تحلیل غیرخطی کمانش نانوتیر کامپوزیتی با درنظر گرفتن نقص هندسی اولیه با استفاده از روش اجزاء محدود
نویسنده
دانشکده مهندسی مکانیک، دانشگاه کاشان، کاشان
چکیده
در این پژوهش، تحلیل کمانش غیرخطی نانوتیرکامپوزیتی مدرج تابعی تقویت شده با توزیع‌های مختلف نانو لولۀ نیترید-بور براساس تئوری الاستیسیته غیرمحلی به‌روش اجزاء محدود بررسی می‌شود. نانوتیرکامپوزیتی تحت بارگذاری‌های الکتروترمومکانیکی و نقص هندسی اولیه درنظر گرفته می‌شود. نانولوله‌های نیترید- بور در راستای ضخامت تیر به‌صورت یکنواخت و مدرج تابعی با چیدمان کاهشی- افزایشی توزیع شده‌اند و از مدل مخلوط توسعه یافته، برای تخمین خواص نانوتیر کامپوزیتی استفاده شده است. محیط الاستیک اطراف نانوتیر کامپوزیتی هوشمند به‌صورت بستر الاستیک مدل‌سازی شده است. معادلات حاکم با استفاده از روش انرژی و تئوری غیرموضعی الاستیسیته استخراج شده و بار کمانش بحرانی برای نانوتیر کامپوزیتی با شرایط مرزی مختلف شامل دو طرف تکیه‌گاه ساده یا دو طرف تکیه‌گاه گیردار به‌روش اجزاء محدود به‌دست می‌آیند. نتایج نشان می‌دهد با افزایش پارامتر نقص هندسی، سفتی نانوتیر کامپوزیتی افزایش یافته، درنتیجه پایداری سازه افزایش می‌یابد. استحکام چیدمان کاهشی- افزایشی، بیشتر از توزیع یکنواخت است. همچنین با اعمال پارامترهای میدان الکتریکی و بستر الاستیک بار کمانش بحرانی نانوتیر کامپوزیتی افزایش می‌یابد.
کلیدواژه‌ها
عنوان مقاله [English]
Nonlinear Buckling Analysis of Nano-composite Beam with Initial Geometrical Imperfection using Finite Element Method
نویسنده [English]
- M. Mohammadimehr
In this research, the nonlinear buckling analysis of Functionally Graded (FG) nano-composite beam reinforced by various distributions of Boron Nitrid Nanotube (BNNT) is investigated under electro-thermodynamical loading with considering initial geometrical imperfection. The analysis is performed based on nonlocal elasticity theory and using the Finite Element Method (FEM). Various distributions of BNNT along the beam’s thickness are considered as uniform and decreasing-increasing functionally graded; and the extended mixture model is used to estimate the properties of nano-composite beam. The elastic medium around the smart nano-composite beam is modeled as elastic foundation. The governing equations of equilibrium are derived using energy method and nonlocal elasticity theory; and the critical buckling load is obtained for various boundary conditions such as simply-simply supported (S-S) and clamped-clamped (C-C) using the FEM. The results indicate that with an increase in the geometrical imperfection parameter, the stiffness of nano-composite beam increases and consequently the stability of the system increases. The effect of FG-X distribution type is more than uniform distributions. Also, the critical buckling load of nano-composite beam increases with an increase in the electric field and elastic foundation.
کلیدواژه‌ها [English]
- Nonlinear buckling analysis
- Initial geometrical imperfection
- Various distributions of BNNTs
- Elastic foundation
- Nonlocal elasticity theory
- finite element method
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2. Liew, K. M., Lei, Z. X., and Zhan, L.W., “Mechanical Analysis of Functionally Graded Carbon Nanotube Reinforced Composites: A Review”, Composite Structures, Vol. 120, pp. 90-97, 2015.
3. Mohammadimehr, M., Mohandes, M., and Moradi, M., “Size Dependent Effect on the Buckling and Vibration Analysis of Double-Bonded Nanocomposite Piezoelectric Plate Reinforced by Boron Nitride Nanotube Based on Modified Couple Stress Theory”, Journal of Vibration and Control, Vol. 22, No. 7, pp. 1790-1807, 2016.
4. Ghorbanpour Arani, A., Maghamikia, S. H., Mohammadimehr, M., and Arefmanesh, A., “Buckling Analysis of Laminated Composite Rectangular Plates Reinforced by SWCNTs using Analytical and Finite Element Method”, Journal of Mechanical Science and Technology, Vol. 25, pp. 809-820, 2011.
5. Li, Z. M., “Thermal Postbuckling Behavior of 3D Braided Beams with Initial Geometric Imperfection under Different Type Temperature Distribution”, Composite Structures, Vol. 108, pp. 924-936, 2014.
6. Simsek, M., and Yurtcu, H. H., “Analytical Solutions for Bending and Buckling of Functionally Graded Nanobeams Based on the Nonlocal Timoshenko Beam Theory”, Composite Structures, Vol. 97, pp. 378-386, 2013.
7. Mohammadimehr, M., Mohammadimehr, M. A., and Dashti, P., “Size-Dependent Effect on Biaxial and Shear Nonlinear Buckling Analysis of Nonlocal Isotropic and Orthotropic Micro-Plate Based on Surface Stress and Modified Couple Stress Theories using Differential Quadrature Method”, Applied Mathematics and Mechanics, Vol. 37, pp. 529-554, 2016.
8. Komijani, M., Esfahani, S. E., Reddy, J. N., Liu, Y. P., and Eslami, M. R., “Nonlinear Thermal Stability and Vibration of Pre/Post-Buckled Temperature and Microstructure- Dependent FGM Beams Resting on Elastic Foundation”, Composite Structures, Vol. 112, No. 1, pp. 292-307, 2014.
9. Mohammadi, H., Mahzoon, M., and Mohammadi, M., “Postbuckling Instability of Nonlinear Nanobeam with Geometric Imperfection Embedded in Elastic Foundation”, Journal of Computational and Nonlinear Dynamics, Vol. 76, No. 4, pp. 2005-2016, 2014.
10. Mohammadimehr, M., and Shahedi, S., “High-Order Buckling and Free Vibration Analysis of Two Types Sandwich Beam Including AL or PVC-Foam Flexible Core and CNTs Reinforced Nanocomposite Face Sheets using GDQM”, Composites Part B: Engineering Available online 29 September 2016. http://dx.doi.org/10.1016/j.compositesb.2016.09.040.
11. Mohammadimehr, M., Rousta Navi, B., and Ghorbanpour Arani, A. “Free Vibration of Viscoelastic Double-Bonded Polymeric Nanocomposite Plates Reinforced by FG-SWCNTs using MSGT, Sinusoidal Shear Deformation Theory and Meshless Method”, Composite Structures, Vol. 131, pp. 654-671, 2015.
12. Mohammadimehr, M., Salemi, M., and Rousta Navi, B., “Bending, Buckling, and Free Vibration Analysis of MSGT MicrocompositeReddy Plate Reinforced by FG-SWCNTs with Temperature-Dependent material Properties under Hydro-Thermo-Mechanical Loadings using DQM”, Composite Structures, Vol. 138, pp. 361-380, 2016.
13. Wang, B., Deng, Z. C., and Zhang, K., “Nonlinear Vibration of Embedded Single-Walled Carbon Nanotube with Geometrical Imperfection under Harmonic Load Based on Nonlocal Timoshenko Beam Theory”, Applied Mathematics and Mechanics. English Edition, Vol. 34, No. 3, pp. 269-280, 2013.
14. Anjomshoa, A., Shahidi, A. R., Hassani, B., and Jomehzadeh, E., “Finite Element Buckling Analysis of Multi-Layered Graphemesheets on Elastic Substrate Based on Nonlocal Elasticity Theory”, Applied Mathematical Modelling, Vol. 38, pp. 5934-5955, 2014.
15. Reddy, J. N., An Introduction to Nonlinear Finite Element Analysis, Oxford University Press, Oxford, New York, 2004.
16. Ghorbanpour Arani, A., Rousta Navia, B., and Mohammadimehr M., “Surface Stress and Agglomeration Effects on Nonlocal Biaxial Buckling Polymeric Nanocomposite Plate Reinforced by CNT using Various Approaches”, Accepted in Advanced Composite Material, DOI: 10.1080/09243046. 2015. 1052189, 2015.
17. Murmu, T., and 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, pp. 1232-1239, 2009.
18. Amabilia, M., Karagiozis, K., and Padoussisa, M. P., “Effect of Geometric Imperfections on Non-Linear Stability of Circular Cylindrical Shells Conveyingfluid”, International Journal of Nonlinear Mecch, Vol. 44, pp. 276-289, 2009.
19. Mohammadimehr, M. and Alimirzaei, S., “Nonlinear Static and Vibration Analysis of Euler-Bernoulli Composite Beam Model Reinforced by FG-SWCNT with Initial Geometrical Imperfection using FEM”, Structural Engineering and Mechanics, Vol. 59, pp. 431-454, 2016.
20. Abdollahi, M., Saidi, A. R., and Mohammadi, M., “Buckling Analysis of Thick Functionally Graded Piezoelectric Plates Based on the Higher-Order Shear and Normal Deformable Theory”, Acta Mechanica, Vol. 226, pp. 2497-2510, 2015.
21. Cady, W. G., Piezoelectricity; an Introduction to the Theory and Applications of Electromechanical Phenomena in Crystals, McGraw-Hill, NewYork, 1946.
22. Moon, W. H., and Hwang, H. J., “Molecular Mechanics of Structural Properties of Boron-Nitride Nanotubes”, Physica E, Vol. 23, pp. 26-30, 2004.
23. Yas, M. H., and Samadi, N., “Free Vibrations and Buckling Analysis of Carbon Nanotube-Reinforced Composite Timoshenko Beams on Elastic Foundation”, Journal of Pressure Vessels and Piping, Vol. 98, pp. 119-128, 2012.
24. Chakraborty, A., Mahaptra, D. R., and Gopalakrishnan, S., “Finite Element Analysis of Free Vibration and Wave Propagationin Asymmetric Composite Beam with Structural Discontinuities”, Composite Structures, Vol. 55, pp. 23-36, 2002.
25. Rafiee, M., He, X. Q., and Liew, K. M., “Non-Linear Dynamic Stability of Piezoelectric Functionally Graded Carbon Nanotube Reinforced Composite Plates with Initial Geometric Imperfection”, International Journal of Non-linear Mechanics, Vol. 59, pp. 37-51, 2014.
26. Reddy, J. N., “Nonlocal Theories for Bending, Buckling and Vibration of Beams”, International Journal of Engineering Science, Vol. 45, pp. 288-307, 2007.
27. Ansari, R., Faraji Oskouie, M., Gholami, R. and Sadeghi, F., “Thermo-Electro-Mechanical Vibration of Postbuckled Piezoelectric Timoshenko Nanobeams Based on the Nonlocal Elasticity Theory”, Composites Part B, Vol. 89, pp. 316-327, 2016.
28. Liu, C., Ke, L. L., Wang, Y., Yang, J., and Kitipornchai, S., “Buckling and Post-buckling of
Size-Dependent Piezoelectric TimoShenko Nanobeams Subject to Thermoelectro-Mechanical Loadings”, International Journal of Structural Stability and Dynamics, Vol. 14, p. 1350067, 2014
2. Liew, K. M., Lei, Z. X., and Zhan, L.W., “Mechanical Analysis of Functionally Graded Carbon Nanotube Reinforced Composites: A Review”, Composite Structures, Vol. 120, pp. 90-97, 2015.
3. Mohammadimehr, M., Mohandes, M., and Moradi, M., “Size Dependent Effect on the Buckling and Vibration Analysis of Double-Bonded Nanocomposite Piezoelectric Plate Reinforced by Boron Nitride Nanotube Based on Modified Couple Stress Theory”, Journal of Vibration and Control, Vol. 22, No. 7, pp. 1790-1807, 2016.
4. Ghorbanpour Arani, A., Maghamikia, S. H., Mohammadimehr, M., and Arefmanesh, A., “Buckling Analysis of Laminated Composite Rectangular Plates Reinforced by SWCNTs using Analytical and Finite Element Method”, Journal of Mechanical Science and Technology, Vol. 25, pp. 809-820, 2011.
5. Li, Z. M., “Thermal Postbuckling Behavior of 3D Braided Beams with Initial Geometric Imperfection under Different Type Temperature Distribution”, Composite Structures, Vol. 108, pp. 924-936, 2014.
6. Simsek, M., and Yurtcu, H. H., “Analytical Solutions for Bending and Buckling of Functionally Graded Nanobeams Based on the Nonlocal Timoshenko Beam Theory”, Composite Structures, Vol. 97, pp. 378-386, 2013.
7. Mohammadimehr, M., Mohammadimehr, M. A., and Dashti, P., “Size-Dependent Effect on Biaxial and Shear Nonlinear Buckling Analysis of Nonlocal Isotropic and Orthotropic Micro-Plate Based on Surface Stress and Modified Couple Stress Theories using Differential Quadrature Method”, Applied Mathematics and Mechanics, Vol. 37, pp. 529-554, 2016.
8. Komijani, M., Esfahani, S. E., Reddy, J. N., Liu, Y. P., and Eslami, M. R., “Nonlinear Thermal Stability and Vibration of Pre/Post-Buckled Temperature and Microstructure- Dependent FGM Beams Resting on Elastic Foundation”, Composite Structures, Vol. 112, No. 1, pp. 292-307, 2014.
9. Mohammadi, H., Mahzoon, M., and Mohammadi, M., “Postbuckling Instability of Nonlinear Nanobeam with Geometric Imperfection Embedded in Elastic Foundation”, Journal of Computational and Nonlinear Dynamics, Vol. 76, No. 4, pp. 2005-2016, 2014.
10. Mohammadimehr, M., and Shahedi, S., “High-Order Buckling and Free Vibration Analysis of Two Types Sandwich Beam Including AL or PVC-Foam Flexible Core and CNTs Reinforced Nanocomposite Face Sheets using GDQM”, Composites Part B: Engineering Available online 29 September 2016. http://dx.doi.org/10.1016/j.compositesb.2016.09.040.
11. Mohammadimehr, M., Rousta Navi, B., and Ghorbanpour Arani, A. “Free Vibration of Viscoelastic Double-Bonded Polymeric Nanocomposite Plates Reinforced by FG-SWCNTs using MSGT, Sinusoidal Shear Deformation Theory and Meshless Method”, Composite Structures, Vol. 131, pp. 654-671, 2015.
12. Mohammadimehr, M., Salemi, M., and Rousta Navi, B., “Bending, Buckling, and Free Vibration Analysis of MSGT MicrocompositeReddy Plate Reinforced by FG-SWCNTs with Temperature-Dependent material Properties under Hydro-Thermo-Mechanical Loadings using DQM”, Composite Structures, Vol. 138, pp. 361-380, 2016.
13. Wang, B., Deng, Z. C., and Zhang, K., “Nonlinear Vibration of Embedded Single-Walled Carbon Nanotube with Geometrical Imperfection under Harmonic Load Based on Nonlocal Timoshenko Beam Theory”, Applied Mathematics and Mechanics. English Edition, Vol. 34, No. 3, pp. 269-280, 2013.
14. Anjomshoa, A., Shahidi, A. R., Hassani, B., and Jomehzadeh, E., “Finite Element Buckling Analysis of Multi-Layered Graphemesheets on Elastic Substrate Based on Nonlocal Elasticity Theory”, Applied Mathematical Modelling, Vol. 38, pp. 5934-5955, 2014.
15. Reddy, J. N., An Introduction to Nonlinear Finite Element Analysis, Oxford University Press, Oxford, New York, 2004.
16. Ghorbanpour Arani, A., Rousta Navia, B., and Mohammadimehr M., “Surface Stress and Agglomeration Effects on Nonlocal Biaxial Buckling Polymeric Nanocomposite Plate Reinforced by CNT using Various Approaches”, Accepted in Advanced Composite Material, DOI: 10.1080/09243046. 2015. 1052189, 2015.
17. Murmu, T., and 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, pp. 1232-1239, 2009.
18. Amabilia, M., Karagiozis, K., and Padoussisa, M. P., “Effect of Geometric Imperfections on Non-Linear Stability of Circular Cylindrical Shells Conveyingfluid”, International Journal of Nonlinear Mecch, Vol. 44, pp. 276-289, 2009.
19. Mohammadimehr, M. and Alimirzaei, S., “Nonlinear Static and Vibration Analysis of Euler-Bernoulli Composite Beam Model Reinforced by FG-SWCNT with Initial Geometrical Imperfection using FEM”, Structural Engineering and Mechanics, Vol. 59, pp. 431-454, 2016.
20. Abdollahi, M., Saidi, A. R., and Mohammadi, M., “Buckling Analysis of Thick Functionally Graded Piezoelectric Plates Based on the Higher-Order Shear and Normal Deformable Theory”, Acta Mechanica, Vol. 226, pp. 2497-2510, 2015.
21. Cady, W. G., Piezoelectricity; an Introduction to the Theory and Applications of Electromechanical Phenomena in Crystals, McGraw-Hill, NewYork, 1946.
22. Moon, W. H., and Hwang, H. J., “Molecular Mechanics of Structural Properties of Boron-Nitride Nanotubes”, Physica E, Vol. 23, pp. 26-30, 2004.
23. Yas, M. H., and Samadi, N., “Free Vibrations and Buckling Analysis of Carbon Nanotube-Reinforced Composite Timoshenko Beams on Elastic Foundation”, Journal of Pressure Vessels and Piping, Vol. 98, pp. 119-128, 2012.
24. Chakraborty, A., Mahaptra, D. R., and Gopalakrishnan, S., “Finite Element Analysis of Free Vibration and Wave Propagationin Asymmetric Composite Beam with Structural Discontinuities”, Composite Structures, Vol. 55, pp. 23-36, 2002.
25. Rafiee, M., He, X. Q., and Liew, K. M., “Non-Linear Dynamic Stability of Piezoelectric Functionally Graded Carbon Nanotube Reinforced Composite Plates with Initial Geometric Imperfection”, International Journal of Non-linear Mechanics, Vol. 59, pp. 37-51, 2014.
26. Reddy, J. N., “Nonlocal Theories for Bending, Buckling and Vibration of Beams”, International Journal of Engineering Science, Vol. 45, pp. 288-307, 2007.
27. Ansari, R., Faraji Oskouie, M., Gholami, R. and Sadeghi, F., “Thermo-Electro-Mechanical Vibration of Postbuckled Piezoelectric Timoshenko Nanobeams Based on the Nonlocal Elasticity Theory”, Composites Part B, Vol. 89, pp. 316-327, 2016.
28. Liu, C., Ke, L. L., Wang, Y., Yang, J., and Kitipornchai, S., “Buckling and Post-buckling of
Size-Dependent Piezoelectric TimoShenko Nanobeams Subject to Thermoelectro-Mechanical Loadings”, International Journal of Structural Stability and Dynamics, Vol. 14, p. 1350067, 2014
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