158
I. S. Nikitin et al.
A large number of stress-based criteria are based on a direct generalization of the
S-N Wöhler-type curves described by Basquin-type relations [1], and based upon the
results of fatigue tests. The main criteria for multiaxial fatigue failure, taking into
account the values of strain amplitudes (strain-based criteria), were proposed in [2–
4]. These criteria are divided into two large groups. The first group includes criteria
that use the amplitudes of the invariant characteristics of the stress state in the loading
cycle such as octahedral stresses, principal stresses, etc. [5, 6], and the second group
includes criteria that take into account the amplitudes of the tangent and/or normal
stresses on the so-called critical plane [7–14]. As a rule, this plane is determined
from the condition of the maximum amplitudes of the tangent, normal stresses, or a
certain combination of them on the planes of various orientations. Reviews on this
topic are given, say, in [15–19].
In order to study the development of fatigue damage zones, there are also two
approaches. The first is based on the classical concepts of fracture mechanics and
relates the conditions for the development of fatigue cracks depending on the amplitudes of stress intensity factors at the crack tip with the increase in the number of
cycles. The basic equation was proposed by Paris and Erdogan [20], there are a large
number of modifications of it [21–23]. The second approach uses representations of
the theory of damage, dating back to [24, 25] and developed in [26–28]. As applied
to the problems of cyclic loading and fatigue failure, it was used in [29–32].
We study the processes of fatigue damage zones development using the damage
theory approach dating back to [24, 25]. In the application to the cyclic loading and
fatigue failure problems, this approach was applied in [27, 28]. We propose a multimode model for the development of fatigue failure based on the evolutionary equation
for the damage function. The model parameters are determined for various modes of
fatigue failure: Low-Cycle Fatigue (LCF) and High-Cycle Fatigue (HCF), as well as,
the regime of Very-High-Cycle Fatigue (VHCF), corresponding to high-frequency
low-amplitude loading.
To distinguish the various modes of fatigue failure, we use the multimode amplitude fatigue curve diagram shown in Fig. 12.1. Up to a value of N ~ 10
3 , the regime of
re-static loading is realized with an amplitude that differs little from the static strength
limit σ B . Further, the left part of the bimodal fatigue curve (Wöhler curve) describes
LCF-HCF modes up to N ~ 10
7 and amplitude values of the order of the fatigue limit
σ u . Then begins the zone of change of fracture mechanisms and a further drop in
fatigue strength, starting from N ~ 10
8 to a new fatigue limit value ˜
σ u in accordance
with the right branch of the bimodal S-N fatigue curve. This branch describes VHCF
mode [33].
It should be noted that at present, the idea of an explicit division of the classic
Wöhler branch into two parts (in fact, LCF and HCF) exists. The boundary of this
transition region is determined not by the value of N, but by the value of the loading
amplitude equal to the yield strength of the material σ T [34] since this changes the
physical mechanism of fatigue failure. In addition, the boundary of the repeatedstatic range N ∼ 10
3 is rather arbitrary. It is also specified in [34] depending on the
strength and plastic characteristics of the material. However, in this chapter, we keep
I. S. Nikitin et al.
A large number of stress-based criteria are based on a direct generalization of the
S-N Wöhler-type curves described by Basquin-type relations [1], and based upon the
results of fatigue tests. The main criteria for multiaxial fatigue failure, taking into
account the values of strain amplitudes (strain-based criteria), were proposed in [2–
4]. These criteria are divided into two large groups. The first group includes criteria
that use the amplitudes of the invariant characteristics of the stress state in the loading
cycle such as octahedral stresses, principal stresses, etc. [5, 6], and the second group
includes criteria that take into account the amplitudes of the tangent and/or normal
stresses on the so-called critical plane [7–14]. As a rule, this plane is determined
from the condition of the maximum amplitudes of the tangent, normal stresses, or a
certain combination of them on the planes of various orientations. Reviews on this
topic are given, say, in [15–19].
In order to study the development of fatigue damage zones, there are also two
approaches. The first is based on the classical concepts of fracture mechanics and
relates the conditions for the development of fatigue cracks depending on the amplitudes of stress intensity factors at the crack tip with the increase in the number of
cycles. The basic equation was proposed by Paris and Erdogan [20], there are a large
number of modifications of it [21–23]. The second approach uses representations of
the theory of damage, dating back to [24, 25] and developed in [26–28]. As applied
to the problems of cyclic loading and fatigue failure, it was used in [29–32].
We study the processes of fatigue damage zones development using the damage
theory approach dating back to [24, 25]. In the application to the cyclic loading and
fatigue failure problems, this approach was applied in [27, 28]. We propose a multimode model for the development of fatigue failure based on the evolutionary equation
for the damage function. The model parameters are determined for various modes of
fatigue failure: Low-Cycle Fatigue (LCF) and High-Cycle Fatigue (HCF), as well as,
the regime of Very-High-Cycle Fatigue (VHCF), corresponding to high-frequency
low-amplitude loading.
To distinguish the various modes of fatigue failure, we use the multimode amplitude fatigue curve diagram shown in Fig. 12.1. Up to a value of N ~ 10
3 , the regime of
re-static loading is realized with an amplitude that differs little from the static strength
limit σ B . Further, the left part of the bimodal fatigue curve (Wöhler curve) describes
LCF-HCF modes up to N ~ 10
7 and amplitude values of the order of the fatigue limit
σ u . Then begins the zone of change of fracture mechanisms and a further drop in
fatigue strength, starting from N ~ 10
8 to a new fatigue limit value ˜
σ u in accordance
with the right branch of the bimodal S-N fatigue curve. This branch describes VHCF
mode [33].
It should be noted that at present, the idea of an explicit division of the classic
Wöhler branch into two parts (in fact, LCF and HCF) exists. The boundary of this
transition region is determined not by the value of N, but by the value of the loading
amplitude equal to the yield strength of the material σ T [34] since this changes the
physical mechanism of fatigue failure. In addition, the boundary of the repeatedstatic range N ∼ 10
3 is rather arbitrary. It is also specified in [34] depending on the
strength and plastic characteristics of the material. However, in this chapter, we keep