PSI - Issue 59

Svitlana Fedorova et al. / Procedia Structural Integrity 59 (2024) 279–284 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

280

2

Table 1. Unified and specific terms.

Evolutionary rupture

Long-term rupture

Fatigue high-cycle rupture

Fatigue low-cycle rupture

stress

stress amplitude number of cycles

strain amplitude number of cycles

L – load

time

ξ , x – duration

lifetime

number of cycles to rupture

number of cycles to rupture

x R – rupture moment

The evolutionary rupture of materials is described by the Kachanov model (1986) with the evolutionary equation

dD

(1)

,

( ( ), ) f L D 



d



with initial condition

(2)

0 ( 0) D D    ,

and with the condition of rupture

(3)

( ) D x D   

R

R

for damage variable D . Condition (3) allows us to predict the value of the rupture moment for any loading protocol ( ) L  . The key problem of this model is the construction of the damageability function ( , ) f L D . Usually the simplest, factorized form is used for it 1 2 ( , ) ( ) ( ) f L D f L f D  . However, in paper by Fedorov (2023) proved that it gives the same rupture moment as the Palmgren-Miner rule (Palmgren (1924), Miner (1945)) for any 1 ( ) f L , 2 ( ) f D , 0 D , R D and ( ) L  . Since this rule does not hold for many materials, the potential benefits of the Kachanov model can only be realized with an unfactorized damage function. This function was proposed Chaboche and Lesne (1988). The paper by Fedorov (2023) sets out the general theory and methods for constructing unfactorized damageability functions, which must satisfy the following requirements:  adequacy of Kachanov model to experimental curves of evolutionary strength under stationary loading: exp R R ( ) x X L   adequacy of Kachanov model to experimental data under non-stationary loadings: exp R R [ ( )] x X L    adequacy of Kachanov model to experimental observations of the evolution of damage variable exp ( ) D  . This order is dictated by the priority of predicting the rupture moment. Thus, the damageability function must approximate a wide range of heterogeneous experimental data through the boundary value problem (1)-(3), which looks dauntingly complex. However, the potential representation of the damageability function in the article by Fedorov (2023) made it possible to decompose the problem into separate subproblems, which is discussed in this paper. Further, unless otherwise indicated, the presentation is based on the indicated work, with possible modification of the notation. 2. Potential representation of the damageability function The formulated problem can be solved by representing the damageability function through the normalized potential ( , ) F L D , after which the evolutionary equation (1) takes the form

1



( ( ), ) F L D 

dD



 

  

exp R f L D X L    ( ( ), )

( ( )) 



(4)

d

D



