نوع مقاله: مقاله پژوهشی

نویسندگان

1 کارشناس ارشد، دانشکده مهندسی مکانیک، دانشگاه علم و صنعت ایران، تهران، ایران

2 استاد، دانشکده مهندسی مکانیک، دانشگاه علم و صنعت ایران، تهران، ایران

چکیده

در این مقاله سعی بر این است که بر اساس یک روند نظام‌مند در حوزه‌ی تئوری الاستیسیته، برای مسئله‌ی مقدار مرزی المان حجمی معرف کامپوزیت الیاف بلند تحت بارگذاری کشش یکنواخت، یک حل بسته‌ی دقیق ارایه شود. برای این منظور المان حجمی معرف کامپوزیت‌های الیاف بلند در نظر گرفته شد که متشکل از دو استوانه‌ی هم‌مرکز با استوانه‌ی درونی از جنس الیاف و استوانه‌ی بیرونی از جنس ماتریس است. بدین ترتیب مسئله‌ی مورد بررسی در این پژوهش در حوزه‌ی مسائل متقارن محوری تئوری الاستیسیته قرار می‌گیرد. با بهره‌گیری از تابع لاو برای مسائل متقارن محوری، دستگاه معادلات تعادل ناویر برای مولفه‌های میدان جابجایی تبدیل به یک معادله‌ی بای‌هارمونیک بر حسب تابع لاو می‌شود. سپس یک عبارت جامع برای حل‌های جداپذیر معادله‌ی بای‌هارمونیک در مختصات استوانه‌ای ارائه می‌شود. در گام بعدی، عبارات مناسب از حل‌های جداپذیر با الهام از شرایط مرزی و فیزیک مسئله انتخاب شده و یک حل پیشنهادی استخراج می‌شود. در گام نهایی ثوابت مجهول موجود در حل پیشنهادی توسط شرایط مرزی به‌دست می‌آیند. بدین ترتیب حل دقیق به‌دست آمده قادر به ارضای تمامی معادلات میدانی تئوری الاستیسیته است. با استفاده از این حل توزیع دقیق مولفه‌های میدان تنش و جابجایی در المان حجمی معرف مشخص شده و برخی نتایج عددی برای آن ارایه می‌گردد. بر اساس این حل دقیق عبارتی برای مدول الاستیک طولی معادل کامپوزیت الیاف بلند به‌دست می‌آید و با رابطه‌ی معروف قانون اختلاط مقایسه می‌شود. این مقایسه نشان می‌دهد که قانون اختلاط می‌تواند تقریب مهندسی خوبی برای محاسبه‌ی مدول الاستیک طولی معادل باشد.

کلیدواژه‌ها

عنوان مقاله [English]

Exact micromechanical stress analysis of long fiber composites under uniform tensile loading

نویسندگان [English]

  • Hosein Rahnama 1
  • Mahmood Mehrdad Shokrieh 2

1 School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran

2 Composite Research Lab., Center of Excellence in Experimental Solid Mechanics, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran

چکیده [English]

In this paper, based on a systematic procedure in the field of theory of elasticity, it is tried to establish an exact closed form solution for the problem of a long-fiber composite representative volume element (RVE) under uniform tension loading condition. To this end, the RVE of long fiber composites is considered. The RVE is composed of two concentric cylinders where the inner and outer ones are of fiber and matrix materials, respectively. Consequently, this study deals with an axially symmetric problem of the theory of elasticity. Using the Love function for axisymmetric problems, the Navier equilibrium equations for displacement components are converted to a single biharmonic equation in terms of the Love function. Then, a general expression for the separable solutions of biharmonic equation in cylindrical coordinates is derived. Next, according to the boundary conditions and physical interpretations, appropriate terms from separable solutions are chosen and a solution is suggested. In the final step, the unknown constants in the suggested solution are computed by using the boundary conditions. The obtained exact solution satisfies the entire field equations of theory of elasticity. Using this solution, the exact distribution of stress and displacement field components of the RVE is determined and related numerical results are presented. Based on the exact solution, an expression for the equivalent longitudinal modulus of composite is obtained and compared with the well-known rule of mixture formula. This comparison reveals that the rule of mixture can be a good engineering approximation for the equivalent longitudinal modulus.

کلیدواژه‌ها [English]

  • Exact closed form solution
  • Elasticity
  • Long-fiber composites
  • Representative volume element
  • Biharmonic equation

 

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