Shear buckling of single layer graphene sheets in hygrothermal environment resting on elastic foundation based on different nonlocal strain gradient theories

Highlights

• Nonlocal strain gradient theory is developed for shear buckling study of nanoplates.

• Various refined plate theories needless of any shear correction factor are used.

• Influence of softening stiffness and stiffness enhancement are considered by two small scale parameters.

• Uniform, linear and nonlinear temperature and moisture rises are included.

Abstract

The present paper is focused on the size-dependent shear buckling of nanoplates embedded in Winkler-Pasternak foundation and hygrothermal environment. Hence, the refined higher-order plate theories (Polynomial, Exponential, and Hyperbolic) needless of any shear correction factor are used in the formulations. The equations of motion are derived based on the mentioned theories in conjunction with the nonlocal strain gradient theory employing Hamilton's principle. The four unknown functions denoting the buckling load of plates are defined in a modal manner, and Navier solution method is used to find the shear buckling response. Results for the shear buckling and thermal buckling analysis of nanoplates are approved by existing literature to demonstrate the accuracy of present formulation and solution method. From our knowledge, it is the first time that the hygrothermal environment and also the nonlocal strain gradient theory are applied to study on shear buckling of nanoplates. Hence, the influence of nanoplate geometry, various hygrothermal conditions, elastic medium, nonlocal parameter and gradient parameter on the shear buckling load are obtained and discussed using different plate theories. The numerical results indicate that the shear buckling of nanoplate in the absence of strain gradient parameter is significantly affected by the temperature and moisture variations.

Introduction

Among solid carbon-based materials, graphene's due to their superiorities such as low weight to strength ratio, flexibility, large elastic modulus, in-plane stiffness, ultra-high strength, unique chemical, optical and electrical properties have intensely stimulated the interest of the scientific scholar's societies in chemistry, physics, and engineering. The mentioned reasons caused that graphene-based materials to be used in the nanoelectromechanical systems (NEMS) containing nanosensors, nano-optomechanical, and nanoactuators. Rectangular nanoplate (single layer graphene sheet) is one of the graphene-based materials that defined as a two-dimensional flat monolayer of the nanoscale size. To design plate structures correctly, understanding their buckling behavior is very vital. Hence, factors that affect the occurrence of initial buckling into that plates buckle including aspect ratio, load conditions (shear, tensile and compressive), material symmetry (isotropic, orthotropic, etc.), thermal or hygroscopic stress distributions, constitutive behavior (viscoelastic, nonlinear elastic, linear elastic, etc.), secondary buckling, support conditions, and initial imperfections (Bloom and Coffin, 2000). In order to analyze these parameters, buckling of microplates has been investigated extensively in the literature using local continuum theories (Kubiak, 2013, Timoshenko and Gere, 2009). On the other hand, these theories to predict the nanostructure behavior are not complete. Because, at the nanometer scale, the roles of size-dependent effects have been seen by costly molecular dynamic simulation and rigorous experimental studies. To fix this drawback, several successful continuum mechanics theories (such as the strain gradient elasticity theory (Lam et al., 2003), the nonlocal elasticity theory (Eringen and Edelen, 1972), and the nonlocal strain gradient theory (Askes and Aifantis, 2009)) have been proposed to consider the size effects. Unlike the classical elasticity theory, in the nonlocal elasticity theory stress at a special point in the domain estimates for not only the strain at that point but also the strains at all points in that domain. There are already theoretical investigations on the nonlocal elasticity theory for wave propagation (Chen et al., 2017) free vibration (Ansari et al., 2010, Bounouara et al., 2016, Ghorbanpour-Arani, 2015, Ghorbanpour-Arani and Shokravi, 2013, Mohammadimehr et al., 2014), bending (Ghorbanpour-Arani et al., 2013, Mohammadimehr and Mahmudian-Najafabadi, 2013, Shahsavari and Janghorban,), and (uniaxial, biaxial, shear and thermal) buckling (Ara et al.,, Mohammadi et al., 2014, Nami et al., 2015, Rahmani et al., 2016, Thai, 2012, Zenkour and Sobhy, 2013) of various nanostructures. In new studies (Bouafia et al., 2017), developed the nonlocal elasticity theory in conjunction with a quasi-3D theory for exact analysis on bending and vibration of nanobeams (Bounouara et al., 2016). analyzed the nonlocal frequency of functionally graded nanoplates resting on elastic foundation by a zeroth-order shear deformation theory. Nevertheless, the capacity of nonlocal continuum models for satisfying other size effects on the mechanical properties of nanoscale structures may exist some limited problems (Lim et al., 2015). For example, the nonlocal elasticity theory can only produce softening stiffness with rising the nonlocal parameter. However, by applying the nonlocal elasticity theory, the stiffness enhancement effects seen from the strain gradient elasticity theory (Lam et al., 2003) as well as experimental investigations (such as (Lin et al., 2013)) cannot be predicted well. To end this limitation (Challamel et al., 2009), calibrated the nonlocal parameter with Born–Kármán model of lattice dynamics. They recommended both dynamics and static gradient elasticity theories (nonlocal elasticity theory and strain gradient elasticity theory) can be merged with such a single theory. In continued, Askes and Aifantis employed nonlocal strain gradient theory for study the wave dispersion in carbon nanotubes (Askes and Aifantis, 2009) (Lim et al., 2015). proposed a higher-order nonlocal strain gradient theory relying on the thermodynamics framework. In that article, with comparing between concluded results by molecular dynamic simulation determined that nonlocal strain gradient models of nanobeam structures can accurately describe size-dependent phenomena, especially in wave propagation problems. Since then, the nonlocal strain gradient theory has been widely employed on wave propagation (Karami et al., 2017, Li et al., 2016a), vibration (Li et al., 2016b, Şimşek, 2016), buckling (Li and Hu, 2015) and bending (Liu et al., 2016, Nami and Janghorban, 2014, Xu et al., 2017) studies of nanostructures. Recently, buckling analysis of orthotropic nanoplates under uniform thermal compression loading using the differential quadrature method presented by (Farajpour et al., 2016) to indicate the influence of nonlocal and strain gradient parameters. Overall, according to Hooke's law, the nonlocal parameter is used on the Laplacian of stress on the left side of the constitutive equation, but the strain gradient parameter is considered on the right side of the equation on the Laplacian of strains.

For the elastic behaviors of structures based on Hooke's law, for decreasing the number of variables applied in the equilibrium equations (in comparison of higher-order shear deformation theories (HSDTs)) and capture the shear deformation effects on the lower and upper surfaces of plates (in comparison of classical plate theory (CPT)), a refined theory including only two variables was proposed by Shimpi (Shimpi and Patel, 2006a, Shimpi and Patel, 2006b) (so-called the refined-plate theory (RPT)) for examination on the mechanical behaviors of isotropic and orthotropic plates. Moreover, RPT does not have any shear correction factor which was required in applying the first-order shear deformation plate theory (FSDT). Then, the various models of RPT under the influence of different shape functions by dividing the transverse displacement into bending and shear components for plates were developed (Bourada et al., 2015, Neves et al., 2012) (Yahia et al., 2015). studied the wave behavior of FG plates using refined higher-order shear deformation theories (RHSDTs) with 4-unknown due to cubic, sinusoidal, hyperbolic, and exponential shear strain shape functions. A refined-trigonometric shear deformation theory (R-TSDT) was used for the thermoelastic bending response of FG sandwich plates by (Tounsi et al., 2013). Recently, due to the rapid progress of plate theories in order to reduce the time and effort of computation, the efficiency of some refined theories including only three unknowns was examined for behavior of FG plates (Houari et al., 2016, Tounsi et al., 2016). However, the classical continuum theories (CPT, HSDT, FSDT, and RHSDTs) neglect the influence of thickness stretching effect due to assuming constant transverse displacements through the thickness. Recently, the thickness stretching effect in FG plates using finite element approximations was investigated (Carrera et al., 2011) to achieve accurate results. The thickness stretching effect becomes very important for thick plates analysis and must be considered. In order to achieve the development in previous theories for measuring the shear deformation and thickness stretching effects together, the multiple quasi-3D theories, which defined on the basis of higher-order variations through the thickness for transverse displacements, were suggested, so far (see some of those in Refs (Bourada et al., 2015, Hebali et al., 2014, Thai and Kim, 2015).).

If we want to be realistic, all structures during the investigation are exposed in hygrothermal environment. This heat and moisture can be induced remaining extensional strains and stresses of that structure as well as can be distributed through the volume (Alzahrani et al., 2013). Accordingly, the stiffness and strength of structures are negatively influenced by the moisture and temperature of environs. Hence, the hygrothermal dependency of structures necessitates considering to study more accurately and carefully even in nanoscale. Thus, the performances of plate structures while subjected to heat and moisture effects have been the topic of analysis interest of many analysts. To this aim (Whitney and Ashton, 1971), employed the Ritz method to illustrate the effect of hygrothermal environment on the vibration, bending and buckling analysis of multilayer composite plates (Sreehari and Maiti, 2015). applied a finite element method for buckling and post-buckling analysis of laminated plates while subjected to linear and nonlinear hygrothermal loading via inverse hyperbolic shear deformation theory. Four distinct theories have applied to investigate the bending response of EGM plate exposed to a hygrothermal environment by (Zenkour, 2013). More recently, due to the Importance of size-dependent effects in the modeling of nanostructures, some nonlocal elasticity theories in conjunction with hygrothermal effect are offered and used in the examination of plate structures. A hygro-thermo-mechanical bending response of embedded single layer graphene sheet in elastic foundation via refined sinusoidal shear deformation plate theory was presented by (Alzahrani et al., 2013). In that research, uniform, linear and nonlinear changes of moisture concentrations and temperature field in the presence of nonlocal parameter were studied. The influence of nonlocal parameter on the vibration response of orthotropic nanoplates while surrounding by Winkler-Pasternak elastic medium in a hygrothermal environment was demonstrated based on nonlocal a refined plate theory by (Sobhy, 2016). Further (Sobhy, 2015), employed the Levy type solution for bending response of orthotropic nanoplates due to hygrothermal effects for various boundary conditions. (Shahsavari et al., 2017), analyzed the dynamic behavior of viscoelastic nanoplates in hygrothermal environment analytically. On basis of their work, the influence of hygrothermal environment for increasing the dynamic transverse deflection is higher than the thermal environment, especially for large values of the nonlocal parameter. It can be understood from the mentioned literature studied that reports works about the buckling behavior of nanoplates while exposed to the hygrothermal environment with considering influences of softening stiffness and stiffness enhancement still limited.

With respond to open literature, due to the best of our knowledge, up to now, none analytical study has not been tried to propose a nonlocal strain gradient model for shear buckling analysis of nanoplates. Also, in view of all that has been reviewed so far, very few studies have tried to provide the analytical solution for nanoplates embedded in an elastic medium including hygrothermal effect. To eliminate mentioned limitations, in the present article, a refined higher-order plate theory including exponential, hyperbolic and parabolic functions is employed to study the influences of temperature and moisture variations due to uniform, linear and nonlinear hygrothermal changes on shear buckling response of nanoplates while surrounded by a Winkler-Pasternak foundation. The nonlocal strain gradient theory is also combined with the constitutive equations to determine the influences of softening stiffness as well as stiffness enhancement. The equations of motion are derived via applying Hamilton's principle. The four unknown functions denoting the buckling load of the nanoplate are defined in a modal manner, and Navier solution method is used to find the shear buckling responses. The influences of various parameters such as geometry (length and width), thermal and hygroscopic distributions (uniform, linear and nonlinear), size-dependence parameters (nonlocal parameter and strain gradient parameter), constitutive elastic foundations (Winkler and Pasternak) on the shear buckling behavior of the isotropic nanoplate are investigated.