Prediction of crack propagation using γ-model for through wall cracked Pipes

Abstract

The pipe installations occasionally experience high amplitude vibration (Seismic Vibration). This vibration may initiate a new crack or propagate an existing crack. The monitoring of crack becomes more significant if the pipes carry hazardous fluids. The compliance technique is one of the commonly used methods to monitor crack growth in small size specimens. Crack monitoring in compact tension (CT), three point bend bar (TPBB) specimens are generally preferred for fracture toughness laboratory tests and crack monitoring is done using compliance technique. Crack compliance correlations are available for simple geometries. One of the primary objectives at present investigation is to develop γ-model for straight pipes. Gamma function is a variant of factorial function with its arguments shifted by 1. That is if n is a positive integer then Γ (n) = (n-1)!. The Gamma function is defined for every complex number whose real part is positive and greater than zero. Generally it is given by an integral given as, Γ(z)= ∫_0^∞▒〖t^(z-1) e^(-t) dt〗 Re (z) >0. This modified γ-model has been proposed to predict crack growth in through wall cracked pipe. Here t is replaced by number of cycles N. The parameter z is chosen in such a way that it becomes a non-dimensional parameter yet representing the properties that affect crack growth and since the integral is finite the value of integral is not Γ(z). The integral was assumed to be equal to a non-dimensional representing crack growth at the end of fixed cycles of loading. Generally fatigue crack growth depends on the initial crack length material properties and dimensions, loading conditions etc. So the non-dimensional parameter is chosen in such a way so as to include all those properties. So the formula for predicting the final crack length at the end of cycle is given as (ma_1)/w =∫_0^N▒〖N^(((ma_0)/w-1)) e^(-N) dN〗. Here m also a non-dimensional parameter whose value remain approximately constant for a given cycle interval. The value of m reduces with increase in the value of ΔK. The value of m changes with change in loading condition as well as crack length so 〖m=(E/σ_ys ×K_c/ΔK×K_min/K_max )〗^e. Hence it is needed to correlate parameter m with parameters like two crack driving forces ΔK and Kmax and with the material parameters plane stress fracture toughness (KC), modulus of elasticity (E) and yield stress (σys). Fatigue crack growth depends on both ΔK and Kmax in order to consider effects of mean stress. However, this may not take care of the large deformation that occurs during the loading of specimens/components. In case of pipes, additional difficulties arise due to geometric softening or hardening during the deformation process. However for pipes no such correlation is available so using γ-model we can predict the next incremented depth of crack for pipe.
γ-model has also been applied on single edge notch (tension) SENT specimen and shows results are in good agreement with the experimental results for the SENT specimen. The variation is primarily due to experimental errors or other errors arising due faulty reading data and human error. This method is easy to interpret and less time consuming in successfully predicting crack with good degree of accuracy.