2018
10
3
0
224
Vibration Analysis of FG Nanoplate Based on ThirdOrder Shear Deformation Theory (TSDT) and Nonlocal Elasticity
2
2
In present study, the thirdorder shear deformation theory has been developed to investigate vibration analysis of FG Nanoplates based on Eringen nonlocal elasticity theory. The materials distribution regarding to the thickness of Nanoplate has been considered based on two different models of power function and exponential function. All equations governing on the vibration of FG Nanoplate have been derived from Hamilton’s principle. It has been also obtained the analytical solution for natural frequencies and corresponding mode shapes of simply supported FG Nanoplates. In addition, the general form of stiffness and mass matrix elements has been expressed based on this theory. The effect of different parameters such as power and exponential indexes of targeted function , nonlocal parameter of Nanoplate, aspect ratio and thickness to length ratio of Nanoplate on nondimensional natural frequencies of free vibration responses have been investigated. The obtained analytical results show an excellent agreement with other available solutions of previous studies. The formulation and analytical results obtained from proposed method can be used as a benchmark for further studies to develop this area of research.
1

464
475


M.M
Najafizadeh
Department of Mechanical Engineering, Islamic Azad University, Arak Branch, Arak, Iran
Department of Mechanical Engineering, Islamic
Iran
mnajafizadeh@iauarak.ac.ir


M
Raki
Department of Mechanical Engineering, Islamic Azad University, Arak Branch, Arak, Iran
Department of Mechanical Engineering, Islamic
Iran


P
Yousefi
Department of Mechanical Engineering, Islamic Azad University, Arak Branch, Arak, Iran
Department of Mechanical Engineering, Islamic
Iran
Nanoplate
Functionally graded material (FGM)
Nonlocal elasticity
Third order of shear deformation theory (TSDT)
Natural frequency
[[1] Hirano T., Yamada T., 1988, Multi paradigm expert system architecture based upon the inverse design concept, Proceedings of the International Workshop on Artificial Intelligence for Industrial Applications 1988: 245250.##[2] Bao G., Wang L., 1995, Multiple cracking in functionally graded ceramic/metal coatings, International Journal of Solids and Structures 32(19): 28532871.##[3] Jin Z.H., Paulino G. H., 2001, Transient thermal stress analysis of an edge crack in a functionally graded material, International Journal of Fracture 107(1): 7398.##[4] Erdogan F., Chen Y. F., 1998, Interfacial cracking of FGM/metal bonds, Ceramic Coating 1998: 2937.##[5] Delale F., Erdogan F., 1983, The crack problem for a nonhomogeneous plane, Journal of Applied Mechanics 50(3): 609614.##[6] Aydogdu M., 2009, A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration, Physica E: LowDimensional Systems and Nanostructures 41(9): 16511655.##[7] Civalek Ö., Demir Ç., 2011, Bending analysis of microtubules using nonlocal EulerBernoulli beam theory, Applied Mathematical Modelling 35(5): 20532067.##[8] Reddy J.N., 2007, Nonlocal theories for bending, buckling and vibration of beams, International Journal of Engineering Science 45(2): 288307.##[9] Reddy J. N., Pang S. D., 2008, Nonlocal continuum theories of beams for the analysis of carbon nanotubes, Journal of Applied Physics 103(2): 23511.##[10] Reddy J. N., 2010, Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates, International Journal of Engineering Science 48(11): 15071518.##[11] Roque C. M. C., Ferreira A. J. M., Reddy J. N., 2011, Analysis of Timoshenko nanobeams with a nonlocal formulation and meshless method, International Journal of Engineering Science 49(9): 976984.##[12] Wang C. M., Kitipornchai S., Lim C. W., Eisenberger M., 2008, Beam bending solutions based on nonlocal Timoshenko beam theory, Journal of Engineering Mechanics 134(6): 475481.##[13] Fu Y., Du H., Zhang S., 2003, Functionally graded TiN/TiNi shape memory alloy films, Materials Letters 57(20): 29952999.##[14] Lu C., Wu D., Chen W., 2011, Nonlinear responses of nanoscale FGM films including the effects of surface energies, Nanotechnology 10(6): 13211327.##[15] Shaat M., Mahmoud F. F., Alieldin S. S., Alshorbagy A. E., 2013, Finite element analysis of functionally graded nanoscale films, Finite Elements in Analysis and Design 74: 4152.##[16] Wang W. L., Hu S. J., 2005, Modal response and frequency shift of the cantilever in a noncontact atomic force microscope, Applied Physics Letters 87(18): 183506.##[17] Shaat M., Abdelkefi A., 2016, Modeling of mechanical resonators used for nanocrystalline materials characterization and disease diagnosis of HIVs, Microsystem Technologies 22(2): 305318.##[18] Salehipour H., Nahvi H., Shahidi A. R., 2015, Exact analytical solution for free vibration of functionally graded micro/nanoplates via threedimensional nonlocal elasticity, Physica E: LowDimensional Systems and Nanostructures 66: 350358.##[19] Nami M. R., Janghorban M., 2014, Resonance behavior of FG rectangular micro/nano plate based on nonlocal elasticity theory and strain gradient theory with one gradient constant, Composite Structures 111: 349353.##[20] Salehipour H., Nahvi H., Shahidi A. R., 2015, Exact closedform free vibration analysis for functionally graded micro/nano plates based on modified couple stress and threedimensional elasticity theories, Composite Structures 124: 283291.##[21] Salehipour H., Shahidi A. R., Nahvi H., 2015, Modified nonlocal elasticity theory for functionally graded materials, International Journal of Engineering Science 90: 4457.##[22] Natarajan S., Chakraborty S., Thangavel M., Bordas S., Rabczuk T., 2012, Sizedependent free flexural vibration behavior of functionally graded nanoplates, Computational Materials Science 65: 7480.##[23] Eringen A. C., 2002, Nonlocal Continuum Field Theories, Springer.##[24] Meirovitch L., 2001, Fundamentals of Vibrations, Mc GrawHill.##[25] Aghababaei R., Reddy J. N., 2009, Nonlocal thirdorder shear deformation plate theory with application to bending and vibration of plates, Journal of Sound and Vibration 326(12): 277289.##[26] HosseiniHashemi S., Kermajani M., Nazemnezhad R., 2015, An analytical study on the buckling and free vibration of rectangular nanoplates using nonlocal thirdorder shear deformation plate theory, European Journal of Mechanics  A/Solids 51: 2943.##[27] Zare M., Nazemnezhad R., HosseiniHashemi S., 2015, Natural frequency analysis of functionally graded rectangular nanoplates with different boundary conditions via an analytical method, Meccanica 50(9): 23912408.##]
Mechanical Buckling Analysis of Composite Annular Sector Plate with BeanShaped CutOut using Three Dimensional Finite Element Method
2
2
In this paper, mechanical buckling analysis of composite annular sector plates with bean shape cut out is studied. Composite material sector plate made of GlassEpoxy and GraphiteEpoxy with eight layers with same thickness but different fiber angles for each layer. Mechanical loading to form of uniform pressure loading in radial, environmental and biaxial directions is assumed. The method used in this analysis is three dimensional (3D) finite elements based on the elasticity relations. With zero first and second variation of potential energy of the entire annular sector plate, we find stability equation. Green nonlinear displacement strain relations to obtain geometric stiffness matrix is used. Unlike many studies, in present work three dimensional finite elements method has been used with an eight node element and meshing in the thickness direction is done, too. The bean shaped cut out in the sector has increased the complexity of the analysis. The continuing, effect of different parameters including cut out dimensions, fiber angles of layers, loading direction and dimensions of the annular sector plate on the mechanical buckling load has been investigated and interesting results have been obtained.
1

476
488


H
Behzad
Faulty of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran
Faulty of Mechanical Engineering, Shahrood
Iran


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


M
Shariyat
Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran
Faculty of Mechanical Engineering, K.N. Toosi
Iran
Annular sector plate
Composite material
3D finite elements method
Mechanical buckling
[[1] Bruno D., Lato S., 1991, Buckling of moderately thick composite plates, Journal of Computers & Structures 18: 6575.##[2] Chai G. B., 1994, Buckling of generally laminated composite plates with various edge support conditions, Journal of Composite Structures 29: 299310.##[3] Kim Y. S., Hoa S. V., 1995, Biaxial buckling behavior of composite rectangular plates, Journal of Composite Structures 31: 247252.##[4] Zhou Y. H., Zheng X., Harik I. E., 1995, A Seminumerical method for buckling of sector plates, Journal of Computers & Structures 57(5): 847854.##[5] Sundaresan P., Singh G., Rao V., 1998, Buckling of moderately thick rectangular composite plates subjected to partial edge compression, Journal of Mechanical Sciences 40(11): 11051117.##[6] Shariyat M., 2007, Thermal buckling analysis of rectangular composite plates with temperaturedependent properties based on a layer wise theory, Journal of ThinWalled Structures 45: 439452.##[7] Özben T., 2009, Analysis of critical buckling load of laminated composites plate with different boundary conditions using FEM and analytical methods, Journal of Computational Materials Science 45: 10061015.##[8] Alipour M. M., Shariyat M., 2011, Semianalytical buckling analysis of heterogeneous variable thickness viscoelastic circular plates on elastic foundations, Journal of Mechanics Research Communications 38: 594601.##[9] Dash P., Singh B. N., 2012, Buckling and postbuckling of laminated composite plates, Journal of Mechanics Research Communications 46: 17.##[10] Jabbarzadeh M., Baghdar M.K. D., 2013, Thermal buckling analysis of FGM sector plates using differential quadrature method, Journal of Modares Mechanical Engineering 13(2): 3345.##[11] Asemi K., Shariyat M., 2013, Highly accurate nonlinear threedimensional finite element elasticity approach for biaxial buckling of rectangular anisotropic FGM plates with general orthotropy directions, Journal of Composite Structures 106: 235249.##[12] Fazzolari F.A., Banerjee J. R., Boscolo M., 2013, Buckling of composite plate assemblies using higher order shear deformation theoryAn exact method of solution, Journal of ThinWalled Structures 71: 1834.##[13] Lopatin A.V., Morozov E.V., 2014, Buckling of a uniformly compressed composite rectangular CCCC sandwich plate, Journal of Composite Structures 108: 332340.##[14] Asemi K., Salehi M., Akhlaghi M., 2014, Postbuckling analysis of FGM annular sector plates based on three dimensional elasticity graded finite elements, Journal of NonLinear Mechanics 67: 164177.##[15] Abolghasemi S., Shaterzadeh A.R., Rezaei R., 2014, Thermomechanical buckling analysis of functionally graded plates with an elliptic cutout, Journal of Aerospace Science and Technology 39: 250259.##[16] Nasirmanesh A., Mohammadi S., 2015, XFEM buckling analysis of cracked composite plates, Journal of Composite Structures 131: 333343.##[17] Rezaei R., Shaterzadeh A.R., Abolghasemi S., 2015, Buckling analysis of rectangular functionally graded plates with an elliptic hole under thermal loads, Journal of Solid Mechanics 7(1): 4157.##[18] Shaterzadeh A. R., 2015, Thermomechanical buckling analysis of FGM plates with circular cut out, Journal of Solid and Fluid Mechanics 5(2): 97107.##[19] Mansouri M. H., Shariyat M., 2015, Biaxial thermomechanical buckling of orthotropic auxetic FGM plates with temperature and moisture dependent material properties on elastic foundations, Journal of Composites Part B 83: 88104.##[20] Asemi K., Shariyat M., Salehi M., Ashrafi H., 2013, A full compatible threedimensional elasticity element for buckling analysis of FGM rectangular plates subjected to various combinations of biaxial normal and shear loads, Journal of Finite Elements in Analysis and Design 74: 921.##[21] Herakovich C. T., 1998, Mechanics of Fibrous Composites, 1st Edition, Amazon.##[22] Zienkiewicz O. C., Taylor R. L., 2000, The Finite Element Method: It’s The Basis, 5th Edition, Amazon.##[23] Darvizeh M., Darvizeh A., Shaterzadeh A.R., Ansari R., 2010, Thermal buckling of spherical shells with cutout, Journal of Thermal Stresses 33: 441458.##]
Influences of SmallScale Effect and Boundary Conditions on the Free Vibration of NanoPlates: A Molecular Dynamics Simulation
2
2
This paper addresses the influence of boundary conditions and smallscale effect on the free vibration of nanoplates using molecular dynamics (MD) and nonlocal elasticity theory. Based on the MD simulations, Largescale Atomic/Molecular Massively Parallel Simulator (LAMMPS) is used to obtain fundamental frequencies of single layered graphene sheets (SLGSs) which modeled in this paper as the most common nanoplates. On the other hand, governing equations are derived using nonlocal elasticity and the firstorder shear deformation theory (FSDT). Afterwards, these equations solved using generalized differential quadrature method (GDQ). The smallscale effect is applied in the governing equations of motion by nonlocal parameter. The effects of different side lengths, boundary conditions, and nonlocal parameter are inspected for the aforementioned methods. The results obtained from the MD simulations are compared with those of nonlocal elasticity theory to calculate appropriate values for the nonlocal parameter. As a result, for the first time, the nonlocal parameter values are suggested for graphene sheets with various boundary conditions. Furthermore, it is shown that nonlocal elasticity approach using classical plate theory (CLPT) assumptions overestimates the natural frequencies.
1

489
501


S.F
Asbaghian Namin
University of Mohaghegh Ardabili, Ardabil, Iran
University of Mohaghegh Ardabili, Ardabil,
Iran


R
Pilafkan
University of Mohaghegh Ardabili, Ardabil, Iran
University of Mohaghegh Ardabili, Ardabil,
Iran
rezapilafkan@um.ac.ir
Nanoplates
Molecular dynamics simulations
Fundamental frequencies
Nonlocal elasticity theory
Nonlocal parameter
[[1] Ramsden J., 2011, Nanotechnology: An Introduction, Elsevier.##[2] Murmu T., Pradhan S.C., 2009, Smallscale effect on the free inplane vibration of nano plates by nonlocal continuum model, Physica E: LowDimensional Systems and Nanostructures 41(8): 16281633.##[3] Baughman R.H., Zakhidov A.A., De Heer W.A., 2002, Carbon nanotubesthe route toward applications, Science 297(5582): 787792.##[4] Liew K.M., Zhang Y., Zhang L.W., 2017, Nonlocal elasticity theory for graphene modeling and simulation: prospects and challenges, Journal of Modeling in Mechanics and Materials 1(1): 0159.##[5] Pradhan S.C., Phadikar J.K., 2009, Nonlocal elasticity theory for vibration of nano plates, Journal of Sound and Vibration 325(1): 206223.##[6] HosseiniHashemi S., Zare M., Nazemnezhad R., 2013, An exact analytical approach for free vibration of Mindlin rectangular nanoplates via nonlocal elasticity, Composite Structures 100: 290299.##[7] Zhang Y., Lei Z.X., Zhang L.W., Liew K.M., Yu J.L., 2015, Nonlocal continuum model for vibration of singlelayered graphene sheets based on the elementfree kpRitz method, Engineering Analysis with Boundary Elements 56: 9097.##[8] Zhang Y., Zhang L.W., Liew K.M., Yu J.L., 2016, Free vibration analysis of bilayer graphene sheets subjected to inplane magnetic fields, Composite Structures 144: 8695.##[9] Ansari R., Rajabiehfard R., Arash B., 2010, Nonlocal finite element model for vibrations of embedded multilayered graphene sheets, Computational Materials Science 49(4): 831838.##[10] Ansari R., Sahmani S., Arash B., 2010, Nonlocal plate model for free vibrations of singlelayered graphene sheets, Physics Letters A 375(1): 5362.##[11] Pradhan S.C., Kumar A., 2011, Vibration analysis of orthotropic graphene sheets using nonlocal elasticity theory and differential quadrature method, Composite Structures 93(2): 774779.##[12] Xing Y.F., Liu B., 2009, New exact solutions for free vibrations of thin orthotropic rectangular plates, Composite Structures 89(4): 567574.##[13] Setoodeh A.R., Malekzadeh P., Vosoughi A.R., 2011, Nonlinear free vibration of orthotropic graphene sheets using nonlocal Mindlin plate theory, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science.##[14] Shahidi A.R., Shahidi S.H., Anjomshoae A., Estabragh E.R., 2016, Vibration analysis of orthotropic triangular nano plates using nonlocal elasticity theory and Galerkin method, Journal of Solid Mechanics 8(3): 679692.##[15] Zhang L.W., Zhang Y., Liew K.M., 2017, Modeling of nonlinear vibration of graphene sheets using a mesh free method based on nonlocal elasticity theory, Applied Mathematical Modelling 49: 691704.##[16] 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.##[17] Reddy J.N., Phan N.D., 1985, Stability and vibration of isotropic, orthotropic and laminated plates according to a higherorder shear deformation theory, Journal of Sound and Vibration 98(2): 157170.##[18] Reddy J.N., 2004, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC press.##[19] Bert C.W., Malik M., 1996, Differential quadrature method in computational mechanics: a review, Applied Mechanics Reviews 49: 128.##[20] Plimpton S., 1995, Fast parallel algorithms for shortrange molecular dynamics, Journal of Computational Physics 117(1): 119.##[21] Haile J.M., Johnston I., Mallinckrodt A.J., McKay S., 1993, Molecular dynamics simulation: elementary methods, Computers in Physics 7(6): 625625.##[22] http://lammps.sandia.gov/doc/Manual.html/, Sandia National Laboratories, 2015.##[23] Seifoori S., Hajabdollahi H., 2015, Impact behavior of singlelayered graphene sheets based on analytical model and molecular dynamics simulation, Applied Surface Science 351: 565572.##[24] Chu E., 2008, Discrete and Continuous Fourier Transforms: Analysis, Applications and Fast Algorithms, CRC Press.##[25] SakhaeePour A., 2009, Elastic properties of singlelayered graphene sheet, Solid State Communications 149(1): 9195.##]
3D Thermoelastic Interactions in an Anisotropic Lastic Slab Due to Prescribed Surface Temparature
2
2
The present paper is devoted to the determination of displacement, stresses and temperature from three dimensional anisotropic half spaces due to presence of heat source. The normal mode analysis technique has been used to the basic equations of motion and generalized heat conduction equation proposed by GreenNaghdi modelII [1]. The resulting equation are written in the form of a vector –matrix differential equation and exact expression for displacement component, stresses, strains and temperature are obtained by using eigen value approach. Finally, temperature, stresses and strain are presented graphically and analyzed.
1

502
521


Gh
Debkumar
Department of Mathematics, Jadavpur University, Kolkata, India
Department of Mathematics, Jadavpur University,
India
debkumarghosh2020@gmail.com


L
Abhijit
Department of Mathematics, Jadavpur University, Kolkata, India
Department of Mathematics, Jadavpur University,
India


R
Kumar
Department of Mathematics, Kurukshetra University, India
Department of Mathematics, Kurukshetra University,
India
rajneesh_kuk@rediffmail.com


R
Surath
Department of Mathematics, Brainware College of Engineering, Barasat, Kolkata, India
Department of Mathematics, Brainware College
India
eigenvalue
Generalized thermoelasticity
Normal mode analysis and vectormatrix
Differential equation
[[1] Chandrasekharaiah D.S., Srinath K.S., 1998, Thermoelastic plane waves without energy dissipation in a rotating body, Mechanics Research Communications 24: 551560.##[2] Green A.E., Lindsay K.A., 1972, Thermoelasticity, Journal of Elasticity 2: 17.##[3] Green A.E., Naghdi P.M., 1991, A re examination of the basic postulate of thermomechanics, Proceedings of the Royal Society of London 432: 171194.##[4] Green A.E., Naghdi P.M., 1992, An unbounded heat wave in an elastic solid, Journal of Thermal Stresses 15: 253264.##[5] Green A.E., Naghdi P.M., 1992, Thermoelasticity without energy dissipation, Journal of Elasticity 31: 189208.##[6] Lord H.W., Shulman Y.A., 1967, Geeneralized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids 15: 299309.##[7] Roychoudhuri S.K., Bandyopadhyay N., 2004, Thermoelastic wave propagation in a rotating elastic medium without energy dissipation, International Journal of Mathematics and Mathematical Sciences 1: 99107.##[8] Roychoudhuri S.K., Dutta P.S., 2005, Thermoelastic interaction without energy dissipation in an infinite solid with distributed periodically varing heat sources, International Journal of Solid and Structures 42: 41924203.##[9] Chattopadhyay A., Rogerson G.A., 2001, Wave reflection in slightly compressible finitely deformed elastic media, Archives of Applied Mathematics 71: 307316.##[10] Sarkar N., Lahiri A., 2013, The effect of gravity field on the plane waves in a fiber reinforced two temperature magneto thermoelastic medium under Lord Shulman theory, Journal of Thermal Stresses 36: 895914.##[11] Sarkar N., Lahiri A., 2012, A three dimensional thermoelastic problem for a half Space without energy dissipation, International Journal of Engineering Science 51: 310325.##[12] Sharma J.N., Chouhan R.S., 1999, On the problem of body forces and heat sources in thermoelasticity without energy dissipation, Indian Journal of Pure and Applied Mathematics 30: 595610.##]
Failure Criteria Analysis of Laminate Composite Materials
2
2
This paper deals with the development of a numerical simulation methodology for estimating damages in laminate composite materials caused by a lowspeed impact. Experimental tests were performed on laminate plates reinforced with woven carbon fibers and epoxy resin. Three thickness plates were evaluated. The impact loads were transversal and punctual. Two lamina failure criteria were evaluated. The first is the maximum stress. The second is a proposed modification of the Hashin failure criterion. Four lamina degradation criteria were evaluated too. The numerical contact loads between the plate and impactor were well represented. The numerical damaged areas and lengths were similar or greater than the experimental results.
1

522
531


L
Nourine
Laboratoire de Mécanique Appliquée, Université des Sciences et de la Technologie d’Oran , Algéira
Laboratoire de Mécanique Appliquée,
Algeria
mechanics151@yahoo.com


A
Sahli
Laboratoire de Recherche des Technologies Industrielles, Université Ibn Khaldoun de Tiaret, Algéira
Laboratoire de Recherche des Technologies
Algeria


S
Sahli
Département de Génie Mécanique, Université d’Oran, Algéira
Département de Génie Mécaniqu
Algeria
Criterion damage
Failure criteria
Impact
Laminate composite materials
Damage strength
[[1] Lakshminarayana H. V., Murthy S. S. A., 1984, Shearflexible triangular finite element model for laminated composite plates, International Journal for Numerical Methods in Engineering 20: 591 623.##[2] Luo R. K., Green E. R., Morrison C. J., 1999, Impact damage analysis of composite plates, International Journal of Impact Engineering 22: 435447.##[3] Zhao G. P., Cho C. D., 2007, Damage initiation and propagation in composite shells subjected to impact, Composite Structures 78: 91100.##[4] Ganapathy S., Rao K. P., 1998, Failure analysis of laminated composite cylindrical/spherical shell panels subjected to lowvelocity impact, Computers and Structures 68: 627641.##[5] Li C. F., Hu N., Yin Y. J., 2002, Lowvelocity impactinduced damage of continuous fiber reinforced composite laminates, Part I: An fem numerical model, Composites Part AApplied Science and Manufacturing 33: 10551062.##[6] Li C. F., Hu N., Cheng J. G., 2002, Lowvelocity impactinduced damage of continuous fiberreinforced composite laminates, Part II: Verification and numerical investigation, Composites Part A: Applied Science and Manufacturing 33: 10631072.##[7] Sahli A., Boufeldja S., Kebdani S., Rahmani O., 2014, Failure analysis of anisotropic plates by the boundary element method, Journal of Mechanics 30: 561570.##[8] NIU M. C. Y. ,1992, Composite Airframe Structure Practical Design Information and Data, Hong Kong, Conmilit Press.##[9] icardi U., Locatto S., Longo A., 2007, Assessment of recent theories for predicting failures of composite laminates, Applied Mechanics Reviews 60(2): 7686.##[10] Matthews F.L., Rawlings R.D., 1994, Composite Materials: Engineering and Science, Chapman & Hall, London.##[11] Mendonça Paulo de Tarso R., 2005, Composite Materials and Sandwich Structures : Design and Analysis, Barueri.##[12] Jenkins C. F., 1920, Report on Materials of Construction Used in Aircraft and in Aircraft Engines, Great Britain Aeronautical Research Committee, London.##[13] Tsai S. W., Wu E. M., 1971, A general theory of strength for anisotropic materials, Journal of Composite Materials 5(1): 5880.##[14] Hashin Z., Rotem A., 1973, A fatigue failure criterion for fiber reinforced materials, Journal of Composite Materials 7: 448464.##[15] Hashin Z., 1980, Failure criteria for unidirectional fiber composites, Journal of Applied Mechanics 47: 329334.##[16] Hinton M.J., Kaddour A.S., Soden P.D., 2002, A comparison of the predictive capabilities of current failure theories for composite laminates, judge against experimental evidence, Composites Science and Technology 62(1213): 17251797.##[17] Hinton M.J., Soden P.D., 1998, Predicting failure in composite laminates: the background to the exercise, Composites Science and Technology 58(7): 10011010.##]
Extraction of Nonlinear ThermoElectroelastic Equations for High Frequency Vibrations of Piezoelectric Resonators with Initial Static Biases
2
2
In this paper, the general case of an anisotropic thermoelectro elastic body subjected to static biasing fields is considered. The biasing fields may be introduced by heat flux, body forces, external surface tractions, and electric fields. By introducing proper thermodynamic functions and employing variational principle for a thermoelectro elastic body, the nonlinear constitutive relations and the nonlinear equation of motion are extracted. The equations have the advantage of employing the Lagrangian strain and second PiolaKirchhoff stress tensor with symmetric characteristics. These equations are used to analyze the high frequency vibrations of piezoelectric resonators under finite biasing fields. A system of three dimensional equations is derived for initial and incremental fields on the body. Capability of the equations in numerical modelling of temperaturefrequency and forcefrequency effects in quartz crystal is demonstrated. The numerical results compare well with the data from experiments. These equations may be used in accurate modelling of piezoelectric devices subjected to thermo electro mechanical loads.
1

532
546


M.M
Mohammadi
School of Mechanical Engineering, College of Engineering, University of Zanjan, Zanjan, Iran
School of Mechanical Engineering, College
Iran
dr.mohamamdi@gmail.com


M
Hamedi
College of Mechanical Engineering, University of Tehran, Tehran, Iran
College of Mechanical Engineering, University
Iran
mhamedi@ut.ac.ir


H
Daneshpajooh
Department of Electrical Engineering, Pennsylvania State University, USA
Department of Electrical Engineering, Pennsylvania
USA
Nonlinear Equations
Thermoelectro elasticity
Initial static bias
Resonators
[[1] Patel M. S., 2008, Nonlinear Behavior in Quartz Resonators and its Stability, PhD dissertation, New Brunswick, New Jersey University.##[2] Ebrahimi F., Rastgoo A.,2009, Temperature effects on nonlinear vibration of FGM plates coupled with piezoelectric actuators, Journal of Solid Mechanics 1(4): 271288.##[3] Ultrasonics, Ferroelectrics, and Frequency Control, 2015, IEEE Transactions on 62 2015(6): 11041113.##[4] Baumhauer J. C., Tiersten H. F., 1973, Nonlinear electro elastic equations for small fields superposed on a bias, The Journal of the Acoustical Society of America 54(4): 10171034.##[5] Tichý J., Jiˇrí E., Erwin K., Jana P.,2010, Fundamentals of Piezoelectric Sensorics: Mechanical, Dielectric, and Thermodynamical Properties of Piezoelectric Materials, Springer Science & Business Media.##[6] Dulmet B., Roger B.,2001, Lagrangian effective material constants for the modeling of thermal behavior of acoustic waves in piezoelectric crystals. I. Theory, The Journal of the Acoustical Society of America 110(4): 17921799.##[7] Tiersten H. F., 1971, On the nonlinear equations of thermoelectro elasticity, International Journal of Engineering Science 9(7): 587604.##[8] Tiersten H. F.,1975, Nonlinear electro elastic equations cubic in the small field variables, The Journal of the Acoustical Society of America 57(3): 660666.##[9] Yang J. S., Batra R. C., 1995, Free vibrations of a linear thermo piezo electric body, Journal of Thermal Stresses 18(2): 247262.##[10] Lee P. C. Y., Wang Y. S., Markenscoff X., 1975, High− frequency vibrations of crystal plates under initial stresses, The Journal of the Acoustical Society of America 57(1): 95105.##[11] Lee P. C. Y., Yong Y. K., 1986, Frequency‐temperature behavior of thickness vibrations of doubly rotated quartz plates affected by plate dimensions and orientations, Journal of Applied Physics 60(7): 23272342.##[12] Yong Y. K., Wu W., 2000, Lagrangian temperature coefficients of the piezoelectric stress constants and dielectric permittivity of quartz, In Frequency Control Symposium and Exhibition, Proceedings of the 2000 IEEE/EIA International.##[13] Ultrasonics, Ferroelectrics, and Frequency Control, IEEE Transactions on 48 2001(5): 14711478.##[14] Yong Y. K., Mihir P., Masako T., 2007, Effects of thermal stresses on the frequencytemperature behavior of piezoelectric resonators, Journal of Thermal Stresses 30(6): 639661.##[15] Mase G., Thomas R., Smelser E., George E. M., 2009, Continuum Mechanics for Engineers, CRC press.##[16] Lee P. C. Y., Yong Y. K., 1984, Temperature derivatives of elastic stiffness derived from the frequency‐temperature behavior of quartz plates, Journal of Applied Physics 56(5): 15141521.##[17] Yang J.,2013, Vibration of Piezoelectric Crystal Plates, World Scientific.##[18] Yang J.,2005, An Introduction to the Theory of Piezoelectricity, Springer Science & Business Media.##[19] Kuang Zh. B.,2011, Some Thermodynamic Problems in Continuum Mechanics, INTECH Open Access Publisher.##[20] Montanaro A., 2010, Some theorems of incremental thermoelectro elasticity, Archives of Mechanics 62(1): 4972.##[21] Yang J., Yuantai H., 2004, Mechanics of electro elastic bodies under biasing fields, Applied Mechanics Reviews 57(3): 173189.##[22] EerNisse E.P., 1980, Temperature dependence of the force frequency effect for the AT, FC, SC and rotated XCuts, 34th Proceedings of the Annual Symposium on Frequency Control.##[23] Beerwinkle A. D., 2011, Nonlinear Finite Element Modeling of Quartz Crystal Resonators, M.S. Thesis, Oklahoma State University, Stillwater, USA.##[24] Mohammadi M. M., Hamedi M.,2016, Experimental and numerical investigation of forcefrequency effect in crystal resonators, Journal of VibroEngineering 18(6): 37093718.##[25] Thurston R. N., McSkimin H. J., Andreatch Jr P.,1966, Third‐order elastic coefficients of quartz, Journal of Applied Physics 37(1): 267275.##[26] Kittinger E., Jan T., Wolfgang F.,1986, Nonlinear piezoelectricity and electrostriction of alpha quartz, Journal of Applied Physics 60(4): 14651471.##[27] Reider Georg A., Erwin K., Tichý J.,1982, Electro elastic effect in alpha quartz, Journal of Applied Physics 53(12): 87168721.##[28] Ratajski J. M., 1968, Forcefrequency coefficient of singly rotated vibrating quartz crystals, IBM Journal of Research and Development 12(1): 9299.##]
Shape Dependent Term Investigation of Khan Liu Yield/ Fracture Criterion as a Function of Plastic Strain for Anisotropic Metals
2
2
The current paper primarily aims to suggest a mathematical model for the shapedependent term of Khan Liu (KL) Yield/ fracture criterion as a function of Plastic Strain for DP590 steel alloy. The shapedependent term in the mention criterion can generalize the application of this criterion in order to predict the behavior of other materials. Plane stress case and the first quarter of the stress plane have been specifically studied. Uniaxial stresses in rolling and transverse directions of sheet and also the tensions caused by equalbiaxial tension have been experimentally used. Then, material constants of KL yield/ fracture criterion and Khan Huang Liang (KHL) constitutive equation are calculated using genetic algorithm (GA) optimization and the value of the shapedependent factor in KL criterion is extracted. The same has been repeated for various plastic strains and finally a polynomial mathematical model based on the plastic strain for the KL shapedependent factor is suggested. Hence, material constants of KL criterion could be calculated using at least tests namely experimental uniaxial stress test, experimental equalbiaxial stress, and one of the optimization models such as GA. Using the given mathematical model based on the plastic strain, correction term can be calculated and the generalized form of KL criterion can be used for various ductile metallic materials.
1

547
560


F
Farhadzadeh
Marine Department, MalekAshtar University of Technology, Isfahan, Iran
Marine Department, MalekAshtar University
Iran


M
Tajdari
Department of Mechanical Engineering, Arak Branch, Islamic Azad University, Arak, Iran
Department of Mechanical Engineering, Arak
Iran
metajdari@iauarak.ac.ir


M
Salmani Tehrani
Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, Iran
Department of Mechanical Engineering, Isfahan
Iran
Yield/ fracture criterion
Constitutive equation
Shapedependent term
Cruciform specimen
DP590 Steel alloy
[[1] Khan A.S., Liu H., 2012, Strain rate and temperature dependent fracture criteria for isotropic and anisotropic metals, International Journal of Plasticity 37: 115.##[2] Khan A.S., Yu S., 2012, Deformation induced anisotropic responses of Ti–6Al–4V alloy. Part I: Experiments, International Journal of Plasticity 38: 113.##[3] Khan A.S., Yu S., Liu H., 2012, Deformation induced anisotropic responses of Ti–6Al–4V alloy Part II: A strain rate and temperature dependent anisotropic yield criterion, International Journal of Plasticity 38: 1426.##[4] Boresi A., Schmidt R., Sidebottom O., 1993, Advanced Strength of Materials,Wiley, New York.##[5] Banabic D., 2010, Sheet Metal Forming Processes: Constitutive Modelling and Numerical Simulation, Springer Science & Business Media.##[6] Lin Y., Chen X. M., 2011, A critical review of experimental results and constitutive descriptions for metals and alloys in hot working, Materials & Design 32(4): 17331759.##[7] Boresi A., Schmidt R., Sidebottom O., 1993, Advanced Mechanics of Materials, NewYork, John Wiley & Sons.##[8] Jacob L., 1990, Plasticity Theory, New York, Macmillan Publishing Company.##[9] Banabic D., 2010, A review on recent developments of MarciniakKuczynski model, Computer Methods in Materials Science 10(4): 113.##[10] Barlat F., Banabic D., Cazacu O., 2002, Anisotropy in sheet metals, International Conference and Workshop on Numerical Simulation of 3D Sheet Forming Processes, Jeju Island, Korea.##[11] Fields D., Backofen W., 1957, Determination of strain hardening characteristics by torsion testing, Proceeding of American Society for Testing and Materials 57: 12591272.##[12] Zhang X., 2003, Experimental and Numerical Study of Magnesium Alloy During HotWorking Process, PhD Thesis, Shanghai Jiaotong University.##[13] Cheng Y.Q., 2008, Flow stress equation of AZ31 magnesium alloy sheet during warm tensile deformation, Journal of Materials Processing Technology 208(1): 2934.##[14] Farhadzadeh F., Tajdari M., Salmani Tehrani M., 2017, Determination of material constants of Khan Huang Liang constitutive criterion by genetic and particle swarm algorithms for Ti6Al4V alloy, International Offshore Industries Conferences, SharifUniversity of Technology: Tehran, Iran.##[15] Deng N., Kuwabara T., Korkolis Y., 2015, Cruciform specimen design and verification for constitutive identification of anisotropic sheets, Experimental Mechanics 55(6): 10051022.##[16] Khan A.S., Huang S., 1995, Continuum Theory of Plasticity, John Wiley & Sons.##[17] Barsoum I., 2008, The Effect of Stress State in Ductile Failure, PhD Thesis was carried out at the Department of Solid Mechanics at the Royal Institute of Technology (KTH) , Stockholm, Sweden.##[18] Altenbach H., Öchsner A., 2014, Plasticity of PressureSensitive Materials, Springer.##[19] Liang R., Khan A.S., 1999, A critical review of experimental results and constitutive models for BCC and FCC metals over a wide range of strain rates and temperatures, International Journal of Plasticity 15(9): 963980.##[20] Khan A.S., Suh Y.S., Kazmi R., 2004, Quasistatic and dynamic loading responses and constitutive modeling of titanium alloys, International Journal of Plasticity 20(12): 22332248.##[21] Khan A.S., Kazmi R., Farrokh B., 2007, Multiaxial and nonproportional loading responses, anisotropy and modeling of Ti–6Al–4V titanium alloy over wide ranges of strain rates and temperatures, International Journal of Plasticity 23(6): 931950.##[22] Khan A.S., Liang R., 1999, Behaviors of three BCC metal over a wide range of strain rates and temperatures: experiments and modeling, International Journal of Plasticity 15(10): 10891109.##[23] Farrokh B., Khan A.S., 2009, Grain size, strain rate, and temperature dependence of flow stress in ultrafine grained and nanocrystalline Cu and Al: synthesis, experiment, and constitutive modeling, International Journal of Plasticity 25(5): 715732.##[24] Holland J.H., Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence, MIT Press Cambridge, USA.##[25] Goldberg D.E., 1989, Genetic Algorithm in Search, Optimization and Machine Learning, AddisonWesley Longman Publishing , Boston, USA.##[26] Janikow C.Z., Michalewicz Z., 1991, An experimental comparison of binary and floating point representations in genetic algorithms, ICGA 1991: 3136.##[27] Geiger M., Hußnätter W., Merklein M., 2005, Specimen for a novel concept of the biaxial tension test, Journal of Materials Processing Technology 167(2): 177183.##[28] Hannon A., Tiernan P., 2008, A review of planar biaxial tensile test systems for sheet metal, Journal of Materials Processing Technology 198(1): 113.##]
New Method for Large Deflection Analysis of an Elliptic Plate Weakened by an Eccentric Circular Hole
2
2
The bending analysis of moderately thick elliptic plates weakened by an eccentric circular hole has been investigated in this article. The nonlinear governing equations have been presented by considering the vonKarman assumptions and the firstorder shear deformation theory in cylindrical coordinates system. Semianalytical polynomial method (SAPM) which had been presented by the author before has been used. By applying SAPM method, the nonlinear partial differential equations have been transformed to the nonlinear algebraic equations system. Then, the nonlinear algebraic equations have been solved by using Newton–Raphson method. The obtained results of this study have been compared with the results of other references and the accuracy of the results has been shown. The effect of some important parameters on the results such as the location of the circular hole, the ratio of major to minor radiuses of elliptical plate, the size of circular hole and boundary conditions have been studied. It is concluded that applying the presented method is very convenient and efficient. So, it can be used for analyzing the mechanical behavior of elliptical plates, instead of relatively complicated formulations in elliptic coordinates system.
1

561
570


Sh
Dastjerdi
Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University, Mashhad, Iran
Young Researchers and Elite Club, Mashhad
Iran
dastjerdi_shahriar@yahoo.com


L
Yazdanparast
Department of Mechanical Engineering, Damavand Branch, Islamic Azad University, Damavand, Iran
Department of Mechanical Engineering, Damavand
Iran
Elliptical plate
Eccentric circular hole
Firstorder shear deformation theory
Semianalytical polynomial method (SAPM)
[[1] Dastjerdi Sh., Jabbarzadeh M., 2016, Nonlocal thermoelastic buckling analysis of multilayer annular/circular nanoplates based on first and third order shear deformation theories using DQ method, Journal of Solid Mechanics 8(4): 859874.##[2] Dastjerdi Sh., Jabbarzadeh M., 2016, Nonlocal bending analysis of bilayer annular/circular nano plates based on first order shear deformation theory, Journal of Solid Mechanics 8(3): 645661.##[3] Sato K., 2006, Bending of an elliptical plate on elastic foundation and under the combined action of lateral load and inplane force, III European Conference on Computational Mechanics.##[4] Datta S., 1976, Large deﬂections of elliptic plates exhibiting rectilinear orthotropy and placed on elasticfoundation, Journal of Applied Mechanics 43(4): 690692.##[5] Kutlu A., Hakkı Omurtag M., 2012, Large deflection bending analysis of elliptic plates on orthotropic elastic foundation with mixed finite element method, International Journal of Mechanical Sciences 65(1): 6474.##[6] Zhong H., Li X., He Y., 2005, Static ﬂexural analysis of elliptic Reissner–Mindlin plates on a Pasternak foundation by the triangular differential quadrature method, Archive of Applied Mechanics 74(10): 679691.##[7] Parnes R., 1989, Bending of simplysupported elliptic plates: B.P.M. solutions with secondorder derivative boundary conditions, Journal of Applied Mechanics 56(2): 356363.##[8] Parnes R., 1988, A BPM solution for elliptical plates subjected to eccentric loads, International Journal of Solids and Structures 24(8): 761776.##[9] Wang Z.Q., Jiang J., Tang B.T., Zheng W., 2014, Numerical solution of bending problem for elliptical plate using differentiation matrix method based on Barycentric Lagrange interpolation, Applied Mechanics and Materials 638: 17201724.##[10] Hsieh M.C., Hwu Ch., 2002, Bending of an anisotropic plate weakened by an elliptical hole, The 3rd AsianAustralasian Conference on Composite Materials, New Zealand.##[11] Leissa A.W., 1967, Vibration of a simplysupported elliptical plate, Journal of Sound and Vibration 6(1): 145148.##[12] Leissa A.W., 1969, Vibration of Plates, Washington, Office of Technology Utilization, SP160, NASA.##[13] Leissa A.W., 1978, Recent research in plate vibrations: Classical theory, The Shock and Vibration Digest 9: 1324.##[14] Leissa A.W., 1978, Recent research in plate vibrations: Complicating effects, The Shock and Vibration Digest 9: 2135.##[15] Leissa A.W., 1981, Plate vibration research, 19761980: Classical theory, The Shock and Vibration Digest 13: 1122.##[16] Leissa A.W., 1981, Plate vibration research: Classical theory, The Shock and Vibration Digest 13: 1936.##[17] Leissa A.W., 1987, Recent studies in plate vibrations: part I, Classical theory, The Shock and Vibration Digest 19: 1118.##[18] Leissa A.W., 1987, Recent studies in plate vibrations: part II, Complicating effects, The Shock and Vibration Digest 19: 1024.##[19] Wang C.M., Wang L., Liew K.M., 1994, Vibration and buckling of super elliptical plates, Journal of Sound and Vibration 171(3): 301314.##[20] Altekin M., Altay G., 2008, Static analysis of pointsupported superelliptical plates, Archive of Applied Mechanics 78(4): 259266.##[21] Altekin M., 2008, Free linear vibration and buckling of superelliptical plates resting on symmetrically distributed pointsupports on the diagonals, ThinWalled Structures 46(10): 10661086.##[22] Altekin M., 2009, Free vibration of orthotropic superelliptical plates on intermediate supports, Nuclear Engineering and Design 239(6): 981999.##[23] Altekin M., 2010, Bending of orthotropic superelliptical plates on intermediate point supports, Ocean Engineering 37(11): 10481060.##[24] Laura P.A., Rossit C., 1998, Thermal bending of thin, anisotropic, clamped elliptic plates, Ocean Engineering 26(5): 485488.##[25] Zhang D.G., 2013, Nonlinear bending analysis of super elliptical thin plates, International Journal of NonLinear Mechanics 55: 180185.##[26] Mc Nitt R.P., 1963, Free vibration of a damped semielliptical plate and a quarterelliptical plate, AIAA Journal 29(9): 11241125.##[27] Hasheminejad S.M., Vaezian S., 2014, Free vibration analysis of an elliptical plate with eccentric elliptical cutouts, Meccanica 49: 3750.##[28] Irie T., Yamada G., 1979, Free vibration of an orthotropic elliptical plate with a similar hole, Bulletin of JSME 22: 14561462.##[29] Ghaheri A., Keshmiri A., TaheriBehrooz F., 2014, Buckling and vibration of symmetrically laminated composite elliptical plates on an elastic foundation subjected to uniform inplane force, Journal of Engineering Mechanics 140 (7).##[30] Biswas P., 1975, Large deflection of a heated elliptical plate under stationary temperature, Defense Science Journal 26: 4146.##[31] Dastjerdi Sh., Lotfi M., Jabbarzadeh M., 2016, The effect of vacant defect on bending analysis of graphene sheets based on the Mindlin nonlocal elasticity theory, Composites Part B 98: 7887.##[32] Guminiak M., Szajek K., 2014, Static analysis of circular and elliptic plates resting on internal flexible supports by the boundary element method, Journal of Applied Mathematics and Computational Mechanics 13(2): 2132.##]
A Contact Problem of an Elastic Layer Compressed by Two Punches of Different Radii
2
2
The elasticity mixed boundary values problems dealing with halfspace contact are generally well resolved. A large number of these solutions are obtained by using the integral transformation method and methods based the integral equations. However, the problems of finite layer thicknesses are less investigated, despite their practical interests. This study resolves a quasistationary problem of an isotropic elastic layer compressed by two rigid cylinders with flat ends. Hankel transformation and auxiliary functions with boundary conditions reduce the differential equation to an algebraic equations system, which can be solved in a numerical way. The contact efforts equations are established. From the general method, solutions of particular cases are also resolved. A particular case is studied, the contact zone pressure and stresses distribution curves are presented.
1

571
580


K
Seghir
Department of Mechanical Engineering, Faculty of Technology, University of Batna2, Algeria
Department of Mechanical Engineering, Faculty
Algeria


M
Bendaoui
Department of Mechanical Engineering, Faculty of Technology, University of Batna2, Algeria
Department of Mechanical Engineering, Faculty
Algeria


R
Benbouta
Department of Mechanical Engineering, Faculty of Technology, University of Batna2, Algeria
Department of Mechanical Engineering, Faculty
Algeria
r_benbouta@yahoo.fr
Keywords: Contact problem
Elastic layer
Cylindrical punches
Flat ends
[[1] Harding J.W., Sneddon I.N., 1945, The elastic stresses produced by the indentation of the plane surface of a semiinfinite elastic solid by a rigid punch, proceedings of the Cambridge Philosophical Society 41: 1626.##[2] Ufliand Ja.S., 1965, Survey of Articles on the Applications of Integral Transforms in the Theory of Elasticity, Defense Technical Information Center, North Carolina State University.##[3] Kuz’min Iu.N., Ufliand Ia.S., 1967, The contact problem of an elastic layer compressed by two punches, Journal of Applied Mathematics and Mechanics 31(4): 711715.##[4] Zakorko V.N., 1978, Contact problem for a layer with two stamps, Journal of Applied Mathematics and Mechanics 42(6): 10681073.##[5] Dhaliwal R.S., Sing B.M., 1977, Axisymmetric contact problem for an elastic layer on a rigid foundation with a cylindrical hole, International Journal of Engineering Science 15: 421428.##[6] Ufljand Ia.S., 1963, Integral Transforms in Elasticity Problems, Nauka. SSR.##[7] Sneddon I.N., 1959, A note on the axially symmetric punch problem, Mathematika 6: 118119.##[8] Sneddon I.N., 1977, Application of Integral Transforms in the Theory of Elasticity, CISM Courses and lectures, SpringerVerlag Vien, New York.##[9] Kuo C.H., Keer L., 1992, Contact stress analysis of a layered transversely isotropic halfspace, ASME Journal of Tribology 114: 253262.##[10] Matnyac S.V., 2003, Contact stresses distribution under a rigid cylinder rolling over a prestressed strip, International Applied Mechanics 39(7): 840847.##[11] Shelestovshii B.G., Gabrusev G.V., 2004, Thermoelastic state of a transversely isotropic layer between two annular punches, International Applied Mechanics 40(4): 6777.##[12] Ruiny C., Dahan M., 2002, Chargements axisymétriques d’un bicouche transversalement isotrope, Mécanique 330 : 469473.##[13] Grilitskii D.V., Kizima Y., 1981, Theory of Deformation Elastic and Thermic, Axisymetric Contact Problem, Nauka, Moscow.##[14] Argatov I.I., Dmitriev N.N., 2003, Fundamentals of the Theory of Elastic Discrete Contact, St. Polytechnics, St. Petersburg, Russian.##[15] Gradshteyn I.I., Ryzhik I.M., 1980, Tables of Integrals, Series, and Products, Academie press, New York.##[16] Seghir K., Benbouta R., Balbacha El., 2010, Analysis of stresses in the contact zone Rigid cylindrical indenter – transversely isotropic elastic layer, Matériaux & Techniques 98: 227232.##[17] Ditkin V.A., Prudnikov V.A., 1965, Integral Transforms and Operational Calculus, Pergamon Press, New York.##]
Torsion in Microstructure Hollow ThickWalled Circular Cylinder Made up of Orthotropic Material
2
2
In this paper, a numerical solution has been developed for hollow circular cylinders made up of orthotropic material which is subjected to twist using micro polar theory. The effect of twisting moment and material internal length on hollow thickwalled circular cylinder made up of micro polar orthotropic material is investigated. Finite difference method has been used to exhibit the influence of shear moduli and material internal length on shear stresses and couple stresses. It is found that the effect of small characteristic length on shear stresses is negligible and couple stresses present its significance when characteristic length is large in solid particle. The behavior of couple stresses are nonlinear for large internal length while for small internal length couple stresses are linear in nature except near the free boundaries. Torsion in hollow cylinder made up of micro polar orthotropic play vital role in the presence of cracks and holes. Therefore, torsional analysis of hollow cylinder plays important role in the field of biomechanics.
1

581
590


S
Yadav
Department of Mathematics, Jaypee Institute of Information Technology, Noida, India
Department of Mathematics, Jaypee Institute
India
sneh.mathematics@gmail.com


S
Sharma
Department of Mathematics, Jaypee Institute of Information Technology, Noida, India
Department of Mathematics, Jaypee Institute
India
Elastic
Orthotropic
Micro polar
Characteristics length
Twist
Couple stress
[[1] Pagano N.J., Sih G.C., 1968, Stress singularities around a crack in a cosserat plate, International Journal of Solids and Structures 4(5): 531553.##[2] Fatemi J., Onck P.R., Poort G., van Keulen F., 2003, Cosserat moduli of anisotropic cancellous bone: A micromechanical analysis, Journal de Physique IV 105: 273280.##[3] Eringen A.C., 1966, Linear theory of micropolar elasticity, Journal of Mathematics and Mechanics 15: 909923.##[4] Eringen A.C., 1999, Microcontinuum Field Theories, Springer, Berlin edition.##[5] Altenbach H., Eremeyev V.A., 2010, Thinwalled structures made of foams, Cellular and Porous Materials in Structures and Processes 2010: 167242.##[6] Gauthier R.D., Jahsman W.E., 1975, A quest for micropolar elastic constants, Journal of Applied Mechanics 42(2): 369374.##[7] Merkel A., Tournat V., Gusev V., 2011, Experimental evidence of rotational elastic waves in granular phononic crystals, Physical Review Letters 107(22): 225502.##[8] Kvasov R., Steinberg L., 2013, Numerical modeling of bending of micropolar plates, ThinWalled Structures 69: 6778.##[9] Hadjesfandiari A.R., Dargush G.F., 2014, Evolution of generalized couplestress continuum theories: A critical analysis, Physics 2014:1501.03112.##[10] Hadjesfandiari A.R., Dargush G.F., 2013, Fundamental solutions for isotropic sizedependent couple stress elasticity, International Journal of Solids and Structures 50(9): 12531265.##[11] Taliercio A., Veber D., 2009, Some problems of linear elasticity for cylinders in micropolar orthotropic material, International Journal of Solids and Structures 46(22): 39483963.##[12] Taliercio A., 2010, Torsion of micropolar hollow circular cylinders, Mechanics Research Communications 37(4): 406411.##[13] Sharma S., Yadav S., Sharma R., 2015, Thermal creep analysis of functionally graded thickwalled cylinder subjected to torsion and internal and external pressure, Journal of Solid Mechanics 9(2): 302318.##[14] Taliercio A., Veber D., 2016, Torsion of elastic anisotropic micropolar cylindrical bars, European Journal of MechanicsA/Solids 55: 4556.##[15] Smith A., 1970, Torsion and vibrations of cylinders of a micropolar elastic solid, Recent Advances in Engineering Science 5:129137.##[16] Reddy G.V.K., Venkatasubramanian N.K., 1976, Saintvenant's problem for a micropolar elastic circular cylinder, International Journal of Engineering Science 14(11): 10471057.##[17] Eringen A.C., 1968,Theory of Micropolar Elasticity, New York, Academic Press.##]
Dispersion of Love Wave in a FiberReinforced Medium Lying Over a Heterogeneous HalfSpace with Rectangular Irregularity
2
2
This paper concerned with the dispersion of Love wave in a fiberreinforced medium lying over a heterogeneous halfspace. The heterogeneity is caused by the consideration of quadratic variation in density and directional rigidity of lower halfspace. The irregularity has been considered in the form of rectangle at the interface of the fiberreinforced layer and heterogeneous halfspace. The dispersion equation of Love wave has been deduced for existing geometry of the problem under suitable boundary conditions using variable separation method. It has also been observed that for a homogeneous layer with rigidity lying over a regular homogeneous isotropic halfspace, the velocity equation coincides with the classical results of Love wave. The effect of the medium characteristics on the dispersion of Love waves has been discussed and the results are displayed with graphs by means of MATLAB programming to clear the physical significance. The study of Love wave dispersion with irregular interface helps civil engineers in building construction, analysis of earthquake in mountain roots, continental margins, and so on. It is also beneficial for the study of seismic waves generated by artificial explosions.
1

591
602


R.M
Prasad
Department of Mathematics, S.N. Sinha College, Tekari, Gaya, Bihar824236, India
Department of Mathematics, S.N. Sinha College,
India
ratanmaniprasad@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
Fiber reinforcement
Rectangular irregularity
Heterogeneous halfspace
Phase velocity
[Raoofian Naeeni M., EskandariGhadi M., 2016, A potential method for body and surface wave propagation in transversely isotropic half and fullspaces, Civil Engineering Infrastructures Journal 49: 263288.##[2] Raoofian Naeeni M., EskandariGhadi M., 2016, Analytical solution of the asymmetric transient wave in a transversely isotropic halfspace due to both buried and surface impulses, Soil Dynamics and Earthquake Engineering 81: 4257.##[3] Achenbach J.D., 1973, Wave Propagation in Elastic Solid, North Holland, Publishing Company American Elsevier, Amsterdam, New York.##[4] Spencer A.J.M., 1972, Deformation of FiberReinforced Material, Oxford University Press, London.##[5] Richter C.F., 1958, Elementary Seismology, W. H. Freeman and Company, San Francisco and Bailey Bros, Swinfen Ltd, London.##[6] Meissner E., 1921, Elastic Oberflachenwellen mit dispersion in einem inhomogeneous medium, Viertelgahrsschriftden Naturforschender GeSellschaft in Zurich 66: 181185.##[7] Chattopadhyay A., Choudhury S., 1990, Propagation, reflection and transmission of magneto elastic shear waves in a selfreinforced medium, International Journal of Engineering Science 28: 485495.##[8] AbdAlla A.M., Nofal T.A., AboDahab S.M., AlMullise A., 2013, Surface waves propagation in fibrereinforced anisotropic elastic media subjected to gravity field, International Journal of Physical Sciences 8: 574584.##[9] Singh B., 2006, Wave propagation in thermally conducting linear fibrereinforced composite materials, Archive of Applied Mechanics 75: 513520.##[10] Singh B., Singh S.J., 2004, Reflection of plane waves at the free surface of a fiberreinforced elastic halfspace, Sadhana 29: 249257.##[11] Kumar R., Gupta R.R., 2009, Deformation due to various sources in a fibrereinforced anisotropic generalized thermoelastic medium, Canadian Journal of Physics 87: 179187.##[12] Chattopadhyay A., Singh A.K., 2012, Propagation of magneto elastic shear waves in an irregular selfreinforced layer, Journal of Engineering Mathematics 75: 139155.##[13] Singh S.S., 2010, Love wave at a layer medium bounded by irregular boundary surfaces, Journal of Vibration and Control 17: 789795.##[14] 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: 402408.##[15] Chattopadhyay A., Gupta S., Sharma V.K., Kumari P., 2010, Effects of irregularity and anisotropy on the propagation of shear waves, International Journal of Engineering, Science and Technology 2: 116126.##[16] Gupta S., Chattopadhyay A., Kundu S., 2010, Influence of irregularity and rigidity on the propagation of torsional wave, Applied Mathematical Sciences 4: 805816.##[17] Gupta S., Majhi D.K., Vishwakarma S.K., Kundu S., 2011, Propagation of torsional surface waves under the effect of irregularity and initial stress, Applied Mathematics 2: 14531461.##[18] Selim M.M., 2007. Propagation of torsional surface waves in heterogeneous halfspace with irregular free surface, Applied Mathematical Sciences 1: 14291437.##[19] Gupta S., Chattopadhyay A., Majhi D.K., 2010, Effect of irregularity on the propagation of torsional surface waves in an initially stressed anisotropic poroelastic layer, Applied Mathematics and Mechanics 31: 481492.##[20] AbdAlla A.M., Ahmed S.M., 1999, Propagation of Love waves in a nonhomogeneous orthotropic elastic layer under initial stress overlying semiinfinite medium, Applied Mathematics and Computation 106: 265275.##[21] Vaishnav P.K., Kundu S., AboDahab S.M., Saha A., 2017, Torsional surface wave propagation in anisotropic layer sandwiched between heterogeneous halfspace, Journal of Solid Mechanics 9: 213224.##[22] Vishwakarma S.K., Gupta S., 2013, Existence of torsional surface waves in an earth’s crustal layer lying over a sandy mantle, Journal of Earth System Science 122: 14111421.##[23] Wang C.D., Lin Y.T., Jeng Y.S, Ruan Z.W., 2010, Wave propagation in an inhomogeneous crossanisotropic medium, International Journal for Numerical and Analytical Methods in Geomechanics 34: 711732.##[24] Dey S., Gupta S., 1987, Longitudinal and shear waves in an elastic medium with void pores, Proceedings of the Indian National Science Academy 53: 554563.##[25] Gupta S., Sultana R., Kundu S., 2015, Influence of rigid boundary on the propagation of torsional surface wave in an inhomogeneous layer, Journal of Earth System Science 124: 161170.##[26] Biot M.A., 1965, Mechanics of Incremental Deformation, John Wiley and Sons, New York.##[27] Love A.E. H., 1911, Some Problems of Geodynamics, Cambridge University Press, London.##[28] Chattopadhyay A., Choudhury S., 1995, Magneto elastic shear waves in an infinite selfreinforced plate, International Journal for Numerical and Analytical Methods in Geomechanics 19: 289304.##[29] Gubbins D., 1990, Seismology and Plate Tectonics, Cambridge University Press, Cambridge, New York.##]
Crack Influences on the Static and Dynamic Characteristic of a MicroBeam Subjected to Electro Statically Loading
2
2
In the present work the pullin voltage of a micro cracked cantilever beam subjected to nonlinear electrostatic pressure was studied. Two mathematical models were employed for modeling the problem: a lumped mass model and a classical beam model. The effect of crack in the lumped mass model is the reduction of the effective stiffness of the beam and in the beam model; the crack is modeled as a massless rotational spring the compliance of which is related to the crack depth. Using these two models the pullin voltage is extracted in the static and dynamic cases. Stability analysis is also accomplished. It has been observed that the pullin voltage decreases as the crack depth increases and also when the crack approaches the clamped support of the beam. The finding of this research can further be used as a nondestructive test procedure for detecting cracks in microbeams.
1

603
620


A.R
Shahani
Department of Applied Mechanics, Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran
Department of Applied Mechanics, Faculty
Iran
shahani@kntu.ac.ir


G
Rezazadeh
Mechanical Engineering Departments, Urmia University, Urmia, Iran
Mechanical Engineering Departments, Urmia
Iran


A
Rahmani
Department of Applied Mechanics, Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran
Department of Applied Mechanics, Faculty
Iran
MEMS
Cracked microbeam
Stability Analysis
[[1] Senturia S. D., 2001, Microsystem Design, Boston, Kluwer.##[2] Younis M. I., Nayfeh A. H., 2003, A study of the nonlinear response of a resonant microbeam to an electric actuation, Journal of Nonlinear Dynamics 31: 91117.##[3] Elata D., Bamberger H., 2006, On the dynamic pullIn of electrostatic actuators with multiple degrees of freedom and multiple voltage sources, Journal of Microelectromechanical Systems 15: 131140.##[4] Zhang Y., Zhao Y., 2006, Numerical and analytical study on the pullin instability of microstructure under electrostatic loading, Sensors and Actuators A: Physical 127: 366380.##[5] Rezazadeh G., Fathaliou M., Sadeghi M., 2011, Pullin voltage of electrostatically actuated microbeams in terms of lumped model pullin voltage using novel design corrective coefficients, Sensing and Imaging 12: 117131.##[6] Zhang W. M., Yan H., Peng Z. K., Meng G., 2014, Electrostatic pullin instability in MEMS/NEMS: a review, Sensors and Actuators A: Physical 214: 187214.##[7] Choi B., Lovell E. G., 1997, Improved analysis of microbeams under mechanical and electrostatic loads, Journal of Micromechanics and Micro engineering 7: 2429.##[8] Chowdhury S., Ahmadi M., Miller W. C., 2005, A closedform model for the pullin voltage of electrostatically actuated cantilever beams, Journal of Micromechanics and Micro engineering 15: 756763.##[9] Chao P. C. P., Chiu C. W., Liu T. H., 2008, DC dynamic pullin predictions for a generalized clamped–clamped microbeam based on a continuous model and bifurcation analysis, Journal of Micromechanics and Micro engineering 18: 114.##[10] Krylov S., 2007, Lyapunov exponents as a criterion for the dynamic pullin instability of electrostatically actuated microstructures, International Journal of NonLinear Mechanics 42: 626642.##[11] VarvaniFarahani A., 2005, Silicon MEMS components: a fatigue life assessment approach, Microsystem Technologies 11: 129134.##[12] Hill M J., Rowcliffe D. J., 1947, Deformation of silicon at low temperatures, Journal of Materials Science 9: 15691576.##[13] Muhlstein C. L, Brown S. B., Ritchie R. O., 2001, Highcycle fatigue and durability of polycrystalline silicon films in ambient air, Sensors and Actuators A: Physical 94: 177188.##[14] Ando T., Shikida M., Sato K., 2001, Tensilemode fatigue testing of silicon films as structural materials for MEMS, Sensors and Actuators A: Physical 93: 7075.##[15] Motallebi A., Fathalilou M., Rezazadeh G., 2012, Effect of the open crack on the pullin instability of an electrostatically actuated microbeam, Acta Mechanica Solida Sinica 25: 627637.##[16] Sourki R., Hoseini S. A. H., 2016, Free vibration analysis of sizedependent cracked microbeam based on the modified couple stress theory, Applied Physics A 413: 111.##[17] Tadi Beni Y., Jafaria A., Razavi H., 2015, Size effect on free transverse vibration of cracked nanobeams using couple stress theory, International Journal of Engineering 28: 296304.##[18] Loya J. A., ArandaRuiz J., Fern J., 2014, Torsion of cracked Nanorods using a nonlocal elasticity model, Journal of Physics D : Applied Physics 47: 115304115315.##[19] Barr A. D. S., Christides S., 1984, Onedimensional theory of crack EulerBernoulli beams, International Journal of Mechanical Sciences 26: 639639.##[20] Rizos P. F., Aspragathos N., Dimarogonas A. D., 1990, Identification of crack location and magnitude in cantilever beam from the vibration modes, Journal of Sound Vibration 138: 381388.##[21] Chondros T. G., Dimarogonas A. D., Yao J., 1998, A continuous cracked beam vibration theory, Journal of Sound Vibration 215: 1734.##[22] Behzad M., Meghdari A., Ebrahimi A 2008 A linear theory for bending stressstrain analysis of a beam with an edge crack, Engineering Fracture Mechanics 75: 46954705.##[23] Li Q. S., 2002, Free vibration analysis of nonuniform beams with an arbitrary number of cracks and concentrated mass, Journal of Sound Vibration 252: 509525.##[24] Binici B., 2005, Vibration of beams with multiple open cracks subjected to axial force, Journal of Sound Vibration 287: 277295.##[25] Zhang G. P., Wang Z. G., 2008, Fatigue of smallscale metal materials: from micro to nanoscale structural integrity and microstructural worthiness, Springer Netherlands 152: 275326.##[26] Sadeghian H., Goosen H., Bossche A., Thijsse B., Van Keulen F., 2011, On the sizedependent elasticity of silicon nanocantilevers: impact of defects, Journal of Physics D : Applied Physics 44: 072001.##[27] Shengli K., Shenjie Z., Zhifeng N., Kai W., 2009, Static and dynamic analysis of micro beams based on strain gradient elasticity theory, International Journal of Engineering Science 47: 487498.##[28] Zamanzadeh M., Rezazadeh G., Jafarsadeghipoornaki I., Shabani R., 2013, Static and dynamic stability modeling of a capacitive FGM microbeam in presence of temperature changes, Applied Mathematical Modeling 37: 69646978.##[29] Akbaş S. D., 2016, Free vibration of edge cracked functionally graded microscale beams based on the modified couple stress theory, International Journal of Structural Stability and Dynamics 16: 1750033.##[30] Torabi K., Dastgerdi J., 2012, An analytical method for free vibration analysis of Timoshenko beam theory applied to cracked nanobeams using a nonlocal elasticity model, Thin Solid Films 520: 65956602.##[31] Gounaris G. D., Papadopoulos C. A., Dimarogonas A. D., 1996, Crack identification in beams by coupled response measurement, Computational and Structural 58: 409423.##[32] Shifrin E. I., Ruotolo R., 1999, Natural frequencies of a beam with an arbitrary number of cracks, Journal of Sound Vibration 223: 409423.##[33] Dimarogonas A. D., Paipettis S. A., 1983, Analytical Methods in Rotor Dynamics, Elsevier Applied Science, London.##[34] Dimarogonas A. D., 1976, Vibration Engineering, Paul, West Publishers.##[35] Rezazadeh G., Tahmasebi A., Zubostow M., 2006, Application of piezoelectric layers in electrostatic MEM actuators: controlling of pullin voltage, Microsystem Technologies 12: 11631170.##[36] Dominicus J., Ijntema Harrie A. C., 1992, Static and dynamic aspects of an airgap capacitor, Sensors and Actuators A: Physical 35: 121128.##[37] Lee K. B., 2011, Principles of MicroElectromechanical Systems, John Wiley & Sons, New Jersey.##[38] Younis M. I., 2011, MEMS Linear and Nonlinear Statics and Dynamics, Springer, New York.##[39] Rochus V., Rixen D. J. and Golinval J.C., 2005, Electrostatic coupling of MEMS structures: transient simulations and dynamic pullin, Nonlinear Analysis, Theory, Methods & Applications 63: 16191633.##[40] Vytautas O., Rolanas D., 2010, Microsystems Dynamics, Springer, New York.##[41] Osterbeg P. M., 1995, ElectroStatically Actuated MicroElectromechanical Test Structures for Material Property Measurements, PhD Dissertation Massachusetts Institute of Technology.##[42] Joglekar M. M., Pawaskar D. N., 2011, Closedform empirical relations to predict the static pullin parameters of electrostatically actuated microcantilevers having linear with variation, Microsystem Technologies 17: 3545.##[43] Abbasnejhad B., Rezazadeh G., Shabani R., 2011, Stability analysis of a capacitive FGM microbeam using modified couple stress theory, Acta Mechanica Solida Sinica 26: 427440.##]
Damping and Frequency Shift in Microscale Modified Couple Stress Thermoelastic Plate Resonators
2
2
In this paper, the vibrations of thin plate in modified couple stress thermoelastic medium by using Kirchhoff Love plate theory has been investigated. The governing equations of motion and heat conduction equation for Lord Shulman (LS) [1] theory are written with the help of Kirchhoff Love plate theory. The thermoelastic damping of microbeam resonators is analyzed by using the normal mode analysis. The solutions for the free vibrations of plates under clampedsimply supported (CS) and clampedfree (CF) conditions are obtained. The analytical expressions for thermoelastic damping of vibration and frequency shift are obtained for couple stress generalized thermoelastic and coupled thermoelastic plates. A computer algorithm has been constructed to obtain the numerical results. The thermoelastic damping and frequency shift with varying values of length and thickness are shown graphically in the absence and presence of couple stress for (i) clampedsimply supported, (ii) clampedfree boundary conditions. Some particular cases are also presented.
1

621
636


S
Devi
Department of Mathematics and Statistics, Himachal Pradesh University, Shimla, India
Department of Mathematics and Statistics,
India
shaloosharma2673@gmail.com


R
Kumar
Department of Mathematics, Kurukshetra University, India
Department of Mathematics, Kurukshetra University,
India
rajneesh_kuk@rediffmail.com
Modified couple stress theory
Thermoelasticity
Thermoelastic damping
Frequency shift
[Lord H.W., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids 15: 299309.##[2] Mindlin R. D., 1963, Influence of couplestresses on stressconcentrations, Experimental Mechanics 3: 17.##[3] Mindlin R. D., Tiersten H. F., 1962, Effects of couplestresses in linear elasticity, Archive for Rational Mechanics and Analysis 11: 415448.##[4] Toupin R. A., 1962, Elastic materials with couplestresses, Archive for Rational Mechanics and Analysis 11(1): 385414.##[5] Yang F., Chong A. C. M., Lam D. C. C., Tong P., 2002, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures 39: 27312743.##[6] Eringen A. C., 1966, Linear theory of micropolar elasticity, Journal of Mathematics and Mechanics 15: 909923.##[7] Tsiatas G. C., 2009, A new Kirchhoff plate model based on a modified couple stress theory, International Journal of Solids and Structures 46: 27572764.##[8] Sun Y., Tohmyoh H., 2009, Thermoelasic damping of the axisymmetric vibration of circular plate resonators, Journal of Sound and Vibration 319: 392405.##[9] Sun Y., Saka M., 2010, Thermoelasic damping in microscale circular plate resonators, Journal of Sound and Vibration 329: 338337.##[10] Sharma J. N., Sharma R., 2011, Damping in microscale generalized thermoelastic circular plate resonators, Ultrasonics 51: 352358.##[11] Ezzat M.A., ElKaramany A.S., Samaan A.A., 2001, Statespace formulation to generalized thermoviscoelasticity with thermal relaxation, Journal of Thermal Stresses 24(9): 823846.##[12] ElKaramany A.S., Ezzat M.A., 2002, On the boundary integral formulation of thermoviscoelasticity theory, International Journal Engineering Sciences 40(17): 19431956##[13] Ezzat M.A., ElKaramany A. S., 2003, On uniqueness and reciprocity theorems for generalized thermoviscoelasticity with thermal relaxation, Canadian Journal of Physics 81(6): 823833.##[14] Ezzat M.A., ElKaramany A.S., ElBary A.A., 2017, Twotemperature theory in Green–Naghdi thermoelasticity with fractional phaselag heat transfer, Microsystem Technologies 24(2): 951961.##[15] Fang Y., Li P., Wang Z., 2013, Thermoelasic damping in the axisymmetric vibration of circular microplate resonators with two dimensional heat conduction, Journal of Thermal Stresses 36: 830850.##[16] Shaat M., Mahmoud F. F., Gao X. L., 2014, Faheem A. F., 2014, Sizedependent bending analysis of Kirchhoff nanoplates based on a modified couplestress theory including surface effects, International Journal of Mechanical Sciences 79: 3137.##[17] Simsek M., Aydm M., Yurtcu H. H., Reddy J. N., 2015, Sizedependent vibration of a microplate under the action of a moving load based on the modified couple stress theory, Acta Mechanica 226: 38073822.##[18] Darijani H., Shahdadi A. H., 2015, A new shear deformation model with modified couple stress theory for microplates, Acta Mechanica 226(8): 27732788.##[19] Gao X. L., Zhan G. Y., 2016, A nonclassical Kirchhoff plate model incorporating microstructure, surface energy and foundation effects, Continuum Mechanics and Thermodynamics 28: 195213.##[20] Reddy J. N., Romanoff J., Loya J. A., 2016, Nonlinear finite element analysis of functionally graded circular plates with modified couple stress theory, European Journal of Mechanics A/Solids 56: 92104.##[21] Chen W., Wang Y., 2016, A model of composite laminated Reddy plate of the globallocal theory based on new modified couplestress theory, Mechanics of Advanced Materials and Structures 23(6): 636651.##[22] Marin M., 1998, A temporally evolutionary equation in elasticity of micropolar bodies with voids, Scientific Bulletin Series A Applied Mathematics and Physics 60: 312.##[23] Marin M., 2010, Harmonic vibrations in thermoelasticity of microstretch materials, Journal of Vibration and Acoustics 132(4): 044501044506.##[24] Sharma K., Marin M., 2013, Effect of distinct conductive and thermodynamic temperatures on the reflection of plane waves in micropolar elastic halfspace, Scientific Bulletin Series A Applied Mathematics and Physics 75(2): 121132.##[25] Marin M., Agarwal R. P., Codarcea L., 2017, A mathematical model for threephaselag dipolar thermoelastic bodies, Journal of Inequalities and Applications 109: 116.##[26] Rao S. S., 2007, Vibration of Continuous Systems, John Wiley & Sons, Inc. Hoboken, New Jersey.##[27] Sharma J. N., Kaur R., 2014, Transverse vibrations in thermoelasticdiffusive thin microbeam resonators, Journal of Thermal Stresses 37: 12651285.##[28] Sharma J. N., 2011, Thermoelastic damping and frequency shift in Micro/Nanoscale anisotropic beams, Journal of Thermal Stresses 34(7): 650666.##[29] Daliwal R.S., Singh A., 1980, Dynamical Coupled Thermoelasticity, Hindustan Publishers, Delhi.##]
Shear Waves Through Non Planar Interface Between Anisotropic Inhomogeneous and ViscoElastic HalfSpaces
2
2
A problem of reflection and transmission of a plane shear wave incident at a corrugated interface between transversely isotropic inhomogeneous and viscoelastic halfspaces is investigated. Applying appropriate boundary conditions and using Rayleigh’s method of approximation expressions for reflection and transmission coefficients are obtained for the first and second order approximation of the corrugation. Further, closed form formulae of these coefficients are presented for a corrugated interface of periodic shape (cosine law interface). Numerical computations for this particular type of corrugated interface are performed and a number of graphs are plotted to illustrate the effect of different parameters of the both halfspaces on the reflection and transmission coefficients. It is found that these coefficients depend upon the amplitude of corrugation of the boundary, angle of incidence and frequency of the incident wave and are strongly influenced by the anisotropy, inhomogeneity and viscoelasticity of the halfspaces. Some special cases are also derived.
1

637
654


B
Prasad
Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad, India
Department of Applied Mathematics, Indian
India
bishwanathprasad92@gmail.com


P
Chandra Pal
Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad, India
Department of Applied Mathematics, Indian
India


S
Kundu
Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad, India
Department of Applied Mathematics, Indian
India
SHwaves
Viscoelastic
Inhomogeneity
Anisotropy
Corrugated boundary
[[1] Rayleigh L., 1907, On the dynamical theory of gratings, The Royal Society Publishing 79: 399416.##[2] Asano S., 1960, Reflection and refraction of elastic waves at a corrugated boundary surface partI: The case of incidence of SHwave, Bulletin of Earthquake Research Institute 38: 177197.##[3] Asano S., 1961, Reflection and refraction of elastic waves at a corrugated boundary surface partII, Bulletin of Earthquake Research Institute 39: 367406.##[4] Asano S., 1966, Reflection and refraction of elastic waves at a corrugated interface, Bulletin of the Seismological Society of America 56: 201221.##[5] Abubakar I., 1962, Scattering of plane elastic waves at rough surfaces, Proceedings of Cambridge Philosophical Society 58: 136157.##[6] Dunkin J.W., Eringen A.C., 1962, The reflection of elastic waves from the wavy boundary of a halfspace, Proceedings of the 4th U. S. National Congress on Applied Mechanics, University of California Press, Berkeley.##[7] Abubakar I., 1962, Reflection and refraction of plane SHwaves at irregular interface, Journal of Physics of the Earth 10: 114.##[8] Abubakar I., 1962, Reflection and refraction of plane SHwaves at irregular interfaces, Journal of Physics of the Earth 10: 1520.##[9] Kaushik V.P., Chopra S.D., 1981, Reflection and transmission of plane SHwaves at an anisotropic elastic viscoelastic interface, Geophysical Research Bulletin 19: 112.##[10] Gogna M.L., Chander S., 1985, Reflection and refraction of SHwaves at an interface between anisotropic inhomogeneous elastic and viscoelastic halfspaces, Acta Geophysica Polonica 33: 357375.##[11] Gupta S., 1987, Reflection and transmission of SHwaves in laterally and vertically heterogeneous media at an irregular boundary, Geophysical Transmission 33: 89111.##[12] Kumar R., Tomar S.K., Chopra A., 2003, Reflection/refraction of SHwaves at a corrugated interface between two different anisotropic and vertically heterogeneous elastic solid halfspaces, Australia and New Zealand Industrial and Applied Mathematics Journal 44: 447460.##[13] Tomar S.K., Kaur J., 2003, Reflection and transmission of SHwaves at a corrugated interface between two laterally and vertically heterogeneous anisotropic elastic halfspaces, Earth Planets and Space 55: 531547.##[14] Kaur J., Tomar S.K., Kaushik V.P., 2005, Reflection and refraction of SHwaves at a corrugated interface between two laterally and vertically heterogeneous viscoelastic solid halfspaces, International Journal of Solids Structures 42(13): 36213643.##[15] Chattopadhyay A., Gupta S., Sharma V.K., Kumari P., 2008, Propagation of SH waves in an irregular monoclinic crustal layer, Archive of Applied Mechanics 78(12): 989999.##[16] Chattopadhyay A., Gupta S., Sharma V.K., Kumari P., 2009, Reflection and refraction of plane quasiP waves at a corrugated interface between distinct triclinic elastic half spaces, International Journal of Solids and Structures 46: 32413256.##[17] Chattopadhyay A., Gupta S., Sharma V.K., Kumari P., 2010, Propagation of shear waves in viscoelastic medium at irregular boundaries, Acta Geophysica 58(2): 195214.##[18] Chattopadhyay A., Gupta S., Sahu S.A., Singh A.K., 2013, Dispersion of horizontally polarized shear waves in an irregular nonhomogeneous selfreinforced crustal layer over a semiinfinite selfreinforced medium, Journal of Vibration and Control 19(1): 109119.##[19] Kumar R., Garg S.K., Ahuja S., 2015, Wave propagation in fibrereinforced transversely isotropic thermoelastic media with initial stress at the boundary, Journal of Solid Mechanics 7(2): 223238.##[20] Prasad B., Pal P.C., Kundu S., 2017, Propagation of SHwaves through non planer interface between viscoelastic and fibrereinforced solid halfspaces, Journal of Mechanics 33(4): 545557.##[21] Kakar R., 2015, Rayleigh waves in a homogeneous magnetothermo Voigttype viscoelastic halfspace under initial surface stress, Journal of Solid Mechanics 7(3): 255267.##[22] Kumar S., Pal P.C., Majhi S., 2017, Reflection and transmission of SHwaves at a corrugated interface between two semiinfinite anisotropic magneto elastic halfspaces, Waves in Random and Complex Media 27(2): 339358.##[23] Vaishnav P.K., Kundu S., AboDahab S.M., Saha A., 2017, Torsional surface wave propagation in anisotropic layer sandwiched between heterogeneous halfspace, Journal of Solid Mechanics 9(1): 213224.##[24] Gupta R.N., 1965, Reflection of plane waves from a linear transition layer in liquid media, Geophysics 30(1): 122132.##[25] Schoenberg M., 1971, Transmission and reflection of plane waves at an elasticviscoelastic interface, Geophysical Journal of the Royal Astronomical Society 25: 3547.##[26] Kaushik V.P., Rana A., 1997, Transmission of SH waves through a linear viscoelastic layer sandwiched between two anisotropic inhomogeneous elastic half spaces, Indian Journal of Pure and Applied Mathematics 28(6): 851863.##[27] Savarensky E., 1975, Seismic Waves, Mir Publication, Moscow.##[28] Gubbins D., 1990, Seismology and Plate Tectonics, Cambridge University Press, Cambridge.##]
Elastic Wave Propagation at Imperfect Boundary of Micropolar Elastic Solid and Fluid Saturated Porous Solid HalfSpace
2
2
This paper deals with the reflection and transmission of elastic waves from imperfect interface separating a micropolar elastic solid halfspace and a fluid saturated porous solid halfspace. Longitudinal and transverse waves impinge obliquely at the interface. Amplitude ratios of various reflected and transmitted waves are obtained and computed numerically for a specific model and results obtained are depicted graphically with angle of incidence of incident waves. It is found that these amplitude ratios depend on angle of incidence of the incident wave, imperfect boundary and material properties of halfspaces. From the present study, a special case when fluid saturated porous halfspace reduces to empty porous solid is also deduced and discussed graphically.
1

655
671


V
Kaliraman
Department of Mathematics, Chaudhary Devi Lal University, Sirsa, Haryana, India
Department of Mathematics, Chaudhary Devi
India
vsisaiya@gmail.com


R.K
Poonia
Department of Mathematics, Chandigarh University, Gharuan, Mohali, Punjab, India
Department of Mathematics, Chandigarh University,
India
Porous solid
Micro polar elastic solid
Reflection
Transmission
Longitudinal wave
Transverse wave
Amplitude ratios
Stiffness
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Free Vibration of Functionally Graded Cylindrical Shell Panel With and Without a Cutout
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2
The free vibration analysis of the functionally graded cylindrical shell panels with and without cutout is carried out using the finite element method based on a higherorder shear deformation theory. A higherorder theory is used to properly account for transverse shear deformation. An eight noded degenerated isoparametric shell element with nine degrees of freedom at each node is considered. The stiffness and mass matrices are derived based on the principle of minimum potential energy. The stiffness and mass matrices of the element are evaluated by performing numerical integration using the Gaussian quadrature. The effect of volume fraction exponent on the fundamental natural frequency of simply supported and clamped functionally graded cylindrical shell panel without a cutout is studied for various aspect ratios and arclength to thickness ratios. Results are presented for variation of the fundamental natural frequency of the cylindrical shell panel with cutout size for simply supported and clamped boundary conditions.
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k.S
Sai Ram
Department of Civil Engineering RVR&JC College of Engineering Chowdavaram Guntur, India
Department of Civil Engineering RVR&JC
India
sairamks@yahoo.com


K
Pratyusha
Department of Civil Engineering RVR&JC College of Engineering Chowdavaram Guntur, India
Department of Civil Engineering RVR&JC
India


P
Kiranmayi
Department of Civil Engineering RVR&JC College of Engineering Chowdavaram Guntur, India
Department of Civil Engineering RVR&JC
India
Functionally Graded Materials
Free vibration
Finite Element Method
Higherorder shear deformation theory
Cylindrical shell panel with a cutout
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