Abstract    In this paper, a size-dependent formulation for the Bernoulli-Euler beam is developed based on a new model of couple stress theory presented by Hadjesfandiari and Dargush. The constitutive equation obtained in this new model, consists of only one length scale parameter that is capable of capturing the micro-structural size effect in predicting the mechanical behavior of the structure. Having one length scale parameter is claimed to be an advantage of the model in comparison with the classical couple stress theory. The governing equations and boundary conditions of the Bernoulli-Euler beam are developed using the variational formulation and the Hamilton principle. The static bending and free vibration problems of a Bernoulli-Euler beam with various boundary conditions are solved. Numerical results demonstrate that the value of deflection predicted by the new model is lower than that of the classical theory. It is also found that natural frequencies obtained by the present couple stress model are higher than those predicted by the classical theory. The differences between results obtained by the present model and the classical theory become significant as the thickness of the beam gets close to the length scale parameter of the beam material.

Keywords    Bernoulli-Euler beam, couple stress theory, microstructural effect, static bending, free vibration

چکیده    در اين مقاله، با استفاده از مدل جديد کوپل تنش ارائه شده توسط Hadjesfandiari و Dargush، فرمول بندي وابسته به اثر اندازه ريزساختار ماده براي يک تير اويلر- برنولي توسعه داده شده است. روابط ساختاري بدست آمده در اين مدل جديد، شامل يک پارامتر مقياس طولي ماده است که اين تئوري را قادر به پيش بيني رفتار مکانيکي مواد ناشي از تاثير اندازه ريزساختار مي سازد. داشتن تنها يک پارامتر مقياس طولي ماده يکي از مزيت هاي اين مدل در مقايسه با تئوري کلاسيک کوپل تنش عنوان شده است. معادلات حاکم و شرايط مرزي تير اويلر- برنولي با استفاده از روش وردشي و اصل هميلتون بدست آمده و مسائل خمش استاتيکي و ارتعاش آزاد يک تير اويلر- برنولي با شرايط مرزي مختلف حل شده اند. نتايج عددي نشان مي دهند که مقدار خيز پيش بيني شده در مدل جديد کمتر از مقدار پيش بيني شده در تئوري الاستيسيته کلاسيک مي باشد. همچنين فرکانس هاي ارتعاش آزاد بدست آمده در اين تئوري بالاتر از تئوري کلاسيک مي باشد. تفاوت بين جوابهاي بدست آمده از تئوري کوپل تنش حاضر و تئوری کلاسيک با نزديک شدن مقدار ضخامت تير به پارامتر مقياس طولي ماده، زياد مي شود.

References      1.        Fleck, N., Muller, G., Ashby, M. and Hutchinson, J., "Strain gradient plasticity: Theory and experiment", Acta Metallurgica et Materialia,  Vol. 42, No. 2, (1994), 475-487. 2.        Ma, Q. and Clarke, D.R., "Size dependent hardness of silver single crystals", Journal of Materials Research,  Vol. 10, No. 04, (1995), 853-863. 3.        Stölken, J. and Evans, A., "A microbend test method for measuring the plasticity length scale", Acta Materialia,  Vol. 46, No. 14, (1998), 5109-5115. 4.        Chong, A. and Lam, D.C., "Strain gradient plasticity effect in indentation hardness of polymers", Journal of Materials Research,  Vol. 14, No. 10, (1999), 4103-4110. 5.        McFarland, A.W. and Colton, J.S., "Role of material microstructure in plate stiffness with relevance to microcantilever sensors", Journal of Micromechanics and Microengineering,  Vol. 15, No. 5, (2005), 1060. 6.        Mindlin, R. and Tiersten, H., "Effects of couple-stresses in linear elasticity", Archive for Rational Mechanics and Analysis,  Vol. 11, No. 1, (1962), 415-448. 7.        Toupin, R.A., "Elastic materials with couple-stresses", Archive for Rational Mechanics and Analysis,  Vol. 11, No. 1, (1962), 385-414. 8.        Koiter, W., "Couple stresses in the theory of elasticity, i and ii", in Nederl. Akad. Wetensch. Proc. Ser. B. Vol. 67, (1964), 17-29. 9.        Anthoine, A., "Effect of couple-stresses on the elastic bending of beams", International Journal of Solids and Structures,  Vol. 37, No. 7, (2000), 1003-1018. 10.     Zhou, S. and Li, Z., "Length scales in the static and dynamic torsion of a circular cylindrical micro-bar", Journal of Shandong University of Technology,  Vol. 31, No. 5, (2001), 401-407. 11.     Asghari, M., Kahrobaiyan, M., Rahaeifard, M. and Ahmadian, M., "Investigation of the size effects in timoshenko beams based on the couple stress theory", Archive of Applied Mechanics,  Vol. 81, No. 7, (2011), 863-874. 12.     Yang, F., Chong, A., Lam, D. and Tong, P., "Couple stress based strain gradient theory for elasticity", International Journal of Solids and Structures,  Vol. 39, No. 10, (2002), 2731-2743. 13.     Park, S. and Gao, X., "Bernoulli–euler beam model based on a modified couple stress theory", Journal of Micromechanics and Microengineering,  Vol. 16, No. 11, (2006), 2355. 14.     Kong, S., Zhou, S., Nie, Z. and Wang, K., "The size-dependent natural frequency of Bernoulli–Euler micro-beams", International Journal of Engineering Science,  Vol. 46, No. 5, (2008), 427-437. 15.     Asghari, M., Ahmadian, M., Kahrobaiyan, M. and Rahaeifard, M., "On the size-dependent behavior of functionally graded micro-beams", Materials & Design,  Vol. 31, No. 5, (2010), 2324-2329. 16.     Ma, H., Gao, X.-L. and Reddy, J., "A microstructure-dependent timoshenko beam model based on a modified couple stress theory", Journal of the Mechanics and Physics of Solids,  Vol. 56, No. 12, (2008), 3379-3391. 17.     Asghari, M., Rahaeifard, M., Kahrobaiyan, M. and Ahmadian, M., "The modified couple stress functionally graded timoshenko beam formulation", Materials & Design,  Vol. 32, No. 3, (2011), 1435-1443. 18.     Asghari, M., Kahrobaiyan, M. and Ahmadian, M., "A nonlinear timoshenko beam formulation based on the modified couple stress theory", International Journal of Engineering Science,  Vol. 48, No. 12, (2010), 1749-1761. 19.     Şimşek, M., Kocatürk, T. and Akbaş, Ş.D., "Static bending of a functionally graded microscale timoshenko beam based on the modified couple stress theory", Composite Structures,  Vol. 95, No., (2013), 740-747. 20.     Wanji, C., Chen, W. and Sze, K., "A model of composite laminated reddy beam based on a modified couple-stress theory", Composite Structures,  Vol. 94, No. 8, (2012), 2599-2609. 21.     Ke, L.-L., Wang, Y.-S., Yang, J. and Kitipornchai, S., "Nonlinear free vibration of size-dependent functionally graded microbeams", International Journal of Engineering Science,  Vol. 50, No. 1, (2012), 256-267. 22.     Wang, L., Xu, Y. and Ni, Q., "Size-dependent vibration analysis of three-dimensional cylindrical microbeams based on modified couple stress theory: A unified treatment", International Journal of Engineering Science,  Vol. 68, No., (2013), 1-10. 23.     Ghayesh, M.H., Farokhi, H. and Amabili, M., "Nonlinear dynamics of a microscale beam based on the modified couple stress theory", Composites Part B: Engineering,  Vol. 50, (2013), 318-324. 24.     Hadjesfandiari, A.R. and Dargush, G.F., "Couple stress theory for solids", International Journal of Solids and Structures,  Vol. 48, No. 18, (2011), 2496-2510. 25.     Şimşek, M., "Dynamic analysis of an embedded microbeam carrying a moving microparticle based on the modified couple stress theory", International Journal of Engineering Science,  Vol. 48, No. 12, (2010), 1721-1732. 26.     Rao, S.S., "Vibration of continuous systems, John Wiley & Sons,  (2007).