2017
9
4
0
225
Viscous Fluid FlowInduced Nonlocal Nonlinear Vibration of Embedded DWBNNTs
2
2
In this article, electrothermo nonlocal nonlinear vibration and instability of viscousfluidconveying double–walled boron nitride nanotubes (DWBNNTs) embedded on Pasternak foundation are investigated. The DWBNNT is simulated as a Timoshenko beam (TB) which includes rotary inertia and transverse shear deformation in the formulation. Considering electromechanical coupling, the nonlinear governing equations are derived using Hamilton’s principle and discretized based on the differential quadrature method (DQM). The lowest four frequencies are determined for clampedclamped boundary condition. The effects of dimensionless small scale parameter, elastic medium coefficient, flow velocity, fluid viscosity and temperature change on the imaginary and real components of frequency are also taken into account. Results indicate that the electric potential increases with decreasing nonlocal parameter. It is also worth mentioning that decreasing nonlocal parameter and existence of Winkler and Pasternak foundation can enlarge the stability region of DWBNNT.
1

680
696


A
Ghorbanpour Arani
Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
Institute of Nanoscience & Nanotechnology, University of Kashan, Kashan, Iran
Faculty of Mechanical Engineering, University
Iran
aghorban@kashanu.ac.ir


Z
Khoddami Maraghi
Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
Faculty of Mechanical Engineering, University
Iran


R
Kolahchi
Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
Faculty of Mechanical Engineering, University
Iran


M
Mohammadimehr
Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
Faculty of Mechanical Engineering, University
Iran
Nonlinear vibration and instability
DWBNNTs
Pasternak foundation
Conveying viscous fluid
Piezoelasticity theory
[[1] Schwartz M., 2002, Smart Materials, John Wiley and Sons, A WileyInterscience Publication Inc., New York.##[2] Vang J., 2006, The Mechanics of Piezoelectric Structures, World Scientific Publishing Co., USA.##[3] Shahzad Khan M.D., Shahid Khan M., 2011, Computational study of hydrogen adsorption on potassiumdecorated boron nitride nanotubes, International Nano Letters 1(8): 103110.##[4] Yan Z., Jiang L.Y., 2011, The vibrational and buckling behaviors of piezoelectric nanobeams with surface effects, Nanotechnology 22(24): 245703.##[5] Wang C.M., Tan V.B.C., Zhang Y.Y., 2006, Timoshenko beam model for vibration analysis of multiwalled carbon nanotubes, Journal of Sound and Vibration 294(45): 10601072.##[6] Wang C.M., Zhang Y.Y., He X.Q., 2007, Vibration of nonlocal Timoshenko beams, Nanotechnology 18(10): 105401105409.##[7] Lu P., Lee H.P., Lu C., Zhang P.Q., 2007, Application of nonlocal beam models for carbon nanotubes, International Journal of Solids and Structures 44(16): 52895300.##[8] Chang W.J., Lee H.L., 2009, Free vibration of a singlewalled carbon nanotube containing a fluid flow using the Timoshenko beam model, Physics Letters A 373(10): 982985.##[9] Ke L.L., Xiang Y., Yang J., Kitipornchai S., 2009, Nonlinear free vibration of embedded doublewalled carbon nanotubes based on nonlocal Timoshenko beam theory, Computational Materials Science 47(2): 409417.##[10] Mohammadimehr M., Saidi A.R., Ghorbanpour Arani A., Arefmanesh A., Han Q., 2010, Torsional buckling of a DWCNT embedded on winkler and pasternak foundations using nonlocal theory, Journal of Mechanical Science and Technology 24(6): 12891299.##[11] Wang B., Zhao J., Zhou S., 2010, A micro scale Timoshenko beam model based on strain gradient elasticity theory, European Journal of Mechanics  A/Solids 29(4): 591599.##[12] Jiang L.Y., Yan Z., 2010, Timoshenko beam model for static bending of nanowires with surface effects, Physica E: Lowdimensional Systems and Nanostructures 42(9): 22742279.##[13] Asghari M., Kahrobaiyan M.H., Ahmadian M.T., 2010, A nonlinear Timoshenko beam formulation based on the modified couple stress theory, International Journal of Engineering Science 48(12): 17491761.##[14] Yang Y., Zhang L., Lim C.W., 2011, Wave propagation in doublewalled carbon nanotubes on a novel analytically nonlocal Timoshenkobeam model, Journal of Sound and Vibration 330(8): 17041717.##[15] Lei X.W., Natsuki T., Shi J.X., Ni Q.Q., 2012, Surface effects on the vibrational frequency of doublewalled carbon nanotubes using the nonlocal Timoshenko beam model, Composites Part B: Engineering 43(1): 6469.##[16] Shen Z.B., Li X.F., Sheng L.P., Tang G.J., 2012, Transverse vibration of nanotubebased micromass sensor via nonlocal Timoshenko beam theory, Computational Materials Science 53(1): 340346.##[17] Yan Z., Jiang L.Y., 2011, Surface effects on the electromechanical coupling and bending behaviors of piezoelectric nanowires, Journal of Physics D: Applied Physics 44(7): 075404.##[18] SalehiKhojin A., Jalili N., 2008, Buckling of boron nitride nanotube reinforced piezoelectric polymeric composites subject to combined electrothermomechanical loadings, Composites Science and Technology 68(6): 14891501.##[19] Ghorbanpour Arani A., Amir S., Shajari A.R., Mozdianfard M.R., Khoddami Maraghi Z., Mohammadimehr M., 2011, Electrothermal nonlocal vibration analysis of embedded DWBNNTs, Proceedings of the Institution of Mechanical Engineers, Part C 224(26): 745756.##[20] Ghorbanpour Arani A., Kolahchi R., Mosallaie Barzoki A.A., 2011, Effect of material inhomogeneity on electrothermomechanical behaviors of functionally graded piezoelectric rotating cylinder, Applied Mathematical Modelling 35(6): 27712789.##[21] Ghorbanpour Arani A., Amir S., Shajari A.R., Mozdianfard M.R., 2012, Electrothermomechanical buckling of DWBNNTs embedded in bundle of CNTs using nonlocal piezoelasticity cylindrical shell theory, Composites Part B: Engineering 43(2): 195203.##[22] Ghorbanpour Arani A., Shokravi M., Amir S., Mozdianfard M.R., 2012, Nonlocal electrothermal transverse vibration of embedded fluidconveying DWBNNTs, Journal of Mechanical Science and Technology 26(5): 14551462.##[23] Eringen A.C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 54(9): 47034710.##[24] Wang L., Ni Q., 2009, A reappraisal of the computational modelling of carbon nanotubes conveying viscous fluid, Mechanics Research Communications 36(7): 833837.##[25] Kuang Y.D., He X.Q., Chen C.Y., Li G.Q., 2009, Analysis of nonlinear vibrations of doublewalled carbon nanotubes conveying fluid, Computational Materials Science 45(4): 875880.##[26] Karami G., Malekzadeh P., 2002, A new differential quadrature methodology for beam analysis and the associated differential quadrature element method, Computer Methods in Applied Mechanics and Engineering 191(32): 35093526.##[27] Ke L.L., Wang Y.S., 2011, Flowinduced vibration and instability of embedded doublewalled carbon nanotubes based on a modified couple stress theory, Physica E: Lowdimensional Systems and Nanostructures 43(5): 10311039.##[28] Mosallaie Barzoki A.A., Ghorbanpour Arani A., Kolahchi R., Mozdianfard M.R., 2012, Electrothermomechanical torsional buckling of a piezoelectric polymeric cylindrical shell reinforced by DWBNNTs with an elastic core, Applied Mathematical Modelling 36(7): 29832995.##[29] Ghavanloo E., Daneshmand F., Rafiei M., 2010, Vibration and instability analysis of carbon nanotubes conveying fluid and resting on a linear viscoelastic Winkler foundation, Physica E: Lowdimensional Systems and Nanostructures 42(9): 22182224.##[30] Wang L., Ni Q., Li M., Qian Q., 2008, The thermal effect on vibration and instability of carbon nanotubes conveying fluid, Physica E: Lowdimensional Systems and Nanostructures 40: 31793182.##[31] Khodami Maraghi Z., Ghorbanpour Arani A., Kolahchi R., Amir S., Bagheri M.R., 2012, Nonlocal vibration and instability of embedded DWBNNT conveying viscose fluid, Composites Part B: Engineering 45 (1): 423432.##[32] Ghorbanpour Arani A., Kolahchi R., Khoddami Maraghi Z., 2013, Nonlinear vibration and instability of embedded doublewalled boron nitride nanotubes based on nonlocal cylindrical shell theory, Applied Mathematical Modelling 37(1415): 76857707.##]
An Experimental Investigation on Fracture Analysis of Polymer Matrix Composite under Different Thermal Cycling Conditions
2
2
Fracture analysis of glass/epoxy composites under different thermal cycling conditions is considered. Temperature difference, stacking sequence, fiber volume fraction and number of thermal cycles are selected as the experimental design factors. The Taguchi method is implemented to design of the experiment and an apparatus is developed for automatic thermal cycling tests. The tensile tests are done to study mechanical behavior of the specimens after the thermal cycling. The results show that the stacking sequence is the main effective factor on the fracture surface behavior of the specimens. Also, long splitting, lateral and angled breakage are the dominate failure mode of [0]8, [02/902]s and [0/±45/90]s layups, respectively. It’s found that the thermal cycling and temperature difference cause to increase the surface matrix loss significantly. This surface matrix loss can be an initial region to matrix debonding and crack propagation. Also, when the angle difference between lamina is increased the mechanical properties are reduced under the thermal cycling, significantly.
1

697
706


A.R
Ghasemi
Faculty of Mechanical Engineering, Department of Solid Mechanics, University of Kashan, Kashan, Iran
Faculty of Mechanical Engineering, Department
Iran
ghasemi@kashanu.ac.ir


M
Moradi
Faculty of Mechanical Engineering, Department of Solid Mechanics, University of Kashan, Kashan, Iran
Faculty of Mechanical Engineering, Department
Iran
Thermal cycling
Fracture of composite
Polymer matrix composites (PMCs)
Taguchi Method
[[1] Kessler S., McManus H., Matuszeski T., 2001, The effects of cryocycling on the mechanical properties of IM7/9772, Proceedings of the American Society for Composites, Blacksburg.##[2] Paillous A., Pailler C., 1994, Degradation of multiply polymermatrix composites induced by space environment, Composite 25(4): 287295.##[3] Ramanujam N., Vaddadi P., Nakamura T., Singh R.P., 2008, Interlaminar fatigue crack growth of crossply composites under thermal cycles, Composite Structures 85: 175187.##[4] Eselun S.A., Neubert H.D., Wolff E.G., 1979, Microcracking effects on dimensional stability, National SAMPE Technical Conference, Proceedings of the 24th Conference.##[5] Rinaldi G., Maura G., 1993, Durable glass–epoxy composites cured at low temperatures– effects of thermal cycling, UV irradiation and wet environmental, Polymer International 31(3): 339345.##[6] Cohen D., Hyer M.W., Tompkins S.S., 1984, The effects of thermal cycling on matrix cracking and stiffness changes in composite tubes, National SAMPE Technical Conference, Proceedings of the 16th Conference.##[7] Ghasemi A.R., Baghersad R., Sereshk M.R.V., 2011, Nonlinear behavior of polymer based composite laminates under cyclic thermal shock and its effects on residual stresses, Journal of Polymer Science and Technology 24(2): 133140.##[8] Adams D.S., Bowles D.E., Herakovich C.T., 1986, Thermally induced transverse cracking in graphiteepoxy crossply laminates, Journal of Reinforced Plastics & Composites 3: 152169.##[9] Rouquie S., Lafarie_Frenot M. C., Cinquin J., Colombaro A. M., 2005, Thermal cycling of carbon/epoxy laminates in neutral and oxidative environments, Composites Science and Technology 65: 403409.##[10] Scida D., Assarar M., Poilâne C., Ayad R., 2013, Influence of hygrothermal ageing on the damage mechanisms of flaxfibre reinforced epoxy composite, Composites Part B:Engineering 48: 5158.##[11] LafarieFrenot M.C., Rouquie S., 2004, Influence of oxidative environments on damage in c/epoxy laminates subjected to thermal cycling, Composite Science and Technology 64(1011): 17251735.##[12] Mouzakis D.E., Zoga H., Galiotis C., 2008, Accelerated environmental ageing study of polyester/glass fiber reinforced composites (GFRPCs), Composites Part B:Engineering 39: 467475.##[13] LafarieFrenot M.C., Ho N. Q., 2006, Influence of free edge intralaminar stresses on damage process in CFRP laminates under thermal cycling conditions, Composites Science and Technology 66(10): 13541365.##[14] Devore J.L., Farnum N.R., 1999, Applied Statistics for Engineering and Scientists, Duxbury Press, Pacific Grove, CA.##[15] Ju J., 2007, Transverse cracking of M40J/PMRII50 composites under thermalmechanical loading: Part I Characterization of main and interaction effects using statistical design of experiments, Journal of Composite Materials 41(9): 10671086.##[16] Lochner R.H., Matar J.E.,1990, Design for Quality–An Introduction to the Best of Taguchi and Western Methods of Statistical Experimental Design, New York.##[17] Vankanti V.K., Ganta V., 2014, Optimization of process parameters in drilling of GFRP composite using Taguchi method, Journal of Material Research & Technology 3(1): 3541.##[18] ASTM D3039/ D 3039 M95a, 1997, Standard test method for tensile properties of polymer matrix composite materials.##[19] Ghasemi A.R., Moradi M., 2015, Surface degradation of polymer matrix composites under different low thermal cycling conditions, Journal of Solid Mechanics 9(1): 5462.##[20] MINITAB 17 statistical software, Minitab Inc, 2013.##]
Nonlocal Piezomagnetoelasticity Theory for Buckling Analysis of Piezoelectric/Magnetostrictive Nanobeams Including Surface Effects
2
2
This paper presents the surface piezomagnetoelasticity theory for sizedependent buckling analysis of an embedded piezoelectric/magnetostrictive nanobeam (PMNB). It is assumed that the subjected forces from the surrounding medium contain both normal and shear components. Therefore, the surrounded elastic foundation is modeled by Pasternak foundation. The nonlocal piezomagnetoelasticity theory is applied so as to consider the small scale effects. Based on Timoshenko beam (TB) theory and using energy method and Hamilton’s principle the motion equations are obtained. By employing an analytical method, the critical magnetic, electrical and mechanical buckling loads of the nanobeam are yielded. Results are presented graphically to show the influences of small scale parameter, surrounding elastic medium, surface layers, and external electric and magnetic potentials on the buckling behaviors of PMNBs. Results delineate the significance of surface layers and external electric and magnetic potentials on the critical buckling loads of PMNBs. It is revealed that the critical magnetic, electrical and mechanical buckling loads decrease with increasing the small scale parameter. The results of this work is hoped to be of use in micro/nano electro mechanical systems (MEMS/NEMS) especially in designing and manufacturing electromagnetoelastic sensors and actuators.
1

707
729


A
Ghorbanpour Arani
Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran 
Institute of Nanoscience & Nanotechnology, University of Kashan, Kashan, Iran
Faculty of Mechanical Engineering, University
Iran
aghorban@kashanu.ac.ir


M
Abdollahian
Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
Faculty of Mechanical Engineering, University
Iran


A.H
Rahmati
Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
Faculty of Mechanical Engineering, University
Iran
Magnetostrictive piezoelectric
Nanobeam
Surface effect
Nonlocal piezomagnetoelasticity theory
Buckling
[[1] Eringen A.C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 54: 47034710.##[2] Aydogdu M., 2009, A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration, Physica E 41: 16511655.##[3] Thai H.T., 2012, A nonlocal beam theory for bending, buckling, and vibration of nanobeams, International Journal of Engineering Science 52: 5664.##[4] Nazemnezhad R., HosseiniHashemi S., 2014, Nonlocal nonlinear free vibration of functionally graded nanobeams, Composite Structure 110: 192199.##[5] Rahmani O., Pedram O., 2014, Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory, International Journal of Engineering Science 77: 5570.##[6] Marotti de Sciarra F., Barretta R., 2014, A gradient model for Timoshenko nanobeams, Physica E 62: 19.##[7] Akgöz B., Civalek Ö., 2013, A sizedependent shear deformation beam model based on the strain gradient elasticity theory, International Journal of Engineering Science 70: 114.##[8] Mohammad Abadi M., Daneshmehr A.R., 2014, Size dependent buckling analysis of microbeams based on modified couple stress theory with high order theories and general boundary conditions, International Journal of Engineering Science 74: 114.##[9] Muralt P., 2001, Piezoelectric thin films for MEMS, Encyclopedia of Materials: Science and Technology 69997008.##[10] Nabar B.P., ÇelikButler Z., Butler D.P., 2014, Piezoelectric ZnO nanorod carpet as a NEMS vibrational energy harvester, Nano Energy 10: 7182.##[11] Falconi C., Mantini G., D’Amico A., Wang Z.L., 2009, Studying piezoelectric nanowires and nanowalls for energy harvesting, Sensors and Actuators B 139: 511519.##[12] Sun C., Shi J., Wang X., 2010, Fundamental study of mechanical energy harvesting using piezoelectric nanostructures, Journal of Applied Physics 108: 034309.##[13] Ghorbanpour Arani A., Abdollahian M., Kolahchi R., Rahmati A.H., 2013, Electrothermotorsional buckling of an embedded armchair DWBNNT using nonlocal shear deformable shell model, Composites Part B 51: 291299.##[14] Chen C., Li S., Dai L., Qian C.Z., 2014, Buckling and stability analysis of a piezoelectric viscoelastic nanobeam subjected to van der Waals forces, Communications in Nonlinear Science and Numerical Simulation 19: 16261637.##[15] Asemi S.R., Farajpour A., Mohammadi M., 2014, Nonlinear vibration analysis of piezoelectric nanoelectromechanical resonators based on nonlocal elasticity theory, Composite Structures 116: 703712.##[16] Pradhan S.C., Reddy G.K., 2011, Buckling analysis of single walled carbon nanotube on Winkler foundation using nonlocal elasticity theory and DTM, Computational Materials Science 50: 10521056.##[17] Han Q., Lu G., 2003, Torsional buckling of a doublewalled carbon nanotube embedded in an elastic medium, European Journal of MechanicsA/Solids 22: 875883.##[18] Rahmati A.H., Mohammadimehr M., 2014, Vibration analysis of nonuniform and nonhomogeneous boron nitride nanorods embedded in an elastic medium under combined loadings using DQM, Physica B 440: 8898.##[19] Abdollahian A., Ghorbanpour Arani A., Mosallaei Barzoki A.A., Kolahchi, R., Loghman A., 2013, Nonlocal wave propagation in embedded armchair TWBNNTs conveying viscous fluid using DQM, Physica B 418: 115.##[20] Zenkour A.M., Sobhy M., 2013, Nonlocal elasticity theory for thermal buckling of nanoplates lying on Winkler–Pasternak elastic substrate medium, Physica E 53: 251259.##[21] Moshtaghin A.F., Naghdabadi R., Asghari M., 2012, Effects of surface residual stress and surface elasticity on the overall yield surfaces of nanoporous materials with cylindrical nanovoids, Mechanics of Materials 51: 7487.##[22] Zhang C.h., Chen W., Zhang C.h., 2012, On propagation of antiplane shear waves in piezoelectric plates with surface effect, Physics Letters A 376: 32813286.##[23] Zhang L.L., Liu J.X., Fang X.Q., Nie G.Q., 2014, Effects of surface piezoelectricity and nonlocal scale on wave propagation in piezoelectric nanoplates, European Journal of MechanicsA/Solids 46: 2229.##[24] HosseiniHashemi S., Nazemnezhad R., Bedroud M., 2014, Surface effects on nonlinear free vibration of functionally graded nanobeams using nonlocal elasticity, Applied Mathematical Modelling 38: 35383553.##[25] Ansari R., Sahmani S., 2011, Bending behavior and buckling of nanobeams including surface stress effects corresponding to different beam theories, International Journal of Engineering Science 49: 12441255.##[26] Reddy K.S.M., Estrine E.C., Lim D.H., Smyrl W.H., Stadler B.J.H., 2012, Controlled electrochemical deposition of magnetostrictive Fe1xGax alloys, Electrochemistry Communications 18: 127130.##[27] Guo J., Moritaa S., Yamagataa Y., Higuchi T., 2013, Magnetostrictive vibrator utilizing iron–cobalt alloy, Sensors and Actuators A 200: 101106.##[28] Wang H., Zhang Z.D., Wu R.Q., Sun L.Z., 2013, Largescale firstprinciples determination of anisotropic mechanical properties of magnetostrictive Fe–Ga alloys, Acta Materialia 61: 29192925.##[29] EspinosaAlmeyda Y., RodríguezRamos R., GuinovartDíaz R., BravoCastillero J., LópezRealpozo J.C., CamachoMontes H., Sabina F.J., Lebon F., 2014, Antiplane magnetoelectroelastic effective properties of threephase fiber composites, International Journal of Solids and Structures 51: 35083521.##[30] Elloumi R., KallelKamoun I., ElBorgi S., Guler M.A., 2014, On the frictional sliding contact problem between a rigid circular conducting punch and a magnetoelectroelastic halfplane, International Journal of Mechanical Sciences 87: 117.##[31] Ma J., Ke L.L., Wang Y.S., 2014, Frictionless contact of a functionally graded magnetoelectroelastic layered halfplane under a conducting punch, International Journal of Solids and Structures 51: 27912806.##[32] Wei J., Su X., 2008, Transientstate response of wave propagation in magnetoelectroelastic square column, Acta Mechanica Solida Sinica 21: 491499.##[33] Lang Z., Xuewu L., 2013, Buckling and vibration analysis of functionally graded magnetoelectrothermoelastic circular cylindrical shells, Applied Mathematical Modelling 37: 22792292.##[34] Li Y.S., 2014, Buckling analysis of magnetoelectroelastic plate resting on Pasternak elastic foundation, Mechanics Research Communications 56: 104114.##[35] Razavi S., Shooshtari A., 2015, Nonlinear free vibration of magnetoelectroelastic rectangular plates, Composite Structures 119: 377384.##[36] Li Y.S., Cai Z.Y., Shi S.Y., 2014, Buckling and free vibration of magnetoelectroelastic nanoplate based on nonlocal theory, Composite Structures 111: 522529.##[37] Ke L.L., Wang Y.S., 2014, Free vibration of sizedependent magnetoelectroelastic nanobeams based on the nonlocal theory, Physica E 63: 5261.##[38] Ghorbanpour Arani A., Abdollahian M., Kolahchi R., 2014, Nonlinear vibration of embedded smart composite microtube conveying fluid based on modified couple stress theory, Polymer Composites 36: 13141324.##[39] Gurtin M.E., Murdoch A.I., 1957, A continuum theory of elastic material surface, Archived for Rational Mechanics Analysis 57: 291323.##[40] Ansari R., Mohammadi V., Faghih Shojaei M., Gholami R., Sahmani S., 2013, Postbuckling characteristics of nanobeams based on the surface elasticity theory, Composites part B 55: 240246.##[41] Lu P., He L.H., Lee H.P., Lu C., 2006, Thin plate theory including surface effects, International Journal of Solids and Structures 43: 46314647.##]
Coupled BendingLongitudinal Vibration of Three Layer Sandwich Beam using Exact Dynamic Stiffness Matrix
2
2
A Newtonian (vectorial) approach is used to develop the governing differential equations of motion for a three layer sandwich beam in which the uniform distribution of mass and stiffness is dealt with exactly. The model allows for each layer of material to be of unequal thickness and the effects of coupled bending and longitudinal motion are accounted for. This results in an eighth order ordinary differential equation whose closed form solution is developed into an exact dynamic member stiffness matrix (exact finite element) for the beam. Such beams can then be assembled to model a variety of structures in the usual manner. However, such a formulation necessitates the solution of a transcendental eigenvalue problem. This is accomplished using the WittrickWilliams algorithm, whose implementation is discussed in detail. The algorithm enables any desired natural frequency to be converged upon to any required accuracy with the certain knowledge that none have been missed. The accuracy of the method is then confirmed by comparison with five sets of published results together with a further example that indicates its range of application. A number of further issues are considered that arise from the difference between sandwich beams and uniform single material beams, including the accuracy of the characteristic equation, coordinate transformations, modal coupling and the application of boundary conditions.
1

730
750


A.
Zare
School of Engineering, Yasouj University, Yasouj, Iran
School of Engineering, Yasouj University,
Iran
zare@yu.ac.ir


B
Rafezy
Sahand University of Technology, Tabriz, Iran
Sahand University of Technology, Tabriz, Iran
Iran


W.P
Howson
Independent Consultant, Gwanwyn, Craig Penlline, CF71 7RT, UK
Independent Consultant, Gwanwyn, Craig Penlline,
England
Sandwich beam
Exact dynamic stiffness matrix
Coupled motion
Transcendental eigenvalue problem
WittrickWilliams algorithm
[[1] Kerwin E. M., 1959, Damping of flexural waves by a constrained viscoelastic layer, Journal of the Acoustical Society of America 31: 952962.##[2] Mead D. J., 1962, The DoubleSkin Damping Configuration, University of Southampton.##[3] DiTaranto R. A., 1965, Theory of vibratory bending for elastic and viscoelastic layered finitelength beams, Journal of Applied Mechanics 32: 881886.##[4] Yin T. P., Kelly T. J., Barry J. E., 1967, A quantitative evaluation of constrained layer damping, Transactions of the American Society of Mechanical Engineers, Journal of Engineering for Industry 89: 773784.##[5] Mead D. J., Markus S., 1969, The forced vibration of a threelayer, damped sandwich beam with arbitrary boundary conditions, Journal of Sound and Vibration 10: 163175.##[6] Rao D. K., 1978, Frequency and loss factors of sandwich beams under various boundary conditions, Journal of Mechanical Engineering Science 20: 271282.##[7] Sakiyama T., Matsuda H., Morita C., 1996, Free vibration analysis of sandwich beam with elastic or viscoelastic core by applying the discrete Green function, Journal of Sound and Vibration 191: 189206.##[8] Yu Y.Y., 1959, A new theory of elastic sandwich plateone dimensional case, Journal of Applied Mechanics 26: 415421.##[9] Rao Y. V. K. S., Nakra B. C., 1970, Influence of rotary and longitudinal translatory inertia on the vibrations of unsymmetrical sandwich beams, Proceeding of the 15th Conference I.S.T.A.M.##[10] Mead D. J., 1982, A comparison of some equations for the flexural vibration of damped sandwich beams, Journal of Sound and Vibration 83: 363377.##[11] Chonan S., 1982, Vibration and stability of sandwich beams with elastic bonding, Journal of Sound and Vibration 85(4): 525537.##[12] Mead D. J., Markus S., 1985, Coupled flexural, longitudinal and shear wave motion in two and threelayered damped beams, Journal of Sound and Vibration 99(4): 501519.##[13] Marur S. R., Kant T., 1996, Free vibration analysis of fiber reinforced composite beams using higher order theories and finite element modeling, Journal of Sound and Vibration 194: 337351.##[14] Kameswara Rao M., Desai Y.M., Chitnis M. R., 2001, Free vibration of laminated beams using mixed theory, Composite Structures 52: 149160.##[15] Silverman I. K., 1995, Natural frequencies of sandwich beams including shear and rotary effects, Journal of Sound and Vibration 183: 547561.##[16] Fasana A., Marchesiello S., 2001, RayleighRitz analysis of sandwich beams, Journal of Sound and Vibration 241: 643652.##[17] Amirani M. C., Khalili S. M. R., Nemati N., 2009, Free vibration analysis of sandwich beam with FG core using the element free Galerkin method, Composite Structures 90: 373379.##[18] Hashemi S. M., Adique E. J., 2009, Free vibration analysis of sandwich beams: A dynamic finite element, International Journal of Vehicle Structures and Systems 1(4): 5965.##[19] Hashemi S. M., Adique E. J., 2010, A quasiexact dynamic finite element for free vibration analysis of sandwich beams, Applied Composite Materials 17(2): 259269.##[20] Banerjee J. R., 2003, Free vibration of sandwich beams using the dynamic stiffness method, Computers and Structures 81: 19151922.##[21] Howson W. P., Zare A., 2005, Exact dynamic stiffness matrix for flexural vibration of threelayered sandwich beams, Journal of Sound and Vibration 282: 753767.##[22] Banerjee J. R., Sobey A.J., 2005, Dynamic stiffness formulation and free vibration analysis of a threelayered sandwich beam, International Journal of Solids and Structures 42(8): 21812197.##[23] Banerjee J. R., Cheung C. W., Morishima R., Perera M., Njuguna J., 2007, Free vibration of a threelayered sandwich beam using the dynamic stiffness method and experiment, International Journal of Solids and Structures 44: 75437563.##[24] Jun L., Xiaobin L., Hongxing H., 2009, Free vibration analysis of thirdorder shear deformable composite beams using dynamic stiffness method, Archive of Applied Mechanics 79: 10831098.##[25] Khalili S. M. R., Nemati N., Malekzadeh K., Damanpack A. R., 2010, Free vibration analysis of sandwich beams using improved dynamic stiffness method, Composite Structures 92: 387394.##[26] Damanpack A. R., Khalili S. M. R., 2012, Highorder free vibration analysis of sandwich beams with a flexible core using dynamic stiffness method, Composite Structures 94: 15031514.##[27] Wittrick W. H., Williams F. W., 1971, A general algorithm for computing natural frequencies of elastic structures, Quarterly Journal of Mechanics and Applied Mathematics 24: 263284.##[28] Howson W. P., Williams F. W., 1973, Natural frequencies of frames with axially loaded Timoshenko members, Journal of Sound and Vibration 26: 503515.##[29] Zare A., 2004, Exact Vibrational Analysis of Prismatic Plate and Sandwich Structures, Ph.D. Thesis, Cardiff University.##[30] Ahmed K. M., 1971, Free vibration of curved sandwich beams by the method of finite elements, Journal of Sound and Vibration 18: 6174.##[31] Ahmed K. M., 1972, Dynamic analysis of sandwich beams, Journal of Sound and Vibration 10: 263276.##[32] Sakiyama T., Matsuda H., Morita C., 1997, Free vibration analysis of sandwich arches with elastic or viscoelastic core and various kinds of axisshape and boundary conditions, Journal of Sound and Vibration 203(3): 505522.##[33] Bozhevolnaya E., Sun J. Q., 2004, Free vibration analysis of curved sandwich beams, Journal of Sandwich Structures & Materials 6(1): 4773.##[34] Petrone F., Garesci F., Lacagnina M., Sinatra R., 1999, Dynamical joints influence of sandwich plates, Proceedings of the 3rd European Nonlinear Oscillations Conference, Copenhagen, Denmark.##[35] Marura S. R., Kant T., 2008, Free vibration of higherorder sandwich and composite arches, Part I: Formulation, Journal of Sound and Vibration 310: 91109.##]
ThreeDimensional Finite Element Analysis of Stress Intensity Factors in a Spherical Pressure Vessel with Functionally Graded Coating
2
2
This research pertains to the threedimensional (3D) finite element analysis (FEA) of the stress intensity factors (SIFs) along the crack front in a spherical pressure vessel coated with functionally graded material (FGM). The vessel is subjected to internal pressure and thermal gradient. The exponential function is adopted for property of FGMs. SIFs are obtained for a wide variety of crack shapes and layer thickness. The reported results clearly show that the material gradation of coating and the crack configuration can significantly affect the variation of SIFs along the crack front. The results are given which are applicable for fatigue life assessment and fracture endurance of FGM coating spherical pressure vessel and can be used in design purposes.
1

751
759


H
Eskandari
Abadan Institute of Technology, Petroleum University of Technology, Abadan, Iran
Abadan Institute of Technology, Petroleum
Iran
eskandari@put.ac.ir
Spherical pressure vessel
Functionally graded coating
Stress intensity factor
3D crack
Finite Element Analysis
[[1] Akis T., 2009, Elastoplastic analysis of functionally graded spherical pressure vessels, Computational Materials Science 46: 545554.##[2] Perl M., Bernshtein V., 2012, Threedimensional stress intensity factors for ring cracks and arrays of coplanar cracks emanating from the inner surface of a spherical pressure vessel, Engineering Fracture Mechanics 94: 7184.##[3] Perl M., Bernshtein V., 2011, 3D stress intensity factors for arrays of inner radial lunular or crescentic cracks in thin and thick spherical pressure vessels, Engineering Fracture Mechanics 78: 14661477.##[4] Hakimi A.E., Le Grognec P., Hariri S., 2008, Numerical and analytical study of severity of cracks in cylindrical and spherical shells, Engineering Fracture Mechanics 75: 10271044.##[5] Perl M., Bernshtein V., 2010, 3D stress intensity factors for arrays of inner radial lunular or crescentic cracks in a typical spherical pressure vessels, Engineering Fracture Mechanics 77: 535548.##[6] You L.H., Zhang J.J., You X.Y., 2005, Elastic analysis of internally pressurized thickwalled spherical pressure vessels of functionally graded materials, International Journal of Pressure Vessels and Piping 82(5): 347354.##[7] Chen Y.Z., Lin X.Y., 2008, Elastic analysis for thick cylinders and spherical pressure vessels made of functionally graded materials, Computational Materials Science 44: 581587.##[8] Choules B.D., Kokini K., Taylor T.A., 2001, Thermal fracture of ceramic thermal barrier coatings under high heat flux with time dependent behavior – Part I: Experimental results, Materials Science and Engineering: A 299:296304.##[9] Rangaraj S., Kokini K., 2004, A study of thermal fracture in functionally graded thermal barrier coatings using a cohesive zone model, Journal of Engineering Materials and Technology 126: 103115.##[10] Eischen J.W., 1987, Fracture of nonhomogeneous materials, International Journal of Fracture 34: 322.##[11] Zhang C., Cui M., Wang J., Gao X.W., Sladek J., Sladek V., 2011, 3D crack analysis in functionally graded materials, Engineering Fracture Mechanics 78: 585604.##[12] Paulino G.H., Kim J.H., 2004, On the poisson’s ratio effect on mixedmode stress intensity factors and Tstress in functionally graded materials, International Journal of Computational Engineering Science 5:833861.##[13] Mohammadi M., Dryden J.R., 2009, Influence of the spatial variation of Poisson’s ratio upon the elastic field in nonhomogeneous axisymmetric bodies, International Journal of Solids and Structures 46:788795.##[14] Moghaddam A.S., Alfano M., Ghajar R., 2013, Determining the mixed mode stress intensity factors of surface cracks in functionally graded hollow cylinders, Materials & Design 43: 475484##[15] Raju I.S., Newman Jr. J.C., 1980, Stress intensity factors for internal surface cracks in cylindrical pressure vessel, ASME Journal of Pressure Vessel Technology 102: 342356.##]
Exact ClosedForm Solution for Vibration Analysis of Truncated Conical and Tapered Beams Carrying Multiple Concentrated Masses
2
2
In this paper, an exact closedform solution is presented for free vibration analysis of EulerBernoulli conical and tapered beams carrying any desired number of attached masses. The concentrated masses are modeled by Dirac’s delta functions which creates no need for implementation of compatibility conditions. The proposed technique explicitly provides frequency equation and corresponding mode as functions with only two integration constants which leads to solution of a two by two eigenvalue problem for any number of attached masses. Using Basic functions which are made of the appropriate linear composition of Bessel functions leads to make implementation of boundary conditions much easier. The proposed technique is employed to study effect of quantity, position and translational inertia of the concentrated masses on the natural frequencies and corresponding modes of conical and tapered beams for all standard boundary conditions. Unlike many of previous exact approaches, presented solution has no limitation in number of concentrated masses. In other words, by increase in number of attached masses, there is no considerable increase in computational effort.
1

760
782


K
Torabi
Department Mechanical Engineering, Faculty of Engineering, University of Isfahan, Isfahan, Iran
Department Mechanical Engineering, Faculty
Iran
k.torabi@eng.ui.ac.ir


H
Afshari
Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Isfahan, Iran
Department of Mechanical Engineering, Khomeinishah
Iran


M
Sadeghi
Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
Department of Solid Mechanics, Faculty of
Iran


H
Toghian
Department of Mechanical Engineering, Islamic Azad University, Najafabad Branch, Najafabad, Iran
Department of Mechanical Engineering, Islamic
Iran
Exact solution
Transverse vibration
Concentrated mass
Conical beam
Tapered beam
[[1] Chen Y., 1963, On the vibration of beams or rods carrying a concentrated mass, Journal of Applied Mechanics 30: 310311.##[2] Low K.H., 2000, A modified Dunkerley formula for eigenfrequencies of beams carrying concentrated masses, International Journal of Mechanical Sciences 42: 12871305.##[3] Laura P.A.A., Pombo J.L., Susemihl E.L., 1974, A note on the vibration of a clamped–free beam with a mass at the free end, Journal of Sound and Vibration 37: 161168.##[4] Dowell E.H., 1979, On some general properties of combined dynamical systems, ASME Journal of Applied Mechanics 46: 206209.##[5] Laura P.A.A., Irassar P.L., Ficcadenti G.M., 1983, A note of transverse vibration of continuous beams subjected to an axial force and carrying concentrated masses, Journal of Sound and Vibration 86: 279284.##[6] Gurgoze M., 1984, A note on the vibrations of restrained beams and rods with point masses, Journal of Sound and Vibration 96: 461468.##[7] Gurgoze M., 1985, On the vibration of restrained beams and rods with heavy masses, Journal of Sound and Vibration 100: 588589.##[8] Liu W.H., Wu J.R., Huang C.C., 1988, Free vibrations of beams with elastically restrained edges and intermediate concentrated masses, Journal of Sound and Vibration 122: 193207.##[9] Torabi K., Afshari H., Najafi H., 2013, Vibration analysis of multistep BernoulliEuler and Timoshenko beams carrying concentrated masses, Journal of Solid Mechanics 5: 336349.##[10] Farghaly S.H., ElSayed T.A., 2016, Exact free vibration of multistep Timoshenko beam system with several attachments, Mechanical Systems and Signal Processing 7273: 525546.##[11] Cranch E.T., Adler A.A., 1956, Bending vibration of variable section beams, Journal of Applied Mechanics 23: 103108.##[12] Conway H.D., Dubil J.F., 1965, Vibration frequencies of truncatedcone and wedge beams, Journal of Applied Mechanics 32: 932934.##[13] Mabie H.H., Rogers C.B., 1968, Transverse vibration of tapered cantilever beams with end support, Journal of Acoustics Society of America 44: 17391741.##[14] Heidebrecht A.C., 1967, Vibration of nonuniform simply supported beams, Journal of the Engineering Mechanics Division 93: 115.##[15] Mabie H.H., Rogers C.B., 1972, Transverse vibration of doubletapered cantilever beams, Journal of Acoustics Society of America 5: 17711775.##[16] Goel R.P., 1976, Transverse vibrations of tapered beams, Journal of Sound and Vibration 47: 17.##[17] Downs B., 1977, Transverse vibration of cantilever beams having unequal breadth and depth tapers, Journal of Applied Mechanics 44: 737742.##[18] Bailey C.D., 1978, Direct analytical solution to nonuniform beam problems, Journal of Sound and Vibration 56: 501507.##[19] Gupta A.K., 1985, Vibration of tapered beams, Journal of Structural Engineering 111: 1936.##[20] Naguleswaran S., 1992, Vibration of an Euler–Bernoulli beam of constant depth and with linearly varying breadth, Journal of Sound and Vibration 153: 509522.##[21] Naguleswaran S., 1994, A direct solution for the transverse vibration of Euler–Bernoulli wedge and cone beams, Journal of Sound and Vibration 172: 289304.##[22] Abrate S., 1995, Vibration of nonuniform rods and beams, Journal of Sound and Vibration 185: 703716.##[23] Laura P.A.A., 1996, Gutierrez R.H., Rossi R.E., Free vibration of beams of bilinearly varying thickness, Ocean Engineering 23: 16.##[24] Datta A.K., Sil S.N., 1996, An analysis of free undamped vibration of beams of varying crosssection, Computers & Structures 59: 479483.##[25] Hoffmann J.A., Wertheimer T., 2000, Cantilever beam vibration, Journal of Sound and Vibration 229: 12691276.##[26] Genta G., Gugliotta A., 1988, A conical element for finite element rotor dynamics. Journal of Sound and Vibration 120(l): 175182.##[27] Attarnejad R., Manavi N., Farsad A., 2006, Exact solution for the free vibration of tapered bam with elastic end rotational restraints, Chapter Computational Methods 19932003.##[28] Torabi K., Afshari H., Zafari E., 2012, Transverse Vibration of Nonuniform EulerBernoulli Beam, Using Differential Transform Method (DTM), Applied Mechanics and Materials 110116: 24002405.##[29] Yan W., Kan Q., Kergrene K., Kang G., Feng X.Q., Rajan R., 2013, A truncated conical beam model for analysis of the vibration of rat whiskers, Journal of Biomechanics 46: 19871995.##[30] Boiangiu M., Ceausu V., Untaroiu C.D., 2014, A transfer matrix method for free vibration analysis of EulerBernoulli beams with variable cross section, Journal of Vibration and Control 22: 25912602.##[31] Lau J.H., 1984, Vibration frequencies for a nonuniform beam with end mass, Journal of Sound and Vibration 97: 513521.##[32] Grossi R.O., Aranda A., 1993, Vibration of tapered beams with one end spring hinged and the other end with tip mass, Journal of Sound and Vibration 160: 175178.##[33] Auciello N.M., 1996, Transverse vibration of a linearly tapered cantilever beam with tip mass of rotatory inertia and eccentricity, Journal of Sound and Vibration 194: 2534.##[34] Wu J.S., Chen C.T., 2005, An exact solution for the natural frequencies and mode shapes of an immersed elastically restrained wedge beam carrying an eccentric tip mass with mass moment of inertia, Journal of Sound and Vibration 286: 549568.##[35] Auciello N.M., Maurizi M.J., 1997, On the natural vibration of tapered beams with attached inertia elements, Journal of Sound and Vibration 199: 522530.##[36] Wu J.S., Hsieh M., 2000, Free vibration analysis of a nonuniform beam with multiple point masses, Structural Engineering and Mechanics 9: 449467.##[37] Wu J.S., Lin T.L., 1990, Free vibration analysis of a uniform cantilever beam with point masses by an analyticaland numerical combined method, Journal of Sound and Vibration 136: 201213.##[38] Wu J.S., Chen D.W., 2003, Bending vibrations of wedge beams with any number of point masses, Journal of Sound and Vibration 262: 10731090.##[39] Caddemi S., Calio I., 2008, Exact solution of the multicracked Euler–Bernoulli column, International Journal of Solids and Structures 45: 13321351.##[40] Caddemi S., Calio I., 2009, Exact closedform solution for the vibration modes of the Euler–Bernoulli beam with multiple open cracks, Journal of Sound and Vibration 327: 473489.##[41] De Silva C.W., 2000, Vibration: Fundamentals and Practice, CRC Press.##[42] Karman T.V., Biot M.A., 1940, Mathematical Methods in Engineering, McGrawHill, New York.##[43] Lighthill M.J., 1958, An Introduction to Fourier Analysis and Generalised Functions, Cambridge University Press, London.##[44] Colombeau J.F., 1984, New Generalized Functions and Multiplication of Distribution, NorthHolland, Amsterdam.##]
Influences of Heterogeneities and Initial Stresses on the Propagation of LoveType Waves in a Transversely Isotropic Layer Over an Inhomogeneous HalfSpace
2
2
In the present paper, we are contemplating the influences of heterogeneities and prestresses on the propagation of Lovetype waves in an initially stressed heterogeneous transversely isotropic layer of finite thickness lying over an inhomogeneous half space. The material constants and prestress have been taken as space dependent and arbitrary functions of depth in the respective media. To simplify the problem, we have used Whittaker’s function and separation of variables method. We present a general dispersion relation to describe the impacts on the propagation of Lovetype waves in the structure. The present dispersion relation is analyzed case wise and also validated by comparison of the standard Love wave equation. Further, numerical computations are demonstrated graphically for the set of dimensionless parameters between dimensionless phase velocity and dimensionless wave number of the wave.
1

783
793


P
Alam
Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad Jharkhand826004, India
Department of Applied Mathematics, Indian
India
alamparvez.amu@gmail.com


S
Kundu
Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad Jharkhand826004, India
Department of Applied Mathematics, Indian
India
Transversely
Isotropic
Heterogeneity
Phase velocity
Initial stress
[[1] Love A.E.H., 1920, Mathematical Theory of Elasticity, Cambridge University Press, UK.##[2] Ewing W.M., Jardetzky W.S., Press F., 1957, Elastic Waves in Layered Media, McGrawHill, New York.##[3] Biot M.A., 1965, Mechanics of Incremental Deformations, Wiley, New York.##[4] Gubbins D., 1990, Seismology and Plate Tectonics, Cambridge University Press, Cambridge.##[5] Ding H., Chen W., Zhang L., 2006, Elasticity of Transversely Isotropic Materials, Springer Science & Business Media.##[6] Singh A.K., Kumari N., Chattopadhyay A., Sahu S.A., 2015, Smooth moving punch in an initially stressed transversely isotropic magnetoelastic medium due to shear wave, Mechanics of Advanced Materials and Structures 23: 774783.##[7] Acharya D.P., Roy I., Sengupta S., 2009, Effect of magnetic field and initial stress on the propagation of interface waves in transversely isotropic perfectly conducting media, Acta Mechanica 202: 3545.##[8] Ahmad F., Khan A., 2001, Effect of rotation on wave propagation in a transversely isotropic medium, Mathematical Problems in Engineering 7(2): 147154.##[9] Singh B., 2016, Wave propagation in a rotating transversely isotropic twotemperature generalized thermoelastic medium without dissipation, International Journal of Thermophysics 37(1): 113.##[10] Kundu S., Gupta S., Manna S., 2014, SHtype waves dispersion in an isotropic medium sandwiched between an initially stressed orthotropic and heterogeneous semiinfinite media, Meccanica 49(3): 749758.##[11] Zhu H., Zhang L., Han J., Zhang Y., 2014, Love wave in an isotropic homogeneous elastic halfspace with a functionally graded cap layer, Applied Mathematics and Computation 231: 9399.##[12] Kakar R., 2015, Dispersion of love wave in an isotropic layer sandwiched between orthotropic and prestressed inhomogeneous halfspaces, Latin American Journal of Solids and Structures 12(10): 19341949.##[13] Dey S., De P.K., 2010, Propagation of channel wave in an incompressible anisotropic initially stressed plate of finite thickness, Tamkang Journal of Science and Engineering 13(2): 127134.##[14] Dhua S., Chattopadhyay A., 2015, Torsional wave in an initially stressed layer lying between two inhomogeneous media. Meccanica 50(7): 17751789.##[15] Kundu S., Manna S., Gupta S., 2014, Love wave dispersion in prestressed homogeneous medium over a porous halfspace with irregular boundary surfaces, International Journal of Solids and Structures 51: 36893697.##[16] Chattaraj R., Samal S.K., Mahanti N., 2013, Dispersion of love wave propagating in irregular anisotropic porous stratum under initial stress, International Journal of Geomechanics 13(4): 402408.##[17] Dey S., Addy S.K., 1978, Love waves under initial stresses, Acta Geophysica Polonica 26(1):4754.##[18] Mahmoud S.R., 2012, Influence of rotation and generalized magnetothermoelastic on Rayleigh waves in a granular medium under effect of initial stress and gravity field, Meccanica 47(7): 15611579.##[19] Kepceler T., 2010, Torsional wave dispersion relations in a prestressed bimaterial compounded cylinder with an imperfect interface, Applied Mathematical Modelling 34(12): 40584073.##[20] Biot M. A., 1940, The influence of initial stress on elastic waves, Journal of Applied Physics 11: 522530.##[21] Bullen K.E., 1940, The problem of the earth’s density variation, Bulletin of the Seismological Society of America 30(3): 235250.##[22] Birch F., 1952, Elasticity and constitution of the earth's interior, Journal of Geophysical Research 57(2): 227286.##[23] Dey S., Gupta A.K., Gupta S., 1996, Torsional surface waves in nonhomogeneous and anisotropic medium, The Journal of the Acoustical Society of America 99(5): 27372741.##[24] Gupta S., Chattopadhyay A., Vishwakarma S.K., Majhi D.K., 2011, Influence of rigid boundary and initial stress on the propagation of love wave, Applied Mathematics 2: 586594.##[25] Gupta S., Majhi D.K., Kundu S., Vishwakarma S.K., 2013, Propagation of love waves in nonhomogeneous substratum over initially stressed heterogeneous halfspace, Applied Mathematics and Mechanics 34(2): 249258.##[26] Whittaker E., Watson G.N., 1990, A Course of Modern Analysis, Universal Book Stall, New Delhi.##]
Mechanical Characteristics and Failure Mechanism of NanoSingle Crystal Aluminum Based on Molecular Dynamics Simulations: Strain Rate and Temperature Effects
2
2
Besides experimental methods, numerical simulations bring benefits and great opportunities to characterize and predict mechanical behaviors of materials especially at nanoscale. In this study, a nanosingle crystal aluminum (Al) as a typical face centered cubic (FCC) metal was modeled based on molecular dynamics (MD) method and by applying tensile and compressive strain loadings its mechanical behaviors were investigated. Embedded atom method (EAM) was employed to represent the interatomic potential of the system described by a canonical ensemble. Stressstrain curves and mechanical properties including modulus of elasticity, Poisson’s ratio, and yield strength were determined. Furthermore, the effects of strain rate and system temperature on mechanical behavior were obtained. It was found that the mechanical properties exhibited a considerable dependency to temperature, but they hardly changed with increase of strain rate. Moreover, nucleation and propagation of dislocations along the plane of maximum shearing stress were the mechanisms of the nanocrystalline Al plastic deformation.
1

794
801


R
Rezaei
Faculty of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran
Faculty of Mechanical Engineering, Shahrood
Iran


H
TavakoliAnbaran
Faculty of Physics, Shahrood University of Technology, Shahrood, Iran
Faculty of Physics, Shahrood University of
Iran


M
Shariati
Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
Department of Mechanical Engineering, Faculty
Iran
mshariati44@um.ac.ir
Nanocrystalline aluminum
Mechanical Properties
Molecular Dynamics
Defromation mechanism
[[1] Bhushan B., 2004, Springer Handbook of Nanotechnology, SpingerVerlag Berlin Heidelberg.##[2] Dao D. V., Nakamura K., Bui T. T., Sugiyama S., 2010, Micro/nanomechanical sensors and actuators based on SOIMEMS technology, Advances in Natural Sciences: Nanoscience and Nanotechnology 1(1):013001013010.##[3] Ekinci K. L., Roukes M. L., 2005, Nanoelectromechanical systems, Review of Scientific Instruments 76: 061101.##[4] Rapaport D., 2004, The Art of Molecular Dynamics Simulation, Cambridge University Press.##[5] Narayan K., Behdinan K., Fawaz Z., 2007, An engineeringoriented embeddedatommethod potential fitting procedure for pure fcc and bcc metals, Journal of Materials Processing Technology 182: 387397.##[6] Nath S. K. D., 2014, Elastic, elastic–plastic properties of Ag, Cu and Ni nanowires by the bending test using molecular dynamics simulations, Computational Materials Science 87: 138144.##[7] Ikeda H., Qi Y., Cagin T., Samwer K., Johnson W. L., Goddard W. A., 1999, Strain rate induced amorphization in metallic nanowires, Physical Review Letters 82: 2900.##[8] Koh S. J. A., Lee H. P., 2006, Molecular dynamics simulations of size and strain rate dependent mechanical response of FCC metallic nanowires, Nanotechnology 17: 34513467.##[9] Wu H. A., 2004, Molecular dynamics simulation of loading rate and surface effects on the elastic bending behavior of metal nanorod, Computational Materials Science 31: 287291.##[10] Rezaei R., Shariati M., TavakoliAnbaran H., Deng C., 2016, Mechanical characteristics of CNTreinforced metallic glass nanocomposites by molecular dynamics simulations, Computational Materials Science 119: 1926.##[11] Rezaei R., Deng C., 2017, Pseudoelasticity and shape memory effects in cylindrical FCC metal nanowires, Acta Materialia 132: 4956.##[12] Rezaei R., Deng C., Shariati M., TavakoliAnbaran H., 2017,The ductility and toughness improvement in metallic glass through the dual effects of graphene interfae, Journal of Materials Research 32: 392403.##[13] Rezaei R., Deng C., TavakoliAnbaran H., Shariati M., 2016, Deformationtwinning mediated pseudoelasticity in metalgraphene nanolaminates, Philosophical Magazine Letter 96: 322329.##[14] Lim M. C. G., Zhong Z. W., 2009, Molecular dynamics analyses of an Al (110) surface, Physica A 388: 40834090.##[15] Khan A., Suh Y. S., Chen X., Takacs L., Zhang H., 2006, Nanocrystalline aluminum and iron: Mechanical behavior at quasistatic and high strain rates, and constitutive modeling, International Journal of Plasticity 22: 195209.##[16] Groh S., Marin E. B., Horstemeyer M. F., Zbib H. M., 2009, Multiscale modeling of the plasticity in an aluminum single crystal, International Journal of Plasticity 25: 14561473.##[17] Yuan L., Shan D., Guo B., 2007, Molecular dynamics simulation of tensile deformation of nanosingle crystal aluminum, Journal of Materials Processing Technology 184: 15.##[18] Li R., Zhong Y., Huang C., Tao X., Ouyang Y., 2013, Surface energy and surface selfdiffusion of Al calculated by embedded atom method, Physica B 422: 5155.##[19] Hachiya K., Ito Y., 2002, Transitionmetallike interatomic potentials for aluminium, Journal of Alloys and Compounds 337: 5357.##[20] Alavi S., Thompson D., 2006, Molecular dynamics simulations of the melting of aluminum nanoparticles, Journal of Physical Chemistry A 110: 15181523.##[21] Ozgen S., Duruk E., 2004, Molecular dynamics simulation of solidification kinetics of aluminium using Sutton–Chen version of EAM, Materials Letters 58: 10711075.##[22] Gao C. Y., Zhang L. C., 2012, Constitutive modelling of plasticity of fcc metals under extremely high strain rates, International Journal of Plasticity 3233: 121133.##[23] Guo Y., Zhuang Z., Li X. Y., Chen Z., 2007, An investigation of the combined size and rate effects on the mechanical responses of FCC metals, International Journal of Solids and Structures 44: 11801195.##[24] Karimzadeh A., Ayatollahi M. R., Alizadeh M., 2014, Finite element simulation of nanoindentation experiment on aluminum 1100, Computational Materials Science 81: 595600.##[25] Field M. J., 2007, A Practical Introduction to the Simulation of Molecular Systems, Cambridge University Press.##[26] Daw M. S., Baskes M. I., 1984, Embeddedatom method: derivation and application to impurities, surfaces, and other defects in metals, Physics Review B 29: 64436453.##[27] Callister W. D., Rethwisch D. G., 2011, Fundamentals of Materials Science and Engineering: an Integrated Approach, John Wiley & Sons.##[28] Plimpton S., 1995, Fast parallel algorithms for shortrange molecular dynamics, Journal of Computational Physics 117: 119.##[29] Mendelev M. I., Srolovitz D. J., Ackland G. J., Han S., 2005, Effect of Fe segregation on the migration of a nonsymmetric sigma5 tilt grain boundary in Al, Journal of Materials Research 20: 208218.##[30] Buehler M. J. ,2008, Atomistic Modeling of Materials Failure, Springer.##[31] Song H. Y., Zha X. W., 2010, Influence of nickel coating on the interfacial bonding characteristics of carbon nanotube–aluminum composites, Computational Materials Science 49: 899903.##[32] Beer F. P., Johnston E. R., Dewolf J. T., 2006, Mechanics of Materials, McGraw Hill.##]
Buckling Study of Thin Tank Filled with Heterogeneous Liquid
2
2
Buckling of imperfect thin shell tank which is subjected to uniform axial compression is analyzed. The effect of internal pressure on the stability of a shell tank filled with a homogeneousheterogeneous liquid was considered. Investigation of the liquid nature effect on reduction of the shell buckling load is performed by using the finite elements method. Calculating results in terms of analytical formula give a good agreement with the numerical results given by Abaqus when using actual measurements. The obtained results show the influence of the physical characteristics of liquid especially in the case of heterogeneous liquid. The study of combination between compression load, lateral pressure and the mechanical properties of liquid filling the tank is recommended for dimensioning the shell tanks to avoid the buckling phenomenon.
1

802
810


J
El Bahaoui
Abdelmalek Essaadi Universty, Faculty of Science, M2SM Group ,93000 M’Hannech ,Tetuan, Morocco
Abdelmalek Essaadi Universty, Faculty of
Morocco
jelbahaoui@yahoo.com


H
Essaouini
Abdelmalek Essaadi Universty, Faculty of Science, M2SM Group ,93000 M’Hannech ,Tetuan, Morocco
Abdelmalek Essaadi Universty, Faculty of
Morocco


L
El Bakkali
Abdelmalek Essaadi Universty, Faculty of Science, M2SM Group ,93000 M’Hannech ,Tetuan, Morocco
Abdelmalek Essaadi Universty, Faculty of
Morocco
Thin Shell tank
Buckling strenght
Homogeneousheterogeneous liquid
Imperfection
Finit elements method
[[1] Donnell L.H., 1933, The Problem of Elastic Stability, Transactions of the American Society of Mechanical Engineers, Aeronautical Division, New York.##[2] Arbocz J., Babcock C.D., 1969, The effect of general imperfections on the buckling of cylindrical shells, Journal of Applied Mechanics 36: 2838.##[3] Gros D., 1999, Flambage des Coques Cylindriques sous Pression Interne et Flexion : Sensibilité aux Imperfections Géométriques, Thèse Institut National de Sciences Appliquées de Lyon, Lyon.##[4] Kim S.E., Kim C.S., 2002, Buckling strength of the cylindrical shell and tank subjected to axially compressive loads, Thin Walled Structures 40(4): 329353.##[5] Jamal M., Midani M., Damil N., PotierFerry M., 1999, Influence of localized imperfections on the buckling of cylindrical shells under axial compression, International Journal of Solids and Structures 36: 330353.##[6] Jamal M., Lahlou L., Midani M., Zahrouni H., Limam A., Damil N., PotierFerry M., 2003, A semianalytical buckling analysis of imperfect cylindrical shells under axial compression, International Journal of Solids and Structures 40: 13111327.##[7] Abaqus, 2006, Standard user’s Manual, Version 6.8., Simulia, Dassault Systems.##[8] Limam A., El Bahaoui J., Khamlichi A., EL Bakkali L., 2011, Effect of multiple localized geometric imperfections on stability of thin axisymmetric cylindrical shells under axial compression, International Journal of Solids and Structures 48: 10341043.##[9] Lo Frano R., Forasassi G., 2008, Buckling of imperfect thin cylindrical shell under lateral pressure, Science and Technology of Nuclear Installations 2008: 685805.##[10] Fan H., Chen Zh., Cheng J., Huang S., Feng W., Liu L., 2016, Analytical research on dynamic buckling of thin cylindrical shells with thickness variation under axial pressure, ThinWalled Structures 101: 213221.##[11] Wu J., Cheng Q. H., Liu B., Zhang Y. W., Lu W.B., Hwang K.C., 2012 ,Study on the axial compression buckling behaviors of concentric multiwalled cylindrical shells filled with soft materials, Journal of the Mechanics and Physics of Solids 60: 803826.##[12] Khamlichi A., Bezzazi M., Limam A., 2004, Buckling of elastic cylindrical shells considering the effect of localized axisymmetric imperfections, ThinWalled Structures 42(7): 10351047.##[13] Bryngelson S.H., Freund J.B., 2016, Buckling and its effect on the confined flow of a model capsule suspension, Rheologica Acta 55: 451464.##[14] Dai H.L., Rao Y.N., Dai T., 2016, A review of recent researches on FGM cylindrical structures under coupled physical interactions, Composite Structures 152: 199225.##[15] Michael Rotter J., Sadowski A. J., 2012, Cylindrical shell bending theory for orthotropic shells under general axisymmetric pressure distributions, Engineering Structures 42: 258265.##[16] Zingoni A., 2015, Liquidcontainment shells of revolution: A review of recent studies on strength, stability and dynamics, ThinWalled Structures 87: 102114.##[17] Bazhenov V.A., Luk’yanchenko O.O., Kostina O.V., Gerashchenko O.V.,2014, Probabilistic approach to derermination of reliability of an imperfect supporting shell, Strength of Materials 46(4): 567574.##]
Numerical and Experimental Study on Ratcheting Behavior of Plates with Circular Cutouts under Cyclic Axial Loading
2
2
In this paper, accumulation of plastic deformation of AISI 1045 steel plates with circular cutouts under cyclic axial loading is studied. Loading was applied under forcecontrol conditions. Experimental tests were performed using a Zwick/Roell servo hydraulic machine. Under forcecontrol loading with nonzero mean force, plastic strain was accumulated in continuous cycles called ratcheting. Numerical analysis was carried out by ABAQUS software using nonlinear isotropic/kinematic hardening model. The results of the numerical simulations were compared to experimental data. The results demonstrated that the ratcheting response of plates with circular cutouts could be numerically simulated with a reasonable accuracy. It was observed that the local and global plastic deformation increase with increasing the notch diameter. Also, maximum principal stress was the main parameter for initiation of crack around the notch. Based on numerical results, at notch root, both ratcheting strain and local mean stress relaxation was occur simultaneously and due to relaxation of local mean stress, plastic shakedown was occurred.
1

811
820


K
Kolasangiani
Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
Department of Mechanical Engineering, Ferdowsi
Iran


M
Shariati
Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
Department of Mechanical Engineering, Ferdowsi
Iran
mshariati44@um.ac.ir


Kh
Farhangdoost
Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
Department of Mechanical Engineering, Ferdowsi
Iran


A
VarvaniFarahani
Department of Mechanicaland and Industrial Engineering, Ryerson University, Toronto, Canada
Department of Mechanicaland and Industrial
Canada
Plate with circular cutout
Numerical and experimental study
Cyclic loading
Nonlinear isotropic/kinematic hardening model
[[1] Dowling E., 1998, Mechanical Behavior of Materials, Prentice Hall, Fourth Edition.##[2] Koe S., Nakamura H., Tsunenari T., 1978, Fatigue life estimation of notched plates based on elastoplastic analysis, Journal of the Society of Materials Science 27(300): 847852.##[3] Sakane M., Ohnami M., 1983, A study of the notch effect on the low cycle fatigue of metals in creep–fatigue interacting conditions at elevated temperatures, Journal of Engineering Materials and Technology 105(2): 7580.##[4] Sakane M., Ohnami M., 1986, Notch effect in lowcycle fatigue at elevated temperaturelife prediction from crack initiation and propagation considerations, Journal of Engineering Materials and Technology 108(4): 279284.##[5] Fatemi A., Zeng Z., Plaseied A., 2004, Fatigue behavior and life predictions of notched specimens made of QT and forged microalloyed steels, International Journal of Fatigue 26: 663672.##[6] VarvaniFarahani A., Kodric T., Ghahramani A., 2005, A method of fatigue life prediction in notched and unnotched components, Journal of Material Processing Technology 169: 94102.##[7] Medekshas H., Balina V., 2006, Assessment of low cycle fatigue strength of notched components, Materials&Design 27: 132140.##[8] Nozaki M., Zhang S., Sakane M., Kobayashi K., 2011, Notch effect on creepfatigue life for Sn3.5Ag solder, Engineering Fracture Mechanics 78: 17941807.##[9] Savaidis A., Savaidis G., Zhang Ch., 2001, Elasticplastic FE analysis of a notched shaft under multiaxial nonproportional synchronous cyclic loading, Theoretical and Applied Fracture Mechanics 36: 8797.##[10] Wang CH., Rose L.R.F., 1998, Transient and steadystate deformation at notch root under cyclic loading, Mechanics of Materials 30: 229241.##[11] Shi D.Q., Hu X.A., Wang J.K., Yu H.C., Yang X.G., Huang J., 2013, Effect of notch on fatigue behavior of a directionally solidified superalloy at high temperature, Fatigue & Fracture of Engineering Materials & Structures 36: 12881297.##[12] ASTM A37005, Standard test method and definitions for mechanical testing of steel products.##[13] Chen G., Chen X., ChangDong N., 2006, Uniaxialratchetingbehaviorof63Sn37Pb solder with loading histories and stress rates, Material Science and Engineering A 421: 238244.##[14] Shariati M., Kolasangiani K., Norouzi G., Shahnavaz, A., 2014, Experimental study of SS316L cantilevered cylindrical shells under cyclic bending load, ThinWalled Structures 82: 124131.##[15] Gaudin C., Feaugas X., 2004, Cyclic creep process in AISI 316L stainless steel in terms of dislocation patterns and internal stresses, Acta Materials 52: 30973110.##[16] ABAQUS 6.10.1 PR11 user’s manual.##]
Biaxial Buckling Analysis of Symmetric Functionally Graded Metal Cored Plates Resting on Elastic Foundation under Various Edge Conditions Using Galerkin Method
2
2
In this paper, buckling behavior of symmetric functionally graded plates resting on elastic foundation is investigated and their critical buckling load in different conditions is calculated and compared. Plate governing equations are derived using the principle of minimum potential energy. Afterwards, displacement field is solved using Galerkin method and the proposed process is examined through numerical examples. Effect of FGM power law index, plate aspect ratio, elastic foundation stiffness and metal core thickness on critical buckling load is investigated. The accuracy of this approach is verified by comparing its results to those obtained in another work, which is performed using Fourier series expansion.
1

821
831


M
Rezaei
Department of Mechanical Engineering, School of Engineering, Yasouj University, Yasouj, Iran
Department of Mechanical Engineering, School
Iran


S
Ziaee
Department of Mechanical Engineering, School of Engineering, Yasouj University, Yasouj, Iran
Department of Mechanical Engineering, School
Iran
ziaee@yu.ac.ir


S
Shoja
Department of Civil Engineering, School of Engineering, Yasouj University, Yasouj, Iran
Department of Civil Engineering, School of
Iran
Functionally graded material
Plate
Buckling analysis
Galerkin Method
Elastic foundation
[Bever M.B., Duwez P.E., 1972, Gradients in composite materials, Materials Science and Engineering 10: 18.##[2] Singh B.N., Lal A., 2010, Stochastic analysis of laminated composite plates on elastic foundation: The cases of postbuckling behavior and nonlinear free vibration, International Journal of Pressure Vessels and Piping 87: 559574.##[3] Li X.Y., Ding H.J., Chen W.Q., 2008, Elasticity solutions for a transversely isotropic functionally graded circular plate subject to an axisymmetric transverse load qrk, International Journal of Solids and Structures 45: 191210.##[4] Li X.Y., Ding H.J., Chen W.Q., 2008, Axisymmetric elasticity solutions for a uniformly loaded annular plate of transversely isotropic functionally graded materials, Acta Mechanica 196: 139159.##[5] Liu G.R., Han X., Lam K.Y., 2001, Material characterization of FGM plates using elastic waves and an inverse procedure, Journal of Composite Materials 35(11): 954971.##[6] Zhong Z., Shang E., 2008, Closedform solutions of threedimensional functionally graded plates, Mechanics of Advanced Materials and Structures 15: 355363.##[7] Kim K.D., Lomboy G.R., Han S.C., 2008, Geometrically nonlinear analysis of functionally graded material (FGM) plates and shells using a fournode quasiconforming shell element, Journal of Composite Materials 42(5): 485511.##[8] Feldman E., Aboudi J., 1997, Buckling analysis of functionally graded plates subjected to uniaxial loading, Composite Structures 38(14): 2936.##[9] Ma L.S., Wang T.J., 2004, Relationships between axisymmetric bending and buckling solutions of FGM circular plates based on thirdorder plate theory and classical plate theory, International Journal of Solids and Structures 41: 85101.##[10] Zhang X., Chen F., Zhang H., 2013, Stability and local bifurcation analysis of functionally graded material plate under transversal and inplane excitations, Applied Mathematical Modelling 37(1011): 66396651.##[11] Mahdavian M., 2009, Buckling analysis of simplysupported functionally graded rectangular plates under nonuniform Inplane compressive loading, Journal of Solid Mechanics 1(3): 213225.##[12] Chen X.L., Liew K.M., 2004, Buckling of rectangular functionally graded material plates subjected to nonlinearly distributed inplane edge loads, Smart Materials and Structures 13: 14301437.##[13] Cheng Z.Q., Batra R.C., 2000, Deflection relationships between the homogeneous Kirchhoff plate theory and different functionally graded plate theories, Archives of Mechanics 52(1): 143158.##[14] Pan E., 2003, Exact solution for functionally graded anisotropic elastic composite laminates, Journal of Composite Materials 37(21): 19031920.##[15] Kashtalyan M., 2004, Threedimensional elasticity solution for bending of functionally graded rectangular plates, European Journal of Mechanics  A/Solids 23(5): 853864.##[16] Zenkour A.M., 2007, Benchmark trigonometric and 3D elasticity solutions for an exponentially graded thick rectangular plate, Archive of Applied Mechanics 77(4): 197214.##[17] Zheng L., Zhong Z., 2009, Exact solution for axisymmetric bending of functionally graded circular plate, Tsinghua Science & Technology 14(2): 6468.##[18] Vel S.S., Batra R.C., 2012, Exact solution for thermoelastic deformations of functionally graded thick rectangular plates, AIAA Journal 40(7): 14211433.##[19] Nguyen T.K., Sab K., Bonnet G., 2008, Firstorder shear deformation plate models for functionally graded materials, Composite Structures 83(1): 2536.##[20] Hopkins D.A., Chamis C.C., 1988, A unique set of micromechanics equations for high temperature metal matrix composites, NASA TM 87154..##[21] Birman V., 1995, Stability of functionally graded hybrid composite plates, Composite Engineering 5(7): 913921.##[22] Chen X.L., Liew K.M., 2004, Buckling of rectangular functionally graded material plates subjected to nonlinearly distributed inplane edge loads, Smart Materials and Structures 13(6): 14301437.##[23] Saidi A.R., Rasouli A., Sahraee S., 2009, Axisymmetric bending and buckling analysis of thick functionally graded circular plates using unconstrained thirdorder shear deformation plate theory, Composite Structures 89(1): 110119.##[24] Mohammadi M., Saidi A., Jomehzadeh E., 2010, Levy solution for buckling analysis of functionally graded rectangular plates, Appl Compos Mater 17(2): 8193.##[25] Dung D.V., Thiem H.T., 2012, On the nonlinear stability of eccentrically stiffened functionally graded imperfect plates resting on elastic foundation, Proceedings of ICEMA2 Conference, Hanoi.##[26] Sobhy M., 2013, Buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions, Composite Structures 99: 7687.##[27] Naderi A., Saidi A.R., 2011, Exact solution for stability analysis of moderately thick functionally graded sector plates on elastic foundation, Composite Structures 93: 629638.##[28] Yaghoobi H., Fereidoon A., 2014, Mechanical and thermal buckling analysis of functionally graded plates resting on elastic foundations: An assessment of a simple refined nthorder shear deformation theory, Composites: Part B 62: 5464.##[29] Thai H.T., Kim S.E., 2013, Closedform solution for buckling analysis of thick functionally graded plates on elastic foundation, International Journal of Mechanical Sciences 75: 3444.##[30] Timoshenko S., WoinowskyKrieger S., 1989, Theory of Shell and Plates, McGrawHill Book Company.##[31] Latifi M., Farhatnia F., Kadkhodaei M., 2013, Buckling analysis of rectangular functionally graded plates under various edge conditions using Fourier series expansion, European Journal of Mechanics  A/Solids 41: 1627.##]
Investigating the Effect of Stiffness/Thickness Ratio on the Optimal Location of Piezoelectric Actuators Through PSO Algorithm
2
2
This article has studied the effect of ratio of stiffness and thickness between piezoelectric actuators and host plat has been explored on optimal pattern for placement of piezoelectric work pieces around a hole in thin isotropic plate under static loading to reduce stress concentration. The piezoelectric actuators reduce directly or indirectly the stress concentration by applying positive and negative strains on the host plate. For this purpose, various modes as the thickness/stiffness ratios of the plate to the piezoelectric patches as ≥1 or ≤1 were considered. Then, a Python code was developed using particle swarm optimization algorithm in order to achieve the best model of piezoelectric actuators around the hole for maximum reduction in stress concentration factor. Also, the maximum stress concentration on the top and bottom of the hole was moved to another point around the edge by changing the location of piezoelectric patches. The results obtained from software solutions were confirmed by experimental tests.
1

832
848


S
Jafari Fesharaki
Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran
Department of Mechanical Engineering, Najafabad
Iran
jafari@pmc.iaun.ac.ir


S.Gh
Madani
Department of Mechanical Engineering, Faculty of Engineering, University of Kashan, Kashan, Iran
Department of Mechanical Engineering, Faculty
Iran


S
Golabi
Department of Mechanical Engineering, Faculty of Engineering, University of Kashan, Kashan, Iran
Department of Mechanical Engineering, Faculty
Iran
Optimization
Piezoelectric patches
Stiffness
particle swarm optimization
[[1] Wang Q., Wu N., 2012, A review on structural enhancement and repair using piezoelectric materials and shape memory alloys, Smart Materials and Structures 21(1): 013001.##[2] Kang Z., Tong L., 2008, Topology optimizationbased distribution design of actuation voltage in static shape control of plates, Computers & Structures 86(19): 18851893.##[3] Kang Z., Wang X., Luo Z., 2012, Topology optimization for static shape control of piezoelectric plates with penalization on intermediate actuation voltage, Journal of Mechanical Design 134(5): 051006.##[4] Mehrabian A. R., YousefiKoma A., 2011, A novel technique for optimal placement of piezoelectric actuators on smart structures, Journal of the Franklin Institute 348(1): 1223.##[5] Huang B., Kim H. S., 2014, Control of freeedge interlaminar stresses in composite laminates using piezoelectric actuators, Smart Materials and Structures 23(7): 074002.##[6] Platz R., Stapp C., Hanselka H., 2011, Statistical approach to evaluating active reduction of crack propagation in aluminum panels with piezoelectric actuator patches, Smart Materials and Structures 20(8): 085009.##[7] Wu N., Wang Q., 2011, An experimental study on the repair of a notched beam subjected to dynamic loading with piezoelectric patches, Smart Materials and Structures 20(11): 115023.##[8] Nguyen Q., Tong L., Gu Y., 2007, Evolutionary piezoelectric actuators design optimisation for static shape control of smart plates, Computer Methods in Applied Mechanics and Engineering 197(1): 4760.##[9] Nguyen Q., Tong L., 2007, Voltage and evolutionary piezoelectric actuator design optimisation for static shape control of smart plate structures, Materials & Design 28(2): 387399.##[10] Nguyen Q., Tong L., 2004, Shape control of smart composite plate with nonrectangular piezoelectric actuators, Composite Structures 66(1): 207214.##[11] Hsu J. C., Tseng C. T., Chen Y. S., 2014, Analysis and experiment of selffrequencytuning piezoelectric energy harvesters for rotational motion, Smart Materials and Structures 23(7): 075013.##[12] Sridharan S., Kim S., 2009, Piezoelectric control of stiffened panels subject to interactive buckling, International Journal of Solids and Structures 46(6): 15271538.##[13] Correia V. M. F., Soares C. M. M., Soares C. A. M., 2003, Buckling optimization of composite laminated adaptive structures, Composite Structures 62(3): 315321.##[14] Chee C. Y. K., Tong L., Steven G. P., 2002, Static shape control of composite plates using a slopedisplacementbased algorithm, AIAA Journal 40(8): 16111618.##[15] Lin J. C., Nien M., 2007, Adaptive modeling and shape control of laminated plates using piezoelectric actuators, Journal of Materials Processing Technology 189(1): 231236.##[16] Zhang H., Lennox B., Goulding P. R., Leung A. Y., 2000, A floatencoded genetic algorithm technique for integrated optimization of piezoelectric actuator and sensor placement and feedback gains, Smart Materials and Structures 9(4): 552.##[17] Qing G., Qiu J., Liu Y., 2006, A semianalytical solution for static and dynamic analysis of plates with piezoelectric patches, International Journal of Solids and Structures 43(6): 13881403.##[18] Da Mota Silva S., Ribeiro R., Rodrigues J. D., Vaz M., Monteiro J., 2004, The application of genetic algorithms for shape control with piezoelectric patches—an experimental comparison, Smart Materials and Structures 13(1): 220.##[19] Nakasone P. H., Silva E. C. N., 2010, Dynamic design of piezoelectric laminated sensors and actuators using topology optimization, Journal of Intelligent Material Systems and Structures 21(16): 16271652.##[20] Rafiee M., He X., Liew K., 2014, Nonlinear analysis of piezoelectric nanocomposite energy harvesting plates, Smart Materials and Structures 23(6): 065001.##[21] Roy T., Chakraborty D., 2009, Optimal vibration control of smart fiber reinforced composite shell structures using improved genetic algorithm, Journal of Sound and Vibration 319(1): 1540.##[22] Wu N., Wang Q., 2010, Repair of vibrating delaminated beam structures using piezoelectric patches, Smart Materials and Structures 19(3): 035027.##[23] Kumar R., Mishra B., Jain S., 2008, Static and dynamic analysis of smart cylindrical shell, Finite Elements in Analysis and Design 45(1): 1324.##[24] Quintero A. V., Besse N., Janphuang P., Lockhart R., Briand D., De Rooij N. F., 2014, Design optimization of vibration energy harvesters fabricated by lamination of thinned bulkPZT on polymeric substrates, Smart Materials and Structures 23(4): 045041.##[25] Kurata M., Li X., Fujita K., Yamaguchi M., 2013, Piezoelectric dynamic strain monitoring for detecting local seismic damage in steel buildings, Smart Materials and Structures 22(11): 115002.##[26] Jadhav P. A., Bajoria K. M., 2013, Free and forced vibration control of piezoelectric FGM plate subjected to electromechanical loading, Smart Materials and Structures 22(6): 065021.##[27] Sensharma P. K., Haftka R. T., 1996, Limits of stress reduction in a plate with a hole using piezoelectric actuators, Journal of Intelligent Material Systems and Structures 7(4): 363371.##[28] Sensharma P. K., Palantera M. J., Haftka R. T., 1993, Stress reduction in an isotropic plate with a hole by applied induced strains, Journal of Intelligent Material Systems and Structures 4(4): 509518.##[29] Shah D., Joshi S., Chan W., 1994, Stress concentration reduction in a plate with a hole using piezoceramic layers, Smart Materials and Structures 3(3): 302.##[30] Fesharaki J. J., Golabi S., 2015, Optimum pattern of piezoelectric actuator placement for stress concentration reduction in a plate with a hole using particle swarm optimization algorithm, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 229(4): 614628.##[31] Eberhart R., Kennedy J., A new optimizer using particle swarm theory, Proceeding of IEEE, Nagoya, Japan.##[32] Rao S. S., Rao S., 2009, Engineering Optimization: Theory and Practice, John Wiley & Sons.##]
NonLinear Analysis of Asymmetrical Eccentrically Stiffened FGM Cylindrical Shells with NonLinear Elastic Foundation
2
2
In this paper, semianalytical method for asymmetrical eccentrically stiffened FGM cylindrical shells under external pressure and surrounded by a linear and nonlinear elastic foundation is presented. The proposed linear model is based on two parameter elastic foundation Winkler and Pasternak. According to the von Karman nonlinear equations and the classical plate theory of shells, straindisplacement relations are obtained. The smeared stiffeners technique and Galerkin method, used for solving nonlinear problem. To finding the nonlinear dynamic response of fourth order RungeKutta method is used. The effect of parameters asymmetrical eccentrically stiffened on the nonlinear dynamic buckling response of FGM cylindrical shells have been investigated.
1

849
864


A.R
Shaterzadeh
Faculty of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran
Faculty of Mechanical Engineering, Shahrood
Iran
a_shaterzadeh@shahroodut.ac.ir


K
Foroutan
Faculty of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran
Faculty of Mechanical Engineering, Shahrood
Iran
FGM cylindrical shells
Nonlinear dynamic analysis
Asymmetrical stiffened
Nonlinear elastic foundation
[[1] Dung D. V., Nam V. H., 2014, Nonlinear dynamic analysis of eccentrically stiffened functionally graded circular cylindrical thin shells under external pressure and surrounded by an elastic medium, European Journal of Mechanics  A/Solids 46: 4253.##[2] Darabi M., Darvizeh M., Darvizeh A., 2008, Nonlinear analysis of dynamic stability for functionally graded cylindrical shells under periodic axial loading, Composite structures 83: 201211.##[3] Sofiyev A. H., Schnack E., 2004, The stability of functionally graded cylindrical shells under linearly increasing dynamic torsional loading, Engineering Structures 26:13211331.##[4] Sheng G. G., Wang X., 2008, Thermo mechanical vibration analysis of a functionally graded shell with flowing fluid, European Journal of Mechanics  A/Solids 27:10751087.##[5] Sofiyev A. H., 2009, The vibration and stability behavior of freely supported FGM conical shells subjected to external pressure, Composite structures 89: 356366.##[6] Hong C. C., 2013, Thermal vibration of magnetostrictive functionally graded material shells, European Journal of Mechanics  A/Solids 40: 114122.##[7] Huang H., Han Q., 2010, Nonlinear dynamic buckling of functionally graded cylindrical shells subjected to a timedependent axial load, Composite structures 92: 593598.##[8] Budiansky B., Roth R. S., 1962, Axisymmetric Dynamic Buckling of Clamped Shallow Spherical Shells, NASA Technical Note D.1510.##[9] Pellicano F. , 2009, Dynamic stability and sensitivity to geometric imperfections of strongly compressed circular cylindrical shells under dynamic axial loads, Communications in Nonlinear Science and Numerical Simulation 14(8): 34493462.##[10] Duc N. D., Thang P. T., 2015, Nonlinear dynamic response and vibration of shear deformable imperfect eccentrically stiffened SFGM circular cylindrical shells surrounded on elastic foundations, Aerospace Science and Technology 40: 115127.##[11] Sofiyev A. H., 2011, Nonlinear buckling behavior of FGM truncated conical shells subjected to axial load, International Journal of NonLinear Mechanics 46: 711719.##[12] Brush D. O., Almroth B. O., 1975, Buckling of Bars, Plates and Shells, McGrawHill. New York.##[13] ShaoWen Y., 1979, Buckling of cylindrical shells with spiral stiffeners under uniform compression and torsion, Composite structures 11: 587595.##[14] Najafizadeh M. M., Hasani A., Khazaeinejad P., 2009, Mechanical, stability of functionally graded stiffened cylindrical shells, Applied Mathematical Modelling 33: 11511157.##[15] Shen H. S., 1998, Postbuckling analysis of imperfect stiffened laminated cylindrical shells under combined external pressure and thermal loading, Applied Mathematics and Mechanics 19: 411426.##[16] Ghiasian S. E., Kiani Y., Eslami M.R., 2013, Dynamic buckling of suddenly heated or compressed FGM beams resting on nonlinear elastic foundation, Composite structures 106: 225234.##[17] Huang H., Han Q., 2010, Research on nonlinear postbuckling of functionally graded cylindrical shells under radial loads, Composite structures 92:13521357.##[18] Volmir A. S., 1972, The Nonlinear Dynamics of Plates and Shells, Science Edition, Russian.##[19] Sewall J. L., Naumann E. C., 1968, An Experimental and Analytical Vibration Study of Thin Cylindrical Shells with and Without Longitudinal Stiffeners, NASA Technical Note D4705.##[20] Sewall J. L, Clary R. R., Leadbetter S. A., 1964, An Experimental and Analytical Vibration Study of a Ringstiffened Cylindrical Shell Structure with Various Support Onditions, NASA Technical Note D2398.##[21] Paliwal D. N., Pandey R. K., Nath T., 1996, Free vibration of circular cylindrical shell on Winkler and Pasternak foundation, International Journal of Pressure Vessels and Piping 69: 7989.##[22] Sofiyev A. H., Avcar M., Ozyigit P., Adigozel S., 2009, The free vibration of nonhomogeneous truncated conical shells on a Winkler foundation, International Journal of Engineering and Applied Sciences 1: 3441.##]
An Investigation of Stress and Deformation States of Rotating Thick Truncated Conical Shells of Functionally Graded Material
2
2
The present study aims at investigating stress and deformation behavior of rotating thick truncated conical shells subjected to variable internal pressure. Material prpperties of the shells are graded along the axial direction by Moritanaka scheme, which is achieved by elemental gradation of the properties.Governing equations are derived using principle of stsionary total potential (PSTP) and shells are subjected to clamped clamped boundary conditions. Aluminumzirconia, metalceramic and ceramicmetal FGM is considered and effects of grading index of material properties and pressure distribution are analyzed. Distribution of Radial displacement and circumferential stress in both radial and axial direction is presented. Further a comparison of behaviors of different FGM shells and homogeneous shells are made which shows, a significant reduction in stresses and deformations of FGM shells as compared to homogeneous shell. FGM shell having value of grading parameter n = 2 is most suitable for the purpose of rotating conical shells having variable pressure distribution as compared to homogeneous shell and shell having other values of grading parameter n.
1

865
877


A
Thawait
Department of Mechanical Engineering, Institute of Technology, Guru Ghasidas Vishwavidyalaya, Bilaspur, 495009, India
Department of Mechanical Engineering, Institute
India
amkthawait@gmail.com


L
Sondhi
Department of Mechanical Engineering, Shri Shankaracharya Technical Campus, SSGI, Bhilai, 490020, India
Department of Mechanical Engineering, Shri
India


Sh
Bhowmick
Department of Mechanical Engineering, National Institute of Technology (NIT), Raipur, 492010, India
Department of Mechanical Engineering, National
India


Sh
Sanyal
Department of Mechanical Engineering, National Institute of Technology (NIT), Raipur, 492010, India
Department of Mechanical Engineering, National
India
Rotating thick truncated conical shell
Functionally graded material
Linear elastic analysis
[[1] Asemi K., Akhlaghi M., Salehi M., Zad S.K.H., 2011, Analysis of functionally graded thick truncated cone with finite length under hydrostatic internal pressure, Archive of Applied Mechanics 81: 10631074.##[2] Bayat M., Sahari B., Saleem M., Dezvareh S., Mohazzab A.H., 2011, Analysis of functionally graded rotating disks with parabolic concave thickness applying an exponential function and the moritanaka scheme, IOP Conference Series: Materials Science and Engineering 17: 111.##[3] Civalek O., 2006, The determination of frequencies of laminated conical shells via the discrete singular convolution method, Journal of Mechanics of Materials and Structures 1: 163182.##[4] Civalek O., Gürses M., 2009, Free vibration analysis of rotating cylindrical shells using discrete singular convolution technique, International Journal of Pressure Vessels and Piping 86: 677683.##[5] Heydarpour Y., Aghdam M.M., 2016, Transient analysis of rotating functionally graded truncated conical shells based on the Lord–Shulman model, ThinWalled Structures 104: 168184.##[6] Hua L., 2000, Frequency analysis of rotating truncated circular orthotropic conical shells with different boundary conditions, Composites Science and Technology 60: 29452955.##[7] Ma X., Jin G., Xiong Y., Liu Z., 2014, Free and forced vibration analysis of coupled conical–cylindrical shells with arbitrary boundary conditions, International Journal of Mechanical Sciences 88: 122137.##[8] Malekzadeh P., Daraie M., 2014, Dynamic analysis of functionally graded truncated conical shells subjected to asymmetric moving loads, ThinWalled Structures 84:113.##[9] Nejad M.Z., Jabbari M., Ghannad M., 2015, Elastic analysis of FGM rotating thick truncated conical shells with axiallyvarying properties under nonuniform pressure loading, Composite Structures 122: 561569.##[10] Nejad M.Z., Jabbari M., Ghannad M., 2014, Elastic analysis of rotating thick truncated conical shells subjected to uniform pressure using disk form multilayers, ISRN Mechanical Engineering 2014: 110.##[11] Nejad M.Z., Jabbari M., Ghannad M., 2014, A semi analytical solution of thick truncated cones using matched asymptotic method and disk form multilayers, Archive of Mechanical Engginering 61: 495513.##[12] Qinkai H., Fulei C., 2013, Effect of rotation on frequency characteristics of a truncated circular conical Shell, Archive of Applied Mechanics 83: 17891800.##[13] Seidi J., Khalili S.M.R., Malekzadeh K., 2015, Temperaturedependent buckling analysis of sandwich truncated conical shells with FG facesheets, Composite Structures 131: 682691.##[14] Seshu P., 2003, A Text Book of Finite Element Analysis, PHI Learning Pvt, Ltd.##[15] Sofiyev A.H., 2015, Buckling analysis of freelysupported functionally graded truncated conical shells under external pressures, Composite Structures 132: 746758.##[16] Sofiyev A.H., Kuruoglu N., 2016, Combined effects of elastic foundations and shear stresses on the stability behavior of functionally graded truncated conical shells subjected to uniform external pressures, ThinWalled Structures 102: 6879.##[17] Zeighampour H., Beni Y.T., 2014, Analysis of conical shells in the framework of coupled stresses theory, International Journal of Engineering Science 81: 107122.##[18] Thawait A. K., Sondhi L., Bhowmick S., Sanyal S., 2017, An investigation of stresses and deformation states of clamped rotating functionally graded disks, Journal of Theoretical and Applied Mechanics 55: 189198.##]
Effect of Submerged Arc Welding Parameter on Crack Growth Energy on St37
2
2
In this article the influence of welding parameters such as electrical current, feed rate and stick out on crack growth energy was investigated. Therefore, prepared specimens were welded in various conditions. Using the Minitab software, 18 states of 36 possible states were chosen and applied. Then a crack was crated on the weld metal and the forcedisplacement diagram was plotted. Comparison of results shows that in high electricity current, the extra heat flux is the dominant factor, which causes coarse grains of microstructures. On the other hand, in low electricity current, lack of fusion and penetration reduce the crack propagation energy. Furthermore, the neural network could predict the amount of energy with high accuracy.
1

878
890


J.A
Khodaii
Department of Mechanical Engineering ,Amirkabir University of Technology, Tehran, Iran
Department of Mechanical Engineering ,Amirkabir
Iran


A
Mostafapour
Department of Mechanical Engineering ,University of Tabriz, Tabriz, Iran
Department of Mechanical Engineering ,University
Iran
amostafapur@tabrizu.ac.ir


M.R
Khoshravan
Department of Mechanical Engineering ,University of Tabriz, Tabriz, Iran
Department of Mechanical Engineering ,University
Iran
Submerged arc welding
Crack growth energy
Welding parameter
Optimization
[[1] Yawar J., Lal H., 2015, Effect of various parameters on flux consumption, carbon and silicon in submerged arc welding (Saw),International Journal on Emerging Technologies 6(2):176180.##[2] Sharma M., Gupta D., Karun., 2014, Submerged arc welding: A review, International Journal of Current Engineering and Technology 4(3): 18141817.##[3] Yang Y., 2008, The Effect of Submerged Arc Welding Parameters on the Properties of Pressure Vessel and Wind Turbine Tower Steels, University of Saskatchevan.##[4] Murugan N., Gunaraj V., 2005, Prediction and control of weld bead geometry and shape relationships in submerged arc welding of pipes, Journal of Materials Processing Technology 168(3): 478487.##[5] Dhas J.E.R., Kumanan S., 2010, Weld quality prediction of submerged arc welding process using a function replacing hybrid system, Advances in Production Engineering & Management 5: 512.##[6] Juang S.C., Tarng Y.S., 2002, Process parameter selection for optimizing the weld pool geometry in the tungsten inert gas welding of stainless steel, Journal of Materials Processing Technology 122(1): 3337.##[7] Yang L.J., Bibby M.J., Chandel R.S., 1993, Linear regression equations for modeling the submergedarc welding process, Journal of Materials Processing Technology 39(12): 3342.##[8] Murugan N., Gunaraj V., 2005, Prediction and control of weld bead geometry and shape relationships in submerged arc welding of pipes, Journal of Materials Processing Technology 168(3): 478487.##[9] Reed R.P., 1983, The Economic Effects of Fracture in the United States, US Department of Commerce, National Bureau of Standards.##[10] Hasson D.F., Zanis C.A., Anderson D.R., 1984, Fracture toughness of HY130 steel weld metals, Welding Journal 63(6):197202.##[11] Om H., Pandey S., 2013, Effect of heat input on dilution and heat affected zone in submerged arc welding process, Sadhana 38(6): 13691391.##[12] Shen S., Oguocha I.N.A., Yannacopoulos S., 2012, Effect of heat input on weld bead geometry of submerged arc welded ASTM A709 Grade 50 steel joints, Journal of Materials Processing Technology 212(1): 286294.##[13] Cretegny L., Saxena A., 1998, Fracture toughness behavior of weldments with mismatched properties at elevated temperature, International Journal of Fracture 92(2): 119130.##[14] Zúñiga D.F., Kalthoff J.F., Canteli A.F., Grasa J., Doblaré M., 2013, Upper shelf fracture toughness of as welded manual metal arc weld metal, ECF15.##[15] Irwin G.R., 1948, Fracture dynamics, Fracturing of Metals 1948: 147166.##]
Extension of VIKOR Method to Find an Optimal Layout for Fixture's Supporting Points in Order to Reduce Work Piece Deformation
2
2
In automotive industry fixtures have a direct effect on product manufacturing quality, productivity and cost, as a result fixtures, particularly welding fixture, play a crucial role in the auto industry. The fixture is a special tool for holding a work piece in proper position during manufacturing operation, so in the phase of the fixture design process positioning pins and surfaces are used to make sure that the work piece is positioned correctly and remain in the same position throughout the operation. The less positioning surfaces leads to the less work piece deformation. The aim of this paper is to find optimal number of positioning surfaces using VIKOR method with Shanon entropy concept to extract and utilize objective weights. VIKOR, means multicriteria optimization and compromise solution, is a modern approach that has preference over other MCDM methods. An empirical example is presented to demonstrate an application of mentioned method.
1

891
904


M
Ehsanifar
Department of Industrial Engineering, Islamic Azad University, Arak Branch, Arak, Iran
Department of Industrial Engineering, Islamic
Iran
mehsanifar@iauarak.ac.ir


M
Hemesy
Department of Industrial Engineering, Islamic Azad University, Arak Branch, Arak, Iran
Department of Industrial Engineering, Islamic
Iran
Fixture design
Vikor method
MCDM
ABAQUS
Supporting points
[[1] Automotive Industry, 2014, Encyclopedia Britannica, Retrieved 25.##[2] Automobile Industry Analysis Essay , 2017.##[3] Silk Road, 2005, China's Automotive Industry.##[4] Morris D., Donnelly T., Donnelly T., 2004, Supplier parks in the automotive industry, Supply Chain Management: An International Journal 9(2): 129133.##[5] Henriksen Erik K. ,1973, Jig and Fixture Design Manual, Industrial Press Inc, New York.##[6] Colvin Fred H., Haas Lucian L., 1938, Jigs and Fixtures: A Reference Book, McGrawHill Book Company, New York and London.##[7] Prabhaharan G., Padmanaban K. P., Krishnakumar R., 2007, Machining fixture layout optimization using FEM and evolutionary techniques, International Journal of Advanced Manufacturing Technology 32: 10901103.##[8] Zeshan A., 2013, Fixture layout optimization for large metal sheets using genetic algorithm, World Academy of Science, Engineering and Technology 7: 994999.##[9] Krishnikumar K., 2000, Machining fixture layout optimization using the genetic algorithm, International Journal of Machine Tools & Manufacture 40: 579598.##[10] Selvakumar S. , Arulshri K. P. , Padmanaban K. P. , Sasikumar K. S. K. , 2013, Design and optimization of machining fixture layout using ANN and DOE, The International Journal of Advanced Manufacturing Technology 65( 912): 15731586.##[11] Yang B. , Wang Z., Yang Y., Kang Y., Li Ch., 2017, Optimization of fixture locating layout for sheet metal part by cuckoo search algorithm combined with finite element analysis, Advances in Mechanical Engineering 9(6): 110.##[12] Zhang X., Yang W., Li M., 2010, An uncertainty approach for fixture layout optimization using monte carlo method, International Conference on Intelligent Robotics and Applications ICIRA, Intelligent Robotics and Applications.##[13] Bai X., Hu F., He G., Ding B., 2014, A memetic algorithm for multiobjective fixture layout optimization, SAGE Journals 229: 30473058.##[14] Wang Z. Q., Yang Y., Kang Y. G., Chang Z. P., 2014, A location optimization method for aircraft weaklyrigid structures, International Journal for Simulation and Multidisciplinary Design Optimization 5: A18A21.##[15] Lu C., 2015, Fixture layout optimization for deformable sheet metal work piece, The International Journal of Advanced Manufacturing Technology 78: 8598.##[16] Tao Z.J., Kumar A.S., Nee A.Y.C., 1999, A computational geometry approach to optimum clamping synthesis of machining fixtures, International Journal of Production Research 37(15): 34953517.##[17] Liao Y.J., Hu S.J., 2000, Flexible multibody dynamics based fixture–work piece analysis model for fixturing stability, International Journal of Machine Tools and Manufacture 40: 343362.##[18] Li B., Melkote S.N., 2001, Optimal fixture design accounting for the effect of work piece dynamics, International Journal of Advanced Manufacturing Technology 18: 701707.##[19] Li B., Shiu B. W., Lau K. J., 2002, Fixture configuration design for sheet metal assembly with laser welding: A case study, The International Journal of Advanced Manufacturing Technology 19: 501509.##[20] Ma J., Wang M.Y., 2011, Compliant fixture layout design using topology optimization method, 2011 IEEE International Conference on Robotics and Automation, Shanghai, China.##[21] Jindo T., Hirasago K., Nagamachi M., 1995, Development of a design support system for office chairs using 3D graphics, International Journal of Industrial Ergonomic 15: 4962.##[22] Chen M. F., Tzeng G. H., Ding C. G., 2003, Fuzzy MCDM approach to select service provider, In IEEE International Conference on Fuzzy Systems.##[23] Wang T. C., Lee H. D., 2009, Developing a fuzzy TOPSIS approach based on subjective weights and objective weights, Expert Systems with Applications 36: 89808985.##[24] Deng H., Yeh C. H., Willis R. J., 2000, Intercompany comparison using modified TOPSIS with objective weights, Computers and Operations Research 27: 963973.##[25] Opricovic S., 1998, Multicriteria optimization of civil engineering systems, Faculty of Civil Engineering 2: 521.##[26] Sayadi M. K., Heydari M., Shahanaghi K., 2009, Extension of VIKOR method for decision making problem with interval numbers, Applied Mathematical Modelling 33(5): 22572262.##[27] Yu P.L., 1973, A class of solutions for group decision problems, Management Science 19(8): 936946.##[28] Zeleny M., 1982, Multiple Criteria Decision Making, McGraw Hill, New York.##[29] Zhang N., Wei G., 2013, Extension of VIKOR method for decision making problem based on hesitant fuzzy set, Mathematical Modelling 37: 49384947.##]