Shear Deformable Shell Model Research Articles

Free nonlinear vibration analysis of nano-truncated conical shells based on modified strain gradient theory

In this study, the nonlinear free vibrations behaviour of nano-truncated conical shells was analysed, using the first-order shear deformable shell model. The analysis took into account the structure size through modified strain gradient theory, and differential quadrature and Fréchet derivative methods in von Kármán-Donnell-type approach to kinematic nonlinearity. The governing equations were obtained, utilizing Hamilton's principle. Partial differential equations plus the non-classical and classical boundary conditions were used to obtain the shells’ equations of motion. Discretizing the boundary conditions and equations of motion were performed based on a generalized differential quadrature analogy. The eigenvalue system was considered based on the harmonic balance technique. The Galerkin and Fréchet derivative approaches were used to determine the nonlinear free vibration behaviour of the carbon nano-cone, which was modelled in the simply- and clamped-supported boundary conditions. Comparisons were made between the findings from the new model versus the couple and classical stress theories, indicating that the classical and modified couple stress theories are distinct representations of modified strain gradient theory. The results also revealed that the degree of hardening of nano-truncated conical shells in the modified strain gradient theory is less than that of modified couple stress and classical theories. This led to a rise in the non-dimensional amplitude and frequency ratios. This study investigated the effect of size on free nonlinear vibrations of nano-truncated conical shells for various apex angles and lengths. Finally, we evaluated and compared our findings versus those reported by previous studies, which confirmed the precision and accuracy of our results.

Microstructural Size Dependency in Nonlinear Lateral Stability of Random Reinforced Microshells via Meshfree-Based Applied Mathematical Modeling

The prime objective of this research work is to develop an efficient small scale-dependent computational framework incorporating microstructural tensors of dilatation gradient, rotation gradient, and deviatoric stretch gradient to analyze nonlinear lateral stability of cylindrical microshells. The numerical strategy is established based upon a mixed formation of the third-order shear deformable shell model and modified strain gradient continuum mechanics. The graphene nanoplatelet reinforcements are assumed to be randomly dispersed in a checkerboard scheme within the resin matrix. Accordingly, to extract the effective material properties, the Monte Carlo simulation together with a probabilistic technique are employed. The numerical solution for the microstructural-dependent nonlinear problem is carried out via the moving Kriging meshfree method having the capability to accommodate accurately the essential boundary conditions using proper moving Kriging shape function. It is represented that the role of the stiffening characters related to the effect of microstructural dilatation gradient, rotation gradient, and deviatoric stretch reduces continuously by going to deeper territory of the load-deflection stability path. Moreover, it is indicated that among various microstructural gradient tensors, the stiffening character of the rotation gradient is higher than deviatoric stretch gradient, and the stiffening character of the latter is more considerable than the dilatation gradient tensor.

Thermal free vibration analysis of functionally graded plates and panels with an improved finite shell element

This article deals with thermal free vibration analysis of functionally graded material (FGM) plates and panels using an improved first-order shear deformable (I-FSDT) shell model. The Effective material properties of the FGMs are estimated according to a power law distribution with temperature dependency. The FGM structure can be subjected to uniform or nonlinear temperature rise. The results obtained from the proposed formulation are validated with those available in the literature. Natural frequencies of FGM plates and cylindrical panels are computed for different power law indexes, thermal loadings, boundary conditions and aspect ratios. The present (I-FSDT) finite shell element model is much like the standard (FSDT) model but does not need shear correction factors and is of high accuracy. This work is the first application of the formulation to free vibration analysis of FG plate and shell structures including thermal effects.

A unified modified couple stress model for size-dependent free vibrations of FG cylindrical microshells based on high-order shear deformation theory

The aim of present work is to propose a unified modified couple stress-based shear deformable shell model that has capability of analyzing size-dependent free vibrations of functionally graded (FG) cylindrical microshells. In the current model, the classical strain tensors and curvature tensor components are given within the framework of a unified high-order shear deformation theory. The trapezoidal shape factor is taken into account for achieving reasonable stress resultants over thickness and makes the proposed model more accurate for thick shell structures. Various classical boundary conditions of the cylindrical microshells are considered using the admissible functions of the displacements containing a beam modal function. The governing equations associated with free vibrations of microshells are derived using a standard Lagrange method. A few comparisons are conducted and verify that the results from the current model coincide favorably to the ones from three-dimensional elasticity solutions. The comparisons also prove that the proposed model has a sufficient capability of dealing with free vibration problems of the thick functionally graded cylindrical microshells. To the end, the size-dependent free vibrations of an FG cylindrical microshell are investigated. As new results, natural frequencies of the FG microshells with various boundary conditions are tabulated, and the effects of some important parameters on vibration characteristics of FG microshells are discussed.

Using a New Size Dependent Orthotropic Elastic Shell Model for the Investigation of Free Vibration of Protein Microtubules

Protein microtubules are one of the most effective intracellular components. The microtubules structure is in a manner that their behaviour is similar to that of the orthotropic materials. Therefore, in this paper, size-dependent vibration of the anisotropic protein microtubule is studied. For this purpose, using a first shear deformable shell model and based on couple stress theory, new equations are developed for the dynamic behavior of anisotropic protein microtubule. After solving the governing equations of microtubule motion, the effects of cytoplasm environment, microtubule dimensions and its mechanical properties, and material length scale parameters on the natural frequency of microtubules are investigated.

An efficient size-dependent shear deformable shell model and molecular dynamics simulation for axial instability analysis of silicon nanoshells

Understanding the size-dependent behavior of structures at nanoscale is essential in order to have an effective design of nanosystems. In the current investigation, the surface elasticity theory is extended to study the nonlinear buckling and postbuckling response of axially loaded silicon cylindrical naoshells. Thereby, an efficient size-dependent shear deformable shell model is developed including the size effect of surface free energy. A boundary layer theory of shell buckling in conjunction with a perturbation-based solution methodology is employed to predict the size dependency in the buckling loads and postbuckling behavior of silicon nanoshells having various thicknesses. After that, on the basis of the Tersoff empirical potential, a molecular dynamics (MD) simulation is performed for a silicon cylindrical nanoshell with thickness of four times of silicon lattice constant, the critical buckling load and critical shortening of which are extracted and compared with those of the developed non-classical shell model. It is demonstrated that by taking the effects of surface free energy into account, a very good agreement is achieved between the results of the developed size-dependent continuum shell model and those of MD simulation.

A nonlocal strain gradient hyperbolic shear deformable shell model for radial postbuckling analysis of functionally graded multilayer GPLRC nanoshells

The present study addresses the size dependency in nonlinear instability of functionally graded multilayer graphene platelet-reinforced composite (GPLRC) nanoshells under hydrostatic pressure including jointly the nonlocal elastic and strain gradients stress fields. For this objective, the new unconventional continuum theory namely as nonlocal strain gradient theory of elasticity is utilized within the framework of a refined hyperbolic shear deformation shell theory. Via stacking up a number of individual layers, the graphene platelet (GPL) nanofillers are distributed uniformly and three different functionally graded patterns based upon a layerwise change of the GPL weight fraction through the shell thickness direction. The effective material properties corresponding to uniform (U-GPLRC) and X-GPLRC, O-GPLRC, A-GPLRC functionally graded patterns of dispersion are extracted by Halpin-Tsai micromechanical scheme. By employing jointly the boundary layer theory of shell buckling and a two-stepped perturbation technique, explicit analytical expressions are achieved for nonlocal strain gradient stability curves of functionally graded multilayer GPLRC nanoshells. It is indicated that by increasing the value of GPL weight fraction for the U-GPLRC and O-GPLRC nanoshells, the significance of the both nonlocal and strain gradient size dependencies reduces, while for the X-GPLRC and A-GPLRC nanoshells, it increases.

Size dependent buckling analysis of functionally graded piezoelectric cylindrical nanoshell

In this paper, the size-dependent formulation of shear deformable functionally graded piezoelectric (FGP) cylindrical nanoshells is developed based on a new modified couple stress theory. After the general formulation, the buckling of FGP cylindrical nanoshells under pressure is investigated by using the first order shear deformable shell model. The material properties are assumed to be varied along thickness direction according to the power law distribution. The equilibrium equations and boundary conditions are obtained by using the minimum potential energy principle. A buckling analysis of simply-supported FGP cylindrical nanoshells under uniform lateral external pressure is carried out and the effects of different parameters on the critical pressure are examined. The effects of geometrical, electrical and material properties, such as material length scale parameter, length, thickness, external electric voltage and material property gradient index, on the critical pressure are illustrated. It is indicated that the critical pressure is significantly size-dependent.

Theoretical study of the effect of shear deformable shell model, elastic foundation and size dependency on the vibration of protein microtubule

In this paper, isotropic cylindrical shell model with first order shear deformation theory based on modified couple stress theory is used to investigate the mechanical behavior of microtubules and to consider the size effect in the nano-scale. Besides, partial equations of shell motion with classical and non-classical corresponding boundary conditions are derived from Hamilton׳s principle. The study models the microtubule as a simply supported shell and the Navier procedure is used to solve the vibration problem. To simulate the environment of microtubule, Pasternak foundation is used. The free vibration of the microtubule is examined in the presence and absence of cytoplasm, and the significant parameter, length dependence flexural rigidity is determined based on the new model. Finally, the effect of different system parameters such as length, size effect parameters, shear modulus ratio and substrate parameters on free vibration microtubule is investigated. Comparison of the results of the new model with the classical model and the results obtained by other researchers reveals that the model presented in this paper is more accurate than other models used in this area and in better agreement with experimental data.

A shear deformable conical shell formulation in the framework of couple stress theory

In this paper, the governing equations of the conical shell are derived by using the first-order shear deformable shell model and taking into account the size parameter through couple stress theory. In order to obtain the governing equations, Hamilton’s principle is used and the equations of shell motion with partial differentials are derived along with classical and non-classical boundary conditions. Finally, the free vibration of the single-walled carbon nanocone (SWCNC) is scrutinized through examples. The SWCNC is modeled as simply supported, and the Galerkin method is used to solve the vibration problem. The results of the new model are compared with those of the classical theory, which point to the conclusion that the classical model is a special case of couple stress theory. Results also reveal that nanoshell rigidity in the couple stress theory is greater than that in the classical theory, which leads to an increase in natural frequencies. Moreover, the study investigates the effect of the size parameter on nanoshell vibration for different lengths and apex angles.

A shear deformable cylindrical shell model based on couple stress theory

In this paper, formulation of the thin cylindrical shell via the modified couple stress theory by taking account of shear deformation and rotary inertia is obtained. To do this, the study developed the first shear deformable cylindrical shell theory by considering the size effects via the couple stress theory and the equations of motion of shell with classical and non-classical boundary conditions were extracted through Hamilton’s principle. In the end, as an example, free vibrations of the single-walled carbon nanotube (SWCNT) were investigated. Here, the SWCNT was modeled as a simply supported shell, and the Navier procedure was used to solve the vibration problem. The results of the new model were compared with those of the classical theory, pointing to the conclusion that the classical model is a special case of the modified couple stress theory. The findings also demonstrate that the rigidity of the nano-shell in the modified couple stress theory compared with that in the classical theory is greater, resulting in the increase in natural frequencies. In addition, the effect of the material length scale parameter on the vibrations of the nano-shell in different lengths and thickness was investigated.

Size-dependent axial buckling analysis of functionally graded circular cylindrical microshells based on the modified strain gradient elasticity theory

In this paper, a size-dependent first-order shear deformable shell model is developed based upon the modified strain gradient theory (MSGT) for the axial buckling analysis of functionally graded (FG) circular cylindrical microshells. It is assumed that the material properties of FG materials, which obey a simple power-law distribution, vary through the thickness direction. The principle of virtual work is utilized to formulate the governing equations and corresponding boundary conditions. Numerical results are presented for the axial buckling of FG circular cylindrical microshells subject to simply-supported end conditions and the effects of material length scale parameter, material property gradient index, length-to-radius ratio and circumferential mode number on the size-dependent critical buckling load are extensively studied. For comparison purpose, the critical buckling loads predicted by modified couple stress theory (MCST) and classical theory (CT) are also presented. Results show that the size effect plays an important role for lower values of dimensionless length scale parameter. Moreover, it is observed that the critical buckling loads obtained based on MSGT are greater than those obtained based on MCST and CT.

Nonlocal shear deformable shell model for torsional buckling and postbuckling of microtubules in thermal environments

Buckling and postbuckling analysis is presented for microtubules subjected to torsion in thermal environments. The microtubule is modeled as a nonlocal shear deformable cylindrical shell which contains small scale effects. The governing equations are based on a higher order shear deformation theory. The thermal effects are included and the material properties are assumed to be temperature-dependent. The small scale parameter e0a is estimated by matching the buckling twist angle of microtubules obtained from the nonlocal shear deformable shell model with the existing result. The results show that the small scale effect plays an important role in the postbuckling of microtubules.

Electro-thermo-torsional buckling of an embedded armchair DWBNNT using nonlocal shear deformable shell model

Electro-thermo-torsional buckling response of a double-walled boron nitride nanotube (DWBNNT) has been investigated based on nonlocal elasticity and piezoelasticity theories. The effects of surrounding elastic medium such as the spring constant of the Winkler-type and the shear constant of the Pasternak-type are taken into account. The van der Waals (vdW) forces are considered between inner and outer layers of nanotube. According to the relationship between the piezoelectric coefficient of armchair boron nitride nanotubes (BNNTs) and stresses, the first order shear deformation theory (FSDT) is used. Energy method and Hamilton’s principle are employed to obtain coupled differential equations containing displacements, rotations and electric potential terms. The detailed parameter study is conducted to investigate the effects of nonlocal parameter, elastic foundation modulus, temperature change, piezoelectric and dielectric constants on the critical torsional buckling load. Results indicate that the critical buckling load decreases when piezoelectric effect is considered.

Natural frequencies and buckling of pressurized nanotubes using shear deformable nonlocal shell model

The small-scale effect on the natural frequencies and buckling of pressurized nanotubes is investigated in this study. Based on the firstorder shear deformable shell theory, the nonlocal theory of elasticity is used to account for the small-scale effect and the governing equations of motion are obtained. Applying modal analysis technique and based on Galerkin’s method a procedure is proposed to obtain natural frequencies of vibrations. For the case of nanotubes with simply supported boundary conditions, explicit expressions are obtained which establish the dependency of the natural frequencies and buckling loads of the nanotube on the small-scale parameter and natural frequencies obtained by local continuum mechanics. The obtained solutions generalize the results of nano-bar and -beam models and are verified by the literature. Based on several numerical studies some conclusions are drawn about the small-scale effect on the natural frequencies and buckling pressure of the nanotubes.

Nonlinear vibration of microtubules in living cells

Large amplitude flexural vibration behavior is presented for microtubules (MTs) embedded in an elastic matrix of cytoplasm. The microtubule is modeled as a nonlocal shear deformable cylindrical shell which contains small scale effect. The surrounding elastic medium is modeled as a two-parameter elastic foundation. Formulations are based on higher order shear deformation shell theory with a von Kármán-Donnell-type of kinematic nonlinearity. The thermal effect is also included and the material properties are assumed to be temperature-dependent. The small scale parameter e 0 a is estimated by matching the fundamental frequencies from existing results with the numerical results obtained from the nonlocal shear deformable shell model. The numerical results show that the fundamental frequencies of MTs are very sensitive to the small scale parameter e 0 a. The results reveal that the small scale parameter e 0 a reduces the natural frequencies of MTs, but the effect of the small scale parameter on the nonlinear vibration response is relatively weaker when compared to the effect of the small scale parameter on the postbuckling response of MTs.

Nonlocal shear deformable shell model for bending buckling of microtubules embedded in an elastic medium

A nonlocal shear deformable shell model is developed for buckling of microtubules embedded in an elastic matrix of cytoplasm under bending in thermal environments. The results reveal that the lateral constraint has a significant effect on the buckling moments of a microtubule when the foundation stiffness is sufficiently large.

Nonlocal shear deformable shell model for thermal postbuckling of axially compressed double-walled carbon nanotubes

An investigation is reported of the thermal buckling and postbuckling of axially compressed double-walled carbon nanotubes (CNTs) subjected to a uniform temperature rise. The double-walled carbon nanotube is modeled as a nonlocal shear deformable cylindrical shell, which contains small-scale effects and van der Waals interaction forces. The governing equations are based on higher order shear deformation shell theory with a von Kármán–Donnell-type of kinematic nonlinearity and include thermal effects. Temperature-dependent material properties, which come from molecular dynamics (MD) simulations, and an initial point defect, which is simulated as a dimple on the tube wall, are both taken into account. The small-scale parameter, e 0 a, is estimated by matching the buckling temperature of CNTs observed from the MD simulation results with the numerical results obtained from the nonlocal shear deformable shell model. The numerical illustrations concern the thermal postbuckling response of perfect and imperfect, single- and double-walled CNTs with different values of compressive load ratio. The results show that buckling temperature and postbuckling behavior of nanotubes are very sensitive to the small-scale parameter. The results reveal that temperature-dependent material properties have a significant effect on the thermal postbuckling behavior of both single- and double-walled CNTs.

Nonlocal Shear Deformable Shell Model for Post-Buckling of Axially Compressed Double-Walled Carbon Nanotubes Embedded in an Elastic Matrix

Buckling and post-buckling analysis is presented for axially compressed double-walled carbon nanotubes (CNTs) embedded in an elastic matrix in thermal environments. The double-walled carbon nanotube is modeled as a nonlocal shear deformable cylindrical shell, which contains small scale effects and van der Waals interaction forces. The surrounding elastic medium is modeled as a tensionless Pasternak foundation. The post-buckling analysis is based on a higher order shear deformation shell theory with the von Kármán–Donnell-type of kinematic nonlinearity. The thermal effects are also included and the material properties are assumed to be temperature-dependent and are obtained from molecular dynamics (MD) simulations. The nonlinear prebuckling deformations of the shell and the initial local point defect, which is simulated as a dimple on the tube wall, are both taken into account. A singular perturbation technique is employed to determine the post-buckling response of the tubes and an iterative scheme is developed to obtain numerical results without using any assumption on the shape of the contact region between the tube and the elastic medium. The small scale parameter e0a is estimated by matching the buckling loads of CNTs observed from the MD simulation results with the numerical results obtained from the nonlocal shear deformable shell model. Numerical solutions are presented to show the post-buckling behavior of CNTs surrounded by an elastic medium of conventional and tensionless Pasternak foundations. The results show that buckling and post-buckling behavior of CNTs is very sensitive to the small scale parameter e0a. The results reveal that the unilateral constraint has a significant effect on the post-buckling response of CNTs when the foundation stiffness is sufficiently large.

Buckling and postbuckling of radially loaded microtubules by nonlocal shear deformable shell model

This paper presents an investigation on the buckling and postbuckling of microtubules (MTs) subjected to a uniform external radial pressure in thermal environments. The microtubule is modeled as a nonlocal shear deformable cylindrical shell which contains small scale effects. The governing equations are based on higher order shear deformation shell theory with a von Kármán–Donnell-type of kinematic nonlinearity and include the extension-twist and flexural-twist couplings. The thermal effects are also included and the material properties are assumed to be temperature-dependent. A singular perturbation technique is employed to determine the buckling pressure and postbuckling equilibrium paths. The small scale parameter e 0 a is estimated by matching the buckling pressure of MTs measured from the experiments with the numerical results obtained from the nonlocal shear deformable shell model. The numerical results show that buckling pressure and postbuckling behavior of MTs are very sensitive to the small scale parameter e 0 a. The results reveal that the 13_3 microtubule has a stable postbuckling path, whereas the 13_2 microtubule has an unstable postbuckling behavior due to the presence of skew angles.