تحلیل خمش و ارتعاش آزاد نانو ورق مدرج تابعی با استفاده از نظریه ورق مرتبه بالای مثلثاتی

نوع مقاله : مقاله پژوهشی

نویسندگان

دانشکده مهندسی مکانیک، دانشگاه صنعتی شیراز، شیراز، ایران

چکیده

در این مقاله خمش و ارتعاش آزاد نانو ورق مدرج تابعی با استفاده از یک نظریۀ ورق مرتبه بالای مثلثاتی جدید بررسی شده است. معادلات حاکمه با استفاده از اصل همیلتون استخراج گردیده و سپس حل دقیق خمش و ارتعاش آزاد نانو ورق مستطیلی با شرط مرزی ساده به کمک روش ناویر به دست آمده است. همچنین ازنظریۀ غیرمحلی برای لحاظ اثرات اندازه استفاده شده است. خواص مکانیکی نانو ورق مدرج تابعی با تابع توانی در راستای ضخامت تغییر می‌کند. به منظور تأیید دقت نظریۀ ارائه شده، نتایج حاصل از حل حاضر با نتایج موجود مقایسه شده است و مطابقت بسیار خوبی حاصل گردیده است. همچنین، اثرات نسبت طول به ضخامت، نسبت ابعادی درون صفحه ای و پارامتر غیرمحلی روی رفتار خمشی و ارتعاش آزاد نانو ورق بررسی شده است. نتایج نشان می‌دهد که لحاظ پارامتر غیرمحلی در معادلۀ حاکمه یا افزایش مقدار شاخص توانی، باعث کاهش بسامد طبیعی و افزایش خیز نانو ورق می‌شود و به عبارت دیگر موجب کاهش سفتی نانو ورق می‌گردد. در ضمن با افزایش طول نانو ورق یا نسبت ابعادی، تأثیر پارامتر غیرمحلی و اثرات اندازه کاهش می‌یابد. همچنین، نظریه ارائه شده علاوه بر ارائه جوا بهای دقیق خمش و ارتعاش آزاد برای نانو ورق ضخیم و نسبتاً ضخیم با هزینه محاسباتی کم، توزیع سهمیوار تنش برشی درون ضخامت را پیش بینی می‌کند.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Bending and Free Vibration Analysis of Functionally Graded Nano-plate Using Trigonometric Higher-Order Plate Theory

نویسندگان [English]

  • A. Azizi
  • A. Setoodeh
Department of Mechanical and Aerospace Engineering, Shiraz University of Technology, Shiraz, Iran
چکیده [English]

In this paper, bending and free vibration analyses of functionally graded nano-plates are investigated using a new trigonometric higher-order plate theory. The governing equations are developed by employing Hamilton’s principle and then a Navier-type analytical solution for bending and free vibration of simply supported rectangular FG nano-plates is obtained. Furthermore, the nonlocal theory of elasticity is used to take into account the small scale effects. The mechanical properties of the functionally graded nano-plates are assumed to vary by a power law function through the thickness. In order to confirm the accuracy of the present theory, the obtained results from the present solution are compared with the existed results, and a very good agreement is achieved. Moreover, the effects of length-to-thickness ratio, aspect ratio and nonlocal parameter on the bending and free vibration solutions are investigated. The results demonstrate that the inclusion of nonlocal parameter in governing equations or increasing the power index, leads to reduction of the natural frequency and increasing of the deflection and in another word the nano-plate stiffness is reduced. Also, the impact of the nonlocal parameter and size effects is reduced by increasing the length of the nano-plate or aspect ratio. Furthermore, the present theory not only provides exact solution for the bending and free vibration of thick and moderately thick functionally graded nano-plates with minimum computational cost, but also exhibits the parabolic distribution of the shear stress through the thickness.

کلیدواژه‌ها [English]

  • Functionally graded nano-plate
  • Trigonometric higher-order theory
  • Nonlocal theory
  • bending
  • free vibration
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