Application of Differential Equations in Predicting Mechanical Fatigue Failure

Abstract

Cyclically-loaded structures, when mechanical fatigue failure occurs, is a major engineering issue since the failure may not be predicted and lead to disastrous failure of the system. To ensure the safety of a structure, efficient use of materials and to reduce maintenance cost, the fatigue life of a structure must be predicted. Differential equations are a powerful mathematical tool which can be used to model fatigue behaviour, such as the evolution of stress-strain, accumulation of damage, initiation of cracks and propagation of cracks with time. The article is a critical examination of the use of models that relies on the use of Differential Equations in foretelling mechanical fatigue. The damage theories, classical theories of fatigue and the crack growth laws are explained in a single framework of mathematics. A combination of the history of stresses and the material properties/constitutive equations is suggested to be used in the estimation of fatigue life. The results show that the behavior of fatigue, including non linearbehavior can adequately be modeled as a differential equation, with the estimates being more realistic than those carried out by strict empirical methods. However, the constraints applied, such as the heterogeneity of the materials, the complicated loading conditions and the uncertainty in parameters estimation still make the prediction less accurate. The paper ends by looking into the future research directions that will enhance the dependability and practicality of the industrial application of fatigue through data-driven hybrid models, multiscale differential models and real-time monitoring of fatigue.