Abstract
We discuss here the bending deformations of a three-layered plate taking into account surface and interfacial stresses. The first-order shear deformation plate theory and the Gurtin-Murdoch model of surface stresses will be considered and the formulae for stiffness parameters of the plate are derived. Their dependence on surface elastic moduli will be analyzed.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Altenbach H (2000) An alternative determination of transverse shear stiffnesses for sandwich and laminated plates. Int J Solids Struct 37(25):3503–3520
Altenbach H, Eremeyev V (2017a) Thin-walled structural elements: Classification, classical and advanced theories, new applications. In: Altenbach H, Eremeyev V (eds) Shell-like Structures: Advanced Theories and Applications, CISM Courses and Lectures, vol 572, Springer, Cham, pp 1–62
Altenbach H, Eremeyev VA (2011) On the shell theory on the nanoscale with surface stresses. Int J Engng Sci 49(12):1294–1301
Altenbach H, Eremeyev VA (2017b) On the elastic plates and shells with residual surface stresses. Proc IUTAM 21:25–32
Altenbach H, Eremeev VA, Morozov NF (2010a) On equations of the linear theory of shells with surface stresses taken into account. Mech Solids 45(3):331–342
Altenbach H, Eremeyev VA, Lebedev LP (2010b) On the existence of solution in the linear elasticity with surface stresses. ZAMM 90(3):231–240
Altenbach H, Eremeyev VA, Lebedev LP (2011) On the spectrum and stiffness of an elastic body with surface stresses. ZAMM 91(9):699–710
Altenbach H, Eremeyev VA, Morozov NF (2012) Surface viscoelasticity and effective properties of thin-walled structures at the nanoscale. Int J Engng Sci 59:83–89
Altenbach H, Eremeyev VA, Morozov NF (2013) On the influence of residual surface stresses on the properties of structures at the nanoscale. In: Altenbach H, Morozov NF (eds) Surface Effects in Solid Mechanics, Adv. Struct. Mat., vol 30, Springer, Heidelberg, pp 21–32
Chróścielewski J, Makowski J, Pietraszkiewicz W (2004) Statics and Dynamics of Multyfolded Shells. Nonlinear Theory and Finite Elelement Method (in Polish). Wydawnictwo IPPT PAN, Warszawa
dell’Isola F, Seppecher P (1995) The relationship between edge contact forces, double forces and interstitial working allowed by the principle of virtual power. C R de l’Académie des sciences Série IIb, Mécanique, physique, astronomie 321:303–308
dell’Isola F, Seppecher P (1997) Edge contact forces and quasi-balanced power. Meccanica 32(1):33–52
dell’Isola F, Seppecher P, Madeo A (2012) How contact interactions may depend on the shape of Cauchy cuts in Nth gradient continua: approach “á la d’Alembert”. ZAMP 63(6):1119–1141
Duan H, Wang J, Karihaloo BL (2009) Theory of elasticity at the nanoscale. Adv Appl Mech 42:1–68
Eremeyev VA (2016) On effective properties of materials at the nano-and microscales considering surface effects. Acta Mech 227(1):29–42
Eremeyev VA, Rosi G, Naili S (2016) Surface/interfacial anti-plane waves in solids with surface energy. Mech Res Comm 74:8–13
de Gennes PG (1981) Some effects of long range forces on interfacial phenomena. J Physique Lettres 42(16):377–379
Gurtin ME, Murdoch AI (1975) A continuum theory of elastic material surfaces. Arch Ration Mech Anal 57(4):291–323
Gurtin ME, Murdoch AI (1978) Surface stress in solids. Int J Solids Struct 14(6):431–440
Javili A, McBride A, Steinmann P (2013) 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(1):010,802
Lebedev LP, Cloud MJ, Eremeyev VA (2010) Tensor Analysis with Applications in Mechanics. World Scientific, New Jersey
Libai A, Simmonds JG (1998) The Nonlinear Theory of Elastic Shells, 2nd edn. Cambridge University Press, Cambridge
Mindlin RD (1965) Second gradient of strain and surface-tension in linear elasticity. Int J Solids Struct 1(4):417–438
Nazarenko L, Stolarski H, Altenbach H (2017) A definition of equivalent inhomogeneity applicable to various interphase models and various shapes of inhomogeneity. Proc IUTAM 21:63–70
Rowlinson JS, Widom B (2003) Molecular Theory of Capillarity. Dover, New York
Ru CQ (2016) A strain-consistent elastic plate model with surface elasticity. Cont Mech Thermodyn 28(1–2):263–273
Wang J, Huang Z, Duan H, Yu S, Feng X, Wang G, Zhang W, Wang T (2011) Surface stress effect in mechanics of nanostructured materials. Acta Mech Solida Sin 24(1):52–82
Wang J, Duan HL, Huang ZP, Karihaloo BL (2006) A scaling law for properties of nano-structured materials. Proc Roy Soc A 462(2069):1355–1363
Wang Z, Zhao Y (2009) Self-instability and bending behaviors of nano plates. Acta Mech Solida Sin 22(6):630–643
Acknowledgements
V.A.E. acknowledges financial support from the Russian Science Foundation under the grant Methods of microstructural nonlinear analysis, wave dynamics and mechanics of composites for research and design of modern metamaterials and elements of structures made on its base (No 15-19-10008).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Altenbach, H., Eremeyev, V.A. (2018). Bending of a Three-Layered Plate with Surface Stresses. In: Altenbach, H., Carrera, E., Kulikov, G. (eds) Analysis and Modelling of Advanced Structures and Smart Systems. Advanced Structured Materials, vol 81. Springer, Singapore. https://doi.org/10.1007/978-981-10-6895-9_1
Download citation
DOI: https://doi.org/10.1007/978-981-10-6895-9_1
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-10-6764-8
Online ISBN: 978-981-10-6895-9
eBook Packages: EngineeringEngineering (R0)