Issue 48

Y. Khalfi et alii, Frattura ed Integrità Strutturale, 48 (2019) 208-221; DOI: 10.3221/IGF-ESIS.48.22 209 I NTRODUCTION omposite materials have important advantages over traditional materials. They bring many functional advantages: lightness, mechanical and chemical resistance, reduced maintenance, freedom of forms. They make it possible to increase the lifespan of certain equipment thanks to their mechanical and chemical properties. They contribute to the reinforcement of safety thanks to a better resistance to shocks and fire. They offer better thermal or sound insulation and, for some of them, good electrical insulation. They also enrich the design possibilities by lightening structures and making complex shapes, able to fulfill several functions. In each of the application markets (automotive, building, electricity, industrial equipment, etc.), these remarkable performances are at the origin of innovative technological solutions [1]. They constitute one of the most advanced class of materials whose popularity in industrial applications keeps growing exponentially [2]. Their advent has been aided by the development of new processing methods, theoretical approaches of homogenization [3, 4] and numerical simulations of heterogeneous materials [5]. This class of materials is commonly divided into three categories [6] : (i) fibrous composites consisting of continuous fibers embedded in a matrix, (ii)laminated composites consisting of various stacked layers, and (iii) particle-reinforced composites composed of particles in a matrix. The buckling of rectangular plates has been a subject of study in solid mechanics for more than a century. Many exact solutions for isotropic and orthotropic plates have been developed, most of them can be found in Timoshenko and Woinowsky-Krieger [7], Timoshenko and Gere [8], Bank and Jin [9], Kang and Leissa [10], Aydogdu and Ece [11], and Hwang and Lee [12]. In company with studies of buckling behavior of plate, many plate theories have been developed. The simplest one is the classical plate theory (CPT) which neglects the transverse normal and shear stresses. This theory is not appropriate for the thick and orthotropic plate with high degree of modulus ratio. In order to overcome this limitation, the shear deformable theory which takes account of transverse shear effects is recommended. The Reissner [13] and Mindlin [14] theories are known as the first-order shear deformation theory (FSDT), and account for the transverse shear effects by the way of linear variation of in-plane displacements through the thickness. However, these models do not satisfy the zero traction boundary conditions on the top and bottom faces of the plate, and need to use the shear correction factor to satisfy the constitutive relations for transverse shear stresses and shear strains. For these reasons, many higher-order theories have been developed to improve in FSDT such as Levinson [15] and Reddy [16]. Shimpi and Patel [17] presented a four variable refined plate theory (RPT) for orthotropic plates.This theory which looks like higher-orde theory uses only four unknown functions in order to derive two governing equations for orthotropic plates. The most interesting feature of this theory is that it does not require shear correction factor, and has strong similarities with the CPT in some aspects such as governing equation, boundary conditions and moment expressions. The accuracy of this theory has been demonstrated for static bending and free vibration behaviors of plates by Shimpi and Patel [17], therefore, it seems to be important to extend this theory to the static buckling behavior. In this paper, the four variable RPT developed by Shimpi and Patel [17] has been extended to the buckling behavior of isotropic and orthotropic plate resting on two-parameter Pasternak’s foundations subjected to the in-plane loading. Using the Navier method, the closed-form solutions have been obtained. Numerical examples involving side-to-thickness ratio, effects of the foundation parameters and modulus ratio are presented to illustrate the accuracy of the present theory in predicting the critical buckling load of isotropic and orthotropic plates. The numerical results obtained by the present theory are compared with solutions of classical theory (CPT) and solutions of first order shear deformation theory (FSDT) and high order shear theory (HSDT). M ATHEMATICAL F ORMULATION onsider a rectangular composite plate of thickness h , length a and width b , referred to the rectangular Cartesian coordinate system ( x, y, z ), as shown in Fig 1. Since in this type of plates there is material symmetry with respect to the median plane (the origin of the coordinate system is appropriately chosen in the direction of the thickness of the composite plate so that it will be confused with the neutral surface) the equations of membranes and bending will be decoupled and therefore equilibrium equations [18]. Based on the refined theory of shear deformation [19], the displacement field can be written as: 0 ( , , ) ( , ) ( )        b s w w u x y z u x y z f z x x (1a) 0 ( , , ) ( , ) ( )        b s w w v x y z v x y z f z y y (1b) C C