Thermal Buckling Analysis of Graphene Nanoplates Based on the Modified Couple Stress Theory using Finite Strip Method and Two-Variable Refined Plate Theory
Authors
Abstract
Graphene is one of the nanostructured materials that has recently attracted the attention of many researchers. This is due to the increasing expansion of nanotechnology and the application of this nanostructure in technology and industry owing to its mechanical, electrical and thermal properties. Thermal buckling behavior of single-layered graphene sheets is studied in this paper. Given the failure of classical theories to consider the scale effects and the limitations of the nano-sized experimental investigations of nano-materials, the small-scale effect is taken into account in this study, by employing the modified couple stress theory which has only one scale parameter. On the other hand, the two-variable refined plate theory, which considers the shear deformations in addition to bending deformations, is used to define the displacement field and to formulate the problem. The developed finite strip method formulation is used to evaluate the critical buckling temperature of the nanoplates. The validity of the proposed method is confirmed by comparing the results of this study with the those in the literature. The effects of different boundary conditions, temperature changing patterns, aspect ratio, and the ratio of length parameter to thickness on the critical buckling temperature are considered and the results are presented in the form of Tables and Figures
Keywords
1. Sobhy, M., “Hygrothermal Vibration of Orthotropic Double-Layered Graphene Sheets Embedded in an Elastic Medium using the Two-Variable Plate Theory”, Applied Mathematical Modelling, Vol. 40, pp. 85-99, 2016.
2. He, L. H., Lim, C. W., and Wu, B. S.,“A Continuum Model for Size-Dependent Deformation of Elastic Films of Nano-Scale Thickness”, International Journal of Solids and Structures, Vol. 41, pp. 847-857, 2004.
3. Lim, C., and He, L., “Size-Dependent Nonlinear Response of Thin Elastic Films with Nano-Scale Thickness”, International Journal of Mechanical Sciences, Vol. 46, pp. 1715-1726, 2004.
4. Kitipornchai, S., He, X. Q., and Liew, K. M., “Continuum Model for the Vibration of Multilayered Graphene Sheets”, Physical Review B, Vol. 72, p. 075443, 2005.
5. XQ. He, S. Kitipornchai, KM. Liew, “Resonance Analysis of Multi-Layered Graphene Sheets used as Nanoscale Resonators”, Nanotechnology, Vol. 16, p. 2086, 2005.
6. Sarrami-Foroushani, S., and 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, pp. 83-95, 2014.
7. Lu, P., Zhang, P. Q., Lee, H. P., Wang, C. M., and Reddy, J. N., “Non-Local Elastic Plate Theories”, in Proceedings of the Royal Society of London a: Mathematical, Physical and Engineering Sciences, pp. 3225-3240, 2007.
8. Pradhan, S., and Phadikar, J., “Small Scale Effect on Vibration of Embedded Multilayered Graphene Sheets Based on Nonlocal Continuum Models”, Physics Letters A, Vol. 373, pp. 1062-1069, 2009.
9. Murmu, T., and Pradhan, S., “Small-Scale Effect on the Free In-Plane Vibration of Nanoplates by Nonlocal Continuum Model”, Physica E: Low-Dimensional Systems and Nanostructures, Vol. 41, pp. 1628-1633, 2009.
10. Aksencer, T., and Aydogdu, M., “Levy Type Solution Method for Vibration and Buckling of Nanoplates using Nonlocal Elasticity Theory”, Physica E: Low-Dimensional Systems and Nanostructures, Vol. 43, pp. 954-959, 2011.
11. Wang, Y. Z., Li, F. M., and Kishimoto, K.,“Thermal Effects on Vibration Properties of Double-Layered Nanoplates at Small Scales”, Composites Part B: Engineering, Vol. 42, pp. 1311-1317, 2011.
12. Shimpi, R. P., “Refined Plate Theory and Its Variants”, AIAA Journal, Vol. 40, pp. 137-146, 2002.
13. Narendar, S., “Buckling Analysis of Micro/Nano-Scale Plates Based on Two-Variable Refined Plate Theory Incorporating Nonlocal Scale Effects”, Composite Structures, Vol. 93, pp. 3093-3103, 2011.
14. Narendar, S., and Gopalakrishnan, S., “Scale Effects on Buckling Analysis of Orthotropic Nanoplates Based on Nonlocal Two-Variable Refined Plate Theory”, Acta Mechanica, Vol. 223, pp. 395-413, 2012.
15. Satish, N., Narendar, S., and Gopalakrishnan, S.,“Thermal Vibration Analysis of Orthotropic Nanoplates Based on Nonlocal Continuum Mechanics”, Physica E: Low-Dimensional Systems and Nanostructures, Vol. 44, pp. 1950-1962, 2012.
16. Ameur, M., Tounsi, A., Mechab, I., and El Bedi, A. A., “A New Trigonometric Shear Deformation Theory for Bending Analysis of Functionally Graded Plates Resting on Elastic Foundations”, KSCE Journal of Civil Engineering, Vol. 15, pp. 1405-1414, 2011.
17. Thai, H. -T., and Vo, T. P., “A New Sinusoidal Shear Deformation Theory for Bending, Buckling, and Vibration of Functionally Graded Plates”, Applied Mathematical Modelling, Vol. 37, pp. 3269-3281, 2013.
18. Tounsi, A., Houari, M. S. A., and Benyoucef, S.,“A Refined Trigonometric Shear Deformation Theory for Thermoelastic Bending of Functionally Graded Sandwich Plates”, Aerospace Science and Technology, Vol. 24, pp. 209-220, 2013.
19. Zenkour, A., and Sobhy, M., “Nonlocal Elasticity Theory for Thermal Buckling of Nanoplates Lying on Winkler-Pasternak Elastic Substrate Medium”, Physica E: Low-Dimensional Systems and Nanostructures, Vol. 53, pp. 251-259, 2013.
20. Sobhy, M., “Generalized Two-Variable Plate Theory for Multi-Layered Graphene Sheets with Arbitrary Boundary Conditions”, Acta Mechanica, Vol. 225, p. 2521, 2014.
21. Sobhy, M., “Levy-Type Solution for Bending of Single-Layered Graphene Sheets in Thermal Environment using the Two-Variable Plate Theory”, International Journal of Mechanical Sciences, Vol. 90, pp. 171-178, 2015.
22. Park, S., and Gao, X., “Bernoulli-Euler Beam Model Based on a Modified Couple Stress Theory”, Journal of Micromechanics and Microengineering, Vol. 16, p. 2355, 2006.
23. Ma, H. M., Gao, X. L., and Reddy, J. N., “A Microstructure-Dependent Timoshenko Beam Model Based on a Modified Couple Stress Theory”, Journal of the Mechanics and Physics of Solids, Vol. 56, pp. 3379-3391, 2008.
24. Tsiatas, G. C., “A New Kirchhoff Plate Model Based on a Modified Couple Stress Theory”, International Journal of Solids and Structures, Vol. 46, pp. 2757-2764, 2009.
25. Ma, H. M., Gao, X. L., and Reddy, J. N., “A Nonclassical Reddy-Levinson Beam Model Based on a Modified Couple Stress Theory”, International Journal for Multiscale Computational Engineering, Vol. 8, 2010.
26. Ke, L. L., Wang, Y. S., Yang, J., and Kitipornchai, S., “Free Vibration of Size-Dependent Mindlin Microplates Based on the Modified Couple Stress Theory”, Journal of Sound and Vibration, Vol. 331, pp. 94-106, 2012.
27. Thai, H. -T., and Kim, S. -E., “A Size-Dependent Functionally Graded Reddy Plate Model Based on a Modified Couple Stress Theory”, Composites Part B: Engineering, Vol. 45, pp. 1636-1645, 2013.
28. Kim, J., and Reddy, J., “Analytical Solutions for Bending, Vibration, and Buckling of FGM Plates using a Couple Stress-Based Third-Order Theory”, Composite Structures, Vol. 103, pp. 86-98, 2013.
29. Mirsalehi, M., Azhari, M., and Amoushahi, H., “Stability of Thin FGM Microplate Subjected to Mechanical and Thermal Loading Based on the Modified Couple Stress Theory and Spline Finite Strip Method”, Aerospace Science and Technology, Vol. 47, pp. 356-366, 2015.
30. 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, Vol. 39, pp. 2731-2743, 2002.
31. Hindiin, R., “Influence of Rotary Inertia and Shear on Flexural Motion of Isotropic Elastic Plates”, JAPME, Vol. 18, pp. 31-38, 1951.
32. Reddy, J., “Nonlocal Theories for Bending, Buckling and Vibration of Beams”, International Journal of Engineering Science, Vol. 45, pp. 288-307, 2007.
33. Zenkour, A. M., and Sobhy, M., “Thermal Buckling of Functionally Graded Plates Resting on Elastic Foundations using the Trigonometric Theory”, Journal of Thermal Stresses, Vol. 34, pp. 1119-1138, 2011.
34. Touratier, M., “An Efficient Standard Plate Theory”, International Journal of Engineering Science, Vol. 29, pp. 901-916, 1991.
35. Hosseini-Hashemi, S., Fadaee, M., Es' Haghi, M., “A Novel Approach for In-Plane/Out-of-Plane Frequency Analysis of Functionally Graded Circular/Annular Plates”, International Journal of Mechanical Sciences, Vol. 52, pp. 1025-1035, 2010.
36. Mechab, I., Mechab, B., and Benaissa, S., “Static and Dynamic Analysis of Functionally Graded Plates using Four-Variable Refined Plate Theory by the New Function”, Composites Part B: Engineering, Vol. 45, pp. 748-757, 2013.
37. Kiani, Y., Bagherizadeh, E., and Eslami, M. R.,“Thermal Buckling of Clamped Thin Rectangular FGM Plates Resting on Pasternak Elastic Foundation (Three Approximate Analytical Solutions)”, ZAMM‐ Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 91, pp. 581-593, 2011
2. He, L. H., Lim, C. W., and Wu, B. S.,“A Continuum Model for Size-Dependent Deformation of Elastic Films of Nano-Scale Thickness”, International Journal of Solids and Structures, Vol. 41, pp. 847-857, 2004.
3. Lim, C., and He, L., “Size-Dependent Nonlinear Response of Thin Elastic Films with Nano-Scale Thickness”, International Journal of Mechanical Sciences, Vol. 46, pp. 1715-1726, 2004.
4. Kitipornchai, S., He, X. Q., and Liew, K. M., “Continuum Model for the Vibration of Multilayered Graphene Sheets”, Physical Review B, Vol. 72, p. 075443, 2005.
5. XQ. He, S. Kitipornchai, KM. Liew, “Resonance Analysis of Multi-Layered Graphene Sheets used as Nanoscale Resonators”, Nanotechnology, Vol. 16, p. 2086, 2005.
6. Sarrami-Foroushani, S., and 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, pp. 83-95, 2014.
7. Lu, P., Zhang, P. Q., Lee, H. P., Wang, C. M., and Reddy, J. N., “Non-Local Elastic Plate Theories”, in Proceedings of the Royal Society of London a: Mathematical, Physical and Engineering Sciences, pp. 3225-3240, 2007.
8. Pradhan, S., and Phadikar, J., “Small Scale Effect on Vibration of Embedded Multilayered Graphene Sheets Based on Nonlocal Continuum Models”, Physics Letters A, Vol. 373, pp. 1062-1069, 2009.
9. Murmu, T., and Pradhan, S., “Small-Scale Effect on the Free In-Plane Vibration of Nanoplates by Nonlocal Continuum Model”, Physica E: Low-Dimensional Systems and Nanostructures, Vol. 41, pp. 1628-1633, 2009.
10. Aksencer, T., and Aydogdu, M., “Levy Type Solution Method for Vibration and Buckling of Nanoplates using Nonlocal Elasticity Theory”, Physica E: Low-Dimensional Systems and Nanostructures, Vol. 43, pp. 954-959, 2011.
11. Wang, Y. Z., Li, F. M., and Kishimoto, K.,“Thermal Effects on Vibration Properties of Double-Layered Nanoplates at Small Scales”, Composites Part B: Engineering, Vol. 42, pp. 1311-1317, 2011.
12. Shimpi, R. P., “Refined Plate Theory and Its Variants”, AIAA Journal, Vol. 40, pp. 137-146, 2002.
13. Narendar, S., “Buckling Analysis of Micro/Nano-Scale Plates Based on Two-Variable Refined Plate Theory Incorporating Nonlocal Scale Effects”, Composite Structures, Vol. 93, pp. 3093-3103, 2011.
14. Narendar, S., and Gopalakrishnan, S., “Scale Effects on Buckling Analysis of Orthotropic Nanoplates Based on Nonlocal Two-Variable Refined Plate Theory”, Acta Mechanica, Vol. 223, pp. 395-413, 2012.
15. Satish, N., Narendar, S., and Gopalakrishnan, S.,“Thermal Vibration Analysis of Orthotropic Nanoplates Based on Nonlocal Continuum Mechanics”, Physica E: Low-Dimensional Systems and Nanostructures, Vol. 44, pp. 1950-1962, 2012.
16. Ameur, M., Tounsi, A., Mechab, I., and El Bedi, A. A., “A New Trigonometric Shear Deformation Theory for Bending Analysis of Functionally Graded Plates Resting on Elastic Foundations”, KSCE Journal of Civil Engineering, Vol. 15, pp. 1405-1414, 2011.
17. Thai, H. -T., and Vo, T. P., “A New Sinusoidal Shear Deformation Theory for Bending, Buckling, and Vibration of Functionally Graded Plates”, Applied Mathematical Modelling, Vol. 37, pp. 3269-3281, 2013.
18. Tounsi, A., Houari, M. S. A., and Benyoucef, S.,“A Refined Trigonometric Shear Deformation Theory for Thermoelastic Bending of Functionally Graded Sandwich Plates”, Aerospace Science and Technology, Vol. 24, pp. 209-220, 2013.
19. Zenkour, A., and Sobhy, M., “Nonlocal Elasticity Theory for Thermal Buckling of Nanoplates Lying on Winkler-Pasternak Elastic Substrate Medium”, Physica E: Low-Dimensional Systems and Nanostructures, Vol. 53, pp. 251-259, 2013.
20. Sobhy, M., “Generalized Two-Variable Plate Theory for Multi-Layered Graphene Sheets with Arbitrary Boundary Conditions”, Acta Mechanica, Vol. 225, p. 2521, 2014.
21. Sobhy, M., “Levy-Type Solution for Bending of Single-Layered Graphene Sheets in Thermal Environment using the Two-Variable Plate Theory”, International Journal of Mechanical Sciences, Vol. 90, pp. 171-178, 2015.
22. Park, S., and Gao, X., “Bernoulli-Euler Beam Model Based on a Modified Couple Stress Theory”, Journal of Micromechanics and Microengineering, Vol. 16, p. 2355, 2006.
23. Ma, H. M., Gao, X. L., and Reddy, J. N., “A Microstructure-Dependent Timoshenko Beam Model Based on a Modified Couple Stress Theory”, Journal of the Mechanics and Physics of Solids, Vol. 56, pp. 3379-3391, 2008.
24. Tsiatas, G. C., “A New Kirchhoff Plate Model Based on a Modified Couple Stress Theory”, International Journal of Solids and Structures, Vol. 46, pp. 2757-2764, 2009.
25. Ma, H. M., Gao, X. L., and Reddy, J. N., “A Nonclassical Reddy-Levinson Beam Model Based on a Modified Couple Stress Theory”, International Journal for Multiscale Computational Engineering, Vol. 8, 2010.
26. Ke, L. L., Wang, Y. S., Yang, J., and Kitipornchai, S., “Free Vibration of Size-Dependent Mindlin Microplates Based on the Modified Couple Stress Theory”, Journal of Sound and Vibration, Vol. 331, pp. 94-106, 2012.
27. Thai, H. -T., and Kim, S. -E., “A Size-Dependent Functionally Graded Reddy Plate Model Based on a Modified Couple Stress Theory”, Composites Part B: Engineering, Vol. 45, pp. 1636-1645, 2013.
28. Kim, J., and Reddy, J., “Analytical Solutions for Bending, Vibration, and Buckling of FGM Plates using a Couple Stress-Based Third-Order Theory”, Composite Structures, Vol. 103, pp. 86-98, 2013.
29. Mirsalehi, M., Azhari, M., and Amoushahi, H., “Stability of Thin FGM Microplate Subjected to Mechanical and Thermal Loading Based on the Modified Couple Stress Theory and Spline Finite Strip Method”, Aerospace Science and Technology, Vol. 47, pp. 356-366, 2015.
30. 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, Vol. 39, pp. 2731-2743, 2002.
31. Hindiin, R., “Influence of Rotary Inertia and Shear on Flexural Motion of Isotropic Elastic Plates”, JAPME, Vol. 18, pp. 31-38, 1951.
32. Reddy, J., “Nonlocal Theories for Bending, Buckling and Vibration of Beams”, International Journal of Engineering Science, Vol. 45, pp. 288-307, 2007.
33. Zenkour, A. M., and Sobhy, M., “Thermal Buckling of Functionally Graded Plates Resting on Elastic Foundations using the Trigonometric Theory”, Journal of Thermal Stresses, Vol. 34, pp. 1119-1138, 2011.
34. Touratier, M., “An Efficient Standard Plate Theory”, International Journal of Engineering Science, Vol. 29, pp. 901-916, 1991.
35. Hosseini-Hashemi, S., Fadaee, M., Es' Haghi, M., “A Novel Approach for In-Plane/Out-of-Plane Frequency Analysis of Functionally Graded Circular/Annular Plates”, International Journal of Mechanical Sciences, Vol. 52, pp. 1025-1035, 2010.
36. Mechab, I., Mechab, B., and Benaissa, S., “Static and Dynamic Analysis of Functionally Graded Plates using Four-Variable Refined Plate Theory by the New Function”, Composites Part B: Engineering, Vol. 45, pp. 748-757, 2013.
37. Kiani, Y., Bagherizadeh, E., and Eslami, M. R.,“Thermal Buckling of Clamped Thin Rectangular FGM Plates Resting on Pasternak Elastic Foundation (Three Approximate Analytical Solutions)”, ZAMM‐ Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 91, pp. 581-593, 2011
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