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On the Mathematical Foundation of a Decoupled Directional Distortional Hardening Model for Metal Plasticity in the Framework of Rational Thermodynamics

Md Mahmudur Rahman, Md Mahmudul Hasan Pathik, Nazrul Islam

Abstract

This study proposes a modification to the yield condition that addresses the mathematical constraints inherent in the Directional Distortional Hardening models developed by Feigenbaum and Dafalias. The modified model resolves both the mathematical inconsistency found in the complete model and the limitations of the r-model. In the complete model, inconsistency arises between the distortional term in the yield surface and the plastic part of the free energy in the absence of kinematic hardening. Additionally, the r-model fails to capture the flattening of the yield surface in the reverse loading direction due to the absence of a fourth-order anisotropic tensor structure in the distortional term. To address these issues, the proposed model introduces a decoupled distortional hardening term in the yield function. This modification enables the simultaneous representation of both flattening and sharpening of the yield surface, and permits isotropic hardening with distortion even without kinematic hardening. A consistent mathematical formulation based on rational mechanics and a corresponding numerical algorithm are also developed, establishing a foundation for future experimental investigations and model validation.

On the Mathematical Foundation of a Decoupled Directional Distortional Hardening Model for Metal Plasticity in the Framework of Rational Thermodynamics

Abstract

This study proposes a modification to the yield condition that addresses the mathematical constraints inherent in the Directional Distortional Hardening models developed by Feigenbaum and Dafalias. The modified model resolves both the mathematical inconsistency found in the complete model and the limitations of the r-model. In the complete model, inconsistency arises between the distortional term in the yield surface and the plastic part of the free energy in the absence of kinematic hardening. Additionally, the r-model fails to capture the flattening of the yield surface in the reverse loading direction due to the absence of a fourth-order anisotropic tensor structure in the distortional term. To address these issues, the proposed model introduces a decoupled distortional hardening term in the yield function. This modification enables the simultaneous representation of both flattening and sharpening of the yield surface, and permits isotropic hardening with distortion even without kinematic hardening. A consistent mathematical formulation based on rational mechanics and a corresponding numerical algorithm are also developed, establishing a foundation for future experimental investigations and model validation.
Paper Structure (11 sections, 63 equations, 5 figures, 1 table)

This paper contains 11 sections, 63 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Framework of Feigenbaum and Dafalias Distortional Hardening Model
  3. Complete Model and Its Inconsistency
  4. r-model and Its Limitation [Feigenbaum and Dafalias, 2008]
  5. Discussion on the Limitations of Feigenbaum’s Distortional Hardening Models
  6. Development of a Modified Distortional Hardening Model
  7. Modification in the Expression of Anisotropy and the Proposed Yield Surface
  8. Hardening Rules based on the Proposed Model
  9. Numerical Implementation
  10. Discussion
  11. Appendix: Bardet and Choucair bardet1991linearized Numerical Procedure

Figures (5)

  • Figure 1: Experiments by Wu and Yeh wu1991experimental on Stainless Steel 304 tubular specimens prestrained axially to loading points (L.P.) 1 and 2 in two-dimensional stress space
  • Figure 2: Schematic of the unit radial tensor ($\mathbf{n}^{r}$) and unit normal ($\mathbf{n}$) to the subsequent yield surface
  • Figure 3: Evolution of internal variables (a) $k$, (b) $\alpha_1$, and (c) $r_1$ over the loading iterations
  • Figure 4: Evolution and distortion of the modeled Yield Surface (Y.S.) in the $\sigma_{11}$--$\sigma_{12}$ plane.
  • Figure 5: Flowchart illustrating the numerical procedure by Bardet and Choucair bardet1991linearized for incremental elastoplastic stress-strain analysis