ارائه یک فرمول بندی المان محدود C1 برای مدل میکرو ورق میندلین-رایسنر

نوع مقاله : مقاله پژوهشی

نویسندگان

دانشکده مهندسی مکانیک، دانشگاه گیلان، رشت، ایران

چکیده

در این مقاله، فرمولبندی المان محدود میکرو ورق میندلین-رایسنر بر اساس نظریه الاستیسیته گرادیان کرنش ارائه می گردد. با توجه به وجود مشتقات مرتبه بالا، جهت ارائه یک فرمولبندی المان محدود استاندارد، باید از توابع درونیابی با پیوستگی  C1 استفاده نمود. براین اساس، شکل کلی ماتریس سفتی و بردار نیروی المان میکرو ورق براساس اصل کار مجازی استخراج شده و سپس فرمولبندی بر روی یک المان چهار ضلعی چهار گرهی با 36درجه آزادی اعمال می گردد. تغییر شکل ورق های مستطیلی میندلین-رایسنر گرادیانی تحت فشار یکنواخت با سه دسته شرایط مرزی مختلف مورد مطالعه قرار می گیرد. بتدا میکرو ورق با لبه های ساده بررسی شده و حل المان محدود گرادیانی بدست آمده با حل دقیق میکرو ورق براساس این نظریه مقایسه می گردد، که تطابق بسیار نزدیکی بین پاسخ های این دو روش مشاهده می شود. سپس میکرو ورق با چهار لبه گیردار و نیز سه لبه ساده و یک لبه آزاد بررسی می گردند. در هر دو مورد، پاسخی همگرا با ریز نمودن اندازه المانها بدست می آید. در صورت نازک بودن ورق و نیز بزرگ بودن نسبت ضخامت به پارامتر مقیاس طول ماده، نتایج بدست آمده در تحلیل المان محدود بر مبنای نظریه گرادیان کرنش با حل ورق کرشهف بر مبنای الاستیسیتهی کلاسیک منطبق می گردند.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

A C1 Finite Element Formulation for Mindlin-Reissner Microplate Model

نویسندگان [English]

  • F. Dadgar-Rad
  • S.R Ashraf
  • A. Imani
Department of Mechanical Engineering, University of Guilan, Rasht, Iran
چکیده [English]

In this paper, a C1 finite element (FE) formulation of Mindlin-Reissner microplate based on strain gradient elasticity theory is developed. The general form of the stiffness matrix and force vector of the microplate element is firstly extracted, and then specialized on a four-node quadrilateral element with 36 degrees of freedom. Deformation of rectangular microplates with simply-supported edges, clamped edges, and three edges simply-supported and the fourth edge free, and under uniform external pressure is then studied. For the case of microplate with simply-supported boundaries, comparison between the FE and the corresponding exact solution is made, which shows extremely close results. For the next two examples, a convergent solution by means of mesh refinement is obtained. Moreover, for the case of thin plates and for large values of the thickness-to-material length ratio, the results of gradient-based FE analysis are coincident with those of the Kirchhoff plate model based on classical elasticity. Numerical simulations show that the introduced element is able to capture the size effect phenomenon at micron scale. When the plate thickness is in the order of the material length parameter, the value of deflection is lower than that predicted by the models based on classical elasticity.

کلیدواژه‌ها [English]

  • Strain gradient elasticity theory
  • finite element method
  • Mindlin-Reissner microplate
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