journal article·2020-03-24Open access

Advances in modelling and analysis of nano structures: a review

Vikram Singh Chandel, Guannan Wang, Mohammad Talha

Nanotechnology Reviews· Volume 9· Issue 1

Abstract

Abstract Nanostructures are widely used in nano and micro-sized systems and devices such as biosensors, nano actuators, nano-probes, and nano-electro-mechanical systems. The complete understanding of the mechanical behavior of nanostructures is crucial for the design of nanodevices and systems. Therefore, the flexural, stability and vibration analysis of various nanostructures such as nanowires, nanotubes, nanobeams, nanoplates, graphene sheets and nanoshells has received a great attention in recent years. The focus has been made, to present the structural analysis of nanostructures under thermo-magneto-electro-mechanical loadings under various boundary and environmental conditions. This paper also provides an overview of analytical modeling methods, fabrication procedures, key challenges and future scopes of development in the direction of analysis of such structures, which will be helpful for appropriate design and analysis of nanodevices for the application in the various fields of nanotechnology.
Source: crossref · Quality: baseline

Research context

OpenAlex topics: Nonlocal and gradient elasticity in micro/nano structures · Carbon Nanotubes in Composites · Composite Structure Analysis and Optimization

SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.

Affiliations

School of Engineering, Indian Institute of Technology Mandi , Himachal Pradesh 175005 , India

Department of Civil Engineering, Zhejiang University , Hangzhou 310058 , China

References (100+)

Show source-provided references
1. Farajpour A., Ghayesh M.H., Farokhi H., A review on the mechanics of nanostructures, Int. J. Eng. Sci., 2018, 133, 12, 231-263.10.1016/j.ijengsci.2018.09.006 ↗
2. Lu L., Guo X., Zhao J., On the mechanics of Kirchhoff and Mindlin plates incorporating surface energy, Int. J. Eng. Sci., 2018, 124, 3, 24-40.10.1016/j.ijengsci.2017.11.020 ↗
3. Rafiee R., Moghadam R.M., On the modeling of carbon nanotubes: a critical review, Compos. Part B: Eng., 2014, 56, 1, 435-449.
4. Yengejeh S.I., Kazemi S.A., Oechsner A., Advances in mechanical analysis of structurally and atomically modified carbon nanotubes and degenerated nanostructures: A review, Compos. Part B: Eng., 2016, 86, 2, 95-107.
5. Ansari R., Rouhi H., Sahmani S., Calibration of the analytical nonlocal shell model for vibrations of double-walled carbon nanotubes with arbitrary boundary conditions using molecular dynamics, Int. J. Mech. Sci., 2011, 53, 9, 786-792.10.1016/j.ijmecsci.2011.06.010 ↗
6. Shahsavari D., Karami B., Li L., Damped vibration of a graphene sheet using a higher-order nonlocal strain-gradient Kirchhoff plate model, Comptes Rendus Mécanique, 2018, 346, 12, 1216-1232.10.1016/j.crme.2018.08.011 ↗
7. Al-Jumaili A., Alancherry S., Bazaka K., Jacob M., Review on the antimicrobial properties of carbon nanostructures, Materials, 2017, 10, 9, 1066.
8. Thess A., Lee R., Nikolaev P., Dai H., Petit P., Robert J., et al., Crystalline ropes of metallic carbon nanotubes, Science, 1996, 273, 5274, 483-487.10.1126/science.273.5274.483 ↗
9. Bethune D., Kiang C.H., De Vries M., Gorman G., Savoy R., Vazquez J., et al., Cobalt-catalysed growth of carbon nanotubes with single-atomic-layer walls, Nature, 1993, 363, 6430, 605-607.10.1038/363605a0 ↗
10. Journet C., Maser W., Bernier P., Loiseau A., de La Chapelle M.L., Lefrant d.S., et al., Large-scale production of single-walled carbon nanotubes by the electric-arc technique, Nature, 1997, 388, 6644, 756.10.1038/41972 ↗
11. Dai H., Carbon nanotubes: opportunities and challenges, Surf. Sci., 2002, 500, 1-3, 218-241.10.1016/S0039-6028(01)01558-8 ↗
12. Liu J., Fan S., Dai H., Recent advances in methods of forming carbon nanotubes, MRS Bulletin, 2004, 29, 4, 244-250.10.1557/mrs2004.75 ↗
13. Javey A., Guo J., Wang Q., Lundstrom M., Dai H., Ballistic carbon nanotube field-effect transistors, nature, 2003, 424, 6949, 654.10.1038/nature01797 ↗
14. Javey A., Kim H., Brink M.,Wang Q., Ural A., Guo J., et al., High-k dielectrics for advanced carbon-nanotube transistors and logic gates, Nature Mater., 2002, 1, 4, 241.10.1038/nmat769 ↗
15. Rosenblatt S., Yaish Y., Park J., Gore J., Sazonova V., McEuen P.L., High performance electrolyte gated carbon nanotube transistors, Nano Lett., 2002, 2, 8, 869-872.10.1021/nl025639a ↗
16. Fuhrer M., Kim B., Dürkop T., Brintlinger T., High-mobility nanotube transistor memory, Nano Lett., 2002, 2, 7, 755-759.10.1021/nl025577o ↗
17. Lau K.T., Hui D., Effectiveness of using carbon nanotubes as nano-reinforcements for advanced Compos. Struct., Carbon, 2002, 9, 40, 1605-1606.10.1016/S0008-6223(02)00157-4 ↗
18. Lau A.K.T., Hui D., The revolutionary creation of new advanced materials-carbon nanotube composites, Compos. Part B: Eng., 2002, 33, 4, 263-27710.1016/S1359-8368(02)00012-4 ↗
19. Lau K.T., Chipara M., Ling H.Y., Hui D., On the effective elastic moduli of carbon nanotubes for nanocomposite structures, Compos. Part B: Eng., 2004, 35, 2, 95-101.10.1016/j.compositesb.2003.08.008 ↗
20. Alizada A., Sofiyev A., Modified Young’s moduli of nanomaterials taking into account the scale effects and vacancies, Meccanica, 2011, 46, 5, 915-92010.1007/s11012-010-9349-1 ↗
21. Wang Q., Arash B., A review on applications of carbon nanotubes and graphenes as nano-resonator sensors, Comput. Mater. Sci., 2014, 82, 350-360.10.1016/j.commatsci.2013.10.010 ↗
22. An K.H., Kim W.S., Park Y.S., Moon J.M., Bae D.J., Lim S.C., et al., Electrochemical properties of high-power supercapacitors using single-walled carbon nanotube electrodes, Adv. Funct. Mater., 2001, 11, 5, 387-392.10.1002/1616-3028(200110)11:5<387::AID-ADFM387>3.0.CO;2-G ↗
23. Niu C., Sichel E.K., Hoch R., Moy D., Tennent H., High power electrochemical capacitors based on carbon nanotube electrodes, Appl. Phys. Lett., 1997, 70, 11, 1480-1482.10.1063/1.118568 ↗
24. Malik S., Maqbool M., Hussain S., Irfan H., Carbon nanotubes: description, properties and applications, J. Pak. Mater. Soc., 2008, 2, 1, 21-26.
25. Baughman R.H., Cui C., Zakhidov A.A., Iqbal Z., Barisci J.N., Spinks G.M., et al., Carbon nanotube actuators, Science, 1999, 284, 5418, 1340-1344.10.1126/science.284.5418.1340 ↗
26. Bandaru P.R., Yamada H., Narayanan R., Hoefer M., The role of defects and dimensionality in influencing the charge, capacitance, and energy storage of graphene and 2D materials, Nanotechnol. Rev., 2017, 6, 5, 421-433.10.1515/ntrev-2016-0099 ↗
27. Majeed S., Zhao J., Zhang L., Anjum S., Liu Z., Xu G., Synthesis and electrochemical applications of nitrogen-doped carbon nanomaterials, Nanotechnol. Rev., 2013, 2, 6, 615-635.10.1515/ntrev-2013-0007 ↗
28. De Heer W.A., Chatelain A., Ugarte D., A carbon nanotube fieldemission electron source, Science, 1995, 270, 5239, 1179-1180.10.1126/science.270.5239.1179 ↗
29. Rinzler A., Hafner J., Nikolaev P., Nordlander P., Colbert D., Smalley R., et al., Unraveling nanotubes: field emission from an atomic wire, Science, 1995, 269, 5230, 1550-1553.10.1126/science.269.5230.1550 ↗
30. Sugie H., TanemuraM., Filip V., Iwata K., Takahashi K., Okuyama F., Carbon nanotubes as electron source in an x-ray tube, Appl. Phys. Lett., 2001, 78, 17, 2578-2580.10.1063/1.1367278 ↗
31. Brand O., Fedder G.K., Carbon Nanotube Devices: Properties, Modeling, Integration and Applications, vol. 8, John Wiley & Sons, 2008.
32. Lee N., Chung D., Han I., Kang J., Choi Y., Kim H., et al., Application of carbon nanotubes to field emission displays, Diamond Related Mater., 2001, 10, 2, 265-270.10.1016/S0925-9635(00)00478-7 ↗
33. Saito Y., Uemura S., Field emission from carbon nanotubes and its application to electron sources, Carbon, 2000, 38, 2, 169-182.10.1016/S0008-6223(99)00139-6 ↗
34. Rosen R., Simendinger W., Debbault C., Shimoda H., Fleming L., Stoner B., et al., Application of carbon nanotubes as electrodes in gas discharge tubes, Appl. Phys. Lett., 2000, 76, 13, 1668-1670.10.1063/1.126130 ↗
35. Zhu W., Bower C., Zhou O., Kochanski G., Jin S., Large current density from carbon nanotube field emitters, Appl. Phys. Lett., 1999, 75, 6, 873-875.10.1063/1.124541 ↗
36. Kwo J.L., Tsou C., Yokoyama M., Lin I.N., Lee C.C., Wang W.C., et al., Field emission characteristics of carbon nanotube emitters synthesized by arc discharge, J. Vacuum Sci. Technol. B: Micro-electr. Nanomet. Struct. Proces., Measurem. Phenom., 2001, 19, 1, 23-26.10.1116/1.1340668 ↗
37. Sorkin V., Zhang Y.W., Graphene-based pressure nano-sensors, J. Molec. Model., 2011, 17, 11, 2825-2830.10.1007/s00894-011-0972-0 ↗
38. Power A.C., Gorey B., Chandra S., Chapman J., Carbon nano-materials and their application to electrochemical sensors: a review, Nanotechnol. Rev., 2018, 7, 1, 19-41.10.1515/nano.0034.00139 ↗
39. Li Z., Xu K., Wei F., Recent Progress on Photodetectors Based on Low Dimensional Nanomaterials, Nanotechnol. Rev., 2018.10.1515/ntrev-2018-0084 ↗
40. Zhe Z., Yuxiu A., Nanotechnology for the oil and gas industry-an overview of recent progress, Nanotechnol. Rev., 2018, 7, 4, 341-353.10.1515/ntrev-2018-0061 ↗
41. Wu S., Lin Q., Yuen Y., Tai Y.C., MEMS flow sensors for nano-fluidic applications, Sensors and Actuators A: Physical, 2001, 89, 1-2, 152-158.10.1016/S0924-4247(00)00541-0 ↗
42. Khanna V.K., Nanosensors: physical, chemical, and biological, CRC Press, 2016.10.1201/b11289 ↗
43. Arash B., Wang Q., Wu N., Gene detection with carbon nanotubes, J. Nanotechnol. Eng. Med., 2012, 3, 2, 020902.10.1115/1.4007388 ↗
44. Modi A., Koratkar N., Lass E., Wei B., Ajayan P.M., Miniaturized gas ionization sensors using carbon nanotubes, Nature, 2003, 424, 6945, 171, 80.10.1038/nature01777 ↗
45. Kong J., Franklin N.R., Zhou C., Chapline M.G., Peng S., Cho K., et al., Nanotube molecular wires as chemical sensors, Science, 2000, 287, 5453, 622-625.10.1126/science.287.5453.622 ↗
46. Malekzad H., Zangabad P.S., Mirshekari H., Karimi M., Hamblin M.R., Noble metal nanoparticles in biosensors: recent studies and applications, Nanotechnol. Rev., 2017, 6, 3, 301-329.10.1515/ntrev-2016-0014 ↗
47. Koyani R., Pérez-Robles J., Cadena-Nava R.D., Vazquez-Duhalt R., Biomaterial-based nanoreactors, an alternative for enzyme delivery, Nanotechnol. Rev., 2017, 6, 5, 405-419.10.1515/ntrev-2016-0071 ↗
48. Saini D., Synthesis and functionalization of graphene and application in electrochemical biosensing, Nanotechnol. Rev., 2016, 5, 4, 393-416.10.1515/ntrev-2015-0059 ↗
49. Arie T., Akita S., Nakayama Y., Growth of tungsten carbide nanoneedle and its application as a scanning tunnelling microscope tip, J. Physics D: Applied Physics, 1998, 31, 14, L4910.1088/0022-3727/31/14/001 ↗
50. Dai H., Franklin N., Han J., Exploiting the properties of carbon nanotubes for nanolithography, Appl. Phys. Lett., 1998, 73, 11, 1508-1510.10.1063/1.122188 ↗
51. Dai H., Hafner J.H., Rinzler A.G., Colbert D.T., Smalley R.E., Nanotubes as nanoprobes in scanning probe microscopy, Nature, 1996, 384, 6605, 147.10.1038/384147a0 ↗
52. Nagy G., Levy M., Scarmozzino R., Osgood Jr R., Dai H., Smalley R., et al., Carbon nanotube tipped atomic force microscopy for measurement of< 100 nm etch morphology on semiconductors, Appl. Phys. Lett., 1998, 73, 4, 529-531.10.1063/1.122069 ↗
53. Wong S.S., Harper J.D., Lansbury P.T., Lieber C.M., Carbon nanotube tips: high-resolution probes for imaging biological systems, J. Amer. Chem. Soc., 1998, 120, 3, 603-604.10.1021/ja9737735 ↗
54. Lau K.t., Gu C., Hui D., A critical review on nanotube and nanotube/nanoclay related polymer composite materials, Compos. Part B: Eng., 2006, 37, 6, 425-436.10.1016/j.compositesb.2006.02.020 ↗
55. Alizada A., Sofiyev A., On the mechanics of deformation and stability of the beam with a nanocoating, J. Reinforced Plastics Compos., 2011, 30, 18, 1583-1595.10.1177/0731684411428382 ↗
56. Alizada A., Sofiyev A., Kuruoglu N., Stress analysis of a substrate coated by nanomaterials with vacancies subjected to uniform extension load, Acta Mechanica, 2012, 223, 7, 1371-1383.10.1007/s00707-012-0649-5 ↗
57. Sofiyev A., Hui D., Najafov A., Turkaslan S., Dorofeyskaya N., Yuan G., Influences of shear stresses and rotary inertia on the vibration of functionally graded coated sandwich cylindrical shells resting on the Pasternak elastic foundation, J. Sandwich Structures & Materials, 2015, 17, 6, 691-720.10.1177/1099636215594560 ↗
58. Sofiyev A., Turkaslan B.E., Bayramov R., Salamci M., Analytical solution of stability of FG-CNTRC conical shells under external pressures, Thin-Walled Structures, 2019, 144, 106338.10.1016/j.tws.2019.106338 ↗
59. Das S., Srivastava V.C., An overview of the synthesis of CuO ZnO nanocomposite for environmental and other applications, Nanotechnol. Rev., 2018, 7, 3, 267-282.10.1515/ntrev-2017-0144 ↗
60. Anwar A., Kanwal Q., Akbar S., Munawar A., Durrani A., Farooq M.H., Synthesis and characterization of pure and nanosized hydroxyapatite bioceramics, Nanotechnol. Rev., 2017, 6, 2, 149-157.10.1515/ntrev-2016-0020 ↗
61. NB R.K., Crasta V., Praveen B., Kumar M., Studies on structural, optical and mechanical properties ofMWCNTs and ZnO nanoparticles doped PVA nanocomposites, Nanotechnol. Rev., 2015, 4, 5, 457-467.10.1515/ntrev-2015-0020 ↗
62. Kalinitchev A., Multicomponent mass transfer kinetics in nanocomposite (NC) bifunctional matrixes: NC selectivity and diffusion concentration waves, Nanotechnol. Rev., 2014, 3, 5, 467-497.10.1515/ntrev-2014-0007 ↗
63. Lau K.T., Lu M., Hui D., Coiled carbon nanotubes: Synthesis and their potential applications in advanced Compos. Struct., Compos. Part B: Eng., 2006, 37, 6, 437-448.
64. Chipara M., Artiaga R., Lau K., Chipara D., Hui D., Dynamical mechanical analysis of multiwall carbon nanotubes-styreneisoprene- styrene block copolymer nanocomposite, Compos. Commun., 2017, 3, 23-27.10.1016/j.coco.2017.01.001 ↗
65. Ma H.L., Jia Z., Lau K.T., Li X., Hui D., Shi S.Q., Enhancement on mechanical strength of adhesively-bonded composite lap joints at cryogenic environment using coiled carbon nanotubes, Compos. Part B: Eng., 2017, 110, 396-401.10.1016/j.compositesb.2016.11.019 ↗
66. Lau K.t., Wong T.T., Leng J., Hui D., Rhee K.Y., Property enhancement of polymer-based composites at cryogenic environment by using tailored carbon nanotubes, Compos. Part B: Eng., 2013, 54, 41-43.10.1016/j.compositesb.2013.03.044 ↗
67. Ma H.L., Lau K.T., Hui D., Shi S.Q., Poon C.K., Theoretical analysis on the pullout behavior of carbon nanotube at cryogenic environment with the consideration of thermal residual stress, Compos. Part B: Eng., 2017, 128, 67-75.10.1016/j.compositesb.2017.07.009 ↗
68. Falvo M.R., Clary G., Taylor Ii R., Chi V., Brooks Jr F.,Washburn S., et al., Bending and buckling of carbon nanotubes under large strain, Nature, 1997, 389, 6651, 58210.1038/39282 ↗
69. Wang Y., Li F., Dynamical properties of nanotubes with nonlocal continuum theory: A review, Science China Physics, Mech. Astron., 2012, 55, 7, 1210-1224.10.1007/s11433-012-4781-y ↗
70. Arash B., Wang Q., A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes, Comput. Mater. Sci., 2012, 51, 1, 303-313.10.1016/j.commatsci.2011.07.040 ↗
71. Iijima S., Brabec C., Maiti A., Bernholc J., Structural flexibility of carbon nanotubes, J. Chem. Phys., 1996, 104, 5, 2089-2092.10.1063/1.470966 ↗
72. Hernandez E., Goze C., Bernier P., Rubio A., Elastic properties of C and Bx Cy N z composite nanotubes, Phys. Rev. Lett., 1998, 80, 20, 4502.
73. Sánchez-Portal D., Artacho E., Soler J.M., Rubio A., Ordejón P., Ab initio structural, elastic, and vibrational properties of carbon nanotubes, Phys. Rev. B, 1999, 59, 19, 12678.10.1103/PhysRevB.59.12678 ↗
74. Yakobson B., Campbell M., Brabec C., Bernholc J., High strain rate fracture and C-chain unraveling in carbon nanotubes, Computational Mater. Sci., 1997, 8, 4, 341-348.10.1016/S0927-0256(97)00047-5 ↗
75. Liew K., Wong C., He X., Tan M., Meguid S., Nanomechanics of single and multiwalled carbon nanotubes, Phys. Rev. B, 2004, 69, 11, 115429.10.1103/PhysRevB.69.115429 ↗
76. Li C., Chou T.W., Elastic wave velocities in single-walled carbon nanotubes, Phys. Rev. B, 2006, 73, 24, 245407.10.1103/PhysRevB.73.245407 ↗
77. Xiang P., Liew K.M., Free vibration analysis of microtubules based on an atomistic-continuum model, J. Sound Vibr., 2012, 331, 1, 213-230.10.1016/j.jsv.2011.08.024 ↗
78. Arghavan S., Singh A., On the vibrations of single-walled carbon nanotubes, J. Sound Vibr., 2011, 330, 13, 3102-3122.10.1016/j.jsv.2011.01.032 ↗
79. Yan J., Liew K., He L., Free vibration analysis of single-walled carbon nanotubes using a higher-order gradient theory, J. Sound Vibr., 2013, 332, 15, 3740-3755.10.1016/j.jsv.2013.02.004 ↗
80. Krishnan A., Dujardin E., Ebbesen T., Yianilos P., Treacy M., Young’s modulus of single-walled nanotubes, Phys. Rev. B, 1998, 58, 20, 14013.10.1103/PhysRevB.58.14013 ↗
81. Yakobson B.I., Brabec C., Bernholc J., Nanomechanics of carbon tubes: instabilities beyond linear response, Phys. Rev. Lett., 1996, 76, 14, 2511.
82. Parnes R., Chiskis A., Buckling of nano-fibre reinforced composites: a re-examination of elastic buckling, J. Mech. Phys. Solids, 2002, 50, 4, 855-879.10.1016/S0022-5096(01)00101-6 ↗
83. Behera L., Chakraverty S., Recent researches on nonlocal elasticity theory in the vibration of carbon nanotubes using beam models: A Review, Arch. Comput. Meth. Eng., 2017, 24, 3, 481-494.10.1007/s11831-016-9179-y ↗
84. Liu Y., Reddy J., A nonlocal curved beam model based on a modified couple stress theory, Int. J. Struct. Stabil. Dynam., 2011, 11, 03, 495-512.10.1142/S0219455411004233 ↗
85. Park S., Gao X., Bernoulli-Euler beam model based on a modified couple stress theory, J. Micromech. Microeng., 2006, 16, 11, 2355.
86. Rahman A., Correlations in the Motion of Atoms in Liquid Argon, Phys. Rev., 1964, 136, A405-A411.10.1103/PhysRev.136.A405 ↗
87. Dauxois T., Fermi, Pasta, Ulam and a mysterious lady, arXiv preprint arXiv:0801.1590, 200810.1063/1.2835154 ↗
88. Alder B.J., Wainwright T.E., Studies in molecular dynamics. I. General method, J. Chem. Phys., 1959, 31, 2, 459-466.10.1063/1.1730376 ↗
89. Jeong B.W., Kim H.Y., Molecular dynamics simulations of the failure behaviors of closed carbon nanotubes fully filled with C60 fullerenes, Comput. Mater. Sci., 2013, 77, 7-12.10.1016/j.commatsci.2013.04.024 ↗
90. Wong C., Vijayaraghavan V., Nanomechanics of imperfectly straight single walled carbon nanotubes under axial compression by using molecular dynamics simulation, Comput. Mater. Sci., 2012, 53, 1, 268-277.10.1016/j.commatsci.2011.08.011 ↗
91. Gurtin M.E., Murdoch A.I., A continuum theory of elastic material surfaces, Archive for Rational Mechanics and Analysis, 1975, 57, 4, 291-323.10.1007/BF00261375 ↗
92. Gurtin M.E., Murdoch A.I., Surface stress in solids, Int. J. Solids Struct., 1978, 14, 6, 431-440.10.1016/0020-7683(78)90008-2 ↗
93. Miller R.E., Shenoy V.B., Size-dependent elastic properties of nanosized structural elements, Nanotechnology, 2000, 11, 3, 13910.1088/0957-4484/11/3/301 ↗
94. Steigmann D., Ogden R., Elastic surface-substrate interactions, Proc. Royal Soc. London Series A: Math., Phys. Eng. Sci., 1999, 455, 1982, 437-474.10.1098/rspa.1999.0320 ↗
95. Yan Z., Jiang L., The vibrational and buckling behaviors of piezoelectric nanobeams with surface effects, Nanotechnology, 2011, 22, 24, 245703.10.1088/0957-4484/22/24/245703 ↗
96. Ansari R., Sahmani S., Surface stress effects on the free vibration behavior of nanoplates, Int. J. Eng. Sci., 2011, 49, 11, 1204-1215.10.1016/j.ijengsci.2011.06.005 ↗
97. Guo J.G., Zhao Y.P., The size-dependent elastic properties of nanofilms with surface effects, J. Appl. Phys., 2005, 98, 7, 074306.10.1063/1.2071453 ↗
98. Lü C., Chen W., Lim C.W., Elastic mechanical behavior of nano scaled FGM films incorporating surface energies, Compos. Sci. Technol., 2009, 69, 7-8, 1124-1130.10.1016/j.compscitech.2009.02.005 ↗
99. Wang G., Chen Q., He Z., Pindera M.J., Homogenized moduli and local stress fields of unidirectional nano-composites, Compos. Part B: Eng., 2018, 138, 265-277.10.1016/j.compositesb.2017.11.029 ↗
100. Chen Q., Wang G., Pindera M.J., Finite-volume homogenization and localization of nanoporous materials with cylindrical voids. Part 1: Theory and validation, Europ. J. Mech. A: Solids, 2018, 70, 141-155.