The US NRC and EPRI have developed a probabilistic fracture mechanics code through a memorandum of understanding. The code is called xLPR (extremely low probability of rupture) and can be used to evaluate a variety of structural integrity problems dealing with fatigue and primary water stress corrosion cracking (PWSCC) degradation. The code version 2 is now officially released and has been used by both NRC, EPRI and their contractors to perform a series of studies.

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# Themes: Probabilistic Aspects of Fatigue Crack Growth and Fracture: Frameworks, Tools, and Applications

# FAVPRO: NRC’S 21ST CENTURY REACTOR PRESSURE VESSEL PROBABILISTIC FRACTURE ANALYSIS TOOL

Planned and unplanned transients in the operation of a nuclear reactor place stresses on the reactor pressure vessel (RPV) that affect its structural integrity. FAVPRO: Fracture Analysis of Vessels – Probabilistic, is a modern, parallel, object-oriented software tool that can provide high-confidence probabilistic assessments of reactor pressure vessel integrity for any population of flaws and any number of transients, with the goal of risk-informing technical and regulatory decision making. FAVPRO is the successor to the NRC’s FAVOR code with higher solution speeds and new features like automated testing and documentation. The updated tool also includes new embrittlement models and other improvements to align with new fracture mechanics standards.

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# ADAPTIVE MULTIPLE IMPORTANCE SAMPLING FOR STRUCTURAL RISK ASSESSMENT [Keynote]

The USAF Airworthiness Bulletin Risk Identification and Acceptance for Airworthiness Determination defines airworthiness in terms of the probability of aircraft loss per flight hour1 with one important component being the aircraft structure. The probability-of-failure of an aircraft component is challenging to compute due to its small size, typically 10-7 or less. As a result, simplified fracture mechanics models are usually used with a small number of random variables. However, these simplifications may lead to an inaccurate probability-of-failure estimate. To address this issue, an adaptive multiple importance sample method was developed that can compute very low probabilities with significant efficiency. This allows one to consider more realistic fracture mechanics models and a larger number of random variables than has been previously possible. The method is adaptive in that it will adjust to the varying relative importance of the random variables for different applications. Convergence is ensured such that the coefficient of variation is below a user-defined threshold. Results to date show efficiency gains of 5 or 6 orders of magnitude over standard Monte Carlo sampling for typical problems of interest. The methodology will be outlined and demonstrated using aircraft example problems.

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# FRAUCTURE MECHANICS-BASED PROBABLISTIC STRUCTURAL INTEGRITY ASSESSMENT FOR AERO-ENGINE TURBINE DISK

Aero-engine turbine disks are safety-relevant components which are operated under high thermal and mechanical stress conditions. The aim of this work is to present part of a fracture mechanics-based probabilistic assessment procedure under development which aims at calculating the critical rotational speed of the turbine disk based on the numerical-analytical solutions and regulations for the failure probability. In particular, the rim-peeling failure mode is considered as case study. A semi-circular surface crack is modelled at the most stressed region at the diaphragm of a turbine disk. In order to design a lab representative specimen, beside the crack driving force, expressed in terms of J-integral, also the constraint to plastic deformation e.g., stress triaxiality, at the crack-tip must be similar for the same crack in the specimen and in the disk. The analytical solutions to calculate the crack driving force for the lab representative specimen are used for the Monte Carlo simulations, the result of which has been assessed in the form of a Failure Assessment Diagram (FAD). The results of the probabilistic structural integrity assessment show good agreement between Monte Carlo simulations and certification values for the disk in terms of expected failure mode and value of the critical speed.

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# PROBABILISTIC FRACTURE MECHANICS FOR HEAVY-DUTY GAS TURBINE ROTOR OPERATIONS IN THE ENERGY SECTOR [Keynote]

We present probabilistic fracture mechanics methodologies and applications from an energy industry perspective. Topics include probabilistic fracture mechanics for heavy-duty rotating equipment, including gas turbine rotor disks, probabilistic modeling of forging flaw crack nucleation, modeling of non-destructive inspection capabilities, and probabilistic crack propagation from low-cycle fatigue-initiated cracks. We will present relevant new design and service applications in which reliable risk quantification and minimization are paramount. We will also illustrate how the developed Monte Carlo scheme harnesses the power of high-performance computing, including Graphics Processing Unit (GPU) utilization, to enable a fast computational turn-around time for the millions of individual fatigue crack growth calculations needed to resolve the low-risk requirements.

The presented methods pave the way for a fast and reliable robust risk quantification of power plant components and systems, including probabilistic digital twins, and support power plants’ efficient, reliable operation. These methods are essential for the energy transition, including intermittent renewable energy sources such as wind turbines and photovoltaic systems, where the start-up flexibility of gas turbines is a crucial requirement.

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# A SOFTWARE FRAMEWORK FOR PROBABILISTIC FATIGUE CRACK GROWTH ANALYSIS OF METALLIC COMPONENTS [Keynote]

A comprehensive software framework has been developed for probabilistic fatigue crack growth (FCG) analysis of safety-critical metallic components. The framework has been implemented in a computer code called DARWIN® (Design Assessment of Reliability With INspection). DARWIN determines the probability of fracture for a component as a function of operating cycles, with and without inspection, by integrating finite element (FE) geometries, stress, and temperature analysis results; fracture mechanics models; material anomaly data; probability of anomaly detection; and uncertain inspection schedules with a user-friendly graphical user interface (GUI). The framework can accommodate anomalies occurring anywhere in the volume of the component (such as material voids or inclusions) or anomalies occurring only on the surface of the component (such as manufacturing or maintenance damage).

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# PROLOCA 7.1 A PROBABILISTIC FRAMEWORK FOR FATIGUE ANALYSIS OF ALUMINUM AND WELD STEEL STRUCTURES [Keynote]

PROLOCA 7.1 is a probabilistic fracture mechanics (PFM) cods developed for the analysis of damage initiation and growth up to the point of structural failure. The PROLOCA (PRObability of Loss Of Coolant Accident) code was formulated to address nuclear piping and was based on past probabilistic analyses of fatigue in aircraft. These methods were integrated into a code, PROLOCA 2.0 originally developed under NRC contract [1], which was developed for nuclear piping analyses. Since that time, PROLOCA was continually improved under contract to an international team of regulators and operators. In this paper we examine some of the key differences between PROLOCA and other frameworks for fatigue analyses. Examples of the application of PROLOCA to dissimilar metal welds and aircraft damage tolerance are given to demonstrate the extremely low risk of failure with and without inspections and leak detection

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# PROBABILISTIC STRUCTURAL INTEGRITY ASSESSMENT OF WELDED JOINTS [Keynote]

The fatigue assessment of welded joints requires several input data, which can be subdivided into three categories: geometry, material and loading. The number of input data depends essentially on the complexity of the models employed and on the level of accuracy of the analysis. It is common practice to use safety factors in design to account for the scatter of the input parameters. Nevertheless, overly-conservative factors lead often to unrealistic estimations of fatigue life. This work presents a fracture mechanics-based model for the structural integrity assessment of welded joints under constant amplitude fatigue loading, in which the local geometry at the weld toe and the fatigue crack growth properties are considered statistically distributed. The approach is validated against a large number of experimental data.

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# PROBABILISTIC CRITICAL FLAW SIZE ASSESSMENTS IN THE CIRCUMFERENTIAL WELDS OF LAYERED PRESSURE VESSELS [Keynote]

The National Aeronautics and Space Administration (NASA) operates approximately 300 aging, carbon steel, layered pressure vessels (LPVs) that were designed and manufactured prior to ASME Boiler and Pressure Vessel (B&PV) code requirements. Fitness-for-service assessments and traditional evaluations of these non-code vessels is a challenge due to unique uncertainties that are not present in code vessels, such as missing construction records and the use of proprietary materials in construction. Furthermore, many of the steels used in these non-code vessels are at a risk of cleavage fracture at low temperatures within the operating temperature ranges of the NASA sites where these vessels are installed. Additionally, the stress state in critical regions of the LPVs, such as the longitudinal seam welds and circumferential welds, is uncertain due to weld residual stresses (WRS), geometric discontinuities, and stress concentrations in weld connections. In order to guide non-destructive evaluation (NDE) and assessment of the circumferential welds and account for uncertainties in these non-code LPVs, probabilistic critical initial flaw size (CIFS) and critical crack size (CCS) analyses were performed for eleven locations of interest within the head-to-shell and shell-to-shell circumferential welds of three demonstration LPVs.

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# INTERMITTENCY IN FATIGUE CRACK GROWTH AND FATIGUE STRIATIONS

The fatigue crack growth rate exhibits apparent self-similarity as it grows as a power of the stress intensity factor (the Paris–Erdogan law). We have studied the fatigue crack growth in two aluminum alloys (Al-1050 and Al-5005) using optical methods and found that the crack tip advances in an intermittent way, characterized by a power-law distribution of crack tip jump sizes. The exponent of the distribution is around two – higher than what is usually observed in avalanching systems – and there is a cutoff that increases with increasing crack velocity. If the generally accepted one-to-one correspondence between the crack tip advancement per cycle and the fatigue striation lines on the fracture surface holds, one should expect a similar distribution for the striation line spacings. We have performed post-mortem fractography using scanning electron microscopy and, by automatically tracking the striation spacings, we indeed see a similar power-law distribution with a cutoff and an exponent around two.

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