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Volume 7, Number 6, November 2019

Putting emphasis on the effect of existence of porosity in the functionally graded materials (FGMs) on the dynamic responses of waves scattered in FG nanobeams resulted in implementation of a novel porosity-based homogenization method for FGMs and show its applicability in a wave propagation problem in the presence of axial pre-load for the first time. In the employed porosity-dependent method, the coupling between density and Young's moduli is included to consider for the effective moduli of the FG nanobeam by the means of a more reliable homogenization technique. The beam-type element will be modeled via the classical theory of beams, namely Euler-Bernoulli beam theory. Also, the dynamic form of the principle of virtual work will be extended for such nanobeams to derive the motion equations. Applying the nonlocal constitutive equations of Eringen on the obtained motion equations will be resulted in derivation of the nanobeam's governing equations. Depicted results reveal that the dispersion responses of FG nanobeams will be decreased as the porosity volume fraction is increased which must be noticed by the designers of advanced nanosize devices who are interested in employment of wave dispersion approach in continuous systems for specific goals.

Key Words
wave propagation; porous materials; functionally graded materials (FGMs); nonlocal elasticity theory

(1) Farzad Ebrahimi:
Department of Mechanical Engineering, Faculty of Engineering, Imam Khomeini International University, Qazvin, Iran;
(2) Ali Dabbagh:
School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran.

In this article the frequency response analysis of curved magneto-electro-viscoelastic functionally graded (CMEV-FG) nanobeams resting on viscoelastic foundation has been carried out. To this end, the study incorporates the Euler-Bernoulli beam model in association with Eringen's nonlocal theory to incorporate the size effects. The viscoelastic foundation in the current investigation is assumed to be the combination of Winkler-Pasternak layer and viscous layer of infinite parallel dashpots. The equations of motion are derived with the aid of Hamilton's principle and the solution to vibration problem of CMEV-FG nanobeams are obtained analytically. The material gradation is considered to follow Power-law rule. This study thoroughly investigates the influence of prominent parameters such as linear, shear and viscous layers of foundation, structural damping coefficient, opening angle, magneto-electrical field, nonlocal parameter, power-law exponent and slenderness ratio on the frequencies of FG nanobeams.

Key Words
curved nanobeaml free vibration; magneto-electro-viscoelastic materials; FGM; nonlocal elasticity

(1) Farzad Ebrahimi:
Department of Mechanical Engineering, Faculty of Engineering, Imam Khomeini International University, Postal code: 3414916818, Qazvin, Iran;
(2) Ramin Ebrahimi Fardshad, Vinyas Mahesh:
Department of Mechanical Engineering, Nitte Meenakshi Institute of Technology, Bangalore, India-560064.

This research is devoted to study post-buckling analysis of functionally graded carbon nanotubes reinforced composite (FG-CNTRC) micro plate with cut out subjected to magnetic field and resting on elastic medium. The basic formulation of plate is based on first order shear deformation theory (FSDT) and the material properties of FG-CNTRCs are presumed to be changed through the thickness direction, and are assumed based on rule of mixture; moreover, nonlocal Eringen\'s theory is applied to consider the size-dependent effect. It is considered that the system is embedded in elastic medium and subjected to longitudinal magnetic field. Energy approach, domain decomposition and Rayleigh-Ritz methods in conjunction with Newton-Raphson iterative technique are employed to trace the post-buckling paths of FG-CNTRC micro cut out plate. The influence of some important parameters such as small scale effect, cut out dimension, different types of FG distributions of CNTs, volume fraction of CNTs, aspect ratio of plate, magnitude of magnetic field, elastic medium and biaxial load on the post-buckling behavior of system are calculated. With respect to results, it is concluded that the aspect ratio and length of square cut out have negative effect on postbuckling response of micro composite plate. Furthermore, existence of CNTs in system causes improvement in the post-buckling behavior of plate and different distributions of CNTs in plate have diverse response. Meanwhile, nonlocal parameter and biaxial compression load on the plate has negative effect on post-buckling response. In addition, imposing magnetic field increases the postbuckling load of the microstructure.

Key Words
nonlocal nonlinear post-buckling; FG-CNTRC micro rectangular cut out plate; first order shear deformation plate theory; Newton-Raphson iterative technique; magnetic field

(1) M. Jamali, T. Shojaee, B. Mohammadi:
School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran;
(2) R. Kolahchi:
Institute of Research and Development, Duy Tan University, Da Nang 550000, Vietnam.

The effect of an increasing percentage of nanofiller (glass cenosphere) with Glass/Epoxy hybrid composite curved panels modeled mathematically using the multiscale concept and subsequent numerical eigenvalues of different geometrical configurations (cylindrical, spherical, elliptical, hyperboloid and flat) predicted in this research article. The numerical model of Glass/Epoxy/Cenosphere is derived using the higher-order polynomial type of kinematic theory in association with isoparametric finite element technique. The multiscale mathematical model utilized for the customized computer code for the evaluation of the frequency data. The numerical model validation and consistency verified with experimental frequency data and convergence test including the experimental elastic properties. The experimental frequencies of the multiscale nano filler-reinforced composite are recorded through the impact hammer frequency test rig including CDAQ-9178 (National Instruments) and LABVIEW virtual programming. Finally, the nano cenosphere filler percentage and different design associated geometrical parameters on the natural frequency data of hybrid composite structural configurations are illustrated through a series of numerical examples.

Key Words
modal responses; nano glass cenosphere; hybrid composite; FEM; experimental analysis

(1) Harsh Kumar Pandey:
Dr. C.V. Raman Institute of Science & Technology, Kargi Road Kota, Bilaspur (C.G.), India;
(2) Chetan Kumar Hirwani:
Department Mechanical Engineering, Aditya Engineering College, Aditya Nagar, Surampalem, Andhra Pradesh, India 533437;
(3) Nitin Sharma:
School of Mechanical Engineering, KIIT (Deemed to be University) Bhubaneswar: 751024, India;
(4) Pankaj V. Katariya, Hukum Chand Dewangan, Subrata Kumar Panda:
Department Mechanical Engineering, NIT Rourkela, Rourkela: 769008, Sundergarh, Odisha, India.

Vibration analysis of carbon nanotubes (CNTs) is very essential field owing to their many promising applications in tiny instruments. In current study, the Eringen\'s nonlocal elasticity theory with clamped-clamped and clamped-free end conditions is utilized for the vibration analysis of armchair and zigzag SWCNTs. The Fourier method is utilized to solve the ordinary differential equation. The motion equation for this system is developed using a novel wave propagation method. Complex exponential functions have been used and the axial model depends on BCs that has been described at the edges of CNTs. The behavior of different nonlocal parameters is considered to find the vibrational frequency of SWCNTs. It is exhibited that the effect of frequencies against aspect ratio by varying the bending rigidity. It has been investigated that by increasing the nonlocal parameter decreases the frequencies and on increasing the aspect ratio increases the frequencies for both the tubes. To generate the fundamental natural frequencies of SWCNTs, computer software MATLAB engaged. The numerical results are validated with existing open text. Since the percentage of error is negligible, the model has been concluded as valid.

Key Words
nonlocal theory; WPA; carbon nanotubes; wave propagation approach; Fourier method

(1) Muzamal Hussain, Muhammad Nawaz Naeem:
Department of Mathematics, Govt. College University Faisalabad, 38000, Faisalabad, Pakistan;
(2) Abdelouahed Tounsi:
Materials and Hydrology Laboratory, University of Sidi Bel Abbes, Algeria Faculty of Technology, Civil Engineering Department, Algeria;
(3) Abdelouahed Tounsi:
Department of Civil and Environmental Engineering, King Fahd University of Petroleum & Minerals, 31261 Dhahran, Eastern Province, Saudi Arabia;
(4) Muhammad Taj:
Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad, 1300, Azad Kashmir, Pakistan.

In this investigation, dynamic and bending behaviors of isolated protein microtubules are analyzed. Microtubules (MTs) can be considered as bio-composite structures that are elements of the cytoskeleton in eukaryotic cells and posses considerable roles in cellular activities. They have higher mechanical characteristics such as superior flexibility and stiffness. In the modeling purpose of microtubules according to a hollow beam element, a novel single variable sinusoidal beam model is proposed with the conjunction of modified strain gradient theory. The advantage of this model is found in its new displacement field involving only one unknown as the Euler-Bernoulli beam theory, which is even less than the Timoshenko beam theory. The equations of motion are constructed by considering Hamilton's principle. The obtained results are validated by comparing them with those given based on higher shear deformation beam theory containing a higher number of variables. A parametric investigation is established to examine the impacts of shear deformation, length scale coefficient, aspect ratio and shear modulus ratio on dynamic and bending behaviors of microtubules. It is remarked that when length scale coefficients are almost identical of the outer diameter of MTs, microstructure-dependent behavior becomes more important.

Key Words
protein microtubules; modified strain gradient theory; single variable beam theory; bending; vibration

(1) Djazia Leila Benmansour, Abdelmoumen Anis Bousahla, Houari Heireche:
Laboratoire de Modélisation et Simulation Multi-échelle, Département de Physique, Faculté des Sciences Exactes, Département de Physique, Université de Sidi Bel Abbés, Algeria;
(2) Abdelhakim Kaci:
Université Dr Tahar Moulay, Faculté de Technologie, Département de Génie Civil et Hydraulique, BP 138 Cité En-Nasr 20000 Saida, Algérie;
(3) Abdelhakim Kaci, Abdelouahed Tounsi:
Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department, Algeria;
(4) Abdelmoumen Anis Bousahla:
Centre Universitaire de Relizane, Algérie;
(5) Abdelouahed Tounsi:
Department of Civil and Environmental Engineering, King Fahd University of Petroleum & Minerals, 31261 Dhahran, Eastern Province, Saudi Arabia;
(6) Afaf S. Alwabli, Alawiah M. Alhebshi, Khalid Al-ghmady:
Department of Biology, Faculty of Sciences, King Abdulaziz University, Jeddah, Saudi Arabia;
(7) S.R. Mahmoud:
GRC Department, Jeddah Community College, King Abdulaziz University, Jeddah, Saudi Arabia.

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