Deviatoric Neuber method for stress and strain analysis at notches under multiaxial loadings


Significant stress concentrations have been found to be localized at notched elements and other geometrical irregularities such as key ways, oil holes, and sharp corners. Increased stress concentration leads to premature fatigue failure. For this reason, it is important to accurately forecast strains and stresses at notch locations. This is particularly important in fatigue failure analysis of notched structural elements.

Several notched elements such as axles and shafts are normally exposed to multiaxial loadings, such as combined torsion and bending loadings, and experience complex strain and stress responses at notched locations. Finite element analysis can be adopted in non-linear elasto-plastic analysis in a move to establish accurately local notch strains and stresses for short loading histories. Unfortunately, finite element based numerical methods are computationally costly and infeasible for computation of complicated element geometries as well as long load histories. Therefore, more efficient approximation approaches are therefore needed to establish notch strains and stresses for engineering notched elements.

Equivalent strain energy density and the Neuber rule methods have been developed to estimate elasto-plastic stress and strain responses. However, the Neuber rule overestimates notch elasto-plastic strains and stresses while the equivalent strain energy density underestimates the same. This is based on the fact that the original Neuber Rule was developed for notched elements under pure shear loading conditions, but it has been extended to other multiaxial loadings where the current stress state can be different from the pure shear stress state.

Researchers Professor Ayhan Ince from Concordia University and Dr. Dongjun Bang at Purdue University proposed an analytical approximation model based on the deviatoric form of the Neuber rule in a bid to forecast more accurate elasto-plastic strain and stress results at notches. Their research work is published in International Journal of Fatigue.

The authors developed the model by coupling the materials constitutive equations as well as the deviatoric form of the Neuber notch correction approach. They correlated this analytical model with the non-linear finite element data obtained from different materials and notch geometries under cyclic non-proportional and monotonic loading conditions.

Ince and Bang compared the estimated notch root stresses and strains with the non-linear finite element solutions for SAE 1070 and SAE 1045 steel notched bars. The SAE 1070 and SAE 1045 steel notched-bars were exposed to monotonic and cyclic non-proportional loadings, respectively. The developed model indicated good agreement with the non-linear finite element data. The proposed analytical model for estimating stresses and strains for notched elements under both non-proportional loadings presents a promising modeling approach as compared to the computationally expensive non-linear finite element analysis.

The precise notch strains and stresses approximations observed from the proposed model are promising, and indicate that the notch stress and strain histories can be successively applied to undertake fatigue analysis for notched elements exposed to complex non-proportional multiaxial loadings. The proposed model impacts design practices in many industries and facilitate a movement towards more extensive model-based approaches to optimize the design of engineering components.

Deviatoric Neuber method for stress and strain analysis at notches under multiaxial loadings. Advances in Engineering

About the author

Prof. Ayhan Ince received his M.Sc. and Ph.D. degrees in Mechanical and Mechatronics Engineering, both from the University of Waterloo. He has more than 10 years of experience in the defense/automotive industry in research and development.  He was an Assistant Professor of Purdue Polytechnic Institute at Purdue University from 2014-2017 and joined Concordia University summer of 2017 as an Associate Professor in the Department of Mechanical, Industrial & Aerospace Engineering.

His research primarily addresses deformation, fatigue, and fracture, and modeling damage mechanics of materials. He has been a member of the Scientific Committee of and a Keynote Speaker at international conferences on fatigue and fracture.


Ayhan Ince and Dongjun Bang. Deviatoric Neuber method for stress and strain analysis at notches under multiaxial loadings. International Journal of Fatigue, volume 102 (2017), pages 229–240.


Go To International Journal of Fatigue

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