Issue 77

A. Sivtseva et alii, Fracture and Structural Integrity, 77 (2026) 138-172; DOI: 10.3221/IGF-ESIS.77.10

and the parameters of the functions g ( β ) have significantly narrower ranges of statistical distribution than when using expressions of the form (Eq. 2). It is assumed that the fatigue damage accumulation does not lead to an increase in the values of mechanical characteristics, i.e., there is no self-healing or significant structural change that could increase the material’s resistance in any direction. Then the values of the integrity K ( β ) and damage D ( β ) lie in the range from 0 to 1. Under the given loading mode, the mechanical characteristic reaches the value p ( β ) ( N = N f ) = p ( β ) ( n = 1) = p f ( β ) at the moment of fatigue failure. It is obvious that the values of these quantities are determined by the loading conditions (parameters φ ( α ) ). A widely used approach is one in which the damage and integrity functions are introduced considering the value of the mechanical characteristic at the moment of fatigue failure:

−

p

p

p

( ) ( ) α ϕ ,

( ) β

( ) β

( ) β

0

0

(4)

( ) β D K *

( ) β = − = * 1

( ) β D g n ( ) β = *

=

=

=

α

β

1, , k

1, . l

,

−

−

p

p

p

p

( ) β

( ) β

( ) β

( ) β

f

f

0

0

In this form of notation, the damage and integrity functions lose their original meaning and begin to reflect the nature of the change in material properties up to the values p f ( β ) . To avoid confusion, the quantities introduced by (Eq. 4) will be referred to as “normalized integrity” K * ( β ) and “normalized damage” D * ( β ) . Despite the loss of their original physical meaning, the use of “normalized integrity” and “normalized damage” is convenient in some cases, since D * ( β ) ( N = N f ) = 1, whereas D ( β ) ( N = N f ) ≤ 1. The functions g ( β ) are examined in detail and the requirements imposed on them are defined. The analysis is limited to continuous and differentiable functions over the domain of definition (i.e., abrupt drops in mechanical characteristics during cyclic loading and the use of piecewise functions with derivative discontinuities are excluded). It is assumed that (Eq. 3) or (Eq. 2) can be written in the following form: Here, a ( β ) , b ( β ) , c ( β ) … are parameters of the function g ( β ) that depend on the cyclic loading parameters φ ( α ) , but do not depend on the number of cycles. It should be noted that seemingly different models may vary only in the form of the dependencies a ( β ) ( φ ( α ) ), b ( β ) ( φ ( α ) )…, and otherwise have an equivalent form. Such models will describe experimental data identically when different loading modes are not considered and, therefore, can be grouped into one set of equivalent models. The values of the parameters a ( β ) , b ( β ) , c ( β ) … must ensure the following properties of the functions g ( β ) : 1. The range of the functions g ( β ) lies within [0; 1], with g ( β ) ( n = 0) = 0, g ( β ) ( n = 1) ≤ 1. Otherwise, the introduced damage loses its physical meaning. 2. The derivative of the function g ( β ) must be non-negative, i.e., dg ( β ) / dn ≥ 0. This requirement is also dictated by physical meaning: the accumulation of fatigue damage does not lead to an improvement in the material’s mechanical properties (at least in the vast majority of experimental studies). Violating these conditions does not strictly invalidate a model but necessitates mathematical domain restrictions. For example, in cases where g ( β ) ( n → 1) → ∞, but g ( β ) ( n = (1–10 –4 )) ≤ 1), an additional restriction on the domain of the function can be introduced that has little effect on its descriptive ability. As noted above, two-stage and three-stage dependencies are most often observed in the experiments investigating the residual mechanical properties of composites. To study the applicability of various functions for describing two-stage and three-stage dependencies, the second partial derivatives of the damage functions can be used: D ′′ ( n , φ ( α ) ) = g ( β ) ′′ ( n , φ ( α ) ) = ∂ 2 g ( β ) /∂ n 2 (hereinafter, the index ( β ) is omitted). Depending on the behavior of the second derivative (assuming it equals zero no more than once over the domain), several cases can be distinguished: 1. D ′′ ≥ 0 – the derivative of the damage function is monotonically increasing. In this case, the model is applicable for describing two-stage dependencies with accelerated damage accumulation (Fig. 2, red line). 2. D ′′ ≤ 0 – the derivative of the damage function is monotonically decreasing. In this case, the model is applicable for describing two-stage dependencies with decelerated damage accumulation (Fig. 2, blue line). 3. D ′′ ( n = 0) < 0, D ′′ ( n = 1) > 0 – the derivative of the damage function decreases, reaches a minimum, and then begins to increase. In this case, the model is applicable for describing the well-known three-stage “fast–slow–fast” dependencies (Fig. 2, green line). ( ) β D g n a ( ) β = , ( ) β ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) α β α β α ϕ ϕ ϕ , , b c ( ) α β = = ,... , 1, , k 1, . l (5)

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