This white paper discusses the use of numerical modeling for fracture and fatigue crack growth assessments in fitness-for-service evaluations. It outlines the limitations of analytical solutions and need for accurate stress intensity factors. Two numerical techniques, finite element analysis and boundary element analysis, are described for modeling complex crack geometries. A case study uses boundary element software to model a crack in a sag mill gear and predict crack growth to determine safe operating conditions.

1. White Paper
Accurate Fracture and Fatigue Crack Growth
Fitness for Service Assessments
Authors
Matt Rudas - Senior FE Analyst, Deacon Engineers
Jurrien de Vos - Principal, Deacon Engineers
All information within this document is copyright and the property of Deacon Engineers - © 2015
Web: http://deaconengineers.com.au
ACN 135 205 408
www.deaconengineers.com.au
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2. Accurate Fracture and Fatigue Crack Growth Fitness for Service Assessments
CONTENTS
Page
1. EXECUTIVE SUMMARY........................................................................................ 3
2. OVERVIEW............................................................................................................ 3
3. FITNESS-FOR-SERVICE ASSESSMENTS OF CRACK-LIKE FLAWS .................. 3
4. NUMERICAL MODELLING OF CRACK-LIKE OF FLAWS ..................................... 5
4.1 Flaw modelling using FEA ......................................................................................7
4.2 Flaw modelling using BEA......................................................................................7
5. CASE STUDY – SAG MILL GIRTH GEAR ASSESSMENT USING BEASY ........... 7
5.1 Background ............................................................................................................7
5.2 Numerical model for SIF evaluation........................................................................9
5.3 Fatigue crack propagation of flaw.........................................................................11
5.4 Outcomes of gear assessment.............................................................................13
6. CONCLUSION ..................................................................................................... 13
7. BIOGRAPHIES..................................................................................................... 13
7.1 Jurrien de Vos, Principal.......................................................................................13
7.2 Matthew Rudas, Senior FE Analyst ......................................................................13
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3. Accurate Fracture and Fatigue Crack Growth Fitness for Service Assessments
1. EXECUTIVE SUMMARY
This whitepaper discusses the application of numerical modelling for the calculation of
fracture parameters for Fitness-For-Service assessments. The shortcomings of analytical
solutions presented in assessment codes, and the requirements for high degrees of
accuracy of calculated fracture parameters, are discussed. The main two solid mechanics
numerical techniques are briefly outlined and two commercially available fracture
mechanics and fatigue crack growth software packages are presented. Finally, the fracture
mechanics component of a Fitness-For-Service assessment is presented as a case study.
The case study is a cracked helical mill gear tooth and the assessment was carried out
using BEASY Boundary Element Analysis software.
2. OVERVIEW
The quality of Fitness-For-Service assessments involving crack-like flaws is highly
dependent on the accuracy of calculated fracture parameters. Analytical solutions
presented in fracture assessment codes are limited to a small number of idealized
situations and extrapolation beyond the validity of the solutions is not recommended.
Numerical analysis, using Boundary Element Analysis (BEA) or Finite Element Analysis
(FEA) software, is essential for structures having complex geometries and loading histories
not covered by the solutions in the codes. Three-dimensional models that include detailed
flaw morphologies generate accurate inputs for crack growth law calculations that are
extremely sensitive to input errors. This provides accurate predictions for remaining life
and equipment de-rating figures and allows the establishment of safe inspection and
monitoring regimes.
3. FITNESS-FOR-SERVICE ASSESSMENTS OF CRACK-LIKE FLAWS
Fitness-For-Service (FFS) assessments are carried out on defective engineering assets with
the aim of answering the following key questions:
• Can the asset continue to be operated in a defective state?
• If not, can the asset be re-rated to ensure continued safe operation?
• What repair and inspection strategy should be adopted?
More specifically, for structures containing crack-like flaws, an FFS can provide answers to
the following questions:
• What are the reasons for initiation of the flaw?
• Is the flaw growing in size, and if so, at what rate?
• What is the required load reduction to prevent the flaw from growing?
• What is the required load reduction to achieve a particular rate of growth?
• What is the critical (failure) size of the flaw?
• What is the growth path of the flaw?
The successful FFS assessment of crack-like flaws is heavily dependent on the accuracy of
the fracture input data, namely the stress intensity factors (SIF’s). Crack growth rates, for
example, are calculated by raising the crack tip SIF range to the power of m, the crack
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4. Accurate Fracture and Fatigue Crack Growth Fitness for Service Assessments
growth exponent. Depending on the selected crack growth law, m can take on values as
high as 8, where an error of 10% in the calculated SIF range can result in an error factor of
2.3 on the crack growth rate.
Figure 1 – The Paris Law for fatigue crack growth. The SIF range at the crack tip is raised to
the power of the crack growth exponent, m in order to calculate the crack growth
increment for a single load cycle, da/dn.
SIF’s can be determined in two ways:
• By analytical solutions presented in FFS and fracture assessment codes and
handbooks, or;
• By numerical modelling of the structure containing the crack.
Calculation of SIF’s by analytical solutions is done using idealized scenarios of structure
and flaw geometry and applied loads. Calculations are generally performed according to
the rules of applicable codes such as BS 7910 - Guide to methods for assessing the
acceptability of flaws in metallic structures. Care needs to be taken to ensure that the
limits of applicability of the codes are not exceeded - extrapolation outside of the limits is
forbidden by the rules of the codes. Simplification of the actual scenarios to allow the use
of the codes may lead to errors that are highly magnified, as shown earlier in the
hypothetical error factor calculation. Also, any changes in the crack geometry as a result
of load re-distribution by crack growth, are not taken into account.
For cases that fall outside of the validity of the solutions within codes, accurate
determination of SIF values can only be made by numerical fracture mechanics modelling
of the structure containing the crack.
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5. Accurate Fracture and Fatigue Crack Growth Fitness for Service Assessments
Figure 2 – Example surface flaw definition from BS7910. Analytical solutions for SIF’s are
presented for specific ratios of a, c, B and W, the plate width in the plane of the flaw.
Figure 3 – Fracture of gear teeth showing complex crack propagation paths. Hypoid gear
tooth loads and stress redistribution due to crack growth makes this scenario difficult to
accurately assess with analytical solutions.
4. NUMERICAL MODELLING OF CRACK-LIKE OF FLAWS
Fracture parameters can be obtained numerically by commercially available software
based on FEA and BEA. Solid mechanics FEA involves discretization of the volume of a
structure into finite elements in order to generate an approximate solution for
displacements, stresses and strains. Generally, as the number of elements increases, the
error of the solution decreases. By using a different mathematical approach to FEA, BEA
only requires that the surface of the model is discretized into boundary elements.
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6. Accurate Fracture and Fatigue Crack Growth Fitness for Service Assessments
Both methods have their strengths and weaknesses, and both are more suited to certain
applications than others. However, when used correctly, both methods have been proven
over time to accurately determine SIF’s for use in crack growth studies and FFS
assessments.
Figure 4 – FEA ANSYS model of rotating equipment with girth gear.
Figure 5 – BEA BEASY model of helical girth gear segment.
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7. Accurate Fracture and Fatigue Crack Growth Fitness for Service Assessments
4.1 Flaw modelling using FEA
Commercially available packages such as Abaqus are able to calculate two- and three-
dimensional SIF’s in a quasi-static fracture model using contour integrals. Solution
dependant two- and three-dimensional crack growth is modelled using the extended
finite element method (XFEM), where re-meshing of the model by the user is not
required.
Figure 6 – Abaqus surface crack mesh and stress results.
4.2 Flaw modelling using BEA
Commercially available packages such as BEASY are able to calculate two- and three-
dimensional SIF’s using both crack tip opening displacement (COD) and the J-integral.
BEASY uses the Dual Boundary Element Method for crack elements, which means that
only one of the crack faces needs to be modelled by the user, with the other being
generated automatically. Automatic two- and three-dimensional crack growth is also
supported by the software.
5. CASE STUDY – SAG MILL GIRTH GEAR ASSESSMENT USING BEASY
5.1 Background
The girth gear on a semi-autogenous grinding (SAG) ball mill used in the mineral
processing industry was found to contain a large crack, originating from the face of one of
the helical gear teeth. Several attempts to repair the crack were made by excavating the
surface of the gear tooth in the vicinity of the crack, however the crack continued to
propagate. With a scheduled maintenance shut of the mill due some months away, the
plant operator commissioned an FFS assessment with the aim of investigating the
possibility of avoiding a costly, unscheduled shut to carry out weld repairs of the gear
tooth. The flow of work for the assessment of the flaw is shown below in Figure 7.
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Figure 7 – Mill gear assessment workflow.
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Figure 8 – Mill girth gear showing excavated tooth face. The crack breakthrough edges
spanned the width of the excavation.
5.2 Numerical model for SIF evaluation
In this instance, the BEA fracture and fatigue crack growth module of BEASY software was
used for the assessment. The helical gear geometry, tooth face excavation and non-planar
flaw geometry were modelled in full detail in BEASY. SIF values were calculated using the
J-contour integral at each mesh point along the crack front.
Figure 9 – BEASY element mesh showing excavation on tooth face.
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Figure 10 – Cutaway BEASY mesh showing the crack originating at the base of the
excavation.
Figure 11 – Scaled deflection image showing opening of the crack under load.
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Figure 12 – Automatically generated BEASY internal points forming circular contour paths
for evaluation of the J-integral at each element along the crack front.
Figure 13 – BEASY mode I SIF results along the crack front for the first crack growth step.
5.3 Fatigue crack propagation of flaw
The automatic crack growth capabilities of BEASY were used to calculate the growth rate
and path of the crack. A two stage crack growth law, in accordance with the requirements
of BS 7910 for steels in air, was adopted for the crack propagation study. Material
properties were taken from the BEASY database for the appropriate material, as follows:
• Ultimate stress
• Yield stress
• Part-through and plane-strain fracture toughness
• Threshold stress intensity factor range at R = 0
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• Crack growth rate coefficient
• Crack growth rate exponent
Figure 14 - Cutaway BEASY mesh showing the crack after several growth steps. BEASY
automatically meshed the crack and component breakthrough surfaces during crack
growth.
Figure 15 – BEASY graph of average crack size on the crack front for 5 growth increments.
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5.4 Outcomes of gear assessment
Initially, the results of the BEASY SIF calculations were used to provide the plant operator
with a reduced load magnitude, in order to eliminate propagation of the crack. The
amount by which the load was required to be reduced was not feasible from a process
perspective, therefore a fatigue crack growth study was carried out.
The crack growth calculations indicated the statistical probabilities of the crack reaching a
critical size prior the planned maintenance shut. In this instance, the most likely rate of
crack growth was low enough to allow the plant operator to continue running the mill. An
NDT monitoring regime was put in place to ensure the growth rate did not exceed the
predictions made using the fatigue crack growth software.
The FFS assessment enabled the plant operator to avoid a costly, unscheduled repair to
the gear.
6. CONCLUSION
Analytical solutions for fracture parameters presented in FFS and flaw assessment codes
are often unable to be applied to structures with complex geometries, loading histories
and flaw morphologies. Simplifying the assessments to enable the use of analytical
solutions can lead to large errors in predicted safe operating lives of defective engineering
assets. In these situations, accurate fracture parameters can only be obtained by
numerical analysis of the flawed structure.
7. BIOGRAPHIES
7.1 Jurrien de Vos, Principal
Jurrien de Vos founded Deacon Engineers in Perth, Western Australia in 2009. Since then
Deacon Engineers has expanded its offices across Australia. Jurrien has had many years’
experience in the design and failure investigation of gearing and tyres for rotating
equipment. He has extensive experience with failure of elements in rolling contact from
his time with Hofmann Engineering and RCR Tomlinson.
7.2 Matthew Rudas, Senior FE Analyst
Deacon Engineers’ specialist FEA and BEA analyst is Matt Rudas who holds a PhD in
fracture mechanics from the University of Western Australia. He has extensive experience
in FEA and BEA, computer aided engineering, mechanical design, fracture, fatigue and
dynamic stress analysis. He is experienced in laboratory and real world experimental
fatigue and stress analysis and strain measurement. His postgraduate research focused on
modelling crack growth in bi-material composites using BEA.
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