PSI - Issue 14

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

203

5

elastically, and the actual elasto-plastic response of the material at the notch tip ( σ max k ; ε max k ): max max { , } { , } k k e f e f         for , e f Y e f Y         .

(12)

The transformation Φ in essence determines the mapping of points A e-f ( σ e-f ; ε e-f ) of the semiaxis CE ∞ of the fictitious elastic pseudo states of the material to points A max k (σ max k ; ε max k ) of the segment CP ∞ of the stress-strain curve ( Makhutov and Reznikov, 2018) :

 

, Y e f Y         . e f

A ( e f 

,    e f

)

max A ( k

,  

)

for

(13)

e f 

max max k

k

A number of approximate analytical methods of that kind (which are commonly known as stress-strain conversion rules) were developed and are widely applied along with numerical and experimental ones. These include: 1) Linear rule that is based on the assumption that strain concentration factor is equal to theoretical stress concentration factor : t K K   . (14) 2) Neuber rule that relates stress and strain concentration factors ( K σ and K ε ) to the theoretical stress concentration factor K t : 2 / 1 t K K K    , (15) 3) Molski and Glinka or (Equivalent Strain Energy Density) method which assumes that the strain energy density in the notch root W ε is related to the energy density due to nominal stress and strain W n by a factor of K t 2 (Molski and Glinka,1981). 4) Hardrath and Ohman method that expresses stress concentration factor K σ as a function of theoretical stress concentration factor K t , elasticity modulus E and secant modulus at the notch root E S : 1 ( 1) / t S K K E E     . (17) Many experimental and numerical investigations have been carried out in order to verify these rules. Results show that Neuber rule predicts an upper bound on maximum local strains, linear rule provides a lower bound and the Equivalent Strain Energy Density method estimates will lie between the two bounds (Adibi-Asl, 2009 ) . 3. Modifications of Neuber rule Neuber rule proved to be the most convenient and widely used approximate analytical method allowing determination of the stress-strain material response at the notch zone in elastoplastic formulation. It 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. 1) that correspond to normal loading regimes. However the accuracy of estimates obtained using equation (15) decreases dramatically as soon as applied strains move to the range III that corresponds to accident and catastrophic loading regimes. Seeger and Heuler proposed a generalized version of Neuber rule allowing local assessment of the material stress-strain response at the notch region for the case of 2 t n W K W   . (16)