On effective properties of materials at the nano- and microscales considering surface effects. (English) Zbl 1382.74093

Eremeyev, Victor A.

Acta Mech. 227, No. 1, 29-42 (2016).

Summary: In the last years, the rapid increase in the technical capability to control and design materials at the nanoscale has pushed toward an intensive exploitation of new possibilities concerning optical, chemical, thermoelectrical and electronic applications. As a result, new materials have been developed to obtain specific physical properties and performances. In this general picture, it was natural that the attention toward mechanical characterization of the new structures was left, in a sense, behind. Anyway, once the theoretically designed objects proceed toward concrete manufacturing and applications, an accurate and general description of their mechanical properties becomes more and more scientifically relevant. The aim of the paper is therefore to discuss new methods and techniques for modeling the behavior of nanostructured materials considering surface/interface properties, which are responsible for the main differences between nano- and macroscale, and to determine their actual material properties at the macroscale. Our approach is intended to study the mechanical properties of materials taking into account surface properties including possible complex inner microstructure of surface coatings. We use the Gurtin-Murdoch model of surface elasticity. We consider the inner regular and irregular surface thin coatings (i.e., ordered or disordered nanofibers arrays) and present few examples of averaged 2D properties of them. Since the actual 2D properties depend not only on the mechanical properties of fibers or other elements of a coating, but also on the interaction forces between them, the analysis also includes information on the geometry of the microstructure of the coating, on mechanical properties of elements and on interaction forces. Further we use the obtained 2D properties to derive the effective properties of solids and structures at the macroscale, such as the bending stiffness or Young’s modulus.

MSC:

74M25

Micromechanics of solids

References:

[1]	Aifantis, E., Update on a class of gradient theories, Mech. Mater., 35, 259-280, (2003)
[2]	Altenbach, H.; Eremeev, V.; Morozov, N.F., On equations of the linear theory of shells with surface stresses taken into account, Mech. Solids, 45, 331-342, (2010)
[3]	Altenbach, H.; Eremeyev, V.A., On the shell theory on the nanoscale with surface stresses, Int. J. Eng. Sci., 49, 1294-1301, (2011) · Zbl 1423.74561
[4]	Altenbach, H.; Eremeyev, V.A.; Lebedev, L.P., On the existence of solution in the linear elasticity with surface stresses, ZAMM, 90, 231-240, (2010) · Zbl 1355.74014
[5]	Altenbach, H.; Eremeyev, V.A.; Lebedev, L.P., On the spectrum and stiffness of an elastic body with surface stresses, ZAMM, 91, 699-710, (2011) · Zbl 1325.74023
[6]	Altenbach, H.; Eremeyev, V.A.; Morozov, N.F., Surface viscoelasticity and effective properties of thin-walled structures at the nanoscale, Int. J. Eng. Sci., 59, 83-89, (2012) · Zbl 1423.74805
[7]	Altenbach, H., Eremeyev, V.A., Morozov, N.F.: Mechanical properties of materials considering surface effects. In: Cocks, A., Wang, J. (eds.) IUTAM Symposium on Surface Effects in the Mechanics of Nanomaterials and Heterostructures, IUTAM Bookseries (closed), vol. 31, pp. 105-115. Springer, Dordrecht (2013)
[8]	Arroyo, M.; Belytschko, T., An atomistic-based finite deformation membrane for single layer crystalline films, J. Mech. Phys. Solids, 50, 1941-1977, (2002) · Zbl 1006.74061
[9]	Ashby M.F., Evans A.G., Fleck N.A., Gibson L.J., Hutchinson J.W., Wadley H.N.G.: Metal Foams: A Design Guid. Butterworth-Heinemann, Boston (2000)
[10]	Askes, H.; Aifantis, E.C., Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results, Int. J. Solids Struct., 48, 1962-1990, (2011)
[11]	Bažant, Z.P., Size effect, Int. J. Solids Struct., 37, 69-80, (2000) · Zbl 1073.74593
[12]	Bhushan, B. (ed.): Springer Handbook of Nanotechnology. Springer, Berlin (2007)
[13]	Bhushan, B.; Jung, Y.C.; Koch, K., Micro-, nano- and hierarchical structures for superhydrophobicity, self-cleaning and low adhesion, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci., 367, 1631-1672, (2009)
[14]	Chen, C.; Shi, Y.; Zhang, Y.; Zhu, J.; Yan, Y., Size dependence of young’s modulus in zno nanowires, Phys. Rev. Lett., 96, 075505, (2006)
[15]	Christensen R.M.: Mechanics of Composite Materials. Dover, New York (2005)
[16]	Contreras, C.B.; Chagas, G.; Strumia, M.C.; Weibel, D.E., Permanent superhydrophobic polypropylene nanocomposite coatings by a simple one-step dipping process, Appl. Surf. Sci., 307, 234-240, (2014)
[17]	Craighead, H.G., Nanoelectromechanical systems, Science, 290, 1532-1535, (2000)
[18]	Cuenot, S.; Frétigny, C.; Demoustier-Champagne, S.; Nysten, B., Surface tension effect on the mechanical properties of nanomaterials measured by atomic force microscopy, Phys. Rev. B, 69, 165410, (2004)
[19]	Dastjerdi, R.; Montazer, M., A review on the application of inorganic nano-structured materials in the modification of textiles: focus on anti-microbial properties, Colloids Surf. B Biointerfaces, 79, 5-18, (2010)
[20]	Davydov, D.; Voyiatzis, E.; Chatzigeorgiou, G.; Liu, S.; Steinmann, P.; Böhm, M.C.; Müller-Plathe, F., Size effects in a silica-polystyrene nanocomposite: molecular dynamics and surface-enhanced continuum approaches, Soft Mater., 12, s142-s151, (2014)
[21]	dell’Isola, F.; Madeo, A.; Seppecher, P., Boundary conditions at fluid-permeable interfaces in porous media: a variational approach, Int. J. Solids Struct., 46, 3150-3164, (2009) · Zbl 1167.74393
[22]	dell’Isola, F.; Rotoli, G., Validity of Laplace formula and dependence of surface tension on curvature in second gradient fluids, Mech. Res. Commun., 22, 485-490, (1995) · Zbl 0845.76006
[23]	dell’Isola, F.; Sciarra, G.; Vidoli, S., Generalized hooke’s law for isotropic second gradient materials, Proc. R. Soc. Lond. A Math. Phys. Eng. Sci., 465, 2177-2196, (2009) · Zbl 1186.74019
[24]	dell’Isola, F.; Seppecher, P., The relationship between edge contact forces, double forces and interstitial working allowed by the principle of virtual power, Comptes rendus de l’Académie des sciences. Série 2, 321, 303-308, (1995) · Zbl 0844.73006
[25]	dell’Isola, F.; Seppecher, P., Edge contact forces and quasi-balanced power, Meccanica, 32, 33-52, (1997) · Zbl 0877.73055
[26]	dell’Isola, F.; Seppecher, P.; Madeo, A., How contact interactions may depend on the shape of Cauchy cuts in \(N\)th gradient continua: approach “à la d’alembert”, ZAMP, 63, 1119-1141, (2012) · Zbl 1330.76016
[27]	Duan, H.L.; Karihaloo, B.L., Thermo-elastic properties of heterogeneous materials with imperfect interfaces: generalized levin’s formula and hill’s connections, J. Mech. Phys. Solids, 55, 1036-1052, (2007) · Zbl 1170.74017
[28]	Duan, H.L.; Wang, J.; Huang, Z.P.; Karihaloo, B.L., Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress, J. Mech. Phys. Solids, 53, 1574-1596, (2005) · Zbl 1120.74718
[29]	Duan, H.L., Wang, J., Karihaloo, B.L.: Theory of elasticity at the nanoscale. In: Aref, H., Van der Giessen, E. (eds.) Advances in Applied Mechanics, vol. 42, pp. 1-68. Elsevier, Amsterdam (2008) · Zbl 1352.74096
[30]	Duan, H.L.; Wang, J.; Karihaloo, B.L.; Huang, Z.P., Nanoporous materials can be made stiffer than non-porous counterparts by surface modification, Acta Mater., 54, 2983-2990, (2006)
[31]	Ekinci, K.L.; Roukes, M.L., Nanoelectromechanical systems, Rev. Sci. Instrum., 76, 061101, (2005)
[32]	Eremeyev, V.; Morozov, N., The effective stiffness of a nanoporous rod, Doklady Phys., 55, 279-282, (2010) · Zbl 1254.74010
[33]	Eremeyev, V.A.; Altenbach, H.; Morozov, N.F., The influence of surface tension on the effective stiffness of nanosize plates, Doklady Phys., 54, 98-100, (2009) · Zbl 1255.74043
[34]	Eremeyev, V.A.; Lebedev, L.P., Existence of weak solutions in elasticity, Math. Mech. Solids, 18, 204-217, (2013)
[35]	Eringen A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002) · Zbl 1023.74003
[36]	Escobar, A.M.; Llorca-Isern, N., Superhydrophobic coating deposited directly on aluminum, Appl. Surf. Sci., 305, 774-782, (2014)
[37]	Ganesh, V.A.; Raut, H.K.; Nair, A.S.; Ramakrishna, S., A review on self-cleaning coatings, J. Mater. Chem., 21, 16304-16322, (2011)
[38]	de Gennes, P.G., Some effects of long range forces on interfacial phenomena, J. Phys. Lett., 42, 377-379, (1981)
[39]	de Gennes P.G., Brochard-Wyart F., Quéré D.: Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer, New York (2004) · Zbl 1139.76004
[40]	Gent, A.; Thomas, A., Mechanics of foamed elastic materials, Rubber Chem. Technol., 36, 597-610, (1963)
[41]	Gibson L.J., Ashby M.F.: Cellular Solids: Structure and Properties, 2nd edn. Cambridge Solid State Science Series. Cambridge University Press, Cambridge (1997)
[42]	Grimm, S.; Giesa, R.; Sklarek, K.; Langner, A.; Gösele, U.; Schmidt, H.W.; Steinhart, M., Nondestructive replication of self-ordered nanoporous alumina membranes via cross-linked polyacrylate nanofiber arrays, Nano Lett., 8, 1954-1959, (2008)
[43]	Guo, J.G., Zhao, Y.P.: The size-dependent elastic properties of nanofilms with surface effects. J. Appl. Phys. 98, 074306-11 (2005) · Zbl 1314.35209
[44]	Gurtin, M.E.; Markenscoff, X.; Thurston, R.N., Effect of surface stress on natural frequency of thin crystals, Appl. Phys. Lett., 29, 529-530, (1976)
[45]	Gurtin, M.E.; Murdoch, A.I., Addenda to our paper A continuum theory of elastic material surfaces, Arch. Ration. Mech. Anal., 59, 389-390, (1975) · Zbl 0349.73008
[46]	Gurtin, M.E.; Murdoch, A.I., A continuum theory of elastic material surfaces, Arch. Ration. Mech. Anal., 57, 291-323, (1975) · Zbl 0326.73001
[47]	He, J.; Lilley, C.M., Surface effect on the elastic behavior of static bending nanowires, Nano Lett., 8, 1798-1802, (2008)
[48]	Heinonen, S.; Huttunen-Saarivirta, E.; Nikkanen, J.P.; Raulio, M.; Priha, O.; Laakso, J.; Storgårds, E.; Levänen, E., Antibacterial properties and chemical stability of superhydrophobic silver-containing surface produced by sol-gel route, Colloids Surf. A Physicochem. Eng. Aspects, 453, 149-161, (2014)
[49]	Huang, G.Y.; Yu, S.W., Effect of surface piezoelectricity on the electromechanical behaviour of a piezoelectric ring, Phys. Status Sol. B, 243, r22-r24, (2006)
[50]	Huang, Z.; Sun, L., Size-dependent effective properties of a heterogeneous material with interface energy effect: from finite deformation theory to infinitesimal strain analysis, Acta Mech., 190, 151-163, (2007) · Zbl 1117.74047
[51]	Huang, Z.; Wang, J., A theory of hyperelasticity of multi-phase media with surface/interface energy effect, Acta Mech., 182, 195-210, (2006) · Zbl 1121.74007
[52]	Huang Z., Wang J.: Micromechanics of nanocomposites with interface energy effect. In: Li, S., Gao, X.L. (eds) Handbook on Micromechanics and Nanomechanics, pp. 303-348. Pan Stanford Publishing, Stanford (2013)
[53]	Ibach, H., The role of surface stress in reconstruction, epitaxial growth and stabilization of mesoscopic structures, Surf. Sci. Rep., 29, 195-263, (1997)
[54]	Javili, A.; dell’Isola, F.; Steinmann, P., Geometrically nonlinear higher-gradient elasticity with energetic boundaries, J. Mech. Phys. Solids, 61, 2381-2401, (2013) · Zbl 1294.74014
[55]	Javili, A.; McBride, A.; Steinmann, P., Numerical modelling of thermomechanical solids with mechanically energetic (generalised) Kapitza interfaces, Comput. Mater. Sci., 65, 542-551, (2012) · Zbl 1352.74096
[56]	Javili, A., McBride, A., Steinmann, P.: Thermomechanics of solids with lower-dimensional energetics: on the importance of surface, interface, and curve structures at the nanoscale. A unifying review. Appl. Mech. Rev. 65, 010802-1-31 (2012)
[57]	Javili, A.; McBride, A.; Steinmann, P.; Reddy, B., Relationships between the admissible range of surface material parameters and stability of linearly elastic bodies, Philos. Mag., 92, 3540-3563, (2012)
[58]	Javili, A.; Steinmann, P., A finite element framework for continua with boundary energies. part I: the two-dimensional case, Comput. Methods Appl. Mech. Eng., 198, 2198-2208, (2009) · Zbl 1227.74075
[59]	Javili, A.; Steinmann, P., A finite element framework for continua with boundary energies. part II: the three-dimensional case, Comput. Methods Appl. Mech. Eng., 199, 755-765, (2010) · Zbl 1227.74074
[60]	Javili, A.; Steinmann, P., On thermomechanical solids with boundary structures, Int. J. Solids Struct., 47, 3245-3253, (2010) · Zbl 1203.74031
[61]	Javili, A.; Steinmann, P., A finite element framework for continua with boundary energies. part III: the thermomechanical case, Comput. Methods Appl. Mech. Eng., 200, 1963-1977, (2011) · Zbl 1228.74082
[62]	Jing, G.Y., Duan, H.L., Sun, X.M., Zhang, Z.S., Xu, J., Wang, Y.D.L.J.X., Yu, D.P.: Surface effects on elastic properties of silver nanowires: Contact atomic-force microscopy. Phys. Rev. B 73, 235409-6 (2006) · Zbl 1073.74593
[63]	Kampshoff, E.; Hahn, E.; Kern; K., Correlation between surface stress and the vibrational shift of CO chemisorbed on cu surfaces, Phys. Rev. Lett., 73, 704-707, (1994)
[64]	Kang, X.; Zi, W.W.; Xu, Z.G.; Zhang, H.L., Controlling the micro/nanostructure of self-cleaning polymer coating, Appl. Surf. Sci., 253, 8830-8834, (2007)
[65]	Kim, C.; Ru, C.; Schiavone, P., A clarification of the role of crack-tip conditions in linear elasticity with surface effects, Math. Mech. Solids, 18, 59-66, (2013)
[66]	Kim, C.I.; Schiavone, P.; Ru, C.Q., Effect of surface elasticity on an interface crack in plane deformations, Proc. R. Soc. A, 467, 3530-3549, (2011) · Zbl 1243.74005
[67]	Kushch, V.I.; Chernobai, V.S.; Mishuris, G.S., Longitudinal shear of a composite with elliptic nanofibers: local stresses and effective stiffness, Int. J. Eng. Sci., 84, 79-94, (2014) · Zbl 06982829
[68]	Kushch, V.I.; Sevostianov, I.; Chernobai, V.S., Effective conductivity of composite with imperfect contact between elliptic fibers and matrix: maxwell’s homogenization scheme, Int. J. Eng. Sci., 83, 146-161, (2014) · Zbl 1423.74048
[69]	Lagowski, J.; Gatos, H.C.; Sproles, E.S., Surface stress and normal mode of vibration of thin crystals: gaas, Appl. Phys. Lett., 26, 493-495, (1975)
[70]	Laplace, P.S.: Sur l’action capillaire. supplément à la théorie de l’action capillaire. In: Traité de mécanique céleste, vol. 4. Supplement 1, Livre X, pp. 771-777. Gauthier-Villars et fils, Paris (1805) · Zbl 1227.74074
[71]	Laplace, P.S.: À la théorie de l’action capillaire. supplément à la théorie de l’action capillaire. In: Traité de mécanique céleste, vol. 4. Supplement 2, Livre X, pp. 909-945. Gauthier-Villars et fils, Paris (1806)
[72]	Liu, K.; Jiang, L., Bio-inspired self-cleaning surfaces, Annu. Rev. Mater. Res., 42, 231-263, (2012)
[73]	Liu, X.; Luo, J.; Zhu, J., Size effect on the crystal structure of silver nanowires, Nano Lett., 6, 408-412, (2006)
[74]	Longley, W.R., Name, R.G.V. (eds.): The Collected Works of J. Willard Gibbs, PHD., LL.D. Vol. I Thermodynamics. Longmans, New York (1928) · Zbl 1110.74611
[75]	Lurie, S.; Belov, P., Gradient effects in fracture mechanics for nano-structured materials, Eng. Fract. Mech., 130, 3-11, (2014)
[76]	Lurie, S.; Volkov-Bogorodsky, D.; Zubov, V.; Tuchkova, N., Advanced theoretical and numerical multiscale modeling of cohesion/adhesion interactions in continuum mechanics and its applications for filled nanocomposites, Comput. Mater. Sci., 45, 709-714, (2009)
[77]	Lurie, S.A.; Belov, P.A., Cohesion field: barenblatt’s hypothesis as formal corollary of theory of continuous media with conserved dislocations, Int. J. Fract., 150, 181-194, (2008) · Zbl 1143.74012
[78]	Lurie, S.A.; Kalamkarov, A.L., General theory of continuous media with conserved dislocations, Int. J. Solids Struct., 44, 7468-7485, (2007) · Zbl 1166.74309
[79]	Ma, X.; Liu, A.; Xu, H.; Li, G.; Hu, M.; Wu, G., A large-scale-oriented zno rod array grown on a Glass substrate via an in situ deposition method and its photoconductivity, Mater. Res. Bull., 43, 2272-2277, (2008)
[80]	Melechko, A.V.; Merkulov, V.I.; McKnight, T.E.; Guillorn, M.; Klein, K.L.; Lowndes, D.H.; Simpson, M.L., Vertically aligned carbon nanofibers and related structures: controlled synthesis and directed assembly, J. Appl. Phys., 97, 041301, (2005)
[81]	Michelitsch, T.; Maugin, G.; Nowakowski, A.; Nicolleau, F.; Rahman, M., The fractional Laplacian as a limiting case of a self-similar spring model and applications to n-dimensional anomalous diffusion, Fract. Calc. Appl. Anal., 16, 827-859, (2013) · Zbl 1314.35209
[82]	Miller, R.E.; Shenoy, V.B., Size-dependent elastic properties of nanosized structural elements, Nanotechnology, 11, 139, (2000)
[83]	Mindlin, R.D., Second gradient of strain and surface-tension in linear elasticity, Int. J. Solids Struct., 1, 417-438, (1965)
[84]	Mishuris, G.S., Interface crack and nonideal interface concept (mode III), Int. J. Fract., 107, 279-296, (2001)
[85]	Mishuris, G.S.: Mode III interface crack lying at thin nonhomogeneous anisotropic interface. Asymptotics near the crack tip. In: Movchan A.B. (ed.) IUTAM Symposium on Asymptotics, Singularities and Homogenisation in Problems of Mechanics, Solid Mechanics and Its Applications, vol. 113, pp. 251-260. Kluwer, New York (2004)
[86]	Mishuris, G.S.; Kuhn, G., Asymptotic behaviour of the elastic solution near the tip of a crack situated at a nonideal interface, ZAMM, 81, 811-826, (2001) · Zbl 1003.35121
[87]	Ostoja-Starzewski, M., Lattice models in micromechanics, Appl. Mech. Rev., 55, 35-59, (2002) · Zbl 1110.74611
[88]	Ostoja-Starzewski, M.; Li, J.; Joumaa, H.; Demmie, P., From fractal media to continuum mechanics, ZAMM, 94, 373-401, (2014) · Zbl 1302.74011
[89]	Özgür, Ü.; Alivov, Y.I.; Liu, C.; Teke, A.; Reshchikov, M.; Doğan, S.; Avrutin, V.; Cho, S.J.; Morkoc, H., A comprehensive review of zno materials and devices, J. Appl. Phys., 98, 041301, (2005)
[90]	Pan, X.H.; Yu, S.W.; Feng, X.Q., A continuum theory of surface piezoelectricity for nanodielectrics, Sci. China Phys. Mech. Astronomy, 54, 564-573, (2011)
[91]	Podio-Guidugli, P.; Caffarelli, G.V., Surface interaction potentials in elasticity, Arch. Ration. Mech. Anal., 109, 345-385, (1990)
[92]	de Poisson S.D.: Nouvelle théorie de l’action capillaire. Bachelier Père et Fils, Paris (1831)
[93]	Povstenko, Y.: Mathematical modeling of phenomena caused by surface stresses in solids. In: Altenbach, H., Morozov, N.F. (eds.) Surface Effects in Solid Mechanics, pp. 135-153. Springer, Berlin (2013) · Zbl 1255.74043
[94]	Rios, P.; Dodiuk, H.; Kenig, S.; McCarthy, S.; Dotan, A., Transparent ultra-hydrophobic surfaces, J. Adhes. Sci. Technol., 21, 399-408, (2007)
[95]	Rosi, G.; Madeo, A.; Guyader, J.L., Switch between fast and slow Biot compression waves induced by “second gradient microstructure” at material discontinuity surfaces in porous media, Int. J. Solids Struct., 50, 1721-1746, (2013)
[96]	Rowlinson J.S., Widom B.: Molecular Theory of Capillarity. Dover, New York (2003)
[97]	Rubin, M.; Benveniste, Y., A Cosserat shell model for interphases in elastic media, J. Mech. Phys. Solids, 52, 1023-1052, (2004) · Zbl 1112.74422
[98]	Sanjay, S.L., Annaso, B.G., Chavan, S.M., Rajiv, S.V.: Recent progress in preparation of superhydrophobic surfaces: a review. J. Surf. Eng. Mater. Adv. Technol. 2(2), 1-19, Art ID:18791 (2012)
[99]	Schiavone, P.; Ru, C.Q., Solvability of boundary value problems in a theory of plane-strain elasticity with boundary reinforcement, Int. J. Eng. Sci., 47, 1331-1338, (2009) · Zbl 1213.35388
[100]	Sciarra, G.; dell’Isola, F.; Coussy, O., Second gradient poromechanics, Int. J. Solids Struct., 44, 6607-6629, (2007) · Zbl 1166.74341
[101]	Seppecher, P.: Les fluides de Cahn-Hilliard. Mémoire d’habilitation á diriger des recherches, Université du Sud Toulon (1996) · Zbl 1167.74393
[102]	Sfyris, D.; Sfyris, G.; Galiotis, C., Curvature dependent surface energy for a free standing monolayer graphene: some closed form solutions of the non-linear theory, Int. J. Non-Linear Mech., 67, 186-197, (2014) · Zbl 06982879
[103]	Shenoy, V.B., Atomistic calculations of elastic properties of metallic fcc crystal surfaces, Phys. Rev. B, 71, 094104, (2005)
[104]	Sigaeva, T., Schiavone, P.: The effect of surface stress on an interface crack in linearly elastic materials. Math. Mech. Solids. (2014). doi:10.1177/1081286514534871 · Zbl 1370.74137
[105]	Sigaeva, T.; Schiavone, P., Solvability of a theory of anti-plane shear with partially coated boundaries, Arch. Mech., 66, 113-125, (2014) · Zbl 1298.74165
[106]	Spinelli, P.; Verschuuren, M.; Polman, A., Broadband omnidirectional antireflection coating based on subwavelength surface mie resonators, Nat. Commun., 3, 692, (2012)
[107]	Steigmann, D.J.; Ogden, R.W., Plane deformations of elastic solids with intrinsic boundary elasticity, Proc. R. Soc. A, 453, 853-877, (1997) · Zbl 0938.74014
[108]	Steigmann, D.J.; Ogden, R.W., Elastic surface-substrate interactions, Proc. R. Soc. A, 455, 437-474, (1999) · Zbl 0926.74016
[109]	Tan, L.K.; Kumar, M.K.; An, W.W.; Gao, H., Transparent, well-aligned tio_{2} nanotube arrays with controllable dimensions on Glass substrates for photocatalytic applications, ACS Appl. Mater. Interfaces, 2, 498-503, (2010)
[110]	Tian, X.; Yi, L.; Meng, X.; Xu, K.; Jiang, T.; Lai, D., Superhydrophobic surfaces of electrospun block copolymer fibers with low content of fluorosilicones, Appl. Surf. Sci., 307, 566-575, (2014)
[111]	Šilhavý, M., A direct approach to nonlinear shells with application to surface-substrate interactions, Math. Mech. Complex Syst., 1, 211-232, (2013) · Zbl 1391.74145
[112]	Wang, G.F.; Feng, X.Q., Effects of surface elasticity and residual surface tension on the natural frequency of microbeams, Appl. Phys. Lett., 90, 231904, (2007)
[113]	Wang, G.F.; Feng, X.Q., Effect of surface stresses on the vibration and buckling of piezoelectric nanowires, EPL Lett. J. Explor. Front. Phys., 91, 56007, (2010)
[114]	Wang, J.; Duan, H.L.; Huang, Z.P.; Karihaloo, B.L., A scaling law for properties of nano-structured materials, Proc. R. Soc. A, 462, 1355-1363, (2006) · Zbl 1149.76652
[115]	Wang, J.; Huang, Z.; Duan, H.; Yu, S.; Feng, X.; Wang, G.; Zhang, W.; Wang, T., Surface stress effect in mechanics of nanostructured materials, Acta Mech. Solida Sin., 24, 52-82, (2011)
[116]	Wang, X.; Wang, X.; Zhou, J.; Song, J.; Liu, J.; Xu, N.; Wang, Z.L., Piezoelectric field effect transistor and nanoforce sensor based on a single zno nanowire, Nano Lett., 6, 2768-2772, (2006)
[117]	Wang, Z.L.; Song, J., Piezoelectric nanogenerators based on zinc oxide nanowire arrays, Science, 312, 242-246, (2006)
[118]	Wang, Z.Q.; Zhao, Y.P.; Huang, Z.P., The effects of surface tension on the elastic properties of nano structures, Int. J. Eng. Sci., 48, 140-150, (2010)
[119]	Yan, Z.; Jiang, L., Electromechanical response of a curved piezoelectric nanobeam with the consideration of surface effects, J. Phys. D Appl. Phys., 44, 365301, (2011)
[120]	Yan, Z.; Jiang, L., Surface effects on the electroelastic responses of a thin piezoelectric plate with nanoscale thickness, J. Phys. D Appl. Phys., 45, 255401, (2012)
[121]	Young, T., An essay on the cohesion of fluids, Philos. Trans. R. Soc. Lond., 95, 65-87, (1805)
[122]	Zhu, H.X.; Wang, J.X.; Karihaloo, B.L., Effects of surface and initial stresses on the bending stiffness of trilayer plates and nanofilms, J. Mech. Mater. Struct., 4, 589-604, (2009)