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Matrix FEM equation describing the large-strain deformation of an incompressible material

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A tensor–matrix FEM equation describing large-strain deformation is derived. The equation is simplified and modified to describe the deformation of incompressible materials. The results of test analysis are presented

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References

  1. Yu. N. Bukharev and Yu. A. Churilov, “Postcritical behavior of shell structures,” Vest. NNGU, Ser. Mekh., No. 2, 161–165 (2000).

  2. A. I. Golovanov and L. U. Sultanov, “Numerical investigation of large elastoplastic strains of three-dimensional bodies,” Int. App. Mech., 41, No. 6, 614–620 (2005).

    Article  Google Scholar 

  3. A. N. Guz, Elastic Waves in Bodies with Initial (Residual) Stresses [in Russian], A.S.K., Kyiv (2004).

    Google Scholar 

  4. A. I. Lurie, Nonlinear Theory of Elasticity, North-Holland, Amsterdam (1990).

    MATH  Google Scholar 

  5. J. T. Oden, Finite Elements of Nonlinear Continua, McGraw-Hill, New York (1971).

    Google Scholar 

  6. A. S. Semenov, “PANTOCRATOR: a finite-element software system for solving nonlinear problems of mechanics,” in: Proc. Int. Conf. on Scientific and Technological Problems of Predicting the Reliability and Life of Structures and Methods to Solve Them (St. Petersburg, October 14–17, 2003) [in Russian], SPbGPU, St. Petersburg (2003), pp. 466–480.

  7. K. F. Chernykh and Z. N. Litvinenkova, Theory of Elastic Large-Strain Deformation [in Russian], Izd. LGU, Leningrad (1988).

    Google Scholar 

  8. R. A. Arciniega and J. N. Reddy, “Tensor-based finite element formulation for geometrically nonlinear analysis of shell structures,” Comp. Meth. Appl. Mech. Eng., 196, 1048–1073 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Bouaricha and R. B. Schnabel, “Tensor methods for large sparse systems of nonlinear equations,” Math. Program., 82, No. 3, 377–400 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  10. V. V. Chekhov, “Tensor-based matrices in geometrically non-linear FEM,” Int. J. Numer. Meth. Eng., 63, No. 15, 2086–2101 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  11. C. A. Felippa, Nonlinear Finite Element Methods (ASEN 5107), Course Material, Ch. 1. Overview, Department of Aerospace Engineering Sciences, University of Colorado at Boulder (2007), http://www.colorado.edu/engineering/cas/courses.d/NFEM.d/NFEM.Ch01.d/NFEM.Ch01.pdf

  12. GNU Scientific Library, http://www.gnu.org/software/gsl/

  13. B. Jeremic and S. Sture, “Tensor objects in finite element programming,” Int. J. Numer. Meth. Eng., 41, 113–126 (1998).

    Article  MATH  Google Scholar 

  14. V. A. Maksimyuk, E. A. Storozhuk, and I. S. Chernyshenko, “Using mesh-based methods to solve nonlinear problems of statics for thin shells,” Int. Appl. Mech., 45, No. 1, 32–56 (2009).

    Article  ADS  Google Scholar 

  15. G. R. Miller, P. Arduino, J. Jang, and C. Choi, “Localized tensor-based solvers for interactive finite element applications using C++ and Java,” Comp. Struct., 81, 423–437 (2003).

    Article  Google Scholar 

  16. R. Mises, “Uber die Stabilitatsprobleme der Elastizitastheorie,” ZAMM, 3, No. 6, 406–462 (1923).

    Article  MATH  Google Scholar 

  17. T. Rojc and B. Stok, “About finite element sensitivity analysis of elastoplastic systems at large strains,” Comp. Struct.,. – 81, 1795–1809 (2003).

    Google Scholar 

  18. N. P. Semenyuk, V. M. Trach, and V. V. Ostapchuk, “Nonlinear axisymmetric deformation of anisotropic spherical shells,” Int. Appl. Mech., 45, No. 10, 1101–1111 (2009).

    Article  ADS  Google Scholar 

  19. A. V. Shimanovskii and V. V. Shalinskii, “Physically and geometrically nonlinear deformation of bars: Numerical analytic problem solving,” Int. Appl. Mech., 45, No. 5, 572–577 (2009).

    Article  ADS  Google Scholar 

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Correspondence to V. V. Chekhov.

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Translated from Prikladnaya Mekhanika, Vol. 46, No. 10, pp. 71–77, October 2010.

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Chekhov, V.V. Matrix FEM equation describing the large-strain deformation of an incompressible material. Int Appl Mech 46, 1147–1153 (2011). https://doi.org/10.1007/s10778-011-0407-8

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  • DOI: https://doi.org/10.1007/s10778-011-0407-8

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