Skip to main content
SpringerLink
Log in

Mixed functionals in the theory of nonlinearly elastic shells

  • Published:
International Applied Mechanics Aims and scope

Abstract

Results obtained on the basis of linearized functionals in the theory of nonlinearly elastic composite shells are analyzed and generalized. The Kirchhoff-Love and Timoshenko hypotheses are used. Possible membrane or shear locking is taken into account. New approaches are proposed to improve the convergence of numerical solution for new classes of nonlinear problems for thin and nonthin shells with a curvilinear (circular, elliptical) hole. The stress-strain state of shells is analyzed using different versions of shell theory. The influence of the nonlinear properties and orthotropy of composite materials on the stress distribution in structural members is studied.

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

Similar content being viewed by others

REFERENCES

  1. N. P. Abovskii, N. P. Andreev, and A. P. Deruga, Variational Principles in the Theory of Elasticity and the Theory of Shells [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  2. N. P. Abovskii, N. P. Andreev, A. P. Deruga, L. V. Endzhievskii, V. B. Idel’son, and N. I. Marchuk, “ Development and application of extreme variational principles in nonlinear design of complex shell systems,” in: Spatial Structures in the Krasnoyarsk Territory [in Russian], Krasnoyarsk (1986), pp. 3–17.

  3. K. Washizu, Variational Methods in Elasticity and Plasticity, Univ. Press, Oxford (1982).

    Google Scholar 

  4. A. I. Golovanov and M. S. Kornishin, An Introduction to the Finite-Element Method in Statics of Thin Shells [in Russian], Izd. Kazanskogo Fiz.-Tekhn. Inst., Kazan (1990).

    Google Scholar 

  5. L. I. Golub, V. A. Maksimyuk, and I. S. Chernyshenko, “Modeling the nonlinearly elastic deformation of orthotropic spherical shells with curvilinear holes,” Visn. Kyiv. Univ., Ser. Fiz.-Mat. Nauky, 5, 256–258 (2001).

    Google Scholar 

  6. E. A. Gotsulyak and K. Pemsing, “Allowing for rigid-body displacements in finite-element solutions for shells,” in: Numerical Methods in Structural Analysis [in Russian], Izd. Kievskogo Inzh.-Stroit. Inst., Kiev (1978), pp. 93–98.

    Google Scholar 

  7. E. A. Gotsulyak, V. N. Ermishev, and N. T. Zhadrasinov, “Convergence of the curvilinear-mesh method in shell problems,” Sopr. Mater. Teor. Sooruzh., 39, 80–84 (1981).

    Google Scholar 

  8. A. N. Guz, I. S. Chernyshenko, V. P. Georgievskii, and V. A. Maksimyuk, “Stress state of thin-walled structural members made of nonlinearly elastic orthotropic composites,” Prikl. Mekh., 24, No.4, 25–32 (1988).

    Google Scholar 

  9. A. N. Guz, V. A. Maksimyuk, and I. S. Chernyshenko, “Boundary-value problems in the theory of thin and nonthin orthotropic composite shells with nonlinearly elastic properties and low shear stiffness,” Mekh. Komp., 37, No.1, 91–100 (2001).

    Google Scholar 

  10. A. N. Guz, V. A. Maksimyuk, and I. S. Chernyshenko, “Problem-oriented functionals in the theory of nonlinearly elastic composite shells,” Mekh. Komp., 38, No.4, 497–506 (2002).

    Google Scholar 

  11. É. U. Dadamukhamedov, V. A. Maksimyuk, and I. S. Chernyshenko, “Influence of the stiffness of reinforcement on the strain state of shells with a hole,” Teor. Prikl. Mekh., 22, 67–71 (1991).

    Google Scholar 

  12. I. P. Ermakovskaya, V. A. Maksimyuk, and I. S. Chernyshenko, Nonlinearly Elastic Two-Dimensional Static Problems for Orthotropic Thin Shells and a Technique for Their Solution [in Russian], Manuscript No. 7526-V 88 dep. at VINITI 10.19.88, Red. Zh. Prikl. Mekh., Kiev (1988).

    Google Scholar 

  13. V. G. Karnaukhov, I. F. Kirichok, and V. I. Kozlov, “Influence of dissipative heating on the vibrations of thin-walled composite structures,” in: Dynamics of Structural Members, Vol. 9 of the 12-volume series Mechanics of Composites [in Russian], A.S.K., Kiev (1999), pp. 144–173.

    Google Scholar 

  14. A. N. Guz, A. S. Kosmodamianskii, V. P. Shevchenko, et al., Stress Concentration, Vol. 7 of the 12-volume series Mechanics of Composites [in Russian], A.S.K., Kiev (1998).

    Google Scholar 

  15. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers. Definitions, Theorems, and Formulas for Reference and Review, McGraw-Hill, New York (1968).

    Google Scholar 

  16. G. M. Kulikov and S. V. Plotnikova, “Mixed finite-element analysis of locally loaded multilayer shells. 2. Geometrically nonlinear formulation,” Mekh. Komp. Mater., 38, No.6, 815–826 (2002).

    Google Scholar 

  17. V. A. Lomakin, “The theory of plasticity of anisotropic media,” Vestn. Mosk. Univ., Mat. Mekh., No. 4, 49–53 (1964).

  18. V. A. Lomakin and M. A. Yumasheva, “Stress-strain relationships in nonlinear deformation of orthotropic fiberglass,” Mekh. Polim., No. 4, 28–34 (1965).

    Google Scholar 

  19. V. A. Maksimyuk, “Mathematical modeling of the nonlinearly elastic deformation of orthotropic composite shells,” Probl. Upravl. Inform., No. 2, 149–151 (1998).

  20. V. A. Maksimyuk, “Solution of physically nonlinear problems of the theory of orthotropic shells using mixed functionals,” Int. Appl. Mech., 36, No.10, 1349–1354 (2000).

    Google Scholar 

  21. V. A. Maksimyuk, “Study of the nonlinearly elastic state of an orthotropic cylindrical shell with a hole, using mixed functionals,” Int. Appl. Mech., 37, No.12, 1602–1606 (2001).

    Google Scholar 

  22. V. A. Maksimyuk, “Applying Lagrange multipliers in static problems for composite shells,” Dop. NAN Ukrainy, No. 11, 75–79 (1998).

  23. V. A. Maksimyuk, “Numerical solution of physically nonlinear stress-concentration problems for curvilinear shells,” Dop. NAN Ukrainy, No. 12, 68–71 (1998).

  24. V. A. Maksimyuk, “Physically nonlinear problems in the theory of orthotropic composite shells with curvilinear boundaries,” Prikl. Mekh., 34, No.9, 28–32 (1998).

    Google Scholar 

  25. V. A. Maksimyuk and I. S. Chernyshenko, “Stress-strain analysis of shells with holes using mixed functionals,” Teor. Prikl. Mekh., 32, 126–132 (2001).

    Google Scholar 

  26. V. A. Maksimyuk and I. S. Chernyshenko, “Nonlinear elastic state of thin-walled toroidal shells made of orthotropic composites,” Int. Appl. Mech., 35, No.12, 1238–1245 (1999).

    Google Scholar 

  27. V. A. Maksimyuk and I. S. Chernyshenko, “Numerical solution of problems for shells of variable stiffness with allowance for transverse shears,” Int. Appl. Mech., 27, No.3, 275–278 (1991).

    Google Scholar 

  28. V. A. Maksimyuk and I. S. Chernyshenko, “Numerical solution of boundary-value problems in the theory of thin shells and plates with curvilinear holes,” Teor. Prikl. Mekh., 30, 117–126 (1999).

    Google Scholar 

  29. V. A. Maksimyuk and I. S. Chernyshenko, “Numerical analysis of the efficiency of the theories of thin and nonthin composite shells in stress-concentration problems,” Teor. Prikl. Mekh., 31, 46–52 (2000).

    Google Scholar 

  30. V. A. Maksimyuk and I. S. Chernyshenko, “Mixed functionals in physically nonlinear static problems for composite shells,” in: Problems of Mechanics (an anniversary collection in honor of A. Yu. Ishlinskii ‘s 90th birthday) [in Russian], Fizmatlit, Moscow (2003), pp. 531–537.

    Google Scholar 

  31. A. O. Rasskazov and A. V. Kozlov, “Numerical study of the nonaxisymmetric vibrations of a shell of revolution under nonstationary loading,” Prikl. Mekh., 34, No.5, 68–75 (1998).

    Google Scholar 

  32. G. N. Savin, Stress Distribution around Openings [in Russian], Naukova Dumka, Kiev (1968).

    Google Scholar 

  33. A. N. Guz, I. S. Chernyshenko, V. N. Chekhov, et al., The Theory of Thin Shells Weakened by Openings, Vol. 1 of the five-volume series Methods of Shell Design [in Russian], Naukova Dumka, Kiev (1980).

    Google Scholar 

  34. I. S. Chernyshenko and V. A. Maksimyuk, “Nonlinear problems of the statics of orthotropic shells with allowance for transverse shear strain,” Int. Appl. Mech., 25, No.8, 802–807 (1989).

    Google Scholar 

  35. G. Alfano, F. Auricchio, L. Rosati, and E. Sacco, “MITC finite elements for laminated composite plates,” Int. J. Numer. Meth. Eng., 50, 707–738 (2001).

    Google Scholar 

  36. R. J. Alves de Sousa, R. M. Natal Jorge, R. A. Fontes Valenle, J. M. A. Cesar de Sa, and P. M. A. Areias, “A new volumetric and shear locking-free 3D enhanced strain element,” Eng. Comput., 20, No.7, 896–925 (2003).

    Google Scholar 

  37. F. Auricchio and E. A. Sacco, “Mixed-enhanced finite element for the analysis of laminated composite plates,” Int. J. Numer. Mech. Eng., 44, 1481–1504 (1999).

    Google Scholar 

  38. Y. Basar and O. Kintzel, “Large rotation analysis of thin-walled shells,” in: S. N. Atluri and F. Brust (eds.), Advances in Computational Engineering and Sciences, Int. Conf. on Computational Engineering Science ICCES2K, Tech. Sci. Press, Los Angeles, 1 (2000), pp. 818–823.

    Google Scholar 

  39. K.-J. Bathe, A. Iosilevich, and D. Chapelle, “An evaluation of the MITC shell elements,” Comput. Struct., 75, 1–30 (2000).

    Google Scholar 

  40. K.-J. Bathe, A. Iosilevich, and D. Chapelle, “An inf-sup test for shell finite elements,” Comput. Struct., 75, 439–456 (2000).

    Google Scholar 

  41. T. Belytschko, W. K. Liu, and B. Moran, Nonlinear Finite Elements for Continue and Structures, John Wiley & Sons, Chichester (2000).

    Google Scholar 

  42. K.-U. Bletzinger, M. Bischoff, and E. Ramm, “A unified approach for shear-locking-free triangular and rectangular shell finite elements,” Comput. Struct., 75, 321–334 (2000).

    Google Scholar 

  43. J. M. A. Cesar de Sa, R. M. Natal Jorge, R. A. Fontes Valente, and P. M. Almeida Areias, “ Development of shear locking-free shell elements using an enhanced assumed strain formulation,” Int. J. Numer. Meth. Eng., 53, 1721–1750 (2002).

    Google Scholar 

  44. V. N. Chekhov and S. V. Zakora, “On the interaction of closely spaced circular openings in a transversely isotropic shallow cylindrical shell,” Int. Appl. Mech., 39, No.4, 479–483 (2003).

    Google Scholar 

  45. J. Y. Cho and S. N. Atluri, “Analysis of shear flexible beams, using the meshless local Petrov-Galerkin method, based on a locking-free formulation,” Eng. Comput., 18, No.1–2, 215–240 (2001).

    Google Scholar 

  46. E. N. Dvorkin and K.-J. Bathe, “A continuum mechanics based on four-node shell element for general nonlinear analysis,” Eng. Comput., 1, 77–88 (1984).

    Google Scholar 

  47. R. A. Fontes Valente, R. M. Natal Jorge, R. P. R. Cardoso, J. M. A. Cesar de Sa, and J. J. A. Gracio, “On the use of an enhanced transverse shear strain shell element for problems involving large rotations,” Comp. Mech., 30, No.4, 286–296 (2003).

    Google Scholar 

  48. F. Gabaldon and J. M. Goicolea, “Linear and non-linear finite element error estimation based on assumed strain fields,” Int. J. Numer. Meth. Eng., 55, 413–429 (2002).

    Google Scholar 

  49. L. I. Golub, V. A. Maksimyuk, and I. S. Chernyshenko, “Numerical nonlinear elastic analysis of orthotropic spherical shells with an elliptic cutout,” Int. Appl. Mech., 38, No.2, 203–208 (2002).

    Google Scholar 

  50. Y. Guo, M. Ortiz, T. Belytschko, and E. A. Repetto, “Triangular composite finite element,” Int. J. Numer. Meth. Eng., 47, 287–316 (2000).

    Google Scholar 

  51. A. N. Guz, V. A. Maksimyuk, and I. S. Chernyshenko, “Numerical stress-strain analysis of shells including the nonlinear and shear properties of composites,” Int. Appl. Mech., 38, No.10, 1220–1228 (2002).

    Google Scholar 

  52. N. H. Kim, K. K. Choi, J.-S. Chen, and M. E. Botkin, “Meshfree analysis and design sensitivity analysis for shell structures,” Int. J. Numer. Meth. Eng., 53, 2116–2087 (2002).

    Google Scholar 

  53. G. M. Kulikov and S. V. Plotnikova, “Non-linear strain-displacement equations exactly representing large rigid body motions. Part I. Timoshenko-Mindlin shell theory,” Comput. Methods Appl. Mech. Eng., 192, No.7–8, 851–875 (2003).

    Google Scholar 

  54. S. Li and W. K. Liu, “Meshfree and particle methods and their applications,” Appl. Mech. Rev., 55, No.1, 1–34 (2002).

    Google Scholar 

  55. R. H. MacNeal, “The evolution of lower order plate and shell elements in MSC/NASTRAN,” Finite Elem. Anal. Des., 5, No.3, 197–222 (1989).

    Google Scholar 

  56. K. Mallikarjuna Rao and U. Shrinivasa, “A set of pathological tests to validate new finite elements,” Sadhana, 26, 549–590 (2001).

    Google Scholar 

  57. D. Mijuca, “A new primal-mixed 3D finite element,” Facta Univ. Ser.: Mech., Autom. Contr. Robot., 3 No.11, 167–178 (2001).

    Google Scholar 

  58. O. K. Morachkovski, Yu. V. Romashov, and V. A. Salo, “The method of R-functions in the solution of elastic problems on the basis of Reissner’s mixed variational principle,” Int. Appl. Mech., 38, No.2, 174–180 (2002).

    Google Scholar 

  59. K. C. Park, “Improved strain interpolation for curved C0 elements,” Int. J. Numer. Meth. Eng., 22, 281–288 (1986).

    Google Scholar 

  60. J. Pitkäranta, “Mathematical and historical reflections on the lowest-order finite element models for thin structures,” Comput. Struct., 81, No.8–11, 895–909 (2003).

    Google Scholar 

  61. G. Prathap, “The finite element method in structural engineering,” in: Solid Mechanics and Its Applications, Vol. 24, Kluwer Acad. Publ., Dordrecht (1993).

    Google Scholar 

  62. R. Schlebusch, J. Matheas, and B. Zastrau, “On the avoidance of the Poisson thickness locking in surface-related shell theories by an extension of the EAS method,” Mach. Dynam. Probl., 26, No.4, 101–114 (2002).

    Google Scholar 

  63. A. C. Scordelis and K. S. Lo, “Computer analysis of cylindrical shells,” Amer. Concr. Inst.J., 61, No.5, 539–560 (1964).

    Google Scholar 

  64. K. J. Shnerenko and V. F. Godzula, “Stress distribution in a composite cylindrical shell with a large circular opening,” Int. Appl. Mech., 39, No.11, 1323–1327 (2003).

    Google Scholar 

  65. J. C. Simo and M. S. Rifai, “A class of mixed assumed strain methods and the method of incompatible modes,” Int. J. Num. Meth. Eng., 29, 1595–1638 (1990).

    Google Scholar 

  66. J. Stegmann, R. L. Jensen, J. M. Rauhe, and E. Lund, “Shell element for geometrically non-linear analysis of composite laminates and sandwich structures,” in: Proc. 14th Nordic Seminar on Computational Mechanics Lund, Sweden, October 19–20 (2001), pp. 83–86.

  67. K. Y. Sze and L. Q. Yao, “A hybrid stress ANS solid-shell element and its generalization for smart structure modeling. Part I. Solid-shell element formulation,” Int. J. Numer. Meth. Eng., 48, 545–564 (2000).

    Google Scholar 

  68. D. G. Talaslidis and A. Ch. Tokatlidis, “Nonlinear finite element analysis of R/C shells,” Facta Univ. Ser. Mech. Autom. Contr. Robot., 2, No. 10, 1329–1348 (2000).

    Google Scholar 

  69. A. T. Vasilenko and G. P. Urusova, “Solving stress problems for elastic systems of anisotropic shells of revolution with regard for transverse shear and reduction,” Int. Appl. Mech., 39, No.5, 587–594 (2003).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev

    V. A. Maksimyuk & I. S. Chernyshenko

Authors
  1. V. A. Maksimyuk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. S. Chernyshenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Prikladnaya Mekhanika, Vol. 40, No. 11, pp. 45–84, November 2004.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Maksimyuk, V.A., Chernyshenko, I.S. Mixed functionals in the theory of nonlinearly elastic shells. Int Appl Mech 40, 1226–1262 (2004). https://doi.org/10.1007/s10778-005-0032-5

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10778-005-0032-5

Keywords

Search

Navigation