Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams Bekir Akgöz, Ömer Civalek ⇑ Akdeniz University, Civil Engineering Department, Division of Mechanics, Antalya, Turkey article info Article history: Available online 12 January 2011 This paper is dedicated to Professor E.S. Suhubi on the occasion of his 76th birthday. Keywords: Strain gradient elasticity Micro-beams Buckling Size effect Modified couple stress Bernoulli–Euler beam abstract A class of higher-order continuum theories, such as modified couple stress, nonlocal elas- ticity, micropolar elasticity (Cosserat theory) and strain gradient elasticity has been recently employed to the mechanical modeling of micro- and nano-sized structures. In this article, however, we address stability problem of micro-sized beam based on the strain gra- dient elasticity and couple stress theories, firstly. Analytical solution of stability problem for axially loaded nano-sized beams based on strain gradient elasticity and modified couple stress theories are presented. Bernoulli–Euler beam theory is used for modeling. By using the variational principle, the governing equations for buckling and related boundary con- ditions are obtained in conjunctions with the strain gradient elasticity. Both end simply supported and cantilever boundary conditions are considered. The size effect on the critical buckling load is investigated. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction Micro- and nano-beams have been widely used in nano- and micro-sized systems and devices such as biosensors, nano- wires, atomic force microscope, microactuators, nano-probes, micro-electromechanical, ultra thin films and nano-electrome- chanical systems (Hung & Senturia, 1999; Li, Bhushan, Takashima, Baek, & Kim, 2003; Moser & Gijs, 2007). In these applications, it is observed that the size effect has a major role on static and dynamic deformation behavior of material and cannot be negligible (McFarland & Colton, 2005; Senturia, 2001). In microstructures, the size effect cannot be interpreted implicitly by beam models based on classical (macro) elasticity theories due to lack of material length scale parameters. Then, higher order continuum (nonlocal) theories, which contain additional material length scale parameters besides the classical material constants (Lame) have been proposed to predict the size dependence of these nano/micro-structures. Mostly generally known higher order theories are the micropolar (Cosserat) elasticity, nonlocal theory of Eringen, strain gra- dient elasticity and couple stress theories. In the theory of micropolar elasticity by Cosserat brothers (Cosserat & Cosserat, 1909) additional rotational degrees of freedom at each material point in the body were considered. The classical couple stress theory is one of the other higher order continuum theories which contain two additional material length scale param- eters besides the classical constants for an isotropic elastic material, elaborated by Mindlin and Tiersten (1962), Toupin (1962), Koiter (1964). After the study proposed by Koiter (1964) some other researchers have made some development on this theory (Eringen, 1983; Eringen & Suhubi, 1964a, 1964b; Mindlin, 1964, 1965; Mindlin & Eshel, 1968; Toupin, 1964; Vardoulakis, Exadaktylos, & Kourkoulis, 1998). Recently, a modified couple stress theory was proposed by Yang, Chong, Lam, and Tong (2002) which contain only one additional material length scale parameter in addition to the classical material constants. Also, the couple stress tensor is symmetric in this theory. Both the strain gradient elasticity and couple 0020-7225/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2010.12.009 ⇑ Corresponding author. Tel.: +90 242 3106319. E-mail address: civalek@yahoo.com (Ö. Civalek). International Journal of Engineering Science 49 (2011) 1268–1280 Contents lists available at ScienceDirect International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci