Issue 75

D. I. Vichuzhanin et alii, Fracture and Structural Integrity, 75 (2026) 220-237; DOI: 10.3221/IGF-ESIS.75.16

During deformation, damage  varies from 0 before the deformation starts to 1 at the moment of crack initiation, and the intermediate values of  characterize the amount of microscopic material damage by strain-induced defects. Assume that, at a fixed test temperature, the dependence   , f W f k    can be described by the approximating linear function:     0 1 f W C k C k     (11) where   0 C k and   1 C k are the unknown functions of the stress triaxiality factor k . To describe the k dependence of f W at fixed values of   , similarly to [7], the study uses the exponential law:   exp f W k     (12)

where  and  are empirical coefficients. Let 0  , 1   , 0  and

1   have the values of the coefficients  and  when

0    and

1    , respectively, then it

follows from Eqn. (12) that:    , 0 exp f W k 0



k    

(13)

0

 f W k







 





 

k

, 1

exp

(14)



1

1

  into Eqn. (11), after some

Substituting the right-hand sides of Eqs. (13) and (14) and the corresponding values of

  0 C k and   1 C k and obtain the interpolation formula:

transformations we find expressions for

 f W k













exp      k

 

 



1 k    

,

1

exp

(15)







0

0

1

For the parametric identification of Eqn. (15), we use the data from Tab. 2 and the dependences   eq k  and   eq    during the testing, which are partially shown in figs. 6, 9, 12, 15, and 18. The functional of squared deviations (16) of the objective function (10) from 1 is minimized by varying the coefficients 0  , 1   , 0  and 1   .

n

      2 1 i 

min

(16)

i

1

where n is the number of types of testing at the given temperature. The coefficients are varied by the Hooke–Jeeves method [25]. As a result of searching, it is possible to hit the local minimum of the functional (16); therefore, we select not one initial search point, but a certain set of points. The Hooke–Jeeves procedure is performed for each initial search point. The found values of the coefficients are presented in Tab. 3. When calculating by Eqn. (10), as the upper limit of integration, we substitute the values of eq  from Tab. 2, which were reached at the site of failure during the tests. The identification error is less than 12%.

Coefficients in Eqn. (15)

Material

T , ºC

0  , MJ/m 3

0 

1   , MJ/m 3

1  

25 -50 25 -50

7.52 10.79 10.47 7.47

1.87 3.23 2.24 1.09

3.92 2.85 2.11 1.48

0.3

Reinforced epoxy resin

0.65 0.31

Pure epoxy resin

0.021

Table 3: Identification results.

233