Advertisement

Lobachevskii Journal of Mathematics

, Volume 39, Issue 9, pp 1478–1483 | Cite as

Analysis of Finite Elasto-Plastic Strains: Integration Algorithm and Numerical Examples

  • L. U. SultanovEmail author
Selected Articles from the Journal Uchenye Zapiski Kazanskogo Universiteta, Seriya Fiziko-Matematicheskie Nauki
  • 5 Downloads

Abstract

The paper is devoted to the development of a calculation technique for elasto-plastic solids with regard to finite strains. The kinematics of elasto-plastic strains is based on the multiplicative decomposition of the total deformation gradient into elastic and inelastic (plastic) components. The stress state is described by the Cauchy stress tensor. Physical relations are obtained from the equation of the second law of thermodynamics supplemented with a free energy function. The free energy function is written in an invariant form of the left Cauchy–Green elastic strain tensor. An elasto-plasticity model with isotropic strain hardening is considered. Based on an analog of the associated rule of plastic flows and the von Mises yield criterion, we develop the method of stress projection onto the yield surface (known as the radial return method) with an iterative refinement of the current stress-strain state. The iterative procedure is based on the introduction of additional virtual stresses to the resolving power equation. The constitutive relations for the rates and increments of the true Cauchy stresses are constructed. In terms of the incremental loading method, the variational equation is obtained on the basis of the principle of possible virtual powers. Spatial discretization is based on the finite element method; an octanodal finite element is used.We present the solution to the problem of tension of a circular bar and give a comparison with results of other authors.

Keywords and phrases

nonlinear elasticity finite strains plasticity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. U. Sultanov, “Analysis of finite elastoplastic deformations. Kinematics and constitutive equations,” Uchen. Zap. Kazan. Univ., Ser. Fiz.-Mat. Nauki 157 (4), 158–165 (2015).Google Scholar
  2. 2.
    B. Eidel and F. Gruttmann, “Elastoplastic orthotropy at finite strains: multiplicative formulation and numerical implementation,” Comput. Mater. Sci. 28, 732–742 (2003). doi 10.1016/j. commatsci. 2003. 08. 027CrossRefGoogle Scholar
  3. 3.
    J. Schröder, F. Gruttmann, and J. Löblein, “A simple orthotropic finite elasto-plasticity model based on generalized stress-strain measures,” Comput. Mech. 30, 48–64 (2002). doi 10.1007/s00466-002-0366-3CrossRefzbMATHGoogle Scholar
  4. 4.
    J. S. Simo, “A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition: Part I. Continuumformulation,” Comput. Methods Appl. Mech. Eng. 66, 199–219 (1988). doi 10.1016/0045-7825(88)90076-XCrossRefzbMATHGoogle Scholar
  5. 5.
    C. Miehe, “A theory of large-strain isotropic thermoplasticity based on metric transformation tensors,” Arch. Appl. Mech. 66, 45–64 (1995). doi 10.1007/BF00786688zbMATHGoogle Scholar
  6. 6.
    Y. Basar and M. Itskov, “Constitutive model and finite element formulation for large strain elasto-plastic analysis of shell,” Comput. Mech. 23, 466–481 (1999). doi 10.1007/s004660050426CrossRefzbMATHGoogle Scholar
  7. 7.
    A. Meyers, P. Schievbe, and O. T. Bruhns, “Some comments on objective rates of symmetric Eulerian tensors with application to Eulerian strain rates,” Acta Mech. 139, 91–103 (2000). doi 10.1007/BF01170184CrossRefzbMATHGoogle Scholar
  8. 8.
    H. Xiao, O. T. Bruhns, and A. Meyers, “A consistent finite elastoplasticity theory combining additive and multiplicative decomposition of the stretching and deformation gradient,” Int. J. Plasticity 16, 143–177 (2000). doi 10.1016/S0749-6419(99)00045-5CrossRefzbMATHGoogle Scholar
  9. 9.
    J. Bonet and R. D. Wood, Nonlinear Continuum Mechanics for Finite Element Analysis (Cambridge Univ. Press, Cambridge, 1997).zbMATHGoogle Scholar
  10. 10.
    M. Rouainia and D. M. Wood, “Computational aspects in finite strain plasticity analysis of geotechnical materials,” Mech. Res. Commun. 33, 123–133 (2006). doi 10.1016/j. mechrescom. 2005. 06. 014CrossRefzbMATHGoogle Scholar
  11. 11.
    J. S. Simo and M. Ortiz, “A unified approach to finite deformation elastoplastic analysis lased on the use of hyperelastic constitutive equations,” Comput. Methods. Appl. Mech. Eng. 49, 221–245 (1985). doi 10.1016/0045-7825(85)90061-1CrossRefzbMATHGoogle Scholar
  12. 12.
    A. L. Eterovic and K.-J. Bathe, “A hyperelastic-based large strain elasto-plastic constitutive formulation with combined isotropic-kinematic hardening using the logarithmic stress and strain measures,” Int. J. Numer. Meth. Eng. 30, 1099–1114 (1990). doi 10.1002/nme. 1620300602CrossRefzbMATHGoogle Scholar
  13. 13.
    R. L. Davydov and L. U. Sultanov, “Numerical algorithm of solving the problem of large elastic-plastic deformation by FEM,” Vestn. Perm. Politekh. Univ., Mekh., No. 1, 81–93 (2013).Google Scholar
  14. 14.
    R. L. Davydov and L. U. Sultanov, “Numerical algorithm for investigating large elasto-plastic deformations,” J. Eng. Phys. Thermophys. 88, 1280–1288 (2015). doi 10.1007/s10891-015-1310-7CrossRefGoogle Scholar
  15. 15.
    A. I. Golovanov and L. U. Sultanov, “Numerical investigation of large elastoplastic strains of threedimensional bodies,” Int. Appl. Mech. 41, 614–620 (2005). doi 10.1007/s10778-005-0129-xCrossRefGoogle Scholar
  16. 16.
    A. I. Abdrakhmanova and L. U. Sultanov, “Numerical modelling of deformation of hyperelastic incompressible solids,” Mater. Phys. Mech. 26, 30–32 (2016).Google Scholar
  17. 17.
    A. I. Golovanov, Yu. G. Konoplev, and L. U. Sultanov, “Numerical investigation of finite deformations of hyperelastic bodies. IV. Finite-element implementation. Examples of the solution of problems,” Uch. Zap. Kazan. Univ., Ser. Fiz.-Mat. Nauki 152, 115–126 (2010).MathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Kazan (Volga Region) Federal UniversityKazanRussia

Personalised recommendations