INTRODUCTION
Modern aircraft structures are designed using a damage tolerance philosophy.
This design philosophy envisions sufficient strength and structural integrity
of the aircraft to sustain major damage and to avoid catastrophic failure. However,
structural aging of the aircraft may significantly reduce the strength below
an acceptable level. This raises many important safety issues (Chen
et al., 1999).
The most likely places for crack initiating and development are the rivet holes
due to the high stress concentration in this area.

Fig. 1: 
Larger crack formed by the linkup of fatigue cracks at adjacent
rivets 
Such cracks may grow in time, leading to a loss of strength and the reduction
of the lifetime of the sheet as shown in Fig. 1. If the structure
is concerned with different loading, the crack behavior 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
(Karlsson and Backlund, 1978).
FRACTURE MECHANICS
Fracture mechanics involves a study of the presence of the cracks on overall
properties and behavior of the engineering component. The process of fracture
may be initiated at defect locations like microcracks, 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 (Broek, 2002).
Cracked body can be subjected to one of the three modes of loads as shown in Fig. 2. In some cases, body may experience combination of the three modes:
• 
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 (Fig. 2a) 
• 
Inplane shear mode: This mode corresponds to inplane shear loading
which tends to slide one crack surface with respect to the other. This is
also referred as Mode II loading (Fig. 2b) 
• 
Outofplane shear mode: This is the tearing and antiplane shear
mode where the crack surfaces move relative to one another and parallel
to the leading edge of the crack (Fig. 2c) 

Fig. 2(ac): 
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, the displacement extrapolation method is employed to calculate the SIF.
Various numerical methods have been used to derive SIF such as Finite Difference
Method (FDM), Finite Element Method (FEM) and Boundary Element Method (BEM).
Among them, FEM has been widely employed for the solution of both fracture problems
linear elastic and elastoplastic. A typical and practical point matching technique,
called Displacement Extrapolation Method (DEM) is chosen for the numerical analysis
method (Guinea et al., 2000).
Solving a fracture mechanics problem involves performing a linear elastic or
elasticplastic static analysis and then using specialized postprocessing commands
or macros to calculate desired fracture parameters. The two main aspects of
in this procedure are modeling the crack region and calculating fracture parameters
(Souiyah et al., 2009).
EVALUATING STRESS INTENSITY FACTOR (K_{I})
The stress or strain state is always there in three dimensional analysis. But in most cases, they can be simplified to either plane strain or plane stress by ignoring either the out of plane strain or plane stress. In a thin body generally, the stress through the thickness (σ_{z}) cannot vary appreciably due to the thin section. Because there can be no stresses normal to a free surface, σ_{z} = 0 throughout the section and a biaxial state of stress results. This is termed as plane stress condition.
Plane stress assumption is valid for very thinwalled structures, while plane strain is predominant condition in structures with large thickness.
The evaluation of SIF (K_{I}) by Displacement Extrapolation Method (DEM) is as discussed bellow for plane stress 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:
Where:
u, v, w 
= 
Displacements in a local Cartesian coordinate system as shown
in Fig. 3 
r, θ 
= 
Coordinates in a local cylindrical coordinate system as shown in Fig.
3 
G 
= 
Shear modulus 
In plane stress:
v = Poisson’s ratio
For Mode 1, SIF at crack tip is expressed as:
where, Δv, are the motions of one crack face with respect to the other.
Then A and B are determined so that:
At points J and K.

Fig. 3: 
Nodes used for the approximate cracktip displacements for
full crack model 
Next, let r approach 0:
Thus, Eq. 5 becomes:
FINITE ELEMENT MODEL DEVELOPMENT
“A through crack emanating from holes” is one among the practical problems
in chapter 14 from the text book “elementary engineering fracture mechanics”
by David Broek Published by Kluwer Academic Publishers reprinted in year 2002
(Broek, 2002).
Bowie has presented the K solution for radial through cracks emanating for unloaded open holes. For the case where the crack is not small compared to the hole, one might assume as a first engineering approach that the combination behave as if the hole were part of the crack is as shown in Fig. 4. The effective crack size is then equal to the physical crack plus the diameter of the hole. The stress intensity factor for the asymmetric case with 2aeff = D+a:
Where:
D 
= 
Diameter of the hole (10 mm) 
a 
= 
Crack length (1 to 30 mm) 
σ 
= 
Tensile load (10 MPa) 
The objective of this study is to determine SIF for a crack emanating from a rivet hole in a plate as shown in Fig. 4. The objective is achieved by developing a 2D finite element model of a plate with rivet holes and a through crack subjected to a tensile load. The SIF is calculated at crack tip for various crack length by generating mesh using crack tip elements.

Fig. 4: 
Geometry of model 

Fig. 5(ab): 
(a) Finite element model and boundary condition and (b) zoomed
view of crack tip 
To achieve the required objective, 2D finite element model is developed and boundary conditions are applied in preprocessor of the ANSYS software. To mesh the model with crack, plane 82 with singular elements is used. In ANSYS, KSCON command is used to generate the singular elements around the crack tip. The model is then solved (Static Analysis) in solution menu. Then the SIF is evaluated in general postprocessor by using KCALC command.
The geometry of the test model created in ANSYS is as shown in the Fig.
4. It contains a through crack emanating from a hole. The meshing of the
model is as shown in the Fig. 5a, b. The
element used to mesh the model is 8node plane 82 quadrilateral element. The
symmetry boundary condition is applied at the both sides of the plate to make
it as infinite length. The load is applied to the top edge and the bottom edge
is fixed in all degree of freedom. The material considered is 2024T3 Aluminium
Alloy (ASME). The material is assumed to be linear elastic with young’s
modulus of 73.1 X 103 MPa and Poisson’s ratio 0.33.
ANSYS preprocessor’s (PREP7) KSCON command is used to generate the singular elements around the crack tip. For this model there are 36 singular elements around the crack tip and the radius of the first row elements is Δa.
Where:
RESULTS AND DISCUSSIONS
The geometry was imposed by plane stress condition and edge load (σ) applied under modeI loading condition. The variation of normalized Stress Intensity Factor (K_{I}/K_{O}) (by plane stress method) with respect to a/D ratio [actual crack length (a) to the Diameter of the rivet hole (D)] is as shown in Table 1.
The normalized SIF (K_{I}/K_{O}) 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).
Where:
The variation of normalized SIF (K_{I}/K_{O}) (by plane stress method) with respect to a/D ratio is as shown in Fig. 8. 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 0.1 there is a steep rise in SIF K_{I}, 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 grow further (for a/D ranging from 0.1 to 3) the crack tip move far from the stressed areas hence the value of SIF drops down and become almost stable.
K_{O} is the stress intensity factor for a crack of length 2a in a
large sheet subject to a remote uniform tensile stress perpendicular to crack
direction and is given by
in a large sheet.
A comparison of the results of Bowie equation and experiment for a through
crack emanating from a hole with the finite element analysis result by using
ANSYS software for a given rage of crack length is as shown in the Fig.
68. The present result which was obtained by using the
finite element method are in good agreement with Bowie equation and experiment
for a through crack emanating from a hole. The percentage deviation calculated
for FEA and theoretical results is as show in column % error in the Table
1.
The Deformed geometries for crack length (a) of 1, 8 and 25 mm is as shown
in Fig. 911. The maximum Von Mises is
found to be at crack tip.

Fig. 6: 
Bowie’s analysis as compared to the engineering method 

Fig. 7: 
Stress intensity factors for crack emanating from circular
hole by using Bowie equation 
Table 1: 
Variation of Stress Intensity Factor (K_{I}) using
plane stress method for different crack lengths (a) 


Fig. 8: 
Stress intensity factors for crack emanating from circular
hole using finite element method 

Fig. 9: 
Von mises stress distribution for a = 1 mm 

Fig. 10: 
Von mises stress distribution for a = 8 mm 

Fig. 11: 
Von mises stress distribution for a = 25 mm 
Table 2: 
YDirectional stresses, Von Mises stresses and SIF at crack
tip for a = 1, 8 and 25 mm 

The value of maximum Ydirectional stresses, Von Mises stresses and Stress
Intensity Factor (K_{I}) at crack tip is as shown in Table
2.
CONCLUSION
The problem of determining stress intensity factors for a crack emanating from a rivet hole in a tensile loaded infinite plate is of prime importance in damage tolerance analysis. The method used in this report 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.