2014
1
1
1
70
1

Mechanical Properties of Graphene/Epoxy Nanocomposites under Static and Flexural Fatigue Loadings
https://macs.semnan.ac.ir/article_274.html
10.22075/macs.2014.274
1
In the present study, the effect of various weight fractions of graphene nanoplatelet (GPL) on flexural fatigue behavior of epoxy polymer has been investigated at room temperature and generally the temperature was monitored on the surface of specimen during each test. The flexural stiffness of grapheme nanoplatelet/epoxy nanocomposites at 0.1, 0.25 and 0.5 wt. % as a main effective parameter on flexural bending fatigue was considered. The samples were implemented to different displacement fatigue amplitudes and it led to the known bending strength ratio. Finally, the flexural fatigue responses of graphene nanoplatelet/epoxy nanocomposites at mentioned graphene contents were taken into account. The experimental results show that the addition of 0.25 wt. % of graphene nanoplatelet on fatigue life was more effective in comparison with 0.1 and 0.5 wt. % epoxy graphene nanocomposites. According to the addition of graphene nanoplatelets, a remarkable increase in fatigue life of epoxy was observed. For instance, at the bending strength ratio equal to 43% by adding 0.1, 0.25 and 0.5 wt. % of graphene into epoxy resin, 22.4, 27.4 and 17 times improvement in flexural bending fatigue life of the neat epoxy were observed, respectively.
0

1
7


Mahmood
Shokrieh
Composites Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, 1684613114, Iran.
Iran
shokrieh@iust.ac.ir


Masood
Esmkhani
Composites Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, 1684613114, Iran.
Iran
esmkhani@iust.ac.ir


Amir
Haghighatkhah
Composites Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, 1684613114, Iran.
Iran
ahaqiqatkhah@mecheng.iust.ac.ir
Mechanical properties
Boundary Layer
Flexural bending fatigue
Displacement control
Graphene nanoplatelets
Nanocomposites
[[1] Paepegem VW, Degrieck J. Experimental Setup for and Numerical Modeling of Bending Fatigue Experiments on Plain Woven Glass/epoxy Composites. Compos Struct 2001; 51: 18.##[2] Paepegem VW, Degrieck J. A New Coupled Approach of Residual Stiffness and Strength for Fatigue of Fibrereinforced Composites, Int J Fatigue 2002; 24: 747762.##[3] Paepegem VW, Degrieck J. Fatigue Degradation Modelling of Plain Woven Glass/epoxy Composites. Compos Part A 2001; 32: 14331441.##[4] Ramkumar A, Gnanamoorthy R. Effect of nanoclay addition on the displacementcontrolled flexural fatigue behaviour of a polymer. Mater Sci 2010; 45: 4180–87.##[5] Rajeesh AK, Gnanamoorthy R, Velmurugan R. Effect of humidity on the indentation hardness and flexural fatigue behavior of polyamide 6 nanocomposite. Mater Sci and Eng 2010; 527: 2826–2830.##[6] Timmaraju MV, Gnanamoorthy R, Kannan K. Effect of environment on flexural fatigue behavior of polyamide 66/hectorite nanocomposites. Int J Fatigue 2011; 33: 541–48.##[7] Timmaraju MV, Gnanamoorthy R, Kannan K. Effect of initial imbibed moisture content on flexural fatigue behavior of polyamide 66/hectorite nanocomposites at laboratory condition. Mater Sci and Eng 2011; 528: 2960–2966.##[8] Shokrieh MM, Esmkhani M, Haghighatkhah AR. Flexural fatigue behaviour of carbon nanofiber/epoxy nanocomposites. Fatigue Fract Eng Mater Struct 2014; 37: 553560.##[9] Shokrieh MM, Esmkhani M, Haghighatkhah AR. Displacementcontrolled flexural bending fatigue behaviour of graphene/epoxy nanocomposites. Compos Mater DOI: 10.1177/0021998313503483 (in press)##[10] Shokrieh MM, Esmkhani M, Haghighatkhah AR. Zhao Z. Flexural fatigue behaviour of synthesized graphene/carbonnanofiber/epoxy hybrid nanocomposites. Mater Des 2014; 62: 401408.##[11] Böger L, Sumfleth J, Hedemann H, Schulte K. Improvement of fatigue life by incorporation of nanoparticles in glass fibre reinforced epoxy. Compos Part AAPPL. S 2010; 41: 1419–1424.##[12] Fritzschen J, Lorenz H, Klüppel M. CNT Based ElastomerHybridNanocomposites with Promising Mechanical and Electrical Properties. Macrmol Mater and Eng 2009; 294: 551–560.##[13] Li J, Wong PS, Kim JK. Hybrid nanocomposites containing carbon nanotubes and graphite nanoplatelets. Mater Sci Eng A 2008; 484: 660–663.##[14] ASTM B59396. Standard Test Method for Bending Fatigue Testing for CopperAlloy Spring Materials (2003).##[15] Berchem K, Hocking MG. A simple plane bending fatigue and corrosion fatigue testing machine, Measurement Sci and Technol 2006;17: 60–66.##[16] ASTM D 790–10, Standard test methods for flexural properties of unreinforced and reinforced plastics and electrical insulating materials, 2010.##[17] Shokrieh MM, M. Esmkhani, Shokrieh Z., Zhao Z. Stiffness prediction of grapheme nanoplatelet/epoxy nanocomposites by a combined molecular dynamics–micromechanics method, Comput Mater Sci 2014; 92: 444–450.##]
1

Predicting Young’s Modulus of Aggregated Carbon Nanotube Reinforced Polymer
https://macs.semnan.ac.ir/article_275.html
10.22075/macs.2014.275
1
Prediction of mechanical properties of carbon nanotubebased composite is one of the important issues which should be addressed reasonably. A proper modeling approach is a multiscale technique starting from nano scale and lasting to macro scale passing inbetween scales of micro and meso. The main goal of this research is to develop a multiscale modeling approach to extract mechanical properties of CNT based nanocomposites emphasizing on mesoscale parameters. Agglomeration and nonstraight shapes of CNTs have to be captured in this specific scale. The representative volume element (RVE) for mesoscale is identified considering local concentration of CNTs as the main source of inhomogeneity in the investigated material region. Irregular tessellation technique on the basis of Voronoi method and Bayes algorithm is employed to partition the RVE at meso scale into constitutive polygons containing one single aggregate. A MATLAB code is written to perform this stage on the basis of random pattern. Mechanical properties of the tasseled regions are extracted by a combination of micromechanics rule addressing local position and aggregates in the material region. A bounding technique accounting for nonstraight shape of CNT is utilized to consider the any arbitrary shape of wavy CNT. Investigated material region at macro scale is divided into constitutive blocks assigning random volume fractions of CNT to each block implying nonuniformed dispersion of CNT. The results demonstrate the importance of considering the position of local aggregates in modeling procedure. The obtained results of modeling are compared with experimentally measured mechanical properties.
0

9
16


Roham
Rafiee
Composites Research Laboratory, Faculty of New Sciences and Technologies, University of Tehran, Tehran, 1439955941 Iran
Iran
roham.rafiee@ut.ac.ir


Vahid
Firouzbakht
Composites Research Laboratory, Faculty of New Sciences and Technologies, University of Tehran, Tehran, 1439955941 Iran
Iran
firouzbakht@ut.ac.ir
Carbon nanotube
Multiscale modeling
Irregular tessellation
Stochastic modeling
[[1] Dai H. Carbon nanotubes: opportunities and challenges. Surf Sci 2002; 500(13): 218–41##[2] Kang, Heung YY, Kim JH, Lee JW, Gollapudi R, Subramaniam S, et al. Introduction to carbon nanotube and nanofiber smart materials, Compos. Part BEngineering 2006; 37(6): 382–94##[3] SalvetatDelmotte JP, Rubio A. Mechanical properties of carbon nanotubes: a fiber digest for beginners. Carbon 2002; 40(10) 1729–734##[4] Lau KT, Gu C, Hui D. A critical review on nanotube and nanotube/nanoclay related polymer composite materials, Compos Part BEngineering 2006; 37(6): 425–36##[5] Qian D, Dickey E, Andrews R, Rantell T. Load transfer and deformation mechanisms in carbon nanotubepolystyrene composites, Appl Phys Lett 2000; 76(20): 2868870##[6] Schadler L, Giannaris SC, Ajayan PM. Load transfer in carbon nanotube epoxy composites, Appl Phys Lett 2000; 73(26): 3842844##[7] Zhu J, Peng H, RodriguezMacias F, Margrave J, Khabashesku V, Imam A, Lozano K, Barrera E. Reinforcing epoxy polymer composites through covalent integration of functionalized nanotubes, Adv Funct Mater 2004; 14(7): 643–48##[8] Odegard GM, Gates TS, Wise KE, Park C, Siochi EJ. Constitutive modeling of nanotube–reinforced polymer composites, Compos Sci Technol 2003; 63(11): 1671–687##[9] Ashrafi B, Hubert P. Modeling the elastic properties of carbon nanotube array/polymer composites, Compos Sci Technology 2006; 66(34): 387–96##[10] Han Y, Elliott J. Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites, Compos Mater Sci 2007; 39 :315–23.##[11] Villoria RG, Miravete A. Mechanical model to evaluate the effect of the dispersion in nanocomposites, Acta Mater 2007; 55(9) 3025–031##[12] Tserpes KI, Panikos P, Labeas G, Panterlakis SpG, Multiscale modeling of tensile behavior of carbon nanotubereinforced composites, Theor Appl Fract mech 2008; 49(1): 5160##[13] Frankland SJV, Harik VM, Odegard GM, Brenner DW, Gates TS. The stress–strain behavior of polymer–nanotube composites from molecular dynamics simulation, Compos Sci Technol 2003; 63(11): 1655–661##[14] Mokashi VV, Qian D, Liu Y. A study on the tensile response and fracture in carbon nanotubebased composites using molecular mechanics, Compos Sci Technol 2007; 67(34): 530–40##[15] Shokrieh MM, Rafiee R. On the tensile behavior of an embedded carbon nanotube in polymer matrix with nonbonded interphase region, Compos Struct 2010; 92(3): 64752##[16] Selmi A, Friebel C, Doghri I, Hassis H. Prediction of the elastic properties of single walled carbon nanotube reinforced polymers: A comparative study of several micromechanical models, Compos Sci Technol 2007; 67(10): 2071–084##[17] Seidel GD, Lagoudas DC. Micromechanical analysis of the effective elastic properties of carbon nanotube reinforced composites, Mech Mater 2006; 38(810): 884–907##[18] Liu Y, Nishimura N, Otani Y. Largescale modeling of carbonnanotube composites by a fast multipole boundary element method, Comput Mater Sci 2005; 34(2): 17387##[19] Thostenson ET, Ren Z, Chou TW. Advances in the science and technology of carbon nanotubes and their composites: a review, Compos Sci Technol 2001; 61(13): 1899–912##[20] Anumandla V, Gibson RF. A comprehensive closed form micromechanics model for estimating the elastic modulus of nanotubereinforced composites, Compos Part A 2006; 37(12):2178–185##[21] Yeh MK, Tai NH, Liu JH. Mechanical peroperties of phenolicBased nanocomposites reinforced by multiwalled carbon nanotubes and carbon fibers, Compos Part A 2008; 39(4):677–84##[22] Shao LH, Luo RY, Bai SL,Wang J. Prediction of effective moduli of carbon nanotubereinforced composites with waviness and debonding. Compos Struct 2009; 87(3): 274281##[23] Montazeri A, Javadpour J, Khavandi A, Tcharkhtchi A, Mohajeri A. Mechanical properties of multiwalled carbon nanotube/epoxy composites, Mater Des 2010; 31: 4202–208##[24] Shady E, Gowayed Y. Effect of nanotube geometry on the elastic properties of nanocomposites, Compos Sci Technol 2010; 70(10): 1476–481##[25] Omidi M, Rokni DTH, S.Milani A, Seethaler RJ, Arasteh R. Prediction of the mechanical characteristics of multiwalled carbon nanotube/epoxy composites using a new form of the rule of mixtures, Carbon 2010; 48(11): 3218–228##[26] Fisher FT, Bradshaw RD, Brinson LC. Fiber waviness in nanotubereinforced polymer composites—I: modulus predictions using effective nanotube properties, Compos Sci Technol 2003; 63(11): 1689–1703##[27] Bradshaw RD, Fisher FT, Brinson LC. Fiber waviness in nanotubereinforced polymer composites: II. Modeling via Numerical approximation of the dilute strain concentration tensor. Compos Sci Technol 2003; 63(11): 1689–1703##[28] Ayatollahi MR, et al. Effect of multiwalled carbon nanotube aspect ratio on mechanical and electrical properties of epoxybased nanocomposites. Polym Test 2011; 30(5): 54856.##[29] Martone A, Faiella G, Antonucci V, Giordano M, Zarrelli M. The effect of the aspect ratio of carbon nanotubes on their effective reinforcement modulus in an epoxy matrix, Compos Sci Technol 2011; 71(8): 1117–1123##[30] Srivastava VK, Singh S. A MicroMechanical model for elastic modulus of multiwalled CarbonNanotube/Epoxy resin composites, Int J of Composite Materials 2012; 2(2): 16.##[31] Li C, Chou TW. Multiscale modeling of carbon nanotube reinforced polymer composites, Nanosci Nanotechnol 2003; 3(5): 42330.##[32] Tsai JL, Tzeng SH, Chiu YT, Characterizing elastic properties of carbon nanotubes/polyimide nancomposites using multiscale simulation, Compos. Part BEngineering 2010; 41: 106115##[33] Luo D, Wang WX, Takao Y. Effects of the distribution and geometry of carbon nanotubes on the macroscopic stiffness and microscopic stresses of nanocomposites, Compos Sci Technol 2007; 67(14): 2947–2958##[34] Spanos PD, Kontsos A. A multiscale monte carlo finite element method for determining mechanical properties of polymer nanocomposites, Probabilis Eng Mech 2007; 23(4): 45670##[35] Shokrieh MM, Rafiee R. Development of a full range multiscale model to obtain elastic properties of CNT/polymer composites. Iran Polym J 2012; 21: 397–402##[36] Okabe A, Boots B, Kokichi Sugihara K, Chiu SN. Spatial Tessellations – Concepts and Applications of Voronoi Diagrams. 2nd edition. John Wiley; 2000##[37] Mishnaevsky Jr. Computational esomechanics of composites. John Wiley; 2007##[38] Shokrieh MM, Rafiee R. Investigation of nanotube length effect on the reinforcement efficiency in carbon nanotube based composites, Compos Struct 2010; 92 (10): 2415420##[39] Shokrieh MM, Rafiee R. Prediction of mechanical properties of an embedded carbon nanotube in polymer matrix based on developing equivalent long fiber, Mech Res Commun 2010; 37(2):235240##[40] Shokrieh MM, Rafiee R. On the tensile behavior of an embedded carbon nanotube in polymer matrix with nonbonded interphase region, Compos Struct 2010; 92(3): 647652##[41] Hashin,Z. The elastic moduli of heterogeneous materials, Appl Mech 1962; 29: 143–150##[42] Christensen, RM. Mechanics of Composite Materials. New York: WileyInterscience; 1979##[43] Shi DL, Feng XQ, Huang YY, Hwang KC, Gao H. The effect of nanotube waviness and agglomeration on the elastic property of carbon nanotubereinforced composite, Eng Mater Technol 2004; 126: 25057##[44] Rafiee R. Influence of carbon nanotube waviness on the stiffness reduction of CNT/polymer composites, Compos Struct 2013; 97: 30409.##[45] Ogasawara T, Ishida Y, Ishikawa T, Yokota R. Characterization of multiwalled carbon nanotube/phenylethynyl terminated polyimide composites. Compos Part A 2004; 35(1): 67–74.##[46] Kanagaraj S, Varanda FR, Zhiltsova TV, Oliveira MSA, Simoes JAO. Mechanical properties of high density polyethylene/carbon nanotube composites. Compos Sci Technol 2007; 67(1516): 30713077##[47] Xiao SP, Zhang LC, Zarudi I. Mechanical and rheological properties of carbon nanotubereinforced polyethylene composites. Compos Sci Technol 2007; 67(2): 17782.##]
1

Biaxial Buckling and Bending of Smart Nanocomposite Plate Reinforced by CNTs using Extended Mixture Rule Approach
https://macs.semnan.ac.ir/article_276.html
10.22075/macs.2014.276
1
In this research, the buckling and bending behaviour of smart nanocomposite plate reinforced by single walled carbon nanotubes (SWCNTs) under electromagnetomechanical loadings is studied. The extended mixture rule approach is used to determine the elastic properties of nanocomposite plate. Equilibrium equations of smart nanocomposite plate are derived using the Hamilton’s principle based on the classical plate theory (CPT). The nonlocal critical biaxial buckling load and the nonlocal deflection of smart nanocomposite plate are obtained by applying the Eringen’s theory and Navier’s method. In this article, the influences of applied voltage, magnetic field, aspect ratios, nonlocal parameter, and elastic foundation coefficients on the critical buckling load and deflection of smart nanocomposite plate are investigated. The nonlocal critical biaxial buckling load of smart nanocomposite plate increases with the increase in applied voltage and magnetic field intensity. The nonlocal deflection of smart nanocomposite plate decreases with an increase in the magnetic field intensity. Also, the stability of smart nanocomposite plate increases in the presence of elastic foundation.
0

17
26


Mehdi
Mohammadimehr
Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
Iran
mmohammadimehr@kashanu.ac.ir


Borhan
RoustaNavi
Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
Iran
borhanrousta@yahoo.com


Ali
GhorbanpourArani
Institute of Nanoscience & Nanotechnology, University of Kashan, Kashan,Iran
Iran
aghorban@kashanu.ac.ir
Biaxial buckling and bending
Smart nanocomposite plate
Electromagnetomechanical loadings
The extended mixture rule approach
Classical plate theory
[[1] Fukuda H, Kawata K, On Young’s modulus of short fibre composites. Sci Technol 1974; 7(20): 227.##[2] Jin Y, Yuong FG, Simulation of elastic properties of singlewalled carbon nanotubes. Compos Sci Technol 2003; 63 (15): 157.##[3] Chang T, Gao H, Sizedependent elastic properties of a singlewalled carbon nanotube via a molecular mechanics model. Mech Phys Sol2003; 51 (10): 7459.##[4] Griebel M, Hamaekers J, Molecular dynamics simulations of the elastic moduli of polymer–carbon nanotube composites. Mech Eng, 2004; 193 (17): 8873.##[5] Han Y, Elliott J, Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites. Comput Mater Sci 2007; 39 (31): 23–5.##[6] Zhu R, Pan E, Roy AK, Molecular dynamics study of the stress–strain behavior of carbonnanotube reinforced, composites. Mater Sci Eng 2007; 447: 51–7.##[7] Li X, Gao H, Scrivens WA, Fei D, Xu X, Sutton MA, Reynolds AP, Myrick ML, Reinforcing mechanisms of singlewalled carbon nanotubereinforced polymer composites. Nanosci Nanotechnol 2007; 7(17): 239.##[8] Joshi P, Upadhyay SH, Evaluation of elastic properties of multi walled carbon nanotube reinforced composite, Comput Mater Sci 2014; 81: 338332.##[9] SalehiKhojin A, Jalili N, Buckling of boron nitride nanotube reinforced piezoelectric polymeric composites subject to combined electrothermomechanical loadings, Compos Sci Technol 2008; 68: 1501–1489.##[10] Vodenitcharova T, Zhang LC, Bending and local buckling of a nanocomposite beam reinforced by a singlewalled carbon nanotube. Int J Sol Struct 2006; 43 (30): 246.##[11] Shen HS, Postbuckling of nanotubereinforced composite cylindrical shells in thermal environments. Part II: Pressureloaded shells. Compos Struct 2011; 93: 2496503.##[12] ShaatM, MahmoudFF, GaoXL, FaheemAF, Sizedependent bending analysis of Kirchhoff nanoplates based on a modified couplestress theory including surface effects, Int J Mech Sci 2014; 79: 3731. [13] Golmalani ME, Rezataleb J, Nonlinear bending analysis of orthotropic nanoscale plates in an elastic matrix based on nonlocal continuum mechanics, Compos Struct2014; 111:97 85. [14] Malekzade P, a Shojaee M, Buckling analysis of quadrilateral laminated plates with CNT reinforced composite layers, ThinWalled Struct 2013; 71: 118108. [15] Mohammadimehr M, Mohandes M, Moradi M, size dependent effect on the buckling and vibration analysis of double bonded nanocomposite piezoelectric plate reinforced by BNNT based on modified couple stress theory, Vib Control, first published on August 5, 2014 as doi:10.1177/1077546314544513.##[16] Mohammadimehr M, Rousta Navi B, Ghorbanpour Arani A, Surface stress effect on the nonlocal biaxial buckling and bending analysis of piezoelectric nano plate reinforced by CNT using EshelbyMoriTanaka approach, Sol Mech, Accepted 6 August 2014.##[17] Ghorbanpour Arani A, Kolahchi R, Vossough H, Buckling analysis and smart control of SLGS using elastically coupled PVDF nanoplate based on the nonlocal Mindlin plate theory. Physica B 2012; 407: 4465–4458.##[18] Murmu T, Pradhan SC, Buckling of biaxially compressed orthotropic plates at small scales, Mech Res Commun2009; 36: 938–933.##[19] Zhu p, Lei ZX, Liew KM, Static and free vibration analyses of carbon nanotubereinforced composite plates using finite element method with first order shear deformation plate theory. Compos Struct 2012; 94: 1460– 1450.##[20] Lei ZX, Liew KM, Yu JL, Buckling analysis of functionally graded carbon nanotubereinforced composite plates using the elementfree kpRitz method. Compos Struct 2013; 98: 168– 160.##[21] Jafari Mehrabadi S, Sobhani Aragh B, Khoshkhahesh V, Taherpour A, Mechanical buckling of rectangular nanocomposite plate reinforced by aligned and straight singlewalled carbon nanotubes. Compos: Part B 2012; 43: 2040– 2031.##[22] Alzahrani EO, Zenkour AM, Sobhy M, Small scale effect on hygrothermomechanical bending of nanoplates embedded in an elastic medium. Compos Struct2013; 105: 172– 163.##[23] Alibeigloo A, Static analysis of functionally graded carbon nanotubereinforced composite plate embedded in piezoelectric layers by using theory of elasticity. Compos Struct 2013; 95: 622– 612.##[24] Mohammadimehr M, Rahmati A, Small scale eﬀect on electrothermomechanical vibration analysis of singlewalled boron nitride nanorods under electric excitation, Turkis J Env Sci 2013; 37: 151.##[25] Malekzadeh P, Setoodeh AR, Alibeygi Beni A, Small scale effect on the thermal buckling of orthotropic arbitrary straightsided quadrilateral nanoplates embedded in an elastic medium. Compos Struct 2011; 93: 2083– 9.##[26] Rahmati A H, Mohammadimehr M, Vibration analysis of nonuniform and nonhomogeneous boron nitride nanorods embedded in an elastic medium under combined loadings using DQM, Physica B: Condensed Matter 2014, 440: 9888.##[27] Mohammadimehr M, Saidi A R, Ghorbanpour Arani A, Arefmanesh A, Han Q, Buckling analysis of doublewalled carbon nanotubes embedded in an elastic medium under axial compression using nonlocal Timoshenko beam theory, Proc. J of IMech E Part C: Mech Eng Sci2011, 225: 506498.##[28] Kraus J, Electromagnetics. USA: McGrawHill Inc., 1984.##[29] Ghorbanpour Arani A, Maghamikia Sh, Mohammadimehr M, Arefmanesh A, Buckling analysis of laminated composite rectangular plates reinforced by SWCNTs using analytical and finite element methods. Mech Sci Technol 2011; 25 (3): 820809##]
1

An Analytical Study on Effects of adding Nanoparticles to Water and Enhancement in Thermal Properties Based on FalknerSkan Model
https://macs.semnan.ac.ir/article_277.html
10.22075/macs.2014.277
1
In the age of technology, it is vital to cool down different parts of a device to use it more beneficially. Using nanofluids is one of the most common methods which has shown very effective results. In this paper, we have rephrased a classic equation in fluid mechanics, i.e. the FalknerSkan boundary layer equation, in order to be used for nanofluid. This nonlinear equation, which was presented by Liao, has been solved by Homotopy Analysis Method (HAM). This method is very capable to solve a wide range of nonlinear equations. The physical interpretation of results which are velocity and temperature profiles are explained in details and they are parallel with experimental outcomes of previous researchers.
0

27
35


Y.
Rostamiyan
Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, Iran
Iran
y.rostamiyan@yahoo.com


M.
Abbasi
Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, Iran
Iran


F.
Aghajani
Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, Iran
Iran


F.
Hedayati
Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, Iran
Iran


S.M.
Hamidi
Department of Electrical and Energy Engineering, University of Vaasa, Vaasa 65200, Finland
Iran
Falknerskan
Boundary Layer
Nanofluids
HAM
[[1] Lee S, Choi SUS, Li S, Eastman JA. Measuring thermal conductivity of fluids containing oxidenanoparticles. ASME J Heat Transfer 1999; 121: 280–9.##[2] Xuan Y, Li Q. Heat transfer enhancement of nanofluids. Int J Heat Fluid Flow 2000;21:58–64.##[3] Das SK, Putra N, Roetzel W. Pool boiling characteristics of nanofluids. Int J Heat Mass Transfer 2003; 46: 851–62.##[4] Lee S, Choi SUS, Li S, Eastman JA. Measuring thermal conductivity of fluids containing oxidenanoparticles. ASME J Heat Transfer 1999; 121: 280–9.##[5] Xuan Y, Li Q.: Heat transfer enhancement of nanofluids. Int J Heat Fluid Flow 2000; 21: 58–64.##[6] Das SK, Putra N, Roetzel W. Pool boiling characteristics of nanofluids. Int J Heat Mass Transfer 2003; 46: 851–62.##[7] Das SK, Putra N, Roetzel W. Pool boiling of nanofluids on horizontal narrow tubes. Int J Multiphase Flow 2003; 29: 1237–47.##[8] Das SK, Putra N, Thiesen P, Roetzel W. Temperature dependence of thermal conductivity enhancement for nanofluids. ASME J Heat Transfer 2003; 125: 567–74.##[9] Wang X, Xu X, Choi SUS. Thermal conductivity of nanoparticle–fluid mixture. J Therm Phys Heat Transfer 1999; 13: 474–80.##[10] Eastman JA, Choi SUS, Li S, Yu W, Thomson LJ. Anomalously increased effective thermal conductivities of ethylene glycolbased nanofluids containing copper nanoparticles. Appl Phys Lett 2001; 78: 718–20.##[11] Xie H, Wang J, Xi T, Liu Y, Ai F, Wu Q. Thermal conductivity enhancement of suspensions containing nanosized alumina particles. Appl Phys 2002; 91: 4568–72.##[12] Dongsheng Wen, Guiping Lin, SaeidVafaei, Kai Zhang. Particuology 2009; 7: 141–150.##[13] Wang BX, Zhou LP, Peng XF. A fractal model for predicting the effective thermal conductivity of liquid with suspension of nanoparticles. Int J Heat Mass Transfer 2003; 46: 2665–72.##[14] Xue QZ. Model for effective thermal conductivity of nanofluids.PhysLett A 2003; 307: 313–7.##[15] Yu W, Choi SUS. The role of interfacial layers in the enhanced thermal conductivity of nanofluids: arenovated Maxwell model. Nanoparticle Res 2003; 5: 167–71.##[16] Falkner, V. M, and Skan, S. W. Solutions of the boundarylayer equations. Philosophical Magazine, 1931; 7(12): 865896.##[17] S. Abbasbandy and T. Hayat. Solution of the MHD FalknerSkan flow by homotopy analysis method, Commun Nonlinear Sci Numer Simul 2009; (14): 3591–3598.##[18] Pade. On the solution of Falkner–Skan equations, J Math Anal Appl 2003; (285): 264–274.##[19] Liao, S. J. On the Homotopy Analysis Method for Nonlinear Problems. Appl Math Comput 2004; 47 (2)##[20] Liao, S. J. An Explicit, Totally Analytic Approximation of Blasius’s Viscous Flow Problems. Int J NonLinear Mech 1999; 34 (4)##[21] Liao, S. J. Proposed Homotopy Analysis Techniques for the Solution of Nonlinear Problems. Ph.D. Dissertation, Shanghai Liao Tong University, China 1992.##[22] Domairry, G. and N. Nadim. Assessment of Homotopy Analysis Method and Homotopy Perturbation Method in NonLinear Heat Transfer Equation. Int Commun Heat Mass Transfer 2008; 35 (1):##[23] Domairry, G. and M. Fazeli. Homotopy Analysis Method to Determine the Fin Efficiency of Convective Straight Fins with TemperatureDependent Thermal Conductivity. Commun Nonlinear Sci Numer Simul 2009; 14 (2)##[24] Domairry, G., A. Mohsenzadeh, and M. Famouri. The Application of Homotopy Analysis Method to Solve Nonlinear Differential Equation Governing Jeffery–Hamel Flow. Commun Nonlinear Sci Numer Simul 2009; 14 (1): 85##[25] Fakhari A, Domairry G, Ebrahimpour. Approximate explicit solutions of nonlinear BBMB equations by homotopy analysis method and comparison with the exact solution. Phys lett A 2007; 368: 6468.##[26] Sohouli AR, Domairry D, Famouri M, Mohsenzadeh A. Analytical solution of natural convection of Darcian fluid about a vertical full cone embedded in porous media prescribed wall temperature by means of HAM, Int Commun Heat and Mass Transfer 2008; 35(10): 13801384.##[27] Ziabakhsh Z, Domairry G. Analytic solution of natural convection flow of a nonNewtonian fluid between two vertical flat plates using homotopy analysis method, Commun Nonlinear Sci Numer Simul 2009; 14(5): 18681880.##[28] Sohouli AR, Famouri M, Kimiaeifar A, Domairry G. Application of homotopy analysis method for natural convection of Darcian fluid about a vertical full cone embedded in pours media prescribed surface heat flux, Commun Nonlinear Sci Numer Simul 2010; 15(7): 16911699##[29] Ziabakhsh Z, Domairry G, Ghazizadeh HR. Analytical solution of the stagnationpoint flow in a porous medium by using the homotopy analysis method, Taiwan Institute Chem Eng 2009; 40(1): 9197.##[30] Rashidi MM, Domairry G, Dinarvand S. Approximate solutions for the Burger and regularized long wave equations by means of the homotopy analysis method, Commun Nonlinear Sci Numer Simul 2009; 14(3): 708717.##[31] Rashidi MM, Mohimanianpour SA, Abbasbandy S. Analytic approximate solutions for heat transfer of a micropolar fluid through a porous medium with radiation, Commun Nonlinear Sci Numer Simul 2011; 16(4): 18741889.##[32] Abdoul R. Ghotbi, H. Bararnia, G. Domairry, A. Barari: Investigation of a powerful analytical method into natural convection boundary layer flow, Commun Nonlinear Sci Numer Simul 2009; 14(5): 22222228##[33] Bararnia H, Ghasemi E, Domairry G, Soleimani S. Behavior of micropolar flow due to linear stretching of porous sheet with injection and suction, Adv Eng Softw 2010; 41(6): 893897.##[34] Jalaal M, Bararnia H, Domairry G. A series exact solution for onedimensional nonlinear particle equation of motion, Powder Technol 2011; 207 (13): 461464.##[35] Kimiaeifar A, Saidi AR, Bagheri GH, Rahimpour M, Domairry DG. Analytical solution for Van der Pol–Duffing oscillators. Chaos Solitons & Fractals 2009; 42(5): 26602666.##[36] Oztop HF, AbuNada E. Numerical study of natural convection in partially heated rectangular enclosures ﬁlled with nanoﬂuids, Int J Heat Fluid Flow 2008; 29: 1326–1336.##]
1

Optimization of Hybrid Composite Laminate Based on the Frequency using Imperialist Competitive Algorithm
https://macs.semnan.ac.ir/article_278.html
10.22075/macs.2014.278
1
Imperialist competitive algorithm (ICA) is a new sociopolitically motivated global search strategy. The ICA is applied to hybrid composite laminates to obtain minimum weight and cost. The approach which is chosen for conducting the multiobjective optimization was the weighted sum method (WSM). The hybrid composite Laminates are made of glass/epoxy and carbon/epoxy to combine the lightness and economical attributes of the first with highstiffness property of the second in order to make tradeoff between the cost and weight as the objective functions and natural flexural frequency as a constraint. The results were evaluated for different weighting factors (a) including optimum stacking sequences, and number of plies made of either glass or carbon fibers using the ICA, and were compared with those using the genetic algorithm (GA) and ant colony system (ACS). The comparisons confirmed the advantages of hybridization and revealed that the ICA outperformed the GA and ACS in terms of function’s value and constraint accuracy.
0

37
48


Hossein
Hemmatian
Departement of Mechanical Engineering, Semnan Branch, Islamic Azad University, Semnan, Iran
Iran
hoseinhemmatian@gmail.com


Abdolhossein
Fereidoon
Faculty of Mechanical Engineering, Semnan University, Semnan, 1911135131, Iran
Iran
afereidoon@semnan.ac.ir


Hadi
Shirdel
Faculty of Mechanical Engineering, Semnan University, Semnan, 1911135131, Iran
Iran
hoseinhemmatian@yahoo.com
Composite laminate
Hybridization
Stacking sequence
Imperialist competitive algorithm
Frequency
[[1] Ghasemi H, Roberto Brighenti R, Xiaoying Zhuang X, Jacob Muthu J, Timon Rabczuk T. Optimization of fiber distribution in fiber reinforced composite by using NURBS functions. Comput Mater Sci 2014; 83: 463473.##[2] Blasques J P, Stolpe M. Maximum stiffness and minimum weight optimization of laminated composite beams using continuous fiber angles. Struct Multidiscip O 2011; 43(4): 573588.##[3] Sadollah A, Bahreininejad A, Hamdi M, Purbolaksono J. Optimum mechanical behavior of calcium phosphate cement/hydroxyl group functionalized multiwalled carbon nanotubes/bovine serum albumin composite using metaheuristic algorithms. Neural Comput Appl 2014; 24(1): 193200.##[4] Zuo Z H, Xie Y M. Maximizing the effective stiffness of laminate composite materials. Comput Mater Sci 2014; 83: 5763.##[5] Hemmatian H, Fereidoon A, Sadollah A, Bahreininejad A. Optimization of laminate stacking sequence for minimizing weight and cost using elitist ant system optimization. Adv Eng Softw 2013; 57: 818.##[6] Sobey AJ, Blake JIR, Shenoi RA. Implications of failure criteria choices on the rapid concept design of composite grillage structures using multiobjective optimisation. Struct Multidiscip O 2013; 47: 735747.##[7] Omkar SN, Venkatesh A, Mudigere M. MPIbased parallel synchronous vector evaluated particle swarm optimization for multiobjective design optimization of composite structures. Eng Appl Artif Intel 2012; 25: 16111627.##[8] Rahul, S G, Chakraborty D, Dutta A. Multiobjective optimization of hybrid laminates subjected to transverse impact. Compos Struct 2006; 73(3): 360369.##[9] Javidrad F, Nouri R. A simulated annealing method for design of laminates with required stiffness properties. Compos Struct 2011; 93: 112735.##[10] Gubran HBH, Gupta K. The effect of stacking sequence and coupling mechanisms on the natural frequencies of composite shafts. Sound Vib 2012; 282: 231248.##[11] Banerjee JR. Explicit analytical expressions for frequency equation and mode shapes of composite beams. Sol Struct 2001; 38: 24152426.##[12] Shun F H, Chao SC. Determination of elastic constants of materials by vibration testing. Compos Struct 2000; 49(2): 183190.##[13] Abachizadeh M, Tahani M. An ant colony optimization approach to multi objective optimal design of symmetric hybrid laminates for maximum fundamental frequency and minimum cost. Struct Multidisc Optim 2009; 37: 367376.##[14] Abdalla MM, Setoodeh S, Gürdal Z. Design of variable stiffness composite panels for maximum fundamental frequency using lamination parameters. Compos Struct 2007; 81: 283291.##[15] Bert CW. Optimal design of a compositematerial plate to maximize its fundamental frequency. Sound Vib 1977; 50: 229237.##[16] Reiss R, Ramachandran S. Maximum frequency design of symmetric angleply laminates. Compos Struct 1987; 4: 14761487.##[17] Grenestedt JL. Layup optimization and sensitivity analysis of the fundamental eigenfrequency of composite plates. Compos Struct 1989. 12: 193209.##[18] Duffy KJ, Adali S. Maximum frequency design of prestressed symmetric, crossply laminates of hybrid construction. Adv Des Auto 1991; 2: 477484.##[19] Adali S. Design of sheardeformable antisymmetric angleply laminates to maximize the fundamental frequency and frequency separation. Compos Struct 1984; 2: 349369.##[20] Apalak MK, Yildirim M, Ekici R. Layer optimization for maximum fundamental frequency of laminated composite plates for different edge conditions. Compos Sci Technol 2008; 68: 537550.##[21] Fukunaga H, Sekine H, Sato M. Optimal design of symmetric laminated plates for fundamental frequency. Sound Vib 1994; 171: 219229.##[22] Narita Y. Layerwise optimization for the maximum fundamental frequency of laminated composite plates. Sound Vib 2003; 263: 10051016.##[23] Narita Y, Hodgkinson JM. Layerwise optimisation for maximizing the fundamental frequencies of pointsupported rectangular laminated composite plates. Compos Struct 2005; 69: 127135.##[24] Adali S, Duffy KJ. Minimum cost design of vibrating laminates by hybridization. Eng Optim 1992; 19: 255267.##[25] Farshi B, Rabiei R. Optimum design of composite laminates for frequency constraints. Compos Struct 2007; 81(4): 87597.##[26] Adali S, Verijenko VE. Optimum stacking sequence design of symmetric hybrid laminates undergoing free vibrations. Compos Struct 2001; 54: 131138.##[27] Tahani M, Kolahan F, Sarhadi A. Genetic algorithm for multiobjective optimal design of sandwich composite laminates with minimum cost and maximum frequency. ICMPM, Sathyamangalam, India, 2005. p. 741748.##[28] Kolahan F, Tahani M, Sarhadi A. Optimal design of sandwich composite laminates for minimum cost and maximum frequency using simulated annealing. TICME, Tehran, Iran, 2005.##[29] Hao W, Ying Y, Yujia L. Reliability Based Optimization of Composite Laminates for Frequency Constraint. Chinese Journal of Aeronautics 2008; 21: 320327.##[30] Grosset L, Venkataraman S, Haftka RT. Genetic optimization of twomaterial composite laminates. In: 16th ASC Technical Meeting, Blacksburg, Virginia, 2001.##[31] Abachizadeh M, Tahani M. Ant colony optimization of hybrid laminates for minimum cost and weight. In: Saravanos DA (ed) Proceedings of 6th international symposium on advanced composites COMP’07, Corfu, Greece, 2007.##[32] AtashpazGargari E, Lucas C. Imperialist competitive algorithm: An algorithm for optimization inspired by imperialistic competition. In: IEEE Congress on Evolutionary Computation, Singapore, 2007. p. 466167.##[33] AtashpazGargari E, Hashemzadeh F, Rajabioun R, Lucas C. Colonial competitive algorithm: A novel approach for PID controller design in MIMO distillation column process. Int J Intel Comput Cyber 2008; 3: 33755.##[34] Kaveh A, Talatahari S. Imperialist competitive algorithm for engineering design problems. Asian J Civil Eng 2010; 11: 675697.##[35] Kaveh A, Talatahari S. Optimum design of skeletal structures using imperialist competitive algorithm. Comput Struct 2010; 88: 12201229.##[36] Abdi B, Mozafari H, Ayob A, Kohandel R. Imperialist Competitive Algorithm and its Application in Optimization of Laminated Composite Structures. European J Sci Res 2011; 55: 174187.##[37] Mozafari H, Abdi B, Ayob A. Optimization of Composite Plates Based on Imperialist Competitive Algorithm. Int J Comput Sci Eng 2010; 2: 28162819.##[38] Mozafari H, Alias A, Kamali F. Optimum Design of Composite Plates under Thermal Buckling Loads using Imperialist Competitive Algorithm. Int J Comput Sci Eng Technol 2010; 1: 5458.##[39] Esmaeilzadeh M. A modified colonial competitive algorithm for optimizing convex functions. Int J Intel Comput Cyber 2013; 6(4): 370385.##[40] Reddy JN. Mechanics of laminated composite plates and shells: Theory and analysis. 2nd ed. CRC Press Inc., Boca Raton, Florida, 2004.##[41] BiabangardOskouyi A, AtashpazGargari E, Soltani N, Lucas C. Application of Imperialist Competitive Algorithm for materials property characterization from sharp indentation test. Int J Eng Simul 2008; 10.##[42] Sepehri Rad H, Lucas C. Application of Imperialistic Competition Algorithm in Recommender Systems. In: 13th Int'l CSI Computer Conference (CSICC'08), Kish Island, Iran, 2008.##[43] Rajabioun R, AtashpazGargari E, Lucas C. Colonial Competitive Algorithm as a Tool for Nash Equilibrium Point Achievement. Lect notes comput sc 2008; 5073: 680695.##[44] Niknam T, Taherian Fard E, Pourjafarian N, Rousta A. An efficient hybrid algorithm based on modified imperialist competitive algorithm and Kmeans for data clustering. Eng Appl Artif Intel 2011; 24: 306317.##[45] Suman B, Kumar P. A survey of simulated annealing as a tool for single and multiobjective optimization. Oper Res Soc 2006; 15: 11431160.##[46] Herakovich CT. Mechanics of fiberous composites, New York: John Whiley and Sons, 1998##]
1

Elasticity Solution of Functionally Graded Carbon Nanotube Reinforced Composite Cylindrical Panel
https://macs.semnan.ac.ir/article_279.html
10.22075/macs.2014.279
1
Based on threedimensional theory of elasticity, static analysis of functionally graded carbon nanotube reinforced composite (FGCNTRC) cylindrical panel subjected to mechanical uniformed load with simply supported boundary conditions is carried out. In the process, stress and displacement fields are expanded according to the Fourier series along the axial and circumferential coordinates. From constitutive law, stressdisplacement relations and equilibrium equations, state space equation is obtained. The obtained first order governing differential equations can be solved analytically. The effects of CNT distribution cases, the volume fraction of CNT, length to mid radius ratio, span of the cylindrical panel, variation of mechanical load and radius to thickness ratio on the bending behaviour of the cylindrical panel are examined. It should be noted that by using Fourier series solution it is possible only to solve the static behaviour of cylindrical panel with simply supported for all of edges and for the nonsimply supported boundary conditions it is possible to solve numerically. The obtained analytical solution can be used to validate the results of approximate two dimensional conventional theories.
0

49
60


Akbar
Alibeigloo
Department of Mechanical Engineering, Tarbiat Modares University, Tehran, 14115143, Iran
Iran
abeigloo@modares.ac.ir
Carbon nanotube
Cylindrical panel
Boundary Layer
Static
Elasticity
[[1] Thostenson, ET, Ren,ZF, Chou, TW. Advances in the science and technology of carbon nanotubes and their composite: a review, Compos Sci Thecnol 2001; 61:1899912.##[2] Gou J, Minaie B, Wang B, Liang Z, Zhang C. Computational and experimental study of interfacial bonding of singlewalled nanotube reinforced composites stiffness, Comp Mater Sci 2004;31:225–236.##[3] Wuite J, Adali S. Deflection and stress behavior of nanocomposite reinforced beams using a multi scale analysis, Compos Struct 2005;71:388–396.##[4] Vodenitcharova T, Zhang C. Bending and local buckling of a nanocomposite beam reinforced by a singlewalled carbon nanotube, Int J Sol Struct 2006; 43:3006–3024.##[5] Shen HS. Nonlinear bending of functionally graded carbon nanotubereinforced composite plates in thermal environments, Compos Struct 2009;91:9–19.##[6] Formica G, Lacarbonara W, Alessi R. Vibrations of carbon nanotubereinforced composites, Sound Vib 2010;329:1875–1889.##[7] Shen HS, Zhang C. Thermal buckling and post buckling behavior of functionally grade carbon nanotubereinforced composite plates, Mater Design 2010;31:3403–3411.##[8] Ke LL, Yang J, Kitipornchai S. Nonlinear free vibration of functionally graded carbon nanotubereinforced composite beams, Compos Struct 2010;92:676683.##[9] Shen HS. Post buckling of nanotubereinforced composite cylindrical shells in thermal environments, Part I: Axiallyloaded shells, Compos Struct 2011; 93: 2096–2108.##[10] Shen HS. Post buckling of nanotubereinforced composite cylindrical shells in thermal environments, Part II: Pressureloaded shells, Compos Struct 2011;93: 2496–2503.##[11] Wang ZX, Shen HS. Nonlinear vibration of nanotubereinforced composite plates in thermal environments, Comp Mater Sci 2011;50: 2319–2330.##[12] Mehrabadi SJ, Sobhani Aragh B, Khoshkhahesh V, Taherpour A. Mechanical buckling of nanocomposite rectangular plate reinforced by aligned and strait singlewalled carbon nanotubes, Compos Part BEng 2012;43(4):20312040.##[13] Zhu P, Lei ZX, Liew KM. Static and free vibration analyses of carbon nanotubereinforced composite plates using finite element method with first order shear deformation plate theory, Compos Struct 2012;94:1450–1460.##[14] Wang ZX, Shen HS. Nonlinear vibration and bending of sandwich plates with nanotubereinforced composite face sheets, Compos Part BEng 2012;43:411–421.##[15] Yas MH, Heshmati M. Dynamic analysis of functionally graded nanocomposite beam reinforced by randomly oriented carbon nanotube under the action of moving load, Appl Math Model 2012;36:1371–1394.##[16] Alibeigloo A. Static analysis of functionally graded carbon nanotubereinforced composite plate embedded in piezoelectric layers by using theory of elasticity, Compos Struct 2013;95:612622.##[17] Bhardwaj G, Upadhyay AK, Pandey R, Shukla KK. Nonlinear flexural and dynamic response of CNT reinforced laminated composite plate, Compos Part BEng 2013;45:89100.##[18] Shen HS. Thermal buckling and post buckling behavior of functionally graded carbon nanotubereinforced composite cylindrical shells, Compos Part BEng 2012; 43:10301038.##[19] Shen HS, Xiang Y. Nonlinear vibration of nanotubereinforced composite cylindrical shells in thermal environments, Comput. Methods Appl. Mech. Eng 2012; 213216:196–205.##[20] MoradiDastjerdi R, Foroutan M, Pourasgha A. Dynamic analysis of functionally graded nanocomposite cylinders reinforced by carbon nanotube by a meshfree method, Mater Design 2013;44:256–266.##[21] MoradiDastjerdi R, Foroutan A, Pourasgha A. Eshelby–Mori–Tanaka approach for vibrational behavior of continuously graded carbon nanotubereinforced cylindrical panels. Compos Part BEng 2012;43:1943–1954.##]
1

Free Vibration Analysis of Composite Plates with Artificial Springs by Trigonometric Ritz Method
https://macs.semnan.ac.ir/article_280.html
10.22075/macs.2014.280
1
In this paper free vibration analysis of two rectangular isotropic plates, which are connected to each other by two translational and rotational springs along the edges, are investigated. The equation of motion and associated boundary and continuity conditions are derived using the extended Hamilton principle. To solve the eigenvalue problem, the Ritz method is utilized. Numerical investigations are presented to show some applications of this method. In this research two types of problems are investigated: first, vibration of a continuous plate and second, free vibration of two hinged plates. This approach is usually referred to as the artificial spring method, which can be regarded as a variant of the classical penalty method. In order to validate the results, the achieved results are compared to results which are presented in literatures.
0

61
70


Hossein
Ghadirian
Faculty of Mechanical Engineering, Tarbiat Modares University, Tehran, Iran
Iran
dhghadirian@yahoo.com


Mohammad
Ghazavi
Faculty of Mechanical Engineering, Tarbiat Modares University, Tehran, Iran
Iran
dhghadirian1@yahoo.com


Korosh
Khorshidi
Department of Mechanical Engineering, Faculty of Engineering, Arak University, 3815688349, Arak, Iran
Iran
kkhorshidi@araku.ac.ir
vibration analysis
Composite plates
Artificial spring
Trigonometric Ritz method
[Leissa AW, Vibration of plates ,DTIC Document, 1969.##[2] B. RD, 1984 Formulas for Natural Frequency and Mode Shape, Malabar, FL: Robert E. Krieger Publishing Company.##[3] Gorman DJ, Vibration analysis of plates by the superposition method, World Scientific, 1999.##[4] Reddy JN, Mechanics of laminated composite plates and shells: theory and analysis, CRC press, 2003.##[5] Timoshenko S, WoinowskyKrieger S, Theory of plates and shells, McGrawhill New York, 1959.##[6] Leissa A, Recent research in plate vibrations, 19731976: complicating effects, Shock and Vibration Inform. Shock Vib Digest 1978; 10: 2135.##[7] Qatu MS, Sullivan RW, Wang W, Recent research advances on the dynamic analysis of composite shells: 2000–2009, Compos Struct 2010; 93: 1431.##[8] Lee H, Ng T, Vibration of symmetrically laminated rectangular composite plates reinforced by intermediate stiffeners, Compos Struct 1994; 29: 405413.##[9] Lee H, Ng T, Effects of Torsional and Bending Restraints of Intermediate Stiffeners on the Free Vibration of Rectangular Plates, Struct Mech 1995; 23: 309320.##[10] Huang C, Leissa A, Chan C, Vibrations of rectangular plates with internal cracks or slits, Int J Mech Sci 2011; 53: 436445.##[11] Grossi RO, Raffo J, Natural vibrations of anisotropic plates with several internal line hinges, Acta Mech 2013; 224: 26772697.##[12] Quintana MV, Grossi RO, Free Vibrations of a Trapezoidal Plate with an Internal Line Hinge, The Scientific World Journal 2014.##[13] Wang C, Xiang Y, Wang C, Buckling and vibration of plates with an internal line hinge via the Ritz method, Proceedings of the First AsianPacific Congress on Computational Mechanics 2001, 16631672.##[14] Du J, Liu Z, Li WL, Zhang X, Li W, Free inplane vibration analysis of rectangular plates with elastically pointsupported edges, Vib Acoust 2010; 132: 031002.##[15] Liew K, Lam K, Effects of arbitrarily distributed elastic point constraints on vibrational behaviour of rectangular plates, sound vibration 1994; 174: 2336.##[16] Kim C, Dickinson S, The flexural vibration of line supported rectangular plate systems, sound and vibration 1987; 114:129142.##[17] Xiang Y, Zhao Y, Wei G, Levy solutions for vibration of multispan rectangular plates, Int J Mech Sci 2002; 44: 11951218.##[18] Zhou D, Vibrations of pointsupported rectangular plates with variable thickness using a set of static tapered beam functions, Int J Mech Sci 2002; 44: 149164.##[19] Zhou D, Cheung Y, Free vibration of line supported rectangular plates using a set of static beam functions, sound vibration 1999; 223: 231245.##[20] Quintana M, Grossi RO, Free vibrations of a generally restrained rectangular plate with an internal line hinge, Appl Acoust 2012; 73: 356365.##[21] Dozio L, On the use of the trigonometric Ritz method for general vibration analysis of rectangular Kirchhoff plates, ThinWalled Struct 2011; 49: 129144.##[22] Dozio L, Inplane free vibrations of singlelayer and symmetrically laminated rectangular composite plates, Compos Struct 2011; 93:17871800.##[23] Young D, Vibration of rectangular plates by the Ritz method, appl mech 1950; 17:448453.##[24] Warburton G, The vibration of rectangular plates, Proceedings of the Institution of Mechanical Engineers 1954; 168: 371384.##[25] Leissa A, The free vibration of rectangular plates, sound vibration, 31 (1973) 257293.##[26] S. Bassily, S. Dickinson, On the use of beam functions for problems of plates involving free edges, appl mech 1975; 42 : 858.##[27] Goncalves P, Brennan M, Elliott S, Numerical evaluation of highorder modes of vibration in uniform Euler–Bernoulli beams, sound vibration 2007; 301: 10351039.##[28] Xiang Y, Reddy J, Natural vibration of rectangular plates with an internal line hinge using the first order shear deformation plate theory, sound vibration 2003; 263: 285297.##[29] Beslin O, Nicolas J, A hierarchical functions set for predicting very high order plate bending modes with any boundary conditions, sound vibration 1997; 202:633655.##[30] Xing Y, Liu B, New exact solutions for free vibrations of thin orthotropic rectangular plates, Compos Struct 2009; 89: 56757.##]