• Structural Engineering and Mechanics
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• v.70 no.5
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• pp.623-637
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• 2019

This paper analyzes the non-stationary vibration and super-harmonic resonances in nonlinear dynamic motion of viscoelastic nano-resonators. For this purpose, a new coupled size-dependent model is developed for a plate-shape nano-resonator made of nonlinear viscoelastic material based on modified coupled stress theory. The virtual work induced by viscous forces obtained in the framework of the Leaderman integral for the size-independent and size-dependent stress tensors. With incorporating the size-dependent potential energy, kinetic energy, and an external excitation force work based on Hamilton's principle, the viscous work equation is balanced. The resulting size-dependent viscoelastically coupled equations are solved using the expansion theory, Galerkin method and the fourth-order Runge-Kutta technique. The Hilbert-Huang transform is performed to examine the effects of the viscoelastic parameter and initial excitation values on the nanosystem free vibration. Furthermore, the secondary resonance due to the super-harmonic motions are examined in the form of frequency response, force response, Poincare map, phase portrait and fast Fourier transforms. The results show that the vibration of viscoelastic nanosystem is non-stationary at higher excitation values unlike the elastic ones. In addition, ignoring the small-size effects shifts the secondary resonance, significantly.

• Structural Engineering and Mechanics
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• v.70 no.5
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• pp.525-534
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• 2019

An analytical research on buckling of simply supported thin plate with variable thickness under bi-axial compression is presented in this paper. Combining the perturbation technique, Fourier series expansion and Galerkin methods, the linear governing differential equation of the plate with arbitrary thickness variation under bi-axial compression is solved and the analytical expression of the critical buckling load is obtained. Based on that, numerical analysis is carried out for the plates with different thickness variation forms and aspect ratios under different bi-axial compressions. Four different thickness variation forms including linear, parabolic, stepped and trigonometric have been considered in this paper. The calculated critical buckling loads and buckling modes are presented and compared with the published results in the tables and figures. It shows that the analytical expressions derived by the theoretical method in this paper can be effectively used for buckling analysis of simply supported thin plates with arbitrary thickness variation, especially for the stepped thickness that used in engineering widely.

• Computers and Concrete
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• v.24 no.5
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• pp.445-452
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• 2019

In this paper, dynamic stress, strain and deflection analysis of concrete pipes conveying nanoparticles-water under the seismic load are studied. The pipe is buried in the soil which is modeled by spring and damper elements. The Navier-Stokes equation is used for obtaining the force induced by the fluid and the mixture rule is utilized for considering the effect of nanoparticles. Based on refined two variables shear deformation theory of shells, the pipe is simulated and the equations of motion are derived based on energy method. The Galerkin and Newmark methods are utilized for calculating the dynamic stress, strain and deflection of the concrete pipe. The influences of internal fluid, nanoparticles volume percent, soil medium and damping of it as well as length to diameter ratio of the pipe are shown on the dynamic stress, strain and displacement of the pipe. The results show that with enhancing the nanoparticles volume percent, the dynamic stress, strain and deflection decrease.

• Ocean Systems Engineering
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• v.11 no.1
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• pp.59-81
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• 2021

The nonlinear formulation using the principle of virtual work-energy for free vibration of a large-sag extensible catenary riser in two dimensions is presented in this paper. A support at one end is hinged and the other is a free-sliding roller in the horizontal direction. The catenary riser has a large-sag configuration in the static equilibrium state and is assumed to displace with large amplitude to the motion state. The total virtual work of the catenary riser system involves the virtual strain energy due to bending, the virtual strain energy due to axial deformation, the virtual work done by the effective weight, and the inertia forces. The nonlinear equations of motion for two-dimensional free vibration in the Cartesian coordinate system is developed based on the difference between the Euler's equations in the static state and the displaced state. The linear and nonlinear stiffness matrices of the catenary riser are obtained and the eigenvalue problem is solved using the Galerkin finite element procedure. The natural frequencies and mode shapes are obtained. The results are validated with regard to the reference research addressing the accuracy and efficiency of the proposed nonlinear formulation. The numerical results for free vibration and the effect of the nonlinear behavior for catenary riser are presented.

• Smart Structures and Systems
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• v.30 no.2
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• pp.121-133
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• 2022

An analytical approach for the free vibration behavior of a sandwich cylindrical shell with a functionally graded (FG) core is presented. It is considered that the FG distribution is in the direction of thickness. The material properties are temperature-dependent. The sandwich cylindrical shell with a FG core is considered with two cases. In the first model, i.e., Ceramic-FGM-Metal (CFM), the interior layer of the cylindrical shell is rich metal while the exterior layer is rich ceramic and the FG material is located between two layers and for the second model i.e., Metal-FGM-Ceramic (MFC), the material distribution is in reverse order. This study develops Carrera's Unified Formulation (CUF) to analyze sandwich cylindrical shell with an FG core for the first time. Considering the Principle of Virtual Displacements (PVDs) according to the CUF, the dependent boundary conditions and governing equations are obtained. The coupled governing equations are derived using Galerkin's method. In order to validate the present results, comparisons are made with the available solutions in the previous researches. The effects of different geometrical and material parameters on the free vibration behavior of a sandwich cylindrical shell with an FG core are examined.

• Steel and Composite Structures
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• v.46 no.5
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• pp.649-658
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• 2023

Although some scholars have studied the thermal post-buckling of graphene platelets strengthened metal foams (GPLRMFs) plates, they have not considered the influence of initial geometrical imperfection. Inspired by this fact, the present paper studies the thermal post-buckling characteristics of GPLRMFs plates with initial geometrical imperfection. Three kinds of graphene platelets (GPLs) distribution patterns including three patterns have been considered. The governing equations are derived according to the first-order plate theory and solved with the help of the Galerkin method. According to the comparison with published paper, the accuracy and correctness of the present research are verified. In the end, the effects of material properties and initial geometrical imperfection on the thermal post-buckling response of the GPLRMFs plates are examined. It can be found that the presence of initial geometrical imperfection reduces the thermal post-buckling strength. In addition, the present study indicates that GPL-A pattern is best way to improve thermal post-buckling strength for GPLRMFs plates, and the presence of foams can improve the thermal post-buckling strength of GPLRMFs plates, the Foam- II and Foam- I patterns have the lowest and highest thermal post-buckling strength. Our research can provide guidance for the thermal stability analysis of GPLRMFs plates.

• Advances in nano research
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• v.14 no.4
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• pp.303-312
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• 2023

In this study, the bifurcation analysis of functionally graded material is done using exponential volume fraction law. Shell theory of Love is used for vibration of shell. The Galerkin's method is applied for the formation of three equations in eigen value form. This eigen form gives the frequencies using the computer software MATLAB. The variations of natural frequencies (Hz) for Type-I and Type-II functionally graded cylindrical shells are plotted for exponential volume fraction law. The behavior of exponent of volume fraction law is seen for three different values. Moreover, the frequency variations of Type-I and -II clamped simply supported FG cylindrical shell with different positions of ring supports against the circumferential wave number are investigated. The procedure adopted here enables to study vibration for any boundary condition but for brevity, numerical results for a cylindrical shell with clamped simply supported edge condition are obtained and their analysis with regard various physical parameters is done.

• Steel and Composite Structures
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• v.48 no.2
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• pp.191-206
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• 2023

The present research investigates how micromechanical models affect the behavior of Functionally Graded (FG) plates under different boundary conditions. The study employs diverse micromechanical models to assess the effective material properties of a two-phase particle composite featuring a volume fraction of particles that continuously varies throughout the thickness of the plate. Specifically, the research examines the vibrational response of the plate on a Winkler-Pasternak elastic foundation, considering different boundary conditions. To achieve this, the governing differential equations and boundary conditions are derived using Hamilton's principle, which is based on a four-variable shear deformation refined plate theory. Additionally, the Galerkin method is utilized to compute the plate's natural frequencies. The study explores how the plate's natural frequencies are influenced by various micromechanical models, such as Voigt, Reuss, Hashin-Shtrikman bounds, and Tamura, as well as factors such as boundary conditions, elastic foundation parameters, length-to-thickness ratio, and aspect ratio. The research results can provide valuable insights for future analyses of FG plates with different boundaries, utilizing different micromechanical models.

• Steel and Composite Structures
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• v.46 no.1
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• pp.15-31
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• 2023

Organic solar cells utilized natural polymers to convert solar energy to electricity. The demands for green energy production and less disposal of toxic materials make them one of the interesting candidates for replacing conventional solar cells. However, the different aspects of their properties including mechanical strength and stability are not well recognized. Therefore, in the present study, we aim to explore the chaotic responses of these organic solar cells. In doing so, a specific type of organic solar cell constructed from layers of material with different thicknesses is considered to obtain vibrational and chaotic responses under different boundaries and initial conditions. A square plate structure is examined with first-order shear deformation theory to acquire the displacement field in the laminated structure. The bounding between different layers is considered to be perfect with no sliding and separation. On the other hand, nonlocal elasticity theory is engaged in incorporating the structural effects of the organic material into calculations. Hamilton's principle is adopted to obtain governing equations with regard to boundary conditions and mechanical loadings. The extracted equations of motion were solved using the perturbation method and differential quadrature approach. The results demonstrated the significant effect of relative glass layer thickness on the chaotic behavior of the structure with higher relative thickness leading to less chaotic responses. Moreover, a comprehensive parameter study is presented to examine the effects of nonlocality and relative thicknesses on the natural frequency of square organic solar cell structure.

• Advances in nano research
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• v.13 no.5
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• pp.465-474
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• 2022

Based on the nonlocal strain gradient (NSG) theory and considering the influence of moment of inertia, the governing equations of motion of porous functionally graded (FG) nanoplates with four edges clamped are established; The Galerkin method is applied to eliminate the spatial variables of the partial differential equation, and the partial differential governing equation is transformed into an ordinary differential equation with time variables. By satisfying the boundary conditions and solving the characteristic equation, the dispersion relations of the porous FG strain gradient nanoplates with four edges fixed are obtained. It is found that when the wave number is very small, the influences of nonlocal parameters and strain gradient parameters on the dispersion relation is very small. However, when the wave number is large, it has a great influence on the group velocity and phase velocity. The nonlocal parameter represents the effect of stiffness softening, and the strain gradient parameter represents the effect of stiffness strengthening. In addition, we also study the influence of power law index parameter and porosity on guided wave propagation.