Determination of stress intensity factor for a crack emanating from a hole in

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME
373
DETERMINATION OF STRESS INTENSITY FACTOR FOR A CRACK
EMANATING FROM A HOLE IN A PRESSURIZED CYLINDER USING
DISPLACEMENT EXTRAPOLATION METHOD
AKASH.D.A (1)
, Anand.A (2)
, G.V.GNANENDRA REDDY(3)
, SUDEV.L.J(4)
(1)
Department of Mechanical Engineering, SJCIT Chickaballapur, 562101
(2)
Department of Mechanical Engineering, Vidyavardhaka College of engineering, Mysore, 570002
(3)
Department of Mechanical Engineering, SJCIT Chickaballapur, 562101
(4)
Department of Mechanical Engineering, Vidyavardhaka College of engineering, Mysore, 570002
ABSTRACT
The machine elements with cylindrical profile such as pressure vessel, cylindrical
shells, which have been used extensively as the structural configuration in aerospace and
shipping industries needs to be leak proof. But however it’s not possible to fabricate 100%
leak proof pressure vessel / cylindrical shell as the industrial materials do not have uniform
composition. Thus defects or cracks are inevitable in their substructure, also during their
service life a crack may initiate on an internal/external boundary of circular cylinder which
has important influence on stress distribution in the structure. Hence the structural
assessments of hallow cylinders ranging from thick walled pressure vessel to thin walled
pipes has to be carried out, that in-turn relay’s on availability of Stress Intensity Factor (
S.I.F) for fracture analysis. The magnitude of the S.I.F determines the propagation of crack.
In this paper, Considerable effort has been devoted for computation of the S.I.F of crack
emanating from a hole in pressurized cylinder. The objective of this work is to determine SIF
(Plane Strain) for a crack emanating from a hole in a Pressurised cylinder using Finite
Element Method (FEM). From this study it was observed that the value of SIF rises suddenly
when the crack tip is near to the hole and it stabilises as the crack tip move far from the hole.
The SIF values evaluated for different crack length using FEM is normalised with the
analytical values obtained from theoretical equation with respect to (a/D) ratio which
provides important information for subsequent studies such as the crack growth rate
determination and prediction of residual strength with plane strain and plane stress
conditions.
INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING
AND TECHNOLOGY (IJMET)
ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online)
Volume 4, Issue 2, March - April (2013), pp. 373-382
© IAEME: www.iaeme.com/ijmet.asp
Journal Impact Factor (2013): 5.7731 (Calculated by GISI)
www.jifactor.com
IJMET
© I A E M E

374
Keywords: Fracture Mechanics, S.I.F and crack emanating from Hole
1. INTRODUCTION
Pressure vessel and cylindrical shells have been used extensively as the structural
configuration in aerospace and shipping industries needs to be leak proof. However new
advancement in computers has made Finite Element Analysis (FEA) a practical tool in the
study of pressure vessels [1], especially in determining stresses in local areas such as
penetrations and service holes. The most likely places for crack initiating and development of
cracks are the service holes. Due to the high stress concentration in this area cracks may grow in
time, leading to a loss of strength and the reduction of the lifetime of the product as shown in
Figure1. If the structure is concerned with different loading, the crack behaviour must be assessed
in order to avoid catastrophic failures. For this, the knowledge of the crack size, service stress,
material properties and Stress Intensity Factor (SIF) is required [2]. Hence the structural
assessments of hallow cylinders /shell ranging from thick walled pressure vessel to thin
walled pipes has to be done, that in-turn relay’s on availability of stress intensity factor for
fracture and fatigue analysis. Thus it has been recognized that the stress intensity factor is an
important parameter to determine the safety of a cracked component, but the basic practical
problem a designer faces, is to make a decision to opt the method for determining stress
intensity factors. It is not easy to strike a balance between the accuracy of the method, time
required to get a solution, and cost. Numerous equations for stress intensity factors are
available in the literature [1 – 6]. These factors represent various geometries and loading
conditions of fundamental importance in the prediction of structural failure of cracked
cylindrical bodies. In all there are probably more than 600 formulas for calculating K values
for different crack configurations, body geometries, and loading situations.
Here the objective of the work is to determine S.I.F for a crack emanating from a hole in
a pressurized cylinder.
Fig.1: Larger crack formed by the link-up of fatigue cracks at adjacent rivets.
2. FRACTURE MECHANICS
Fracture mechanics involves a study of the presence of the cracks on overall properties
and behaviour of the engineering component. The process of fracture may be initiated at defect
locations like micro-cracks, voids, and the cavities at the grain boundaries. These defects can
lead to the formation of a crack due to the rupture and disentanglement of molecules, rupture of
atomic bonds or dislocation slip [3].
Cracked body can be subjected to three modes of loads as shown in Figure 2. In some
cases, body may experience combination of the three modes:

375
1. Opening mode: The principal load is applied normal to the crack surfaces, which tends
to open the crack. This is also referred as Mode I loading (Figure 2a).
2. In-plane shear mode: This mode corresponds to in-plane shear loading which tends to
slide
One crack surface with respect to the other. This is also referred as Mode II loading
(Figure2b).
3. Out-of-plane shear mode: This is the tearing and anti-plane shear mode where the crack
surfaces move relative to one another and parallel to the leading edge of the crack (Figure 2c).
(a) (b) (c)
Fig. 2: Three modes of loading that can be applied to a crack
The Stress Intensity Factor (SIF) is one the most important parameters in fracture
mechanics analysis. It defines the stress field close to the crack tip and provides fundamental
information of how the crack is going to propagate. In this study, A typical and practical point
matching technique, called Displacement Extrapolation Method (DEM) is chosen for the
numerical analysis method. Plane strain assumption is valid for very thin-walled structures; the
evaluation of S.I.F (KI) by Displacement Extrapolation Method (DEM) is as discussed bellow for
plane strain condition.
The stress intensity factors at a crack for a linear elastic fracture mechanics analysis may
be computed using the KCALC command. The analysis uses a fit of the nodal displacements in
the vicinity of the crack. The actual displacements at and near a crack for linear elastic materials
are
)1(
22
11
k
r
G
K
u ++=
π
(1)
)k1(
2
r
G2
K
v 1
+
π
+= (2)
π
+=
2
r
G
K2
w 111
(3)

376
Where:
u, v, w = displacements in a local Cartesian coordinate system as shown in figure 3
r, θ = coordinates in a local cylindrical coordinate system as shown in figure 3.
G = shear modulus
v
K
+
=
1
ν
In Plane Stress (4)
K= 3 - 4 In Plane strain…………………………..(5)
‫ݒ‬ = Poisson's ratio
For Mode-1, SIF at crack tip is expressed as
r
V
k1
G
2K1
∆
+
π= (6)
Where ∆v, are the motions of one crack face with respect to the other.
Then A and B are determined so that
BrA
r
V
+= (7)
At points J and K.
Next, let r approach 0
A
r
V
lim 0r =→ (8)
Fig. 3: Nodes Used for the Approximate Crack-Tip Displacements for Full crack Model
Thus, Equation 5 becomes:
mm
mm
N
k
GA
K 21
1
2
2
+
= π (9)
3. OBJECTIVE OF WORK AND METHODOLOGY
The objective of this work is to determine S.I.F for a longitudinal crack emanating from a
hole in a pressurized cylinder as shown in figure 4. The objective is achieved by developing a
model of a cylinder with hole and a through crack using CATIA V5 software .The CATIA model
is imported to ANSYS.The FE model is meshed using 8-node quadrilateral doubly curved
SHELL 93 elements in the pre-processor of the ANSYS software . Further as a part of the finite
element work, a mesh sensitivity Study was conducted.

377
A shell with a longitudinal crack was meshed using three different mesh densities.
Mainly, the area around the imperfection was modelled with a finer mesh. Further the crack
tip singular elements were created using KSCON command. For this model there are 36
singular elements around the crack tip and the radius of the first row elements is ∆a (Where ∆a =
a/100).The model is then solved (Static Analysis) by subjecting it to an internal pressure of 1MPa
load with appropriate boundary conditions. Then the S.I.F is evaluated in general postprocessor
by using KCALC command.
The geometry of the meshed test model with crack tip singular elements in ANSYS 12 is
as shown in the Figure 5.The material considered is 304 steel (ASME). The material is assumed
to be linear elastic with young’s modulus of 2.5GPa and poisons ratio 0.3
Where,
D= Diameter of the hole (20mm),
a= Half Crack length
σ=applied hoop stress (Pr/t)
P= Internal pressure 1MPa
t=Thickness of the cylindrical shell
10mm
Fig.4: Geometry of model
(a) (b) (c)
Fig. 5: (a) Finite Element Model meshed with Boundary Condition (b) Zoomed View of
elements near crack tip (c) Zoomed View of Crack Tip Singular Elements

378
4. VALIDATION OF PROPOSED METHODOLOGY
The methodology proposed to determine Mode-1 S.I.F in previous section is validated
using the practical problem “Determination of S.I.F of longitudinal cracks in a pressurized
cylindrical shell” from reference 3.A cylindircal shell with varying longitudinal crack length given
in reference 3 is as shown in Figure 6.
Fig.6: Longitudinal crack in internally pressurized cylinder
Mode I S.I.F (KI) is given by
KI(Theo)= )(f. 1 απσ a ………………………………………………(10)
Where
)(
)07.029.152.01()(f 32
1
Rt
a
x
xxx
=
−++=α
The half crack length was varied from 20mm to maximum half crack length of 439.53mm .The
maximum crack length in a given dimensions of cylindrical shell was determined using curvature
parameter β [9]
4 2
)1(12
Rt
a
ν−=β ……………………….(11)
if β =8 for longitudinal cracks , thickness of the cylindrical shell is 10mm and radius of the
cylindrical shell is 1000mm then the maximum crack length for given set of cylindrical shell
dimension is 439.53mm.
The values of S.I.F obtained by the theoretical and FEA is given in table 1.

379
Table 1: Mode-1 S.I.F (KI) Using FEA and theoretical KI(Theo) for different Half Crack
lengths (a)
Fig.7: variation of theoretical and FEA values of S.I.F Vs crack length for longitudinal
crack in Pressurized cylinder
From the Fig 7 it is indicated that the results which were obtained by using the finite
element method are in good agreement with theoretical equation for a longitudinal through crack
emanating in internally pressurized cylindrical shell with an average percentage of error 1.53%
which is negligible. Thus the proposed methodology to determine the Mode-1 S.I.F for
longitudinal cracks in pressurised cylindrical shell is validated against a standard Procedure.
Half Crack
Length (a)
mm
Mode -I SIF by
FEA
mmMPa
Mode -I SIF
by Analytical
mmMPa
% error
20 821.49 851.322 3.50
40 1282.6 1330.62 3.61
60 1749.6 1828.751 4.33
80 2265.4 2352.63 3.71
100 2831.4 2931.63 3.42
120 3448.9 3555.117 2.99
140 4114.8 4221.655 3.42
180 5570.8 5669.381 1.74
220 7168.4 7247.292 1.09
260 8886.5 8933.523 0.53
300 10711 10710.546 0.00
340 12634 12561.166 -0.58
380 14608 14471.544 -0.94
400 15685 15446.58 -1.54
439.532 17798 17398.911 -2.29

380
5. RESULTS AND DISCUSSIONS
For the problem “Determination of SIF for a Crack Emanating from a Hole in a
Pressurized cylinder” the test models containing a through crack emanating from a hole are
meshed and with plane strain condition it was internally pressurized and respective SIF’s are
calculated.Theoritical Mode-1 is calculated using the relation
mm
mm
N
a)Theo(K 2eff1 πσ= …………………… (12)
The variation of normalized Stress Intensity Factor (KI/KO) (By Plane Strain Method)
with respect to a/D ratio [actual crack length (a) to the Diameter of the hole (D)] is as shown in
Figure 8. The normalized SIF (KI/KO) is used to obtain the characteristic curve of SIF which
depends only on the geometrical factor and its variation within the given domain (a/D).
It’s observed that as the crack is near to the hole the stress concentration around holes has
a strong influence on the SIF value. For a/D ratio 0.5 there is a steep rise in SIF KI, this is due to
crack is small and the crack tip is near to stress concentration at the hole from which crack in
emanating. As the crack grows further (for a/D ranging from 0.1 to 25) the crack tip moves far
from the stressed areas hence the value of SIF increases the system will fail with increase in crack
length.
Table 2: The normalized Stress Intensity Factor (KI/KO) with respect to a/D ratio
Half
Crack
Length (a)
mm
Hole
dia(D)
mm
a/D Mode -I SIF by
FEA(KI)
mmMPa
aeff
mm
Mode -I SIF by
Theoritical(Ko)
mmMPa
KI / Ko
10 20 0.50 743.96 15 686.51 1.08368
20 20 1.00 853.01 20 792.72 1.076059
40 20 2.00 1085.8 30 970.88 1.118372
60 20 3.00 1310.4 40 1121.07 1.168882
80 20 4.00 1540.1 50 1253.40 1.228742
140 20 7.00 2302.5 80 1585.43 1.452284
180 20 9.00 2881.6 100 1772.57 1.625663
200 20 10.00 3191.3 110 1859.09 1.716596
220 20 11.00 3514.8 120 1941.75 1.810118
260 20 13.00 4200.3 140 2097.33 2.002688
300 20 15.00 4932.1 160 2242.14 2.199727
320 20 16.00 5313.5 170 2311.15 2.299075
380 20 19.00 6519.8 200 2506.79 2.600855
400 20 20.00 6938.3 210 2568.70 2.701098

381
Fig.8: Variation of normalized S.I.F for a crack emanating from circular hole in
pressurized cylinder Vs a/D
The Deformed Geometries for crack length (a) of 20 mm is as shown in Figures 9, The
maximum Von-Misses stress is found to be at crack tip.
Fig 9: (a) VonMises Stress Distribution for Pressurised cylinder containing hole dia
D=20mm and crack length a=20mm
6. CONCLUSION
The problem of determining stress intensity factors for a crack emanating from a hole in a
pressurized cylinder is of prime importance in damage tolerance analysis. In the present study
ANSYS12, unified FEA software is chosen. It has the required pre-processing capabilities
for finite element modeling and analysis of cracked shell structures as demonstrated in this
paper.

382
The variation of normalized S.I.F (Ki/Ko) with respect to a/D ratio is used to obtain the
characteristic curve of SIF which depends only on the geometrical factor and its variation within
the given domain (a/D).
The fracture mechanics analysis on “effects of pressure bulging near the crack on the
stress intensity factor” is described in this paper that provides an explanation for the lower
strength of cracked cylinders.
The presented stress intensity factors in this paper are essential to predict
(1) Mixed mode fracture under static, dynamic and sustained loads
(2) Residual strength
(3) Crack growth life under cyclic loading conditions.
However there is a clear need to verify the predictions using experimental
investigations, but the method used in this paper can be utilized for calculating the stress
intensity factor for many other loading cases and many values of the crack length. This provides
important information for subsequent studies, especially for fatigue loads, where stress intensity
factor is necessary for the crack growth rate determination.
REFERENCES
[1] Heckman david: Finite element analysis of pressure vessels, MBARI 1998
[2] Chuin-Shan Chen, Paul A. Wawrzynek, and Anthony R. Ingraffea, 1999, “Crack Growth
Simulation and Residual Strength Prediction in Airplane Fuselages”, NASA/CR-1999-209115.
[3] “Fracture mechanics-Fundamentals and Applications” by T.L.Anderson published by Taylor
and Francis group 2005
[4] Gustavo V. Guinea, Jaime Planas and Manuel Elices, 2000, “KI Evaluation by the
Displacement Extrapolation Technique”, Engineering Fracture Mechanics 66 (2000), pp. 243-
255.
[5] Miloud Souiyah, A. Muchtar, Abdulnaser Alshoaibi and A.K. Ariffin, 2009, “Finite Element
Analysis of the Crack Propagation for Solid Materials” American Journal of Applied Sciences 6
(7), pp. 1396-1402.
[6] Craig A. Barwell, Lorenz.,A Study of Failure in Small Pressurized Cylindrical Shells
Containing a Crack., NASA/CR-1998-208454
[7] Yao-Chen Li, 1984, “The Finite Element Method By Employing The Singular Element With
Concordant Displacement At The Crack Tip” Engineering Fracture Mechanics, Vol. 19, No. 5,
pp. 959-972.
[8] Agne Karlsson and Jan Backlund, 1978, “Summary of SIF Design Graphs for Cracks
Emanating From Circular Holes” International Journal of Fracture, Vol. 14, No. 6
[9] Richard D. Young“Non-linear local bending response and bulging factors for longitudinal
and circumferential cracks in pressurised cylindrical shells”.
[10] Kannan.P, K.Balamurugan and K. Thirunavukkarasu, “Experimental Investigation on the
Influence of Silver Interlayer in Particle Fracture of Dissimilar Friction Welds”, International
Journal of Mechanical Engineering & Technology (IJMET), Volume 3, Issue 2, 2012,
pp. 32 - 37, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359.
[11] I.M.Jamadar, S.M.Patil, S.S.Chavan, G.B.Pawar and G.N.Rakate, “Thickness Optimization of
Inclined Pressure Vessel using Non Linear Finite Element Analysis using Design by Analysis
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