IJE TRANSACTIONS A: Basics Vol. 28, No. 10 (October 2015) 1533-1542   

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S. Dastjerdi and M. Jabbarzadeh
( Received: August 15, 2015 – Accepted: October 16, 2015 )

Abstract    In this paper, it is tried to find an approximate single layer equivalent for multi-layer graphene sheets based on third order non-local elasticity theory. The plates are embedded in two parameter Winkler-Pasternak elastic foundation, and also the thermal effects are considered. A uniform transverse load is imposed on the plates. Applying the non-local theory of Eringen based on third order shear deformation theory and considering the van der Waals interaction between the layers, the governing equations are derived for a multi-layer graphene sheet. The governing equations for single layer graphene sheet are obtained by eliminating the van der Waals interaction. In this study, two different methods are applied to solve the governing equations. First, the results are obtained applying the differential quadrature method (DQM), which is a numerical method, and then a new semi-analytical polynomial method (SAPM) is presented. The results from DQM and SAPM are compared and it is concluded that the SAPM results are satisfactorily accurate in comparison with DQM. Since analyzing a multi-layer graphene sheet needs a time-consuming computational process, it is investigated to find an appropriate thickness for a single layer sheet to equalize the maximum deflections of multi-layer and single layer sheets. It is concluded that by considering a constant value of the van der Waal interaction between the layers, the maximum deflections of multi and single layer sheets are equal in a specific thickness of the single layer sheet.


Keywords    Single and multi-layer graphene sheet, Non-local elasticity theory of Eringen, Differential Quadrature method (DQM), Semi-Analytical Polynomial Method (SAPM), Winkler-Pasternak elastic foundation, Thermal environment


چکیده    در این پژوهش تلاش شده است ورق چند لایه گرافن با یک ورق تک لایه بر اساس تئوری الاستیسیته غیرموضعی با به کار گیری تئوری مرتبه سوم تغییر شکل برشی معادل¬سازی شود. ورق مورد بررسی بر روی پایه الاستیک وینکلر-پسترناک و در محیط حرارتی تحت بار عرضی یکنواخت قرار گرفته است. روابط تعادل برای ورق چند لایه گرافن بر اساس تئوری الاستیسیته غیرموضعی مرتبه سوم برشی و با در نظر گرفتن نیروی واندروالس بین لایه¬ها به دست آمده است. با نادیده گرفتن نیروی واندروالس، روابط حاکم برای ورق تک لایه حاصل می¬شوند. در این تحقیق از دو روش مختلف برای حل معادلات حاکم استفاده شده است. ابتدا نتایج با استفاده از روش عددی مربعات دیفرانسیلی(DQM) به دست آمده و سپس یک روش نیمه تحلیلی (SAPM) ارائه شده است. نتایج به دست آمده از دو روش با یکدیگر مقایسه و ملاحظه گردید نتایج روش SAPM با روابط ساده¬تر نسبت به DQM از دقت و همگرایی مناسبی برخوردار می­باشد. از آنجا که تحلیل ورق¬های چندلایه، از حجم محاسبات گسترده برخوردار بوده و در نتیجه زمان بسیار طولانی را صرف می¬کند، در این تحقیق با تغییر ضخامت ورق تک لایه، یک ضخامت معادل محاسبه گردیده است، بطوری که خیز بیشینه دو ورق چند و تک لایه برابر هم شوند. نتایج حاصل نشان می¬دهد که با ثابت انگاشتن نیروی واندروالس بین لایه¬ها، خیز بیشینه ورق¬های چند و تک لایه در یک ضخامت مشخص برای ورق تک لایه با یکدیگر برابر می¬باشند.


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