and the extent of the plastic zone at the overload. The Willenborg model Crack growth equations Threshold equation To predict the crack growth rate at the near threshold region, the following relation has been used Paris–Erdoğan equation To predict the crack growth rate in the intermediate regime, the Paris–Erdoğan equation is used Forman equation In 1967, Forman proposed the following relation to account for the increased growth rates due to stress ratio and when approaching the fracture toughness McEvily–Groeger equation McEvily and Groeger proposed the following power-law relationship which considers the effects of both high and low values of . NASGRO equation The NASGRO equation is used in the crack growth programs AFGROW, FASTRAN and NASGRO software. It is a general equation that covers the lower growth rate near the threshold and the increased growth rate approaching the fracture toughness , as well as allowing for the mean stress effect by including the stress ratio . The NASGRO equation is where , , , , , and are the equation coefficients. McClintock equation In 1967, McClintock developed an equation for the upper limit of crack growth based on the cyclic crack tip opening displacement where is the flow stress, is the Young's modulus and is a constant typically in the range 0.1–0.5. Walker equation To account for the stress ratio effect, Walker suggested a modified form of the Paris–Erdogan equation where, is a material parameter which represents the influence of stress ratio on the fatigue crack growth rate. Typically, takes a value around , but can vary between . In general, it is assumed that compressive portion of the loading cycle has no effect on the crack growth by considering which gives This can be physically explained by considering that the crack closes at zero load and does not behave like a crack under compressive loads. In very ductile materials like Man-Ten steel compressive loading does contribute to the crack growth according to . Elber equation Elber modified the Paris–Erdogan equation to allow for crack closure with the introduction of the opening stress intensity level at which contact occurs. Below this level there is no movement at the crack tip and hence no growth. This effect has been used to explain the stress ratio effect and the increased rate of growth observed with short cracks. Elber's equation is Ductile and brittle materials equation The general form of the fatigue-crack growth