ORIGINAL_ARTICLE
Modeling of the intermolecular Force-Induced Adhesion in Freestanding Nanostructures Made of Nano-beams
Among the intermolecular interactions, the Casimir and van der Waals forces are the most important forces that highly affect the behavior of nanostructures. This paper studies the effect of such forces on the adhesion of cantilever freestanding nanostructures. The nanostructures are made of a freestanding nano-beam which is suspended between two upper and lower conductive surfaces. The linear spring model is applied to derive the elastic force. The Lumped Parameter Model (LPM) is used to obtain constitutive equations of the systems. The maximum length of the nano-beam which prevents the adhesion is computed. Results of this study are useful for design and development of miniature devices.
http://jacm.scu.ac.ir/article_12264_032be72b744fbed8aecf4e6958fd85dd.pdf
2016-08-15T11:23:20
2019-03-23T11:23:20
1
7
10.22055/jacm.2016.12264
Adhesion
Casimir force
Van der Waals force
Freestanding nano-beam
Lumped Parameter Model (LPM)
Alireza
Yekrangi
yekrangi_ali@yahoo.com
true
1
Department of Engineering, Ramsar Branch, Islamic Azad University, Ramsar, Iran
Department of Engineering, Ramsar Branch, Islamic Azad University, Ramsar, Iran
Department of Engineering, Ramsar Branch, Islamic Azad University, Ramsar, Iran
AUTHOR
Rahman
Soroush
eng.soroush322@yahoo.com
true
2
Department of Engineering, Lahijan Branch, Islamic Azad University, Lahijan, Iran
Department of Engineering, Lahijan Branch, Islamic Azad University, Lahijan, Iran
Department of Engineering, Lahijan Branch, Islamic Azad University, Lahijan, Iran
LEAD_AUTHOR
[1] Abadian, N., Gheisari, R., Keivani, M., Kanani, A., Mokhtari J., Rach, R., and Abadyan, M., "Effect of the centrifugal force on the electromechanical instability of U-shaped and double-sided sensors made of cylindrical nanowires", Journal of the Brazilian Society of Mechanical Sciences and Engineering, pp. 1-20, 2016.
1
[2] Keivani, M., Kanani, A., Mardaneh, M. R., Mokhtari, J., Abadyan, N., and Abadyan, M., "Influence of Accelerating Force on the Electromechanical Instability of Paddle-Type and Double-Sided Sensors Made of Nanowires", International Journal of Applied Mechanics,Vol. 8, No. 01, pp. 1650011, 2016.
2
[3] Keivani, M., Khorsandi, J., Mokhtari, J., Kanani, A., Abadian, N., and Abadyan, M., "Pull-in instability of paddle-type and double-sided NEMS sensors under the accelerating force", Acta Astronautica, Vol. 119 pp. 196-206, 2016.
3
[4] Beni, Y. T., Koochi, A., Kazemi, A. S., and Abadyan, M., "Modeling the influence of surface effect and molecular force on pull-in voltage of rotational nano–micro mirror using 2-DOF model", Canadian Journal of Physics, Vol. 90, No. 10, pp. 963-974, 2012.
4
[5] Keivani, M., Mokhtari, J., Kanani, A., Abadian, N., Rach, R., and Abadyan, M., "A size-dependent model for instability analysis of paddle-type and double-sided NEMS measurement sensors in the presence of centrifugal force", Mechanics of Advanced Materials and Structures just-accepted, pp. 1-40, 2016.
5
[6] Benic, Y. T., Noghreh Abadid, A. R., and Noghreh Abadie, M., "A Deﬂection of Nano-Cantilevers Using Monotone Solution", 2011.
6
[7] Duan, J. S., Rach, R., and Wazwaz, A. M., "Solution of the model of beam-type micro-and nano-scale electrostatic actuators by a new modified Adomian decomposition method for nonlinear boundary value problems", International Journal of Non-Linear Mechanics, Vol. 49, pp. 159-169, 2013.
7
[8] Zhang, L., Golod, S. V., Deckardt, E., V. Prinz, V., and Grützmacher, "Free-standing Si/SiGe micro-and nano-objects", Physica E: Low-dimensional Systems and Nanostructures, Vol. 23, No. 3, pp. 280-284, 2004.
8
[9] Koochi, A., Kazemi A. S., and Abadyan, M. R., "Simulating deflection and determining stable length of freestanding carbon nanotube probe/sensor in the vicinity of graphene layers using a nanoscale continuum model", Nano 6, No. 05, pp. 419-429, 2011.
9
[10] Lin, W-H., and Zhao Y-P., "Casimir effect on the pull-in parameters of nanometer switches", Microsystem Technologies, Vol. 11, No. 2-3, pp. 80-85, 2005.
10
[11] Lin, W. H., and Zhao, Y. P., "Nonlinear behavior for nanoscale electrostatic actuators with Casimir force", Chaos, Solitons & Fractals, Vol. 23, No. 5, pp. 1777-1785, 2005.
11
[12] Farrokhabadi, A., Abadian, N., Kanjouri, F., and Abadyan, M., "Casimir force-induced instability in freestanding nanotweezers and nanoactuators made of cylindrical nanowires", International Journal of Modern Physics B, Vol. 28, No. 19, pp. 1450129, 2014.
12
[13] Farrokhabadi, A., Abadian, N., Rach, R., and Abadyan, M., "Theoretical modeling of the Casimir force-induced instability in freestanding nanowires with circular cross-section", Physica E: Low-dimensional Systems and Nanostructures, Vol. 63, pp. 67-80, 2014.
13
[14] Soroush, R., Koochi, A., Kazemi, A. S., Noghrehabadi, A., Haddadpour, H., and Abadyan, M., "Investigating the effect of Casimir and van der Waals attractions on the electrostatic pull-in instability of nano-actuators", Physica scripta, Vol. 82, No. 4, pp. 045801, 2010.
14
[15] Abadyan, M., Novinzadeh, A., and Kazemi, A. S., "Approximating the effect of the Casimir force on the instability of electrostatic nano-cantilevers", Physica Scripta, Vol. 81, No. 1, pp. 015801, 2010.
15
[16] Abdi, J., Koochi, A., Kazemi, A. S., and Abadyan, M. "Modeling the effects of size dependence and dispersion forces on the pull-in instability of electrostatic cantilever NEMS using modified couple stress theory", Smart Materials and Structures, Vol. 20, No. 5, pp. 055011, 2011.
16
[17] Fu, Y., Z. Jin., and Wan. L.,"Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS)", Current applied physics, Vol. 11, No. 3, pp. 482-485, 2011.
17
[18] Ke, C., "Resonant pull-in of a double-sided driven nanotube-based electromechanical resonator", Journal of Applied Physics, Vol. 105, No. 2, pp. 024301, 2009.
18
[19] Sedighi, H. M., and Shirazi, K. H., "Dynamic pull-in instability of double-sided actuated nano-torsional switches", Acta Mechanica Solida Sinica, Vol. 28, No. 1, pp. 91-101, 2015.
19
[20] Israelachvili, J. N., Intermolecular and surface forces: with applications to colloidal and biological systems, London: Academic Press, 1985.
20
[21] Bordag, M., Mohideen, U., and Mostepanenko, V. M., "New developments in the Casimir effect", Physics reports, Vol. 353, No. 1, pp. 1-205, 2001.
21
[22] Lamoreaux, S. K., "The Casimir force: background, experiments, and applications", Reports on progress in Physics, Vol. 68, No. 1, pp. 201, 2004.
22
[23] Gusso, A., and Delben, G. J. "Dispersion force for materials relevant for micro-and nanodevices fabrication." Journal of Physics D: Applied Physics, Vol. 41, No. 17, pp. 175405, 2008.
23
ORIGINAL_ARTICLE
Investigation of the vdW Force-Induced Instability in Nano-scale Actuators Fabricated form Cylindrical Nanowires
The presence of van der Waals (vdW) force can lead to mechanical instability in freestanding nano-scale actuators. Most of the previous researches in this area have exclusively focused on modeling the instability in actuators with one actuating components. While, less attention has been paid to actuators consist of two actuating components. Herein, the effect of the vdW force on the instability of freestanding actuators with two parallel actuating components is investigated. Conventional configurations including cantilever and double-clamped geometries are investigated. A continuum mechanics theory in conjunction with Euler-beam model is applied to obtain governing equations of the systems. The nonlinear governing equations of the actuators are solved using two different approaches, i.e. the modified Adomian decomposition and the finite difference method. The maximum length of the nanowire and minimum initial gap which prevents the instability is computed.
http://jacm.scu.ac.ir/article_12265_bc43214d247d95d2a78118aad27820b2.pdf
2016-08-07T11:23:20
2019-03-23T11:23:20
8
20
10.22055/jacm.2016.12265
Freestanding nanoactuators
van der Waals (vdW) force
instability
modified Adomian decomposition
Finite difference method
Rahman
Soroush
eng.soroush322@yahoo.com
true
1
Department of Engineering, Lahijan Branch, Islamic Azad University, Lahijan, Iran
Department of Engineering, Lahijan Branch, Islamic Azad University, Lahijan, Iran
Department of Engineering, Lahijan Branch, Islamic Azad University, Lahijan, Iran
LEAD_AUTHOR
Alireza
Yekrangi
yekrangi_ali@yahoo.com
true
2
Department of Engineering, Ramsar Branch, Islamic Azad University, Ramsar, Iran
Department of Engineering, Ramsar Branch, Islamic Azad University, Ramsar, Iran
Department of Engineering, Ramsar Branch, Islamic Azad University, Ramsar, Iran
AUTHOR
References
1
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2
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[5]. R. C. Batra, M. Porfiri and D. Spinello. Review of modeling electrostatically actuated microelectromechanical systems. Smart Mater. Struct., 16, pp R23-R31, 2004.
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[6] Abadian, N., Gheisari, R., Keivani, M., Kanani, A., Mokhtari, J., Rach, R., & Abadyan, M. Effect of the centrifugal force on the electromechanical instability of U-shaped and double-sided sensors made of cylindrical nanowires. Journal of the Brazilian Society of Mechanical Sciences and Engineering, pp. 1-20, 2016.
7
[7] Keivani, M., Kanani, A., Mardaneh, M. R., Mokhtari, J., Abadyan, N., & Abadyan, M. Influence of Accelerating Force on the Electromechanical Instability of Paddle-Type and Double-Sided Sensors Made of Nanowires. International Journal of Applied Mechanics, 8(01), 1650011, 2016.
8
[8] Keivani, M., Khorsandi, J., Mokhtari, J., Kanani, A., Abadian, N., & Abadyan, M. Pull-in instability of paddle-type and double-sided NEMS sensors under the accelerating force. Acta Astronautica, 119, pp. 196-206, 2016.
9
[9] Keivani, M., Mokhtari, J., Kanani, A., Abadian, N., Rach, R., & Abadyan, M. A size-dependent model for instability analysis of paddle-type and double-sided NEMS measurement sensors in the presence of centrifugal force. Mechanics of Advanced Materials and Structures, (just-accepted), 1-40, 2016.
10
[10]. A. Noghrehabadi, Y. T. Beni, A. Koochi, A. S. Kazemi, A. Yekrangi, M. Abadyan and et al. Closed-form Approximations of the Pull-in Parameters and Stress Field of Electrostatic Cantilever Nano-actuators Considering van der Waals Attraction. Procedia Engineering, 10, 3750-3756, 2011.
11
[11]. A. Koochi, N. fazli, R. Rach and M. Abadyan. Modeling the pull-in instability of the CNT-based probe/actuator under the Coulomb force and the van der Waals attraction. Latin American Journal of solids and structures. 11, 1315-1328, 2014.
12
[12]. A. Koochi, H. Hosseini-Toudeshky, H. R. Ovesy and M. Abadyan. Modeling the Influence Of Surface Effect On Instability Of Nano-Cantilever In Presence Of Van Der Waals Force. International Journal of Structural Stability and Dynamics, 13(04), 1250072, 2013.
13
[13] Farrokhabadi, A., Mokhtari, J., Koochi, A., & Abadyan, M. A theoretical model for investigating the effect of vacuum fluctuations on the electromechanical stability of nanotweezers. Indian Journal of Physics,89(6), 599-609, 2015.
14
[14]. A. Farrokhabadi, R. Rach and M. Abadyan. Modeling the static response and pull-in instability of CNT nanotweezers under the Coulomb and van der Waals attractions. Physica E: Low-dimensional Systems and Nanostructures, 53, 137-145, 2013.
15
[15]. Mokhtari, J., Farrokhabadi, A., Rach, R., & Abadyan, M. Theoretical modeling of the effect of Casimir attraction on the electrostatic instability of nanowire-fabricated actuators. Physica E: Low-dimensional Systems and Nanostructures, 68, 149-158, 2015.
16
[16]. J. Duan, R. Rach and A. Wazwaz. Solution of the model of beam-type micro- and nano-scale electrostatic actuators by a new modified Adomian decomposition method for nonlinear boundary value problems. International Journal of Non-Linear Mechanics, 49, 159-169, 2013.
17
[17]. L. Zhang, S. V. Golod, E. Deckardt, V. Prinz and D. Grützmacher. Free-standing Si/SiGe micro- and nano-objects. Physica E, 23(3-4), 280-284, 2004.
18
[18]. A. Koochi, A. Kazemi and M. Abadyan. Simulating Deflection and Determining Stable Length of Freestanding Carbon Nanotube Probe/Sensor In The Vicinity Of Graphene Layers Using A Nanoscale Continuum Model. Nano Vol. 6, No. 5, 419–429, 2011.
19
[19]. A. Koochi, A. Kazemi, A. Noghrehabadi, A. Yekrangi and M. Abadyan. New approach to model the buckling and stable length of multi walled carbon nanotube probes near graphite sheets. Materials & Design, 32(5), 2949-2955, 2012.
20
[20]. W. M. van Spengen, R. Puers and I. DeWolf. A physical model to predict Stiction in MEMS. J. Micromech. Microeng., 12, 702-713, 2002.
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[21]. J. M. Dequesnes, S. V. Rotkin and N. R. Aluru. Calculation of pull-in voltages for carbon-nanotube-based nanoelectromechanical switches. Nanotechnology, 13, 120-131, 2002.
22
[22]. S. V. Rotkin, Microfabricated Systems and MEMS VI: Proceedings of the International Symposium, Hesketh P. J., Ang S. S., Davidson J. L., Hughes H. G. and Misra D. Ed, Electrochemical Society Inc., Penningtone, New Jersey, USA, 90, 2002.
23
[23]. W. H. Lin and Y. P. Zhao. Dynamics behavior of nanoscale electrostatic actuators. Chin. Phys. Lett., 20, 2070-2073, 2003.
24
[24]. H. M. Sedighi and K.H. Shirazi. Dynamic pull-in instability of double-sided actuated nano-torsional switches. Acta Mechanica Solida Sinica, 2013.
25
[25]. Y. Fu, J. Zhang and L. Wan. Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS). Current Applied Physics, 11, 482-485, 2011.
26
[26]. C. Ke. Resonant pull-in of a double-sided driven nanotube-based electromechanical resonator. Journal of Applied Physics, 105, 024301, 2009.
27
[27]. A. Farrokhabadi, A. Koochi and M. Abadyan. Modeling the instability of CNT tweezers using a continuum model. Microsyst Technol 20, 291–302, 2014.
28
[28]. J.N. Israelachvili, Intermolecular and Surface Forces, 3rd ed.; Elsevier: Amsterdam, The Netherlands, Chapter 13, 2011.
29
[29]. H. M. Sedighi Size-dependent dynamic pull-in instability of vibrating electrically actuated micro-beams based on the strain gradient elasticity theory, Acta Astronautica, 2014.
30
[30]. R. Soroush, A. Koochi, A. S. Kazemi, A. Noghrehabadi, H. Haddadpour and M. Abadyan. Investigating the effect of Casimir and van der Waals attractions on the electrostatic pull-in instability of nanoactuators. Phys. Scripta. 82, 045801, 2010.
31
[31]. R. Rach. A convenient computational form for the Adomian polynomials. Journal of Mathematical Analysis and Applications, 102, 415-419, 1984.
32
[32]. R. Rach, A bibliography of the theory and applications of the Adomian decomposition method, 1961-2011. Kybernetes, 41(7,8), 1087-1148, 2012.
33
[33]. J. Abdi, A. Koochi, A. S. Kazemi and M. Abadyan. Modeling the Effects of Size Dependency and Dispersion Forces on the Pull-In Instability of Electrostatic Cantilever NEMS Using Modified Couple Stress Theory. Smart Materials and Structures, 20, 055011, 2011.
34
ORIGINAL_ARTICLE
Jeffery Hamel Flow of a non-Newtonian Fluid
This paper presents the Jeffery Hamel flow of a non-Newtonian fluid namely Casson fluid. Suitable similarity transform is applied to reduce governing nonlinear partial differential equations to a much simpler ordinary differential equation. Variation of Parameters Method (VPM) is then employed to solve resulting equation. Same problem is solved numerical by using Runge-Kutta order 4 method. A comparison between both the solutions is carried out to check the efficiency of VPM. Effects of emerging parameters are demonstrated both for diverging and converging channels using graphical simulation.
http://jacm.scu.ac.ir/article_12266_b96694dbc7861d9aa77036add0ac91a5.pdf
2016-08-08T11:23:20
2019-03-23T11:23:20
21
28
10.22055/jacm.2016.12266
Jeffery-Hamel flow
Casson fluid
Variation of Parameters Method (VPM)
converging and diverging channels
Runge-Kutta order 4 method
Umar
Khan
h.msedighi@scu.ac.ir
true
1
Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan
Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan
Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan
AUTHOR
Naveed
Ahmed
hmsedighi@yahoo.com
true
2
Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan
Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan
Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan
AUTHOR
Waseem
Sikandar
hmsedighi@hotmail.com
true
3
Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan
Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan
Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan
AUTHOR
Syed Tauseef
Mohyud-Din
syedtauseefs@hotmail.com
true
4
HITEC University Taxila Cantt Pakistan
HITEC University Taxila Cantt Pakistan
HITEC University Taxila Cantt Pakistan
LEAD_AUTHOR
[1] G. Hamel, Spiralförmige Bewgungen Zäher Flüssigkeiten, Jahresber. Deutsch.Math. Verein., 25 (1916) 34-60.
1
G. Hamel, SpiralförmigeBewgungenZäherFlüssigkeiten, Jahresber. Deutsch.Math. Verein., 25 (1916) 34-60.
2
[2] L. Rosenhead, The steady two-dimensional radial flow of viscous fluid between two inclined plane walls, Proc. R. Soc. A 175 (1940) 436-467.
3
[3] K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1967.
4
[4] R. Sadri., Channel entrance flow, PhD thesis, Dept. Mechanical Engineering, the University of Western Ontario, 1997.
5
[5] M. Hamadiche, J. Scott,D. Jeandel.Temporal stability of Jeffery–Hamel flow. J Fluid Mech, 268 (1994) 71–88.
6
[6] E. Fraenkel, Laminar flow in symmetrical channels with slightly curved walls. I:On the Jeffery-Hamel solutions for flow between plane walls, Proc. R. Soc. Lond, A 267 (1962) 119-138.
7
[7] Hermann Schlichting, Boundary-layer Theory, McGraw-Hill Press, New York, 2000.
8
[8] E. W. Mrill, A. M. Benis, E. R. Gilliland, , T. K. Sherwood, E. W. Salzman, Pressure flow relations of human blood hollow fibers at low flow rates. Journal of Applied Physiology, 20 (1965), 954–967.
9
[9] D. A. McDonald, Blood Flows in Arteries, 2nd ed., Arnold, London (1974).
10
[10] S. Nadeem, R.Ul Haq, C. Lee, MHD flow of a Casson fluid over an exponentially shrinking sheet, Scientia Iranica, 19 (2012) 1150-1553.
11
[11] N. Ahmed, U. Khan, S. I. U. Khan, Y. X. Jun, Z. A. Zaidi, S. T. Mohyud-Din, Magneto hydrodynamic (MHD) Squeezing Flow of a Casson Fluid between Parallel Disks, International Journal of Physical Sciences, 8 (2013) 1788-1799.
12
[12] S. Nadeem, R. Ul Haq, N. S. Akbar, Z. H. Khan, MHD three-dimensional Casson fluid flow past a porous linearly stretching sheet, Alexandria Engineering Journal, In Press.
13
[13] S. Abbasbandy, A new application of He's variational iteration method for quadratic Riccati differential equation by using Adomian's polynomials, Journal of Computational and Applied Mathematics 207 (2007), 59-63.
14
[14] S. Abbasbandy, Numerical solutions of nonlinear Klein-Gordon equation by variational iteration method, International Journal of Numerical Methods in Engineering, 70 (2007), 876-881.
15
[15] M. A. Abdou and A. A. Soliman, Variational iteration method for solving Burger's and coupled Burger's equations, Journal of Computational and Applied Mathematics 181 (2005), 245-251.
16
[16] M. A. Noor and S. T. Mohyud-Din, Variational iteration technique for solving higher order boundary value problems, Applied Mathematics and Computation, 189 (2007) 1929—1942.
17
[17] M. A. Abdou and A. A. Soliman, New applications of variational iteration method, Physica D, 211 (1-2) (2005), 1-8.
18
[18] M. Asadullah, U. Khan, N. Ahmed, R. Manzoor, S.T. Mohyud-Din, International Journal of Modern Mathematical Sciences, 6 (2013), 92-106.
19
[19] R. Ellahi, The effects of MHD and temperature dependent viscosity on the flow of non-Newtonian nanofluid in a pipe: Analytical solutions, Applied Mathematical Modeling, 37 (2013) 1451-1467.
20
[20] R. Ellahi, M. Raza, K. Vafai, Series solutions of non-Newtonian nanofluids with Reynolds’ model and Vogel’s model by means of the homotopy analysis method, Mathematical and Computer Modelling, 55 (2012) 1876–1891.
21
[21] M. A. Noor, S. T. Mohyud-Din and A. Waheed, Variation of parameters method for solving fifth-order boundary value problems. Applied Mathematics and Information Sciences, 2 (2008), 135 -141.
22
[22] S. T. Mohyud-Din, M. A. Noor, A. Waheed, Variation of parameter method for solving sixth-order boundary value problems, Communication of the Korean Mathematical Society, (2009), 24, 605-615.
23
[23] S. T. Mohyud-Din, M. A. Noor, A. Waheed, Variation of parameter method for initial and boundary value problems, World Applied Sciences Journal, 11 (2010) 622-639.
24
[24] T. Mohyud-Din, M. A. Noor, A. Waheed, Modified Variation of Parameters Method for Second-order Integro-differential Equations and Coupled Systems, World Applied Sciences Journal, 6 (2009) 1139-1146.
25
[25] U. Khan, N. Ahmed, Z. A. Zaidi, S. U. Jan, S. T. Mohyud-Din, On Jeffery-Hamel Flows, International Journal of Modern Mathematical Sciences, 7 (2013), 236-247.
26
[26] U. Khan, N. Ahmed, Z. A. Zaidi, M. Asadullah, S. T. Mohyud-Din, MHD Squeezing Flow between Two Infinite Plates, Ain Shams Engineering Journal, In Press.
27
ORIGINAL_ARTICLE
An Analytical Technique for Solving Nonlinear Oscillators of the Motion of a Rigid Rod Rocking Bock and Tapered Beams
In this paper, a new analytical approach has been presented for solving strongly nonlinear oscillator problems. Iteration perturbation method leads us to high accurate solution. Two different high nonlinear examples are also presented to show the application and accuracy of the presented method. The results are compared with analytical methods and with the numerical solution using Runge-Kutta method in different figures. It has been shown that the iteration perturbation approach doesn't need any small perturbation and is accurate for nonlinear oscillator equations.
http://jacm.scu.ac.ir/article_12268_6b4ae7952064eecdfe9ab67bbef9e769.pdf
2016-08-01T11:23:20
2019-03-23T11:23:20
29
34
10.22055/jacm.2016.12268
Periodic solution
Nonlinear oscillators
Motion of a rigid rod rocking back
Tapered beams
Gamal
Ismail
gamalm2010@yahoo.com
true
1
Mathematics Department
Faculty of Science
Sohag University
Sohag, Egypt
Mathematics Department
Faculty of Science
Sohag University
Sohag, Egypt
Mathematics Department
Faculty of Science
Sohag University
Sohag, Egypt
LEAD_AUTHOR
[1] He, J. H., “Homotopy perturbation method for bifurcation of nonlinear problems”, International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 6, No. 2, pp. 207-208, 2005.
1
[2] Bayat, M., Pakar, I., Domairry, G., “Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: A review”, Latin American Journal of Solids and Structures, Vol. 1, pp. 1-93, 2012.
2
[3] Nofal, T. A., Ismail, G. M., Abdel-Khalek, S., “Application of homotopy perturbation method and parameter expanding method to fractional Van der Pol damped nonlinear oscillator”, Journal of Modern Physics, Vol. 4, pp. 1490-1494, 2013.
3
[4] He, J. H., “Preliminary report on the energy balance for nonlinear oscillations, Mechanics Research Communications”, Vol. 29, No. (2-3), pp. 107-111, 2002.
4
[5] Khan, Y., Mirzabeigy, A., “Improved accuracy of He’s energy balance method for analysis of conservative nonlinear oscillator”, Neural Computing and Applications, Vo. 25, pp. 889-895, 2014.
5
[6] Bayat, M., Pakar, I., Bayat, M., “High conservative nonlinear vibration equations by means of energy balance method, Earthquakes and Structures”, Vol. 11, No. 1, pp. 129-140, 2016.
6
[7] He, J. H., “Some asymptotic methods for strongly nonlinear equations”, International Journal of Modern Physics B, Vol. 20, No. 2, pp. 1141-1199, 2006.
7
[8] El-Naggar, A. M., Ismail, G. M., “Applications of He's amplitude frequency formulation to the free vibration of strongly nonlinear oscillators”, Applied Mathematical Sciences, Vol. 6, No. 42, pp. 2071-2079, 2012.
8
[9] Sedighi, H. M., Daneshmand, F., Abadyan, M., “Dynamic instability analysis of electrostatic functionally graded doubly-clamped nano-actuators”, Composite Structures, Vol. 124, pp. 55-64, 2015.
9
[10] He, J. H., “Variational iteration method-Some recent results and new interpretations”, Journal of Computational and Applied Mathematics, Vol. 207, pp. 3-17, 2007.
10
[11] Herisanu, N., Marinca, V., “A modified variational iteration method for strongly nonlinear problems”, Nonlinear Science Letters A, Vol. 1, pp. 183-192, 2010.
11
[12] Ganji, S. S., Barari, A., Ganji, D. D., “Approximate analyses of two mass-spring systems and buckling of a column”, Computers and Mathematics with Applications, Vol. 61, pp. 1088-1095, 2011.
12
[13] Sedighi, H. M., Reza, A., Zare, J., “Using parameter expansion method and min-max approach for the analytical investigation of vibrating micro-beams pre-deformed by an electric field”, Advances in Structural Engineering, Vol. 16, No. 4, pp. 693-699, 2013.
13
[14] He, J. H., “Hamiltonian approach to nonlinear oscillators”, Physics Letters A, Vol. 374, No. 23, pp. 2312-2314, 2010.
14
[15] Bayat, M., Pakar, I., Bayat, M., “On the large amplitude free vibrations of axially loaded Euler-Bernoulli beams”, Steel and Composite Structures, Vol. 14, No. 1, pp. 73-83, 2013.
15
[16] Yildirim, A., Saadatnia, Z., Askari, H., Khan, Y., KalamiYazdi, M., “Higher order approximate periodic solutions for nonlinear oscillators with the Hamiltonian approach”, Applied Mathematics Letters, Vol. 24, pp. 2042-2051, 2011.
16
[17] Bayat, M., Pakar, I., Bayat, M., “Nonlinear vibration of conservative oscillator's using analytical approaches”, Structural Engineering and Mechanics, Vol.59, No.4, pp. 671-682, 2016.
17
[18] Bayat, M., Pakar, I., “Nonlinear vibration of an electrostatically actuated microbeam”, Latin American Journal of Solids and Structures, Vol. 11, 534-544, 2014.
18
[19] Ghalambaz, M., Ghalambaz, M., Edalatifar, M., “Nonlinear oscillation of nano electromechanical resonators using energy balance method: considering the size effect and the van der Waals force”, Applied Nanoscience, Vol. 6, pp. 309-317, 2016.
19
[20] Khan, Y., Vazquez-Leal, H., Faraz, N., “An auxiliary parameter method using Adomian polynomials and Laplace transformation for nonlinear differential equations”, Applied Mathematical Modelling, Vol. 37, No. 5, pp. 2702-2708, 2013.
20
[21] Akbarzade, M., Khan, Y., “Dynamic model of large amplitude non-linear oscillations arising in the structural engineering: analytical solutions”, Mathematical and Computer Modelling, Vol. 55, No. (3-4), pp. 480-489, 2012.
21
[22] Jun-Fang., “He’s variational approach for nonlinear oscillators with high nonlinearity”, Computers and Mathematics with Applications, Vol. 58, No. (11-12), pp. 2423-2426, 2009.
22
[23] El-Naggar, A. M., Ismail, G. M., “Analytical solution of strongly nonlinear Duffing oscillators”, Alexandria Engineering Journal, Vol. 55, No. 2, pp. 1581-1585, 2016.
23
[24] El-Naggar, A. M., Ismail, G. M., “Solution of a quadratic non-Linear oscillator by elliptic homotopy averaging method”, Mathematical Sciences Letters, Vol. 4, No. 3, pp. 313-317, 2015.
24
[25] El-Naggar, A. M., Ismail, G. M., “Analytical solutions of strongly non-linear problems by the iteration perturbation method”, Journal of Scientific Research and Reports, Vol. 5, No. 4, pp. 285-294, 2015.
25
[26] Sadeghzadeh, S., Kabiri, A., “Application of higher order Hamiltonian Approach to the nonlinear vibration of micro electro mechanical systems”, Latin American Journal of Solids and Structures, Vo. 13, pp. 478-497, 2016.
26
[27] Nofal, T. A., Ismail, G. M., Mady, A. M., Abdel-Khalek, S., “Analytical and Approximate Solutions to the Fee Vibration of Strongly Nonlinear Oscillators”, Journal of Electromagnetic Analysis and Applications, Vol. 5, pp. 388-392, 2013.
27
[28] He, J. H., “An improved amplitude-frequency formulation for nonlinear oscillators”, International Journal of Nonlinear Science and Numerical Simulation, Vol. 9, pp. 211-212, 2008.
28
[29] Davodi, A. G., Gangi, D. D., Azami, R., Babazadeh, H., “Application of improved amplitude-frequency formulation to nonlinear differential equation of motion equations”, Modern Physics Letters B, Vol. 23, pp. 3427-3436, 2009.
29
[30] Zhang, Hui-Li, “Periodic solutions for some strongly nonlinear oscillators by He's energy balance method”, Computers and Mathematics with Applications, Vol. 58, pp. 2423-2426, 2009.
30
[31] Wu, B. S., Lim, C. W., He, L. H., “A new method for approximate analytical solutions to nonlinear oscillations of nonnatural systems”, Nonlinear Dynamics, Vol. 32, pp. 1-13, 2003.
31
[32] Ganji, S. S., Ganji, D. D., Babazadeh, H., Sadoughi, N., “Application of the amplitude-frequency formulation to nonlinear oscillation system of the motion of a rigid rod rocking back”, Mathematical Methods in the Applied Sciences, Vol. 33, pp. 157-166, 2010.
32
[33] Khah, H. E., Ganji, D. D., “A Study on the Motion of a Rigid Rod Rocking Back and Cubic-Quintic Duffing Oscillators by Using He’s energy balance method”, International Journal of Nonlinear Science, Vol.10, No.4, pp. 447-451, 2010.
33
[34] Ganji, D. D., S. Karimpour, S., Ganji, S. S., “Approximate analytical solutions to nonlinear oscillators of nonlinear systems using He’s energy balance method”, Progress in Electromagnetics Research M, Vol. 5, pp. 43-54, 2008.
34
[35] Hibbeler, R. C., Engineering Mechanics Dynamics, Prentice-Hall, New Jersey, 2001.
35
[36] Hoseini, S. H., Pirhodaghi, T., Ahmadian, M. T. Farrahi, G. H., “On the large amplitude free vibrations of tapered beams: an analytical approach”, Mechanics Research Communications, Vol. 36, pp. 892-897, 2009.
36
[37] Gangi, D. D., Azimi, M., “Application of max min approach and amplitude frequency formulation to nonlinear oscillation systems”, U.P.B. Sci. Bull., Series A, Vol. 74, pp. 131-140, 2012
37
[38] Gangi, D. D., Azimi, M., Mostofi, M., “Energy balance method and amplitude frequency formulation based simulation of strongly non-linear oscillators”, Indian journal of Pure and Applied Physics, Vol. 50, pp. 670-675, 2012.
38
ORIGINAL_ARTICLE
Periodic Solutions of the Duffing Harmonic Oscillator by He's Energy Balance Method
Duffing harmonic oscillator is a common model for nonlinear phenomena in science and engineering. This paper presents He´s Energy Balance Method (EBM) for solving nonlinear differential equations. Two strong nonlinear cases have been studied analytically. Analytical results of the EBM are compared with the solutions obtained by using He´s Frequency Amplitude Formulation (FAF) and numerical solutions using Runge-Kutta method. The results show the presented method is potentially to solve high nonlinear oscillator equations.
http://jacm.scu.ac.ir/article_12269_93df035a219e7c855ade51e2844c40bc.pdf
2016-08-10T11:23:20
2019-03-23T11:23:20
35
41
10.22055/jacm.2016.12269
Energy balance method
Nonlinear oscillator
Duffing-harmonic oscillator
Periodic solutions
A. M.
El-Naggar
hmsedighi@yahoo.com
true
1
Department of Mathematics, Faculty of Science, Benha University, Egypt
Department of Mathematics, Faculty of Science, Benha University, Egypt
Department of Mathematics, Faculty of Science, Benha University, Egypt
AUTHOR
Gamal
Ismail
gamalm2010@yahoo.com
true
2
Mathematics Department
Faculty of Science
Sohag University
Sohag, Egypt
Mathematics Department
Faculty of Science
Sohag University
Sohag, Egypt
Mathematics Department
Faculty of Science
Sohag University
Sohag, Egypt
LEAD_AUTHOR
[1] He, J. H., “Variational iteration method: a kind of nonlinear analytical technique: some examples”, International Journal of Non-Linear Mechanics, Vol. 34, No. 4, pp. 699-708, 1999.
1
[2] He, J. H., “Variational approach for nonlinear oscillators”, Chaos Solitons and Fractals, Vol. 34, No. 5, pp. 1430-1439, 2007.
2
[3] He, J. H., “Variational iteration method - some recent results and new interpretations”, Journal of Computational and Applied Mathematics, Vol. 207, No. 1, pp. 3-17, 2007.
3
[4] He, J. H., Wu, X. H., “Construction of solitary solution and compaction-like solution by variational iteration method”, Chaos Solitons and Fractals, Vol. 29, No. 1, pp. 108-113, 2006.
4
[5] Mickens, R. E., “Mathematical and numerical study of the Duffing-harmonic oscillator”, Journal of Sound and Vibration, Vol. 244, No. 3, pp. 563-567, 2000.
5
[6] Hu, H., Tang, J. H., “Solution of a Duffing-harmonic oscillator by the method of harmonic balance”, Journal of Sound and Vibration, Vol. 294, No. 3, pp. 637-639, 2006.
6
[7] Lim, C. W., Wu, B. S., “A new analytical approach to the Duffing-harmonic oscillator”, Physics Letters A, Vol. 311, No. 4-5, pp. 365-373, 2003.
7
[8] Guo, Z., Leung, A. Y. T., Yang, H. X., “Iterative homotopy harmonic balancing approach for conservative oscillator with strong odd-nonlinearity”, Applied Mathematical Modelling, Vol. 35, No. 4, pp. 1717-1728, 2011.
8
[9] Leung, A. Y. T., Guo, Z., “Residue harmonic balance approach to limit cycles of non-linear jerk equations”, International Journal of Non-Linear Mechanics, Vol. 46, No. 6, pp. 898-906, 2011.
9
[10] Ozis, T., Yildirim, A., “Determination of the frequency-amplitude relation for a Duffing harmonic oscillator by the energy balance method”, Computers and Mathematics with Applications, Vol. 54, No. 7-8, pp. 1184-1187.
10
[11] Ganji, D. D., Esmaeilpour, M., Soleimani, M., “Approximate solutions to Van der Pol damped nonlinear oscillators by means of He's energy balance method”, International Journal of Computer Mathematics, Vol. 87, No. 9, pp. 2014-2023, 2010.
11
[12] Yazdi, M. K., Khan, Y., Madani, M., Askari, H., Saadatnia, Z., Yildirim, A., “Analytical solutions for autonomous conservative nonlinear oscillator”, International Journal Nonlinear Sciences and Numerical Simulation, Vol. 11, No. 11, pp. 979-984, 2010.
12
[13] Yildirim, A., Saadatinia, Z., Askari, H., Khan, Y., Yazdi, M. K., “Higher order approximate periodic solutions for nonlinear oscillators with the Hamiltonian approach”, Applied Mathematics Letters, Vol. 24, No. 12, pp. 2042-2051, 2011.
13
[14] Khan, Y., Wu, Q., “Homotopy perturbation transform method for nonlinear equations using He's polynomials”, Computers and Mathematics with Applications, Vol. 61, No. 8, pp. 1963-1967, 2011.
14
[15] Belendez, A., Gimeno, E., Alvarez, M. L., Mendez, D. I., Hernandez, A., “Application of a modified rational harmonic balance method for a class of strongly nonlinear oscillators”, Physics Letters A, Vol. 372, No. 39, pp. 6047-6052, 2008.
15
[16] Belendez, A., Mendez, D. I., Fernandez, E., Marini, S., Pascual, I., “An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method”, Physics Letters A, Vol. 373, No. 32, pp. 2805-2809, 2009.
16
[17] Sedighi, H. M., Shirazi, K. H., “Vibrations of micro-beams actuated by an electric field via Parameter Expansion Method”, Acta Astronautica, Vol. 85, pp. 19-24. 2013.
17
[18] Sedighi, H. M., Shirazi, K. H., Zare, J., “Novel equivalent function for deadzone nonlinearity: applied to analytical solution of beam vibration using He’s Parameter Expanding Method”, Latin American Journal of Solids and Structures, Vol. 9, pp. 443-451, 2012.
18
[19] He, J. H., “Solution of nonlinear equations by an ancient Chinese algorithm”, Applied Mathematics and Computation, Vol. 151, No. 1, pp. 293-297, 2004.
19
[20] El-Naggar., A. M., Ismail, G. M., “Applications of He’s amplitude-frequency formulation to the free vibration of strongly nonlinear oscillators”, Applied Mathematical Sciences, Vol. 6, No. 42, pp. 2071-2079, 2012.
20
[21] He, J. H., “Variational iteration method a kind of non-linear analytical technique: some examples”, International Journal of Non-Linear Mechanics, Vol. 34, No. 4, pp. 699-708, 1999.
21
[22] He, J. H., “Variational approach for nonlinear oscillators”, Chaos Solitons and Fractals, Vol. 34, No. 5, pp. 1430-1439, 2007
22
[23] Ozis, T., Yildrm, A., “A study of nonlinear oscillators with force by He's variational iteration method”, Journal of Sound and Vibration, Vol. 306, No. 1-2, pp. 372-376, 2007.
23
[24] Sedighi, H. M., Shirazi, K. H., Noghrehabadi, A., “Application of Recent Powerful Analytical Approaches on the Non-Linear Vibration of Cantilever Beams”, International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 13, No. 7-8, pp. 487-494, 2012.
24
[25] Khan, Y., Vazquez-Leal, H., Faraz, N., “An auxiliary parameter method using Adomian polynomials and Laplace transformation for nonlinear differential equations”, Applied Mathematical Modelling, Vol. 37, No. 5, pp. 2702-2708, 2013.
25
[26] Akbarzade, M., Khan, Y., “Dynamic model of large amplitude non-linear oscillations arising in the structural engineering: analytical solutions”, Mathematical and Computer Modelling, Vol. 55, No. 3-4, pp. 480-489, 2012.
26
[27] Ganji, S. S., Barari, A., Karimpour, S., Domairry, G., Motion of a rigid rod rocking back and forth and cubic-quintic Duffing oscillators”, Journal of Theoretical and Applied Mechanics, Vol. 50, No. 1, pp. 215-229, 2012.
27
[28] El-Naggar, A. M., Ismail, G. M., “Analytical solution of strongly nonlinear Duffing oscillators”, Alexandria Engineering Journal, Vol. 55, No. 2, pp. 1581-1585, 2016.
28
[29] El-Naggar, A. M., Ismail, G. M., “Solution of a quadratic non-Linear oscillator by elliptic homotopy averaging method”, Mathematical Sciences Letters, Vol. 4, No. 3, pp. 313-317, 2015.
29
[30] El-Naggar, A. M., Ismail, G. M., “Analytical solutions of strongly non-linear problems by the iteration perturbation method”, Journal of Scientific Research and Reports, Vol. 5, No. 4, pp. 285-294, 2015.
30
[31] Fan, J., He's frequency-amplitude formulation for the Duffing harmonic oscillator”, Computers and Mathematics with Applications, Vol. 58, No. 11-12, pp. 2473-2476, 2009.
31
ORIGINAL_ARTICLE
Saint-Venant torsion of non-homogeneous anisotropic bars
The BEM is applied to the solution of the torsion problem of non-homogeneous anisotropic non-circular prismatic bars. The problem is formulated in terms of the warping function. This formulation leads to a second order partial differential equation with variable coefficients, subjected to a generalized Neumann type boundary condition. The problem is solved using the Analog Equation Method (AEM). According to this method, the governing equation is replaced by a Poisson’s equation subjected to a fictitious source under the same boundary condition. The fictitious load is established using the Boundary Element Method (BEM) after expanding it into a finite series of radial basis functions. The method has all the advantages of the pure BEM, since the discretization and integration are limited only on to the boundary. Numerical examples are presented which illustrate the efficiency and accuracy of the method.
http://jacm.scu.ac.ir/article_12270_5ba4dad84d8c4e4072ae8097ad43c3dc.pdf
2016-08-20T11:23:20
2019-03-23T11:23:20
42
53
10.22055/jacm.2016.12270
Anisotropic materials
Non-homogeneous media
Elasticity
Bars
Torsion
John
Katsikadelis
jkats@central.ntua.gr
true
1
School of Civil Engineering
National Technical University of Athens (NTUA)
Zografou Campus
Athens 15773
Greece
School of Civil Engineering
National Technical University of Athens (NTUA)
Zografou Campus
Athens 15773
Greece
School of Civil Engineering
National Technical University of Athens (NTUA)
Zografou Campus
Athens 15773
Greece
AUTHOR
George
Tsiatas
gtsiatas@gmail.com
true
2
Department of Mathematics, University of Patras, Rio
Department of Mathematics, University of Patras, Rio
Department of Mathematics, University of Patras, Rio
LEAD_AUTHOR
[1] Lekhnitskii, S.G., Theory of Elasticity of an Anisotropic Elastic Body, Holden-Day Inc., San Francisco, Calif., 1963.
1
[2] Muskhelishvili, N.I., Some Basic Problems of the Mathematical Theory of Elasticity, P. Noordhoff, Groningen, Holland, 1953.
2
[3] Sokolnikoff, I.S., Mathematical Theory of Elasticity, McGraw-Hill Book Co., Inc., New York, N.Y., 1956.
3
[4] Timoshenko, S. and Goodier, J.N., Theory of Elasticity, McGraw-Hill Book Co., Inc., New York, N.Y., 1970.
4
[5] Martynovich, B.T. and Martynovich T.L., “Use of Integral Equations in the Solution of Problems of Torsion of Rectilinear-Anisotropic Rods”, Izv. AN SSR, Mekhanika Tverdogo Tela, Vol. 19, pp. 112‑118, 1984.
5
[6] Shaw, F.S., “The Torsion of Solid and Hollow Prisms in the Elastic and Plastic Range by Relaxation Methods”, Report No. ACA-11, Australian Council for Aeronautics, 1944.
6
[7] Ely, J.F. and Zienkiewicz, O.C., “Torsion of Compound Bars-A Relaxation Solution”, International Journal of Mechanical Science, Vol. 1, pp. 356‑365, 1960.
7
[8] Zienkiewicz, O.C. and Cheung, Y.K., “Finite Elements in the Solution of Field Problems”, The Engineer, Vol. 220, pp. 507‑510, 1965.
8
[9] Hermann, L.R., “Elastic Torsional Analysis of Irregular Shapes”, Journal of the Engineering Mechanics Division, ASCE, Vol. 91, No. EM6, Proc. Paper 4562, pp. 11‑19, 1965.
9
[10] Krahula, J.L. and Lauterbach, G.F., “A Finite Element Solution for Saint-Venant Torsion”, American Institute of Aeronautics and Astronautics Journal, Vol. 7, pp. 2200‑2203, 1969.
10
[11] Valliappan, S. and Pulmano, V.A., “Torsion of Nonhomogeneous Anisotropic Bars”, Journal of the Structural Division, Proc. of the ASCE, Vol. 100, No. ST1, pp. 286‑295, 1974.
11
[12] Jaswon, M.A. and Ponter, A.R.S., “An Integral Equation Solution of the Torsion Problem”, Proceedings of the Royal Society (London) A, Vol. 273, pp. 237‑246, 1963.
12
[13] Ponter, A.R.S., “An Integral Equation Solution of the Inhomogeneous Torsion Problem”, SIAM Journal of Applied Mathematics, Vol. 14, pp. 819‑830, 1966.
13
[14] Katsikadelis, J.T. and Sapountzakis, E.J., “Torsion of Composite Bars by Boundary Element Method”, Journal of the Engineering Mechanics Division, ASCE, Vol. 111, pp. 1197‑1210, 1985.
14
[15] Chou, S.I. and Mohr, J.A., “Boundary Integral Method for Torsion of Composite Shafts”, Res. Mechanica, Vol. 29, pp. 41‑56, 1990.
15
[16] Hromadka II, T.V. and Pardoen, G., “Application of the CVBEM to nonuniform St. Venant Torsion”, Computer Methods in Applied Mechanics in Engineering, Vol. 53, pp. 149‑161, 1985.
16
[17] Chou S.I. and Shams-Ahmadi, M., “Complex Variable Boundary Element Method for Torsion of Hollow Shafts, Nuclear Engineering Design”, Vol. 136, pp. 255‑263, 1992.
17
[18] Shams-Ahmadi, M. and Chou, S.I., “Complex Variable Boundary Element Method for Torsion of Composite Shafts”, International Journal for Numerical Methods in Engineering, Vol. 40, pp. 1165‑1179, 1997.
18
[19] Dumir, P.C. and Kumar, R., “Complex Variable Boundary Element Method for Torsion of Anisotropic Bars”, Applied Mathematics Modelling, Vol. 17, pp. 80‑88, 1993.
19
[20] Sapountzakis, E.J., “Solution of non-uniform torsion of bars by an integral equation method”, Computers and Structures, Vol. 77, pp. 659‑667, 2000.
20
[21] Sapountzakis, E.J., “Nonuniform Torsion of multi-material composite bars by the boundary element Method”, Computers and Structures, Vol. 79, pp. 2805‑2816, 2001.
21
[22] Ecsedi, I., “Some analytical solutions for Saint-Venant torsion of non-homogeneous cylindrical bars”, European Journal of Mechanics A/Solids, Vol. 28, pp. 985–990, 2009.
22
[23] Rongqiao, X., Jiansheng, H. and Weiqiu, C., “Saint-Venant torsion of orthotropic bars with inhomogeneous rectangular cross section”, Composite Structures, Vol. 92, pp. 1449–1457, 2010.
23
[24] Ecsedi, I., “Some analytical solutions for Saint-Venant torsion of non-homogeneous anisotropic cylindrical bars”, Mechanics Research Communications, Vol. 52, pp. 95– 100, 2013.
24
[25] Katsikadelis, J.T., “The analog equation method. A boundary-only integral equation method for nonlinear static and dynamic problems in general bodies”, Theoretical and Applied Mechanics, Vol. 27, pp. 13–38, 2002.
25
[26] Katsikadelis, J.T., The Boundary Element Method for Engineers and Scientists, 2nd Edition, Academic Press, Elsevier, UK, 2016.
26
[27] Tsiatas, G.C. and Katsikadelis, J.T., “Large deflection analysis of elastic space membranes”, International Journal for Numerical Methods in Engineering, Vol. 65, pp. 264‑294, 2006.
27
[28] Golberg, M.A., Chen, C.S. and Karur, S.P., “Improved multiquadric approximation for partial differential equations”, Engineering Analysis with Boundary Elements, Vol. 18, pp. 9‑17, 1996.
28