PSI - Issue 28

NikolayA. Makhutov et al. / Procedia Structural Integrity 28 (2020) 1347–1359 N.Makhutov, D.Reznikov/ Structural Integrity Procedia 00 (2019) 000–000

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- low-temperature hardening which leads to an increase in the yield strength σ y -dynamic hardening that also leads to a further increase in the yield strength ( ) e y   (Fig.1, curve 3); - stress triaxiality in the notch zone causing an increase in the magnitude of the first principal stress when plastic strains occur (Fig. 1, curve 4); - stress concentration characterized by the theoretical stress concentration factor K t in the notch of depth l and radius ϱ. A stress-strain diagram that consists of two regions was proposed (Makhutov, 1981; Makhutov, 2008; Makhutov, Matvienko, Romanov, 2018). It includes the region of elastic strains where the linear relationship between stresses and strains holds, and the region of plastic strains with the power law stress-strain relationship. This relationship may be written in absolute « e     » (Fig.1a) and normalized e     » coordinates ( / y eq eq     , / y eq eq e e e  Fig. 1 b ): ( t ) (Fig. 1, curve 2);

e   for

for

or

(8)

y   

E e  

1   ;

  m е   for

y    or

for

(9)

( / ) m

y y е е   

1  

where m is the strain hardening exponent for the stress-strain diagram written in terms of true stresses and strains. For a large number of metal structural materials (steels and alloys) 0.05≤ m ≤0.35. The theoretical stress concentration factor K t is used to relate the nominal stresses σ n and strains e n with the values of the maximum local stresses σ max c and strains e max c in the notch zone in the elastic region:

1

max c n K    t

max c n K    t

max c y    or





for

for

(10)

max

c

1  .

e

K e 

for max c e

e  or

e

K e 

for max e

(11)

max c

max c

t n

y

t n

c

Since the notch zone, like the entire component under consideration is in the elastic region, then in accordance with (8) there is a linear relationship between local stresses and strains:

or

.

(12)

max c Ee





max e   c

max

c

max c

With increasing σ n , the value of K t remains constant until the magnitude of the maximum stress at the notch reaches the yield strength value σ y . After this (at σ max c >σ y ) local stresses σ max c and strains e max c are no longer linearly dependent. In addition, equations (10) and (11) also cease to hold. After yielding occurs, local values of stresses and strains are no longer related to the nominal values by K t . In plastic region nominal and local values of stresses and strains are related in terms of stress K σ and strain K e concentration factors ( K σ ≠ K e ≠const):

max c n K    

max c n K     .

or

(13)

e

K e 

e

e n K e  .

or

(14)

max c

max c

e n

For an inelastic material with a power-law hardening the constitutive equation is written as:

m e   for

0   .

(15)

Neyber equation provides a relationship between maximum local stresses and strains at the notch zone for the region of limited elastoplastic strains ( e <0.03%) (Neiber and Khan, 1975):