Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-23T17:02:01.287Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability Analysis on the Algorithm of Constitutive Relation in Viscoplastic Materials

Published online by Cambridge University Press:  03 June 2016

M.-T. Liu ,
Y.-C. Li ,
X.-Z. Hu ,
J. Zhang  and
T.-G. Tang
M.-T. Liu
Affiliation:
Institute of Fluid PhysicsChina Academy of Engineering PhysicsSichuan, China
Y.-C. Li
Affiliation:
Department of Modern MechanicsUniversity of Science and Technology of ChinaAnhui, China
X.-Z. Hu
Affiliation:
Department of Modern MechanicsUniversity of Science and Technology of ChinaAnhui, China
J. Zhang
Affiliation:
Department of Modern MechanicsUniversity of Science and Technology of ChinaAnhui, China
T.-G. Tang*
Affiliation:
Institute of Fluid PhysicsChina Academy of Engineering PhysicsSichuan, China
*
*Corresponding author (ttg1974@163.com)
Get access

Abstract

The numerical stability of the explicit precise algorithm, which was developed for the viscoplastic materials, was analyzed. It was found that this algorithm is not absolutely stable. A necessary but not sufficient condition for the numerical stability was deduced. It showed that the time step in numerical calculation should be less than a certain value to guarantee the stability of explicit precise algorithm. Through a series of numerical examples, the stability analysis on the explicit precise algorithm was proved to be reliable. At last, an iterative algorithm was presented for viscoplastic materials. Both of the theoretical and numerical results showed that the iterative algorithm is unconditionally stable and its convergence rate is rapid. In practice, the explicit precise algorithm and iterative algorithm can be combined to obtain reliable results with the minimum computing costs.

Keywords

Viscoplastic constitutive relationshipNumerical stabilityExplicit precise algorithmIterative algorithm
Type
Research Article
Information
Journal of Mechanics , Volume 33 , Issue 2 , April 2017 , pp. 173 - 181
Copyright
Copyright © The Society of Theoretical and Applied Mechanics 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Drucker, D. C., “A More Fundamental Approach to Plastic Stress-Strain Relations,” Journal of Applied Mechanics Transactions of the ASME, 18, pp. 323–323 (1951).Google Scholar
2. Drucker, D. C., Prager, W. and Greenberg, H. J., “Extended Limit Design Theorems for Continuous Media,” Quarterly of Applied Mathematics, 9, pp. 381–389 (1952).CrossRefGoogle Scholar
3. Palmer, A. C., Maier, G. and Drucker, D. C., “Normality Relations and Convexity of Yield Surfaces for Unstable Materials or Structural Elements,” Journal of Applied Mechanics, 34, pp. 464–470 (1967).Google Scholar
4. Li, Y. C., Tang, Z. J. and Hu, X. Z., “Further Study on the Drucker Postulate and Plastic Constitutive Relations,” Journal of University of Science and Technology of China, 18, pp. 339–345 (1988).Google Scholar
5. Hageman, L. J. and Walsh, J. M., HELP, A Multi-Material Eulerian Program for Compressible Fluid and Elastic-Plastic Flows in Two Space Dimensions and Time, Ballistic Research Laboratory Report No. 39, 1, pp. 29–36 (1971).Google Scholar
6. Hallquist, J. O., LS-DYNA Theory Manual, Livermore Software Technology Corporation, California, pp. 18.1–18.16 (2006).Google Scholar
7. Lee, J., et al., “Stress Update Algorithm for Enhanced Homogeneous Anisotropic Hardening Model,” Computer Methods in Applied Mechanics and Engineering, 286, pp. 63–86 (2015).Google Scholar
8. Peng, Q. and Chen, M. X., “An Efficient Return Mapping Algorithm for General Isotropic Elastoplasticity in Principal Space,” Computers and Structures, 3, pp. 173–184 (2012).Google Scholar
9. Xu, Y., et al., “An Adaptive Updating Full-Newton Step Interior-Point Algorithm with Modified Newton Direction,” Applied Numerical Mathematics, 91, pp. 98–106 (2015).Google Scholar
10. Ji, H. K., et al., “An Elasto-Plastic Constitutive Model with Plastic Strain Rate Potentials for Anisotropic Cubic Metals,” International Journal of Plasticity, 12, pp. 2298–2334 (2008).Google Scholar
11. Rui, P. R. C. and Yoon, J. W., “Stress Integration Method for a Nonlinear Kinematic/Isotropic Hardening Model and Its Characterization Based on Polycrystal Plasticity,” International Journal of Plasticity, 9, pp. 1684–1710 (2009).Google Scholar
12. Yoon, J. W., Yang, D. Y. and Chung, K., “Elasto-Plastic Finite Element Method Based on Incremental Deformation Theory and Continuum Based Shell Elements for Planar Anisotropic Sheet Materials,” Computer Methods in Applied Mechanics and Engineering, 1-2, pp. 23–56 (1999).Google Scholar
13. Ding, K. Z., Qin, Q. H. and Cardew-Hall, M., “Substepping Algorithms with Stress Correction for the Simulation of Sheet Metal Forming Process,” International Journal of Mechanical Sciences, 11, pp. 1289–1308 (2007).Google Scholar
14. Chen, X., Zhang, J. and Liu, J. K., “An Explicit Sub-Stepping Stress Integration Method and Its Applications in Numerical Simulations of SMA,” Applied Mathematics and Mechanics, 6, pp. 576–585 (2013).Google Scholar
15. Szabó, L. and Kossa, A., “A New Exact Integration Method for the Drucker-Prager Elastoplastic Model with Linear Isotropic Hardening,” International Journal of Solids and Structures, 1, pp. 170–190 (2012).Google Scholar
16. Yamakawa, Y., Hashiguchi, K. and Ikeda, K., “Implicit Stress-Update Algorithm for Isotropic Cam-Clay Model Based on the Subloading Surface Concept at Finite Strains,” International Journal of Plasticity, 5, pp. 634–658 (2010).Google Scholar
17. Rezania, M., Sivasithamparam, N. and Nezhad, M. M., “On the Stress Update Algorithm of an Advanced Critical State Elasto-Plastic Model and the Effect of Yield Function Equation,” Finite Elements in Analysis and Design, 6, pp. 74–83 (2014).Google Scholar
18. Pedroso, D. M., Sheng, D. and Sloan, S. W., “Stress Update Algorithm for Elastoplastic Models with Nonconvex Yield Surfaces,” International Journal for Numerical Methods in Engineering, 13, pp. 2029–2062 (2008).CrossRefGoogle Scholar
19. Li, Y. C., Tan, F. L., Yao, L. and Hu, X. Z., “Thermo-Viscoplastic Constitutive Relation of Damaged Materials with Application,” Explosion and Shock Waves, 24, pp. 289–298 (2004).Google Scholar
20. Johnson, G. R. and Cook, W. H., “A Constitutive Model and Data for Metals Subjected to Large Strains, High Strain Rates and High Temperatures,” Proceedings of the 7th International Symposium on Ballistics, the Netherlands (1983).Google Scholar