Issue 66

R. B. P. Nonato, Frattura ed Integrità Strutturale, 65 (2023) 17-37; DOI: 10.3221/IGF-ESIS.66.02

In this model, structural dimensions, crack semi-width, and stress characterize the overall behavior of a cracked body. The effects of these parameters define the stress intensity factor, K , applied to the case of a structural pipe, given by Eqn. 1,     , , K Y a t D S a (1) where a is the current crack semi-width (see Fig. 1), S is the stress acting on the body, and   , , Y a t D is the geometrical function depending on the dimensions of the body ( a , pipe thickness ( t ), and pipe outside diameter ( D ) ), which from now on will be represented by Y , for simplification purposes. Fig. 1 shows the pipe cross-section corresponding to the cutting plane with the largest crack dimensions, in which D , t , crack width ( 2 a ), crack depth ( b ), and point ( P ) are represented.

Figure 1: Dimensions of a circumferential internal surface crack at the pipe cross-section with the largest crack dimensions.

In other words, the elastic-stress field magnitude can be represented by the parameter K ( I K , III K , which correspond to fracture modes I , II , and III , respectively). When K reaches a threshold value known as fracture toughness (which depends on constraint level, loading rate, and service temperature), theoretically the failure occurs. For example, if the situation involves the mode I , I K has to be compared with C I K (mode I fracture toughness). Thenceforth, the governing crack-growth model of the crack propagation rate, also known as Paris-Erdogan law, can be described by Eqn. 2, II K , and

da

   m r C K

(2)

dN

in which C is the intercept constant and m is the slope of the curve  / r da dN K on a log-log scale, which are experimentally obtained as a function of frequency, environment, stress ratio, temperature, and material [22], and N is the fatigue life (number of stress cycles). Eqn. 2 is a function of stress intensity factor range, r K , which is simply given by   r max min K K K , where max K and min K are the maximum and minimum stress intensity factors, respectively. Eqn. 1 can be transformed into Eqn. 3 to express the relation between the ranges of both stress intensity factor and applied stress:  

  r r K YS a

(3)

in which r S is the far field stress range with both variable amplitude and frequency. As the crack does not propagate below a threshold value of stress intensity factor range ( rth K ), Eqn. 2 can now be rewritten in terms of Eqn. 4 to discern between the stress cycles that effectively contribute to crack propagation and those that does not, i.e.

19