**Paris' law** (also known as the **Paris-Erdogan law**) relates the stress intensity factor range to sub-critical crack growth under a fatigue stress regime. As such, it is the most popular *fatigue crack growth model* used in materials science and fracture mechanics. The basic formula reads

where *a* is the crack length and *N* is the number of load cycles. Thus, the term on the left side, known as the *crack growth rate*, denotes the infinitesimal crack length growth per increasing number of load cycles. On the right hand side, *C* and *m* are material constants, and

where

## History and use

The formula was introduced by P.C. Paris in 1961. Being a power law relationship between the crack growth rate during cyclic loading and the range of the stress intensity factor, the Paris law can be visualized as a linear graph on a log-log plot, where the x-axis is denoted by the range of the stress intensity factor and the y-axis is denoted by the crack growth rate.

Paris' law can be used to quantify the residual life (in terms of load cycles) of a specimen given a particular crack size. Defining the stress intensity factor as

where *
σ
* is a uniform tensile stress perpendicular to the crack plane and

*Y*is a dimensionless parameter that depends on the geometry, the range of the stress intensity factor follows as

where
*Y* takes the value 1 for a center crack in an infinite sheet. The remaining cycles can be found by substituting this equation in the Paris law

For relatively short cracks, *Y* can be assumed as independent of *a* and the differential equation can be solved via separation of variables

and subsequent integration

where
*Y* strongly depends on *a*, numerical methods might be required to find reasonable solutions.