Fracture mechanics regards the initiation and propagation of cracks. The impact of material fracture varies depending on the specific application but the results can be catastrophic. Therefore, gaining an understanding of fracture and failure is very important. The ability to predict when a crack will initiate and fail, and thus the resulting fatigue life of the component must be understood to ensure the safe design and utilisation of structural components. To this end, this project has involved the development of novel techniques for crack growth modelling and fatigue life assessment, including crack initiation and propagation.

Crack modelling

Reversed Plasticity Domain Method for crack initiation

The ability to predict a component’s fatigue life is vitally important and a number of assessment methods have been developed which are in routine use in industry. In addition to understanding the fatigue life of engineering components, the ability to predict the precise load at which failure will occur is also vitally important in order to ensure their integrity during operational life. For this reason, a thorough understanding of different failure behaviours is crucial both in the component design stage and also during service for condition monitoring purposes. The analysis of the steady state response of engineering structures provides invaluable information about the integrity of components when subject to cyclic loading. Few analytical methods exist for this type of investigation and numerical Finite Element modelling can provide much needed information. Direct cyclic analysis (DCA) methods provide a method of determining the steady state shakedown and ratchet response of structures without the need to model the transition period. This leads to significantly reduced computational expense and analysis times, compared to modelling each individual cycle. The Linear Matching Method (LMM) is such a direct method and it provides a numerical procedure for the calculation of the shakedown and ratchet limits. The LMM also allows the generation of a Bree-like diagram through a series of shakedown and ratchet analyses. The Bree Interaction diagram is an efficient tool to produce a plot of primary and secondary stress range which displays the elastic, plastic cyclic, shakedown and ratcheting behaviour regions and the boundaries between them.

A concept has been proposed that utilises the Linear Matching Method and the Bree Interaction diagram in a novel combination that allows the design of an experimental testing programme for low cycle fatigue crack initiation. This method is referred to as the Reversed Plasticity Domain Method (RPDM). It comprises a series of Linear Matching Method analyses for the precise calculation of the shakedown and ratchet limit boundaries in order to identify the reversed plasticity and low cycle fatigue region, in which crack initiation will occur. A Bree-Interaction diagram can then be plotted for visualisation of the critical regions. Identification of this zone allows the calculation of the range of loads in which reversed plasticity and crack initiation will occur. A low cycle fatigue analysis can then be performed which calculates the number of cycles to crack initiation for all loads within this range. The user may then select a single specific design load based on the fatigue life requirements of the component. The Reversed Plasticity Domain Method provides a suite of analysis tools to provide a very efficient technique that encompasses the identification of loads ranges for causing specific damage mechanisms as well as the calculation of the low cycle fatigue life for crack initiation.

Reversed Plasticity Domain Method for crack initiation

Crack simulation is vitally important and there are a number of methods of using fracture mechanics in order to evaluate fracture and fatigue life including R5 and R6 codes and stress intensity factor analysis. However, the focus of this investigation has been with the J-Integral, an alternative to SIF when considering elastic plastic fracture mechanics, and how this can be extended to allow for a cyclic loading history and the evaluation of fatigue life. The cyclic J-Integral was first proposed and implemented by Dowling and Begley. A power law behaviour, similar to that of the Paris equation was developed and so the fatigue crack growth rate of cyclic loading EPFM can be written as:

Where A and m are material constants. The cyclic J-Integral, ΔJ is a function of the stress and strain range, Δσ and Δε and as a result, unlike the cyclic stress intensity factor, is not simply equal to Jmax-Jmin. For this reason, calculating the cyclic J-Integral is inherently more difficult than for monotonic loading and no standard techniques have yet been developed to determine the cyclic J-integral. In order to determine the EPFM fatigue life, the J-Integral must be extended to allow for cyclic loading conditions, much like the SIF range is used in LEFM cyclic fatigue in the Paris Law. The GE/EPRI and Reference Stress Method (RSM) offer simplified methods of approximating the cyclic J-Integral. However, due to the nature of these methods they exhibit considerable limitations and thus produce overly conservative results. A method has been proposed which provides a reasonable approximation for the calculation of the cyclic J-Integral which addresses the known issues in the existing technologies. This can be achieved through modification of a monotonic loading analysis by replacing σy with 2σy and replacing the cyclic load range with a single monotonic load equal to the range. This allows such a modified monotonic analysis to replicate the conditions of a cyclic loading analysis. This method is referred to as the Modified Monotonic Loading (MML) Method. It was discovered that in an un-cracked body subjected to variable loading conditions, the differences between this MML method and the equivalent cyclic analysis were relatively small, highlighting the potential for this technique as a method of determining the cyclic J-Integral. Since the cyclic J-Integral is a function of stress and strain range, for this method to be viable and the hypothesis that it is capable of accurately replicating a cyclic loading analysis to hold true, then the stress range and strain range data from a cyclic loading analysis must match the stress and strain data from the modified monotonic loading analysis. Using this assumption will then allow the determination of the cyclic J-Integral through the MML method within ABAQUS. Following such a hypothesis, the cyclic J-Integral values under fatigue loading can be assumed to be equal to the J-Integral values from the Modified Monotonic Loading method analysis.

Extensive analysis was performed to assess the technique, and the MML results were compared to the cyclic loading analysis to assess the suitability of the technique as a method of calculating the cyclic J-Integral. The stress range and strain range data from the cyclic loading analysis was compared to the stress and strain data of the MML method. It was found that the differences between cyclic and MML analyses were minimal and the contour plots matched very closely. It is assumed that if the stress and strain data of the MML method matches the stress range and strain range data of the cyclic analysis, then the MML can be assumed to be a reasonable approximation of the cyclic J-Integral.  This technique means that a simple monotonic analysis is capable of determining the cyclic J-Integral. This offers a quick and computationally inexpensive method of determining the ΔJ, which would otherwise be difficult and time consuming to calculate. Using the Dowling & Begley law, it can offer an invaluable method of predicting the EPFM fatigue life of a component.

Illustrative process flow of the Reversed Plasticity Domain Method applied to a complex industrial specimen.

Comparison of the stress contours of a standard cyclic analysis (a) and the MML concept (b).

Comparison of the strain contours of a standard cyclic analysis (a) and the MML concept (b).


75 Montrose Street

Glasgow G1 1JX, UK

James Weir Building


T: +44 (0)141 548 2036

All content Copyright © 2020 by SILA research group.