PSI - Issue 28

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

1351

5

where eq eq e e e    . In (Makhutov, 1981) it was proposed to use a power law expression as a functional describing strain hardening in the plastic region for σ>σ y (curves 1 in Fig. 1): 1 ( ) m eq eq f e Ke  (3) / y eq eq     , / eq y e e e  , eq / y

where K and m are material constants determined experimentally. The following equation was proposed in order to take into account the influence of strain rate hardening:

eq e     e

et 

(4)

  eq 

2 f e

0       eq

where et   is the material constant determined experimentally. Other types of functions were also proposed including the exponential ones. In the range of low temperatures t < t 0 exponential forms are used in the function f 3 ( t ) to describe the influence of temperature factor:

  

0        1 1 t t   

(5)

  3

exp

f t

t 



where t 0 =20

0 C ( t 0 =293 K ) is the room temperature, t is the ambient operating temperature, and β t is the material

characteristic determined experimentally. According to (1) - (5) the governing equation can be written in the form:

e 

0   eq     eq e    e 

  

  

  

  

1 1

(6)

exp

m   Ke



T 





et 

et 

t t

0

where et   and et e  are stress (equivalent stress) and strains (equivalent strain) at the strain rate е  and temperature t ; eq e is the equivalent plastic strain; eq e  is the rate of the equivalent plastic strain; 0 1 eq e   s -1 is strain rate during static tests; t 0 is the room temperature. Expression (6) is in a good agreement with the Johnson–Cook model that is widely used in dynamic calculations:

   

0   eq      eq e     e   

  

  

  

  

1 1

(7)

m

eq   

 

1 ln C

exp

K e

  

  





T 



0

y

eq

t t

0

where σ y0 is the yield strength under static loading; K is the isotropic (static) hardening, m is the strain hardening exponent. 3. Assessment of the state of stresses and strains under high-speed low-temperature loading The problem of assessment of the stress-strain response of the notched specimen (Fig. 2c) subjected to high-speed low-temperature loading requires accounting for a number of factors: - elastoplastic stress-strain diagrams of the standard specimen under static tension with yield strength σ y (Fig. 1, curve 1);