The Stress Intensity Factor in the Presence of a Hole in the Plate Rouhollah Basirat1

The Stress Intensity Factor in the Presence of a Hole in the Plate
Rouhollah Basirat1, Morteza Ahmadi*1, Mohammad Fatehi Marji2
1Faculty of Engineering, Tarbiat Modares University, Tehran, Iran
2Department of Mining Engineering, Yazd University, Yazd, Iran
e-mail: [email protected]

Abstract: In this research the effect of single hole in a plate, the crack angle, and the plate thickness on the SIFs were investigated. The Stress Intensity Factors (SIF) were calculated using ABAQUS software, J-integral technique. The numerical results were verified using two examples. The results showed that The SIFs are enhanced with increasing of the distant between hole and crack tip till distance from crack tip to hole is less than 2.5 times of hole radius. After that, it is reduced and the distance effect is neutralized at a distance of more than four times the hole radius. The results also indicated that The KI is reduced with increasing the crack angle and KII has a maximum value in ?=45 ? in the pure-tension condition.
Keywords: SIF, Crack, Hole, ABAQUS, Integral J.

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1. Introduction
In the resent years, many studies have been done on the behavior of the cracked object in engineering design; because, mechanical behavior of the body containing crack is different from the body without crack. The main objective of all methods in the mechanics of computational failure is to determine the stress and strain space/contours in the cracked objects. Meanwhile, determining the stress intensity factor (SIF) is one of the important parameters in fracture mechanics.
In linear elastic fracture mechanics (LEFM), SIFs describe the stress field in the vicinity of the crack tip. These factors depend on the fracture geometry, applied stresses, and the initial fracture length. The presence of defects in the form of cracks and holes in plates and samples creates areas of high stress variation. These areas are the most important place for the appearance of different modes of failure in structures including opening/extension mode (Mode I), in-plane shear (Mode II), and out-of-plane shear (Mode III); even if the loading is moderate.
Irwin (1957) introduced the SIF for explaining displacements and stresses in the vicinity of crack tip. After that, several methods including analytical and numerical methods are introduced for calculating SIFs under different condition. The analytical methods have been developed for a variety of common crack configurations; however, these analytical solutions are limited to simple crack geometries and loading situations (Sheibani and Olson., 2013). The different numerical methods such as Finite Element Method (FEM), extend FEM, Displacement Discontinuity Method (DDM), and etc. opened new doors for LEFM problems as well as fracture propagation analysis with emphasize on growing speed of computational calculation.
There are vast research about calculating SIF with different situation using three methods of experimental, analytical, and numerical methods. For instance, the following studies in the field of fracture mechanics:
– the center slant crack problem (Guo et al., 1990; Marji and Dehghani, 2010; Haeri et al., 2013)
– developing several computer codes including FROCK code (Park, 2008) Rock Failure Process Analysis (RFPA2D) code (Tang and Hudson, 2010; Yang, 2011) and 2D Particle Flow Code (PFC2D, 2010) for determining crack propagation path
– the specimens with cracks or holes for brittle solids such as rocks in uniaxial and Brazilian test (Ayatollahi et al., 2008 and 2011, Wang, 2010)
Haeri et al. (2015a,b) investigated the cracks coalescence paths in Brazilian disc and rectangular specimens made from rock-like material containing multiple parallel cracks and multi-holes. They conducted that breaking load of the pre-cracked disc specimens is measured showing that as the number of cracks increases, the final breakage load of the specimen decreases.
Abdollahipour et al. (2015) presented the higher order DDM utilizing special crack tip elements in the solution of LEFM problems. They also studied the crack propagation process in pre-cracked rock like specimens using this method (Abdollahipour et al. 2017). Farahpour et al. (2015) assessed the effect of functionally graded material (FGM) coatings on the fracture behavior of semi-elliptical cracks in cylinders. They calculated the SIF of a longitudinal semi-elliptical crack on the wall of an aluminum cylinder with FGM coating. They observed that there is a particular crack aspect ratio in which the maximum value of SIFs changes from the deepest point to the surface point of the crack. Thomas et al. (2017) provided several contributions toward analyzing and understanding fracture interaction processes through study of the stress intensity factor perturbations by a 3-D finite element-based fracture mechanics model. Akhondzadeh et al. (2017) presented an efficient enrichment strategy for simulating the stress singularities in multimaterial problems and crack-tips terminating at a bi-material interface within the XFEM framework.
In this research the effect of single hole in a plate, the crack angle, and the plate thickness on the SIFs are investigated. The SIFs are calculated using ABAQUS software, J-integral technique.

2. Problem Definition
2.1. Model Geometry
In this research, a hole with different radius and distance from the crack tip is created in the model with ?=45. Then, the effect of changing in the two ratio of r/D and D/W on the SIFs are studied. Fig 1 shows the different parameters and geometry. In this study, this question is answered how much is the distance between the crack tip and the circumference of the hole to have no effect on the SIFs.

Fig 1. The different parameters and geometry in numerical models
2.2. Solve Method
Rice (1968) introduced J-integral approach for two-dimensional (2D) domains containing cracks. Consider a 2-D linear body of linear or nonlinear elastic material free of body forces and subjected to a 2-D deformation field (plane strain, plane stress) so that all stresses ?ij depend only on two Cartesian coordinates (x, y). Suppose the body contains an edge crack as shown in Fig 2.

Fig 2. Crack tip coordinate system and typical line integral contour
In the arbitrary path around a crack tip:

(1)
where W is the strain energy density, Ti are components of the traction vector, ui are the displacement vector components, and ds is an increment of length along the contour (?). The strain energy density and the traction vector components are:

(2)
and

(3)
where ?ij is the stress tensor, ?ij is the strain tensor, and nj are unit vector components normal to ?.
Rice (1968) also showed that the J-integral is equivalent to the energy release rate in a nonlinear elastic material containing a crack:

(4)
where ? is the potential energy and A is the area of the crack. For linear elastic deformation:

(5)
where, for plane strain

(6)
and, for plane stress
E’ = E (7)
In an elastic material, the potential energy is released as the crack grows (Anderson, 1991).
3. Numerical Modeling
3.1. Problem Condition
The J-integral method is applied to perform fracture-mechanics analysis. The singularity elements are used in the crack tips. Ten different contour (?) are applied for calculating the SIFs in the numerical models. Fig 3 presents the model geometry and dimensions. The meshes are fined around the crack for more accurate.

Fig 3. The meshing, geometry and dimensions of models
3.2. Verification
In this section, two examples of single (mode I) and mixed mode (modes I and II) were considered for verification of numerical modelling. Then, the SIFs are calculated in the plate with the presence of a hole.
a) Central horizontal joint under mode I
Fig. 4 presents a center horizontal cracked plate under a tensile uniform load, with length H=200mm, width W =100 mm and the crack length of 2a=24 mm. The far-field tension stress is assumed ?=1 MPa. The elastic modulus (E) and Poison’s ratio (?) were E =72.4×103 MPa and ?=0.3, respectively.

Fig 4. Geometry of the Center-Cracked Tension Specimen
Ten different paths, are considered to calculate the J-integral (Fig. 2). Table 1 illustrates the obtained values for KI, via different paths and various numbers of scattered nodes, as well as the other research results. Aliabadi (1996) has also computed the stress intensity factor for this problem and has proposed the following formula:

(8)
This issue was also solved by Javidrad (2004), using the finite element method together with displacement and stress extrapolation techniques 24. Mirzaei et al (2008) also were solved this problem using element free Galerkin method. They calculated Mode-I SIF using different domain size and various node numbers. The comparison of different methods is illustrated in Table 1.
Table 1. The comparison of different methods in the Center-Cracked Tension Specimen
Researcher Method SIF (MPa?mm)
Aliabadi (1996) Polynomial formula 6.63
Javidrad (2004) Displacement extrapolation method 6.6
Stress extrapolation method 6
Mirzaei et al (2008) Element free Galerkin method 6.73-6.81
In this research J-integral 6.36
b) Inclined joint under mixed mode
Fig. 5 display a central inclined-cracked plate which is subjected to far-field compression stress ?=1MPa in both sides. The plate has length L=200 mm, width W =100 mm and crack length of 2a=30mm. A plain stress condition was assumed with E =50×103 MPa and ?=0.25.

Fig 5. Geometry of the center-inclined-cracked Plate
The mode I and mode II SIFs were calculate according to Eq. (9). In this example (?=45o) stress intensity factors, KI and KII are equal and can be simply determined from Eq. (9). Mirzaei et al (2008) calculated the values of KI and KII for different domains and various numbers of scattered nodes on integral domains. The following equations are used to obtain the theoretical SIFs:

(9)
In the above formulation a=half the flaw length, ?is the flaw inclination angle. The results show in Table 2.
Table 2. The comparison of different methods in the inclined joint under mixed mode
Researcher Method KI (MPa?mm) KII (MPa?mm)
– Theoretical 3.43 3.43
Mirzaei et al (2008) element free Galerkin method 3.54 3.54
In this research J-integral 3.4 3.4
According to the Table 1 and 2, the J-integral method has a meaningful agreement with other techniques.
3.3. Result surveying of numerical models
In this part the effect of crack angle is investigated. The analytical (Eq. 9) and numerical (J-integral) methods are used. The results are shown in Fig 6. Acording to this figure, the below findings are visible:
• the KII=0 and KI has a maximum value in ?=0, because the crack is located in the predencular direction of principle stress (horzontal direction) and it is created the pure mode I condition.
• the KII= KI and KII has a maximum value in ?=45, because there is no stress in the vertical boundaries of model and it is created the mixed mode I and II condition.
• the KII= KI=0 in ?=90, because the crack and stress (vertical direction) is located in the same direction and it’s approved that sress in the same direction with crack direction has no effect on the SIFs. This effect is indicated in Fig 7 based on the state of stress.
• The J-integral technique has a good agreement with analytical method.

Fig 6. The effect of crack angle
4. The effect of single hole on the SIF
Stress intensity solutions are normally expressed in non-dimensional form:

(10)

(11)

(12)
where ? and a are the characteristic stress and crack length, respectively.
Fig. 8 shows the ratio of D/r versus Y1 and Y2. Fig. 8 also shows the state of Mises stress in the around of hole and crack tips. Fig 8 is included three parts:
• D/r ; 2.5: in this part the Y1 and Y2 are increased with increasing of the ratio of D/r. It means that the SIFs are enhanced with increasing of the distant between hole and crack tip till D