Issue 75

P.V. Trusov et al., Fracture and Structural Integrity, 75 (2026) 463-477; DOI: 10.3221/IGF-ESIS.75.31

ε 2 ε 2  

ε 2 

= –

–

+ ε 

L

k k k k k k

1 1

2 2

3

tension x

3

3

(6)

ε 2 

= –

+ ε 

–

L

k k k k k k

1 1

2 2

3

tension x

3

2

Exponent m in Hutchinson’s relation

Number of experiment

Time step in EVP model, s

Type of loading

Loading complexity

30 50

10 –3 10 –3

1 2 3 4 5 6 7

Quasi-uniaxial tension

Simple

100 300

5×10 –4

10 –4 10 –3 10 –3

30 50

Quasi-uniaxial tension along the x3 axis of the laboratory coordinate system until 25% of the strain is reached, then along the x2 axis until 50% is reached

100

5×10 –4

Complex, two-step

8

300

10 –4

Table 1: Plan of numerical experiments. The results of calculation are compared using the L 2 metric for grain displacements

0 1 1    S N K i j

1



 2 s ds

( , ) i j

( , ) i j

 

evp   ep









F S

( ) s

(7)

( )

( )

evp

ep

KN

[0, ] S

where () F S is the estimate of deviation determined by the norm of the difference between the accumulated shears in slip systems calculated using the compared models, i is the number of the grain included in the representative macrovolume under consideration , j is the number of the slip system, , evp ep   are the column vectors of the shears accumulated by the current time t for all slip systems of all grains, S is the prescribed magnitude of strain accumulated by the time of completion of the deformation process, s is the intensity of strain accumulated by time t, defined as

t

 1 = + 2 D L L .  T

 

s

) dt

(2/3

, where D ´ is the deviator of the strain rate tensor

D':D'

t

0

The estimate determines the absolute discrepancy in the solutions obtained using the two models under consideration. The calculation of the relative error also involves an estimate of the relative difference (the ratio of the norm of the difference to the norm of the vector of accumulated shears obtained by use of the elastoviscoplastic model):

1 1 j  S N K i  

1





2

( , ) i j

( , ) i j







( ) s

s ds

( )

evp     evp

evp

ep

KN

ep

[0, ] S

0





G S

(8)

( )

0 1 1    S N K i j

1





2

( , ) i j



s ds

( )

[0, ] S

evp

KN

The results of numerical experiments are given in Tab. 2. With an increase in the exponent m in the EVP model, it is necessary to reduce the time step: the higher the exponent, the more strict the requirements for the time step allowing to maintain the specified accuracy and stability, which leads to an increase in the calculation time. A comparison of the results of calculation (experiments 1–4) were also made in terms of stress intensity (Fig. 1). A comparison of the results of calculating effective stresses (according to von Mises) for experiments 5-8 is presented in Figs. 2-4. From the results presented, it is readily seen that with an increase of the exponent in the viscoplastic law, there is

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