7 th GRACM International Congress on Computational Mechanics Athens, 30 June – 2 July 2011 A MICROSTRUCTURE-DEPENDENT ORTHOTROPIC PLATE MODEL BASED ON A MODIFIED COUPLE STRESS THEORY George C. Tsiatas 1 and Aristophanes J. Yiotis 2 1 School of Civil Engineering National Technical University of Athens Athens, GR-15773, Greece e-mail: gtsiatas@gmail.com 2 School of Civil Engineering National Technical University of Athens Athens, GR-15773, Greece e-mail: fgiotis@otenet.gr Dedicated to Professor Emeritus John T. Katsikadelis on the occasion of his 72th birthday. Keywords: Couple Stress Elasticity, Gradient Elasticity, Orthotropic Plate, Analog Equation Method, Meshless Methods. Abstract. In this paper a modified couple stress model containing only one material length scale parameter is developed for the static analysis of orthotropic micro-plates with arbitrary shape. The proposed model is capable of handling plates with complex geometries and boundary conditions. From a variational procedure the governing equilibrium equation of the micro-plate and the most general boundary conditions are derived, in terms of the deflection, using the principle of minimum potential energy. The resulting boundary value problem is of the fourth order (instead of existing gradient theories which is of the sixth order) and it is solved using the Analog Equation Method (AEM), which is a boundary-type meshless method. Several plates of various shapes, aspect and Poisson’s ratios are analyzed to illustrate the applicability of the developed micro-plate model and to reveal the differences between the current model and the classical plate model. Moreover, useful conclusions are drawn from the micron-scale response of this new orthotropic plate model. 1 INTRODUCTION Since the classical continuum theory is inadequate to predict the behaviour of micron-scaled structures, which has been proven experimentally to be size dependent, the utilization of strain gradient (higher order) theories is inevitable. Although, these general theories encounter the physical problem, they contain additional constants which are difficult to determine even in their simplified form of only two constants. Thus, gradient elasticity theories involving only one additional material constant are very attractive. The couple stress theory is a special case of these higher-order theories in which the effects of the dilatation gradient and the deviatoric stretch gradient are assumed to be negligible. An analytic presentation of the aforementioned theories can be found in [1, 2, 3]. The work that has been reported on the subject is restricted only to the vibration and buckling problems of orthotropic nano-plates of graphene sheet. More specifically, Sakhaee-Pour [4] studied the elastic buckling problem of single-layered graphene sheet by an atomistic modelling approach, while Pradhan and Phadikar [5] carried out the vibration analysis of embedded multilayered graphene sheets and Murmu and Pradhan [6] solved the buckling problem of single-layered graphene sheet employing the nonlocal elasticity theory of Eringen [7]. In this work the simplified couple stress theory of Yang et al. [8] is developed for the static solution of orthotropic Kirchhoff micro-plates with arbitrary shape. Yang et al. [8] modifying the classical couple stress theory proposed a modified couple stress model in which only one material length parameter is needed to capture the size effect. This simplified couple stress theory is based on an additional equilibrium relation, which forces the couple stress tensor to be symmetric. So far it has been developed for the static bending [9] and free vibration [10] problems of a Bernoulli-Euler beam, for the static bending and free vibration problems of a Timoshenko beam [11] and for the solution of a simple shear problem [12] after the derivation of the boundary conditions and the governing differential equation of the theory in terms of the displacement. Moreover, Tsiatas [13] studied the static bending problem of Kirchhoff plates and Tsiatas and Katsikadelis [14] the Saint-Venant torsion problem of bars. The proposed model is capable to handle plates with complex geometries and boundary conditions. To the