GENERALIZED FORCES AND DISSIPATION FUNCTIONS IN THE CONTEXT OF AN INTERNAL VARIABLE APPROACH APPLIED TO THE SOLUTION OF ELASTIC-PLASTIC PROBLEMS

  • Alfonso Nappi Professor, Department of Engineering and Architecture University of Trieste, P.le Europa 1, 34127 Trieste, Italy
  • Daniele Zaccaria Associate Professor, Department of Engineering and Architecture University of Trieste, P.le Europa 1, 34127 Trieste, Italy
Keywords: Backward-difference, Convergence, Convex analysis, Discrete models, Dissipation functions, Elastic-plastic materials, Finite element method, Internal variables, Iterative schemes

Abstract

A non-traditional approach to the numerical analysis of elastic-plastic systems is discussed by focusing on a formulation that makes use of internal variables and dissipation functions. These functions are used in order to enforce the constitutive law, so that they play the role of the yield functions in the framework of the classical theory of plasticity.

With reference to finite element discrete models, it is shown that the solution of an elastic-plastic problem corresponds to the minimum point of a convex function (when the material is stable in Drucker’s sense) and that convergence is guaranteed when a convenient time integration method (usually known as backward-difference scheme) is applied. As a matter of fact, it can be proved that the value of that function progressively decreases (iteration by iteration) when a proper time integration strategy is implemented.

Elastic-plastic systems will be considered, which are subjected to uniaxial and multiaxial stress states (by assuming Mises’ yield condition for two-dimensional and three-dimensional finite elements). In all cases, it will be easily noticed that the dissipation functions depend on convenient generalized forces, whose features are obvious in the presence of uniaxial stress states. Instead, when the structural system is subjected to multiaxial stress states, the actual meaning of the generalized forces must be properly understood in order to define convenient dissipation functions and/or yield functions: this is the main issue of the present paper and represents a topic which, to the authors’ knowledge, has not been adequately investigated, yet.

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References

J.B. Martin, An Internal Variable Approach to the Formulation of Finite Element Problems in Plasticity, in Physical Non-Linearities in Structural Analysis (edited by J. Hult and J. Lemaitre). Springer-Verlag, Berlin, 1981.
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A. Nappi, Application of convex analysis concepts to the numerical solution of elastic plastic problems by using an internal variable approach, Engineering Optimization, 18, 79-92, 1991.
S. Rajgelj, C. Amadio, A. Nappi, An internal variable approach applied to the dynamic analysis of elastic-plastic structures, Earthquake Engineering and Structural Dynamics, 22, 885-903, 1993.
Published
2018-12-19
How to Cite
Nappi, A., & Zaccaria, D. (2018). GENERALIZED FORCES AND DISSIPATION FUNCTIONS IN THE CONTEXT OF AN INTERNAL VARIABLE APPROACH APPLIED TO THE SOLUTION OF ELASTIC-PLASTIC PROBLEMS. IJRDO - Journal Of Mechanical And Civil Engineering (ISSN: 2456-1479), 4(11), 01-17. Retrieved from https://ijrdo.org/index.php/mce/article/view/2551