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Corner Singularities and Regularity Results for the Reissner/Mindlin Plate Model

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Abstract

The theory for elliptic boundary value problems for general elliptic systems is used in order to investigate systematically corner singularities and regularity for weak solutions to a broad class of boundary value problems for the Reissner/Mindlin plate model in polygonal domains. The regularity results for the deflection of the midplane and for the rotation of fibers normal to the midplane are formulated in Sobolev spaces H s, where s>1 is a real number. The number s depends on the geometry, the material parameters and the boundary conditions in general and is related to a decomposition of the fields in a singular and a regular part. The leading singular terms are calculated for a wide class of boundary conditions (36 combinations). The results are critically compared with those known from a stress potential approach.

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References

  1. Agranovic, M.S., Vishik, M.I.: Elliptic problems with a parameter and parabolic problems of general type. Russ. Math. Surv. 19, 53–157 (1964)

    Article  Google Scholar 

  2. Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I. Comm. Pure Appl. Math. 12, 623–727 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  3. Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Comm. Pure Appl. Math. 17, 35–92 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  4. Babuška, I., Li, L.: Hierarchic modelling of plates. Comput. Struct. 40, 419–430 (1991)

    Article  Google Scholar 

  5. Babuška, I., Li, L.: The problem of plate modeling: Theoretical and computational results. Comput. Methods Appl. Mech. Eng. 100, 249–273 (1992)

    Article  MATH  ADS  Google Scholar 

  6. Blum, H., Rannacher, R.: On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Methods Appl. Sci. 2, 556–581 (1980)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  7. Burton, W.S., Sinclair, G.B.: On the singularities in Reissner’s theory for the bending of elastic plates. J. Appl. Mech. 53, 220–222 (1986)

    Article  ADS  Google Scholar 

  8. Grisvard, P.: Singularités en elasticité. Arch. Ration. Mech. Anal. 107, 157–180 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  9. Lagnese, J.E.: Boundary stabilization of thin plates. SIAM Stud. Appl. Math., vol. 10. SIAM, Philadelphia (1989)

    Book  MATH  Google Scholar 

  10. Kondrat’ev, V.A.: Boundary value problems for elliptic equations in domains with conical or angular points. Trans. Mosc. Math. Soc. 16, 227–312 (1967)

    MATH  Google Scholar 

  11. Kozlov, V.A., Maz’ya, V.G., Rossmann, J.: Elliptic Boundary Value Problems in Domains with Point Singularities. American Mathematical Society, Providence (1997)

    MATH  Google Scholar 

  12. Kozlov, V.A., Maz’ya, V.G., Rossmann, J.: Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations. American Mathematical Society, Providence (2001)

    MATH  Google Scholar 

  13. Melzer, H., Rannacher, R.: Spannungskonzentrationen in Eckpunkten der Kirchhoffschen Platte. Bauingenieur 55, 181–184 (1980)

    Google Scholar 

  14. Mindlin, R.D.: Influence of rotary inertia and shear on flexural motions of elastic plates. Trans. ASME Ser. E 18, 31–38 (1951)

    MATH  Google Scholar 

  15. Morozov, N.F.: Matematicheskye voprosy teorii treshchin (Mathematical Methods in the Theory of Cracks). Nauka, Moscow (1984), in Russian

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  17. Rössle, A.: Corner singularities and regularity of weak solutions for the two-dimensional Lamé equations on domains with angular corners. J. Elast. 60, 57–75 (2000)

    Article  MATH  Google Scholar 

  18. Rössle, A., Sändig, A.-M.: Corner singularities and regularity results for the Reissner/Mindlin plate model. Report 2001/04 of the SFB 404 at the University Stuttgart (2001)

  19. Rössle, A., Bischoff, M., Wendland, W., Ramm, E.: On the mathematical foundation of the (1,1,2)-plate model. Int. J. Solids Struct. 36, 2143–2168 (1999)

    Article  MATH  Google Scholar 

  20. Sändig, A.-M., Richter, U., Sändig, R.: The regularity of boundary value problems for the Lamé equations in a polygonal domain. Rostock. Math. Kolloqui. Heft 36, 21–50 (1989)

    MATH  Google Scholar 

  21. Schwab, C.: Hierarchic models of elliptic boundary value problems on thin domains—a-posteriori error estimation and Fourier analysis. Habilitation thesis, University of Stuttgart, Mathematical Institute A (1995)

  22. Schwab, C.: A-posteriori modeling error estimation for hierarchic plate models. Num. Math. 74, 221–259 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  23. Schwab, C., Wright, S.: Boundary layers of hierarchical beam and plate models. J. Elast. 38, 1–40 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  24. Wloka, J.T., Rowley, B., Lawruk, B.: Boundary Value Problems for Elliptic Systems. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  25. Yen, D.H.Y., Zhou, M.: On the corner singularities in Reissner’s theory for the bending of elastic plates. J. Appl. Mech. 56, 726–729 (1989)

    Article  MATH  ADS  Google Scholar 

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  1. IANS, Universität Stuttgart, Pfaffenwaldring 57, 70569, Stuttgart, Germany

    Andreas Rössle & Anna-Margarete Sändig

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  1. Andreas Rössle
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  2. Anna-Margarete Sändig
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Correspondence to Anna-Margarete Sändig.

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Rössle, A., Sändig, AM. Corner Singularities and Regularity Results for the Reissner/Mindlin Plate Model. J Elast 103, 113–135 (2011). https://doi.org/10.1007/s10659-010-9258-5

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