# A Coupled Armstrong-Frederick Type Plasticity Correction Methodology for Calculating Multiaxial Notch Stresses and Strains

## Abstract

Based on the pseudo-strain method, a computational modeling technique coupling with Armstrong-Frederick type nonlinear kinematic hardening rule (Jiang-Sehitoglu model) is developed to calculate the multiaxial stress-strain responses of notched components. The pseudo-strain-true notch stress curve is determined using Neuber’s rule. The material constants in Jiang-Sehitoglu model are calculated using the Ramberg-Osgood curve. The presented method is applied to simulate the notch-tip deformations of circumferentially notched 1070 steel and S460N steel shafts subjected to various loadings, including box, circle, V-shape, zigzag-shape, and butterfly-shape loading paths. The calculated strain loops are in accord with experimental data and show reasonable accuracy.

## Keywords

Notch-tip stress-strain Plasticity model Pseudo-strain Multiaxial cyclic loading## Notes

### Acknowledgment

The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (Nos. 51601221 and 51575524), the Natural Science Basic Research Plan in Shaanxi Province of China (No. 2015JM5240) and the Doctoral Scientific Research Foundation of Air Force Engineering University.

## References

- 1.D. Ye, O. Hertel, M. Vormwald, A unified expression of elastic-plastic notch stress-strain calculation in bodies subjected to multiaxial cyclic loading. Int. J. Solids Struct.
**45**, 6177–6189 (2008)CrossRefGoogle Scholar - 2.H. Chen, D.G. Shang, J. Xiong, A coupled plasticity correction approach to estimating notch root strains under multiaxial cyclic loading. Int. J. Fatigue
**52**, 39–48 (2013)CrossRefGoogle Scholar - 3.G. Glinka, Energy density approach to calculation of inelastic strain-stress near notches and cracks. Eng. Fract. Mech.
**22**, 485–508 (1985)CrossRefGoogle Scholar - 4.M. Hoffmann, T. Seeger, A generalized method for estimating multiaxial elastic-plastic notch stresses and strains. Part I: Theory/Part II: Application and general discussion. J. Eng. Mater. Technol. (Trans. ASME)
**107**, 250–260 (1985)Google Scholar - 5.V.B. Köttgen, M.E. Barkey, D.F. Socie, Pseudo stress and pseudo strain based approaches to multiaxial notch analysis. Fatigue Fract. Eng. Mater. Struct.
**18**, 981–1006 (1995)CrossRefGoogle Scholar - 6.K. Molski, G. Glinka, A method of elastic-plastic stress and strain calculation at a notch root. Mater. Sci. Eng.
**50**, 93–100 (1981)CrossRefGoogle Scholar - 7.H. Neuber, Theroy of stress concentration for shear-strained prismatic bodies with arbitrary nonlinear stress-strain law. J. Appl. Mech. (Trans. ASME)
**28**, 544–549 (1961)CrossRefGoogle Scholar - 8.T.H. Topper, R.M. Wetzel, J.D. Morrow, Neuber’s rule applied to fatigue of notched specimens. J. Mater.
**1**, 200–209 (1969)Google Scholar - 9.A. Moftakhar, A. Buczynski, G. Glinka, Calculation of elastic-plastic strains and stresses in notches under multiaxial loading. Int. J. Fract.
**70**, 357–372 (1995)CrossRefGoogle Scholar - 10.A. Ince, G. Glinka, A. Buczynski, Computational modeling of multiaxial elasto-plastic stress-strain response for notched components under non-proportional loading. Int. J. Fatigue
**62**, 42–52 (2014)CrossRefGoogle Scholar - 11.Y.L. Lee, Y.J. Chiang, H.H. Wong, A constitutive model for estimating multiaxial notch strains. J. Eng. Mater. Technol. (Trans. ASME)
**117**, 33–40 (1995)CrossRefGoogle Scholar - 12.Y.S. Garud, A new approach to the evaluation of fatigue under multiaxial loadings. J. Eng. Mater. Technol. (Trans. ASME)
**103**, 118–125 (1981)CrossRefGoogle Scholar - 13.Y.Y. Jiang, H. Sehitoglu, Comments on the Mróz multiple surface type plasticity models. Int. J. Solids Structures
**33**, 1053–1068 (1996)CrossRefGoogle Scholar - 14.O. Hertel, M. Vormwald, T. Seeger et al., Notch stress and strain approximation procedures for application with multiaxial nonproportional loading. MP Mater. Testing
**47**, 268–277 (2005)CrossRefGoogle Scholar - 15.Y.Y. Jiang, H. Sehitoglu, Modeling of cyclic ratcheting plasticity, Part I: Development of constitutive equations/Part II: Implement of the new model and comparison of theory with experiments. J. Appl. Mech. (Trans. ASME),
**63**, 720–733 (1996b)Google Scholar - 16.M.E. Barkey, Calculation of notch strains under multiaxial nominal loading. Ph.D. dissertation. University of Illinois at Urbana-Champaign (1993)Google Scholar
- 17.Y.Y. Jiang, P. Kurath, Characteristics of the Armstrong-Frederick type plasticity models. Int. J. Plast.
**12**, 387–415 (1996)CrossRefGoogle Scholar - 18.J. Li, Z.P. Zhang, C.W. Li, An improved method for estimation of Ramberg-Osgood curves of steels from monotonic tensile properties. Fatigue Fract. Engng. Mater. Struct.
**39**, 412–426 (2016)CrossRefGoogle Scholar - 19.H. Lang, The difference of the solutions of the elastic and elastoplastic boundary value problem and an approach to multiaxial stress-strain correlation. Ph.D. dissertation. Technische Universität Kaiserslautern (2007)Google Scholar