#### Universality behind Basquin’s Law of Fatigue

An asphalt roadway will crack after thousands of trucks pass, even if none is especially heavy. Engineers have long observed that the number of repeated stress cycles before failure is mathematically related to the stress in each cycle. Now in the 7 March *Physical Review Letters* theorists relate this law to accumulating damage at the microscale. Researchers hope that an improved understanding of fatigue will guide the future design of reliable mechanical parts, from bridges to railroad cars.

The eventual breakage, or “fatigue,” of repetitively stressed materials places stringent limits on how they are used. To predict failure, engineers measure how the number of cycles before breakage increases as the amount of stress (applied force) in each cycle is lowered. Almost a century ago, O. H. Basquin of Northwestern University in Illinois noted that the measured number of cycles decreases as a mathematical power of the stress, although the numerical exponent differed for different materials.

Ferenc Kun of the University of Debrecen, Hungary, and his collaborators realized that this law follows directly from the accumulation of microscopic damage–such as tiny cracks–caused by stress. The longer the material is stressed, the more damage builds up. Like previous researchers, they imagine a disordered material like asphalt to consist of numerous parallel fibers that share the load. Individual fibers vary in two ways: first, in how much stress they can take before breaking, and second, in how much accumulated damage will cause them to break, even without any stress. That is, eventually there are so many cracks or other kinds of microscopic damage that the fiber simply falls apart on its own.

Although these ideas have appeared in other theories of fatigue, no one has connected them with Basquin’s law before. To show the connection, Kun and his colleagues assume that the amount of microscopic damage in each fiber increases with time at a rate that is a power of the stress. This assumption, which is based on experimental data from others, means that if the stress increases, then damage accumulates faster. To simplify the math, they also consider the time until failure (lifetime) of a material held under constant stress to be equivalent to the number of cycles the material would survive if subjected to repeated stresses.

Under steady and prolonged stress, microscopic fibers constantly break as they reach their individual damage thresholds, so the material’s lifetime is directly connected with the way in which damage accumulates over time. Since the rate of accumulation increases as a power of the stress, it turns out that the lifetime decreases with the same exponent, except with a minus sign. So the team concludes that the exponent in Basquin’s law comes directly from the way in which damage accumulates.

The researchers also analyze other aspects of their fatigue model, such as properties of the “cascades” of fiber breakages that other models have also predicted. These groups of fibers fail nearly simultaneously when one fiber reaches its damage threshold, and the rest are overwhelmed by the extra stress caused by losing one of the “team.” The model directly relates fatigue failure to cascades of failure, which could be measured directly in the elongation of the material or the sound waves the cascades emit.

Mikko Alava of the Helsinki University of Technology in Finland says that this research and other recent papers are at last establishing “a bit of connection between the models and the experimental reality” of fatigue failure. “I would hope that there are some experimentalists who come out with ways of testing them,” he says, adding that accurate models could help engineers design more durable materials.