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

1354

8

The effects of strain rates and operating temperatures are taken into account in the expressions for the stress and strain concentration factors (19) - (22) (see Fig. 1 and Fig.2) through the yield strength and strain hardening exponent m included in expressions (1) - (6). The yield strength implicitly included in these expressions since / n n y     . In accordance with (Makhutov, 1981), one can write down an expression for estimating the yield strength for wide ranges of strain rates, working temperatures, and also taking into account stress triaxiality in the notch zone:



  

0     e        e   

  

1 1

eT 





, ,     e t

exp

,

(23)

I

 

 



T 

0

y

k

y

t t

0

where σ y0 is the yield strength under static loading at room temperature t 0 , 0 e  = 10

-3 s -1 is the strain rate under static

loading, β Т = 130 and eT   =0.04 are material constants, t 0 = 20 0 C = 293 K is the room temperature, t is the operating temperature at which loading is carried out; I c is a stress triaxiality factor that characterizes the increase in the first principal stress in the notch zone at which the first plastic strains occur due to stress triaxiality in the notch zone:       2 2 2 1 2 2 3 3 1 2 / 1 1 c c c c c c uni triax ax c I                  , (24)

where

1  ,

,

are the relative principal stresses in the notch zone,

2   c

1   c

3   c

1 / c c

 

1 1 c   /

1 / c c

 

2

3

c

triax c  are the values of the first principal stress in case of uniaxial

1 c y     and 1 1 / uniax uniax c

1 / triax c y     , 1 triax c

uniax c  and 1

and triaxial states of stress upon which plastic yielding occurs. The change of the strain hardening exponent m in the process of plastic deformation (from initiation of plastic strains at σ = σ y until the moment of failure) is determined by equation (9) expressed in true stresses and strains: lg(S / ) 1 1 lg ln 1 y y f f m e           for σ = S f , (25) where ψ f is the relative narrowing of the cross-section area at fracture; S f is the fracture stress in the neck. For the temperature t and the strain rate 0 e  the relationship between the mechanical characteristics of the material can be approximated by the expression (Makhutov, 1981):

n



   

  



 

0 0 y        y

0 1     f

,

(26)

 

y

f

f S

 

here ψ f0 is a relative narrowing of the cross-sectional area at fracture under the temperature t 0 and strain rate 0 e  , n ψ is the material characteristic which for low-carbon steels can be taken equal to 2 (Makhutov, 1981; Makhutov, 2008). Then using equations (14), (19) and (20) the actual strain rate in the notch zone can be estimated as:

e 

K e  

 .

(27)

max c

e n