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 :