Hostname: page-component-848d4c4894-8kt4b Total loading time: 0 Render date: 2024-06-30T22:50:57.075Z Has data issue: false hasContentIssue false
  • English
  • Français

Effective constitutive algorithms in elastoplasticity and elastoviscoplasticity

Published online by Cambridge University Press:  26 September 2008

Sia Nemat-Nasser  and
Luqun Ni
Sia Nemat-Nasser
Affiliation:
Center of Excellence for Advanced Materials, Department of Applied Mechanics and Engineering Sciences, University of California, San Diego, La Jolla, CA 92093, USA
Luqun Ni
Affiliation:
Center of Excellence for Advanced Materials, Department of Applied Mechanics and Engineering Sciences, University of California, San Diego, La Jolla, CA 92093, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The basic constitutive relations for elastoplasticity and elastoviscoplasticity are shown to form a typical boundary layer-type stiff system of ordinary differential equations. Three numerical algorithms are discussed: (i) The singular perturbation method (O'Malley, 1971a, b; Hoppensteadt, 1971; Miranker, 1981; Smith, 1985), which yields accurate results for both the rate-independent and rate-dependent cases, where in the former case, the algorithm is explicit, whereas in the latter case, it is implicit and requires the solution of a nonlinear equation; therefore it is impractical as a constitutive algorithm for large-scale finite-element applications, where the constitutive algorithm is used a great number of times at each finite-element node. (ii) The new constitutive algorithm (Nemat-Nasser, 1991; Nemat-Nasser & Chung, 1989, 1992) which is explicit and accurate for both the rate-independent and rate-dependent cases; the underlying mathematical feature of this new method is investigated, and it is shown that it can be classified as a simplified perturbation method; computable error bounds for this algorithm are obtained, and when the flow rule is given by the commonly used power law, it is shown that the errors are very small, (iii) A modified outer-solution method, which combines the above two techniques, and is simple, explicit, and accurate.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 5 , Issue 3 , September 1994 , pp. 313 - 335
Copyright
Copyright © Cambridge University Press 1994

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

Hoppensteadt, F. 1971 Properties of solutions of ordinary differential equations with small parameters. Comm. Pure Appl. Math. 24, 807–840.CrossRefGoogle Scholar
Miranker, W. L. 1981 Numerical Methods for Stiff Equations and Singular Perturbation Problems. Reidel.Google Scholar
Nayfeh, A. H. 1980 Introduction to Perturbation Techniques. Wiley.Google Scholar
Nemat-Nasser, S. 1991 Rate-independent finite-deformation elastoplasticity: a new explicit constitutive algorithm. Mech. Materials 11, 235–249.Google Scholar
Nemat-Nasser, S. & Chung, D.-T. 1989 CEAM. UCSD, internal communication.Google Scholar
Nemat-Nasser, S. & Chung, D.-T. 1992 An explicit constitutive algorithm for large-strain, large-strain rate elastoviscoplasticity. Computer Methods Appl. Mech. Eng. 95, 205–219.Google Scholar
Nemat-Nasser, S. & Li, Y.-F. 1992 A new explicit algorithm for rate-dependent finite plasticity: performance evaluation. Comput. Struct. 44 (5), 937–963.CrossRefGoogle Scholar
Olmstead, W. E. & Gautesen, A. K. 1989 Asymptotic solution for some singularity perturbed Fredholm equations. Z. Angew. Math. Phys. 40, 230–244.Google Scholar
O'Malley, R. E. Jr, 1971 a Boundary layer methods for nonlinear initial value problems. SIAM Rev. 13, 425–434.Google Scholar
O'Malley, R. E. Jr, 1971 b On the initial value problems for nonlinear system of differential equations with two small parameters. Arch. Rational Mech. Anal. 40, 209–222.Google Scholar
Rashid, M. M. & Nemat-Nasser, S. 1990 Modeling very large flows at very large strain rates for large-scale computation. Comput. Struct. 37 (2), 119–132.Google Scholar
Smith, D. R. 1985 Singular-Perturbation Theory. Cambridge University Press.Google Scholar
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics. The Parabolic Press, Stanford, CA.Google Scholar
Willis, J. R. & Nemat-Nasser, S. 1990 Singular perturbation solution of a class of singular integral equations. Q. Appl. Math. 48 (4), 741–753.Google Scholar