In linear elastic fracture mechanics (LEFM), several models under constant amplitude loading describe the propagation of a crack, and these models are formulated by an initial value problem (IVP). However, for most applications, it is not possible to obtain exact numerical solutions for IVP due to the mathematical formulation of the stress intensity factor. From this, approximate numerical solutions are used for the IVP solution, which may reflect in aspects such as time and high computational cost. Thus, this work presents a new method called Fast Crack Bounds (FCB), to improve the way to obtain the IVP solution. This method was applied to the Paris-Erdogan, Forman, Walker, McEvily and Priddle models by establishing two functions, the upper and lower bounds, for the crack size function. Also, this work presents, for the same models, two new possible solutions through the arithmetic and geometric means of the bounds. For both, bounds and bounds means, the results were compared with the numerical solution obtained by the fourth-order Runge-Kutta method, applied to two numerical examples. As a result, the study presented, for all the models analyzed, an efficient and accurate way to obtain the propagation of an initial crack, reflecting in a considerable computational improvement.
In linear elastic fracture mechanics, the rate of crack propagation is proportional to the range of stress intensity factors. The most popular model relating these quantities is the Paris-Erdogan law. Crack growth computation is an initial value problem whose solution cannot be obtained in closed form, as stress intensity factors, hence crack growth rates, depend on the accumulated growth. For complex geometries, stress intensity factors are evaluated numerically, and crack growth computations can become computationally intensive. This paper presents a theoretical result establishing upper and lower bounds for the crack size function for any number of cycles. The bounds are very narrow, hence accurate crack size approximations can be obtained from only two stress intensity factor evaluations. This leads to a huge gain in computational effort for numerical crack growth computations. Two examples are used herein to explore the accuracy and efficiency of the proposed solution for the crack growth initial value problem.
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