Abstract
We carry out dimension reduction in the homogenization theory 3D periodicity cell problem for the plate with a unidirectional system of channel cuts. We demonstrate that the original 3D problem may be reduced to several 2D problems. The main attention is paid to the solution near the top and the bottom surfaces of the plate. Our numerical analysis indicates the existence of a new type of boundary layer at the upper and lower surfaces of the plate. We estimate the thickness of the found boundary layer. We also find a wrinkling effect on the top and bottom surfaces of the plate.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
Andrianov, I.V., Danishevskyy, V.V., Weichert, D.: Boundary layers in fibre composite materials. Acta Mech. 216(1), 3–15 (2011)
Caillerie, D.: Thin elastic and periodic plates. Mathematical Methods in the Applied Sciences 6(1), 159–191 (1984)
Cioranescu D, Donato P (1999) An Introduction to Homogenization. Oxford University Press, Oxford Cioranescu D, Paulin JSJ: Homogenization in open sets with holes. J. Math. Anal. Appl. 71(2), 590–607 (1979)
Cioranescu D, Damlamian A, Griso G (2018) The Periodic Unfolding Method, Series in Contemporary Mathematics, vol 3, Springer, Singapore, chap Homogenization in perforated domains, pp 199–235
Dryga´s P, Gluzman S, Mityushev V, Nawalaniec W, : Applied Analysis of Composite Media Analytical and Computational Results for Materials Scientists and Engineers. Elsevier, Amsterdam (2020)
van DykeM(1994) Nineteenth-century roots of the boundary-layer idea. SIAM Review 36(3):415–424
Grigolyuk, E.I., Fil’shtinskij LA (1992) Periodic Piecewise Homogeneous Elastic Structures (in Russ.). Nauka, Moscow Grigolyuk EI, Kovalev YD, Fil’shtinskii LA, : Bending of a layer weakened by through tunnel cuts. Dokl. Akad. Nauk SSSR 317(1), 51–53 (1991)
Kalamkarov AL, Kolpakov AG (1997) Analysis, Design and Optimization of Composite Structures. Wiley, Chichester
Kohn, R.V., Vogelius, M.: A new model for thin plates with rapidly varying thickness. Int. J. Solids Struct. 20(4), 333–350 (1984)
Kolpakov, A.A., Kolpakov, A.G.: Capacity and Transport in Contrast Composite Structures: Asymptotic Analysis and Applications. CRC Press, Boca Raton, FL (2009)
Love, A.E.H.: A treatise on the Mathematical Theory of Elasticity. Cambridge University Press, Cambridge (2013)
Lu, J.K.: Complex Variable Methods in Plane Elasticity. World Scientific, Singapore (1995)
Mityushev, V., Rogozin, S.V.: Constructive Methods for Linear and Nonlinear Boundary Value Problems of Analytic Function Theory. Chapman & Hall/CRC, Boca Raton, FL (2000)
Pipes, R.B., Pagano, N.: Interlaminar stresses in composite laminates under uniform axial extension. J. Compos. Mater. 4(4), 538–548 (1970)
Sedov LI (1971) A Course in Continuum Mechanics. Wolters-Noordho
Sendeckyj GP (1974) Mechanics of Composite Materials, vol 2, Academic Press, New York, London, chap Elastic behavior of composites, pp 45–83
Thompson, M.K., Thompson, J.M.: ANSYS Mechanical APDL for Finite Element Analysis. Butterworth-Heinemann, Oxford (2017)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Kolpakov, A.G., Rakin, S.I. (2022). Dimension Reduction in the Plate with Tunnel Cuts. In: Altenbach, H., Bauer, S., Eremeyev, V.A., Mikhasev, G.I., Morozov, N.F. (eds) Recent Approaches in the Theory of Plates and Plate-Like Structures. Advanced Structured Materials, vol 151. Springer, Cham. https://doi.org/10.1007/978-3-030-87185-7_12
Download citation
DOI: https://doi.org/10.1007/978-3-030-87185-7_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-87184-0
Online ISBN: 978-3-030-87185-7
eBook Packages: EngineeringEngineering (R0)