2017
6
2
12
0
A comparative study between two numerical solutions of the NavierStokes equations
2
2
The present study aimed to investigate two numerical solutions of the NavierStokes equations. For this purpose, the mentioned flow equations were written in two different formulations, namely (i) velocitypressure and (ii) vorticitystream function formulations. Solution algorithms and boundary conditions were presented for both formulations and the efficiency of each formulation was investigated by considering a twodimensional low laminar flow around a square pile in a rectangular computational domain. Simulations under the same conditions were conducted to assess the difference between results generated by both formulations. Furthermore, the accuracy of the results was analyzed through a comparison of the results with the available reference data. In addition, computational efficiency of both formulations was investigated in term of computation time. The corresponding results indicated that both formulations are adequate to the case used in the present study. Moreover, performed simulations showed that solving the vorticitystream function form of the flow equations is faster than solving the velocitypressure form of those equations for simulating a twodimensional laminar flow around a square pile.
1

1
12


M.
Alemi
Departamento de Engenharia Civil, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, s/n, Porto 4200465, Portugal
Departamento de Engenharia Civil, Faculdade
Iran
m.alemi@fe.up.pt


R.
Maia
Departamento de Engenharia Civil, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, s/n, Porto 4200465, Portugal
Departamento de Engenharia Civil, Faculdade
Iran
CFD
Laminar
NavierStokes
Square Pile
VelocityPressure
VorticityStream function
[[1] S. V. Patankar, and D. B. Spalding, “A calculation procedure for heat, mass and momentum transfer in threedimensional parabolic flows,” Int. J. Heat Mass Transf., Vol. 15, No. 10, pp. 17871806, (1972). ##[2] H. K. Versteeg, and W. Malalasekera, An Introduction to Computational Fluid Dynamics  The Finite Volume Method. Longman Scientific & Technical, (1995). ##[3] A. J. Chorin, “Numerical solution of the NavierStokes equations,” J. Math. Comput.,Vol. 22, No. 104, pp. 745762, (1968). ##[4] J. Kim, and P. Moin, “Application of a Fractionalstep method to incompressible NavierStokes equations,” J. Comput. Phys., Vol. 59, No. 2, pp. 308323, (1985). ##[5] P. Majander, and T. Siikonen, “A comparison of time integration methods in an unsteady lowReynoldsnumber flow,” Int. J. Numer. Methods Fluids, Vol. 39, No. 5, pp. 361390, (2002). ##[6] J.G. Liu, and C. Wang, “High order finite difference methods for unsteady incompressible flows in multiconnected domains,” J. Comput. Fluids, Vol. 33, No. 2, pp. 223255, (2004). ##[7] S. Biringen, and C.Y. Chow, An introduction to computational fluid mechanics by example. John Wiley & Sons, (2011). ##[8] A. Sohankar, C. Norberg, and L. Davidson, “LowReynoldsnumber flow around a square cylinder at incidence: Study of blockage, onset of vortex shedding and outlet boundary condition,” Int. J. Numer. Methods Fluids, Vol. 26, No. 1, pp. 3956, (1998). ##[9] J. F. Ravoux, a. Nadim, and H. HajHariri, “An embedding method for bluff body flows: interactions of two sidebyside cylinder wakes,” Theor. Comput. Fluid Dyn., Vol. 16, No. 6, pp. 433466, (2003). ##[10] A. Sharma, and V. Eswaran, “Heat and fluid flow across a square cylinder in the twodimensional laminar flow regime,” Numer. Heat Transf. Part A Appl., Vol. 45, No. 3, pp. 247269, (2004). ##[11] B. S. Carmo, and J. R. Meneghini, “Numerical investigation of the flow around two circular cylinders in tandem,” J. Fluids Struct., Vol. 22, No. 67, pp. 979988, (2006). ##[12] E. Weinan, and L. JianGuo, “Finite difference methods for 3D viscous incompressible flows in the vorticityvector potential formulation on nonstaggered grids,” J. Comput. Phys., Vol. 138, No. 1, pp. 5782, (1997). ##[13] T. Hou, and B. Wetton, “Stable fourthorder streamfunction methods for incompressible flows with boundaries,” J. Comput. Math., Vol. 27, No. 4, pp. 441458, (2009).##]
Influence of heat generation on the phase transformations and impact responses of composite plates with embedded SMA wires
2
2
In the present research, in contrast to the available papers, not only the superelasticity but also the shape memory effects are taken into account in determination of the impact responses. At the same time, in addition to modifying Brinson’s model for the shape memory alloys (SMAs), to include new parameters and loading events, and Hertz contact law, distributions of the SMA phases are considered to be both localized and timedependent. Furthermore, effects of the impactinduced heat generation and mechanical energy on the resulting histories of the martensite phase volume fraction, stressstrain, temperature, lateral deflection, and contact force are investigated. The generated heat in the SMA wires during the impact is determined through using a Helmholtz free energy function including the latent heat of the phase transformation. The resulting governing equations are solved by the finite element method. The nonlinear refined constitutive laws are solved through a returnmapping NewtonRaphson procedure. Results reveal that incorporation of the heat generation effects is significant in medium/highvelocity impacts or when the stress field is almost uniform.
1

13
26


A.
Niknami
Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran
Faculty of Mechanical Engineering, K.N. Toosi
Iran
aniknami@gmail.com


M.
Shariyat
Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran
Faculty of Mechanical Engineering, K.N. Toosi
Iran
shariyat@kntu.ac.ir
Shape memory alloy
Impact
Phase transformation
Heat generation
Hybrid composite plate
[[1] S .M. R. Khalili, A. Shokuhfar, and F. Ashenai Ghasemi, “Effect of smart stiffening procedure on lowvelocity impact response of smart structures”, J. Mater Proc. Tech., Vol. 190, No. 13, pp. 142152, (2007). ##[2] A. Shokuhfar, S. M. R. Khalili, F. Ashenai Ghasemi, K. Malekzadeh, and S. Raissi, “Analysis and optimization of smart hybrid composite plates subjected to lowvelocity impact using the response surface methodology (RSM)”, ThinWall Struct, Vol. 46, No. 11, pp. 12041212, (2008). ##[3] M. Meo, F. Marulo, M. Guida, and S. Russo, “Shape memory alloy hybrid composites for improved impact properties for aeronautical applications”, Compos. Struct., Vol. 95, pp.756766, (2013). ##[4] E. H. Kim, I. Lee, J. H. Roh, J. S. Bae, I. H. Choi, and K. N. Koo, “Effects of shape memory alloys on low velocity impact characteristics of composite plate”, Compos. Struct., Vol. 93, No. 11, pp. 29032909, (2011). ##[5] J. H. Roh and J. H. Kim, “Adaptability of hybrid smart composite plate under low velocity impact”, Compos. Part B, Vol. 34, No. 2, pp.117125, (2003). ##[6] M. Shariyat and R. Jafari, “Nonlinear lowvelocity impact response analysis of a radially preloaded twodirectionalfunctionally graded circular plate: A refined contact stiffness approach”, Compos. Part B,Vol. 45, No .1, pp. 981994, (2013). ##[7] M. Shariyat and F. Farzan, “Nonlinear eccentric lowvelocity impact analysis of a highly prestressed FGM rectangular plate, using a refined contact law”, Arch. Appl. Mech., Vol. 83, No .4, pp. 623641, (2013). ##[8] M. Shariyat and F. Farzan Nasab, “Lowvelocity impact analysis of the hierarchical viscoelastic FGM plates, using an explicit shearbending decomposition theory and the new DQ method”, Compos. Struct., Vol. 113, No.1, pp. 6373, (2014). ##[9] M. Shariyat and M. Moradi, “Enhanced algorithm for nonlinear impact of rectangular composite plates with SMA wires, accurately tracing the instantaneous and local phase changes”, Compos. Struct.,Vol. 108, pp. 834847, (2014). ##[10] M. Shariyat and S. H. Hosseini, “Accurate eccentric impact analysis of the preloaded SMA composite plates, based on a novel mixedorder hyperbolic global–local theory”, Compos. Struct.,Vol. 124, pp. 140151, (2015). ##[11] D. Helm and P. Haupt, “Shape memory behaviour: modeling within continuum thermomechanics”, Int. J. Solids Struct., Vol. 40, No. 4, pp. 827849, (2003). ##[12] H. Tobushi and Y. Shimeno, T. Hachisuka, and K. Tanaka, “Influence of strain rate on super elastic properties of TiNi shape memory alloy”, Mech. Mater., Vol. 30, No. 2, pp. 141150, (1998). ##[13] M. Kadkhodaei, RKND. Rajapakse, M. Mahzoon and M. Salimi, “Modeling of the cyclic thermomechanical response of SMA wires at different strain rates”, Smart Mater. Struct., Vol. 16, No. 6, pp. 20912101, (2007). ##[14] P. C. C. Monteiro, M. A. Savi , T. A. Netto, and P. M. C. Pacheco, “A Phenomenological description of the thermomechanical coupling and the ratedependent behavior of shape memory alloys”, J. Intell. Mater. Sys. Struct., Vol. 20, No. 14, pp. 16751687, (2009). ##[15] C. Morin, Z. Moumni and W. Zaki, “A constitutive model for shape memory alloys accounting for thermomechanical coupling”, Int. J. Plast., Vol. 27, No. 5, pp. 748767, (2011). ##[16] J. H. Roh, “Thermomechanical Modeling of shape memory alloys with rate dependency on the pseudoelastic behavior”, Math. Prob. Eng., Vol. 20, No. 1, pp.4165, (2014). ##[17] J. Ignaczak and M. OstojaStarzewski, Thermoelasticity with finite wave speeds, Oxford University Press, United States, New York, (2010). ##[18] M. R. Eslami, R. B. Hetnarski, J. Ignaczak , N. Noda, N. Sumi and Y. Tanigawa, Theory of elasticity and thermal stresses, Springer, (2013). ##[19] L. C. Brinson, “One dimensional constitutive behavior shape memory alloys: Thermomechanical derivation with nonconstant material functions and redefined martensite internal variable”, J. Intell. Mater Syst. Struct., Vol. 4, No. 2, pp. 229242, (1993). ##[20] D. C. Lagoudas, Shape Memory Alloys: Modeling and Engineering Applications, Springer, (2008). ##[21] J. N. Reddy, Mechanics of Laminated Composite Plates and ShellsTheory and Analysis, CRC Press, Boca Raton, (2003). ##[22] V. Birman, “An approach to optimization of shape memory alloy hybrid composite plates subjected to lowvelocity impact”, Composites: Part B, Vol. 27, No. 5, pp. 439446, (1996). ##[23] J. R. Turner, “Contact on a transversely isotropic halfspace, or between two transversely isotropic bodies”, Int. J. Solids Struct., Vol. 16, No. 5, pp. 40919, (1980). ##[24] A. Niknami and M. Shariyat, “Refined constitutive, bridging, and contact laws for including effects of the impactinduced temperature rise in impact responses of composite plates with embedded SMA wires”, ThinWalledStruct., Vol. 106, pp. 166178, (2016). ##[25] S. H. Yang and C.T. Sun, “Indentation law for composite laminates”, In: Composite materials: testing and design (6th conference), ASTM STP787, pp. 425 49, (1982). ##[26] M. R. Eslami, Finite elements methods in mechanics, Springer, (2014). ##[27] E. Serra and M. Bonaldi, “A finite element formulation for thermoelastic damping analysis”, Int. J. Numer. Meth. Eng., Vol. 78, No. 6, pp. 671691, (2009). ##[28] M. Shariyat and A. Niknami, “Impact analysis of strainratedependent composite plates with SMA wires in thermal environments: Proposing refined coupled thermoelasticity, constitutive, and contact models”, Compos. Struct., Vol. 136, pp. 191120, (2016). ##[29] M. Shariyat and A. Niknami, “Layerwise numerical and experimental impact analysis of temperaturedependent transversely flexible composite plates with embedded SMA wires in thermal environments”, Compos. Struct., Vol. 153, pp. 692–703, (2016). ##[30] R. Tiberkak, M. Bachene, S. Rechak and B. Necib, “Damage prediction in composite plates subjected to low velocity impact”, Compos. Struct., Vol. 83, No. 1, pp.7382, (2008). ##[31] L. C. Brinson and R. Lammering, “Finite element analysis of the behavior of shape memory alloys and their applications”, Int. J. Solids Struct., Vol. 30, No. 23, pp.32613280, (1993).##]
Chemical reaction and thermal radiation effects on MHD micropolar fluid past a stretching sheet embedded in a nonDarcian porous medium
2
2
The paper aims at investigating the effects of chemical reaction and thermal radiation on the steady twodimensional laminar flow of viscous incompressible electrically conducting micropolar fluid past a stretching surface embedded in a nonDarcian porous medium. The radiative heat flux is assumed to follow Rosseland approximation. The governing equations of momentum, angular momentum, energy, and species equations are solved numerically using RungeKutta fourth order method with the shooting technique. The effects of various parameters on the velocity, microrotation, temperature and concentration field as well as skin friction coefficient, Nusselt number and Sherwood number are shown graphically and tabulated. It is observed that the micropolar fluid helps the reduction of drag forces and also acts as a cooling agent. It was found that the skinfriction coefficient, heat transfer rate, and mass transfer rate are decreased, and the gradient of angular velocity increases as the inverse Darcy number, porous medium inertia coefficient, or magnetic field parameter increase. Increases in the heat generation/absorption coefficient caused increases in the skinfriction coefficient and decrease the heat transfer rate. It was noticed that the increase in radiation parameter or Prandtl number caused a decrease in the skinfriction coefficient and an increase in the heat transfer rate. In addition, it was found that the increase in Schmidt number and chemical reaction caused a decrease in the skinfriction coefficient and an increase in the mass transfer rate.
1

27
46


L.
Ramamohan Reddy
Department of Mathematics, Mekapati Rajamohan Reddy Institute of Technology and Science, Udayagiri, Nellore District, A.P, India
Department of Mathematics, Mekapati Rajamohan
Iran


M. C.
Raju
Department of Humanities and Sciences, Annamacharya Institute of Technology and Sciences (Autonomous), Rajampet – 516126, A.P., India
Department of Humanities and Sciences, Annamachary
Iran
mcrmaths@yahoo.co.in


G. S. S.
Raju
Department of Mathematics, JNTUA College of Engineering Pulivendula, Pulivendula, A.P, India
Department of Mathematics, JNTUA College
Iran


S. M.
Ibrahim
Department of Mathematics, GITAM University, Vishakhaptanam, A.P.  530045 India
Department of Mathematics, GITAM University,
Iran
MHD
Micropolar fluid
chemical reaction
thermal radiation
VUMAT subroutine
Porous medium
[[1] A. C. Eringen, “Theory of micropolar fluids”, Journal of Mathematics and Mechanics, Vol. 16, pp. 118, (1966). ##[2] T. Y. Na, and I. Pop, “Boundarylayer flow of micropolar fluid due to a stretching wall”, Archives of Applied Mechanics, Vol. 67, No. 4, pp. 229236, (1977). ##[3] A. Desseaux, and N. A. Kelson, “Flow of a micropolar fluid bounded by a stretching sheet”, Anziam J., Vol. 42, pp. 536560, (2000). ##[4] F. M. Hady, On the solution of heat transfer to micropolar fluid from a nonisothermal stretching sheet with injection, Int. J. Numer. Methods for Heat and Fluid Flow, Vol. 6, No. 6, pp. 99104, (1966). ##[5] O. Aydin and A. Kaya, “NonDarcin forced convection flow of a viscous dissipating fluid over a flat plate embedded in a porous medium”, Trans Porous Media, Vol. 73, No. 2, pp. 173186, (2008). ##[6] S. S. Das, A. Satapathy, J. K. Das and J. P. Panda, “Mass transfer effects on MHD flow and heat transfer past a vertical porous plate through a porous medium under oscillatory suction and heat source”, Int. J. of Heat and Mass Transfer., Vol. 52, No. 2526, pp. 59625969, (2009). ##[7] J. Anand Rao and S. Shivaiah, “Chemical reaction effects on an unsteady MHD free convective flow past an infinite vertical porous plate with constant suction and heat source”, Int. J. of Appl. Math and Mech., Vol. 7, No. 8, pp. 98118,(2011). ##[8] S. Y. Ibrahim and O. D. Makinde, “Radiation effect on chemically reacting magneto hydrodynamics (MHD) boundary layer flow of heat and mass transfer through a porous vertical flat plate,” Int. J. Physical Sciences, Vol. 6, No. 6, pp. 15081516, (2011). ##[9] D. Pal and B. Talukdar, “Combined effects of Joule heating and chemical reaction on unsteady magneto hydrodynamic mixed convection of a viscous dissipating fluid over a vertical plate in porous media with thermal radiation,” Mathematical and Computer Modelling, Vol. 54, No. 1112, pp. 30163036, (2011). ##[10] K. Jhansi Rani and Ch. V. Ramana Murthy, “MHD Flow over a Moving Infinite Vertical Porous Plate with Uniform Heat Flux in the presence of Thermal Radiation”, Advanced in Theoretical and Applied Mathematics, Vol. 6, No. 1, pp. 5163, (2011). ##[11] G. V. Ramana Reddy, N. Bhaskar Reddy and Ch. V. Ramana Murthy, “Heat and mass transfer effects on MHD free convection flow past an oscillating plate embedded in porous medium”, International Journal of Physical Sciences, Vol. 22. No. 2M, pp. 375380, (2010). ##[12] T. S. Reddy, M. C. Raju and S. V. K .V Varma “ unsteady MHD Radiative and Chemically reactive free convection flow near a moving vertical plate in porous medium” Journal of Applied Fluid Mechanics ,Vol. 6, No. 3, pp. 443451, (2013). ##[13] M. C. Raju, S. V. K. Varma, N. A. Reddy, “MHD Thermal diffusion natural convection flow between heated inclined plates in porous medium”, Journal on future engineering and technology, Vol. 6, No. 2, pp. 4548, (2011). ##[14] P.Chandrakala, “Radiation Effects on Flow Past an Impulsively Started Vertical Oscillating Plate with Uniform Heat Flux”, International Journal of Dynamics of Fluids, Vol. 7, No. 1, pp. 18, (2011). ##[15] R. Choudhury and U. J. Das, “MHD mixed convective heat and mass transfer in a viscoelastic boundary layer slip flow past a vertical permeable plate with thermal radiation and chemical reaction”, Int. J. of statistika and mathematika, Vol. 3, No. 3, pp. 93101, (2012). ##[16] S. Abzal, G. R. Reddy and S. V. K. Varma, “MHD free convection flow and mass transfer unsteady near a moving vertical plate in the presence of thermal radiation”, Annals of faculty engineering hunedoara International journal of engineering, Tom IX, pp. 2934, (2011). ##[17] V. Ravikumar, M. C. Raju and G. S. S. Raju, “Magnetic field and radiation effects on a double diffusive free convective flow bounded by two infinite impermeable plates in the presence of chemical reaction”, IJSER, Vol. 4, No. 7, pp. 19151923, (2013). ##[18] R. A. Mohamed, AbdelNasser A. Osman, S.M. AboDahab, “Unsteady MHD double diffusive convection boundarylayer flow past a radiate hot vertical surface in porous media in the presence of chemical reaction and heat sink”, Meccanica, Vol. 48, No. 4, pp 931942, (2013). ##[19] M. Y. Malik, T. Salahuddin, Arif Hussain and S Bilal, “MHD flow of tangent hyperbolic fluid over a stretching cylinder: Using Keller box method”, Journal of Magnetism and Magnetic Materials, Vol. 395, pp. 271276, (2015). ##[20] T. Salahuddin, M. Y. Malik, Arif Hussain, S. Bilal and M. Awais, “The effects of transverse magnetic field with variable thermal conductivity on tangent hyperbolic fluid with exponentially varying viscosity”, AIP Advances, Vol. 5, Article ID: 127103, (2015). ##[21] T. Salahuddin, Md. Yousaf Malik, Arif Hussain and M. Awais, “MHD flow of CattanneoChristov heat flux model for Williamson fluid over a stretching sheet with variable thickness: Using numerical approach”, Journal of Magnetism and Magnetic Materials, Vol. 401, pp. 991997, (2015). ##[22] B. Seshaiah, S. V. K. Varma, M. C. Raju, “The effects of chemical reaction and radiation on unsteady MHD free convective fluid flow embedded in a porous medium with timedependent suction with temperature gradient heat source”, International Journal of Scientific Knowledge, Vol. 3 No. 2, pp. 1324, (2013). ##[23] R. Rout and H. B. Pattanayak, “Chemical reaction and radiation effects on MHD flow past an exponentially accelerated vertical plate in presence of heat source with variable temperature embedded in a porous medium”, Annals of faculty engineering hunedoara Int. Jou. Of Engg, Vol. 4, pp. 253259, (2013). ##[24] W. A. Khan, and I. Pop, “The ChengMinkowycz problem for the triple–diffusive natural convection boundary layer flow past a vertical plate in a porous medium”, J. Porous Media, Vol.16, No. 7, pp. 637646, (2013). ##[25] G. S. Seth, R. Nandkeolyar and M. S. Ansari, “Effects of thermal radiation and rotation on unsteady hydro magnetic free convection flow past an impulsively moving vertical plate with ramped temperature in a porous medium”, J. Appl. Fluid Mech, Vol.6, No.1, pp. 2738, (2013). ##[26] D. Ch. Kesavaiah, P. V. Satyanarayana, S. Venkataramana, “Effects of the chemical reaction and radiation absorption on an unsteady MHD convective heat and mass transfer flow past a semiinfinite vertical permeable moving plate embedded in a porous medium with heat source and suction”, Int. J. of Appl. Math and Mech., Vol. 7, No. 1, pp. 5269, (2011). ##[27] U. S. Rajput, S. Kumar, “Radiation effects on MHD flow past an impulsively started vertical plate with variable heat and mass transfer”, Int. J. of Appl. Math. and Mech., Vol. 8, No. 1, pp. 6685, (2012). ##[28] R. Muthucumaraswamy, N. Dhanasekar, G. Easwara Prasad, “Effects on first order chemical reaction on flow past an accelerated isothermal vertical plate in a rotating fluid with variable mass diffusion”, Int. J. Math., Vol. 4, No. 41, pp. 2835, (2013). ##[29] B. Devika, P. V. Satya Narayana, S. Venkataramana, “MHD oscillatory flow of a visco elastic fluid in a porous channel with chemical reaction”, Int. J. Engg. Sci. Inv., Vol. 2, No. 2, pp. 2635, (2013). ##[30] K. Chand, K. D. Singh, S. Kumar, “Hall effect on radiating and chemically reacting MHD oscillatory flow in a rotating porous vertical channel in slip flow regime”, Advances in Applied Sciences Research, Vol. 3, No. 4, pp. 24242437, (2012). ##[31] S. Mukhopadhyay, R. S. R. Gorla, “Effects of partial slip on boundary layer flow past a permeable exponential stretching sheet in presence of thermal radiation”, Heat Mass Transfer, Vol. 48, pp. 17731781, (2012). http://dx.doi.org/10.1007/s0023101210248. ##[32] P. K. Kameswaran, S. Shaw, P. Sibanda, P. V. S. N. Murthy, “Homogeneous–heterogeneous reactions in a nanofluid flow due to a porous stretching sheet”, Int. J. Heat and Mass Transfer, Vol. 57, No. 2, pp. 465472 (2013). ##[33] S. Shaw, P.K. Kameswaran, P. Sibanda, “Homogeneous–heterogeneous reactions in micropolar fluid flow from a permeable stretching or shrinking sheet in a porous medium”, Boundary Value Problems, Vol. 77, No. 1, (2013). ##[34] R. Ellahi, S. Aziz, A. Zeeshan, “NonNewtonian nanofluid flow through a porous medium between two coaxial cylinders with heat transfer and variable viscosity”, Journal of Porous Media, Vol. 16, No. 1, pp. 205216, (2013). ##[35] N. Bachok, A. Ishak, I. Pop, “Boundary layer stagnationpoint flow and heat transfer over an exponentially stretching/shrinking sheet in a nanofluid”, Int. J. f Heat Mass transfer, Vol. 55, No. 2526, pp. 81228128, (2013). ##[36] N. S. Akbar, S. Nadeem, R.U. Haq, Z.H. Khan, “Radiation effects on MHD stagnation point flow of nanofluid towards a stretching surface with convective boundary condition”, Chinese Journal of Aeronautics, Vol. 26, No. 6, pp. 13891397, (2013), DOI: http://dx.doi.org/10.1016/j.cja.2013.10.008. ##[37] M. Sheikholeslami, D. D. Ganji, M. Y. Javed, R. Ellahi, “Effect of thermal radiation on magneto hydrodynamics nanofluid flow and heat transfer by means of two phase model”, J. Magn..Mater. Vol. 374, pp. 3643, (2015). ##[38] M. Turkyilmazoglu, I. Pop, “Heat and mass transfer of unsteady natural convection flow of some nanofluids past a vertical infinite flat plate with radiation effect”, Int. J. Heat Mass Transfer, Vol. 59, pp. 167171, (2013). ##[39] M. Sheikholeslami, D. D. Ganjia, M. M. Rashidib, “Ferro fluid flow and heat transfer in a semi annulus enclosure in the presence of magnetic source considering thermal radiation”, J. Taiwan Inst. Chem. Eng., Vol. 47, pp. 617, (2015). http://dxdoi.org/10.1016/j.jtice.2014.09.026. ##[40] D. Pal, “Combined effects of nonuniform heat source/sink and thermal radiation on heat transfer over an unsteady stretching permeable surface”, Commun. Nonlinear Sci. Numer. Simulat., Vol. 16, pp. 1890–1904, (2011). ##[41] G. C. Shit, R. Haldar, “Effects of thermal radiation on MHD viscous fluid flow and heat transfer over nonlinear shrinking porous sheet”, Appl. Mathematics Mech. (English Edition), Vol. 32, No. 6, pp. 677688, (2011). ##[42] K. Das, “Impact of thermal radiation on MHD slip flow over a flate plate with ##variable fluid properties”, Heat Mass Transfer, Vol. 48, pp. 767778, (2012). ##[43] F. T. Akyildiz, H. Bellout, K. Vajravelu, R.A. Van Gorder, “Existence results for third order nonlinear boundary value problems arising in nano boundary layer fluid flows over stretching surfaces”, Nonlinear Anal.: Real World Appl., Vol. 12, pp. 29192930, (2011). ##[44] A. J. Chamkha, R. S. R. Gorla, K. Ghodeswar, “Nonsimilar solution for natural convective boundary layer flow over a sphere embedded in a porous medium saturated with a nanofluid”, Transp. Porous Media, Vol. 86, No. 1, pp. 1322, (2011). ##[45] N. Bachok, A. Ishak, I. Pop, “Stagnationpoint flow over a stretching/shrinking sheet in a nanofluid”, Nanoscale Res. Lett., Vol. 6, pp. 623632, (2011). ##[46] O. D. Makinde, A. Aziz, “Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition”, Int. J. Thermal Sci., Vol. 50, pp. 13261332, (2011). ##[47] E. M AboEldahaband and M. A ElAziz, “Flow and heat transfer in a micropolar fluid past a stretching surface embedded in a nonDarcian porous medium with uniform free steam”, Mathematics and Computation, Vol. 162, No. 2, pp. 881899, (2005).##]
Prediction of earing in deep drawing of anisotropic aluminum alloy sheet using BBC2003 yield criterion
2
2
This paper investigates the earing phenomenon in deep drawing of AA3105 aluminum alloy, experimentally and numerically. Earing defect is mainly attributed to the plastic anisotropy of sheet metal. In order to control such defect, predicting the evolution of ears in sheet metal forming analyses becomes indispensable. In this regard, the present study implements the advanced yield criterion BBC2003. Based on this yield function and the associated flow rule of plasticity, the constitutive model is derived. Accordingly, a user material VUMAT subroutine is developed and adopted in the commercial finite element software ABAQUS/Explicit. Several plane stress loading problems are designed, through which, the accuracy of the developed subroutine is verified. In addition, cylindrical cups of AA3105 aluminum alloy are fabricated using a deep drawing die. The earing defect was clearly observed on the recovered parts. Using the experimentally obtained constants of BBC2003 yield criterion for this alloy in VUMAT, deep drawing of the cylindrical cups was simulated. The results demonstrate that the earing profile can successfully be predicted using BBC2003 yield function.
1

47
55


S.
Izadpanah
School of Mechanical Engineering, Shahrood University of Technology, Shahrood 3619995161, Iran
School of Mechanical Engineering, Shahrood
Iran


S. H.
Ghaderi
School of Mechanical Engineering, Shahrood University of Technology, Shahrood 3619995161, Iran
School of Mechanical Engineering, Shahrood
Iran
s.h.ghaderi@shahroodut.ac.ir


M.
Gerdooei
School of Mechanical Engineering, Shahrood University of Technology, Shahrood 3619995161, Iran
School of Mechanical Engineering, Shahrood
Iran
Plastic anisotropy
Earing
Sheet metal forming
Advanced yield criterion
VUMAT subroutine
[[1] D. Banabic, Sheet metal forming processes: constitutive modelling and numerical simulation, Springer Science & Business Media, pp. 45120, (2010). ##[2] M. Vrh, M. Halilovič, B. Starman, B. Štok, D.S. Comsa, and D. Banabic, “Capability of the BBC2008 yield criterion in predicting the earing profile in cup deep drawing simulations”, Eur. J Mech. ASolid, Vol. 45, pp. 5974, (2014). ##[3] K. Chung, and K. Shah. “Finite element simulation of sheet metal forming for planar anisotropic metals”, Int. J. Plasticity, Vol. 8, No. 4, pp. 453476, (1992). ##[4] J. W. Yoon, F. Barlat, R. E. Dick, and M. ##E. Karabin, “Prediction of six or eight ears in a drawn cup based on a new anisotropic yield function”, Int. J. Plasticity, Vol. 22, No. 1, pp.174193, (2006). ##[5] J. Gawad, D. Banabic, A.Van Bael, D.S. Comsa, M. Gologanu, P. Eyckens, P. Van Houtte, and D. Roose, “An evolving plane stress yield criterion based on crystal plasticity virtual experiments”, Int. J. Plasticity, Vol. 75, pp.141169, (2015). ##[6] D. Banabic, H. Aretz, D. S. Comsa, and L. Paraianu, “An improved analytical description of orthotropy in metallic sheets”, Int. J. Plasticity, Vol. 21, No. 3, pp. 493512, (2005). ##[7] M. Vrh, M. Halilovič, and B. Štok. “Improved explicit integration in plasticity”, Int. J. Numer. Meth. Eng., Vol. 81, No. 7, pp. 910938 (2010). ##[8] M. Halilovič, M. Vrh, B. Štok, “NICEh: a higherorder explicit numerical scheme for integration of constitutive models in plasticity”, Eng. Comput., Vol 29, No. 1, pp. 55−70, (2013). ##[9] K. Lange, Handbook of metal forming, McGrawHill Book Company, pp. 20.1620.19, (1985). ##[10] S. IzadpanahNajmabad, M. Gerdooei, S. H. Ghaderi, “Determination of BBC2003 yield criterion constants for anisotropic aluminum alloy sheets based on plane strain tensile test”, Modares Mech. Eng., Vol. 15 , No. 11, pp. 127135, (2015). ##[11] B. Starman, M. Vrh, M. Halilovič, and B. Štok, “Advanced modelling of sheet metal forming considering anisotropy and Young’s modulus evolution”, Stroj. Vestn. J. Mech. E., Vol. 60, No. 2, pp. 8492, (2014).##]
Assessment of different methods for fatigue life prediction of steel in rotating bending and axial loading
2
2
Generally, fatigue failure in an element happens at the notch on a surface where the stress level rises because of the stress concentration effect. The present paper investigates the effect of a notch on the fatigue life of the HSLA100 (high strength low alloy) steel which is widely applicable in the marine industry. Tensile test was conducted on specimens and mechanical properties were obtained. Rotating bending and axial fatigue tests were performed at room temperature on smooth and notched specimens and SN curves were obtained. Using the obtained SN curve for smooth specimens, the fatigue strength factor for the notched specimens were predicted by Weibull's weakestlink, Peterson, Neuber, stress gradient and critical distance methods and compared with experimental results. It was found that the critical distance and also Weibull’s weakestlink methods have the best agreement with experimental results.
1

57
68


J.
Amirian
Department of Mechanical Engineering, Isfahan University of Technology, 8415683111, Iran
Department of Mechanical Engineering, Isfahan
Iran


H.
Safari
Subsea Research and Development Center, Isfahan University of Technology, 8415683111, Iran
Subsea Research and Development Center, Isfahan
Iran


M.
Shirani
Subsea Research and Development Center, Isfahan University of Technology, 8415683111, Iran
Subsea Research and Development Center, Isfahan
Iran
mehdi.shirani@cc.iut.ac.ir


M.
Moradi
Department of Mechanical Engineering, Isfahan University of Technology, 8415683111, Iran
Department of Mechanical Engineering, Isfahan
Iran


S.
Shabani
Subsea Research and Development Center, Isfahan University of Technology, 8415683111, Iran
Subsea Research and Development Center, Isfahan
Iran
Fatigue failure
SN curve
Rotating bending
[[1] Y. L. Lee, J. Pan, R. Hathaway and M. Barkey, Fatigue testing and analysis  theory and practice, 1st ed., Elsevier, Oxford, (2005). ##[2] N. E. Dowling, Mechanical behavior of materials  engineering methods for deformation, fracture, and fatigue, 3rd ed., Prentice Hall, New Jersey, (2007). ##[3] E. Siebel and M. Stieler, “Ungleichförmige Spannun gsverteilungbeischwingender Beanspruchung”, VDIZ . Vol. 97, No. 5, pp.121126, (1955). ##[4] D. Taylor, The theory of critical distances  a new perspective in fracture mechanics,1st ed., Elsevier, London, (2007). ##[5] A. Wormsen, B. Sjödin, G. Härkegård and A. F. jeldstad,“Nonlocal stress approach for fatigue assessment based on weakestlink theory and statistics of extremes”,Fatigue Eng. Mater. Struct, Vol. 30, No. 12, pp.12141227, (2007). ##[6] W. Weibull, “A statistical theory of the strength of materials”, IVA Handlingar,Vol. 151, pp.145, (1939). ##[7] W. Weibull,“The phenomenon of rupture in solids”, IVA Handlingar,Vol. 153, pp.155160, (1939). ##[8] W. Weibull,“A statistical distribution function of wide applicability”,Appl. Mech. Eng, Vol. 18, No. 3, pp.293297, (1951). ##[9] M. Shirani and G. Härkegård,“Large scale axial fatigue testing of ductile cast iron for heavy section wind turbine components”, Eng. Failure Anal,Vol. 18, No. 6, pp.14961510, (2011). ##[10] M. Shirani and G. Härkegård,“Fatigue life distribution and size effect in ductile cast iron for wind turbine components”,Eng. Failure Anal, Vol. 18, No. 1, pp.1224, (2011). ##[11] M. Shirani and G. Härkegård,“Damage tolerant design of cast components based on defects detected by 3D Xray computed tomography”,Int. J. Fatigue, Vol. 41, pp.188198, (2012). ##[12] M. Shirani and G. Härkegård, “Casting defects and fatigue behaviour of ductile cast iron for wind turbine components: A comprehensive study”, Materialwiss. Werkstofftech,Vol. 42, No. 12, pp.10591074, (2011). ##[13] M. Shirani and G. Härkegård, “Fatigue crack growth simulation in components with random defects”, J. ASTM In, Vol. 6, No. 9, pp.10891121, (2009). ##[14] T. Montemarano, B. Sach, J. Gudas, M. Vassilaros and H. Vandervelt, J. Ship Prod, Vol. 2, No. 3, pp.145, (1986). ##[15] HSLA Steel, 20021115, archived from the original on 20100103, retrieved 20081011. ##[16] J. Davis, Alloying: Understanding the Basics. ASM International, (2001). ##[17] S. Mikalac and M. Vassilaros, Proc. Of Int. Conf. on Processing, Microstructure and Properties of Microalloyed and Other Modern High Strength Low Alloy Steels, Iron and Steel Society, Pittsburgh, PA, June 36. p. 331 (1991). ##[18] A. Coldren and T. Cox, “Technical Report”, David Taylor Research Laboratory, DTNSRDCN0016785C006, (1985). ##[19] E. Czyryca, Proc. Conf. on Advances in Low Carbon High Strength Ferrous Steels LCFA92, O.N. Mohanty, B.B. Rath, M.A. Imam, C.S. Sivaramakrishnan (Eds.), IndoUS Pacific Rim Workshop, Trans Tech Pub., Jamshedpur, India, March 2528, p. 490, (1992). ##[20] ASTM Standard E8. Standard Test Methods for Tension Testing of Metallic Materials. West Conshohocken (PA, USA): ASTM International, (2007). ##[21] ASTM Standard E294814. Standard Test Method for Conducting Rotating Bending Fatigue Tests of Solid Round Fine Wire. West Conshohocken (PA, USA): ASTM International, (2007). ##[22] ASTM Standard E 46607. Standard practice for conducting force controlled constant amplitude axial fatigue tests of metallic materials. West Conshohocken (PA, USA): ASTM International, (2007). ##[23] H. Belmonte, M. Mulheron, P. Smith,“Weibull analysis, extrapolations and implications for condition assessment of cast iron water mains”,Fatigue Eng. Mater. Struct. Vol. 30, No. 10, pp. 96490, (2007). ##[24] W. Weibull,Fatigue testing and analysis of results, Pergamon, New York, (1961). ##[25] S. Nishijima,“Statistical fatigue properties of some heattreated steels for machine structural use”,ASTM Spec Technol Publ. Vol. 744, pp.7588, (1981). ##[26] J. Wilson,“Statistical comparison of fatigue data”, J. Mater. Sci. Lett, Vol. 7, No. 3, pp. 307308, (1988). ##[27] G. Härkegård and G. Halleraker, “Assessment of methods for prediction of notch and size effects at the fatigue limit based on test data by Böhm and Magin”,Int. J. Fatigue.Vol. 32, No. 10, pp.17011709, (2010). ##[28] R. E. Peterson,Notch sensitivity. In: Sines G, Waisman JL, editors. Metal fatigue. McGraw Hill, New York, (1959). ##[29] H. Neuber,Theory of notch stresses  principles for exact calculation of strength with reference to structural form and material, Springer, Berlin, (1958).##]
Water hammer simulation by explicit central finite difference methods in staggered grids
2
2
Four explicit finite difference schemes, including LaxFriedrichs, NessyahuTadmor, LaxWendroff and LaxWendroff with a nonlinear filter are applied to solve water hammer equations. The schemes solve the equations in a reservoirpipevalve with an instantaneous and gradual closure of the valve boundary. The computational results are compared with those of the method of characteristics (MOC), and with the results of Godunov''s scheme to verify the proposed numerical solution. The computations reveal that the proposed LaxFriedrichs and NessyahuTadmor schemes can predict the discontinuities in fluid pressure with an acceptable order of accuracy in cases of instantaneous and gradual closure. However, LaxWendroff and LaxWendroff with nonlinear filter schemes fail to predict the pressure discontinuities in instantaneous closure. The independency of time and space steps in these schemes are allowed to set different spatial grid size with a unique time step, thus increasing the efficiency with respect to the conventional MOC. In these schemes, no Riemann problems are solved; hence fieldbyfield decompositions are avoided. As provided in the results, this leads to reduced run times compared to the Godunov scheme.
1

69
77


F.
Khalighi
Civil Engineering Department, Shahrood University of Technology, Shahrood, 009823, IRAN
Civil Engineering Department, Shahrood University
Iran


A.
Ahmadi
Civil Engineering Department, Shahrood University of Technology, Shahrood, 009823, IRAN
Civil Engineering Department, Shahrood University
Iran
a.ahmadi@shahroodut.ac.ir


A.
Keramat
Civil Engineering Department, JundiShapur University of Technology, Dezful, 009861, IRAN
Civil Engineering Department, JundiShapur
Iran
Water hammer
LaxFriedrichs
NessyahuTadmor
LaxWendroff
Method of Characteristics
Godunov’s method
[[1] A. S. Tijsseling, “Fluidstructure interaction in liquidfilled pipe systems: a review”, Journal of Fluids and Structures, Vol. 10, No. 2, pp. 109146, (1996). ##[2] H. J. Kwon and J. J Lee, “Computer and experimental models of transient flow in pipe involving backflow preventers”, Journal of Hydraulic Engineering, Vol. 134, No. 4, pp. 426434, (2008). ##[3] M. H. Afshar and M. Rohani, “Exploring the Versatility of the implicit method of characteristic (MOC) for Transient simulation of pipeline systems”, Twelfth International Water Technology Conference, Alexandria, Egypt, (2008). ##[4] S. R. Sabbaghyazdi, A. Abbasi and N. Mastorakis, “Water hammer modeling using 2nd order Godunov finite volume method”. Proceeding of European Computing Conference. Vol. 2, pp. 215223, (2009). ##[5] M. Zhao, M. S. Ghidaoui, “GodunovType Solutions for Water Hammer Flows”, Journal ofHydraulic Engineering, Vol. 130, No. 4, pp. 341348, (2004). ##[6] M. H. Chaudhry and M. Y. Hussaini, “Secondorder accurate explicit finitedifference schemes for water hammer analysis”, Journal of Fluids Engineering, Vol. 107, No. 4, pp. 523529, (1985). ##[7] A. S. Tijsseling and A. Bergant, “Meshless computation of water hammer”, 2nd IAHR International Meeting of the Work groupon Cavitation and Dynamic Problems in Hydraulic Machinery and Systems, Timisoara, Vol. 52, No. 66, pp. 6576 (2007). ##[8] H. Hou, A. C. H. Kruisbrink, A. S. Tijsseling and A. Keramat, “Simulating water hammer with corrective smoothed particle method”, Eindhoven University of Technology, Eindhoven, (2012). ##[9] H. Chaudhry, Applied hydraulic transients, Van Nostrand Reinhold Company, New York, (1979). ##[10] E. Wylie and V. Streeter, Applied hydraulic transients, Fluid Transient in Systems. PrenticeHall, New York, (1993). ##[11] A. Bergant, A. S. TIJSSELING and J. P. VÍTKOVSKÝ, “Parameters affecting waterhammer wave attenuation, shape and timingpart1: Mathematical tools”, Journal of Hydraulic Research, Vol. 46, No. 3, pp. 373381, (2003). ##[12] K. A. Hoffmann and S. T. Chiang, “Computational fluid dynamics for engineers”. Engineering Education Systems, Austin, Texas, (1993). ##[13] A. V. Chikitkin, B. V. Rogov and S. V. Utyuzhnikov, “Highorder accurate monotone compact running scheme for multidimensional hyperbolic equations”. Applied Numerical Mathematics, Vol. 93, No. 3, pp. 150163, (2015). ##[14] L. F. Shampine, “Twostep LaxFriedrichs method”, Applied Mathematics Letters, Vol. 18, No. 10, pp. 11341136, (2004). ##[15] L. F. Shampine, “Solving hyperbolic PDEs in MATLAB”, Applied Numerical ##Analysis & Computational Mathematics, Vol. 2, No. 3, pp. 346358, (2005). ##[16] A. S. Tijsseling and C. S. W. Lavooij, “Water hammer with fluidstructure interaction”, Applied Scientific Research, Vol. 47, No. 3, pp. 273285, (1990). ##[17] A. S. Tijsseling and A. Bergant, “Meshless computation of water hammer”, 2nd IAHR International Meeting of the Work groupon Cavitation and Dynamic Problems in Hydraulic Machinery and Systems, Timisoara, Vol. 52, No. 6, pp. 6576, (2007).##]
Dynamical formation control of wheeled mobile robots based on fuzzy logic
2
2
In this paper, the important formation control problem of nonholonomic wheeled mobile robots is investigated via a leaderfollower strategy. To this end, the dynamics model of the considered wheeled mobile robot is derived using Lagrange equations of motion. Then, using ADAMS multibody simulation software, the obtained dynamics of the wheeled system in MATLAB software is verified. After that, in order to generate and keep the desired formation, a Fuzzy Logic Controller is designed. In this regard, the leader mobile robot is controlled to follow a reference path and the follower robots use the Fuzzy Logic Controller to keep constant relative distance and constant angle with respect to the leader. The efficiency of the suggested dynamicsbased formation controller has been proved using several computer simulations under different situations and desired trajectories. Also, the performance of the follower robot in path tracking is checked in the presence of receiving noisy data from the leader robot.
1

79
91


k.
Alipour
Department of Mechatronics Engineering, Faculty of New Sciences and Technologies, University of Tehran, Tehran, Iran
Department of Mechatronics Engineering, Faculty
Iran
k.alipour@ut.ac.ir


m.
Ghiasvand
Department of Electrical, Biomedical and Mechatronics Engineering, Islamic Azad University, Qazvin Branch, Qazvin, Iran
Department of Electrical, Biomedical and
Iran


B.
Tarvirdizadeh
Department of Mechatronics Engineering, Faculty of New Sciences and Technologies, University of Tehran, Tehran, Iran
Department of Mechatronics Engineering, Faculty
Iran
Wheeled mobile robots
Nonholonomic constraints
Fuzzy logic controller
Formation Control
An efficient finite difference time domain algorithm for band structure calculations of Phononic crystal
2
2
In this paper, a new algorithm for studying elastic wave propagation in the phononic crystals is presented. At first, the displacementbased forms of elastic wave equations are derived and then the forms are discretized using finite difference method. So the new algorithm is called the displacementbased finite difference time domain (DBFDTD). Three numerical examples are computed with this method and the results are compared with experimental measurements and the conventional FDTD method. Also, the computational cost of the new approach is compared with the conventional FDTD method. The comparison showed that the calculation time of the DBFDTD method is 37.5 percent less than that of the FDTD method.
1

93
101


M.
Moradi
Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, 8415683111, Iran
Department of Mechanical Engineering, Isfahan
Iran


M.
Bagheri Nouri
Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, 8415683111, Iran
Department of Mechanical Engineering, Isfahan
Iran
m.bagherinouri@me.iut.ac.ir
Phononic crystal
Wave propagation
Finite difference time domain
Displacementbased formulation
[[1[ M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. DjafariRouhani, “Acoustic band structure of periodic elastic composites”, Phys. Rev. Lett., Vol. 71, No. 13, pp. 20222025, (1993). ##[2[ R. MartinezSala, J. Sancho, J. V. Sanchez, V. Gomez, J. Llinares, and F. Meseguer, “Sound attenuation by sculpture”, Nature., Vol. 378, No. 6554, pp. 241241, (1995). ##[3[ F. R. Montero de Espinosa, E. Jime´nez, and M. Torres, “Ultrasonic Band Gap in a Periodic TwoDimensional Composite”, Phys. Rev. Lett., Vol. 80, No. 6, pp. 12081211, (1998). ##[4] A. Khelif, A. Choujaa, R. laihem, M. Wilm, S. Ballandras, and V. Laude, “Experimental study of band gaps and defect modes in a twodimensional ultrasonic crystal”, IEEE Ultrasonics Symposium, pp. 377380, (2003). ##[5] Y. Pennec, B. DjafariRouhani, J. O. Vasseur, A. Khelif, and P. A. Deymier, “Tunable filtering and demultiplexing in phononic crystals with hollow cylinders”, Phys. Rev. E., Vol. 69, 046608, (2004). ##[6[ W. Liu, J. W. Chen, and X. Y. Su, “Local resonance phononic band gaps in modified twodimensional lattice materials”, Acta Mech. Sin., Vol. 28, pp. 659669, (2012). ##[7[ M. Kafesaki, M. M. Sigalas, and N. García, “Frequency Modulation in the Transmittivity of Wave Guides in ElasticWave BandGap Materials”, Phys. Rev. Lett., Vol. 85, No. 19, pp. 40444047, (2000). ##[8] A. Khelif, B. DjafariRouhani, J. O. Vasseur, and P. A. Deymier, “Transmission and dispersion relations of perfect and defectcontaining waveguide structures in phononic band gap materials”, Phys. Rev. B., Vol. 68, No. 2, 024302, (2003). ##[9[ Y. Yao, Z. Hou, and Y. Liu, “The twodimensional phononic band gaps tuned by the position of the additional rod”, Phys Let. A., Vol. 362, No. 56, pp. 494499, (2007). ##[10[ B. Wu, R. Wei, H. Zhao, and C. He, “Phononic Band Gaps in TwoDimensional Hybrid Triangular Lattice”, Acta Mech. Solida Sin., Vol. 23, No. 3, pp. 255259, (2010). ##[11[ Y. Tanaka, Y. Tomoyasu, and S. Tamura, “Band structure of acoustic waves in phononic lattices: Twodimensional composites with large acoustic mismatch”, Phys. Rev. B., Vol. 62, No. 11, pp. 73877392, (2000). ##[12[ P. Hsieh, T. Wu, and J. Sun, “ThreeDimensional Phononic Band Gap Calculations Using the FDTD Method and a PC Cluster System”, Ieee T. Ultrason. Ferr., Vol. 53, No. 1, pp. 148158, (2006). ##[13] D. GarcíaPablos, M. Sigalas, F. R. Montero de Espinosa, M. Torres, M. Kafesaki, and N. García, “Theory and Experiments on Elastic Band Gaps”, Phys. Rev. Lett., Vol. 84, No. 19, pp. 4349 4352, (2000). ##[14] A. Khelif, P. A. Deymier, B. DjafariRouhani, J. O. Vasseur, and L. Dobrzynski, “Twodimensional phononic crystal with tunable narrow pass band: Application to a waveguide with selective frequency”, J. Appl. Phys., Vol. 94, No. 3, pp. 13081311, (2003). ##[15] J. H. Sun, and T. T. Wu, “Analyses of mode coupling in joined parallel phononic crystal waveguides”, Phys. Rev. B., Vol. 71, 174303, (2005). ##[16] Y. Pennec, B. DjafariRouhani, H. Larabi, J. Vasseur, and A. C. HladkyHennion, “Phononic crystals and manipulation of sound”, Phys. Status Solidi C., Vol. 6, No. 9, pp. 20802085, (2009). ##[17] H. F. Gao, T. Matsumoto, T. Takahashi, and H. Isakari, “Analysis of Band Structure for 2D Acoustic Phononic Structure by BEM and the Block SS Method”, CMESComp. Model. Eng., Vol. 90, No. 4, pp. 283301, (2013). ##[18] M. Kafesaki, and E. N. Economou, “Multiplescattering theory for three dimensional periodic acoustic composites”, Phy. Rev. B., Vol. 60, No.17, 11993, (1999). ##[19] Z. Z. Yan, and Y. S. Wang, “Waveletbased method for calculating elastic band gaps of twodimensional phononic crystals”, J. Comput. Phys., Vol. 74, 224303, (2006). ##[20] B. DjafariRouhani, J. O. Vasseur, A. C. HladkyHennion, P. Deymier, F. Duval, B. Dubus, and Y. Pennec, “Absolute band gaps and waveguiding in free standing and supported phononic crystal slabs”, Photonic Nanostruct., Vol. 6, No. 1, pp. 3237, (2008). ##[21] M. Liu, J. Xiang, Y. Zhong, “The band gap and transmission characteristics investigation of local resonant quaternary phononic crystals with periodic coating”, Appl. Acoust., Vol. 100, pp. 1017, (2015). ##[22] M. Liu, P. Li, Y. Zhong, and J. Xiang, “Research on the band gap characteristics of twodimensional phononic crystals microcavity with local resonant structure”, Shock. Vib., Vol. 2015, 239832, (2015). ##[23] Y. Cao, Z. Hou, and Y. Liu, “Finite difference time domain method for bandstructure calculations of twodimensional phononic crystals”, Solid State Commun., Vol. 132, No. 8, pp. 539543, (2004). ##[24] A. Taflove, Advances in Computational Electrodynamics, Artech House, London, (1999). ##[25] T. T. Wu, J. H. Sun, “4G3 Guided Surface Acoustic Waves in Phononic Crystal Waveguides”, IEEE Ultrasonics Symposium, pp. 673676, (2006).##]
Vibration analysis of functionally graded cylindrical shells with different boundary conditions subjected to thermal loads
2
2
In the present work, study of the vibration of a functionally graded (FG) cylindrical shell made up of stainless steel, zirconia, and nickel is presented. Free vibration analysis is presented for FG cylindrical shells with simply supportedsimply supported and clamped–clamped boundary condition based on temperature independent material properties. The equations of motion are derived by Hamilton’s principle. Material properties assume to be graded in the thickness direction according to a simple power law distribution in terms of the volume fraction of the constituents. Effects of boundary conditions and volume fractions (power law exponent) on the natural frequencies of the FG cylindrical shell are studied. Frequency characteristics of the FG shell are found to be similar to those of isotropic cylindrical shells. Furthermore, natural frequencies of these shells are observed to be dependent on the constituent volume fractions and boundary conditions. Strain displacement relations from Love's and firstorder shear deformation theories are employed. Galerkin method is used to derive the governing equations for clamped boundary conditions. Further, analytical results are validated with those reported in the literature and excellent agreement is observed. Finally, in order to investigate the effects of the temperature gradient, functionally graded materials cylindrical shell with high temperature specified on the inner surface and outer surface at ambient temperature,1D heat conduction equation along the thickness of the shell is applied and the results are reported.
1

103
114


M.
Talebitooti
Department of Mechanical Engineering, Qom University of Technology, Qom, 151937195, Iran
Department of Mechanical Engineering, Qom
Iran
talebi@qut.ac.ir


M.
Ghasemi
Department of Mechanical Engineering, Qom University of Technology, Qom, 151937195, Iran
Department of Mechanical Engineering, Qom
Iran


S. M.
Hosseini
Department of Mechanical Engineering, Qom University of Technology, Qom, 151937195, Iran
Department of Mechanical Engineering, Qom
Iran
Functionally graded materials
Cylindrical shells
Natural frequency
Firstorder shear deformation theory (FSDT)
Thermal load
[[1] M. Yamanouchi, M. Koizumi, T. Hirai, and I. Shiota. Proceedings of the First International Symposium on Functionally Gradient Materials, Japan, (1990).##[2] M. Koizumi, “The concept of FGM” ,Ceramic Transactions, Functionally Gradient Materials, Vol. 34, pp. 310,(1993). ##[3] Anon, FGM components: PM meets the challenge. Metal Powder Report; Vol. 51, pp. 2832 ,(1996). ##[4] N. Sata, ”Characteristic of SiCTiB_ composites as the surface layer of SiCTiB_Cu functionally gradient material produced by selfpropagating hightemperature synthesis”, Ceramic Transactions, Functionally Gradient Materials, Vol. 34, pp. 109116,(1993).##[5] H. Yamaoka, M. Yuki, K. Tahara, T. Irisawa, R. Watanabe, and A. Kawasaki. “Fabrication of Functionally Gradient Material by slurry stacking and sintering process”, Ceramic Transactions, Functionally Gradient Materials, Vol. 34, pp. 72165, (1993).##[6] B. H. Rabin, and R. J. Heaps, “Powder processing of Ni/Al2O3 FGM”, Ceramic Transactions, Functionally Gradient Materials, Vol. 34, pp. 173180, (1993).##[7] N. Noda, “Thermal stresses in functionally graded materials”, Journal of Thermal Stresses Vol. 22, pp. 477512, (1999).##[8] T. Fuchiyama, and N. Noda, “Analysis of thermal stress in a plate of functionally gradient material”, Journal of Science and Engineering, Vol. 16, pp. 263268, (1995).##[9] Y. Obata, and N. Noda,” Steady thermal stresses in a hollow circular cylinder and hollow sphere of a functionally gradient material”, Journal of Thermal Stresses, Vol. 17, pp. 471487, (1994).##[10] J. N. Reddy, and C. D. Chin, “Thermo mechanical analysis of functionally graded cylinders and plates”, Journal of Thermal Stresses, Vol. 21, pp. 593626, (1998).##[11] M. Jabbari, S. Sohrabpour, and M. R. Eslami,”Mechanical and thermal stresses in a functionally graded hollow cylinder due to radially symmetric load”, International Journal of Pressure Vessels and Piping, Vol. 79, pp. 493497, (2002).##[12] H. Awaji, and R. Sivakumar, “Temperature and stress distribution in a hollow cylinder of functionally graded material; the case of temperatureindependent material properties”, Journal of the American Ceramic Society, Vol. 84, pp. 10591065, (2001).##[13] S. Takezono, K. Tao, E. Inamura, and M. Inoue, “Thermal stress and deformation in functionally graded material shells of revolution under thermal loading due to fluid”, JSME International Journal Series, Vol. 39, pp. 573581, (1996).##[14] G. R. Ye, W. Q. Chen, and J. B. Cai, “A uniformly heated functionally graded cylindrical shell with transverse isotropy”, Mechanics Research Communications, Vol. 28, pp. 535542, (2001). ##[15] K. M. Liew, S. Kitipornchai, X. Z. Zhang, and C. W. Lim, “Analysis of the thermal stress behavior of functionally graded hollow circular cylinders”, International Journal of Solids and Structures, Vol. 40, pp. 23552380, (2003).##[16] R. N. Arnold, and G. B. Warburton, “Flexural vibrations of the walls of thin cylindrical shells having freely supported ends”, Proceedings of the Royal Society London A, Vol. 197, pp. 238256, (1949).##[17] A. Ludwig, and R. Krieg,”An analytical quasiexact method for calculating Eigen vibrations of thin circular cylindrical shells”, Journal of Sound and Vibration, Vol. 74, pp. 155174, (1981).##[18] H. Chung,”Free vibration analysis of circular cylindrical shells”, Journal of Sound and Vibration, Vol. 74, pp. 331350, (1981).##[19] W. Soedel, ”A new frequency formula for closed circular cylindrical shells for a large variety of boundary conditions”, Journal of Sound and Vibration, Vol. 70, pp. 309317, (1980).##[20] A. Bhimaraddi, “A higher order theory for free vibration analysis of circular cylindrical shells”, International Journal of Solids and Structures, Vol. 20, pp. 623630, (1984).##[21] K. P. Soldatos, and V. P. Hajigeoriou, “Threedimensional solution of the free vibration problem of homogeneous isotropic cylindrical shells and panels”, Journal of Sound and Vibration, Vol. 137, pp. 369384, (1990).##[22] K. Y. Lam, and C. T. Loy, ”Effects of boundary conditions on frequencies characteristics for a multilayered cylindrical shell”, Journal of Sound and Vibration, Vol. 188, pp. 363384, (1995).##[23] C. T. Loy, K. Y. Lam, and C. Shu, ”Analysis of cylindrical shells using generalized differential quadrature”, Shock and Vibration, Vol. 4, pp. 193198, (1997).##[24] MM. Najafizadeh, MR. Isvandzibaei, Vibration of functionally graded cylindrical shells based on higher order shear deformation plate theory with ring support. Acta Mechanica 2007; 191: 7591.##[25] M. M. Najafizadeh, and M. R. Isvandzibaei, "Vibration of functionally graded cylindrical shells based on different shear deformation shell theories with ring support under various boundary conditions", Journal of Mechanical Science and Technology, Vol. 23, pp. 20722084, (2009).##[26] F. Tornabene, "Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a fourparameter powerlaw distribution", Comput. Methods Appl. Mech. Engrg., Vol. 198, pp. 29112935, (2009).##[27] P. Malekzadeh, and Y. Heydarpour, "Free vibration analysis of rotating functionally graded cylindrical shells in thermal environment", Composite Structures, Vol. 94, pp. 29712981, (2012).##[28] M. J. Ebrahimi, and M. M. Najafizadeh, "Free vibration of twodimensional functionally graded circular cylindrical shells on elastic foundation", Modares Mechanical Engineering, Vol. 38, No. 1, pp. 308324, (2013).##[29] R. Bahadori, and M. M. Najafizadeh "Free vibration analysis of twodimensional functionally graded axisymmetric cylindrical shell on Winkler–Pasternak elastic foundation by Firstorder Shear Deformation Theory and using Navierdifferential quadrature solution methods", Applied Mathematical Modelling, Vol. 39, pp. 48774894, (2015).##[30] G. G. Sheng and X. Wang, “Effects of Thermal Loading on the Buckling and Vibration of RingStiffened Functionally Graded Shell”, J. Therm. Stresses, Vol. 30, pp. 12491267, (2007). ##[31] K. Y. Lam, and W. Qian, “Vibrations of Thick Rotating Laminated Composite Cylindrical Shells”, J. Sound Vibr., Vol. ##225, No. 3, pp. 483501, (1999).##[32] R. Naj, M. Sabzikar Boroujerdy and M. R. Eslami, “Thermal and mechanical instability of functionally graded truncated conical shells”, ThinWalled Structures, Vol. 46, pp. 6578, (2008).##[33] H.S. Shen, and N. Noda, “Postbuckling of FGM Cylindrical Shells under Combined Axial and Radial Mechanical Loads in Thermal Environments”, Int. J. Solids Struct., Vol. 42, pp. 46414662, (2005).##[34] M. S. Qatu, “Vibration of Laminated Shells and Plates”, Elsevier, The Netherlands, (2004).##[35] A. V. Lopatin, and E. V. Morozov, "Buckling of the composite sandwich cylindrical shell with clamped ends under uniform external pressure", Compos. Struct., Vol. 122, pp. 209216, (2015). ##[36] M. Talebitooti, “Vibration and critical speed of orthogonally stiffened rotating FG cylindrical shell under thermomechanical loads using differential quadrature method” J. Term. Stresses, Vol. 36, pp.160188, (2013).##]
Enhancing the low cycle fatigue strength of AA6061 aluminum alloy by using the optimized combination of ECAP and precipitation hardening
2
2
In the present study, mechanical properties and low cycle fatigue behavior of a solidsolutionized AA6061 aluminum alloy produced by equal channel angular pressing (ECAP) process were investigated. The grain refinement after two passes of ECAP significantly increased the yield stress and ultimate tensile stress and decreased the ductility of the alloy. However, the improvement of low cycle fatigue strength was not as remarkable as expected. PostECAP aging heat treatment to the peakaging condition imposed a notable change in the strength and ductility of the alloy so that its fatigue strength partly enhanced. An optimized combination of grain refinement and distributed fine precipitates in the matrix of the alloy was achieved by conducting aging heat treatment between passes of ECAP. The proposed procedure was proved to yield the best combination of strength and ductility, better distribution and size of precipitates, and thus a remarkable improvement in the low cycle fatigue response of the investigated material.
1

115
127


M.
Jooybari
School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran
School of Mechanical Engineering, College
Iran


J.
Shahbazi Karami
Faculty of Mechanical Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran
Faculty of Mechanical Engineering, Shahid
Iran


M.
Sheikhi
Faculty of Mechanical Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran
Faculty of Mechanical Engineering, Shahid
Iran
m.sheikhi@sru.ac.ir
Low cycle fatigue
Equal channel angular Al alloy
Ultrafine grained microstructure
Precipitation hardening
[[1] G. E. Totten, C. E. Bates and G. M. Webster, “Physical Metallurgy and Processes”, Handbook of Aluminum, Eds. G. E. Totten, and D. S. MacKenzie, Marcel Dekker Inc., New York, Vol. 1, (2003).##[2] I. J. Polmear, Light Alloys; Metallurgy of the Light Metals, 3rd ed., Arnold, London, (1995).##[3] Y. Estrin, M. Yu. Murashkin, R. Z. Valiev, “Ultrafine grained aluminium alloys: processes, structural features and properties”, Fundamentals of Aluminium Metallurgy: Production, Processing and Applications, Eds. R. Lumley, Woodhead Publishing Limited, Cambridge, pp. 468503, (2010).##[4] R. Z. Valiev, R. K. Islamgaliev, and I. V. Alexandrov, “Bulk nanostructured materials from severe plastic deformation”, Prog. Mater. Sci., Vol. 45, pp. 103189, (2000).##[5] D. B. Witkin, E. J. Lavernia, ”Synthesis and mechanical behavior of nanostructured materials via cryomilling”, Prog. Mater. Sci., Vol. 51, pp. 160, (2006).##[6] R. Z. Valiev, A. V. Korznikov, and R. R. Mulyukov, “Structure and properties of ultrafinegrained materials produced by severe plastic deformation”, Mater. Sci. Eng., Vol. A 168, pp. 141148, (1993).##[7] Z. Horita, T. Fujinami, and T. G. Langdon, “The potential for scaling ECAP: effect of sample size on grain refinement and mechanical properties”, Mater. Sci. Eng., Vol. A 318, pp. 3441, (2001).##[8] J. Y. Huang, Y. T. Zhu, H. Jiang, and T. C. Lowe, “Microstructures and dislocation configurations in nanostructured Cu processed by repetitive corrugation and straightening”, Acta Mater., Vol. 49, pp. 14971505, (2001).##[9] Z. Horita, T. Fujinami, M. Nemoto, and T.G. Langdon, “Improvement of mechanical properties for Al alloys using equalchannel angular pressing”, J. Mater. Process. Technol., Vol. 117, pp. 288292, (2001).##[10] A. Shan, I. G. Moon, and J. W. Park, “Estimation of friction during equal channel angular (ECA) pressing of aluminum alloys”, J. Mater. Process. Technol., Vol. 122, pp. 255259, (2002).##[11] S. Ferrasse, V. M. Segal, K. T. Hartwig, and R .E. Goforth, “Microstructure and properties of copper and aluminum alloy 3003 heavily worked by equal channel angular extrusion”, Metall. Mater. Trans. Vol. A 28, pp. 10471057, (1997).##[12] V. M. Segal, “Materials processing by simple shear”, Mater. Sci. Eng. Vol. A 197, pp. 157164, (1995).##[13] V. M. Segal, “Equal channel angular extrusion: from macromechanics to structure formation”, Mater. Sci. Eng., Vol. A 271, pp. 322333, (1999).##[14] A. Gholinia, P. B. Prangnell, and M. V. Markushev, “The effect of strain path on the development of deformation structures in severely deformed aluminium alloys processed by ECAE”, Acta Mater., Vol. 48, pp. 11151130, (2000).##[15] R. Z. Valiev, and T. G. Langdon, “Principles of equalchannel angular pressing as a processing tool for grain refinement”, Prog Mater Sci., Vol. 51, pp. 881981, (2006).##[16] B. Huarte, C. J. Luis, I. Puertas, J. León, and R. Luri, “Optical and mechanical properties of an Al–Mg alloy processed by ECAE”, J. Mater. Process. Technol., Vol. 162, pp. 317326, (2005).##[17] Y. W. Tham, M. W. Fu, H. H. Hng, M.S. Yong, and K. B. Lim, “Bulk nanostructured processing of aluminum alloy”, J. Mater. Process. Technol., Vol. 192, pp. 575581, (2007).##[18] A. S. M. Agena, “A study of flow characteristics of nanostructured Al6082 alloy produced by ECAP under upsetting test”, J. Mater. Process. Technol., Vol. 209 , pp. 856863, (2009).##[19] S. Malekjani, P. D. Hodgson, P. Cizek, I. Sabirov, and T. B. Hilditch, “Cyclic deformation response of UFG 2024 Al alloy”, Int. J. Fatigue, Vol. 33, pp. 700709 (2011).##[20] Y. Estrin, and A. Vinogradov, “Fatigue behaviour of light alloys with ultrafine grain structure produced by severe plastic deformation: An overview”, Int. J. Fatigue, Vol. 32, pp. 898907, (2010).##[21] C. S. Chung, J. K. Kim, H. K. Kim, and W. J. Kim, “Improvement of highcycle fatigue life in a 6061 Al alloy produced by equal channel angular pressing”, Mater. Sci. Eng., Vol. A 337, pp. 3944, (2002).##[22] P. Huebner, R. Kiessling, H. Biermann, and A. Vinogradov, “Fracture behaviour of ultrafinegrained materials under static and cyclic loading”, Int. J. Mater. Res., Vol. 97, pp. 15661570, (2006).##[23] P. Huebner, R. Kiessling, H. Biermann, T. Hinkel, W. Jungnickel, R. Kawalla, H. W. Hoeppel, and J. May, “Static and Cyclic Crack Growth Behavior of UltrafineGrained Al Produced by Different Severe Plastic Deformation Methods”, Metall. Mater. Trans., Vol. A 38, pp. 19261933, (2007).##[24] R. Lapovok, C. Loader, F. H. Dalla Torre, and S. L. Semiatin, “Microstructure evolution and fatigue behavior of 2124 aluminum processed by ECAE with back pressure”, Mater. Sci. Eng., Vol. A 425, pp. 3646, (2006).##[25] H. W. Hoeppel, and M. Goeken, “Fatigue behaviour in nanostructured metals”, Nanostructured Metals and Alloys: Processing, Microstructure, Mechanical Properties and Applications, Eds. S. H. Whang, Elsevier Science, pp. 507541, (2011).##[26] A. Vinogradov, A. Washikita, K. Kitagawa, and V. I. Kopylov, “Fatigue life of finegrain AlMgSc alloys produced by equalchannel angular pressing”, Mater. Sci. Eng., Vol. A 349, pp. 318326, (2003).##[27] A. Vinogradov, and S. Hashimoto, “Multiscale Phenomena in Fatigue of UltraFine Grain Materialsan Overview”, Mater. Trans., Vol. 42, pp. 7484, (2001).##[28] W. J. Kim, J. K. Kim, T. Y. Park, S. I. Hong, D. I. Kim, Y. S. Kim, and J. D. Lee, “Enhancement of strength and superplasticity in a 6061 Al alloy processed by equalchannelangularpressing”, Metall. Mater. Trans. Vol. A 33, pp. 31553164, (2002).##[29] H.W. Höppel, M. Kautz, C. Xu, M. Murashkin, T.G. Langdon, R.Z. Valiev, and H. Mughrabi, “An overview: Fatigue behaviour of ultrafinegrained metals and alloys”, Int. J. Fatigue, Vol. 28, pp. 10011010, (2006).##[30] H. W. Höppel, and R. Z. Valiev, “On the possabilities to enhance the fatigue properties of ultrafinegrained metals”, Zeitschrift für Metallkunde., Vol. 93, pp. 641648, (2002).##[31] V. Patlan, A. Vinogradov, K. Higashi, and K. Kitagawa, “Overview of fatigue properties of fine grain 5056 AlMg alloy processed by equalchannel angular pressing”, Mater. Sci. Eng., Vol. A 300, pp. 171182, (2001).##[32] H. Mughrabi, and R. Wang, “Cyclic stress–strain response and highcycle fatigue behaviour of copper polycrystals”, Basic mechanisms in fatigue of metals, Eds. P. Lukas, J. Polak, Elsevier Academia, pp. 116, (1988).##[33] H. Mughrabi, H.W. Höppel, and M. Kautz, “Fatigue and microstructure of extrusion with subsequent hightemperature shorttime aging”, Mater. Sci. Eng., Vol. 503, pp. 167171, (2009).ultrafinegrained metals produced by severe plastic deformation”, Scripta Mater., Vol. 51, pp. 807812, (2004).##[34] H. Mughrabi, H. W. Höppel, M. Kautz, and R. Z. Valiev, “Annealing treatments to enhance thermal and mechanical stability of ultrafinegrained metals produced by severe plastic deformation”, Zeitschrift für Metallkunde., Vol. 94, pp. 10791083, (2003).##[35] K. Hockaufa, T. Niendorf, S. Wagnera, T. Hallea, and L. W. Meyer, “Cyclic behavior and microstructural stability of ultrafinegrained AA6060 under straincontrolled fatigue”, Procedia Engineering, Vol. 2, pp. 21992208, (2010).##[36] V. D. Sitdikov, P. S. Chizhov, M. Yu. Murashkin, A. A. Goidenko, R. Z. Valiev, “Xray studies of dynamic aging in an aluminum alloy subjected to severe plastic deformation”, Mater. Charact., Vol. 110, pp. 222227, (2015).##[37] A. Vinogradov, S. Nagasaki, V. Patlan, K. Kitagawa, and N. Kawazoe, Nanostruct. Mater., Vol. 11, pp. 925–934, (1999).##[38] M. Furukawa, Y. Iwahashi, Z. Horita, M. Nemoto, and T. G. Langdon,”The shearing characteristics associated with equalchannel angular pressing”, Mater. Sci. Eng., Vol. A 257, pp. 328332, (1998).##[39] S. S. Manson, and M. H. Hirschberg, “fatigue an Interdisciplinary approach”, Eds. J. J. Burke, V. Weiss, Syracuse University Press, p 231, (1970).##[40] M. Hockauf, L. W. Meyer, B. Zillmann, ##M. Hietschold, S. Schulze, and L. Krüger, ##“Simultaneous improvement of strength and ductility of AlMgSi alloys by ##combining equalchannel angular ##[41] P. Cavaliere, “Mechanical Properties of Nanocrystalline Materials”, Ed. M. Aliofkhazraei, Handbook of Mechanical Nanostructuring, Wiley–VCH, Weinheim, pp. 35, (2015).##[42] MILHDBK5H,“Metallic materials and elements for aerospace vehicle structures”, Military handbook, , Department of defense of the USA, Chapter 3, P. 277, (1998).##[43] K. S. Kumar, H. Van Swygenhoven, and S. Suresh, “Mechanical behavior of nanocrystalline metals and alloys”, Acta. Mater., Vol. 51, pp. 5743–5774, (2003).##[44] D. Steiner, and V. Gerold, “The fatigue behaviour of agehardened Cu2at.%Co alloy”, Mater. Sci. Eng., Vol. 84, pp. 7788, (1986).##[45] C. Calabrese, and C. Laird, “Cyclic stress–strain response of twophase alloys part I. Microstructures containing particles penetrable by dislocations”, Mater. Sci. Eng., Vol. 13, pp. 141157, (1974).##[46] C. Calabrese, and C. Laird, “Cyclic stress–strain response of twophase alloys part II. Particles not penetrated by dislocations”, Mater. Sci. Eng., Vol. 13, pp. 159174, (1974).##[47] S. Horibe, C. Laird, “Orientation and history dependence of cyclic deformation in Al–Cu single crystals containing θ′ precipitates”, Acta. Metall. Vol. 31, pp. 15671579, (1983).##]