PSI - Issue 22

N. Makhutov et al. / Procedia Structural Integrity 22 (2019) 93–101 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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Neuber rule tends to overestimate local strains, but provides adequate assessments of the maximum local stresses and strains for the range of limited plastic strains (range II, fig. 2) that correspond to normal loading regimes. However the accuracy of estimates obtained using equation (14) decreases dramatically as soon as applied strains move to the range III that corresponds to accident and catastrophic loading regimes. 2. Modification of the Neuber rule A variety of experiments on specimens with various notch geometries were carried out (Makhutov, 1981). These experiments showed that in the case of extensive plastic deformations the right side of the Neuber equation is not equal to 1 and is not a constant value. That is why it was proposed to introduce a correction function into the right side of Neuber rule. This function should depends on K t , strain hardening exponent m and the magnitude of nominal stress σ n . These experiments also showed that the correction function F should have the following properties: 1) When the material response is elastic ( K σ = K ε = K t ), the values of F should be equal to 1; 2) Due to the changes of the geometry of the notch as the values of elasto-plastic strains increase the values of the function F should decrease to a certain minimum value that corresponds to the moment of loss of stability in the process of plastic deformation in the notch zone. 3) As plastic strains increase after the loss of stability of plastic deformation in the notch zone and the area of the cross section decrease the values of the function F should rise. On the basis of generalization of available numerical and analytical solutions as well as the estimation of a huge volume of experimental data such function F ( K t ,σ n ,σ ( ε )) was proposed (Makhutov, 1981) a modification of Neuber rule was developed in which a correction function was introduced into Neuber equation (14). This function allows accounting for the dependence of stress K σ and strain K ε concentration factors from theoretical stress concentration factor K t , nominal stresses σ n and strain hardening exponent m . The following phenomenological function was proposed that satisfies these three requirements and fits well to available experiment data in a wide range of applied strains:   0.5(1 ) 1 1 ( , , ( )) n Y t m K Y t n t n F K K                      , (17) Substituting equation (17) into equation (16) gives:   2 0.5(1 ) 1 1 . n t Y Y t t n m K K K K K                      (18) Taking into account equations (1) and (10), equation (18) may be rewritten in the form of transformation Φ M that carries out a mapping of the curve of pseudoplastic states A p-f ( σ p-f ; ε p-f ) to the actual notch states ( ) max M A ( σ max ; ε max ) located on the stress-strain curve 1 (fig.3). Here instead of pseudoelastic states A e-f ( σ e-f ; ε e-f ) located on semiaxis 2 whose coordinates are determined by equations (6) and (9) in the assumption of nominally elastic behavior, the so-called pseudoplastic states A p-f ( σ p-f ; ε p-f ) of the curve 3 whose coordinates are determined by equations (7) and (10) in the assumption of nominally plastic material response are used as preimage points of the transformation. This transformation characterizes redistribution of stresses and strains at the notch zone in the process of plastic deformation: 2 K K F K K    ( , , ( )) t n    t . (16)