Land Sliding Analysis on Red Clay soil using Fracture Criteria

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Master's Degree Thesis ISRN: BTH-AMT-EX--2011/D-15--SE

Supervisor: Sharon Kao-Walter, BTH

Department of Mechanical Engineering Blekinge Institute of Technology

Karlskrona, Sweden 2011

Mohamed Riyazdeen M.G Yisho ji

Jin ji

Land Sliding Analysis on Red Clay

Soil using Fracture Criteria

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Land Sliding Analysis on Red Clay soil using Fracture Criteria

Mohamed Riyazdeen M.G Yisho ji

Jin ji

Department of Mechanical Engineering Blekinge Institute of Technology

Karlskrona, Sweden 2011

Thesis submitted for completion of Master of Science in Mechanical Engineering with emphasis on Structural Mechanics at the Department of Mechanical Engineering, Blekinge Institute of Technology, Karlskrona, Sweden.

Abstract:

In order to assess the safe and functional design of road ways, buildings, bridges, dams, etc. Also to protect from structural damages and geological disasters. The landslide analysis is to conduct based on force resisting method by applying uniform loads on top of the hill to determine slope stability, instability, for with and without crack on clay soil is to be experimented and evaluate the mechanical properties such as elastic modulus, poison ratio, and shear strength. In addition, the numerical model is implemented to the crack initiation model such as tensile, bending and shear to identify the fracture behavior for Mode I and Mode II and determine the criteria of a stress concentration factor [Kt], stress intensity factor [KI], and crack mouth opening displacement values are analyzed theoretically and verify using ABAQUS. Experimental model shows good agreement with the simulation result.

Keywords: Slope stability, Geological disaster, Stress intensity factor, Land slide, Crack initiation, structural damages.

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Acknowledgement

Our deepest gratitude goes to Dr. Sharon Kao-Walter, our supervisor, for her constant encouragement and guidance. She has walked us through all the stages of this thesis. Besides that, we also want to thank to Huang Ying, professor at Kunming University of Science and Technology, China. Prof Huang has guided and supported in Experiment.

Special thanks to Lic.Sc. Leon, Armando for guided in application of ABAQUS FEM method.

Last, our thanks would go to our beloved family for their loving considerations and great confidence in us all through these years.

We also owe our sincere gratitude to our friends and classmates who gave us their time in listening and helped us work out during the difficult course of the thesis.

Karlskrona, November 2011 Mohamed Riyazdeen M.G Yisho ji

Jin ji

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Abbreviation

3PB 3 point bending test

CMOD Crack mouth opening displacement SENT Single edge notched test

SENB Single edge notched bend SF Safety Factor

SIF Stress Intensity Factor

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1. Notations

Crack length

Thickness of the Specimen in mm

[D] Elastic stiffness matrix or stress-strain matrix Young‘s modulus for plane strain and Plane stress

Geometric function G Shear modulus

Area moment of inertia about neutral axis x J J-Integral

Stress Intensity Factor Stress Concentration Factor K (a) Stiffness

L Length of the beam (mm) The moment about neutral axis P Force (N)

S is the span length of beam

Crack mouth opening displacement v Poisson‘s ratio

W Width of the beam (mm)

Perpendicular distance from the neutral axis

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5 Stress component

Total strain vector, Thermal strain vector

Deflection in mm

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2. Introduction

The phenomenon of cracking and damage simulation is a popular research concern. Materials have different damage phenomena. Rock and concrete are quasi-brittle material. The cracking zone created the formation and performance of shear bands of flexible material production, brittle materials for the formation of discrete cracks.

Since 1960s, a series of dams, bridges and road accidents occur frequently, the brittle structures are concerned as the problem of national security, has become a research topic. From the component of a fracture process to see, the cracks of structural damage often begin from the component surface, has been formed during the process of construction. Because of properties that the weak pull-brittle materials and civil engineering components of the large volume, micro-cracks generated in the construction inevitable. The existence of the initial micro-cracks reduces the security of a component. Crack increase over the time and further development of these cracks may eventually cause severe breakages. A lot of breakages that components are due to the internal fracture with various types of cracks, the existence and expansions of these cracks make the structure carrying capacity as weakened, thus affecting the quality and safety of engineering structures. Therefore, studying crack initiation and crack extension, was important to engineering design and construction.

Fig 1.1 the cracking of road [36]

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2.1 Aim and Scope

In order to achieve safe and functional of road ways, buildings bridges, dams, etc. and to protect from geological disaster the land sliding analysis is experimented. Moreover, better understanding of a factor which causes to take place slope instability.

Indoor land sliding model is created to determine the slope stability and instability under 2 conditions for with and without crack initiation should be examined.

Carry out stability analysis and monitor the displacement of clay soil to establish the elastic modulus, poison ratio, and shear strength of the material.

Crack initiation model is created on ABAQUS to determine the fracture criteria like stress intensity factor (SIF) of two failure modes KI and KII, stress concentration factor Kt, and CMOD etc.

Determine the safety factor (SF) for crack and no-crack model under varying load should be observed and examined from the experimental results. Furthermore compare with numerical results.

Research of slope instability is confined to geological and geotechnical aspects of land sliding.

The study developed here can be applied for various geo-engineering projects and Government projects.

2.2 Background Research

Landslide, as shown in figure 2.1, was one of the frequent occurrences of geological disasters in the mountainous area. China is a mountainous country, landslides happen frequently across the country.

Several landslides occur every year in China, especially in the rainy season, landslides reported are encountered with many times.

According to ―Chinese geology environment information network‖

statistical investigation data, [The national geology disaster notifies

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(January-June 2010)] [1]. Record of China's January-June 2010 occurred in a variety of geological disasters: The country from January to June altogether has geological disaster 19553, landslide 14,614, accounting for 74.7% in the total. The country from January to June there were 21 large and from large-scale geological disasters(the disaster died above 30 people or the direct economic loss above 10 million Yuan large-scale geological disasters have 6,there were 15 cases of geological disasters which more than 10 people following 30 persons died or more than 5 million below 10 million Yuan direct economic losses), landslide 12, account for the total 57.1%.Yuan Li [2] (2004) conducted statistical surveys for relatively serious geological disasters, and a wide variety of geological disasters, huge volumes, are seriously harmed by 290 and County (City) region, the results show: It investigates each kind of geological disaster altogether 56,112 and landslides 28,738 51% per cent of total disaster hidden the danger in geological disaster altogether is 47,832 in which landslide 24,898, account for the total 52%.

The number of missing population and the direct economic loss has increased. (Table 1.1)

Fig 2.1 Landslide of slope [35]

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Table 1.1 Comparison with geological disasters in china between 2009 and 2010.

Year Number

occurred

Death or missing

Direct economic loss/ten thousand Yuan

2010 30,670 2,915 6,38,508.5

2009 10,840 486 1,76,548.8

Volume increase or decrease compared

with 2009

+19,830 +2,429 +4,61,959.7

Percentage increase or decrease compared with

2009/%

+182.9 +499.8 +7,261

In the event of various types of landslides the soil slope landslide accounts a large proportion of landslide. In the literature [2] have 23,466 geotechnical landslides to be statistical for all kinds landslides the soil slope landslide have 16,143 accounting for 69% of the total number of landslides. Yunnan is a mountainous province, the mountainous area accounts for 94%, and red clay widely distributed.

Wet and dry season is obviously, every year May to October.

Precipitation is rich, precipitation concentration. The ecological environment is a fragile; landslide in China suffered one of the most affected provinces [3]. Hongbing Zhang [4] (2004) according to the Yunnan Province 1989~2002 year landslide data, the statistics obtain Yunnan Province have 70﹪ Since 1980s while ―west development‖

and ―west to east the electricity delivers‖ in-depth implementation, the Yunnan mountainous area construction development speed speeds up, the scale enlarged, large-scale civil engineering construction projects have been increasing. Such as highway, water Conservancy and hydropower projects, there are a large number of red soil in slope, the

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stability and social security of the whole project, have a significant economic impact, and the initial crack existence in the soil slope, this is disadvantageous to the red soil slope stable, therefore, launching the research to the red soil slope critical characteristic and the destruction mechanism, specially for initial crack effect on stability of the slope, is of great theoretical significance and engineering application.

Fig. 2.2 (a) Comparison of Economic loss

Fig 2.2 (b) Land sliding statistics 0

100000 200000 300000 400000 500000 600000 700000

2009 2010

Ten thousand Yuan

Ecomnomic Loss

0 5000 10000 15000 20000 25000 30000 35000

Death Number occurred

Numbers

Death and Occurance

Land sliding Statistic for 2009-2010

2009 2010

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3. Theory

3.1 Stress-strain

In the linear elasticity theory

(3-1) Shear strains in the software (



_xy、



_yz and



_zx) are engineering strain, which are twice as the tensile strain



, EPTO will be used to express the total strain vector in material nonlinear analysis.

Figure 3.1: Unit stress vector [21] [22]

In the ANSYS, program provides normal stress and normal strain

are positive, compression is negative.

(3-2) In the three dimension

(3-3) The direction of the secant coefficient of thermal expansion

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1

1 / / / 0 0 0

/ 1 / / 0 0 0

/ / 1 / 0 0 0

[ ] 0 0 0 1 /

0 0 0 1 /

x xy x xz x

yx y y yz y

zx z zy z z

xy yz

xz

E E E

D G

G G

 



 

 

  

 

  

  

 

 

(3-4)

Furthermore, is symmetrical matrix.

(3-5) (3-6) (3-7)

(2-2) can be unfolded by (2-3) ～ (2-7)

(3-8)

(3-9)

(3-10)

(3-11)

(3-12)

(3-13) Where

We will get six equations from (3-1), (3-4), (3-3), (3-5) ～ (3-7)

/ (1 2 / )( ) / ( / )( )

/ ( )( )

x x yz z y x x y xy xz yz z y y y

z xz yz xy z z

E h E E T E h E E T

E h T

        

    

       

   

(3-14)

/ (1 2 / )( ) / ( / )( )

/ ( / )( )

y y xz z y y y y xy xz yz z y x x

z yz xz xy y x z z

E h E E T E h E E T

E h E E T

        

    

       

   

(3-15)

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/ (1 2 / )( ) / ( / )( )

/ ( )( )

z z xy y x x z z yz xz xy y x y y

z xz yz xy z z

E h E E T E h E E T

E h T

        

    

       

   

(3-16)

(3-17)

(3-18)

(3-19) Where

(3-20) Assumed that shear modulus is simplified as in the following equation will be use in Isotropic material calculation

(3-21)

3.2 Three Point Bending

Three point bending test is used to measure the force required to bend a beam. It is often used to select the material for optimum load without bending. [24]

3.3 Simply supported beam

In a Simply supported beam one end of the beam is fixed with pin support and another end with roller support. Pin supports prevent from translation at the end of the beam but not the rotation. More over in roller support prevents translation in the vertical direction but not in the horizontal direction. Hence roller support can resist vertical force but not a horizontal direction. [25]

Bending is also known as flexure, beam bending is often analyzed with Euler-Bernoulli beam equation. The classic formulae for determining the bending stress is [24]

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Where: is the bending stress

The moment about neutral axis

The perpendicular distance to the neutral axis The area moment of inertia about neutral axis x

Fig 3.2 Tension & compression [24]

3.4 Shear and Moment Diagram

Fig 3.3 Shear and moment diagram [24].

Moment M = =

Moment of Inertia I = = 91.54 Distance to the neutral axis = 3.25

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3.5 Bending Stress

An alternative way of calculating Bending stress for 3 point bending is [31].

Where

P-load of the specimen in N

S-span length of the specimen in mm W-width of the specimen in mm B-thickness of the specimen in mm

By substituting the values of load and geometry in the above equation, bending stress for 3 point bending is determined

3.6 Crack Initiation

The initial crack occurs may be caused by surfaces scratched, handling or tooling of the material. For ductile materials, crack is stable until the applied stress is increased and extensive plastic deformation, but for soil material it is un-stable and little plastic deformation [21][22].

3.7 Stress Intensity Factor

In fracture mechanics, the stress intensity factor play an important role, it is more accurate to predict the crack stress near to the crack tip caused by remote load [14]. If you know it is possible to identify the components of stress strain displacement and j-integral. The stress intensity depends on geometric configuration such as, crack length, crack location, physical geometry of the specimen, and also with loading conditions. [22] [32]

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The load at which failure occurs is called as fracture strength, this load types are categorized as Mode I, Mode II, and Mode III. The most commonly used engineering design parameters are under Mode I stress intensity factor [34].

3.8 Three Modes of Failure

Fig 3.4Three modes of failure [21]

Stresses and deformation in the material in front of crack depends on how the cracked structure is loaded. A crack may be loaded in 3 different ways:

Mode I: the crack is opened and the crack surfaces are separated from each other.

Mode II: the crack is sheared in the plane of the crack so that the crack surfaces move relative to each other in shear in the x direction.

Mode III: the crack is sheared in the plane of the crack so that the crack surfaces move relative to each other in shear in the z direction [22].

Here we are focusing on Mode I and Mode II failures. In Mode I Mode II loading of the crack, stresses and deformation in the plane of the plate are of main interest. If the plate is very thin the stress component s perpendicular to the plane of the plate to be zero ( ) in this state of stress is called as plane stress. On the other hand if the plate of stress is not very thin stress ( this state is called as plane strain [22].

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3.9 Geometric function

The geometric function is used to determine the stress intensity factor for Mode I. It varies according to the geometric and loading conditions [21].

Fig 3.5 Geometric function verification [21]

SENT

SENB

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Where

–Span length of the beam –Width of the beam –Crack length

P – Load of the specimen in N –Thickness of the beam in mm

3. 10 J –Integral

An alternating way of calculating stress intensity factor is by using J integral, it is the strain energy release rate. Under Mode I loading condition crack propagation in an elastic plastic material absorbed most of the strain energy by a material and it left a plastic wake even when the sample is unloaded. The J- integral value varies according to plane stress and plane strain [21].

Fig 3.6 J integral [21]

3. 11 Crack Mouth Opening Displacement

Crack mouth opening displacement is the length of crack opening at the front of specimen; it is measured by using a clip gauge either attached to integral knife edges machined to the notch or knife edges mounted at the crack mouth [21].

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Fig 3.7 Crack mouth opening displacement for 3 point bending [21]

Fig 3.8 Detailed view of Crack mouth opening displacement [21]

CMOD

Where

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22 Bending stress

Crack length Young‘s modulus

CMOD plane stress CMOD plane strain

By substituting all the values in the equation analytical CMOD is determined and it varies according to plane stress and plane strain.

Using CMOD (crack mouth opening displacement) stress intensity factor and J integral are also determined [21].

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4. Analysis of Fracture Criteria

4.1 Stress concentration

4.1.1 Stress concentration for circular holes

If it is supposed that stress elastic body has a micro pores, then the hole-edge stress will far overweight the centralized stress when the body without holes and also greater than the stress that in a long distance. This phenomenon is called the hole-edge stress concentration.

The increasing of the holes stress concentration is not due to the decrease of the section. Even, if the section is only reducing a few percentages or a few parts per thousand than the body without a hole, the stress will also concentrate to several times. Moreover the holes with the same shape, the multiple of concentration stress almost have nothing with the size of the hole. As it is, the existence of the hole makes the stress state and deformation state which near the hole completely change.

The hole-edge stress concentration is a local phenomenon. Outside a few times of bore diameter, stress is not affect by the influence of a hole; the distribution and numerical size of stress are same as the body with no holes. Generally speaking, the higher degree of concentration stress, the concentration of the phenomenon is more local. That is saying, the stress following the distance of holes and approaching the stress that without a hole.

The stress concentration is related to the shape of a hole. In general, the edge of circular hole has the minimum degree. Therefore, if it is necessarily to dig hole or keep a hole in the component, it is better to use circular hole instead of other shapes as far as possible. If it is not possible to adopt circular hole, it is better to adopt approximate

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circular hole (such as an elliptic hole) rather than the hole with pointed angle.

Since the hole-edge stress can be analyzed by a simple mathematical tool, so here take the circular hole as an example to discuss the question of hole-edge stress concentration.

First of all, suppose that there is a rectangular plate (or long column) with a small round hole (the radius is ―a‖) far from the boundary. In addition, this round hole suffers uniform tension all around and the intensity is ―q‖. As shown in figure 4.1, the origin of the coordinates is in the center of the round hole, and coordinate axis is parallel to the boundary [10].

q q

a

x y o

¦ ¨

q q

qq q A

r

q q

a

x

y o

¦ ¨

q q

qq q A

r

Fig 4.1.1: Uniform tension and Mix tension in a circular hole [10].

In the condition of the straight edge, it is appropriately using rectangular coordinates; in the condition of the hole‘ edges, it is better to use polar coordinates. By using polar coordinates to solve the question, because here is mainly inspects the stress nearby the round hole, and first of all is transforming straight edge to round edge. Thus, taking the distance ―b‖ which is much greater than ―a‖ as the radius, taking origin of coordinates as the centre of a circle. Drawing a big circle that shown in the dotted line in the figure. Seen from the local phenomenon of stress concentration, in the circumference, for example

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in point A, the condition of stress as same as the body without holes, that is to say,



_x

 q

，



_y q，



_xy 0. Substituting coordinate transformation type from rectangular coordinates to polar coordinates, it can get an outcome that the components of stress of polar coordinates are ^^r ^^q，^^r^ ^⁰. Hence, the primary question changes to a new question: circular ring or cylinder with inside radius

a

and outside radius b suffers uniform distribution q at the outside boundary.

In order to get the new solutions, it is only need to order By Suffering the outer pressure, it can get:

[12]

Since b is much greater than a, so it can take , and get the solution

Secondly, suppose that this rectangular sheet suffers uniform stress q from left and right side and suffers uniform stress q from top and bottom side as shown in figure 4.2. By means of the same treatment and analysis as before, it can be seen that at the circumference, for example at point A, the condition of stress as same as that without hole, that is

,

. Using polar coordinates transform type can get:

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[9] (4-1) This is the boundary conditions of outer boundary. At the edge of hole, the boundary conditions is

（4-2）

So it can suppose that

(4-3) Will type (3-3) into the compatible equation

Delete factor: cos 2 solve the differential equation,

(4-4) Thereby gets the formula

(4-5)

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[12] (4-6)

Typing (d) into the boundary conditions (a) and (b), obtain

Solve A、B、C、D， then order , obtain

Type given values into (3-4), obtain the final expression of components of stress

[17] (4-7）

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There is a stress sheet (length: 20mm, width: 6.5mm) has a small hole（5e-2mm）. This sheet suffers equal uniformly distributed load q=1Mpa both from the top and bottom side as shown in figure 4.3.

Figure 4.4 is its stress analysis plan.

q q

Fig 4.1.2 Stress diagrammatic drawing of sheet.

a

x y

o

¦ ¨

x y

r r

Fig 4.1.3 Stress analysis diagram of sheet [17]

This rectangular sheet suffers the uniform distribution q at left and right side, as shown in figure 4.3. From the above superposition method can draw the answer:

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Along with the edge of hole r=a surrounded normal stress is

Its many necessary numerical values are shown in follows table:

Table 4.1.1 several important values in different angles



0⁰ 30⁰ 45⁰ 60⁰ 90⁰



 ^^q 0 q 2q 3q

Along with y axle, surrounded normal stress is

Table 4.1.2 several important data in different length

r a 2a 3a 4a

3q 1.22q 1.07q 1.04q

It is thus clear that at the boundary of the hole the stress achieves triple of the uniform tension. As far away from the edge, the stress approaches to q, as shown in figure 4.4:

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x

y o 3q

q

3q

q

-q

Figure 4.1.4 Diagram of the round hole surrounding force [17].

Along with the X axle, , surrounded normal stress is:

When ; when , Such as shown in Figure 4.4, tensile stress will arises between r=a and , and the resultant force is 0.1924qa .

From this Figure 4.4 it can be seen the stress distribution of the round hole perimeter. And the final data can be compared with the Ansys analysis fringe in general. [9] [17]

Table 4.1.3 Comparison of Analytical and Numerical results for circular hole

Analytical Simulation Stress

Concentration for Circular hole

3 Mpa 2.888 Mpa

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4.1.2 Stress concentration for elliptical holes

q q

Figure 4.1.5 No crack sample

q q

Figure 4.1.6 Crack sample

Fig 4.1.7 the rate of main axis and minor axis (a: b= 2) )

( )

(  

 R ^m

z  

l

<

m O O,

>

R m), - R(l b m), + R(l

a  

After mapping to the unit circle, then

  

   ^m^m

m m



 



 

2 2

1 '

1 1 '



 

















Boundary condition under mapping

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 



 

 

   

 



 

 f

m

m  



 ² ₂ ₁ ₁

1 '

1 1

Divide by 2πi (σ-ζ)

   

_





 









 







 









i d 2 d 1 i

2 1

' d 1

1 i 2 d 1 i

2 1

1

1 2 2 1



 

 



 



L L

f m

m

Without loading

Firstly assuming that there is no loading around the hole, which means external force acting on the circular line is also zero.

Then

       

   

^ς ^σ ⁽ ⁾ ^σ

   

^ςR ^ψ

 

^ς

κ π

P κ P

ς ψ

ς σ ςR

) σ ( κ ς

π P ς P

x y

y x

0 f 1

2 i 1 ln

2 i

ln 4 1 2

i

 



 



     

 

 

 



 



 







And

   

    

















1 0

f 1

1 0

f 1

n n n n

n n

ς b ψ ς ψ

ς a ς





   

_



 



 





 d

1 1 i

2 d 1 i

2

1 ₁ ₁





 



 

 

 _L

L π

π

On the unit circle



 

 1,^d ^d

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 

_ _

 





 d

1 1 i

2 d 1 i

2 1

1

 

1



^_ ^



 



 

  n

n n L

L

π a π



^



 1

1 n

n

an

Is analytic in the circle, so:

 

a

 

ς

π _n

n

n L

1 1

1

1 1 1

i d 2

1 



 





 __



 



 

 





^





We also get

 

d 0

1 1 i 2

1

' d 1

1 i 2

1

1 2 2



 



 





 





 n

n n

L L

n m a

m π

m m π



 







 







 

1

 

d 0

i 2 d 1 i

2 1

1

1  1 

 



 



^



 n

n n

L L

n π b

ψ

π  



 





   

_





  d

i 2

1

1 ^^



_

L

f ς π

We start to get

ψ

₁

  z

       

 



 



 



 



 







 



 



ψ f m

m

ψ f m

m



 

 



 

 

1 2 1

2 1

1 2 1

2 1

1 ' 1 ' 1

Divide by 2πi (σ-ζ)

   

_





 





 







 

 





i d 2 d 1 i

2 1

' d 1

i 2 d 1 i

2 1

1

1 2

2 1



 

 



 



L L

f π ψ

π

m m π

π

(36)

34 Then

 

_



_^ ^_^d^ i

2

1 ₁

π L 0

 



 





 

 







  1 '

' d 1

i 2

1

2 1 2 1

2 2

m m m

m

π _L 

 

 







 

ψ

 

ς ψ

π _L ¹

1 d

i 2

1 



_ ^_ ^ We get

   



 



 

 



 1 '

i d 2

1

2 1 2

1 m

m f

ς π ψ

L 

 

 





So under the condition without loading, we get

   

    

















1 f0 1

n n n n

n n

z b ψ z

ψ z

z a z

z 



Then

   

_





  ^d

i 2

1

1 ^^



_

L

f ς π

   



 



 

 



 1 '

i d 2

1

2 1 2

1 m

m f

ς π ψ

L 

 

 





After getting that, we loading on the edge of hole, then

       

 

ς ψ

σ ςR ) σ ( κ ς

π iP κ P

ς ψ

ς σ ςR

) σ ( κ ς

π iP ς P

xy x

y y

x

y x

0 f 1

2 i 1 ln

2

ln 4 1 2





 



     

 

 

 



 



 







Taking it into the boundary condition

(37)

35

 



 



   

 



 

 ψ f

m

m  



 ² ₂ ₁ ₁

1 '

1 1

 



 



 



 



 

 ₀ ² ₂ ₀ ₀

' 0

1

1 ψ f

m m

f f

f  



 

Where

     

 

     



²



2

2 2 0

1 1

i 2 2

1 ln 4

2 i





 

 



 



m iP m

R P σ

) σ (

m R m

) σ σ ( π

P f P

f

y x xy

x y

y x





 



 



     





 







 





 

 



The new boundary condition

 



 



 



 



 

 ₀ ² ₂ ₀ ₀

' 0

1

1 ψ f

m m

f f

f  



 

The original boundary condition

 



 



   

 



 

 ψ f

m

m  



 ² ₂ ₁ ₁

1 '

1 1

We will use the same way and get

   

_





  d

i 2

1 ₀

0 ^^



_

L f

f ς π

   



 



 

 



 1 '

i d 2

1

2 0 2 0

0 f

L

f m

m f

ς π

ψ 

 

 





We will get the K-M function of infinite plate with elliptic hole by using above results ^f

  ^

^⁰^,^P_x ^^P_y ^⁰

Fig 4.1.8 Relation between loading and difference of angle

(38)

36

When φ=θ, far from the middle of hole

 

⁰

0























 q

In the polar coordinates

y

x



_  _  



y x xy



e   





_  _ 2i _  ²ⁱ^  2i

 

_y

     

^q

x  



 



_  



_  



     

 







 











i 2 i

2 2i

i 2

























 qe e

x y

Then

   

 



 



 

_

 





 

i 2 1

1 2 2

i i 2

4 4

 

 







 



 



 

qe q

xy x

y y x

   

 



 



 





2 Re 1

4

2 Re 1

4

i 2 2

2 0

i 2 2

2 0







 







 









 







 





q m m f qR

q m

m f qR

Land Sliding Analysis on Red Clay soil using Fracture Criteria

Mohamed Riyazdeen M.G Yisho ji

Jin ji

Land Sliding Analysis on Red Clay

Soil using Fracture Criteria

Land Sliding Analysis on Red Clay soil using Fracture Criteria

Mohamed Riyazdeen M.G Yisho ji

Jin ji

Acknowledgement

Abbreviation

1. Notations

Contents

2. Introduction

2.1 Aim and Scope

2.2 Background Research

Land sliding Statistic for 2009-2010

3. Theory

3.1 Stress-strain









3.2 Three Point Bending

3.3 Simply supported beam

3.4 Shear and Moment Diagram

3.5 Bending Stress

3.6 Crack Initiation

3.7 Stress Intensity Factor

3.8 Three Modes of Failure

3.9 Geometric function

3. 10 J –Integral

3. 11 Crack Mouth Opening Displacement

4. Analysis of Fracture Criteria

4.1 Stress concentration



 q





a

q q





 

 

   

   

   









       

   

   

 

   

    



   





 

 





 

 





 

 

 



 

 

 



   



ψ

  ^