Skip to main content
SpringerLink
Log in

A Reissner-type plate theory for monoclinic material derived by extending the uniform-approximation technique by orthogonal tensor decompositions of nth-order gradients

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

The uniform-approximation approach is an a-priori assumption free structured approach for the derivation of hierarchies of lower-dimensional theories for thin structures with increasing approximation accuracy. In this publication, we derive a second-order consistent plate theory for monoclinic material and investigate several theories that arise from the original theory by a pseudo-reduction approach which aims to reduce the number of PDEs that are to solve. A one-variable model that governs only the interior solution is presented and, in addition, an extended two-variable model that also covers edge effects. Since the second introduced variable is a rotation of a vector field, we have to uniquely identify the rotation dependent parts in general gradients of the vector field, which is resolved by the introduction of an orthogonal decomposition. The final two-variable model is equivalent to the Reissner–Mindlin theory for the special case of isotropic material, whereas the one-variable model is equivalent to the first Reissner PDE. In contrast to this special case, the two-variable model is a coupled system of two PDEs for general monoclinic material.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Altenbach H, Altenbach J, Naumenko K (1998) Ebene Flächentragwerke: Grundlagen der Modellierung und Berechnung von Scheiben und Platten. Springer, Berlin

    Book  MATH  Google Scholar 

  2. Altenbach J, Altenbach H, Eremeyev V (2010) On generalized Cosserat-type theories of plates and shells: a short review and bibliography. Arch Appl Mech 80(1):73–92. doi:10.1007/s00419-009-0365-3

    Article  MATH  Google Scholar 

  3. Ambartsumyan SA (1970) Theory of anisotropic plates: strength, stability, vibration. Technomic Publishing Company, Stamford

    Google Scholar 

  4. Arnold DN, Falk RS (1989) Edge effects in the Reissner-Mindlin plate theory. In: Analytic and computational models of shells, vol 1. ASME, New York

  5. Auffray N (2013) Geometrical picture of third-order tensors. In: Altenbach H, Forest S, Krivtsov V (eds) Generalized continua as models for materials, volume 22 of advanced structured materials. Springer, Berlin, pp 17–40. doi:10.1007/978-3-642-36394-8_2

    Chapter  Google Scholar 

  6. Bronstein IN, Semendjajew KA, Musiol G, Mühling H (2001) Taschenbuch der Mathematik, 5th edn. Verlag Harri Deutsch, Thun

    MATH  Google Scholar 

  7. Cauchy AL (1828) Sur l'èquilibre et le mouvement d'une plaque solide. Exercises de mathématique, vol 3. pp 328–355

  8. Ciarlet P, Destuynder P (1979) Approximation of three-dimensional models by two-dimensional models in plate theory. In: Glowinski E, Rodin Y, Zienkiewicz OC (eds) Energy methods in finite element analysis. (A 79-53076 24-39), Wiley, Chichester, pp 33–45

  9. Cousteix J, Mauss J (2007) Asymptotic analysis and boundary layers. Springer, Berlin

    MATH  Google Scholar 

  10. Friedrichs K, Dressler R (1961) A boundary-layer theory for elastic plates. Commun Pure Appl Math 14(1):1–33

    Article  MathSciNet  MATH  Google Scholar 

  11. Häggblad B, Bathe K (1990) Specifications of boundary conditions for Reissner/Mindlin plate bending finite elements. Int J Numer Methods Eng 30(5):981–1011

    Article  MATH  Google Scholar 

  12. Haimovici M (1966) On the bending of elastic plates. Bull Acad Polon Sci 14:1047–1057

    MathSciNet  MATH  Google Scholar 

  13. Huber MT (1921) Theory of plates, Tow. Naukowe, L’vow

    Google Scholar 

  14. Huber MT (1926) Einige Anwendungen der Biegungstheorie orthotroper Platten. ZAMM J Appl Math Mech 6(3):228–231. doi:10.1002/zamm.19260060306

    Article  MATH  Google Scholar 

  15. Huber MT (1929) Probleme der Statik technisch wichtiger orthotroper Platten. Gebethner & Wolff, Warsaw

    Google Scholar 

  16. Khoma I (1974) General solution of equilibrium equations for the deflection of plates and shells of constant thickness. Sov Phys Dokl 18:756–758

    Google Scholar 

  17. Kienzler R (2002) On consistent plate theories. Arch Appl Mech 72:229–247. doi:10.1007/s00419-002-0220-2

    Article  MATH  Google Scholar 

  18. Kienzler R (2004) On consistent second-order plate theories. In: Kienzler R, Ott I, Altenbach H (eds) Theories of plates and shells: critical review and new applications. Springer, Berlin, pp 85–96. doi:10.1007/978-3-540-39905-6

    Chapter  Google Scholar 

  19. Kienzler R, Schröder R (2009) Einführung in die Höhere Festigkeitslehre. Springer, Berlin

    Book  Google Scholar 

  20. Kirchhoff GR (1850) Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. J Reine Angew Mathe 39:51–88. http://eudml.org/doc/147439

  21. Koiter W (1966) On the nonlinear theory of thin elastic shells. K Ned Akad Wet Proc Ser B 69(1):1–54

    MathSciNet  Google Scholar 

  22. Koiter W (1970a) On the foundations of the linear theory of thin elastic shells. K Ned Akad Wet Proc Ser B 73(3):169–195

    MathSciNet  MATH  Google Scholar 

  23. Koiter W (1970b) On the mathematical foundation of shell theory. Proc Int Congr Math Nice 3:123–130

    Google Scholar 

  24. Krätzig W (1980) On the structure of consistent linear shell theories. In: Koiter W, Mikhailov G (eds) Theory of shells. North-Holland, Amsterdam, pp 353–368

    Google Scholar 

  25. Lekhnitskii S (1968) Anisotropic plates. Gordon and Breach, Cooper Station

    Google Scholar 

  26. Marsden J, Hughes T (1983) Mathematical foundations of elasticity. Dover Publications, Inc, New York

    MATH  Google Scholar 

  27. Mindlin RD (1951) Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates. J Appl Mech 18:31–38

    MATH  Google Scholar 

  28. Naghdi PM (1963) Foundations of elastic shell theory. In: Sneddon I, Hill R (eds) Progress in solid mech, vol 4. North-Holland, Amsterdam, pp 1–90

    Google Scholar 

  29. Palmow WA, Altenbach A (1982) Über eine Cosseratsche Theorie für elastische Platten. Tech Mech 3:5–9

    Google Scholar 

  30. Poisson SD (1829) Mémoire sur l'équilibre et le mouvement des élastiques, vol 8. Mémoire de l'académie des sciences de l'institut national, Paris, pp 357–627

  31. Poniatovskii V (1962) Theory for plates of medium thickness. J Appl Math Mech 26(2):478–486. doi:10.1016/0021-8928(62)90077-1

    Article  MathSciNet  Google Scholar 

  32. Pruchnicki E (2014) Two-dimensional model of order h5 for the combined bending, stretching, transverse shearing and transverse normal stress effect of homogeneous plates derived from three-dimensional elasticity. Math Mech Solids 19(5):477–490. doi:10.1177/1081286512469981

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Reissner E (1945) The effect of transverse shear deformation on the bending of elastic plates. J Appl Mech 12:69–77

    MathSciNet  MATH  Google Scholar 

  35. Schneider P (2010) Eine konsistente Plattentheorie zweiter Ordnung für monotropes Material. Master’s thesis, Universität Bremen. http://nbn-resolving.de/urn:nbn:de:gbv:46-00103412-11

  36. Schneider P (2015) On the mathematical justification of the consistent-approximation approach and the derivation of a shear-correction-factor free refined beam theory. PhD thesis, University of Bremen. http://nbn-resolving.de/urn:nbn:de:gbv:46-00104458-18

  37. Schneider P, Kienzler R (2011) An algorithm for the automatisation of pseudo reductions of PDE systems arising from the uniform-approximation technique. In: Altenbach H, Eremeyev V (eds) Shell-like structures, volume 15 of advanced structured materials. Springer, Berlin, pp 377–390. doi:10.1007/978-3-642-21855-2_25

    Chapter  Google Scholar 

  38. Schneider P, Kienzler R (2014) Comparison of various linear plate theories in the light of a consistent second-order approximation. Math Mech Solids 20:871–882. doi:10.1177/1081286514554352

    Article  MathSciNet  MATH  Google Scholar 

  39. Schneider P, Kienzler R (2015) On exact rod/beam/shaft-theories and the coupling among them due to arbitrary material anisotropies. Int J Solids Struct 56–57:265–279. doi:10.1016/j.ijsolstr.2014.10.022

    Article  Google Scholar 

  40. Schneider P, Kienzler R, Böhm M (2014) Modeling of consistent second-order plate theories for anisotropic materials. ZAMM J Appl Math Mech 94(1–2):21–42. doi:10.1002/zamm.201100033

    Article  MathSciNet  MATH  Google Scholar 

  41. Soler A (1969) Higher-order theories for structural analysis using Legendre polynomial expansions. J Appl Mech 36(4):757–762. doi:10.1115/1.3564767

    Article  MATH  Google Scholar 

  42. Steigmann DJ (2008) Two-dimensional models for the combined bending and stretching of plates and shells based on three-dimensional linear elasticity. Int J Eng Sci 46(7):654–676. doi:10.1016/j.ijengsci.2008.01.015

    Article  MathSciNet  MATH  Google Scholar 

  43. Ting T (1996) Anisotropic elasticity: theory and applications. Oxford University Press, New York

    MATH  Google Scholar 

  44. Vekua I (1985) Shell theory: general methods of construction. In: Brezis H, Douglas RG, Jeffrey A (eds) Monographs, advanced texts and surveys in pure and applied mathematics. Wiley, New York

  45. Zhgenti V, Gyuntner A, Meunargiya T, Tskhadaya F (1980) Solution of problems of the theory of plates and shells by I. N. Vekua’s method. In: Proceedings of 3rd IUTAM symposium on shell theory, North-Holland, pp 669–684

  46. Zhilin P (1992) On the Poisson and Kirchhoff plate theories from the point of view of the modern plate theory. Izvesta Akad Nauk Ross Mekhanica Averdogotela Mosc 3:48–64 (in Russian)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Production Engineering, University of Bremen, Am Biologischen Garten 2, 28359, Bremen, Germany

    Patrick Schneider & Reinhold Kienzler

Authors
  1. Patrick Schneider
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Reinhold Kienzler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Reinhold Kienzler.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Schneider, P., Kienzler, R. A Reissner-type plate theory for monoclinic material derived by extending the uniform-approximation technique by orthogonal tensor decompositions of nth-order gradients. Meccanica 52, 2143–2167 (2017). https://doi.org/10.1007/s11012-016-0573-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-016-0573-1

Keywords

Search

Navigation