PSI - Issue 14

Nikolay A.Makhutov et al. / Procedia Structural Integrity 14 (2019) 199–206 N.Makhutov et al./Procedia Structural Integrity 00 (2018) 000 – 000

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6

extreme loading that is based on introduction of modified nominal stresses σ n,M to account for general yielding of the notched cross-section (Seeger and Heuler, 1980). N.Makhutov in (Makhutov, 1981) proposed a different approach to assessment of material response under the extreme loading regimes that cause extensive plastic deformation and general yielding of the structural component cross section. On the basis of generalization of available numerical and analytical solutions as well as the estimation of a huge volume of experimental data a modification of Neuber rule was developed in which a correction function F ( K t ,σ n ,σ ( ε )) was introduced into Neuber equation (15). 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:

2 K K F K K   

( , , ( )) t n   

.

(18)

t

Assessing the experiment results and available closed form and numerical solutions one may come to the conclusion 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. 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 t n Y t n F K K                , (19)

Substituting equation (18) into equation (17) gives:





0.5(1 ) 1 m 

K

1

 

n   Y



 





2 K K K K     t

/   Y t n

.

t



(20)

Taking into account equations (1) and (14), equation (20) may be rewritten in the form of transformation Φ M that carries out a mapping of the curve of pseudoplastic states A extr ( σ p-f ; ε p-f ) to the actual notch states A max ( σ maxk ; ε maxk ) located on the stress-strain curve 1 (fig.1). Here instead of pseudoelastic states A norm ( 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 extr ( 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:   0.5(1 ) 1 n Y n p f m          

    

   

 

 

Y 

.

(21)

  

p f  

 

max max k

k

p f

 



p f 

Thus, t ransformation Φ M is analogous to Neuber transformation Φ N . It maps the points A extr ( σ p-f ; ε p-f ) of the curve of pseudoplastic states to the points A max (σ max k ; ε max k ) of the stress-strain curve. But in contrast to transformation Φ N the proposed transformation Φ M allows accounting the values of theoretical stress concentration factors K t , nominal stresses σ n , and power hardening exponent m . Thus, determining the relationship between fictitious pseudoplastic states ( σ p-f ; ε p-f ) and maximum local stress and strains ( σ max k , ε max k ) at the notch zone for the wide