Fatigue damage
Damage that occurs in materials subject to fatigue strains is a very complex phenomenon that is fundamentally non-linear. To examine this phenomenon there are today a number of highly sophisticated models that are in fact rather little used. Miner proposed a model that is simple, not to say simplistic, but is nonetheless very widely used. It consists of breaking down a complex loading comprised of variable amplitude cycles between differing levels of stresses, into partial cycles each corresponding to a given level of stress. Damage for a partial cycle is defined by the ratio between the number of cycles effected and the number of cycles leading to a fracture for this loading (following figure). Global damage is considered to be additive, we refer to linear cumulative damage, and the fracture occurs once the damage parameter attains the unit value. In the example in the figure, the material has been subjected to \(N_1\) cycles at maximum stress \(\sigma_1\), \(N_2\) cycles at maximum stress \(\sigma_2\) and \(N_3\) at maximum stress \(\sigma_3\). As a rule, if a material is subject only to cycles at stress level \(\sigma_i\) (\(i = 1\), 2 or 3), then the fracture occurs with a number of cycles \(N_{R_i}\) (\(i = 1\), 2 or 3).
Damage for a level of \(\sigma_i\) après \(N_i\) cycles is
\(d_i = \frac{N_i}{N_{R_i}}\),
and total damage is
\(d = \sum_i d_i =\sum_i \frac{N_i}{N_{R_i}}\).
The fracture occurs when \(d = 1\). In this model, it is assumed that the damage suffered during a given cycle in no way depends on the cycle or cycles that preceded it. The influence of cycle chronology is therefore ignored, which is not at all realistic. It is well known that the damage generated by a specific cycle can be heavily dependent on the loading history of the structure. For example, when we subject, during a cyclical loading, a cracked material to overstress for a cycle, the crack responsible for the damage will see its propagation slowed. What happens is that the plastically strained zone at the crack tip expands instantly the moment the overstress is applied. During the following cycle, imposed at nominal stress, elastic spring-back in the material surrounding the plastic zone exerts compression stresses that confine the crack and delay its propagation.
