1. Introduction
Although some pipelines exhibit certain levels of defect tolerance, it is fundamental to accurately detect, examine and monitor the growth of these defects at an early stage to aid the prediction of the residual operating life of the structure and subsequently minimise the risk of unexpected failures through risk-based management that determine proper inspection intervals and maintenance procedures [
1]. The defect detection and examination stage of a pipeline’s lifecycle typically involves obtaining defect information through acceptable inspection, monitoring, testing and analysis techniques. These techniques include hydrostatic testing [
2], non-destructive evaluation (NDE) and in-line inspection (ILI) [
1].
Fatigue assessment where stable micro-crack growth occurs can be obtained based on the stress-life (S-N) curves-based approaches or the fracture mechanics approach. The S-N curve data are obtained from experiments and are typically used to assess non-planar flaws, e.g., corrosion, while the fracture mechanics approach can be applied to describe the fatigue behaviour of planar flaws such as cracks, and lack of fusion or penetration using an engineering critical assessment (ECA) [
3]. This procedure can be used to assess the criticality of flaws found in service pipelines and to make informed decisions whether the flaws are to be accepted or rejected, and if repair or replacement of the pipeline component is necessary. The application of ECA acceptance criteria in the assessment of pipeline girth welds can significantly reduce the costs for pipeline integrity management by minimizing unnecessary repairs.
Previous researchers have also revealed that the factors significantly influencing the initiation and/or propagation of crack-like defects in pipes and pipeline welds include installation and operating conditions (i.e., high pressure and high temperature) [
4,
5], material properties of the pipe and pipeline weld, weld geometry, and crack size, location and orientation [
6,
7]. Advancements in pipeline materials and fracture mechanisms are, therefore, of great importance leading to an increased trend in the application of computer software to solve crack problems for defect assessment and growth in pipelines to predict residual lifetime [
8,
9,
10,
11,
12,
13], with the possibility of reducing the need for repairs and delays imposed. Today, the basic criteria for fracture mechanics have been formulated and with the help of analytical and numerical models, solutions to various problems can be obtained.
The stress intensity factor (SIF) is used in crack growth prediction to assess whether a specific crack causes a structural component to fracture [
6,
8,
11,
14,
15,
16,
17,
18]. Most of the numerical studies on surface cracked offshore metallic pipes readily available in open literature focus on the growth of shallower surface cracks (
a/
c < 1.0) due to cracks detected at inspection having small aspect ratios [
19]. For example, in [
8] the SIFs of surface cracks up to aspect ratios of
a/
c < 1.0 were presented based on contour integral evaluation on each mesh element node along the crack front. However, cracks with high aspect ratio (
a/
c > 1.0) may exist in offshore metallic pipes owing to corrosion or a coalescence of multiple cracks. One of the objectives of this paper is to, therefore, study the circumferential crack growth of surface cracks in an offshore metallic pipe up to an aspect ratio of
a/
c = 1.6. Furthermore, most of the previous work, for example, [
6,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19], address either the effects of longitudinal cracks or external circumferential cracks under different loading (e.g., internal pressure, bending, and/or torsion) on SIF estimations. Analytical solutions to calculate the SIFs for circumferential surface cracks in pipes subjected to tension and bending loads were developed by [
20] for
a/
c ratios up to 1.0 and later adopted in flaw assessment guides e.g., [
3]. In addition to a crack aspect ratio limit of 1.0, the study by [
20] only considered a range of external surface cracks to estimate geometry correction factors presented in tabular form. This can prove infeasible especially when the need for a continuous evaluation of SIFs during crack propagation arises. To address this, the current work in this paper through FEM proposes a modified version of Raju–Newman method by introducing a weighted function to estimate the geometry correction factor for internal and external surface cracks in metallic pipes subjected to tension.
Difficulties arise in the direct evaluation of the SIF from the local state at a crack tip due to the high stress singularity in that region or substantial plasticity. To account for this, more indirect energy-based methods are employed instead. A fracture parameter used to calculate crack driving force is the crack tip opening displacement (CTOD) [
13,
21]. This is a measure of the distance between crack faces before unstable crack propagation occurs and it is the preferred method of determining crack driving forces experimentally. Nonetheless, there are several ways to define CTOD [
22], which may lead to different values for the same specimen with respect to the adopted definition in fitness for service assessments. Therefore, unlike the J-integral approach, which is theoretically sound and robust, a CTOD definition needs to be established. Another example of such methods is the J-integral. The J-integral, a path independent integral [
23], due to its robustness and simplicity is used to evaluate stress intensity factors (SIFs) and characterise stable crack growth [
24]. Originally developed for 2D cases, the J-integral is computed along a line integral path which encloses a crack tip. A distinct advantage of using the J-integral to define crack tip severity in computational fracture mechanics is its ability to be accurately estimated far away from the crack tip region of high stress and strain gradients. The J-integral is also known to represent the energy release rate per unit crack extension in elastic-plastic materials. The J-integral and SIF are the most widely used in investigating the fracture response of structures. However, most structural components are of complex geometries and under realistic loading, therefore, a general analytical solution does not exist.
Engineering critical assessment of a 3D crack loading provide a means of associating the severity of cracks or crack-like defects with the operating conditions of the pipeline by assuming that fracture occurs when the fracture mechanics parameter reaches a critical value. For simplicity, we assume half of the pipe thickness as the final crack depth in this paper. According to a research on fatigue crack shape evolution by [
25], cracks generally exhibit irregular crack fronts on initiation. However, these cracks develop rapidly into approximate semi-elliptical shapes in response to cyclic loading due to fatigue. Therefore, this study assumes cracks of semi-elliptical shape. A method for the evaluation of 3D J-integral, by applying the second derivative of displacement field to calculate a surface integral term which constitutes the total 3D J-integral is investigated in [
26]. In comparison to the method applied in [
8], this method uses an arbitrary 2D domain surrounding and perpendicular to the 3D crack front to evaluate the J-integral at any point on the crack front. Another objective of this paper is to apply the method presented in [
26] to develop accurate 3D SIF solutions for a range of circumferential semi-elliptical cracks in pipes using COMSOL Multiphysics software [
27]. To our knowledge, the capability of the software to evaluate 3D J-integral and SIFs to include the effect of an additional surface integral term has not been clearly established. This paper therefore demonstrates the application of the software to evaluate the 3D J-integral and SIFs for circumferential semi-elliptical cracks based on the output results of stress, strain and displacement. The SIF solutions are subsequently used for the fatigue crack growth assessments of pipes in fitness-for-service assessment based on the BS 7910 standard. To validate the COMSOL Multiphysics model in the estimation of J-integral and SIF values, a study was performed on a cylindrical model containing an internal and external through-wall circular crack. Results from the analyses were compared with results obtained from previous works [
26,
28] and showed good agreement. Furthermore, due to the lack of study on circumferential internal surface crack in metallic pipe under tension which is also suggested in [
19], analyses on a typical offshore pipeline containing internal and external surface cracks, respectively, are presented herein.
4. Fatigue Crack Propagation Analysis for Semi-Elliptical Surface Cracks in Plain Pipe
Fatigue cracks in offshore structures are undesirable and if not properly monitored can grow leading to failure of the structure due to cyclic loading. The prediction of fatigue crack growth and the evolution of the crack shape is, therefore, of great importance in the offshore industry. However, investigating the fatigue lives of offshore structural components experimentally is costly and requires time. Therefore, it is can be particularly useful to estimate the remaining lives of structures by applying principles such as fracture mechanics.
This section presents the fatigue assessment of an offshore pipe containing semi-elliptical cracks which are dependent on the stress intensity factor (SIF). The SIF values used are the maximum local values evaluated on the extreme points of the crack front i.e., the maximum crack depth and crack length along the pipe surface. A study of the change in crack aspect ratio is also presented. It is assumed that the pipe is subjected to constant amplitude loading where the stress ratio,
R, which is the ratio of applied minimum stress to applied maximum stress in the fatigue cycle is zero:
The most widely used model for predicting stable fatigue growth is the Paris model developed by Paris and Erdogan [
34]. It states that the rate of crack growth is proportional to the SIF range through the power law as shown in Equation (6):
Considering the effect of stress ratio, Equation (6) can be rewritten as:
where
da is the change in crack depth,
dc is the change in crack surface length,
dN is the change in the number of cycles,
C and
m are material properties given as
and
, respectively [
3], and
is the change in stress intensity factor defined by Equation (8):
Substituting for
in Equation (6) and integrating yields:
By integrating using the upper and lower limits of
a, Equation (9) becomes:
where
is the number of load cycles required to grow a crack from an initial crack depth, (
ai) to a larger crack size
ai >
ai+1. Unstable crack growth and fracture eventually occurs when the crack depth reaches a critical value (
ac) and
Kmax =
KIc;
Y is the dimensionless shape factor related to the crack size;
is the cyclic stress range. The integral in Equation (9) is solved numerically because
Y is also dependent on the crack size. It should be noted that a maximum crack depth of
was chosen in this paper to enable direct comparison of stable crack growth with other researchers.
Previous researchers, such as [
35,
36], employ a more rigorous approach to account for the crack shape evolution during fatigue crack growth. This scheme propagates the crack on all the nodes along the crack front by evaluating the local stress intensity factor at each node. A new crack shape and size is then determined after each fatigue load cycle. Although this procedure is likely to provide more practical results, it is generally not employed due to high computational costs. A much simpler scheme which accounts for the local stress intensity factors at only the deepest and surface points of the crack is used in the current work for the fatigue crack growth evaluation. A new crack shape is developed after each loading cycle. However, because the SIFs are only evaluated at these points, the subsequently developed cracks will maintain a semi-elliptical shape. The crack is schematically presented in
Figure 16.
The crack propagation is simulated using the following procedures:
The stress intensity factors,
K, for a surface crack are estimated using Equations (2)–(4) at the deepest point
Ka and at the crack surface
Kc. The validation of the current method uses the pipe geometry in
Figure 9.
By assuming a small linear increment in ai, Δai, and substituting for da in addition to substituting the values of Ka and Kc in Equation (6), the value of dc or Δci for the present cycle is evaluated. Note that the stress ratio, R, is assumed to be zero. Therefore, and .
The crack geometry and shape are updated for the next propagation step by adding Δai and Δci to ai and 2ci, respectively.
The procedures above are repeated until the crack depth reaches a prescribed size, in this case the mid-thickness of the pipe.
Figure 17 shows the procedure for predicting fatigue crack growth using the Paris law, until the ratio of the crack depth to thickness ratio (
ai+1/
t) reaches
.
Using the model developer tool in COMSOL Multiphysics [
27], a method was created in the form of written code to automate the looped process of evaluating the SIFs, updating the crack geometry and calculating the fatigue life.
Since the stress intensity factors at the crack deepest point,
and surface,
differ, the Paris law governing their values can be rewritten as shown in Equation (11):
Assuming
and using the initial crack size, the relationship between the change in crack depth and crack half-length can be derived from Equation (11) to yield:
Equation (12) is employed in COMSOL Multiphysics by dividing the final crack extension ( into a large number of equal increments , to accurately capture the growth trend and assuming each increment is created at a constant crack growth rate. The post-processing stage was automated by developing a script that handles the global evaluations of J-integral, SIFs, and corresponding fatigue life in the previous incremental study and updates the geometry for the next study. For each increment of crack depth, a new crack half-length is evaluated and the crack geometry was updated until the crack depth reached half of the pipe thickness.
Figure 18 shows the predicted fatigue crack growth pattern for a set of surface cracks subjected to tension. The curves demonstrate the change in aspect ratio as the crack grows from an initial size to a final size, in this case the mid-thickness of the pipe. The initial geometries of the cracks are defined by
and
with the values of
swept from 0.05 to 0.5, while the
values are chosen as
.
For all the initial crack shapes considered, the final
a/
c ratios were predicted to tend towards 0.8 when the crack depth became equal to half of the pipe thickness, showing good agreement with previous studies [
8,
37].
According to Annex M.7.3.4 of BS 7910 [
3], it is recommended that the flat plate solution for stress intensity factors may be applied to external circumferential surface flaws in pipe. As such, the theoretical stress intensity factor for tension load may be determined using Equation (13) given by:
for
and
.
Where
is the complete elliptic integral of the second kind given by:
A new set of equations for estimating the stress intensity factors for a semi-elliptical crack in a pipe are proposed considering that the crack aspect ratio converges to
. This study only evaluates the maximum values of SIF at the extreme points of the crack front where the SIF at the crack depth,
and the SIF at the surface length,
, considering the symmetry boundary condition in
Figure 8.
The thickness and internal radius in this study were kept constant at
mm and 160.5 mm, respectively, therefore, the boundary correction factor,
, was normalized using the equation below:
where:
and
can be written in matrix form as:
Assuming the matrix operation above is represented as
, where the elements of
are coefficients (presented in
Table 5) obtained from curve-fitting (see
Appendix A). Therefore:
where:
while
,
and
are given by:
Note that these equations are valid for and at the deepest and surface point of the crack are radians and , respectively.
Figure 19,
Figure 20,
Figure 21 and
Figure 22 show the stress intensity factor results obtained for initial values of
a/
t,
and
a/
c,
when compared with the literature [
3,
8] to gain confidence in the methodology applied in this study. The stress intensity factors obtained from 3-D finite element computation were compared with analytical results obtained from [
3]. For the initial crack aspect ratio,
, maximum absolute percentage errors of 2% and 4% in the stress intensity factor solutions at the deepest point and surface point of the crack were achieved, respectively. For the initial crack aspect ratio,
, maximum absolute errors of 2% and 1% at the deepest and surface point of the crack were achieved, respectively. Additionally, for the initial crack aspect ratio,
, maximum absolute errors of 2% and 2% at the deepest and surface point of the crack were achieved, respectively.
The stress intensity factors were also compared with numerical results from [
8]. For a crack with an initial aspect ratio of
, the maximum absolute errors at the deepest and surface points are 2% and 2%, respectively. For an initial crack aspect ratio of
, the absolute errors at the deepest and surface points are 2% and 2%, respectively, while the absolute errors for an initial crack aspect ratio of
are 2% and 3% at the deepest and surface points, respectively. These results compare well with published literature, therefore, demonstrating the applicability of the current methodology in the determination of stress intensity factor solutions for cracks in pipes.
The SIFs obtained are used to predict the fatigue lives of external surface breaking semi-elliptical cracks under tensile cyclic load and assuming a stress ratio of . The cracks are propagated from an initial crack depth of 1.045 mm ( to a final crack depth of 10.45 mm ( considering different aspect ratios.
The predicted fatigue life of an external surface crack in a plain pipe growing from an initial size of 1.045 mm (
) to a final size of 10.45 mm (
) is shown in
Figure 23. It is clear that the predicted fatigue life increases with the initial crack aspect ratio. For example, the fatigue life increases from about 6.1 million cycles for a crack aspect ratio of
to about 17.2 million cycles when
. This behaviour demonstrates that “short” cracks tend to propagate faster than long cracks, subjected to the same nominal driving force. The fatigue life results were also compared with the work in [
8]. In their work, when the initial crack aspect ratio is
, the number of cycles results in about 5.8 million cycles. Likewise, when
and
, the number of cycles equal 7.5 million cycles and 16.3 million cycles, respectively. After a comparison of the number of cycles predicted using the current method and computational studies by [
8], the errors came up to be 3%, 4%, and 5% for
respectively. Thus, the current model estimates a longer fatigue life by up to 5%.
Similarly, the stress intensity factors, and crack growth of internal surface cracks (
Figure 24) were studied.
The boundary correction factor,
, was normalized using Equation (23):
where:
and
can be written in matrix form as:
Similarly, the matrix operation above can be represented by
, with the elements of
obtained through curve-fitting and presented in
Table 6. Therefore:
where:
while
,
and
are given by:
.
The equations presented here are valid for a/c values of and at the deepest and surface points of the crack where is radians and 0, respectively.
The 3-D stress intensity factors were computationally evaluated and compared with results obtained from BS 7910. The stress intensity magnification factors provided in Annex M of [
3] were linearly interpolated to enable comparison.
The stress intensity factors obtained from 3-D finite element computation were compared with the values obtained from [
3] in
Figure 25,
Figure 26 and
Figure 27. For the initial crack aspect ratio,
, average absolute errors of 10% and 11% in the stress intensity factor solutions at the deepest point and surface point of the crack were achieved, respectively. For the initial crack aspect ratio of
, average absolute errors of 5% and 2% in the stress intensity factor solutions at the deepest point and surface point of the crack were achieved, respectively. The reason for the large error is not quite certain but it may be due to the presence of significant curvature in the geometry for the inner surface crack as compared to the external crack or maybe a lack of consideration of crack growth. However, there is a similar trend in the stress intensity factors for internal and external surface cracks evaluated using the current method. This supports the conclusion made by Bergman [
33] that the influence functions are of the same order for both internal and external surface cracks, indicating the absence of basic errors between the solutions and no significant difference.
The predicted fatigue crack growth of an internal surface crack growing from an initial depth of 1.045 mm to a final depth of 10.45 mm is shown in
Figure 28.
For the initial aspect ratio,
, the number of cycles required to propagate an internal crack with an initial depth of 1.045 mm to a final depth of 10.45 mm is about 6.3 million cycles, while for
, the number of cycles required is about 18.3 million. The relative difference between the fatigue crack growth rate for initial internal cracks of
and
when compared to external cracks are about 4% and 6% more, respectively. This may indicate that the external cracks grow faster than the internal surface cracks hereby requiring earlier assessment. Additionally, the fatigue crack growth pattern for the internal cracks is presented in
Figure 29.
The curves demonstrate the change in aspect ratio as the crack grows from an initial size to a final size, in this case the mid-thickness of the pipe. The initial geometries of the cracks are defined by and with the values of swept from 0.05 to 0.5 while the values chosen as . For all the initial crack shapes considered, the final a/c ratios were predicted to tend towards are maximum value of 0.88 when the crack depth became equal to half of the pipe thickness. A conservative assumption can therefore be made demonstrating that the internal and external cracks maintain similar aspect ratios or shape under the same fatigue loading and initial geometry.