PSI - Issue 42

D.I. Fedorenkov et al. / Procedia Structural Integrity 42 (2022) 537–544 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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hardening, to which the stresses tend up to the true fracture stress k S (Fig. 2, a). Parameter  is controlling the curvature of the hardening law. An algorithm (Fig. 2. b) for determining the isotropic hardening constants is presented using the exponential hardening law as an example. This principle can be applied to other isotropic hardening laws. To implement the algorithm, it is necessary to select a small linear region in the initial section (Fig. 2, a) of the uniaxial tension hardening diagram in the coordinates of true stress-plastic strain near the yield point. It is also necessary to specify an approximate value of inf R . The iterative process is repeated until the stresses coincide with the value of the true tensile stress with a necessary accuracy.

Fig. 2. (a) schematic representation of the exponential isotropic hardening law in true stress-plastic strain coordinates; (b) the iteration algorithm for determining the constants

4.2. Lemaitre damage model Lemaitre (1994) describes the damage parameter evolution using the uniaxial tension diagram as follows: there is no damage in the area of elastic deformations. As soon as plastic strain occurs, the damage level starts to increase. However, this level will be negligible before the ultimate stress u  . After overcoming u  , the damage parameter ω increase in the material (this moment is accompanied by the formation of a neck in the specimen) until failure of the fracture stress R  and the critical damage parameter c  . Despite the theoretical limit 1 c  = , in reality, fracture occurs at 0.2 0.5 c    . The critical damage parameter is a material property at a given temperature. It is possible to determine the critical damage parameter, knowing the ultimate stress and fracture stress:



.

(6)

1 = −

R

c 

u 

Lemaitre's law describes the kinetics of damage accumulation using the exponent s and the amplitude parameter r . In the monotonic loading case after transforming equation (3), the number of cycles needed to initiate a mesocrack is finally expressed as:

m

2

s

( 1 1

)

2 1 s +

   

   



  −

− −

c 

 

 

pl

u 

u 

(

)

 

u

f

pl

.

(7)

2 1 1

N

=

+

− − −

 

 

(

)



R

u

max    − pl f 

pl   

2

2 2 1 s

 

+  

max

c

To determine the constant s of Lemaitre's model, it is necessary to take two points on the fatigue curve, at each of which the applied stress, the number of cycles before failure and the loop width of the first cycle are known. There exists some paired value of the parameters s and r that satisfy a constant critical damage parameter c  . Consequently, r can be found: