ABSTRACT II (KII) for a finite plate

ABSTRACT

This paper
deals with the influenceof crack oblique and its location on the stress
intensity factor mode I (KI) and II (KII) for a finite plate subjected to
uniaxial tension stress. The problem is solved numerically using finite element
software ANSYS R15 and theor.ly using mathematically equations. A good agreement
is observed between the theor. and Num.solutions in all studied cases. We show that increasing the crack oblique ? leads
to decreasing the value of KI and the max.value of KII occurs at
?=45o. Furthermore, KII equal to zero at ? = 0o and 90o
while KI equal to zero at ? = 90o. However, there is no sensitive influenceto the crack location on the stress intensity
factor while there is a considerable influenceof the crack oblique.

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Key Words: Crack, oblique, location, tension,
KI, KII, ANSYS R15.

 

 

 

1.      INTRODUCTION

Fracture can be defined as the process of
fragmentation of a solid into two or more parts under the stresses action. Fracture analysis deals with the
computation of parameters that help to design a structure within the
limits of catastrophic failure. It assumes the presence of a crack in the structure. The study of crack
behavior in a plate is a considerable importance in the design to avoid the
failure the Stress intensity
factor involved in fracture mechanics to describe the elastic stress field
surrounding a crack tip.

Hasebe
and Inohara 1 analyzed the relations between
the stress intensity factors and the oblique of the oblique edge crack for a
semi-infinite plate. Theocaris
and Papadopoulos 2 used the experimental method of reflected
caustics to study the influence of the geometry of an edge-cracked plate on
stress intensity factors KI and KII. Kim and Lee 3
studied KI and
KII for an oblique crack under normal and shear traction and remote
extension loads using ABAQUS software and analytical approach a semi-infinite
plane with an oblique  edge crack and an
internal  crack acted  on by a pair 
of  concentrated  forces 
at  arbitrary  position 
is  studied by Qian  and Hasebe 4. Kimura and Sato 5 calculated
KI and KII of the oblique crack initiated under fretting fatigue conditions. Fett and Rizzi 6 described
the stress intensity factors under various crack surface tractions using an
oblique crack in a semi-infinite body. Choi 7 studied the influenceof crack orientation oblique
for various material and geometric combinations of the coating/substrate system
with the graded interfacial zone. Gokul
et al 8 calculated
the stress intensity factor of multiple straight and oblique cracks in a rivet hole.
Khelil et al 9 evaluated
KI numerically using line strain method and theor.ly. Recentllty, Mohsin 10
and11 studied theor.ly and numerically the stress intensity factors mode I for
center ,single edge and double edge cracked finite plate subjected to tension
stress .

Patr ??ci and Mattheij 12 mentioned that, we can distinguish
several manners in which a force may be applied to the plate which might enable
the crack to propagate. Irwin proposed a classification corresponding to the
three situations represented in Fig. 1. Accordingly, we consider three distinct
modes: mode I, mode II and mode III. In the mode I, or opening mode, the body
is loaded by tensile forces, such that the crack surfaces are pulled apart in
the y direction. The mode II, or sliding mode, the body is loaded by shear
forces parallel to the crack surfaces, which slide over each other in the x
direction. Finally, in the mode III , or tearing mode, the body is loaded by
shear forces parallel to the crack front the crack surfaces, and the crack
surfaces slide over each other in the z direction,

Fig. 1: Three standard loading modes of a crack 12.

The stress fields ahead of a crack tip (Fig. 2) for mode I and mode II
in a linear elastic, isotropic material are as in the follow, Anderson 13

Mode I:

 
……………..(1)

 
……………..(2)

 
………………..…..(3)

Mode II:

 
…….….……..(4)

 
……………….……..(5)

 
……..………..(6)

Fig. 2: Definition
of the coordinate axis ahead of a crack tip 13

In many
situations, a crack is subject to a combination of the three different modes of
loading, I, II and III. A simple example is a crack located at an oblique other
than 90º to a tensile load: the tensile load ?o, is resolved into
two component perpendicular to the crack, mode I, and parallel to the crack, mode
II as shown in Fig. 3. The stress intensity at the tip can then be assessed for
each mode using the appropriate equations, Rae 14.

Fig. 3: Crack
subjected to a combination

of two modes of
loading I and II 14.

                       

Stress intensity solutions are
given in a variety of forms, K can always be related to the through crack
through the appropriate correction factor, Anderson 13

 ,     ……….……….(7)

where ?: characteristic stress, a:
characteristic crack dimension and Y: dimensionless constant that depends on
the geometry and the mode of loading.

We can
generalize the obliqued through-thickness crack of Fig. 4 to any planar crack
oriented 90° ? ? from the applied normal stress. For uniaxial loading, the
stress intensity factors for mode I and mode II are given by

  ……………….(8)

 , …………..(9)

where KI0 is the mode I stress
intensity when ? = 0.

Fig.
4:  Through crack in an infinite plate for the
general case where the

principal stress is not perpendicular
to the crack plane13.

2.     
Materials and Methods

Based
on the assumptions of Linear Elastic Fracture Mechanics LEFM and plane strain
problem, KI and KII to a finite cracked plate for different obliques and
locations under uniaxial tension stresses are studied numerically and theor.ly.

2.1  Specimens Material

The plate specimen material is Steel
(structural) with
modulus of elasticity 2.07E5 Mpa and poison’s ratio 0.29, Young and Budynas 15.
The models of plate specimens with dimensions are shown in Fig. 5. 

Fig. 5: Cracked plate specimens.  

2.2 
 Theor. Solution

Values
of KI and KII are theor.ly calculated based on the following procedure

a)     
Determination
of the KIo (KI when ? = 0) based on (7), where (Tada et al 16 )

…………………….(10)

b)     
 Calculating KI and KII to any planer crack
oriented (?) from the applied normal stress                 using (8) and (9).

2.3  Num.Solution

KI and KII are calculated numerically using
finite element software ANSYS R15 with PLANE183
element as a discretization element. ANSYS models at ?=0o are shown
in Fig. 6 with the mesh, elements and boundary conditions.

 

Fig. 6: ANSYS models with mesh, elements and boundary conditions.

 

 

2.4  PLANE183 Description

PLANE183
is used in this paper as a discretization element with quadrilateral shape,
plane strain behavior and pure displacement formulation. PLANE183 element type
is defined by 8 nodes ( I, J, K, L, M, N, O, P 
) or 6 nodes ( I, J, K, L, M, N) for quadrilateral and trioblique
element, respectively having two degrees of freedom (Ux , Uy) at each node
(translations in the nodal X and Y directions) 17. The geometry, node
locations, and the coordinate system for this element are shown in Fig. 7.

 

Fig. 7: The geometry, node locations, and the coordinate system

for element PLANE183 17.

 

2.5 
The Studied Cases

To explain the influenceof
crack oblique and its location on the KI and KII, many cases (reported in Table
1) are studied theor.ly and numerically.

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 1: The
cases studied with the solution types, models and parameters.

 

 

3.     
Results and Discussions

KI and
KII values are theor.ly calculated by (7 – 10) and numerically using ANSYS R15
with three cases as shown in Table 1.

3.1 Case Study I 

Fig.s
8a, b, c, d, e, f, g, h and i explain the Num.and theor. Differences of KI and
KII with different values of a/b ratio when ? = 0o, 15o,
30o, 40o, 45o, 50o, 60o,
70o and 75o, respectively. From these Fig.s, it is too
easy to see that the KI > KII when ? < 45o while KI < KII when ? > 45o               and
KI ? KII at ? = 45o.

3.2 
Case Study II

A compression between KI and KII values for different crack
locations (models b, e and h) at ?=30o, 45o and 60o
with Differences of a/b ratio are shown in Fig.s 9a, b, c, d, e, f, g, h and i.
From these Fig.s, it is clear that the crack oblique has a considerable influenceon
the KI and KII values but the influenceof crack location is insignificant.

3.3 Case Study
III

Fig.s 10a, b, c and d explain the Differences of KI and KII with
the crack oblique ? = 0o, 15o, 30o, 45o,
60o, 75o and 90o for models b, e and h. From
these Fig.s, we show that the max.KI and KII values appear at ?=0o
and ?=45o, respectively. Furthermore, KII equal to zero at ? = 0o
and ? = 90o. Generally,
the max.values of the normal and shear stresses occur on surfaces where the ?=0o
and ?=45o, respectively.

From all Fig.s, it can be seen that there is no significant difference
between the theor. and Num.solutions.

Furthermore, Fig.s
11 and 12 are graphically illustrated Von._Mises stresses countor plots with
the Difference of location and oblique of the crack, respectively. From these Fig.s,
it is clear that the influenceof crack oblique and the influenceof crack
location are incomparable.

 

Fig. 8: Difference of KI Num., KI Th., KII Num. and KII Th. with the Difference
of a / b and ? for model e .

 

Fig. 9: Difference of KI Num., KI Th., KII Num. and KII Th. with the Difference
of a / b for b, e and h model at ? = 30, 45 and 60.

   

Fig. 10: Difference of KI and KII with the crack oblique: a and b) for
model b, e, h and theor..

                                                                                    
c and d) for model d, e, f and theor..

Fig.
11: Countor plots of Von._Mises stress with the Difference of crack location at
? = 45o.

 

Fig.
12: Countor plots of Von._Mises stress with the Difference of crack oblique at
specific location.

 

 

4.      Conclusions

1)      A good agreement is observed between the theor.
and Num.solutions in all studied cases.

2)     
Increasing the crack oblique ? leads to decrease
the value of KI and the max.value of KII   occurs at ?=45.

3)     
KII vanished at ? = 0o and 90o
while KI vanished at ? = 90o.

4)     
There
is no Clear influence.to the crack location but there is a considerable influenceo f the crack oblique.

 

5.      References

1-     
N. Hasebe and S. Inohara.  Stress Analysis of a Semi-Infinite Plate with
an Oblique Edge Crack.  Ingenieur-Archiv,
Volume 49(1), pp. 51-62, 1980.  

2-     
P. S. Theocaris and G. A. Papadopoulos. The Influence
of Geometry of Edge-Cracked Plates on KI and KII Components of the Stress Intensity
Factor. Journal of Physics D: Applied Physics. Vol. 17(12), pp. 2339-2349, 1984.

3-     
H.K. Kim and S.B. Lee. Stress intensity
factors of an oblique edge crack subjected to normal and shear tractions. Theor.
and Applied Fracture Mechanics, Volume 25(2), pp. 147–154, 1996.

4-     
J. Qian
and N. Hasebe. An  Oblique  Edge  Crack  and 
an  Internal  Crack 
in  a Semi-Infinite Plane  Acted 
on  by  Concentrated 
Force  at  Arbitrary 
Position. Engineering Analysis with Boundary Elements, Vol. 18,   pp. 155-16, 1996.

5-     
T. Kimura and K. Sato.  Simplified Method to Determine Contact Stress Distribution
and Stress Intensity Factors in Fretting Fatigue. International Journal of
Fatigue, Vol. 25, pp.  633–640, 2003.

6-     
T. Fett
and G. Rizzi. Weight Functions for Stress Intensity Factors and T-Stress for
Oblique Cracks in a Half-Space.  International
Journal of Fracture, Vol. 132(1), pp. L9-L16, 2005.

7-     
H. J.
Choi. Stress Intensity Factors for an Oblique Edge Crack in a Coating/Substrate
System with a Graded Interfacial Zone under Antiplane Shear. European Journal
of Mechanics A/Solids. Vol. 26, pp. 337–3, 2007.

8-     
Gokul.R
, Dhayananth.S , Adithya.V, S.Suresh Kumar.  Stress Intensity Factor Determination of
Multiple Straight and Oblique Cracks in Double Cover Butt Riveted Joint. International
Journal of Innovative Research in Science, Engineering and Technology, Vol. 3(3),
2014.

9-     
F. Khelil, M. Belhouari, N. Benseddiq, A.
Talha. A Num.Approach for the Determination of Mode I Stress Intensity Factors
in PMMA Materials. Engineering, Technology and Applied Science Research, Vol. 4(3),
2014.

10-  N. R. Mohsin.  Static and
Dynamic Analysis of Center Cracked Finite Plate Subjected to Uniform Tensile
Stress using Finite Element Method. International Journal of Mechanical
Engineering and Technology (IJMET), Vol. 6, (1), pp. 56-70, 2015.

11- 
N. R.
MOHSIN. Comparison between Theor. and Num.Solutions for Center, Single Edge and
Double Edge Cracked Finite Plate Subjected to Tension Stress. International
Journal of Mechanical and Production Engineering Research and Development
(IJMPERD), Vol. 5(2), pp. 11-20, 2015.

12-  M. Patr ??ci and R. M. M. Mattheij.
Crack Propagation Analysis, http://www.win.tue.nl/analysis/reports/rana07-23.pdf .  

13-  T.L.Anderson. Fracture Mechanics
Fundamentals and Applications. Third Edition, Taylor Group, CRC
Press, 2005.

14- 
C. Rae.
Natural Sciences Tripos Part II- MATERIALS SCIENCE- C15: Fracture and Fatigue.
https://www.msm.cam.ac.uk/teaching/partII/courseC15/C15H.pdf .  

15- 
W.
C. Young and R. G. Budynas. Roark’s Formulas for Stress and Strain. McGraw-Hill
companies, Seventh Edition, 2002.

16- 
 H. Tada, P. C. Paris and G. R. Irwin. The
Stress Analysis of Cracks Handbook. Third edition, ASME presses, 2000.

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