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http://dx.doi.org/10.12989/scs.2021.38.5.583

Static analysis of cutout microstructures incorporating the microstructure and surface effects  

Alazwari, Mashhour A. (Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University)
Abdelrahman, Alaa A. (Mechanical Design & Production Dept., Faculty of Engineering, Zagazig University)
Wagih, Ahmed (Mechanical Design & Production Dept., Faculty of Engineering, Zagazig University)
Eltaher, Mohamed A. (Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University)
Abd-El-Mottaleb, Hanaa E. (Structural Engineering Department, Faculty of Engineering, Zagazig University)
Publication Information
Steel and Composite Structures / v.38, no.5, 2021 , pp. 583-597 More about this Journal
Abstract
This article develops a nonclassical model to analyze bending response of squared perforated microbeams considering the coupled effect of microstructure and surface stress under different loading and boundary conditions, those are not be studied before. The corresponding material and geometrical characteristics of regularly squared perforated beams relative to fully filled beam are obtained analytically. The modified couple stress and the modified Gurtin-Murdoch surface elasticity models are adopted to incorporate the microstructure as well as the surface energy effects. The differential equations of equilibrium including the Poisson's effect are derived based on minimum potential energy. Exact closed form solution is obtained for bending behavior of the proposed model considering the classical and nonclassical boundary conditions for both uniformly distributed and concentrated loads. The proposed model is verified with results available in the literature. Influences of the microstructure length scale parameter, surface energy, beam thickness, boundary and loading conditions on the bending behavior of perforated microbeams are investigated. It is observed that microstructure and surface parameters are vital in investigation of the bending behavior of perforated microbeams. The obtained results are supportive for the design, analysis and manufacturing of perforated nanobeams that commonly used in nanoactuators, nanoswitches, MEMS and NEMS systems.
Keywords
equivalent geometrical model; microstructure and surface effects; perforated beam; bending behavior; classical and nonclassical boundary conditions; exact closed form;
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1 Arefi, M., Mohammad-Rezaei Bidgoli, E. and Civalek, O. (2020), "Bending response of FG composite doubly curved nanoshells with thickness stretching via higher-order sinusoidal shear theory", Mech. Based Des. Struct. Machines, 1-29. https://doi.org/10.1080/15397734.2020.1777157.   DOI
2 Asrari, R., Ebrahimi, F., Kheirikhah, M.M. and Safari, K.H. (2020), "Buckling analysis of heterogeneous magneto-electro-thermo-elastic cylindrical nanoshells based on nonlocal strain gradient elasticity theory", Mech. Based Des. Struct. Machines, 1-24. https://doi.org/10.1080/15397734.2020.1728545.   DOI
3 Babaei, H. and Eslami, M.R. (2020), "Nonlinear bending analysis of size-dependent FG porous microtubes in thermal environment based on modified couple stress theory", Mech. Based Des. Struct. Machines, 1-22. https://doi.org/10.1080/15397734.2020.1784202.   DOI
4 Civalek, O. (2020), "Vibration of functionally graded carbon nanotube reinforced quadrilateral plates using geometric transformation discrete singular convolution method", Int. J. Numer. Method. Eng., 121(5), 990-1019. https://doi.org/10.1002/nme.6254.   DOI
5 Civalek, O. and Jalaei, M.H. (2020), "Buckling of carbon nanotube (CNT)-reinforced composite skew plates by the discrete singular convolution method", Acta Mechanica, 231, 2565-2587. https://doi.org/10.1007/s00707-020-02653-3.   DOI
6 Daikh, A.A., Houari, M.S.A. and Eltaher, M.A. (2020a), "A novel nonlocal strain gradient Quasi-3D bending analysis of sigmoid functionally graded sandwich nanoplates", Compos. Struct., 113347. https://doi.org/10.1016/j.compstruct.2020.113347.   DOI
7 Daikh, A.A., Drai, A., Houari, M.S.A. and Eltaher, M.A. (2020b), "Static analysis of multilayer nonlocal strain gradient nanobeam reinforced by carbon nanotubes", Steel Compos. Struct., 36(6), 643-656. https://doi.org/10.12989/scs.2020.36.6.643.   DOI
8 Daw, M.S. and Baskes, M.I. (1984), "Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals", Physical Review B, 29(12), 6443. https://doi.org/10.1103/PhysRevB.29.6443.   DOI
9 Demir, C., Mercan, K., Numanoglu, H.M. and Civalek, O. (2018), "Bending response of nanobeams resting on elastic foundation", J. Appl. Comput. Mech., 4(2), 105-114. https://doi.org/10.22055/JACM.2017.22594.1137.   DOI
10 Eltaher, M., Abdraboh, A. and Almitani, K. (2018), "Resonance frequencies of size dependent perforated nonlocal nanobeam", Microsyst. Technol., 24(9): 3925-3937. https://doi.org/10.1007/s00542-018-3910-6.   DOI
11 Eltaher, M., Hamed, M., Sadoun, A. and Mansour, A. (2014), "Mechanical analysis of higher order gradient nanobeams", Appl. Math. Comput., 229, 260-272. https://doi.org/10.1016/j.amc.2013.12.076.   DOI
12 Eltaher, M., Kabeel, A., Almitani, K. and Abdraboh, A. (2018), "Static bending and buckling of perforated nonlocal size-dependent nanobeams", Microsyst. Technol., 24(12), 4881-4893. https://doi.org/10.1007/s00542-018-3905-3.   DOI
13 Akbas, S.D. (2016a), "Forced vibration analysis of viscoelastic nanobeams embedded in an elastic medium", Smart Struct. Syst., 18(6), 1125-1143. https://doi.org/10.12989/sss.2016.18.6.1125.   DOI
14 Akbas, S.D. (2018a), "Bending of a cracked functionally graded nanobeam", Adv. Nano Res., 6(3), 219. http://dx.doi.org/10.12989/anr.2018.6.3.219.   DOI
15 Akbas, S.D. (2016b), "Analytical solutions for static bending of edge cracked micro beams", Struct. Eng. Mech., 59(3), 579-599. http://dx.doi.org/10.12989/sem.2016.59.3.579.   DOI
16 Akbas, S.D. (2017a), "Free vibration of edge cracked functionally graded microscale beams based on the modified couple stress theory", Int. J. Struct. Stab. Dynam., 17(03), 1750033. https://doi.org/10.1142/S021945541750033X.   DOI
17 Akbas, S.D. (2017b), "Forced vibration analysis of functionally graded nanobeams", Int. J. Appl. Mech., 9(7), 1750100. https://doi.org/10.1142/S1758825117501009.   DOI
18 Akbas, S.D. (2018b), "Forced vibration analysis of cracked nanobeams", J. Braz. Soc. Mech. Sci. Eng., 40(8), 392. https://doi.org/10.1007/s40430-018-1315-1.   DOI
19 Akbas, S.D. (2018c), "Forced vibration analysis of cracked functionally graded microbeams", Adv. Nano Res., 6(1), 39. http://dx.doi.org/10.12989/anr.2018.6.1.039.   DOI
20 Akbas, S.D. (2019a), "Axially Forced Vibration Analysis of Cracked a Nanorod", J. Comput. Appl. Mech., 50(1), 63-68. https://doi.org/10.22059/jcamech.2019.281285.392.   DOI
21 Akbas, S.D. (2019b), "Axially Forced Vibration Analysis of Cracked a Nanorod", J. Comput. Appl. Mech., 50(1), 63-68. https://doi.org/10.22059/jcamech.2019.281285.392.   DOI
22 Akbas, S.D. (2020), "Modal analysis of viscoelastic nanorods under an axially harmonic load", Adv. Nano Res., 8(4), 277. https://doi.org/10.12989/anr.2020.8.4.277.   DOI
23 Li, L., Tang, H. and Hu, Y. (2018), "The effect of thickness on the mechanics of nanobeams", Int. J. Eng. Sci., 123, 81-91. https://doi.org/10.1016/j.ijengsci.2017.11.021.   DOI
24 Almitani, K.H., Abdelrahman, A.A. and Eltaher, M.A. (2020a), "Influence of the perforation configuration on dynamic behaviors of multilayered beam structure", Structures, 28, 1413-1426. https://doi.org/10.1016/j.istruc.2020.09.055.   DOI
25 Almitani, K.H., Abdelrahman, A.A. and Eltaher, M.A. (2020b), "Stability of perforated nanobeams incorporating surface energy effects", Steel Compos. Struct., 35(4), 555-566. https://doi.org/10.12989/scs.2020.35.4.555.   DOI
26 Lata P. and Kaur. H, (2021), "Interactions in a homogeneous isotropic modified couple stress thermoelastic solid with multi-dual-phase-lag heat transfer and two temperature", Steel Compos. Struct., 38(2), 213-221. http://dx.doi.org/10.12989/scs.2021.38.2.213   DOI
27 Lenci, S., Pieri, F., Haspeslagh, L., De Coster, J., Decoutere, S., Caro, A.M. and Witvrouw, A. (2011), "Stiction-free poly-SiGe resonators for monolithic integration of biosensors with CMOS", Paper presented at the 2011 16th International Solid-State Sensors, Actuators and Microsystems Conference.
28 Li, C. (2017), "Nonlocal thermo-electro-mechanical coupling vibrations of axially moving piezoelectric nanobeams", Mech. Based Des. Struct. Machines, 45(4), 463-478. https://doi.org/10.1080/15397734.2016.1242079.   DOI
29 Arefi, M. and Arani, A.H. (2018), "Higher order shear deformation bending results of a magnetoelectrothermoelastic functionally graded nanobeam in thermal, mechanical, electrical, and magnetic environments", Mech. Based Des. Struct. Machines, 46(6), 669-692. https://doi.org/10.1080/15397734.2018.1434002.   DOI
30 Ansari, R. and Sahmani, S. (2011), "Bending behavior and buckling of nanobeams including surface stress effects corresponding to different beam theories", Int. J. Eng. Sci., 49(11), 1244-1255. https://doi.org/10.1016/j.ijengsci.2011.01.007.   DOI
31 Abdelrahman, A.A. and Eltaher, M.A. (2020), "On Bending and Buckling Responses of Perforated Nanobeams including Surface Energy for Different Beams Theories", Eng. with Comput., https://doi.org/10.1007/s00366-020-01211-8.   DOI
32 Abo-bakr, H.M., Abo-bakr, R.M., Mohamed, S.A. and Eltaher, M.A. (2020a), "Weight Optimization of Axially Functionally Graded Microbeams under Buckling and Vibration Behaviors", Mech. Based Des. Struct. Machines, https://doi.org/10.1080/15397734.2020.1838298.   DOI
33 Abo-Bakr, R.M., Eltaher, M.A. and Attia, M.A. (2020b), "Pull-in and freestanding instability of actuated functionally graded nanobeams including surface and stiffening effects", Eng. with Comput., 1-22. https://doi.org/10.1007/s00366-020-01146-0.   DOI
34 Agwa, M.A. and Eltaher, M.A. (2016), "Vibration of a carbyne nanomechanical mass sensor with surface effect", Appl. Phys. A, 122(4), 335.   DOI
35 Eltaher, M., Mahmoud, F., Assie, A. and Meletis, E. (2013), "Coupling effects of nonlocal and surface energy on vibration analysis of nanobeams", Appl. Math. Comput., 224, 760-774. https://doi.org/10.1016/j.amc.2013.09.002   DOI
36 Luschi, L. and Pieri, F. (2015), "Design of MEMS mass sensors based of flexural phononic crystals", Paper presented at the 2015 XVIII AISEM Annual Conference.
37 Liu, C. and Rajapakse, R.K.N.D. (2009), "Continuum models incorporating surface energy for static and dynamic response of nanoscale beams", IEEE T. Nanotechnol., 9(4), 422-431. https://doi.org/10.1109/TNANO.2009.2034142.   DOI
38 Luschi, L. and Pieri, F. (2012), "A simple analytical model for the resonance frequency of perforated beams", Procedia Eng., 47, 1093-1096. https://doi.org/10.1016/j.proeng.2012.09.341.   DOI
39 Luschi, L. and Pieri, F. (2014), "An analytical model for the determination of resonance frequencies of perforated beams", J. Micromech. Microeng., 24(5), 055004. https://doi.org/10.1088/0960-1317/24/5/055004.   DOI
40 Luschi, L. and Pieri, F. (2016), "An analytical model for the resonance frequency of square perforated Lame-mode resonators", Sensor. Actuat. B, Chem., 222, 1233-1239. https://doi.org/10.1016/j.snb.2015.07.085.   DOI
41 Mahmoud, F., Eltaher, M., Alshorbagy, A. and Meletis, E. (2012), "Static analysis of nanobeams including surface effects by nonlocal finite element", J. Mech. Sci. Technol., 26(11), 3555-3563. https://doi.org/10.1007/s12206-012-0871-z.   DOI
42 Miller, R.E. and Shenoy, V.B. (2000), "Size-dependent elastic properties of nanosized structural elements", Nanotechnology, 11(3), 139. https://doi.org/10.1088/0957-4484/11/3/301.   DOI
43 Mohammadi, S., Eftekhar, A.A., Hunt, W.D. and Adibi, A. (2009), "High-Q micromechanical resonators in a two-dimensional phononic crystal slab", Appl. Phys. Lett., 94(5), 051906. https://doi.org/10.1063/1.3078284.   DOI
44 Oseev, A., Zubtsov, M. and Lucklum, R. (2013), "Gasoline properties determination with phononic crystal cavity sensor", Sensor. Actuat. B Chem., 189, 208-212. https://doi.org/10.1016/j.snb.2013.03.072   DOI
45 Najafzadeh, M., Adeli, M.M., Zarezadeh, E. and Hadi, A. (2020), "Torsional vibration of the porous nanotube with an arbitrary cross-section based on couple stress theory under magnetic field", Mech. Based Des. Struct. Machines, 1-15. https://doi.org/10.1080/15397734.2020.1733602.   DOI
46 Ng, E.J., Yang, Y., Chen, Y. and Kenny, T.W. (2014), "An etch hole-free process for temperature-compensated high Q encapsulated resonators", Paper presented at the Proc. Solid-State Sens., Actuators, Microsyst. Workshop.
47 Nguyen, C.T.C. (2007), "MEMS technology for timing and frequency control. IEEE transactions on ultrasonics", Ferroelec. Frequency Control, 54(2), 251-270.   DOI
48 Eltaher, M.A., Omar, F.A., Abdraboh, A.M., Abdalla, W.S. and Alshorbagy, A. E. (2020), "Mechanical behaviors of piezoelectric nonlocal nanobeam with cutouts", Smart Struct. Syst., 25(2), 219-228. http://dx.doi.org/10.12989/sss.2020.25.2.219.   DOI
49 Eltaher, M.A., Mohamed, N. and Mohamed, S.A. (2020), "Nonlinear buckling and free vibration of curved CNTs by doublet mechanics", Smart Struct. Syst., 26(2), 213-226. http://dx.doi.org/10.12989/sss.2020.26.2.213.   DOI
50 Eltaher, M.A. and Mohamed, N. A. (2020), "Vibration of nonlocal perforated nanobeams with general boundary conditions", Smart Struct. Syst., 25(4), 501-514. http://dx.doi.org/10.12989/sss.2020.25.4.501.   DOI
51 Esen, I, Abdelrahman, A.A. and Eltaher, M.A., (2020), "Dynamics Analysis of Timoshenko Perforated Microbeams under Moving Loads", Eng. with Comput., https://doi.org/10.1007/s00366-020-01212-7.   DOI
52 Esen, I., Ozarpa, C. and Eltaher, M.A. (2021), "Free Vibration of a Cracked FG Microbeam Embedded in an Elastic Matrix and Exposed to Magnetic Field in a Thermal Environment", Compos. Struct., 113552. https://doi.org/10.1016/j.compstruct.2021.113552.   DOI
53 Fanget, S., Hentz, S., Puget, P., Arcamone, J., Matheron, M., Colinet, E. and Roukes, M. (2011), "Gas sensors based on gravimetric detection-A review", Sensor. Actuat. B Chem., 160(1), 804-821. https://doi.org/10.1016/j.snb.2011.08.066.   DOI
54 Gao, X.L. (2015), "A new Timoshenko beam model incorporating microstructure and surface energy effects", Acta Mechanica, 226(2) ,457-474. https://doi.org/10.1007/s00707-014-1189-y.   DOI
55 Gao, X.L. and Mahmoud, F.F. (2014), "A new Bernoulli-Euler beam model incorporating microstructure and surface energy effects", Zeitschrift fur angewandte Mathematik und Physik, 65(2) ,393-404. https://doi.org/10.1007/s00033-013-0343-z.   DOI
56 Shenoy, V. B. (2005), "Atomistic calculations of elastic properties of metallic FCC crystal surfaces", Phys. Review B, 71(9), 094104. https://doi.org/10.1103/PhysRevB.74.149901.   DOI
57 Park, K.K., Lee, H., Kupnik, M., Oralkan, O., Ramseyer, J.P., Lang, H.P. and Khuri-Yakub, B.T. (2011), "Capacitive micromachined ultrasonic transducer (CMUT) as a chemical sensor for DMMP detection", Sensor. Actuat. B Chem., 160(1), 1120-1127. https://doi.org/10.1016/j.snb.2011.09.036   DOI
58 Rottenberg, X., Jansen, R., Cherman, V., Witvrouw, A., Tilmans, H., Zanaty, M. and Abbas, M. (2013), "Meta-materials approach to sensitivity enhancement of MEMS BAW resonant sensors", Paper presented at the SENSORS, 2013 IEEE.
59 Safarpour, M., Rahimi, A.R. and Alibeigloo, A. (2020), "Static and free vibration analysis of graphene platelets reinforced composite truncated conical shell, cylindrical shell, and annular plate using theory of elasticity and DQM", Mech. Based Des. Struct. Machines, 48(4), 496-524. https://doi.org/10.1080/15397734.2019.1646137.   DOI
60 Shokravi, M. (2018), "Forced vibration response in nanocomposite cylindrical shells-Based on strain gradient beam theory", Steel Compos. Struct., 28(3), 381-388. https://doi.org/10.12989/scs.2018.28.3.381.   DOI
61 Sivakumar, N., Kanagasabapathy, H. and Srikanth, H. (2018), "Analysis of Perforated Piezoelectric Sandwich Smart Structure Cantilever Beam Using COMSOL", Materials Today, Proceedings, 5(5), 12025-12034.   DOI
62 Sun, F., Wang, P., Li, W., Fan, H. and Fang, D. (2017), "Effects of circular cutouts on mechanical behaviors of carbon fiber reinforced lattice-core sandwich cylinder", Compos. Part A, Appl. Sci. Manufact., 100, 313-323. https://doi.org/10.1016/j.compositesa.2017.05.029   DOI
63 Van Beek, J. and Puers, R. (2011), "A review of MEMS oscillators for frequency reference and timing applications", J. Micromech. Microeng., 22(1), 013001. https://doi.org/10.1088/0960-1317/22/1/013001.   DOI
64 Gurtin, M.E. and Murdoch, A.I. (1978), "Surface stress in solids", Int. J. Solid. Struct., 14(6), 431-440.   DOI
65 Ghanbari, B., Ghadiri, M. and SafarPour, H. (2020), "A modified strain gradient shell model for vibration analysis of DWCNT conveying viscous fluid including surface effects", Mech. Based Des. Struct. Machines, 1-31. https://doi.org/10.1080/15397734.2020.1753533.   DOI
66 Ghandourh, E.E. and Abdraboh, A.M. (2020), "Dynamic analysis of functionally graded nonlocal nanobeam with different porosity models", Steel Compos. Struct., 36(3), 293-305. https://doi.org/10.12989/scs.2020.36.3.293.   DOI
67 Gurtin, M.E. and Murdoch, A.I. (1975), "A continuum theory of elastic material surfaces", Archiv. Rational Mech. Anal., 57(4), 291-323.   DOI
68 Hamed, M.A., Sadoun, A.M. and Eltaher, M.A. (2019), "Effects of porosity models on static behavior of size dependent functionally graded beam", Struct. Eng. Mech., 71(1), 89-98. http://dx.doi.org/10.12989/sem.2019.71.1.089.   DOI
69 Hamed, M.A., Mohamed, N.A. and Eltaher, M.A. (2020), "Stability buckling and bending of nanobeams including cutouts", Eng. with Comput., https://doi.org/10.1007/s00366-020-01063-2.   DOI
70 Hashemi, M. and Asghari, M. (2017), "On the size-dependent flexural vibration characteristics of unbalanced couple stress-based micro-spinning beams", Mech. Based Des. Struct. Machines, 45(1), 1-11. http://dx.doi.org/10.1080/15397734.2015.1125298.   DOI
71 Hashemi Kachapi, S.H. (2020), "Surface/interface approach in pull-in instability and nonlinear vibration analysis of fluid-conveying piezoelectric nanosensor", Mech. Based Des. Struct. Machines, 1-26. https://doi.org/10.1080/15397734.2020.1725566   DOI
72 Hussein, M.I., Leamy, M.J. and Ruzzene, M. (2014), "Dynamics of phononic materials and structures, Historical origins, recent progress, and future outlook", Appl. Mech. Rev., 66(4).
73 Wang, S., Popa, L.C. and Weinstein, D. (2014), "GaN MEMS resonator using a folded phononic crystal structure", Paper presented at the Proc. Solid-State Sens., Actuators, Microsyst. Workshop (Hilton Head)
74 Heidari, A., Yoon, Y.J., Park, M.K., Park, W.T. and Tsai, J.M.L. (2012), "High sensitive dielectric filled Lame mode mass sensor", Sensor. Actuat. A, Phys., 188, 82-88.   DOI
75 Xiao, Y., Wen, J. and Wen, X. (2012), "Broadband locally resonant beams containing multiple periodic arrays of attached resonators", Phys. Lett. A, 376(16), 1384-1390. https://doi.org/10.1016/j.physleta.2012.02.059.   DOI
76 Yang, F.A.C.M., Chong, A.C.M., Lam, D.C.C. and Tong, P. (2002). "Couple stress-based strain gradient theory for elasticity", Int. J. Solid. Struct., 39(10), 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X.   DOI
77 Yuan, Y. and Xu, K. (2019), "Postbuckling analysis of axially loaded nanoscaled shells embedded in elastic foundations based on Ru's surface elasticity theory", Mech. Based Des. Struct. Machines, 1-21. https://doi.org/10.1080/15397734.2019.1665543.   DOI
78 Yuksel, S.B. (2019), "Experimental investigation of retrofitted shear walls reinforced with welded wire mesh fabric", Struct. Eng. Mech., 70(2), 133-141. https://doi.org/10.12989/sem.2019.70.2.133.   DOI
79 Zhou, D.Y., Liu, L.F. and Zhu, L.M. (2016), "Lateral load-carrying capacity analyses of composite shear walls with double steel plates and filled concrete with binding bars", J. Cent. South Univ., 23(8), 2083-2091. https://doi.org/10.1007/s11771-016-3264-0.   DOI
80 Hozhabrossadati, S.M., Challamel, N., Rezaiee-Pajand, M. and Sani, A.A. (2020), "Free vibration of a nanogrid based on Eringen's stress gradient model", Mech. Based Des. Struct. Machines, 1-19. https://doi.org/10.1080/15397734.2020.1720720.   DOI
81 Jena, S.K., Chakraverty, S., Malikan, M. and Tornabene, F. (2019), "Stability analysis of single-walled carbon nanotubes embedded in winkler foundation placed in a thermal environment considering the surface effect using a new refined beam theory", Mech. Based Des. Struct. Machines, 1-15. https://doi.org/10.1080/15397734.2019.1698437   DOI
82 Jena, S.K., Chakraverty, S., Malikan, M. and Tornabene, F. (2020), "Effects of surface energy and surface residual stresses on vibro-thermal analysis of chiral, zigzag, and armchair types of SWCNTs using refined beam theory", Mech. Based Des. Struct. Machines, 1-15. https://doi.org/10.1080/15397734.2020.1754239   DOI
83 Jeong, K.H. and Amabili, M. (2006), "Bending vibration of perforated beams in contact with a liquid", J. Sound Vib., 298(1-2), 404-419. https://doi.org/10.1016/j.jsv.2006.05.029.   DOI
84 Karami, B., Janghorban, M., Shahsavari, D. and Tounsi, A. (2018), "A size-dependent quasi-3D model for wave dispersion analysis of FG nanoplates", Steel Compos. Struct., 28(1), 99-110. https://doi.org/10.12989/scs.2018.28.1.099.   DOI
85 Kerid, R., Bourouina, H., Yahiaoui, R., Bounekhla, M. and Aissat, A. (2019), "Magnetic field effect on nonlocal resonance frequencies of structure-based filter with periodic square holes network", Physica E, Low-dimensional Syst. Nanostruct., 105, 83-89. https://doi.org/10.1016/j.physe.2018.05.021.   DOI