• Advances in nano research
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• v.4 no.2
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• pp.85-111
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• 2016

In this paper, the classical and non-classical boundary conditions effect on free vibration characteristics of functionally graded (FG) size-dependent nanobeams are investigated by presenting a semi analytical differential transform method (DTM) for the first time. Three kinds of mathematical models, namely; power law (P-FGM), sigmoid (S-FGM) and Mori-Tanaka (MT-FGM) distribution are considered to describe the material properties in the thickness direction. The nonlocal Eringen theory takes into account the effect of small size, which enables the present model to become effective in the analysis and design of nanosensors and nanoactuators. Governing equations are derived through Hamilton's principle and they are solved applying semi analytical differential transform method. The good agreement between the results of this article and those available in literature validated the presented approach. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of the several parameters such as small scale effects, spring constant factors, various material compositions and mode number on the normalized natural frequencies of the FG nanobeams in detail. It is explicitly shown that the vibration of FG nanobeams is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.

• Computers and Concrete
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• v.20 no.6
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• pp.671-682
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• 2017

In this research, seismic response of pipes is examined by applying nanotechnology and piezoelectric materials. For this purpose, a pipe is considered which is reinforced by carbon nanotubes (CNTs) and covered with a piezoelectric layer. The structure is subjected to the dynamic loads caused by earthquake and the governing equations of the system are derived using mathematical model via cylindrical shell element and Mindlin theory. Navier-Stokes equation is employed to calculate the force due to the fluid in the pipe. Mori-Tanaka approach is used to estimate the equivalent material properties of the nanocomposite and to consider the effect of the CNTs agglomeration on the scismic response of the structure. Moreover, the dynamic displacement of the structure is extracted using harmonic differential quadrature method (HDQM) and Newmark method. The main goal of this research is the analysis of the seismic response using piezoelectric layer and nanotechnology. The results indicate that reinforcing the pipeline by CNTs leads to a reduction in the displacement of the structure during an earthquake. Also the negative voltage applied to the piezoelectric layer reduces the dynamic displacement.

• Computers and Concrete
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• v.20 no.6
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• pp.711-718
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• 2017

Nanotechnology is a new filed in concrete structures which can improve the mechanical properties of them in confronting to impact and blast. However, in this paper, a mathematical model is introduced for the concrete models subjected to impact load for wave propagation analysis. The structure is simulated by the sinusoidal shear deformation theory (SSDT) and the governing equations of the concrete model are derived by energy method and Hamilton's principle. The silicon dioxide ($SiO_2$) nanoparticles are used as reinforcement for the concrete model where the characteristics of the equivalent composite are determined using Mori-Tanaka approach. An exact solution is applied for obtaining the maximum velocity of the model. In order to validate the theoretical results, three square models with different impact point and Geophone situations are tested experimentally. The effect of different parameters such as $SiO_2$ nanoparticles volume percent, situation of the impact, length, width and thickness of the model as well as velocity, diameter and height of impactor are shown on the maximum velocity of the model. Results indicate that the theoretical and experimental dates are in a close agreement with each other. In addition, using from $SiO_2$ nanoparticles leads to increase in the stiffness and consequently maximum velocity of the model.

• Advances in nano research
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• v.7 no.2
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• pp.109-123
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• 2019

In this article, the free vibration analysis of annular sandwich plates with various functionally graded (FG) porous cores and carbon nanotubes reinforced composite (CNTRC) facesheets is investigated based on modified couple stress theory (MCST) and first order shear deformation theories (FSDT). The annular sandwich plate is composed of two face layers and a functionally graded porous core layer which contains different porosity distributions. Various approaches such as extended mixture rule (EMR), Eshelby-Mori-Tanaka (E-M-T), and Halpin-Tsai (H-T) are used to determine the effective material properties of microcomposite circular sandwich plate. The governing equations of motion are extracted by using Hamilton's principle and FSDT. A Ritz method has been utilized to calculate the natural frequency of an annular sandwich plate. The effects of material length scale parameters, boundary conditions, aspect and inner-outer radius ratios, FG porous distributions, pore compressibility and volume fractions of CNTs are considered. The results are obtained by Ritz solutions that can be served as benchmark data to validate their numerical and analytical methods in the future work and also in solid-state physics, materials science, and micro-electro-mechanical devices.

• Steel and Composite Structures
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• v.39 no.5
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• pp.493-509
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• 2021

There is not enough mixed finite element method (MFEM) model developed for static and dynamic analysis of functionally graded material (FGM) beams in the literature. The main purpose of this study is to develop a reliable and efficient computational modeling using an efficient functional in MFEM for free vibration and static analysis of FGM composite beams subject to high order shear deformation effects. The modeling of material properties was performed using mixture rule and Mori-Tanaka scheme which are more realistic determination techniques. This method based on the assumption that a two phase composite material consisting of matrix reinforced by spherical particles, randomly distributed in the beam. To explain the displacement components of the shear deformation effects, it was accepted that the shear deformation effects change sinusoidal. Partial differential field equations were obtained with the help of variational methods and then these equations were transformed into a novel functional for FGM beams with the help of Gateaux differential derivative operator. Thanks to the Gateaux differential method, the compatibility of the field equations was checked, and the field equations and boundary conditions were reflected to the function. A MFEM model was developed with a total of 10 degrees of freedom to apply the obtained functional. In the numerical applications section, free vibration and flexure problems solutions of FGM composite beams were compared with those predicted by other theories to show the effects of shear deformation, thickness changing and boundary conditions.

• Smart Structures and Systems
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• v.29 no.3
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• pp.395-420
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• 2022

A new hybrid quasi-3D and 2D high-order shear deformation theory is studied in this mathematical formulation, for an investigation of the bending, free vibrations and buckling influences on a functionally graded material plate. The theoretical formulation has been begun by a displacement field of five unknowns, governing the transverse displacement across the thickness of the plate by bending, shearing and stretching. The transverse shear deformation effect has been taken into consideration, satisfying the stress-free boundary conditions, especially on plate free surfaces as parabolic variation through its thickness. Thus, the mechanical properties of the functionally graded plate vary across the plate thickness, following three distributions forms: the power law, exponential form and the Mori-Tanaka scheme. The mechanical properties are used to develop the equations of motion, obtained from the Hamilton principle, and solved by applying the Navier-type solution for simply supported boundary conditions. The results obtained are compared with other solutions of 2D, 3D and quasi-3D plate theories have been found in the literature.

• Steel and Composite Structures
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• v.48 no.4
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• pp.419-428
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• 2023

In this paper, dynamic and static bending of ball modelled by nanocomposite microbeam by nanoparticles seeing agglomeration is presented. The structural damping is considered by Kelvin-Voigt model. The agglomeration effects are assumed using Mori-Tanaka model. The football ball is modeled by third order shear deformation theory (TSDT). The motion equations are derived by principle of Hamilton's and energy method assuming size effects on the basis of Eringen theory. Using differential quadrature method (DQM) and Newmark method, the static and dynamic deflections of the structure are obtained. The effects of agglomeration and CNTs volume percent, damping of structure, nonlocal parameter, length and thickness of micro-beam are presented on the static and dynamic deflections of the nanocomposite structure. Results show that with increasing CNTs volume percent, the maximum dimensionless dynamic deflection is reduced about 17%. In addition, assuming CNTs agglomeration increases the dimensionless dynamic deflection about 14%. It is also found that with increasing the CNTs volume percent from 0 to 0.15, the static deflection is decreased about 3 times due to the enhance in the stiffness of the structure. In addition, with enhancing the nonlocal parameters, the dynamic deflection is increased about 3.1 times.

• Steel and Composite Structures
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• v.44 no.5
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• pp.633-649
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• 2022

This paper is first implemented with the bending behavior of three-dimensional functionally graded (3DFG) plates in the framework of level set-based topology optimization (LS-based TO). Besides, due to the suitable properties of the current design domain, the thin-plate spline (TPS) is recognized as a RBF to construct the LS function. The overall mechanical properties of the 3DFG plate are assessed using a power-law distribution scheme via Mori-Tanaka micromechanical material model. The bending response is obtained using the first-order shear deformation theory (FSDT). The mixed interpolation of four elements of tensorial components (MITC4) is also implemented to overcome a well-known shear locking problem when the thickness becomes thinner. The Hamilton-Jacobi method is utilized in each iteration to enforce the necessary boundary conditions. The mathematical formulas are expressed in great detail for the LS-based TO using 3DFG materials. Several numerical examples are exhibited to verify the efficiency and reliability of the current methodology with the previously reported literature. Finally, the influences of FG materials in the optimized design are explained in detail to illustrate the behaviors of optimized structures.

• Transactions of Materials Processing
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• v.33 no.1
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• pp.18-35
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• 2024

In the present work, the evolution rules for the internal variables including continuum damage factors are obtained using the thermodynamic framework, which are in turn facilitated to derive the elastic-plastic constitutive relation for the particulate composites. Using the Mori-Tanaka scheme, the homogenization on state and internal variables such as back-stress and damage factors is carried out to procure the rate independent plasticity relations. Moreover, the degradation of mechanical properties of constituents is depicted by the distinctive damages such that the phase and interfacial damages are treated individually accordingly, whereas the kinematic hardening is depicted by combining the Armstrong-Frederick and Phillips' back-stress evolutions. On the other hand, the present constitutive relation for each phase is expressed in terms of the respective damage-free effective quantities, then, followed by transformation into the damage affected overall nominal relations using the aforementioned homogenization concentration factors. An emphasis is placed on the qualitative analyses for strain localization by observing the perturbation growth instead of the conventional bifurcation analyses. It turns out that the proposed constitutive model offers a wide range of strain localization behavior depending on the evolution of various internal variable descriptions.

• Geomechanics and Engineering
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• v.14 no.6
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• pp.519-532
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• 2018

In this work, two dimensional (2D) and quasi three-dimensional (quasi-3D) HSDTs are proposed for bending and free vibration investigation of functionally graded (FG) plates using hyperbolic shape function. Unlike the existing HSDT, the proposed theories have a novel displacement field which include undetermined integral terms and contains fewer unknowns. The material properties of the plate is inhomogeneous and are considered to vary continuously in the thickness direction by three different distributions; power-law, exponential and Mori-Tanaka model, in terms of the volume fractions of the constituents. The governing equations which consider the effects of both transverse shear and thickness stretching are determined through the Hamilton's principle. The closed form solutions are deduced by employing Navier method and then fundamental frequencies are obtained by solving the results of eigenvalue problems. In-plane stress components have been determined by the constitutive equations of composite plates. The transverse stress components have been determined by integrating the 3D stress equilibrium equations in the thickness direction of the FG plate. The accuracy of the present formulation is demonstrated by comparisons with the different 2D, 3D and quasi-3D solutions available in the literature.