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Laminated Plates with Non-linear Visco-elastic Interlayer: The Governing Equations

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Book cover Analysis of Shells, Plates, and Beams

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 134))

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Abstract

Laminated glass panels are widely used in civil, automotive and photovoltaic industries. Polymeric interlayers exhibit time-dependent deformation even at room temperature. Therefore, inelastic deformation of the core layer should be identified from appropriate bending tests and taken into account in the analysis of laminated structures. The aim of this paper is to derive governing differential equations to describe non-linear visco-elastic behaviour of the panel based on the layer-wise plate theory. To this end equilibrium conditions, kinematical relations and constitutive equations for individual layers are introduced. With appropriate compatibility conditions, a system of linear twelfth order partial differential equations is derived.

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References

  • Altenbach H (2000) On the determination of transverse shear stiffnesses of orthotropic plates. Journal of Applied Mathematics and Physics (ZAMP) 51:629 – 649

    Google Scholar 

  • Altenbach H, Naumenko K (2002) Shear correction factors in creep-damage analysis of beams, plates and shells. JSME International Journal Series A, Solid Mechanics and Material Engineering 45:77–83

    Google Scholar 

  • Altenbach H, Naumenko K, L’vov G, Pilipenko S (2003) Numerical estimation of the elastic properties of thin-walled structures manufactured from short-fiberreinforced thermoplastics. Mechanics of Composite Materials 39(3):221–234

    Google Scholar 

  • Altenbach H, Naumenko K, Zhilin PA (2005) A direct approach to the formulation of constitutive equations for rods and shells. In: Pietraszkiewicz W, Szymczak C (eds) Shell Structures: Theory and Applications, Taylor & Francis, Leiden, pp 87–90

    Google Scholar 

  • Altenbach H, Eremeyev VA, Naumenko K (2015) On the use of the first order shear deformation plate theory for the analysis of three-layer plates with thin soft core layer. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 95(10):1004–1011

    Google Scholar 

  • Altenbach H, Altenbach J, Naumenko K (2016) Ebene Flächentragwerke. Springer, Berlin

    Google Scholar 

  • Aßmus M (2018) Globale Strukturanalyse an Photovoltaik-Modulen - Theorie, Numerik, Anwendung. Dissertation, Fakultät Maschinenbau, Otto-von-Guericke- Universität Magdeburg, Magdeburg

    Google Scholar 

  • Aßmus M (2019) Structural Mechanics of Anti-Sandwiches. SpringerBriefs in Continuum Mechanics, Springer

    Google Scholar 

  • Aßmus M, Naumenko K, Altenbach H (2016) A multiscale projection approach for the coupled global–local structural analysis of photovoltaic modules. Composite Structures 158:340–358 Carrera E (2003) Historical review of Zig-Zag theories for multilayered plates and shells. Applied Mechanics Review 56(2):287–308

    Google Scholar 

  • Corrado M, Paggi M (2013) A multi-physics and multi-scale numerical approach to microcracking and power-loss in photovoltaic modules. Composite Structures 95:630–638

    Google Scholar 

  • Eisenträger J, Naumenko K, Altenbach H, Köppe H (2015a) Application of the first-order shear deformation theory to the analysis of laminated glasses and photovoltaic panels. International Journal of Mechanical Sciences 96:163–171

    Google Scholar 

  • Eisenträger J, Naumenko K, Altenbach H, Meenen J (2015b) A user-defined finite element for laminated glass panels and photovoltaic modules based on a layer-wise theory. Composite Structures 133:265–277

    Google Scholar 

  • Filippi M, Carrera E, Valvano S (2018) Analysis of multilayered structures embedding viscoelastic layers by higher-order, and zig-zag plate elements. Composites Part B: Engineering 154:77–89 Foraboschi P (2012) Analytical model for laminated-glass plate. Composites Part B: Engineering 43(5):2094–2106

    Google Scholar 

  • Gao Y, Oterkus S (2019) Fully coupled thermomechanical analysis of laminated composites by using ordinary state based peridynamic theory. Composite Structures 207:397–424

    Google Scholar 

  • Gruttmann F,Wagner W (2017) Shear correction factors for layered plates and shells. Computational Mechanics 59(1):129–146

    Google Scholar 

  • Ivanov IV (2006) Analysis, modelling, and optimization of laminated glasses as plane beam. International Journal of Solids and Structures 43:6887–6907

    Google Scholar 

  • Lebedev LP, Cloud MJ, Eremeyev VA (2010) Tensor Analysis with Applications in Mechanics. World Scientific, New Jersey

    Google Scholar 

  • Libai A, Simmonds JG (2005) The Nonlinear Theory of Elastic Shells. Cambridge University Press

    Google Scholar 

  • Mindlin RD (1951) Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. Trans ASME J Appl Mech 18(2):31 – 38

    Google Scholar 

  • Mulliken A, Boyce M (2006) Mechanics of the rate-dependent elastic–plastic deformation of glassy polymers from low to high strain rates. International Journal of Solids and Structures 43(5):1331–1356

    Google Scholar 

  • Nase M, Rennert M, Naumenko K, Eremeyev VA (2016) Identifying traction– separation behavior of self adhesive polymeric films from in situ digital images under T-peeling. Journal of the Mechanics and Physics of Solids 91:40–55

    Google Scholar 

  • Naumenko K, Eremeyev VA (2014) A layer-wise theory for laminated glass and photovoltaic panels. Composite Structures 112:283–291

    Google Scholar 

  • Naumenko K, Eremeyev VA (2017) A layer-wise theory of shallow shells with thin soft core for laminated glass and photovoltaic applications. Composite Structures 178:434–446

    Google Scholar 

  • Naumenko K, Altenbach J, Altenbach H, Naumenko VK (2001) Closed and approximate analytical solutions for rectangular Mindlin plates. Acta Mechanica 147:153–172

    Google Scholar 

  • Nordmann J,Naumenko K, AltenbachH(2019)Adamage mechanics based cohesivezone model with damage gradient extension for creep-fatigue-interaction. Key Engineering Materials 794:253–259

    Google Scholar 

  • Paggi M, Kajari-Schröder S, Eitner U (2011) Thermomechanical deformations in photovoltaic laminates. The Journal of Strain Analysis for Engineering Design 46(8):772–782

    Google Scholar 

  • Reissner E (1944) On the theory of bending of elastic plates. J Math Phys 23:184 – 191

    Google Scholar 

  • Schulze S (2011) Charakterisierung polymerer Zwischenschichten in Verbundglas- Solarmodulen. Dissertation, Zentrum für Ingenieurwissenschaften, Martin- Luther-Universität Halle-Wittenberg, Halle

    Google Scholar 

  • Schulze S, Pander M, Naumenko K, Altenbach H (2012) Analysis of laminated glass beams for photovoltaic applications. International Journal of Solids and Structures 49(15 - 16):2027–2036

    Google Scholar 

  • Weps M (2012) Ein Beitrag zur Charakterisierung unsymmetrischer Dreischichtverbunde mit schubweicher Zwischenschicht. Dissertation, Zentrum für Ingenieurwissenschaften, Martin-Luther Universität Halle-Wittenberg, Halle

    Google Scholar 

  • Weps M, Naumenko K, Altenbach H (2013) Unsymmetric three-layer laminate with soft core for photovoltaic modules. Composite Structures 105:332–339

    Google Scholar 

  • Zemanová A, Zeman J, ŠejnohaM(2017) Comparison of viscoelastic finite element models for laminated glass beams. International Journal of Mechanical Sciences 131:380–395

    Google Scholar 

  • Zemanova A, Zeman J, Janda T, Sejnoha M (2018) Layer-wise numerical model for laminated glass plates with viscoelastic interlayer. Structural Engineering and Mechanics 65(4):369–380

    Google Scholar 

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Authors and Affiliations

  1. Lehrstuhl für Technische Mechanik, Institut für Mechanik, Otto-von-Guericke-Universität Magdeburg, 39106, Magdeburg, Germany

    Konstantin Naumenko & Holm Altenbach

Authors
  1. Konstantin Naumenko
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  2. Holm Altenbach
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Corresponding author

Correspondence to Konstantin Naumenko .

Editor information

Editors and Affiliations

  1. Chair of Engineering Mechanics, Faculty of Mechanical Engineering, Institute of Mechanics, Otto von Guericke University Magdeburg, Magdeburg, Sachsen-Anhalt, Germany

    Holm Altenbach

  2. Institute of Applied Mathematics, Tbilisi State University, Tbilisi, Georgia

    Natalia Chinchaladze

  3. FB 4 / FG 15 / IW 3, Universität Bremen, Bremen, Germany

    Reinhold Kienzler

  4. Institut für Mechanik, MS2, TU Berlin, Berlin, Berlin, Germany

    Wolfgang H. Müller

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Naumenko, K., Altenbach, H. (2020). Laminated Plates with Non-linear Visco-elastic Interlayer: The Governing Equations. In: Altenbach, H., Chinchaladze, N., Kienzler, R., Müller, W. (eds) Analysis of Shells, Plates, and Beams. Advanced Structured Materials, vol 134. Springer, Cham. https://doi.org/10.1007/978-3-030-47491-1_15

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  • DOI: https://doi.org/10.1007/978-3-030-47491-1_15

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-47490-4

  • Online ISBN: 978-3-030-47491-1

  • eBook Packages: EngineeringEngineering (R0)

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