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.

Appendix: The Derivation for Deviatoric Stress Increments

Mathematically, the condition that plastic flow occurs is defined by $\Delta \hat{s}:\hat{n} \ge 0$. Then, the plastic strain increments (Eq. 2) can be rewritten as

$$ d\hat{\varepsilon }_{p} = \frac{1}{H}\left( {d\hat{s}:\hat{n}} \right)\hat{n} $$

(19)

where the deviatoric stress increments are

$$ \Delta \hat{s} = 2G\Delta \hat{e} = 2G\left[ {\Delta \hat{\varepsilon }_{t} - \Delta \hat{\varepsilon }_{p} - \frac{1}{3}\left( {\Delta \hat{\varepsilon }_{t} :\hat{I}} \right)\hat{I}} \right] $$

(20)

where $\Delta \hat{e}$ is the deviatoric elastic strain increments. Using relations (19), (20) and noting the fact that $\hat{n}:\hat{I} = 0$, the following relations can be obtained

$$ \Delta \hat{s}:\hat{n} = 2G\left[ {\Delta \hat{\varepsilon }_{t} - \frac{1}{H}\left( {d\hat{s}:\hat{n}} \right)\hat{n} - \frac{1}{3}\left( {\Delta \hat{\varepsilon }_{t} :\hat{I}} \right)\hat{I}} \right]\hat{n} = 2G\left[ {\Delta \hat{\varepsilon }_{t} :\hat{n} - \frac{1}{H}\left( {d\hat{s}:\hat{n}} \right)} \right] $$

(21)

From Eq. 21, the following relation can be obtained as

$$ \Delta \hat{s}:\hat{n} = \frac{2GH}{2G + H}\left( {\Delta \hat{\varepsilon }_{t} :\hat{n}} \right) $$

(22)

Inserting Eq. 22 into Eq. 19, the plastic strain increments can be rewritten as

$$ \Delta \hat{\varepsilon }_{p} = \frac{2G}{2G + H}\left( {\Delta \hat{\varepsilon }_{t} :\hat{n}} \right)\hat{n} $$

(23)

Then, the deviatoric stress increments expressed in terms of total strain increments can be obtained.