L.U. Sultanov
Kazan Federal University, Kazan, 420008 Russia
E-mail: Lenar.Sultanov@kpfu.ru
Received August 29, 2017
Abstract
An algorithm for investigation of elastic-plastic solids with regard to finite deformations has been developed. The kinematics of elastic-plastic deformations is based on multiplicative decomposition of the deformation gradient into elastic and inelastic parts. The stress state is determined by the Cauchy stress tensor. The constitutive equations have been obtained from the second law of thermodynamics with the introduction of the elastic free energy function. The elastic free energy function is formulated in an invariant form of the left Cauchy–Green tensor. The von Mises yield criterion with isotropic hardening has been used. The radial return method with an iterative refinement of the current state of deformation has been applied for dividing the elastic and plastic deformations. The principle of virtual work in terms of the virtual velocity has been used. The numerical implementation is based on the finite element method. The solution of the necking of a circular bar has been presented.
Keywords: nonlinear elasticity, finite deformations, plasticity
Acknowledgments. The study was supported by the Russian Science Foundation (project no. 16-11-10299).
Figure Captions
Fig. 1. Force-displacement graph. Solid line – a dependence obtained with the help of the developed algorithm, □ – [20], 
– [22].
Fig. 2. Plastic deformation intensity.
References
1. Sultanov L.U. Analysis of finite elastoplastic deformations. Kinematics and constitutive equations. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2015, vol. 157, no. 4, pp. 158–165. (In Russian)
2. Eidel B., Gruttmann F. Elastoplastic orthotropy at finite strains: Multiplicative formulation and numerical implementation. Comput. Mater. Sci., 2003, vol. 28, nos. 3–4, pp. 732–742. doi: 10.1016/j.commatsci.2003.08.027.
3. Schröder J., Gruttmann F., Löblein J. A simple orthotropic finite elasto-plasticity model based on generalized stress-strain measures. Comput. Mech., 2002, vol. 30, no. 1, pp. 48–64. doi: 10.1007/s00466-002-0366-3.
4. Simo J.S. A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition: Part I. Continuum formulation. Comput. Methods Appl. Mech. Eng., 1988, vol. 66, no. 2, pp. 199–219. doi: 10.1016/0045-7825(88)90076-X.
5. Miehe C. A theory of large-strain isotropic thermoplasticity based on metric transformation tensors. Arch. Appl. Mech., 1995, vol. 66, nos. 1–2, pp. 45–64. doi: 10.1007/BF00786688.
6. Basar Y., Itskov M. Constitutive model and finite element formulation for large strain elasto-plastic analysis of shell. Comput. Mech., 1999, vol. 23, nos. 5–6, pp. 466–481. doi: 10.1007/s004660050426.
7. Meyers A., Schievbe P., Bruhns O.T. Some comments on objective rates of symmetric Eulerian tensors with application to Eulerian strain rates. Acta Mech., 2000, vol. 139, nos. 1–4, pp. 91–103. doi: 10.1007/BF01170184.
8. Xiao H., Bruhns O.T., Meyers A. A consistent finite elastoplasticity theory combining additive and multiplicative decomposition of the stretching and deformation gradient. Int. J. Plast., 2000, vol. 16, no. 2, pp. 143–177. doi: 10.1016/S0749-6419(99)00045-5.
9. Bonet J., Wood R.D. Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge, Cambridge Univ. Press, 1997. 283 p.
10. Rouainia M., Wood D.M. Computational aspects in finite strain plasticity analysis of geotechnical materials. Mech. Res. Commun., 2006, vol. 33, no. 2, pp. 123–133. doi: 10.1016/j.mechrescom.2005.06.014.
11. Simo J.S., Ortiz M. A unified approach to finite deformation elastoplastic analysis lased on the use of hyperelastic constitutive equations. Comput. Methods Appl. Mech. Eng., 1985, vol. 49, no. 2, pp. 221–245. doi: 10.1016/0045-7825(85)90061-1.
12. Eterovic A.L., Bathe K.-J. 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., 1990, vol. 30, no. 6, pp. 1099–1114. doi: 10.1002/nme.1620300602.
13. Davydov R.L., Sultanov L.U. Numerical algorithm of solving the problem of large elastic-plastic deformations by FEM. Vestn. PNIPU. Mech., 2013, no. 1, pp. 81–93. (In Russian)
14. Davydov R.L., Sultanov L.U. Numerical algorithm for investigating large elasto-plastic deformations. J. Eng. Phys. Thermophys., 2015, vol. 88, no. 5, pp. 1280–1288. doi: 10.1007/s10891-015-1310-7.
15. Golovanov A.I., Sultanov L.U. Numerical investigation of large elastoplastic strains of three-dimensional bodies. Int. Appl. Mech., 2005, vol. 41, no. 6, pp. 614–620. doi: 10.1007/s10778-005-0129-x.
16. Abdrakhmanova A.I., Sultanov L.U. Numerical modelling of deformation of hyperelastic incompressible solids. Mater. Phys. Mech., 2016, vol. 26, no. 1, pp. 30–32.
17. Golovanov A.I., Konoplev Yu.G., Sultanov L.U. Numerical investigation of finite deformations of hyperelastic bodies. IV. Finite-element implementation. Examples of the solution of problems. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2010, vol. 152, no. 3, pp. 115–126. (In Russian)
For citation: Sultanov L.U. Analysis of large elastic-plastic deformations: Integration algorithm and numerical examples. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2017, vol. 159, no. 4, pp. 509–517. (In Russian)
The content is available under the license Creative Commons Attribution 4.0 License.