Third-order nonlocal elasticity in buckling and vibration of functionally graded nanoplates on Winkler-Pasternak media

Based on the third-order shear deformation theory (TSDT), the investigation of the free vibration response of a continuously graded carbon nanotube-reinforced (CGCNTR) cylindrical shell is presented. The volume fractions of randomly oriented straight single-walled carbon nanotubes are assumed to be graded in the thickness direction. An embedded carbon nanotube in a polymer matrix and its surrounding inter-phase is replaced with an equivalent fiber for predicting the mechanical properties of the carbon nanotube/polymer composite. The Mori–Tanaka scheme as an accurate micromechanics model is used for estimating the homogenized material properties of nanocomposites reinforced with equivalent fibers. The equations of motion and the associated boundary conditions are derived using the Hamilton’s principle based on TSDT. The discretization of the system by means of the Generalized Differential Quadrature Method leads to a standard linear eigenvalue problem. Detailed parametric studies have been carried out to study the impacts of the various types of equivalent fiber distribution, different boundary conditions and geometrical parameters on the vibration characteristics of CGCNTR cylindrical shells. The interesting finding of the present study is that the graded CNT volume fractions with symmetric distribution through the shell thickness have high capabilities to reduce or increase the natural frequency in comparison with uniformly and asymmetric CNT distribution.

This paper investigates the dynamic buckling of bi-directional (BD) functionally graded (FG) porous cylindrical shells for various boundary conditions, where the FG material is modeled by means of power law functions with even and uneven porosity distributions of ceramic and metal phases. The third-order shear deformation theory (TSDT) is adopted to derive the governing equations of the problem via the Hamilton’s principle. The generalized differential quadrature (GDQ) method is applied together with the Bolotin scheme as numerical strategy to solve the problem, and to draw the dynamic instability region (DIR) of the structure. A large parametric study examines the effect of different boundary conditions at the extremities of the cylindrical shell, as well as the sensitivity of the dynamic stability to different thickness-to-radius ratios, length-to-radius ratios, transverse and longitudinal power indexes, porosity volume fractions, and elastic foundation constants. Based on results, the dynamic stability of BD-FG cylindrical shells can be controlled efficiently by selecting appropriate power indexes along the desired directions. Furthermore, the DIR is highly sensitive to the porosity distribution and to the extent of transverse and longitudinal power indexes. The numerical results could be of great interest for many practical applications, as civil, mechanical or aerospace engineering, as well as for energy devices or biomedical systems.

Natural frequency analysis of imperfect GNPRN conical shell, cylindrical shell, and annular plate structures resting on Winkler-Pasternak Foundations under arbitrary boundary conditions

In this paper, a new and an efficient solution method based on local gradient smoothing method has been applied to free vibration problem of open composite laminated cylindrical and conical shells with elastic boundary conditions. The theoretical model is formulated by the first-order shear deformation theory, and the motion equation is obtained by the Hamilton’s principle. The motion equation is discretized by meshless shape function; in this process, the derivatives of the shape function are approximated by local gradient smoothing method. The accuracy, applicability and efficiency of this method are demonstrated for free vibrations of open composite laminated cylindrical and conical shells with different geometric, material parameters and boundary condition. The numerical results show good convergence characteristics and good agreement between the present method and the existing literature. And through several numerical examples, some useful results for free vibration results of open composite laminated cylindrical and conical shells are obtained, which may serve as a benchmark solutions for researchers to check their analytical and numerical methods.

The vibration modals of composite sandwich cylindrical shells with auxetic honeycomb cells arrayed in the circumferential direction and axial direction were studied theoretically and numerically. The equivalent moduli and relative density of auxetic cores were derived and validated. Subsequently, the differences in these physical quantities caused by the number of layers were taken into consideration. Based on these works, a verified third order shear deformation theory (TSDT) was applied to deduce the free vibration governing equations of sandwich cylindrical shells. Specially, the effects of deepness terms were not ignored. To prove the accuracy of the theory, four finite element modellings of composite sandwich cylindrical shells with different numbers of auxetic cells arrayed in the circumferential direction were established for vibration analysis. Additionally, cylindrical shells with auxetic cells arrayed in the axial direction were modeled to verify the theory. The frequencies of composite sandwich cylindrical shells calculated by theory were a good fit with those obtained from the numerical method. Thus, the influences of dimensions on vibration modals of composite sandwich cylindrical shells were discussed by TSDT. The sensitivity analysis showed that the number of layers, the radius of cylindrical shells, and the thickness of auxetic cores had a great influence on the frequencies of composite auxetic cylindrical shells. Highlights The governing equations of cylindrical shells were deduced by TSDT. Effects of layers on the equivalent stiffnesses of the auxetic core were considered. Effects of dimensions on vibration modal of cylindrical shells were studied.

Buckling of non-Lévy-type rectangular thick plates:New benchmark solutions in the symplectic framework

Modeling aerothermoelastic properties and active flutter control of nanocomposite cylindrical shells in supersonic airflow under thermal environments

Thermal vibration analysis of FG-GPLRC doubly curved shells partially resting on Kerr foundation based on higher-order shear deformation theory

In this work, different shear deformation theories are used for the first time to investigate the effect of shear deformation on the dynamic response of laminated plates and track slab on a Pasternak foundation under moving load. The Pasternak formulation was used to simulate the interaction between the slab and the elastic foundation. Using Hamilton’s principle, the governing equations are derived based on the third-order shear deformation theory (TSDT) in the higher-order shear deformation theory (HSDT). Additionally, governing equations based on the first-order shear deformation theory (FSDT) and the classical laminate plate theory (CLPT) are also derived. The analytical solution of the classical plate theory (CPT) is derived as an example for isotropic thin plates and used as a benchmark solution to verify the accuracy of the governing equations based on TSDT, FSDT, and CLPT. Then, the effect of the thickness-to-width ratio on the dynamic response of the laminated slab is investigated, along with the effects of the Pasternak foundation coefficient, the vertical load movement speed, and load eccentricity on the dynamic response of the laminated plate and the track slab. The effect of uncertainty in the modulus of elasticity on the analytical results is also examined. The results indicate that considering shear deformation is essential in the dynamic analysis of CRTSII plate ballast slabs, but higher-order shear deformation theory should be avoided due to its computational cost and complexity. This study provides a benchmark solution for further work.

Semi-analytical and experimental studies on travelling wave vibrations of a moderately thick cylindrical shell subject to a spinning motion

A fractional nonlocal elastic model for lattice wave analysis

In the present study, time-dependent thermo-elastic creep behavior and life assessment of rotating thick-walled cylindrical shells made of 304L austenitic stainless steel (304L SS) are investigated based on the third-order shear deformation theory (TSDT). Loading is composed of a uniform internal pressure, distributed temperature field, and a centrifugal body force due to rotating speed. Norton’s law is utilized as the material creep constitutive model. Using the minimum total potential energy principle, a system of differential equations in terms of displacement and boundary conditions are derived. Then, the governing equations are solved with an analytical approach, which leads to an accurate solution. Subsequently, an iterative procedure is also proposed to determine the stresses and deformations at different creep times. Larson-Miller Parameter (LMP) and Robinson's linear life fraction damage rule are employed for assessing the creep damages and the remaining life of cylindrical shells. To the best of the researcher’s knowledge, in the previous studies, there is no study carried out into third-order shear deformation theory for thermo-elastic creep analysis of cylinders. To validate the accuracy of the suggested method based on TSDT, a comparison among analytical results and those of the finite element method (FEM) is performed and very good agreement is found. The results indicate that the present analysis is accurate and computationally efficient.

This research investigates the free vibrational behavior of a functionally graded porous (FGP) nanoplate resting on an elastic Pasternak foundation in a hygrothermal environment. The nanoplate is modeled based on the nonlocal strain gradient theory (NSGT) and considering several plate theories including the CPT (classical plate theory), the FSDT (first-order shear deformation theory), and the TSDT (third-order shear deformation theory). Several patterns are investigated for the dispersion of pores, and the surface effects are incorporated to enhance the precision of the model. The governing equations and boundary conditions are derived via Hamilton's principle and an exact solution is provided via the Navier method. The impacts of several parameters on the natural frequencies are inspected such as length scale and nonlocal parameters, surface effects, porosity parameter, hygrothermal environment, and coefficients of the foundation. The results show that the impact of the porosity parameter on the natural frequencies of nanoplates is significantly dependent on the porosity distribution pattern. It is discovered that by increasing the porosity parameter from 0 to 0.6, the relative changes of natural frequencies vary from a decrease of 30 % to an increase of 6 %.

In this paper, Reddy’s third-order shear deformable plate theory is employed for the analysis of centrosymmetric anisotropic plate structures within strain gradient elasticity. The general three-dimensional anisotropic gradient theory is reduced to a two-dimensional formulation for the analysis of thick plate structures. The third-order shear deformation theory (TSDT) takes into account quadratic variation of the transverse shear strains of the plate and does not require shear correction factors. In order to investigate the case of small strains but moderate rotations, the von Karman strains are considered. The TSDT is also simplified to anisotropic Kirchhoff plate theory within gradient elasticity. To study specific material properties in more detail, the (Kirchhoff and TSDT) gradient plate theory of general anisotropy is simplified to the more practical case of orthotropic plates. It is observed that the gradient theory provides the capability to capture the size effects in anisotropic plate structures. As case studies, the bending and buckling behaviors of the simply supported orthotropic (Kirchhoff and TSDT) plates are studied. Variationally consistent boundary conditions are also discussed. Finally, analytical solutions are presented for the bending and buckling of simply supported orthotropic Kirchhoff plates. The effects of internal length scales on deflections and buckling loads are presented.

Nonlinear bending analysis of FG-CNTRC annular plates with variable thickness on elastic foundation