static and dynamic analysis of center cracked finite plate subjected to uniform tensile stress using...

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),

ISSN 0976 – 6359(Online), Volume 6, Issue 1, January (2015), pp. 56-70 © IAEME

56

STATIC AND DYNAMIC ANALYSIS OF CENTER

CRACKED FINITE PLATE SUBJECTED TO UNIFORM

TENSILE STRESS USING FINITE ELEMENT METHOD

Najah R.Mohsin

Southern Technical University, Technical Institute-Nasiriya,

Mechanical Technics Department

ABSTRACT

The study of crack behavior in a plate is a considerable importance in the design to avoid the

failure. This paper deals with investigation of stress intensity factor, Von-Misesstress (ϬVon-mises),

natural frequency, mode shape and the effect of excitation frequency on the finite center cracked

plate subjected to uniform tensile loading depends on the assumptions of Linear Elastic Fracture

Mechanics (LEFM) and plane strain problem. The stress intensity factors mode I (KI) are

numerically calculated by finite element solution using ANSYS (ver. 15) software and theoretically

using standard equations for different crack lengths and plate dimensions. Generally, the results

show that there are no major differences between the two methods. However, the difference between

the two methods occur if we take the plate length parameter in considerate. Furthermore, ϬVon-mises at

crack tip region, 10th natural frequencies and the effect of excitation frequency on the crack tip

stresses are studied for three different materials.

Keywords: Crack Tip, Stress Intensity Factor, Natural Frequency, Finite Element Method (FEM),

Harmonic Analysis, Linear Elastic Fracture Mechanics (LEFM).

1- INTRODUCTION

In general, a fracture is defined as the local separation of an object or material into two or

more pieces under the action of stress. Usually, the fracture of a plate occurs due to the development

of certain displacement discontinuity surfaces within the plate.

Recent development in engineering structures shows that fracture can be caused by small

cracks in the body of structures despite the authenticity of elasticity theory and strength of materials.

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As a result, fracture mechanics filed which is concerned with the propagation of cracks in materials

has developed to study more about this subject, Ali and et al. [1].

According to the types of load, there are three linearly independent cracking modes are used

in fracture mechanics (Fig.1) as follow:

Fig.(1): Mode I, Mode II

• Mode I : a tensile stress normal to the plane of the crack (tensile mode),

• Mode II: a shear stress acting parallel to the plane of the crack and perpendicular to the crack

front (in-plane shear mode).

• Mode III : a shear stress acting parallel to the plane of the crack and parallel to the crack front

(out- of- plane shear mode)

Where Mode I is the most common load type encountered in engineering design.

The problem of determining the stress intensity factors of cracks in a plate is of considerable

importance in the design of safe structures because of stress intensity factors the main key value

defining the stresses around the crack tip arising from that crack.

An approach based on the continuous dislocation technique was formulated by Huang and

Kardomateas[2]to obtain the Mode I and II stress intensity factors KI and KII in a fully anisotropic

infinite strip with a central crack. The elastic solution was applied to calculate KI for a center crack

in an anisotropic strip with the effects of crack length and material anisotropy. The problem of a

crack in a general anisotropic material under conditions of LEFM was examined by Banks-Sills and

et at.[3]. General material anisotropy was considered in which the material and crack coordinates at

arbitrary angles. A three-dimensional treatment was required for this situation in which there may be

two or three modes present. Azevedo[4] was study the stress intensity factors KI and KII for an

inclined central crack on a plate subjected to uniform tensile loading were calculated for different

crack orientations (angles) using FEM analysis, which was carried out in ABAQUS software. The

stress intensity factors were obtained using the J integral method and the modified Virtual Crack

Closure Technique (VCCT). Both methods produced results for KI and KII which were close to the

analytical solution. The effects of the boundary conditions were discussed. Ergun and et at.[5]were

used the FEM to analyze the behavior of repaired cracks in 2024-T3 aluminum with bonded patches

made of unidirectional composite plates. The KI was calculated by FE Musing displacement

correlation technique. Jweeg and et at.[6] were studied natural frequencies of composite plates with



58

the effect of crack orientation, crack position, crack size and based on the shape of the fibers . They

made a comparison between the analytical results and the results get by finite element solution using

ANSYS (ver.14) software. Bhagat and et at.[7] were studied a finite rectangular plate of unit

thickness with two inclined cracks (parallel and non-parallel) under biaxial mixed mode condition

were modeled using FEM. The FEM was used for determination of stress intensity factor by ANSYS

software. Effects of crack inclination angle on stress intensity factor for two parallel and non-parallel

cracks are investigated. The significant effects of different crack inclination parameters on stress

intensity factor were seen for lower and upper crack in two inclined crack. Al-Ansari [8] was a

comparison between six models, calculating KI for central cracked plate with uniform tensile stress,

was made in order to select the suitable model. These models were three theoretical models and three

numerical models. The three numerical models are half ANSYS model, quarter ANSYS model and

weight function model. Crack geometry, crack length, plate length and the applied stresses are the

parameters that used to compare between the models. He concludes that the three theoretical models

can recognize the effect of the width of the plate, the crack length and applied stress but they failed

to recognize the effect of the length of the plate. Ali and et at.[1] were attempts to analyze the stress

intensity factor in various edge cracks along the length of a finite plate which was under a uniform

tension. FEM was utilized for the analysis. In addition, Neural Network Method (NNM) was used to

predict the correlation of stress intensity factor and the position of edge crack along the length of a

finite plate.

For certain cracked configurations subjected to external forces, it is possible to derive closed-

form expressions for the stresses in the body, assuming isotropic linear elastic material behavior. If

we define a polar coordinate axis with the origin at the crack tip Fig.(2), it can be shown that the

stress field in any linear elastic cracked body is given by Anderson[9]

σ�� = � �√� f��θ� + ∑ A�.∞�� r�� . g��θ�, …………….(1)

where Ϭij stress tensor, r and θ are as defined in Fig.(2) , kconstant, fij dimensionless function

of θ in the leading term, Am is the amplitude for the higher-order terms and gij(m)

is a dimensionless

function of θ for the mth

term.

Fig.(2): Definition of the coordinate Mode III crack loading axis ahead of a crack tip



59

The higher-order terms depend on geometry, but the solution for any given configuration

contains a leading term that is proportional to 1/√r.As r → 0, the leading term approaches infinity,

but the other terms remain finite or approach zero. Thus, stress near the crack tip varies with 1/√r,

regardless of the configuration of the cracked body. It can also be shown that displacement near the

crack tip varies with √r. Eq.(1) describes a stress singularity, since stress is asymptotic to r =0.

Most cracks are long and sharp tips. These can be of atomic dimensions in brittle materials.

In 1938, Westergaard solved the stress field for an infinitely sharp crack in an infinite plate. The

elastic stresses were given by the equations, Rae [10].

σ�� = ��√�π� cos �θ� �1 − sin �θ� sin �#θ� $ ……………..(2)

σ%% = ��√�π� cos �θ� �1 + sin �θ� sin �#θ� $ ……………..(3)

σ�% = ��√�π� cos �θ� sin �θ� cos �#θ� ………………..…..(4)

Similar expressions for displacements u

u� = ��µ' �

�π cos �θ� �k − 1 + 2sin� �θ�$ …….……..(5)

u% = ��µ' �

�π sin �θ� �k + 1 − 2cos� �θ�$, …………..(6)

Where µ denotes the shear modulus, k the small difference in formulas for plane stress and plane

strain which is equal to

k = * #+,-., …………….�PlaneStress�3 − 4v …………….�PlaneStrain�9….… (7)

Based on the above equations, we show that, the KI defines the amplitude of the crack-tip

singularity. That is, stresses near the crack tip increase in proportion to KI. Moreover, the stress

intensity factor completely defines the crack tip conditions, if KIis known, it is possible to solve for

all components of stress, strain, and displacement as a function of r and θ. This single-parameter

description of crack tip conditions turns out to be one of the most important concepts in fracture

mechanics, Anderson [9].

Table (1): Mechanical properties used for selected material, Kulkani [12]

Material type Modulus of elasticity

(Mpa) Poisons ratio

density

(Kg/m3

)

Aluminum(alloy) 0.71e5 0.334 2730

Carbon Steel 2.02 e5 0.292 7820

Nickel Silver 1.275 e5 0.322 8690



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2- MATERIALS AND METHODS

2.1- Based on the assumptions of LEFM and plane strain problem, KI to a center cracked plate

under static load were numerically calculated using FEM for three materials Carbon Steel ,

Aluminum(alloy) and Nickel Silver materials are shown in Table (1)and theoretically using

the two standard equations as follows

a) From Ergun and et at.[5] and Rae[10]

K; = σ√π ∗ a =Sec �π∗>�?- �@ A B1 − 0.025 �>?

� + 0.06 �>?FG…..………. (8)

b) From Al-Ansari[8]

K; = σ√π ∗ a B1 + 0.128 �>? − 0.288 �>?� + 1.523 �>?

FG ………….. (9)

2.2- For practical considerations, any kinds of stress, strains and deformations in single parts or

assemblies can be better approximated using FEM. FEM is a numerical technique in which

the governing equations are represented in matrix form, which is to be solved by computer

software. The solution region is represented as an assemblage of small sub-regions called

finite element. The element is the basic building unit with a predetermined number of degrees

of freedom (d.o.f) and can take various forms, e.g. beam, plate, shell or solid elements. The

selection of the best element depends on the type of problem, geometry of boundaries,

boundary conditions, accuracy required, size of the available computer and the maximum

allowable computing cost.

The main purpose of this paper is to observe the behave or of finite plate with a central crack

under the effect of some parameters such as a ratio of crack length to width plate, ratio of width to

length plate, applied stress for different materials by calculating the stress intensity factors KI and

ϬVon-mises to the crack tip. Natural frequencies, mode shapes and also the effect of excitation

frequencies are studied in this paper. Quarter model is selected to represent the finite plate with

center crack in the FEM software (ANSYS ver.15) because the geometry and load applied for

specimen is symmetry (Fig.3).

To compute the required results in a faster and accuracy way, programs are written with

APDL (Ansys Parameter Design Language).The first step of these softwares is to discretize the

structure into finite elements connected at nodes. It is necessary to discretize the plate structure into a

sufficient number of elements in order to obtain a reasonable accuracy, on the other side, the more

elements that are used, the more costly it will be. In this paper, PLANE183 is used as a discretization

element.

International Journal of Mechanical Engineering and Technology (IJMET), I

ISSN 0976 – 6359(Online), Volume 6, Issue 1, January (2015), pp.

Fig. (3): Center cracked plate

Fig. (4): Geometry, node locations and

3- PLANE183 ELEMENT DESCRIPTI

PLANE183 is an isoperimetric

displacement behavior element which is better suited to

be used as a plane element (plane stress, plane strain and general

axisymmetric element. It has two degree of freedom (translation in X and Y directions) at each

Fig.(4)shows the geometry, node

model used in this paper with elements, nodes,

in Fig.(5).

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976

6359(Online), Volume 6, Issue 1, January (2015), pp. 56-70 © IAEME

61

Center cracked plate specimen with dimensions

Geometry, node locations and the co-ordinate system for PLANE183 Element,

ANSYS help [11]

PLANE183 ELEMENT DESCRIPTION

isoperimetric eight or six nodes(I,J,K,L,M,N,O,P)

displacement behavior element which is better suited to modeling irregular meshes .The element may

be used as a plane element (plane stress, plane strain and generalized plane strain) or as an

two degree of freedom (translation in X and Y directions) at each

locations and the co-ordinate system for PLANE183

ements, nodes, boundary conditions and mesh generation

SSN 0976 – 6340(Print),

© IAEME

with dimensions

ordinate system for PLANE183 Element,

(I,J,K,L,M,N,O,P) quadratic or triangle

modeling irregular meshes .The element may

ized plane strain) or as an

two degree of freedom (translation in X and Y directions) at each node.

PLANE183 element. The

boundary conditions and mesh generation are shown



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Fig. (5): Mesh generation for quarter model with 556 elements, 1709 nodes and boundary conditions

4- APDL PROGRAM

In the written APDL program, there are four important processors are used

4.1- Preprocessor (/PREP7):This command contains what you need to use to build a model

such as define element types, real constant ,material properties ,create model geometry

and mesh the object created.

4.2- Solution Processor (/SOLU):This command allows to apply boundary conditions, loads

and create the concentration key point (crack tip) using the(KSCON)command .This

processor contents different analysis such as:

4.2.1- Static analysis (ANTYPE, STATIC): To determine the structure analysis under

static loads.

4.2.2- Mode Analysis (ANTYPE, MODAL): It is used to determine the magnitude of

natural frequencies and mode shapes for the structure.

4.2.3- Harmonic Analysis (ANTYPE, HARMIC): Harmonic response analysis is a

technique used to determine the steady-state response of a linear structure to loads

that vary sinusoidally (harmonically) with time.

4.3- Postprocessor (/Post1): This command used to display the results of stress intensity factor

in lists and display ϬVon-mises in lists, plots or curves.

4.4- The Time-History Postprocessor (/POST26): This command is used to evaluate solution

results at specific points in the model as a function of time, frequency, or some other

change in the analysis parameters that can be related to time. Fig.(6) shows the APDL

flow chart.

Carbon Steel, Aluminum (alloy) and Nickel Silver materials (Table 1) are studied in this paper

to calculate KI, ϬVon-mises with range of a/b at b=0.3 m, range of b/hat b=0.3m, range of b/h at

h=0.3m, range of applied stress, first 10th

natural frequencies, mode shapes and range of excitation

frequency, where a, b and h are defined in Fig.(3).



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Fig.(6): Flow chart of APDL program

Define element type with plane strain option

Great the model area

Great crack point using KSCON command

Discretize lines into suitable division number

Define crack face path

Model analysis

Define material properties

Calculate KI&ϬVon-misesusing

KCALC& PRNSOL command

General Postproc

Select freq. range

and no. of substeps

Select no. of

mode to extract

Define keypoints and lines for model geometry with center crack

Change active CS to

specified CS

Solve

Calculate natural

frequency and

plot mode shape

Apply boundary conditions (displacement & applied stresses)

Time History Variable

Define local crack tip CS

Mesh the model

Harmonic analysis Static analysis

Solution

Solve

General Postproc

Solve

Plot ϬVon-Mises

with the reang of

Exit

Start



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5- RESULTS AND DISCUSSIONS

5.1- Stress Intensity Factors KI

KI are numerically calculated using ANSYS ver. (15) (STATIC Analysis) for three different

materials (Nickel Silver, Aluminum (alloy) and Carbon Steel) and theoretically by Eq.(8) and Eq.(9).

Fig. (7) to Fig.(10) show that the variation of stress intensity factor with different values of

a/b, tensile stresses Ϭt, b/h at specified h=0.3m and b/h at specified b=0.3m. From these figures, it

can be seen that increasing the ratio of a/b and applied stresses leads to increasing the value of KIin a

high level for all the selected values. Fig.(9) shows that small effect for b/h ratio at h=0.3mon the KI

magnitude. From Fig.(10), it is clear that at a specific value b=0.3m in the numerical solution, the KI

increases with ratio of b/h increases but remain unchanged in theoretical solution because the

theoretical equations don’t take the parameter h in a consideration.

In all mentioned figures, it is found that the material type variation is not important to

calculate KI and also we show that KI calculated from theoretical solution either equal or less (with

small ratio) than that of numerical solution except when b/h ratio increasing at specific bas mention

above.

Fig.(7): Variation of stress intensity factor with Fig.(8): Variation of stress intensity factor with

(a/b) ratio numerically for different materials tensile stress numerically for different materials and theoretically using Eq.(8) and Eq.(9) and theoretically using Eq.(8) and Eq.(9)

Fig.(9): Variation of stress intensity factor with (b/h) Fig.(10): Variation of stress intensity factor with

ratio for h=0.3m numerically for different materials (b/h) ratio at b=0.3m numerically for different

and theoretically using Eq.(8) and Eq.(9) materials and theoretically using Eq. (8) and Eq.(9)



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5.2- ϬVon-mises (SEQV) (STATIC Analysis)

Fig.(11) to Fig.(14) show that the variation of ϬVon-mises (in Mpa) at crack tip with different

values of a/b ratio , tensile stress , b/h ratio at specified h=0.3m and b/h ratio at specified b=0.3m.

From these figures, it can be seen that increasing a/b ratio and magnitude of tensile stress leading to

increase ϬVon-mises in a large value and in small value in case of increasing b/h ratio at fixed

b =0.3m. Increasing b/h ratio at fixed h=0.3m from 0.5 to 2.5m with step 0.25m doesn’t change

ϬVon-mises in a considerable value. ϬVon-mises in these figures are numerically approached only due to

difficulty of the theoretical approach. It can be observed that there is a small difference between the

Von-Misses stress when we use a different types of materials. Generally, ϬVon-mises in Carbon Steel is

greater than of Nickel Silver and both of them greater than of Aluminum (alloy) due to the difference

in mechanical properties (modulus of elasticity and poisons ratio).Fig. (15) to Fig. (18) show a plot

results for different states three for quarter model and one for half model.

Fig.(11): Variation of Ϭ Von-mises with (a/b) Fig.(12): Variation of Ϭ Von-mises with

ratio numerically for different materials tensile stress numerically for different materials

Fig.(13): Variation of ϬVon-mises with (b/h) ratio Fig.(14): Variation of ϬVon-mises with (b/h) ratio at h=0.3m numerically for different materials at b=0.3m numerically for different materials



66

Fig.(15): ϬVon-mises distribution for quarter Fig.(16): ϬVon-mises distribution for quarter

model with b = 0.3 m and Ϭt =70 Mpa model with b = 0.15 m and Ϭt =100 Mpa

Fig. (17): ϬVon-mises distribution for quarter Fig.(18): ϬVon-mises distribution for half

model with b = 0.3 m and Ϭt =250 Mpa model with b = 0.3 m and Ϭt =250 Mpa

5.3- Free Vibration Analysis (MODAL Analysis)

This analysis consists of studying the vibration characteristics such as natural frequencies and

mode shapes for the central cracked plate. The first 10th

natural frequencies were numerically

calculated and reported in Table (2) for three different materials. We can see that the magnitudes of

natural frequencies (ω) are different from one material to another because the natural frequencies of

the structure depend on the material stiffness and density.

Furthermore, four mode shapes are shown in Fig.(19) to Fig.(22) to the quarter model to

explain the plate deformation with respect to class of mode shape.



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Table (2): First 10th

natural frequencies for central cracked plate

No. of

natural

frequency

Natural frequency (Hz)

h = 0.3 m

b = 0.3 m

a = 0.09 m

Nickel silver Aluminum(alloy) Carbon steel

1 2.411 3.2052 3.2111

2 3.2539 4.3438 4.2911

3 3.8457 5.1891 4.9466

4 5.27679 7.0115 7.0397

5 6.3736 8.508 8.4098

6 7.4387 9.9238 9.8301

7 8.1193 10.7891 10.83

8 9.1324 12.199 12.059

9 10.196 13.562 13.563

10 10.547 14.025 14.0168

Fig.(19): 4th mode shape for quarter model to Fig.(20): 7th mode shape for quarter model to

Nikel-Silver material with Aluminum (alloy) material

b=h=0.3m and a=0.09 m with b=h=0.3m and a=0.09 m

Fig.(21): 9th mode shape for quarter model to Fig.(22): 10th mode shape for quarter model to

Nikel-Silver material with Carbon Steel material

b=h=0.3m and a=0.09 m with b=h=0.3m and a=0.09 m



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5.4- Harmonic Analysis (HARMIC Analysis)

In this analysis, an external tensile stresses and wide range of excitation frequency was

applied in order to illustrate the behavior of crack tip stresses under deferent values of excitation

frequencies.

Fig.(23) to Fig.(25) illustrate the variation of crack tip ϬVon-mises with wide range of excitation

frequencies 0 to 10Hz to cover more than the first 5th

natural frequencies with the parameters

a=0.09m, h=0.3m,b=0.3m, Ϭt =130Mpa for three different materials Nickel Silver, Aluminum(alloy)

and Carbon Steel. From these figures, it is clear that the magnitude of ϬVon-Mises take a huge value in

crack tip when the excitation frequencies equal to any frequency from plate natural frequencies

(resonance phenomenon) especially at fundamental frequency in Carbon Steel plate (ϬVon-mises =

25850 Mpa) and Aluminum (alloy) plate (ϬVon-mises = 21201 Mpa) and at the second natural

frequency in Nickel Silver plate (ϬVon-mises = 43866 Mpa).

Fig.(23): Variation of ϬVon-mises with excitation frequencies for Carbon steel material

When b = 0.3 m, h = 0.3 m and a = 0.09 m and Ϭt = 130 Mpa

Fig.(24): Variation of ϬVon-mises with excitation frequencies for Aluminum (alloy) material

When b = 0.3 m , h = 0.3 m and a = 0.09 m and Ϭt = 130 Mpa



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Fig.(25): Variation of Ϭ Von-mises with excitation frequencies for Nickle-Silver material

When b = 0.3 m, h = 0.3 m and a = 0.09 m and Ϭt = 130 Mpa

6- CONCLUSIONS

The main conclusions of this work are reported below

6.1- Increasing the crack length and applied stresses lead to increasing the value of KI. In the other

hand, KIvalue change with the change of plate length in the numerical solution but remains

constant in theoretical solution as the theoretical equations don’t take this parameter in a

consideration.

6.2- There is no sensitive effect of the material type on the value of KI.

6.3- The first 10th

natural frequencies for three material types are shown to be different for the

same plate dimensions and boundary conditions because the natural frequency depends on the

stiffness and density of the material.

6.4- ϬVon-mises value takes a huge value at crack tip region when the excitation frequency equals to

any frequency from plate natural frequencies (resonance phenomenon) especially at

fundamental frequency in Carbon Steel and Aluminum (alloy) plate) and at the second natural

frequency in Nickel Silver plate .

7- REFERENCES

[1] Z. Ali, K. Esfahan, S. Meysam, A. Iman, B. Aydin and B. Yashar, “FEM Analysis of

Stress Intensity Factor in Different Edge Crack Positions, and Predicting their Correlation

using Neural Network Method”, Research Journal of Recent Sciences, Vol.3(2), p.p. 69-

73, 2014.

[2] H. Huang and G.A. Kardomateas, “Stress intensity factors for a mixed mode center crack

in an anisotropic strip”, International Journal of Fracture, Vol. 108, p.p. 367–381, 2001.

[3] L. Banks-Sills, P.A. Wawrzynek, B. Carter, A.R. Ingraffea and I. Hershkovitz, “Methods

for calculating stress intensity factors in anisotropic materials: Part II—Arbitrary

geometry”, Engineering Fracture Mechanics, Vol.74, p.p. 1293-1307, 2007.



70

[4] P.C.M. Azevedo, “Stress intensity factors determination for an inclined

central crack on a plate subjected to uniform tensile loading using FE

analysis”, http://paginas.fe.up.pt/~em02115/index_files/r1.pdf, 2008.

[5] E. Ergun, S. Tasgetiren and M. Topcu, “ Stress intensity factor estimation of repaired

aluminum plate with bonded composite patch by combined genetic algorithms and FEM

under temperature effects”, Indian Journal of Engineering & Materials Sciences, Vol. 19,

P.P. 17-23, 2012.

[6] M. J. Jweeg, A, S. Hammood and M. Al-Waily, “Analytical Solution to Oblique Crack

Effect for Difference Composite Material Plates”, ARPN Journal of Science and

Technology, Vol. 2, NO 8, p.p. 697-716, 2012.

[7] R. K. Bhagat, V. K. Singh, P. C. Gope and A.K. Chaudhary, “Evaluation of stress intensity

factor of multiple inclined cracks under biaxial loading “, Fratturaed Integrità Strutturale ,

Vol. 22, p.p. 5-11, 2012.

[8] Dr. L. S. Al-Ansari, “Calculating Stress Intensity Factor (Mode I) for Plate with Central

Crack: Review and Comparison between Several Techniques of Calculations”, Asian

Transactions on Engineering, Vol. 2, p.p. 44-56,2012.

[9] T.L.Anderson, “Fracture Mechanics Fundamentals and Applications”, Third Edition,

Taylor &Francis Group, CRC Press, 2005.

[10] Dr. C.Rae, “ Fracture and Fatigue”, Natural Sciences Tripos Part II, Material Science,

Easter Term 2013-14, Department of Materials Science and Metallurgy, C15, University

of Cambridge, 2014.

[11] ANSYS help.

[12] S.G.Kulkani, “Machine Design”, Sixth reprint, McGraw-Hill companies, 2012.