Calculations based on a “Fracture mechanics” approach can be used to predict fatigue crack growth and to predict the expected service life prior to brittle fracture.

Therefore, to protect against the fatigue failure of a part or a structure that contains a crack the calculation of fatigue crack growth provides a method to predict how many cycles of stress the structure can survive before a crack grows to a critical length followed by brittle fracture. The procedure was initially proposed by P C Paris and was initially met with strong scepticism by the scientific media.

**P C Paris developed a method to predict crack growth that was initially controversial**

**“***In a 1961 paper, P. C. Paris introduced the idea that the rate of crack growth may depend on the stress intensity factor. In a 1963 paper, Paris and Erdogan after reviewing data on a log-log plot of crack growth versus stress intensity range developed what we now refer to as “The Paris equation” with the fixed exponent of 4 which now accepted as between 2 and 5”*

The growth of a crack under cyclic load is controlled by the crack tip stress field, which depends on the stress range (Δσ) or maximum stress minus minimum stress. Since the stress intensity K = σ√(πa), with a cyclic stress there will be a varying stress intensity at the crack tip, so that ΔK = Δσ√(πa). Expressing crack growth rate (da/dN) as a function of crack tip stress intensity range (ΔK) provides results that are independent of specimen geometry.

The results of fatigue crack growth rate tests show that crack growth per cycle (da/dN) increases with ΔK as shown in the above diagram. There are three distinct regions of behaviour. In an inert environment, there is a lower limit ΔKth probably related to the metallurgical structure, below which the propagation rate is negligibly slow, so that the crack merely opens and closes without growing. This is similar to the endurance limit is conventional stress vs cycles diagram. There is also an upper limit that corresponds to the onset of unstable fracturing where the maximum value of stress intensity factor approaches the limiting fracture toughness of the material, KIc. Between these two there is a steady-state region where crack growth depends on some power of ΔK or K of the order of 2 to 10, usually between 2 and 5 with higher values found for materials of lower toughness. That is

da/dN = C(ΔK)^{m}

where C and m are constants for a given material. This is known as the Paris law. In this region, crack growth rate increases with increasing stress intensity and rates are not greatly affected by the usual metallurgical variations. If the initial crack length is known, and the final crack length at which fast fracture occurs is known, or can be calculated, the safe number of cycles can be determined by integrating this equation.

In the slow crack growth region (Region I), the rate of crack growth is much slower than the Paris law would predict. However, because of the limited understanding of threshold effects and the need to err on the side of caution, in life calculations it is usual to assume that the Paris law holds down to the lowest values of ΔK. In fast fracture Region III, as K_{max} approaches the limiting facture toughness of the material, the Paris law underestimates fatigue crack propagation rate. In critical cases, it is simplest and safest to effectively eliminate this region by imposing and upper limit on K_{max}, such as 0.7K_{IC}

**Example calculation**

A pressure vessel made of steel of K_{IC} = 55 MPa√m has a through thickness defect of 2a = 20 mm and is subject to a cyclic load of 0 to 100 MPa.

(i) Will it fail by fast fracture when subject to maximum load?

(ii) Calculate the crack growth rate if the Paris equation is assumed to operate and the constants in the Paris equation are C = 2 x 10^{-13} and m = 6.

(i) For fast fracture, K= σ√(πa) = 100√(π x 0.010) = 18 MPa√m which is much less than KIc so that there is no risk of fast fracture.

(ii) ΔK = Δσ√πa. Therefore,

= 100√(π x 0.010) = 18 MPa√m

da/dN = C x (ΔK)^{m} = 2 x 10^{-13}x (18)^{6} = 6.8 x 10^{-3} mm/cycle. = 6.8 microns per cycle

*Note that as the crack grows, the value of **ΔK will change, and a new calculation should be carried out. In practice, new calculations are recommended when the crack has grown more than 10 per cent.*