Abstract
We discuss a new class of antiplane surface waves in an elastic half space with surface stresses. Here we consider a surface elasticity within stress gradient model , that is when the surface stresses relate to surface strains through an integral constitutive dependence. For antiplane motions the problem is reduced to the wave equation with nonclassical dynamic boundary condition. The dispersion relation is derived.
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
Achenbach, J.D.: Wave Propagation in Elastic Solids. North-Holland, Amsterdam (1973)
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(13), 1962–1990 (2011). https://doi.org/10.1016/j.ijsolstr.2011.03.006
Berezovski, A., Engelbrecht, J., Berezovski, M.: Waves in microstructured solids: a unified viewpoint of modeling. Acta Mechanica 220(1-4), 349–363 (2011). https://doi.org/10.1007/s00707-011-0468-0
Berezovski, A., Giorgio, I., Corte, A.D.: Interfaces in micromorphic materials: wave transmission and reflection with numerical simulations. Math. Mech. Solids 21(1), 37–51 (2016). https://doi.org/10.1177/1081286515572244
dell’Isola, F., Madeo, A., Placidi, L.: Linear plane wave propagation and normal transmission and reflection at discontinuity surfaces in second gradient 3D continua. ZAMM 92(1), 52–71 (2012). https://doi.org/10.1002/zamm.201100022
Duan, H.L., Wang, J., Karihaloo, B.L.: Theory of elasticity at the nanoscale. Adv. Appl. Mech., 42, 1–68 (2008). https://doi.org/10.1016/S0065-2156(08)00001-X
Engelbrecht, J., Berezovski, A.: Reflections on mathematical models of deformation waves in elastic microstructured solids. Math. Mech. Complex Systems 3(1), 43–82 (2015). https://doi.org/10.2140/memocs.2015.3.43
Engelbrecht, J., Berezovski, A., Pastrone, F., Braun, M.: Waves in microstructured materials and dispersion. Phil. Mag. 85(33-35), 4127–4141 (2005). https://doi.org/10.1080/14786430500362769
Engelbrecht, J., Pastrone, F., Braun, M., Berezovski, A.: Hierarchies of waves in nonclassical materials. In: Delsanto P.P. (ed.) Universality of Nonclassical Nonlinearity: Applications to Non-destructive Evaluations and Ultrasonic, pp. 29–47. Springer, New York (2006). https://doi.org/10.1007/978-0-387-35851-2_3
Eremeyev, V.A.: On effective properties of materials at the nano-and microscales considering surface effects. Acta Mechanica 227(1), 29–42 (2016). https://doi.org/10.1007/s00707-015-1427-y
Eremeyev, V.A., Rosi, G., Naili, S.: Surface/interfacial anti-plane waves in solids with surface energy. Mech. Res. Commun. 74, 8–13 (2016). https://doi.org/10.1016/j.mechrescom.2016.02.018
Eremeyev, V.A., Cloud, M.J., Lebedev, L.P.: Applications of Tensor Analysis in Continuum Mechanics. World Scientific, New Jersey (2018a). https://doi.org/10.1142/10959
Eremeyev, V.A., Lebedev, L.P., Cloud, M.J.: Acceleration waves in the nonlinear micromorphic continuum. Mech. Res. Commun. 93, 70–74 (2018b). https://doi.org/10.1016/j.mechrescom.2017.07.004
Eremeyev, V.A., Rosi, G., Naili, S.: Comparison of anti-plane surface waves in strain-gradient materials and materials with surface stresses. Math. Mech. Solids 24(8), 2526–2535 (2019). https://doi.org/10.1177/1081286518769960
Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002). https://doi.org/10.1007/b97697
de Gennes, P.G.: Some effects of long range forces on interfacial phenomena. J. Phys. Lettr. 42(16), 377–379 (1981). https://doi.org/10.1051/jphyslet:019810042016037700
de Gennes, P.G., Brochard-Wyart, F., Quéré, D.: Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer, New York (2004). https://doi.org/10.1063/1.1878340
Giorgio, I., Della Corte, A., dell’Isola, F.: Dynamics of 1D nonlinear pantographic continua. Nonlin. Dyn. 88(1), 21–31 (2017). https://doi.org/10.1007/s11071-016-3228-9
Gourgiotis, P., Georgiadis, H.: Torsional and {SH} surface waves in an isotropic and homogenous elastic half-space characterized by the Toupin–Mindlin gradient theory. Int. J. Solids Struct. 62, 217–228 (2015). https://doi.org/10.1016/j.ijsolstr.2015.02.032
Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Rat. Mech. Anal. 57(4), 291–323 (1975). https://doi.org/10.1007/bf00261375
Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14(6), 431–440 (1978)
Israelachvili, J.N.: Intermolecular and Surface Forces, 3rd edn. Academic Press, Amsterdam (2011). https://doi.org/10.1016/b978-0-12-391927-4.10024-6
Lebedev, L.P., Cloud, M.J., Eremeyev, V.A.: Tensor Analysis with Applications in Mechanics. World Scientific, New Jersey (2010). https://doi.org/10.1142/7826
Maugin, G.A.: Non-Classical Continuum Mechanics: A Dictionary. Springer, Singapore (2017). https://doi.org/10.1007/978-981-10-2434-4
Misra, A., Poorsolhjouy, P.: Granular micromechanics based micromorphic model predicts frequency band gaps. Cont. Mech. Thermodyn. 28(1-2), 215–234 (2016). https://doi.org/10.1007/s00161-015-0420-y
Rosi, G., Auffray, N.: Anisotropic and dispersive wave propagation within strain-gradient framework. Wave Motion 63, 120–134 (2016). https://doi.org/10.1016/j.wavemoti.2016.01.009
Rosi, G., Nguyen, V.H., Naili, S.: Surface waves at the interface between an inviscid fluid and a dipolar gradient solid. Wave Motion 53, 51–65 (2015). https://doi.org/10.1016/j.wavemoti.2014.11.004
Rowlinson, J.S., Widom, B.: Molecular Theory of Capillarity. Dover, New York (2003)
Simmonds, J.G.: A Brief on Tensor Analysis, 2nd edn. Springer, New York (1994)
Steigmann, D.J., Ogden, R.W.: Plane deformations of elastic solids with intrinsic boundary elasticity. Proc. Roy. Soc. A 453(1959), 853–877 (1997). https://doi.org/10.1098/rspa.1997.0047
Steigmann, D.J., Ogden, R.W.: Elastic surface-substrate interactions. Proc. Roy. Soc. A 455(1982), 437–474 (1999). https://doi.org/10.1098/rspa.1999.0320
Vardoulakis, I., Georgiadis, H.G.: SH surface waves in a homogeneous gradient-elastic half-space with surface energy. J. Elasticity 47(2), 147–165 (1997)
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). https://doi.org/10.1016/s0894-9166(11)60009-8
Acknowledgements
The author gratefully thanks the Reviewer for the helpful constructive comments and recommendations.
The author 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-P).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Eremeyev, V.A. (2019). Antiplane Surface Wave Propagation Within the Stress Gradient Surface Elasticity. In: Berezovski, A., Soomere, T. (eds) Applied Wave Mathematics II. Mathematics of Planet Earth, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-030-29951-4_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-29951-4_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-29950-7
Online ISBN: 978-3-030-29951-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)