Fractal and probabilistic analysis on fatigue crack growth rate of metallic materials

Abstract

Load-bearing and complex geometry structures such as aircraft wing spars, thick-walled chemical processing vessels, offshore platforms and jacket structures are designed based on damage-tolerant design philosophy. The design employs fracture mechanics and test data to ensure that structural cracks nucleating during the operation will not propagate before they are detected by periodic inspections. The fracture mechanics equation describing the crack tip stress field (fr-field) is expressed in terms of the far-field stress and relies on the crack geometry factor. Closed-form equations for the far-field stress and the crack geometry factors have been established for standard fracture test coupons and relatively simple structures. The unavailability of the crack geometry factor for complex structures and loading renders the use of the fracture mechanics equation impractical. Inaccurate assessment of the fatigue crack and crack growth rates could jeopardize the safety and integrity of the structures. An alternative approach employing fractal analysis to quantify the fatigue crack growth rates of single-phase metallic material is proposed and examined. The fractal approach avoids the need for the crack geometry factor when calculating the crack tip driving force. The fractal analysis is carried out on digital images of the crack with a precision of 1.19 pixel/^m2 employing the box-counting algorithm to determine the fractal dimension (dF) along the edge of the crack length. The analysis is confined to the power law crack growth rate stage (Paris crack growth regime). Compact tension, C(T) specimens fabricated from AISI 410 martensitic stainless steel provide the reference fatigue crack growth response. Results show that the crack initially exhibits a Euclidean nature (dF-1.0). The fractal dimension increases steadily with increasing crack length in Paris region with 1.05<dF <1.24. The corresponding extent of disparity in the crack tip driving force range is between 18<Afr<40 MPaVm. The fractal dimension (dF) correlates linearly with the normalized crack tip driving force range (Afr/frIC) within the Paris region. The coefficient of fractality (CF) is identified as a characteristic material parameter. This enables the multifractal crack growth rate semiempirical model to be established in terms of Paris coefficient and exponent, fractal characteristics, and fatigue fracture properties of the material. A significant statistical dispersion is noted which is typical of a fatigue response. Given this, a probabilistic model based on Walker’s crack growth rate equation considering the variability in the crack tip driving force range, AK and stress ratio, R is developed. The model's validity is examined using selected sets of fatigue crack growth curves of AI-7075-T6, Al- 2024-351 and Ti-6Al-4V alloys. A good fit of the experimental data is noted. The model variance shows a convergent trend with an increasing number of test coupons, thus providing the statistical means of establishing sample sufficiency. The probabilistic model is annexed to the fractal analysis to yield an integrated probabilistic-fractal fracture model. The application of the integrated model to the general structures that lack the crack geometry factor for fatigue crack growth analysis is demonstrated on a bell crack structure. The results are contrasted with AK estimate established through the contour integral (CI) approach using Abaqus software and a close resemblance is noted. Thus, the model could be employed for the prediction of the fatigue crack growth response of engineering structures where the crack geometry factor is not readily available.