4.1 Typical cumulative damage models
In 1945, the hypothesis that fatigue damage is equal to the accumulated
life fraction was proposed by Miner based on Palmgren’s work, referred
to as Miner’s rule, expressed as:
where is the fatigue damage (the damage at failure is assumed to be
unity), and are the number of preload cycles and the number of cycles to
failure for the constant amplitude stress level , respectively. In the
two-level test, the residual life fraction can be expressed as:
where and are the preload cycle life fraction and the residual life
fraction, respectively. Due to its simplicity, Miner’s rule has become
the standard method for fatigue design of metallic structures in
engineering63. However, numerous
experiments46,64,65 revealed significant deviations
between the predictions of Miner’s rule and the experimental results.
The main drawback of Miner’s rule is its load sequence independence.
Various nonlinear cumulative damage models are proposed accounting for
loading sequence. Kwofie and Rahbar66 proposed the
concept of fatigue driving stress(FDS) based on the S-N curve, the
expression of FDS is:
where is fatigue driving stress, is the applied stress at the stress
level, and have their usual meaning, b is the fatigue strength
exponent. The fatigue driving stress increases with cyclic loading, and
failure occurs when the fatigue driving stress reaches a critical value.
The critical fatigue driving stress is independent of the applied load.
Under variable amplitude loading, the fatigue driving stress is assumed
to remain constant as the load changes. The expression for the
cumulative damage based on the Kwofie and Rohbar model (K-R model for
short) is:
where is the number of failure cycles corresponding to the first stress
level. The residual life fraction in the two-level test can be expressed
as:
From Equation , it can be seen that the K-R model takes into account the
loading sequence by multiplying the log-life ratio on Miner’s rule. When
in the low-high sequence and the loading life fraction is small enough,
the residual life fraction predicted by Equation will be higher than
unity. This phenomenon implies that the residual fatigue strength is
improved due to low amplitude loading. Li et al.44modified the K-R model and proposed a new nonlinear cumulative damage
model that is expressed as:
where b 1 and b 2 are the
fatigue strength exponent of the first and second level stresses,
respectively. The Li model can account for cumulative damage under block
loading at different temperatures. In addition, this model allows
describing the significant increase of residual fatigue life due to
loading history.
Peng et al.67 developed a cumulative fatigue damage
model by combining the S-N curve and the material memory concept. The
S-N curve is assumed to translate and rotate clockwise with cyclic
loading. The fatigue damage is then described by the change in the S-N
curve slope and quantified by the material memory parameter. The Peng
model predicts the residual life fraction for the two-level test as:
Peng model has a high sensitivity to loading history and can accurately
predict fatigue life under multi-level loading. In continuum damage
mechanics, the fatigue behavior of materials is regarded as a continuous
process in that damage accumulates progressively until macroscopic
cracks initiate. Chaboche and Lesne18 developed the
nonlinear continuous damage model based on continuum damage mechanics.
The differential relationship between the damage D and the
applied cycles n is expressed as:
where is the maximum stress, is the mean stress, , and are material
constants. The function is expressed as:
where is ultimate tensile strength, is the fatigue limit for fully
reversed conditions, is the material parameter determined by the damage
process. The Symbol is defined as if and if . For constant load loading,
the cumulative damage can be expressed as:
Under variable amplitude loading, the accumulated damage from the
previous loading step can be transferred to the stress level, and the
equivalent cycle number that can yield the same damage is defined as:
where is the equivalent cycle number, is Equation corresponding to the
loading level. Therefore, the cumulative damage under the loading level
is:
Failure is considered to occur when equals unity. The residual life
fraction under two load levels can be obtained by substituting Equation
and Equation into Equation :