ارايه‌ي يك حل دقيق تحليلي براي ارتعاشات صفحات مستطيلي نانو براساس نظريه‌ي مرتبه‌ي اول برشي ورق‌ها در فضاي غيرمحلي كشسان

AN EXACT SOLUTION FOR FREE VIBRATION OF NONLOCAL MINDLIN RECTANGULAR NANO-PLATES

در اين نوشتار، ارتعاشات آزاد صفحات مستطيلي نانو براساس نظريه‌ي مرتبه‌ي اول برشي ورق‌ها (ميندلين) در فضاي نظريه‌ي كشساني غيرمحلي با استفاده از يك حل دقيق تحليلي بررسي شده است. به‌منظور حل دقيق معادلات حركت، ابتدا معادلات بي‌بعد شده و با استفاده از يك تابع كمكي از يكديگر دي‌كوپله شده‌اند. نهايتاً، معادلات جديدي برحسب يك‌سري توابع پتانسيل به دست مي‌آيد كه به‌صورت دقيق و تحليلي قابل حل‌اند و به اين ترتيب پاسخ ارتعاشات آزاد و فركانس‌هاي طبيعي براي مسيله به دست مي‌آيد. در حل معادلات فوق، از شرايط مرزي لوي استفاده شده است. به‌منظور تاييد صحت روش حل حاضر، نتايج به دست آمده با ديگر روش‌هاي عددي و تقريبي مقايسه شده است. به‌منظور ارايه‌ي نتايج بيشتر، اثرات مودهاي مختلف فركانسي و پارامترهاي مختلف نانو ورق ــ شامل نسبت طول به عرض و نسبت ضخامت به طول ــ بر فركانس طبيعي نانوورق بررسي شده است.

Due to the rapid development of nanotechnology, nano-plates are used in MEMS or NEMS for their superior mechanical, thermal and electrical properties. The dynamic behavior of nano-plates used as thin film elements requires a two-dimensional nano-structure analysis. Hence, one must consider small scale effects in order to refine classical theories and derive the governing equations for these structures. The scale effects are accounted for by considering internal size as a material parameter. The local (classic) continuum theory neglects the effects of long-range load on the motion of the body, and long range inter atomic interactions. Therefore, the internal scale is neglected. Nonlocal linear theory, which has both features of lattice parameter and classical elasticity, could be considered a superior theory for modeling nano-materials. The nonlocal theory of Eringen is a well-known continuum mechanics theory to account for small scale effects by specifying stress at a reference point as a function of the strain field at every point in the body. Many papers dealing with the analysis of nano-structures have been published on this topic, but, in many of the papers, the solutions of the governing equation are based on numerical methods and approximate analytical methods, like the Navier type solution method. Hence, no exact closed-form solution is available in the literature for the free vibration analysis of nano-plates under various boundary conditions. In this paper, the exact analytical solution proposed by Hosseini-Hashemi et al. is employed to solve the governing equations of motion of a rectangular nano-plate for nonlocal Mindlin theory. To this end, equations of motion are derived via equations of momentum balance. Introducing a set of auxiliary and potential functions, the governing equations are decoupled for transverse vibration analysis. By transforming the displacement variables into known functions, the problem leads to a soluble form without any approximations. The solution of natural frequencies is obtained for Levy-type boundary conditions, and, in order to confirm the reliability of the method considered, the results are compared with those reported in the literature. Also, the effects of nonlocal parameters, aspect ratio, thickness to length ratios of the plate and different boundary conditions on vibration frequencies are investigated.