IJE TRANSACTIONS A: Basics Vol. 23, No. 2 (April 2010) 153-168   

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H. Ashrafi and M. Farid
( Received: April 24, 2009 – Accepted in Revised Form: May 20, 2010 )

Abstract    The analysis of viscoelastic materials is one of the most important subjects in engineering structures. Several works have been so far made for the integral equation methods to viscoelastic problems. From the basic assumptions of viscoelastic constitutive equations and weighted residual techniques, a simple but effective Boundary Element (BE) formulation is developed for the Kelvin viscoelastic solid models. This formulation needs only Kelvin’s fundamental solution of isotropic elastostatics with material constants prescribed as explicit functions of time. It is able to solve the quasistatic problems with any load time-dependence and boundary conditions. A system of time-dependent equations is derived by imposing the convenient approximations and adopting the kinematical relations for strain rates. This approach avoids the use of relaxation functions and mathematical transformations. The main feature of the proposed formulation is the absence of domain discretizations, which simplifies the treatment of problems involving infinite domains. A computer code has been developed in the programming environment of MATLAB software. At the end of this paper, two numerical examples have been provided to validate this formulation.


Keywords    Viscoelastic Solids, Boundary Element Approach, Kelvin Solid Model


چکیده    تحلیل مسائل ویسکوالاستیک یکی از مهمترین موضوعات در سازه­های مهندسی می­باشد. تلاش­هایی تاکنون به منظور حل مسائل ویسکوالاستیک با رهیافت­های مختلف معادلات انتگرالی انجام گرفته است. در این مقاله، یک فرمول­بندی المان­های مرزی ساده ولی بسیار مؤثر با استفاده از فرضیه­های بنیادین معادلات متشکله ویسکوالاستیک و اصول پسماند وزنی برای مدل­های ویسکوالاستیک جامد کلوینی توسعه یافته است. این فرمول­بندی جدید تنها به جواب اساسی کلوین مورد استفاده در مسائل الاستواستاتیک ایزوتروپیک نیاز دارد که ثوابت مادی در آن به صورت توابع صریحی از زمان توصیف شده­اند. قابلیت اعمال هر نوع بارهای وابسته به زمان و شرایط مرزی در این رهیافت برای حل مسائل شبه استاتیک وجود دارد. مجموعه­ای از معادلات وابسته به زمان با اعمال فرضیات مناسب و تعمیم روابط سینماتیکی برای نرخ کرنش­ها استخراج شده است. در این رهیافت از توابع وارهیدگی یا خزشی و تبدیلات ریاضی پیچیده استفاده نمی­شود. مهمترین ویژگی فرمول­بندی ارائه شده در عدم نیاز به گسسته­سازی­های دامنه­ای است که آن را برای استفاده در مسائل با دامنه بینهایت نیز مطلوب ساخته است. یک کد کامپیوتری در محیط برنامه­نویسی نرم افزار ماتلب (MATLAB) توسعه یافته است. دو مثال عددی کاربردی به منظور معتبرسازی این فرمول­بندی در پایان مقاله ارائه شده اند.


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