved with just three to four repetitions. With regard to the fact that no research has yet been conducted in this regard, the results, considering the nonlocal coefficient equal to zero, have been compared with other articles in order to check the validity. In the end, the effect of nonlocal coefficient variations on the results in terms of thickness, boundary conditions, hardness of elastic base and difference between nonlocal and local elasticity analysis has been studied.

Keywords


[1] Gibson R.F., Ayorinde E.O., Wen Y.F., Vibrations of Carbon Nanotubes and their Composites: A Review, Composites science and technology, Vol. 67, 2007, pp. 1-28.

[2] Geim A.K., Graphene: Status and Prospects, Science, Vol. 324, 2009, pp. 1530-1534.

[3] Kroto H.W., Heath J.R., O’Brien S.C, Curl R.F., Buckminster Fullerene, Nature, Vol. 318,  1985, pp. 162-163.

[4] Iijima S., Helical Microtubules of Graphitic Carbon, Nature, Vol. 8, 1991, pp. 354-356.

[5] Kong X.Y., Ding Y., Single-Crystal Nano-rings Formed by Epitaxial Self-Coiling of Polar Nano-belts, Science, Vol. 303, 2004, pp. 1348-1351.

[6] Chunyu Li, Atomistic Simulations on Multilayer Graphene Reinforced Epoxy Composites, Composites: Part A, Vol. 43, 2012, pp. 1293-1300.

[7] Kuilla T., Bhadra S., Yao D., Kimc N.M., Bosed S., Leea J.H., Recent Advances in Graphene Based Polymer Composites, Progress in polymer science, Vol. 35, 2010, pp. 1350-1375.

[8] Arash B., Wang Q., A Review on the Application of Nonlocal Elastic Models in Modeling of Carbon Nanotubes and Graphene, Computational materials science, Vol. 51, 2012, pp. 303-313.

[9] Pradhan S.C., Kumar A., Vibration Analysis of Orthotropic Graphene Sheets Using Nonlocal Elasticity Theory and Differential Quadrature Method, Composite structures, Vol. 93, 2004, pp. 774-779.

[10] Haftbaradaran H, Shodja H., Elliptic In homogeneities and Inclusions in Anti-Plane Couple Stress Elasticity with Application to Nano-Composites, International journal of solids and structures, Vol. 46, 2009, pp. 2978-2987.

[11] Fleck N.A, Hutchinson J.W., Strain Gradient Plasticity, Advance applied mechanics, Vol. 33, 1997, pp.295-361.

[12] Yang F, Chong A. C. M, Lam D.C.C, Tong P., Couple Stress Based Strain Gradient Theory for Elasticity, International journal of solid structs, Vol. 39, 2002, pp. 2731-2743.

[13] Eringen A.C., Nonlocal Continuum Field Theories, New york, Springer-Verlag, 2002.

[14] Pradhan S.C., Murmu T., Small Scale Effect on the Buckling of Single-Layered Graphene Sheets under Biaxial Compression via Nonlocal Continuum Mechanics, Computational materials science, Vol. 47, 2009, pp. 268-274.

[15] Boehm H.P., Clauss A., Fischer G. O and Hofmann U., Das Adsorptions Verhalten Sehr Dünner Kohlenstof-ffolien, Zeitschrift für anorganische und allgemeine chemie, Vol. 316, 2004,  pp. 119-127.

[16] Behfar K., Naghdabadi R., Nanoscale Vibrational Analysis of Multi-Layered Graphene Sheet Embedded in an Elastic Medium, Composites science and technology, Vol. 65, 2005,  pp. 1159-1164.

[17] Kitipornchai S., He X. Q., Liew K. M., Continuum Model for the Vibration of Multilayered Graphene Sheets, Physical Review B, Vol. 72, 2005, pp. 1-7.

[18] Liew K.M., He X.Q., Kitipornchai S., Predicting Nanovibration of Multi-Layered Graphene Sheets Embedded in an Elastic Matrix, Acta materialia, Vol. 54, 2006,  pp. 4229-4236.

[19] Duan W.H., Wang C.M., Exact Solutions for Axisymmetric Bending of Micro/Nanoscale Circular Plates Based on Nonlocal Plate Theory, Nanotechnology, Vol. 18, 2007, pp. 1-5.

[20] Pradhan S.C., Phadikar J.K., Small Scale Effect on Vibration of Embedded Multilayered Graphene Sheets Based on Nonlocal Continuum Models, Physics letters A, Vol. 373, 2009, pp. 1062-1069.

[21] Shen H., Shen L., Zhang, Chen-Li., Nonlocal Plate Model for Nonlinear Vibration of Single Layer Graphene Sheets in Thermal Environments, Computational materials science, Vol. 48, 2010, pp. 680-685.

[22] Jomehzadeh E., Saidi A. R., A Study on Large Amplitude Vibration of Multilayered Graphene Sheets, Computational materials science, Vol. 50, 2011, pp. 1043-1051.

[23] Mohammadi M., Ghayour M., Farajpour A., Free Transverse Vibration Analysis of Circular and Annular Graphene Sheets with Various Boundary Conditions Using the Nonlocal Continuum Plate Model, Composites, Vol. 45, 2013, pp. 32-42.

[24] Kerr A.D., An extension of the Kantorovich method, Q Appl Math, 26, 1968, pp. 219.

[25] Fereidoon A., Mohyeddin A., Sheikhi M., Rahmani H., Bending analysis of functionally graded annular sector plates by extended Kantorovich method, Composites Part B, Vol. 43, No.5, 2012, pp. 2172-2179.

 

[26] Aghdam M.M., Mohammadi M., Erfanian V., Bending analysis of thin annular sector plates using extended Kantorovich method, Thin Walled Structures, Vol. 45, No. 12, 2007, pp. 983-990.

[27] Salehi M., Turvery G.J., Elastic large deflection response of annular sector plates—a comparison of DR finite difference, finite element and other numerical solutions. composite structures, Vol. 40, No. 5, 1991, pp. 1267–78.

[28] Harik I.E., Analytical solution to orthotropic sector, Journal of Engineering Mechanics, 110, 1984, pp. 554-68.

[29] Cheung M.S., Chan M.Y.T., Static and dynamic analysis of thin and thick sectorial plates by the finite strip method, composite structures, Vol. 14, No.1-2, 1981, pp.  79-88.

[30] M. E. Golmakani, J. Rezatalab, Nonlinear bending analysis of orthotropic  nanoscale plates in an elastic matrix based on nonlocal continuum mechanics, composite structures, Vol. 111, 2014,  pp. 85-97.