کاربرد روش بدون المان SSPH در تحلیل ارتعاشات آزاد گرافن

نوع مقاله : مقاله پژوهشی

نویسندگان

مهندسی مکانیک، دانشکده فنی و مهندسی، دانشگاه خوارزمی

چکیده

در این مقاله روش هیدرودینامیک ذرات هموار متقارن (SSPH)، برای بررسی ارتعاشات آزاد گرافن دولایه با در نظر گرفتن اثر برشی بین لایه‌ها استفاده می‌شود. شرایط مرزی گیردار- آزاد و فاصله بین دو لایه بیرونی در حین تغییر شکل ثابت فرض می‌شود. معادلات حاکم با در نظر گرفتن تئوری تیر اویلر- برنولی برای هر لایه و در نظر گرفتن جابجایی داخل صفحه علاوه بر جابجایی عرضی، به دست می‌آید. با حل معادلات حاکم به روش SSPHفرکانس‌های اول و دوم محاسبه شده است. این نتایج با نتایج شبیه­سازی دینامیک مولکولی و نتایج به دست آمده از حل عددی معادله حاکم به روش GDQ که از مراجع دیگر استخراج شده است، مقایسه شده است و دقّت و درستی روش حل بدون مش SSPH نشان داده شده است.
 

کلیدواژه‌ها


عنوان مقاله [English]

Application of SSPH Method in Free Vibration Analysis of Graphene

نویسندگان [English]

  • Kourosh Mohammadi
  • Hasan Shokrollahi
Department of Engineering, Kharazmi University
چکیده [English]

In this paper, the symmetric smoothed particle hydrodynamics (SSPH) as a meshless method is explained in detail. Free vibration analysis of bilayer graphenes with interlayer shear effect is modeled. The bilayer graphene is modeled as a sandwich beam with free-clamp end condition. To obtain the governing equations, each graphene layer is modeled based on the Euler-Bernoulli theory and in-plane displacements are also considered in addition to the transverse displacement. It is also assumed that the graphene layers do not have relative displacement during vibration. The results obtained by the sandwich beam model solved by SSPH method, include the first two natural frequencies of the bilayer graphenes. These results are validated by the molecular dynamic and compared with the GDQM results reported in the literature.

کلیدواژه‌ها [English]

  • Meshless method
  • Bilayer Graphene
  • Free Vibration
  • Sandwich Beam Model
  • SSPH Method
[1] G.-R. Liu, Mesh free methods: moving beyond the finite element method. CRC press, 2002.
[2] R. Vignjevic and J. Campbell, "Review of development of the smooth particle hydrodynamics (SPH) method," in Predictive modeling of dynamic processes: Springer, 2009, pp. 367-396.
[3] L. B. Lucy, "A numerical approach to the testing of the fission hypothesis," The astronomical journal, vol. 82, pp. 1013-1024, 1977.
[4] R. A. Gingold and J. J. Monaghan, "Smoothed particle hydrodynamics: theory and application to non-spherical stars," Monthly notices of the royal astronomical society, vol. 181, no. 3, pp. 375-389, 1977.
[5] J. J. Monaghan and J. C. Lattanzio, "A refined particle method for astrophysical problems," Astronomy and astrophysics, vol. 149, pp. 135-143, 1985.
[6] D. S. Balsara, "Von Neumann stability analysis of smoothed particle hydrodynamics—Suggestions for optimal algorithms," Journal of Computational Physics, vol. 121, no. 2, pp. 357-372, 1995.
[7] M. Liu, G. Liu, and K. Lam, "Constructing smoothing functions in smoothed particle hydrodynamics with applications," Journal of Computational and applied Mathematics, vol. 155, no. 2, pp. 263-284, 2003.
[8] W. K. Liu, S. Jun, and Y. F. Zhang, "Reproducing kernel particle methods," International journal for numerical methods in fluids, vol. 20, no. 8‐9, pp. 1081-1106, 1995.
[9] L. D. Libersky and A. G. Petschek, "Smooth particle hydrodynamics with strength of materials," in Advances in the free-Lagrange method including contributions on adaptive gridding and the smooth particle hydrodynamics method: Springer, 1991, pp. 248-257.
[10] L. D. Libersky, A. G. Petschek, T. C. Carney, J. R. Hipp, and F. A. Allahdadi, "High strain Lagrangian hydrodynamics: a three-dimensional SPH code for dynamic material response," Journal of computational physics, vol. 109, no. 1, pp. 67-75, 1993.
[11] G. Zhang and R. Batra, "Symmetric smoothed particle hydrodynamics (SSPH) method and its application to elastic problems," Computational mechanics, vol. 43, no. 3, pp. 321-340, 2009.
[12] Y. Liu, Z. Xu, and Q. Zheng, "The interlayer shear effect on graphene multilayer resonators," Journal of the Mechanics and Physics of Solids, vol. 59, no. 8, pp. 1613-1622, 2011.
[13] R. Nazemnezhad and H. Shokrollahi, "Free vibration analysis of bilayer graphenes with interlayer shear effect," Modares Mechanical Engineering, vol. 14, no. 7, pp. 131-138, 2014.
[14] P. Liu and Y. Zhang, "Temperature-dependent bending rigidity of graphene," Applied Physics Letters, vol. 94, no. 23, p. 231912, 2009.
[15] J. W. Kang and S. Lee, "Molecular dynamics study on the bending rigidity of graphene nanoribbons," Computational Materials Science, vol. 74, pp. 107-113, 2013.