Influence of Winkler-Pasternak Foundation on the Vibrational Behavior of Plates and Shells Reinforced by Agglomerated Carbon Nanotubes
This paper aims to investigate the effect of the Winkler-Pasternak elastic foundation on the natural frequencies of Carbon Nanotube (CNT)-reinforced laminated composite plates and shells. The micromechanics of reinforcing CNT particles are described by a two-parameter agglomeration model. CNTs are gradually distributed along the thickness direction according to various functionally graded laws. Elastic foundations are modeled according to the Winkler-Pasternak theory. The theoretical model
... oretical model considers several Higher-order Shear Deformation Theories (HSDTs) based on the so-called Carrera Unified Formulation (CUF). The theory behind CNTs is explained in detail. The theoretical model presented is solved numerically by means of the Generalized Differential Quadrature (GDQ) method. Several parametric studies are conducted, and their results are discussed. based on the Eshelby-Mori-Tanaka scheme for granular composite materials  , and mechanical properties are obtained. This scheme uses the so-called Hill's elastic moduli [34, 35] to describe the constitutive relations of the CNT particles. For the sake of completeness, some examples concerning the agglomeration of CNTs can be found in the papers    . The present paper aims to use this approach to study the effect of agglomeration on the natural frequencies of functionally graded carbon nanotube-reinforced laminated composite plates and shells resting on the elastic foundation. Although the high structural performance of plates and shells made from conventional composites has been proved by a number of papers                   , by employing CNTs as a reinforcing phase, their performance can be improved even further. A gradual variation of the volume fraction of the CNT particles trough the thickness of the composite has been employed, which is characteristic to functionally graded materials (FGMs). FGMs are a recent class of composite materials designed to deal with problems of stress concentration and mechanical discontinuity . Therefore, the term Functionally Graded Carbon Nanotubes (FG-CNTs) was introduced to refer on this type of CNT-reinforced composite. For the sake of completeness, it should be mentioned that several papers concerning structural models suitable for the mechanical analysis of these kinds of structure have been published recently          . In particular, the gradient elasticity theory was proposed by Barretta et al.  to this end. Alternatively, a nonlocal model can be used for this purpose, as proved in the papers by Romano and Barretta , Romano et al. , Marotti de Sciarra and Barretta , and Apuzzo et al. . To capture the proper mechanical behavior of these structures, adequate structural models must be considered. The use of classical shell theories may result in inaccurate results; therefore, the use of Higher-order Shear Deformation Theories (HSDTs) is required. Recent developments in the area of HSDTs are found in the works by Carrera      , introducing the so-called Carrera Unified Formulation (CUF). This formulation is explained in detail in the books by Tornabene et al. [99, 100] . CUF represents one of the most efficient and complete approaches when studying the mechanical behavior of multilayered composite beams, plates, and shells           . In this paper, various HSDTs based on the CUF Equivalent Single Layer (ESL) approach are employed to investigate the effect of agglomeration on the natural frequencies of FG-CNT-reinforced laminated composite plates and shells resting on the elastic foundation. The elastic foundation is modeled according to the Winkler-Pasternak theory. Papers describing this type of linear elastic foundation model can be found in                   . As far as nonlinear analyses are concerned, the reader can find further details in the works        . Due to its complexity, the problem is solved numerically by the means of the Generalized Differential Quadrature (GDQ) method. The GDQ is an accurate, reliable, and stable numerical technique developed by Shu in the nineties  . This numerical technique is described in detail in the review paper by Tornabene et al.  . Further details concerning this numerical approach, as well as several numerical applications, can be found in the papers    . Finally, it should be mentioned that the present approach was implemented in MATLAB code  .