COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Stress Intensity Factor of a Wide Range of Semi-Elliptical Partly
Through-Wall Crack in a Finite-Thickness Plate
K. P. Kou *
Department of Civil and Environmental Engineering, University of Macau, Macao SAR, China
Email: [email protected]
Abstract In practice, cracks of any awkward shape initiated accidentally, e.g. fabrication errors, exist in structures
such as offshore steel structure. Existing of these flaws could lead to disastrous accidence due to fatigue effect.
Therefore, a safe interval of maintenances is substantial in order to keep a reliable residual life and strength of the
structure. To assess the fatigue effect, one approach is to make use of the stress intensity factor (SIF) of the cracks in
fracture mechanics. These flaws are usually treated as semi-elliptical surface cracks (with a and c as the minor and
major axis) on the member of the structure e.g. BS7910. For plates with semi-elliptical surface crack, Newman, Raju
and Kou have reported the stress intensity factor. However, for plates with partly through-wall cracks, namely an
semi-elliptical crack with an imaginary crack depth a which is greater than the thickness (T) of the plate, the stress
intensity factor was not reported. In the current work, semi-elliptical partly through-wall cracks (of sizes 1.0 ≤ a/T ≤ 1.3,
0.01 ≤ a/(2c) ≤ 0.2) on a finite plate subjected to remote tension were analysed based on linear elastic fracture
mechanics. This work finally produced a comprehensive equation of SIF as a function of a/T and a/(2c) which was
obtained by doing a curve fitting on the SIF which is determined in this work.
Key words: partly-through-wall crack, J-contour integral, SIF
INTRODUCTION
Imperfections always happen in structures especially in welded joints of steel structures. These imperfections may take
the form of cracks or other planar defects. In the case of cyclic loading applied on the structures, these cracks may
propagate under fatigue and this can lead to the failure of structural member and finally to a catastrophic collapse of a
structure. Consequently, prediction of the propagation of an existing of postulated crack becomes an important part of
a safe design nd maintenance of a structure. To predict the crack propagation life and the residual life of structure, it is
necessary to know the severity of the crack, especially in terms of the crack tip conditions. In fracture mechanics, this
severity can be measured by several parameters of which the most widely used is the stress intensity factor (SIF) which
depends on the crack size, geometry of the cracked member and mode of loading. Therefore, with the stress intensity
factor known, prediction of crack propagation can be done.
Figure 1: Partly-through-wall crack
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For semi-elliptical surface cracks in plates, Newman and Raju[1, 2], Kuok and Kou[3, 4] reported the SIF with wide
range aspect ratio (0.01 < a/(2c) < 0.5) and crack depth ratio (0.2 < a/T < 0.99). However, when the crack depth a is
greater than the plate thickness T as shown in Fig. 1. The SIF was not reported in the literature. In the current work,
semielliptical partly through-wall cracks (of sizes 1.0 ≤ a/T ≤ 1.3, 0.01 ≤ a/(2c) ≤ 0.2) on a finite plate subjected to
remote tension were analysed based on linear elastic fracture mechanics. This work finally produced a comprehensive
equation of SIF, at the outer surface point and at the reference point as shown in Fig. 1, as a function of a/T and a/(2c)
which was obtained by doing a curve fitting on the SIF which is determined in this work and was obtained by some
previous workers.
FINITE ELEMENT ANALYSIS
The finite-thickness plates with partly-through-wall crack as shown in Fig. 2(a) have been analysed. The normalized
stress intensity factor (NSIF, Fp) of different cracks configuration were investigated. To simulate the semi-infinite
extent of the plate, assumption[2] of w/c = h/c = 4 for plate width and plate height was made. Due to the awkward
geometry formed by the inner surfaces of the plates and the cracks the quadrilateral isoparametric element, as shown in
Fig. 2(b) with mid-side nodes moved to the quarter points was used as the crack tip element. Moreover, a triangular
mesh as shown in Fig. 2(c) near the intersection of the crack front and the inner surface was made to cope with this
awkward geometry as shaded in Fig. 1. As required for the evaluation of stress intensity factor[2, 5], elements around
the crack tip were made to be orthogonal to the crack front.
(a) configuration of the crack
(b) crack tip element
(c) Triangular mesh near inner surface
Figure 2: The finite-thickness plates with partly-through-wall crack.
In the FE analyses, the J-integral along the crack front was first calculated. Based on the location of each J-integral, the
stress intensity factor along the crack front was then calculated. At the intersection of the crack and the outer surface of
the plate, the plane stress condition was applicable and the stress intensity factor, K, was calculated by Eq. (1). For all
other location inside the plate, the plane strain condition was applicable and the stress intensity factor, K was calculated
according to Eq. (2). The stress intensity factor, K, was then normalised as Eq. (3) in which the Q is the elliptical
integral of the second kind and defined as Eq. (4).
RESULTS AND DISCUSSION
Three dimensional finite element analysis of cracked plates were carried out to investigate the NSIF along the crack
front. A wide array of partly-through-wall crack geometries was included. The selected geometries in this analysis
were as: a/T =1.01, 1.03, 1.05, 1.10, 1.20, 1.30 and a/(2c) = 0.20, 0.10, 0.05, 0.02, 0.01. Based on the selected
parameters, a total of 30 finite element models with partly-through-wall cracks were generated and analysed.
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Selected result of NSIF are shown in Fig. 4 in which the parametric angle φ is illustrated in Fig. 3. In the vicinity of the
inner surface, the factors were found to increase very rapidly because of the sharp region formed by the crack front and
the inner surface as shown in Fig. 1. Due to the nature of the mesh used in this region and of the J-integral, the factor at
the precise intersection of the crack front and the inner surface was not reliable. Consequently, three locations close to
that intersection point were chosen as the reference points which are 0.90T, 0.95T and 0.99T measured from the outer
surface as shown in Fig. 1.
Figure 3: Parametric angle defined in elliptical crack
a/T = 1.01
a/T = 1.30
Figure 4: Distribution of NSIF along the crack front
Aspect ratio a/(2c) Although the partly-through-wall crack was not a whole semi-ellipse , the aspect ratio was still
defined as the ratio of the imaginary crack depth (a) and the crack length (2c). The over all distribution is higher for
lower aspect ratios as shown in Fig. 4. The NSIF at the outer surface was extracted and is shown in Fig. 5. The variation
is within a narrow margin. For aspect ratios higher than 0.05, the NSIF varies between 1.5 and 1.8 and starts to
decrease for aspect ratios lower than 0.05. At the particular reference points, variations of NSIF show a clear trend that
as the aspect ratio gets high, the NSIF decreases in a diminishing way. All NSIF at the outer surface and reference point
show a picture that for a crack with low aspect ratio (Fig. 1), the crack will grow quickly to break the inner surface
while keeping the growth at the outer surface much slower. During this process, the aspect ratios will be getting higher
and higher and finally, a fully-through-wall crack is formed.
Crack depth ratio a/T As shown in Fig. 5, it is clear that the NSIF at both surface point and the reference point increase
with increase of crack depth ratio. However, this increase is not unlimited. It is believed that a maximum value of NSIF
will be achieved at a certain crack depth ratio. Having passed this particular crack depth ratio, the NSIF starts
decreasing and approaches the values for the case of a fully-through-wall crack.
The NISF at the outer surface point and reference points for partly-through-wall crack has been extracted from the
results of finite element analyses. For the convenience of making use the obtained result, these NSIF were fitted into
Eq. (5) which is a function of crack depth ratio and aspect ratio. The coefficients di j were generated by curve fitting for
each set of NSIF and are shown in Table 1.
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(a) surface point
(b) reference point at 0.99T
(c) reference point at 0.95T
(d) reference point at 0.90T
Figure 5: Variation of NSIF at surface point and at the three reference points
Table 1 dij for Eq. (5)
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CONCLUSION
Normalised SIFs for partly-through-wall cracks have been investigated in this analysis. The range of aspect ratio and
crack depth ratio was 0.01 < a/(2c) < 0.2 and 1.01 < a/T < 1.30. For a partly-through-wall crack, the SIFs in the region
near the inner surface is much higher than that at the outer surface point. This implies that, once a surface crack has
grown through the plate thickness, a partly-through-wall crack forms and will continue this growth with a high rate in
the direction of becoming of a through thickness crack. Based on the SIF obtained in this work, fatigue analysis of a
partly-through-wall crack is possible to carry.
REFERENCES
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Centre, Hampton, Virgina, USA, 1973.
2. Raju IS, Newman JC Jr. Stress-intensity factor for a wide range of semi-elliptical surface cracks in finitethickness plates. Engineering Fracture Mechanics, 1979; 11: 817-829.
3. Kuok KM, Kou KP. The SIf for deep semi-elliptical surface crack in finite thickness plates determined by the
nodal displacement method. Proceedings of 15th ASCE Engineering Mechanics Division Conference, 2002.
4. Kou KP. The SIF for long-deep semi-elliptical surface crack in finite thickness plates. Proceedings of
ABAQUS’s Users Conference, 2005.
5. Banks-Sills L. Application of the finite element method to linear elastic fracture mechanics. Applied Mechanics
Reviews, 1991; 44: 447-461.
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