Land Sliding Analysis on Red Clay Soil Using Fracture Criteria

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

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

Supervisor: Sharon Kao-Walter, BTH

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. 1

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

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

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

Stress component

Total strain vector,

Thermal strain vector

Deflection in mm

Contents

Acknowledgement ...... 2

Abbreviation ...... 3

1. Notations ...... 4

2. Introduction ...... 8 2.1 Aim and Scope ...... 9 2.2 Background Research ...... 9

3. Theory ...... 13 3.1 Stress-strain ...... 13 3.2 Three Point Bending ...... 15 3.3 Simply supported beam ...... 15 3.4 Shear and Moment Diagram ...... 16 3.5 Bending Stress ...... 17 3.6 Crack Initiation ...... 17 3.7 Stress Intensity Factor ...... 17 3.8 Three Modes of Failure ...... 18 3.9 Geometric function ...... 19 3. 10 J –Integral ...... 20 3. 11 Crack Mouth Opening Displacement ...... 20

4. Analysis of Fracture Criteria ...... 23 4.1 Stress concentration ...... 23 4.1.1 Stress concentration for circular holes ...... 23 4.1.2 Stress concentration for elliptical holes ...... 31 4.2 Stress Intensity Factor ...... 39 4.2.1 Single Edge Notch Tensile ...... 39 4.2.2 Single Edge Notch Bend ...... 41 4.4.3 Shear Model Test ...... 45

4.3 Numerical Results ...... 47 4.3.1 Simulation Result for Stress concentration...... 47 4.3.2 Simulation Result for Stress Intensity Factor ...... 51

5. Experimental Work ...... 55 5.1 Present situation of landslide ...... 55 5.2. Experimental Setup ...... 57 5.3 Slope-forming process ...... 59 5.3.1 Experimental slope formation method ...... 59 5.3.2 Experimental slope formation process ...... 59 5.4. Experimental development...... 61 5.4.1 Implementation of the load ...... 61 5.4.2 Observation of the experiment process ...... 62 5.4.3 The judgment of experiment finished ...... 62 5.5 Experiment results and analysis ...... 63 5.5.1 Indoor landslide experiment validation ...... 63 5.6. Determine the model parameters ...... 64 5.6.1 Determine of Poisson's ratio ...... 64 5.6.2 Determine of Elastic Modulus ...... 65 5.6.3 Determine of Shear Strength ...... 66 5.7 Landslide simulation and experiment validation ...... 68

6. Conclusion ...... 73

7. Future Work ...... 74

8. References ...... 75

Appendices: ...... 78 ABAQUS File ...... 78 Ansys file...... 94

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]

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

(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]

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

Year Number Death or Direct economic occurred missing loss/ten thousand Yuan

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

2009 10,840 486 1,76,548.8

Volume increase or +19,830 +2,429 +4,61,959.7 decrease compared with 2009 Percentage increase +182.9 +499.8 +7,261 or decrease compared with 2009/%

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

11 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.

700000 600000 500000 400000 300000 200000 100000 Ten thousand YuanthousandTen 0 2009 2010

Ecomnomic Loss

Fig. 2.2 (a) Comparison of Economic loss

Land sliding Statistic for 2009-2010 35000 30000 25000 20000 15000 2009 Numbers 10000 5000 2010 0 Death Number occurred Death and Occurance

Fig 2.2 (b) Land sliding statistics

3. Theory

3.1 Stress-strain In the linear elasticity theory

(3-1)

Shear strains in the software (  、  and  ) are xy yz zx 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

1/EEEx xy / x xz / x 0 0 0  /EEE 1/ / 0 0 0 yx y y yz y (3-4) /EEE / 1/ 0 0 0 []D 1  zx z zy z z 0 0 0 1/ Gxy  0 0 0 1/ Gyz  0 0 0 1/ Gxz 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)

E/(1 h  2 E /)( E     T )/(  E h     E /)( E     T ) x x yzzyx x y xyxzyzzyy y

Ez/ h ( xz   yz  xy )(  z   z  T ) (3-14)

2 yE y/(1 h   xzzyy E /)( E    y  T )/(  E y h  xyxzyzzyx    E /)( E    x  T )

Ez/ h ( yz   xzxyy  E / E x )(  z   z  T ) (3-15)

E/(1 h  2 E /)( E     T )/(  E h     E /)( E     T ) z z xyyxx z z yz xzxyyxy y

Ez/ h ( xz   yz  xy )(  z   z  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]

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

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]

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].

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

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].

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

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].

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

23 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 o q ¦¨q x q q  r y A

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

24 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   q   0 are r ， r . 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:

[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)

Obtain components of stress

[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）

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.

o ¦¨ y x x r  r y 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:

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 q  0 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:

3q o x 3q -q

q y

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 3 Mpa 2.888 Mpa for Circular hole

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) m z  ( )  R(  )  a  R(l +m), b  R(l-m), R > O, O  m < l After mapping to the unit circle, then   1  2  m  '   1 m 2   1 m 2   2 '    m Boundary condition under mapping

1  2  m      '     f   1  1 m 2 1 1 Divide by 2πi (σ-ζ) 1    1 1  2  m  '  1 d  1 d 2 i    2 i   1 m 2   L L 1    1 f     1 d   d 2 i L   2 i L   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 P  iP σ () σ   ς   x y ln ς  x y ςR  ς 1 2π1 κ 4 f 0

Px  iPy σ y() σ x   ψ1 ς  κ ln ς    i ςR ψf 0 ς 2π1 κ  2  And

 n 1ς  f 0 ς  an n1  n ψ1ς  ψf 0 ς  bn n1 1    1     1 d   1 d 2πi L   2πi L  1 1         On the unit circle d d  1,    

1    1 1  1 d  a  n1d    n 2πi L   2πi L  1  n1      

 n1 an n1 Is analytic in the circle, so:

n1 1    1   1  1 d   a     ς   n   1  2πi L    n1   We also get 1 1  2  m  '  1 d  2 2πi L  1 m  

1 1  2  m   a  n nd  0  2  n 2πi L   1 m n1 1 ψ   1 1  1 d  b  n  n1d  0    n   2πi L   2πi L   n1 1 f    ς   d 1   2πi L  

We start to get ψ1z 1  2  m 1  2 1'  ψ1   f    1 m 1 m 2     '  ψ    f   1  2  m 1 1 Divide by 2πi (σ-ζ) 1    1 1 m 2  '  1 d   1 d   2 2πi L   2πi L   m  

1 ψ   1 f     1 d   d 2πi L   2πi L  

Then 1     1 d  0 2πi L   1 1 m 2  '  1 m 2  1 d    '   2 2 1 2πi L   m     m 1 ψ   1 d  ψ ς  1  2πi L   We get 1 f   1 m 2 ψ ς   d   '  1  2 1 2πi L     m So under the condition without loading, we get

 n 1z  f0 zan z n1  n ψ1z  ψf0 zbn z n1 Then 1 f    ς   d 1   2πi L   1 f   1 m 2 ψ ς   d   '  1  2 1 2πi L     m After getting that, we loading on the edge of hole, then P  iP σ () σ   ς   x y ln ς  x y ςR  ς 1 2π1 κ 4 f 0

Px  iPy σ y() σ x   ψ1ς  κ ln ς    i xy ςR 2π1 κ  2 

 ψf 0 ς Taking it into the boundary condition

1  2  m     ' ψ    f   1  1 m 2 1 1 1  2  m     ' ψ    f   f 0  1 m 2 f 0 f 0 0 Where

P  iP σ () σ    2  m  f   f   x y ln   x y R   0      2  2π 4   1 m  2 σ y() σ x   R Px  iPy   m    i xy   2  2   2 1   1 m  The new boundary condition 1  2  m     ' ψ    f   f 0  1 m 2 f 0 f 0 0 The original boundary condition 1  2  m     ' ψ    f   1  1 m 2 1 1 We will use the same way and get 1 f    ς   0 d f 0    2πi L   1 f   1 m 2 ψ ς   0 d   '  f 0  2 f 0 2πi L     m We will get the K-M function of infinite plate with elliptic hole by using above results f    0, Px  Py  0

Fig 4.1.8 Relation between loading and difference of angle

When φ=θ, far from the middle of hole

    q

    0

    0

In the polar coordinates

    x  y

2i     2i   e  y  x  2i xy 

 x  y       q

 y  x  2i   2i  e     2i    qe2i

Then

   q   x y  4 4    2i  q   i  y x xy   e2i 1 1 2 2

qR   2  m  q Re2i f        0    2  4   1 m  2

qR  1 m 2  q Re2i f         0    2  4    m  2

1  2  m 1 m d    2 2πi L  1 m     1 1 m 2 1 1 m2   d     2πi   2  m     2  m L 1   d  0 2πi L    1 1 1  d   2πi L       We get 1  2  m 1 z 2  m 1 m d  lim z    2 z0 2 2πi L  1 m   z1 mz  z   1 1 m 2 1 1 mz2 1  d  lim (z  m)z  2 z m 2 2πi L   m   z  m z  1 mz2 1 1 m2   lim (z  m)z   z m z 2  m z   2  m Take into following functions 1 f    ς   0 d f 0    2πi L   1 f   1 m 2  ς   0 d   '  f 0  2 f 0 2πi L     m Obtain mqR PR 2e2i  m  ς    e2i  qR f 0 4 2 4 qR qR1 m2   1 m 2   ς      'ς f 0 4 4 2  m  2  m f 0

qR  2e2i  m    At last 1ς      4   

2i 2 2i qR  2i e 1 m e  m    1ς   e    2  2  m m   m  '   2  m  2e2i    4Re 'z  4Re 1  Re q x y 1 '   2  m  4  2 2 cos 2   m2  2mcos 2    q    4  2m 2 cos 2  m2

On the boundary  1, p  0

1 2mcos 2  m2  2cos 2     q  1 2mcos 2  m2

When   00 1 2m  m2  2mcos 2   q  1 2mcos 2  m2

1 2m  m2 When  900 ,  q ,  1 m2 1 Taking m  ，then  1.4q ，and q 1Mpa so 2

 1.94Mpa  . Table 4.1.4 Comparison of Analytical and Numerical results for Elliptical hole

Analytical Simulation Stress Concentration 1.94 Mpa 2.019 Mpa for Elliptical hole

4.2 Stress Intensity Factor

4.2.1 Single Edge Notch Tensile

Fig4.2.1 Single Edge notched Test Specimen Geometric dimensions

Table 4.2.1 SENT dimensions.

Length Width Load Crack Length

L=40 mm W=6.5 mm P= 150 N a= 3.25 mm

The given model is axially loaded in both the sides which having a crack length of 3.25 mm whose material properties are young‘s modulus of the steel material is 210 Gpa and poison ratio is 0.3, and for soil material young‘s modulus is 1Pa and poison is 0.3

By using the seam crack SENT is analyzed with both steel and soil materials and compared with the finite element using abaqus.

SENT with-out crack (Steel)

Table 4.2.2 SENT Stress results for Steel without crack

Steel material Analytical Simulation Stress Strain Displacement

SENT with-out crack (Clay Soil)

Table 4.2.3 SENT Stress results for clay soil without crack

Clay Soil material Analytical Simulation

Stress Strain 0.05769 0.05769 Displacement 2.308

SENT with Crack for Steel

Table 4.2.4 SIF results for SENT of steel with crack

Stress Intensity 52.107Mpa 51.06Mpa Factor KI J integral 0.0129 0.01283

SENT with Crack for Soil

Table 4.2.5 SIF results for SENT of clay soil with crack

Stress Intensity Factor KI 52.107Mpa 51.91Mpa J integral

SENT KI Comparison for Steel & Clay Soil 60 50 40 30 20 10 Steel Clay soil

Analytical Vs Simulation Stress IntensityFactor Stress KI

Fig 4.2.2 Analytical and Numerical SIF Comparison for SENT

CMOD comparison for SENT

Table 4.2.6 SENT CMOD

Material CMOD Steel Clay Soil

4.2.2 Single Edge Notch Bend

Fig4.2.3 Single Edge notched Bend Test Specimen Geometric dimensions

Table 4.2.7 SENB Geometric Dimensions

Width Crack length Thickness Load Length Span length W a t F L S 6.5mm 3.25 mm 4 mm 150N 40 mm 25 mm

The given model is fixed in a simply supported beam, in which one end with fixed support and another end with roller support and it is loaded centre of the beam with crack length of 3.25 mm and a load of 150 N is applied at the centre of the beam.

Validation of Result for SENB Steel with-out crack

Table 4.2.8 SENB Stress comparison for Steel without crack

Analytical Simulation Stress

Strain Displacement on y

SENB Comparison for Steel and Soil

Table 4.2.9 SENB Stress comparison for clay soil without crack

Steel Soil Stress Strain Displacement on y

Comparision of Stress for Steel & Clay Soil 35

Stress N/mm^2 Stress 20

10 1 2

Fig4.2.4 Stress analysis for steel and clay soil

Table 4.2.10 SENB Stress Intensity factor validation for steel without crack

J integral 0.108231 0.1058 CMOD

Table 4.2.11 SENB Stress Intensity factor validation for clay soil without crack

J integral 226.45 CMOD

SENB KI Comparison for Steel & Clay soil 160

110 Analytical 60 Simulation

Stress IntensityFactor Stress KI Steel Clay soil

Fig 4.2.5 Analytical and Numerical Stress Intensity Factor validation for SENB

Comparison of and J

Table 4.2.12 Comparison of KI and J

Load for plane stress J-integral J-integral Plane and strain Plane Stress strain 150 N 149.99 0.1058 0.09627 450 N 447.2 0.9522 0.8665

CMOD comparison for plane stress and plane strain

Table 4.2.13 Comparison of CMOD under Plane Stress and Strain

Load Plane Stress Plane Strain Analytical Simulation Analytical Simulation 150 N 6.1257 6.0917 5.5759 5.543 450 N 18.373 18.275 16.72 16.623

Plane Stress

Plane stress exist when one of the principal stress ( ) is zero. It is usually ocurrs in structural elements, when one of the dimension is comparitively smaller than other two. [38]

Plane Strain

When one of the dimension is large compared to the others, the principal strain in the direction of the longest dimension is constrained and can be assumed as zero.eg: dam analysed at a cross section loaded by the reservoir.[38]

KI Based on CMOD

Table 4.2.14 Comparison of KI under Plane stress and strain

Load Plane Stress Mpa Plane Strain Mpa Analytical Simulation Analytical Simulation

150 N 149.99 149.30 136.66 135.85

Comparison of Plane Stress and Plane Strain 155 150 145

140 factor KI factor 135 Analytical 130 Simulation 125 Plane stress Plane strain

Stress Intensity Intensity Stress Verifiication of Theoritical and Numerical Result

Fig4.2.6 Plane Stress and Strain Comparison

4.4.3 Shear Model Test

Fig 4.2.7 Shear Model

Half of the shear model is fixed at the bottom and another half is loaded by uniformly distributed load, which has the crack length of 3 mm from the top of the specimen

Table 4.2.15 Shear model Geometric Dimension

Height Breadth Crack Thickness Load length Stress H = 6.5 b= 6 a = 3 t = 3 mm 15 mm mm mm N/

Stress intensity factor for shear model

Table 4.2.16 Shear model result validation for KII and J

Analytical Simulation Stress intensity factor J-integral steel 0.0118 0.0133 Stress intensity factor J-integral red clay soil

KI Comparison for Mode II Shear Model 60 55 50 45 40 35 Analytical 30 Simulation 25

Stress Intensityfactor Stress KI 20 15 10 Steel Clay soil

Fig 4.2.8 Shear model Analytical and Simulation Results for Mode II

Stress Intensity Factor

4.3 Numerical Results

4.3.1 Simulation Result for Stress concentration

Figure 4.3.1 Stress Concentration for circular-hole

Figure 4.3.2 Stress Concentration for Elliptical-hole

Steel without crack

Fig4.3.3 Composite and Tensile stress

Steel with Crack

Fig4.3.4 Composite and Tensile stress

Fig 4.3.5 the point of maximum tensile stress

Clay soil analysis without crack

Fig 4.3.6 Composite and tensile stress cloud

Clay soil analysis with Crack

Fig4.3.7 Composite and tensile stress cloud

4.3.2 Simulation Result for Stress Intensity Factor

Fig 4.3.8 Fine Meshed Specimen model

Single Edge Notched Tensile

Fig 4.3.9 Stress Vs Displacement for SENT

Fig 4.3.10 Simulation result of SENT specimen

Single Edged Notched Bend

Fig 4.3.11 Simulation result of SENB specimen without crack

Fig 4.3.12 Stress Vs Displacement for SENB

Fig 4.3.13 Simulation result of SENB specimen with crack

Shear Model Test

Fig 4.3.14 Simulation result of Shear model without crack

Fig 4.3.15 Simulation result of Shear model with crack

5. Experimental Work

5.1 Present situation of landslide

The process of a landslide experiment is difficult to observe in the outdoors, and then repeated in the laboratory to simulate in the short term. Therefore, in the 20th century 60s, the overseas scholar starts with the landslide experiment to research on landslide; it has made the remarkable progress in the experimental technology, and the research results in aspect. Landslide experiments conducted earlier the former Soviet Union countries, like Japan, Italy, Czech Republic, the United States and some European countries. The experiment mainly uses to study the landslide in the formation mechanism, the landside mass pore-water pressure and the earth stress distribution changes the sliding surface occurrence developing process, the slippery body movement characteristic, the rainfall landslide and the earthquake landslide mechanism and so on. It obtains the research results have the extremely important scientific value in the landslide preventing and controlling aspect. In domestic, developing landslide experiment is real beginning in the 1980 of the 20th century. Northwest hard branch landslide laboratory, Chengdu University of Science and Technology, Three Gorges University, Kunming University of Science and Technology, all have carried out the landslide experiment. In 1980, northwest hard branch landslide laboratory carried out in a simple test bed landslide crack formation mechanism studied. In 2003, State Key Laboratory of Chengdu University of geological hazard prevention completed experimental study on prediction of Bank slope in three Gorge‘s reservoir determine the cause of Bank slope collapses leading and secondary factors, with different angle of soil, rock slope collapse also collapse of the structure of Bank width for quantitative and semi- quantitative study. In 2005, geological disasters in Three Gorges University, through laboratory test method to simulate the landslide

Qianjiangping in rainfall and water under the action of deformation and failure process, and collected through the test in the moisture content, soil pressure, pour water pressure and displacement changes in test data, research Qianjiangping landslide formation mechanism and sliding mode. In 2008, Kunming University of Science and Technology have the research subject, which is named as ―laboratory test research on laterite type landslide in Yunnan province‖, is presented. Aiming to the typical laterite in Yunnan province, laboratory landslide tests were carried out with contents such as manufacture of laboratory landslide test apparatus, calibration of rainfall capacity, development of landslide test, research of physical characteristics and shear strength characteristic of laterite and analysis of rainfall induced landslide‘s mechanisms. The acquisitive research fruits are of great theory and realization significance for the sliding mechanism and prevention of laterite type landslides.

Fig5.2. landslide test apparatus

Economic development in China increases the civil engineering projects such as highway, water Conservancy and hydropower projects. It occupies a large number of slopes. The whole project has a significant economic impact for the stability also for social security. It is essential to analyze the slope that had the initial crack, systematic study for instability process and mechanisms, by taking the landslide experiment as important study method and in the development direction.

5.2. Experimental Setup

Prepare the red clay type slope in the indoor, through grade loading at the top of the hill, to make slope landslide, the slope of the observed deformation for the loading process. Design the basic dimensions of slope Power Engineering Kunming University of science and technology, soil mechanic‘s laboratory has done a lot of slide experiments for study the impact of rainfall on the slope, and get a lot of practical value data. Based on the results and references the dimension of the slope is desired. Slope size diagram shown in Figure 5.1.

Fig5.1 the basic sizes schemes of the slope [1]

Specific values that the size of slope to see Table 5.1：

Table 5.1 the sizes of the slope

N(The length L(Length that in M（Slope H (slope top bottom β（Slope width）/ height)/cm side of side of angle）/° the the cm slope ) slope)/cm /cm

58 18 58 40 61.1

Preparation of material for the slope Soil material of slope preparation process is divided into two stages: the calculation and soakage. （1）Calculation：

According to the slope size, slope volume, and pre-dry density, quality of soil is determined. Dry soil quality is obtained by loss of soil in the slope forming process. And the measure of moisture content is by adding 100g-200g in the computation to determine the dry soil texture quantity to do the slope, it retains decimal.

Table 5.2 the results of the calculation

Dry Dry soil Wet soil Water density quality quality 3 ratio（%） g/cm （kg）（kg）

0.95 27.5 75.0 95.54

（2）Soakage：

The water evaporation sprays make the loss in a wet trough, and the factor influence will prearrange the moisture content value to enhance

2%. Determine the obtained water to join to do in the earth, afterwards will be even the earth material agitation, after settles for 12 hours to stir again the earth material one time, will settle the total time many in 24 hours. After settles the full 24 hours, takes the representative wet soil sample to determine its moisture content. The side slope model carries on the formation take the actual moisture content and the predetermined dry weight density as the control condition.

5.3 Slope-forming process

5.3.1 Experimental slope formation method

Slope soil is completed, and then produces a model of landslide in model slot. Slope forming methods are commonly used masonry method, rolling method, compaction method and the accumulation method. This article expected the formation the red clay slope can have the glide under the load function. Therefore, choice slope formation method not only considers the feasibility, but also pays attention to the formation slope able to occur glide. The masonry method is difficult for forming; The rolling method needs the big working surface, but this article mold die width is only 40cm, the equipment that suits such working surface difficult to purchase, so the feasibility is not strong; Compaction method simple, but a later period process is hard to deal with the steep slope; Accumulation method is that all soil of the slope to accumulate on the slope size range, with the compaction tool gently compaction to side with slope size. This method is suitable for the low dense degree side slope formation. It has strong feasibility, working space is small. Based on the above analysis, this article chooses accumulation method.

5.3.2 Experimental slope formation process The soil material standing greater than 24 hours sketched the model outline in the outer surface of the tank wall by using chalk before brushes the oil in extension of the contour parallel to the surface area within 1cm -2cm.

Fig5.2 Slope forming steps of one

Secondly, according to pre-dry density, moisture content measured and model volume to calculate the required wet soil quality, Then all wet soil by calculate to pour into in the mold die to pile up densely, and suitably adjusts the wet soil material which outline scope model size.

Fig 5.3 Slope forming steps of two

Finally, the obtained slope face to compaction the compaction to the predetermined model contour line, then suitable fix it up, cause the slope face to be smooth.

Fig.5.4 Slope forming steps of three

5.4. Experimental development

5.4.1 Implementation of the load

Fig5.5 Force Loading

The experimental slope formation over 24 hours carries on the load implementation to the slope. In order to ensure uniform implementation of the load to the slope of the hill, first, the top of the slope divided into an even number of the grid of the same size, then on the top of the slope to place a flat piece of wood, place the weight evenly.

Add load every 24 hours until the slope damage occur.

5.4.2 Observation of the experiment process

Initially load 10.2kg is applied. After an hour measures the slope dimensions (slope high and top of slope breadth) change. An Observation result is shown in Table 5.3:

Table 5.3 Deviation of the slope

Time（h） H/cm N/cm

0h 58 18

1h 57.65 17.9

2h 57.64 17.9

4h 57.42 17.84

10h 57.27 17.80

5.4.3 The judgment of experiment finished Judgment as the end of the experiment is the slope landslide occurred, as shown in the figure5.6:

Fig5.6 Slope failure 5.5 Experiment results and analysis

Indoor landslide experiment results are shown in table 5.4:

Table 5.4 the result of LAB

Load 10.2 20.4 30.6 40.8 51 61.2 [Kg]

No crack stable stable stable stable stable instable

Crack stable stable stable stable instable

5.5.1 Indoor landslide experiment validation

Correctness of the landslide experiment can be verified by numerical simulation. The Parameters of slope soil convert to model parameters, which determine the model parameters.

5.6. Determine the model parameters

5.6.1 Determine of Poisson's ratio

Before the load implementation, first records the experimental slope size. Then implement the load, after every hour measure the size of slope. According to the concept of Poisson's ratio (Materials in one- way tension or compression, transverse normal strain and axial strain are the ratio of the absolute value), get the equation for the Poisson's ratio:

N v  H

Table 5.5 calculation of Poisson‘s Ratio

No crack Crack

Time Poisson Poisson H/cm N/cm H/cm N/cm ‘s Ratio ‘s Ratio

0h 58 18 58 18

1h 57.65 17.9 0.29 57.65 17.9 0.29

2h 57.64 17.9 57.64 17.88 0.33

4h 57.42 17.84 0.27 57.6 17.87 0.33

10h 57.27 17.80 0.27 57.46 17.82 0.33

Average 0.28 0.32 value

Finally, obtain the experimental soil poison‘s ratio value is: No crack: 0.28 Crack: 0.32

5.6.2 Determine of Elastic Modulus

Elastic Modulus is: Material in the elastic deformation stage, the normal stress and the corresponding normal strain‘s ratio. According to the concept of Elastic Modulus, get the equation for the Elastic P FN Modulus: E  0  NNA0 

F —The force acting on the top of the hill surface

N0 —The initial length in the top of the hill A —The area of top hill N —The actual length in the top of the hill Specific modulus of elasticity calculations see tables 5.6:

Table 5.6 calculation of Modulus‘s elasticity

Mod Modul ulus‘ us‘s F / N / s / / elastici N cm elasti ty / city / N 0 KPa ：18c m No crack Crack

199.92 17.80 25.0 199.92 17.66 14.7

299.88 17.51 25.9 299.88 17.24 16.7

A : 399.84 17.13 26.3 399.84 16.64 16.8

720 499.8 16.64 25.5 499.8 16.24 19.7 cm3

599.76 16.05 25.4

Finally, obtain the experimental soil average value for Modulus‗s elasticity is: No crack: 25.6 Crack: 17.0

5.6.3 Determine of Shear Strength

The main indicators of shear strength are internal friction angle and cohesion. Cohesion is mutual attraction in the same substances within the adjacent attraction between the various parts, this kind of mutual attraction is molecular force that between the material elements. Internal friction represents is linking strength that between the soil particle skin-friction force and the soil particle. It can be seen that shear strength refers to the force in soil internal. Power Engineering, Kunming University of science and technique, soil mechanics laboratory, through several experiments found that shear strength values has related with the dry density and moisture content, therefore, this article by measuring dry density and moisture content to determine the shear strength.

Table 5.7 Moisture content of no crack slope

Box weight+ Box weight+ Moisture Box weight Wet soil Dry soil content weight weight （g）（%）（g）（g）

20.691 25.723 24.56 23

22.609 31.02 28.373 31

22.46 30.517 28.623 24

22.688 27.451 26.007 30

28.502 34.405 32.937 25

24.332 30.193 28.607 27

From Table 5.7, obtain the moisture content of the experimental soil sample is (no crack): 26.67% Table 5.8 Moisture content of no crack slope Box weight+ Box weight+ Moisture Box weight in Wet soil Dry soil content weight in weight in （g）（%）（g）（g）

20.691 25.723 22.56 37

22.609 31.02 28.073 35

22.46 30.517 27.633 36

22.688 27.451 25.967 31

28.502 34.405 32.637 30

According to Table 5.8, obtain the moisture content of the experimental soil sample is (crack): 33.8% Table 5.9 Shear Strength index

Dry density/(g/cm3) :1.0 Moisture content/% Internal friction Cohesion /kpa angle /°

23.9 0.47 25.7

25.8 0.98 24.3

27.6 1.55 23.1

29.9 1.35 22.9

31.9 0.80 21.6

33.7 0.30 20.3

Comparison of the conclusion that Power Engineering, Kunming University of science and technique, soil mechanics laboratory (see Table: 5.9) get the indicators of shear strength which experiment soil samples: No Crack: o Cohesion ( c ):1.55 kpa, Internal friction angle ( ):23.1 . Crack: Cohesion ( ):0.3 kpa, Internal friction angle ( ):20.3o.

5.7 Landslide simulation and experiment validation According to the thesis result Conducted in Kunming University of science and technique, select the point C that in top of the hill to study the safety factor by numerical simulation.

Fig5.7Fs of no crack and crack slope (Load: 10.2kg)

Fig5.8Fs of no crack and crack slope (Load: 20.4kg)

Fig5.9Fs of no crack and crack slope (Load: 30.6kg)

Fig5.10Fs of no crack and crack slope (Load: 40.8kg)

Fig5.11Fs of no crack and crack slope (Load: 51kg)

Fig5.12Fs of no crack and crack slope (Load: 61.2kg)

Table 5.10 Simulation result with LAB result

Load Safety The results Safety The Kg factor of of factor of results of point C experiment point experimen no crack s no crack ts C crack crack

10.2 1.58 Stable 1.97 Stable

20.4 1.29 Stable 1.22 Stable

30.6 1.16 Stable 1.10 Stable

40.8 1.07 Stable 1.02 Stable

51 1.01 Stable 0.96 Instable

61.2 0.98 Instable

From the experimental results in Table 5.10 to see: as the load increases, the two slopes (no crack and crack) in the C-point safety factor gradually reduced, When the load increased to 61.2Kg, the safety factor of slope (no crack) less than 1, the slope occurrence glide; When the load increased to 51Kg, the safety factor of slope (crack) less than 1, the slope occurrence glide.

Safety Factor

If SF >1 is Safe, and Slope is Stable.

If SF <1 is Un-Safe and Slope is Instable.

Instability of Slope 2.5

Without crack 2

WithCrack

1.5

1 Safety Factor C Factor Safety

0.5

0 1 2 3 4 5 6 7 Load 1 X 10 kg

Fig5.13 Instability of the slope for with and without crack

6. Conclusion

The experimental indoor land slide physical model was built to examine and gain knowledge of slope stability, instability, and safety factor [SF]. Finite element model was created for crack initiation model and calculated fracture criteria like stress concentration factor [Kt], stress intensity factor [KI and KII], and crack mouth opening displacement [CMOD] are examined and verified [Fig 4.2 to Fig 4.3.15]. The Indoor landslide model was experimented based on force resisting method by applying uniform loads on the top of the hill and obtained instability of the model for with, and without crack was examined and verified with numerical results [Table 5.10 & Fig 5.7 – 5.13]. Slope stability also depends on strength of the materials. Therefore, the slope deviation for with and without crack due to loading was observed carefully from the experiment and determines the elastic modulus, poison ratio, and shear strength of the soil material [Table 5.5, 5.6 & 5.9]. Loading on the top of a slope will increase the driving force and loading on the toe of a slope will increase the resisting force. Due to applying a load on the top of a slope the driving force overcomes the resisting force that makes the stability of the slope to start collapse. As the load increases, the safety factor [C] for crack and no crack gradually reduced [Table 5.10 & Fig 5.13]. Safety factor for the slope (no crack) is more stable than the slope (crack).

7. Future Work

Here, we focus on an Indoor landslide simulation experiment with red clay soil, lots of work to be done in the future. Such as:  Relation between rainfall and land sliding, also the effects of earth quakes on the deviation of land sliding have to be considered. . Improve the design of the experiment by including crack propagation also improve the ABAQUS model with XFEM to match the real situation of the crack propagation in the slope. . By applying impact load and frequency the dynamic analyzes of the model can be determined. . Analyze the indoor model with different materials to determine risk and hazard zones.

8. References

1. Yi shao-ji,Huang-Ying,etal.Research on Influence of Soil Behavior Parameters to Slope Stability[J].Journal of Water Resources and Architectural Engineering,2011,Vol.9,No.3:1-8 2. KIM JY, LEE S R. An improved search strategy for the critical slip surface using finite element stress fields[J].Computers and Geotechnics,1997,22(4): 295-313 3. ZIENKIEWICZ O C, HUMPHESON C, LEWIS R W. Associated and non associated visco-plasticity and plasticity in soil mechanics[J].Géotechnique,1975,25(4):671–89 4. GRIFFITHS D V, LANE P A. Slope stability analysis by finite elements [J]. Géotechnique,1999,49(3):387–403 5. MATSUIT,SAN KC.Finite element slope stability analysis by shear strength reduction technique[J].Soils and Foundations,1992, 32(1):59-70 6. UGAI K. A method of calculation of total factor of safety of slopes byelasto- plastic FEM [J]. Soils and Foundations, 1989,29(2):190—195 7. Ching-Chuan Huang,Chien-Li Lo b,Jia-Shiun Jang et al. Internal soil moisture response to rainfall-induced slope failures and debris discharge [J]. Engineering Geology, 2008, 101:134-145. 8. Lourenco,S.D.N., M.D.Bolton et al. Failure process and hydrologic response of a two layer physical model: implications for rainfall-induced landslides [J]. Geomorphology, 2006, 73:115-130. 9. Xue Erpeng, Dynamic characteristics research on the valve of electromagnetic impactor, Master Thesis work in Kunming University of Science and Technology, China. 10. Fengchun Jiang, Aashish Rohtagi, Kenneth S. Vecchio and Justin L. cheney, 1996, Analysis of dynamic responses for a pre-cracked three point bend specimen, International Journal of Fracture 127: pp. 367-379. 11. I.V. Orynyak and A. Ja Krasowsky, 1998, The modeling of elastic response of a three-point bend specimen under impact loading, Engineering Fracture Mechanics Vol. 60, No. 5-6, pp. 563±575. 12. Xiangping Hu, Shreenidhi R Kulkarni, Fracture Analysis on a 3 point Bend Specimen by Cyclic Impact Loading, Master‘s Degree Thesis ISRN: BTH- AMT-EX—2009/D-05—se. 13. G. Bertolino a, A. Constantinescu a, M. Ferjani a, P. Treiber b, A multiscale approach of fatigue and shakedown for notched structures, Theoretical and

Applied Fracture Mechanics 48 (2007) 140–151. 14. H. R. Kao, Correlation between J-Integral and CMOD of an Imapct Loaded 3-Point Bend Specimen, Lund Institute of Technology. 15. Han Shichang Shen Yujie, Fracture Analysis of the Working Part of an Impactor under Impact Loading, Department of Mechanical Engineering Blekinge Institute of Technology Karlskrona. 16. Yin YanShu, Single symmetry section profiles three point bending and straightening theory[D], YanShan Univerisry,(2010) 45-67 17. Xu ZhiLun, Elastic mechanics brief tutorial[M] BeiJing higher education press,(1997) 62-85 18. Zhang ChaoHui, ANSYS8.0 structure analysis and example analysis[M] BeiJing mechanical industry press, (2005) 43-80 19. Zhang ChaoHui, ANSYS12.0 structure analysis and analytical examples of engineering application (third edition) Beijing mechanical industry press,(2010)47-68 20. Xu Hao, Fatigue strength Beijing higher education press,(1987) 33-49 21. T.L Anderson, Fracture Mechanics: Fundamentals and Applications.CRC Press, ISBN: 0-8493-1656-1. 22. Tore Dahlberg Anders Ekberg, Failure Fracture Fatigue an Introduction. Studentlitteratur ISBN: 91-44-02096-1 23. Arun Shukla, Dynamic Fracture Mechanics. World Scientific Publishing, ISBN: 981-256-840-9 24. Yeditepe University. Department of Mechanical Engineering. ME 402 Experimental ME II, Three Point Bending Test. http://me.yeditepe.edu.tr/courses/me402/three%20point%20bending%20test. pdf 25. James M.Gere, Stephen P.Timoshenko, Mechanics of Materials, PWS Publishing Company, ISBN: 0-534-93429-3 26. All metals and forge group, ferrous and non-ferrous materials, a service centre and forge facility, Parsippany NJ 07054 USA http://www.steelforge.com/metaltidbits/tensilestrength.htm 27. Armando Leon, Non-linear vibration and dynamic fracture mechanics of bridge cables, Blekinge Institute of Technology. 28. Sharon Kao-Walter.,‖on the fracture of Thin Laminates‖, Department of mechanical Engineering, Blekinge Institute of Technology. 29. Rubio, L., J. Fernandez-Saez and C. Navarro, ―Determination of dynamic fracture-initiation toughness using three-point bending test in a modified Hopkinson pressure bar.‖ 30. Khurmi R.S., J.K. Gupta, A text book of machine design, Eurasia publishing house New Delhi.

31. Alan T. Zehnder, Ph.D. Fracture Mechanics, Sibley School of Mechanical and Aerospace Engineering Cornell University, Ithaca, NY 14853. 32. Gdoutos E.E., Democritus University of Thrace, Greece, Fracture mechanics an introduction, published by Springer,ISBN1-4020-3153-X 33. Callister, W. Materials Science and Engieering. John Wiley and Sons, New York1994. 34. Hertzberg, R.,"Deformation and Fracture Mechanics of Engineering Materials‖ John Wiley and Sons, New York 1996. http://www.sv.vt.edu/classes/MSE2094_NoteBook/97ClassProj/anal/kim/inte nsity.html (August 2011). 35. http://news.163.com/11/0824/07/7C73HC6P00014AED.html(Verifiedon Nov 2011). 36. http://news.xinmin.cn/rollnews/2010/01/04/3243370.html (Verified on Nov 2011). 37. Paul D. Ronney, department of aerospace and mechanical engineering, university of California, Basics of mechanical engineering. http://www.ami.ac.uk/courses/topics/0123_mpm/index.html 38. Jaeger, John Conrad; Cook, N.G.W, & Zimmerman, R.W. (2007). ―Fundamentals of rock mechanics‖ (Fourth ed). Wiley-Blackwell. pp. 9–41. ISBN 0632057599.

Appendices:

ABAQUS File

SENT

*Heading ** Job name: TNswithCrk Model name: Model-1 ** Generated by: Abaqus/CAE 6.10-1 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** ** PARTS ** *Part, name=Part-1 *End Part ** ** ** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, name=Part-1-1, part=Part-1 *Node 1, -5.5, 0. 2, -5.5, 3.25 3, -22.5, 3.25

4, -22.5, 0. ------Note:-Due to large data it is not included ------5112, -2.49874997, 3.24783492 5113, -2.49823236, 3.24823236 5114, -2.49783492, 3.24874997 5115, -2.4975853, 3.24935293 *Element, type=CPS8R 1, 1, 28, 408, 101, 1741, 1742, 1743, 1744 2, 28, 29, 409, 408, 1745, 1746, 1747, 1742 3, 29, 30, 410, 409, 1748, 1749, 1750, 1746 4, 30, 31, 411, 410, 1751, 1752, 1753, 1749 ------Note:-Due to large data it is not included ------1632, 27, 27, 393, 392, 27, 5111, 4833, 5112 1633, 27, 27, 392, 391, 27, 5112, 4829, 5113 1634, 27, 27, 391, 390, 27, 5113, 4825, 5114 1635, 27, 27, 390, 389, 27, 5114, 4821, 5115 1636, 27, 27, 389, 26, 27, 5115, 4817, 4839 *Nset, nset=_PickedSet2, internal, generate 1, 5115, 1 *Elset, elset=_PickedSet2, internal, generate 1, 1636, 1

*Nset, nset=_PickedSet3, internal, generate 1, 5115, 1 *Elset, elset=_PickedSet3, internal, generate 1, 1636, 1 *Orientation, name=Ori-1 1., 0., 0., 0., 1., 0. 3, 0. ** Section: Section-1 *Solid Section, elset=_PickedSet2, orientation=Ori-1, material=Material-1 4., *End Instance ** *Nset, nset=_PickedSet11, internal, instance=Part-1-1 27, *Nset, nset=_PickedSet12, internal, instance=Part-1-1 27, *Nset, nset=_PickedSet25, internal, instance=Part-1-1 19, *Nset, nset=_PickedSet28, internal, instance=Part-1-1 3, *End Assembly ** ** MATERIALS ** *Material, name=Material-1

*Density 7800., *Elastic 210000., 0.3 ** ------** ** STEP: Step-1 ** *Step, name=Step-1 *Static 1., 1., 1e-05, 1. ** ** LOADS ** ** Name: Load-1 Type: Concentrated force *Cload _PickedSet25, 1, 150. _PickedSet25, 2, 0. ** Name: Load-2 Type: Concentrated force *Cload _PickedSet28, 1, -150. ** ** OUTPUT REQUESTS ** *Restart, write, frequency=0

** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-2 ** *Contour Integral, crack name=H-Output-2_Crack-1, contours=5, crack tip nodes, type=K FACTORS, direction=KII0 _PickedSet11, _PickedSet12, 0., 1., 0. *End Step

SENB

*Heading ** Job name: KforS253PB Model name: Model-1 ** Generated by: Abaqus/CAE 6.10-1 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** ** PARTS ** *Part, name=Part-1

*End Part ** ** ** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, name=Part-1-1, part=Part-1 *Node 1, 6., 7. 2, 6., 3.75 3, 13.5, 3.75 4, 13.5, 7. ------Note:-Due to large data it is not included ------2839, 13.5, 2.734375 2840, 12.75, 2.125 2841, 13.5, 2.328125 2842, 12.75, 1.71875 2843, 13.5, 1.921875 2844, 12.75, 1.3125 2845, 13.5, 1.515625 2846, 12.75, 0.90625 2847, 13.5, 1.109375

2848, 12.75, 0.5 2849, 13.5, 0.703125 *Element, type=CPE8R 1, 1, 30, 592, 59, 37, 2520, 2521, 2522 2, 30, 31, 593, 592, 38, 2523, 2524, 2520 3, 31, 32, 594, 593, 39, 2525, 2526, 2523 4, 32, 33, 595, 594, 40, 2527, 2528, 2525 5, 33, 34, 596, 595, 41, 2529, 2530, 2527 ------Note:-Due to large data it is not included ------908, 29, 29, 554, 553, 29, 2515, 561, 2516 909, 29, 29, 553, 552, 29, 2516, 560, 2517 910, 29, 29, 552, 551, 29, 2517, 559, 2518 911, 29, 29, 551, 550, 29, 2518, 558, 2519 912, 29, 29, 550, 28, 29, 2519, 557, 565 *Nset, nset=_PickedSet2, internal, generate 1, 2849, 1 *Elset, elset=_PickedSet2, internal, generate 1, 912, 1 *Nset, nset=_PickedSet3, internal, generate 1, 2849, 1 *Elset, elset=_PickedSet3, internal, generate 1, 912, 1 *Orientation, name=Ori-1

1., 0., 0., 0., 1., 0. 3, 0. ** Section: Section-1 *Solid Section, elset=_PickedSet2, orientation=Ori-1, material=Material-1 4., *End Instance ** *Nset, nset=_PickedSet81, internal, instance=Part-1-1 21, *Nset, nset=_PickedSet101, internal, instance=Part-1-1 29, *Nset, nset=_PickedSet102, internal, instance=Part-1-1 29, *Nset, nset=_PickedSet131, internal, instance=Part-1-1 5, *Nset, nset=_PickedSet132, internal, instance=Part-1-1 11, *End Assembly ** ** MATERIALS ** *Material, name=Material-1 *Elastic 100000. 0.3 ** ------

** STEP: Step-1 ** *Step, name=Step-1, Inc=1000 *Static 0.1, 1., 1e-05, 1. ** ** BOUNDARY CONDITIONS ** ** Name: BC-1 Type: Displacement/Rotation *Boundary _PickedSet131, 1, 1 _PickedSet131, 2, 2 _PickedSet131, 6, 6 ** Name: BC-2 Type: Symmetry/Antisymmetry/Encastre *Boundary _PickedSet132, YSYMM ** ** LOADS ** ** Name: Load-1 Type: Concentrated force *Cload _PickedSet81, 2, -150. ** ** OUTPUT REQUESTS **

*Restart, write, frequency=0 ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT *Output, history, frequency=0 ** ** HISTORY OUTPUT: H-Output-1 ** *Contour Integral, crack name=H-Output-1_Crack-1, contours=5, crack tip nodes _PickedSet101, _PickedSet102, 0., 1., 0. ** ** HISTORY OUTPUT: H-Output-2 ** *Contour Integral, crack name=H-Output-2_Crack-1, contours=5, crack tip nodes, type=K FACTORS, direction=KII0 _PickedSet101, _PickedSet102, 0., 1., 0. *End Step

SLOPE

Heading ** Job name: Job-8 Model name: Model-1 ** Generated by: Abaqus/CAE 6.10-1 *Preprint, echo=NO, model=NO, history=NO, contact=NO *Part, name=Slope

*Node 1, 15., 0.2, 15.,15.3, 0.,15.4, 0., 0.5, 50.,15.6, 50., 0.7, 50., 25.8, 25., 25.9, 15., 1.510, 15., 3. 11, 15., 4.512,15., 6.13,15.,7.514,15., 9.15,15.,10.516,15., 12.17, 15., 13.518, 13.5,15.19, 12., 15.20, 10.5, 15. 21, 9., 15.22, 7.5, 15. 23,6., 15.24, 4.5, 15.25, 3., 15.26, 1.5, 15.27, 0., 13.528, 0., 12.29, 0.,10.530, 0., 9. 31, 0., 7.532, 0., 6.33, 0., 4.534, 0., 3.35, 0.,1.536, 1.5, 0.37, 3.,0.38, 4.5, 0.39, 6., 0.40,7.5, 0. *Element, type=CPE4 1, 1, 9, 142, 44 2, 9, 10, 143, 142 3, 10, 11, 144, 143 4, 11, 12, 145, 144 5, 12, 13, 146, 145 6, 13, 14, 147, 146 7, 14, 15, 148, 147 8, 15, 16, 149, 148 9, 16, 17, 150, 149 10, 17, 2, 18, 150 *Nset, nset=slope, generate 1, 630, 1 *Elset, elset=slope, generate 1, 575, 1 *Nset, nset=_PickedSet3, internal, generate 1, 630, 1

*Elset, elset=_PickedSet3, internal, generate 1, 575, 1 ** Section: soil *Solid Section, elset=_PickedSet3, material=soil 1., *End Part ** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, name=Slope-1, part=Slope *End Instance ** *Nset, nset=_PickedSet4, internal, instance=Slope-1 3, 4, 5, 6, 7, 27, 28, 29, 30, 31, 32, 33, 34, 35, 93, 94 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109 *Elset, elset=_PickedSet4, internal, instance=Slope-1 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 126, 151, 176, 201, 226 251, 276, 301, 326, 375, 400, 425, 450, 475, 500, 525, 550, 575 *Nset, nset=_PickedSet5, internal, instance=Slope-1 1, 4, 6, 36, 37, 38, 39, 40, 41, 42, 43, 44, 69, 70, 71, 72 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88 89, 90, 91, 92 *Elset, elset=_PickedSet5, internal, instance=Slope-1

1, 11, 21, 31, 41, 51, 61, 71, 81, 91, 326, 327, 328, 329, 330, 331 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347 348, 349, 350 *Nset, nset=_PickedSet6, internal, instance=Slope-1, generate 1, 630, 1 *Elset, elset=_PickedSet6, internal, instance=Slope-1, generate 1, 575, 1 *End Assembly ** ** MATERIALS ** *Material, name=soil *Elastic 17000., 0.35 *Mohr Coulomb, dependencies=1 36.49, 0., , 0.5 26.25, 0., , 0.75 20.3, 0., , 1. 16.49, 0., , 1.25 13.85, 0., , 1.5 11.94, 0., , 1.75 10.48, 0., , 2. *Mohr Coulomb Hardening, dependencies=1 0.6, 0., , 0.5

0.4, 0., , 0.75 0.3, 0., , 1. 0.24, 0., , 1.25 0.2, 0., , 1.5 0.17, 0., , 1.75 0.78, 0., , 2. ** ------*initial conditions, type=field, variable=1 slope-1.to slope,0.5 ** ** STEP: load ** *Step, name=load, unsymm=YES *Static 0.1, 1., 1e-05, 1. ** ** OUTPUT REQUESTS ** *Restart, write, frequency=0 ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1

** *Output, history, variable=PRESELECT *End Step ** ------** STEP: reduce ** *Step, name=reduce, unsymm=YES *Static 0.1, 1., 1e-05, 1. *field, variable=1 slope-1.slope,2 ** ** BOUNDARY CONDITIONS ** ** Name: BC-1 Type: Displacement/Rotation *Boundary _PickedSet4, 1, 1 ** Name: BC-2 Type: Displacement/Rotation *Boundary _PickedSet5, 1, 1 _PickedSet5, 2, 2 ** ** LOADS ** ** Name: Load-1 Type: Body force

*Dload _PickedSet6, BY, -1.69 ** ** OUTPUT REQUESTS ** *Restart, write, frequency=0 ** ** FIELD OUTPUT: F-Output-1 ** *Output, field *Node Output CF, RF, U *Element Output, directions=YES FV, LE, PE, PEEQ, PEMAG, S *Contact Output CDISP, CSTRESS ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step

Ansys file

PREP7 a=1.5e-3 w=6.25e-3 l=4e-2 beta=0 P=7.63E6 R=8E-5 N=20 E=2.06E11 nu=0.3 H=0.005 M=10

*AFUN,DEG

ET,1,SHELL163 ET,2,SOLID164 KEYOPT,1,3,0

MP,DENS,1,7.85E3 MP,EX,1,E MP,PRXY,1,nu TB,BISO,1,,,,

TBDATA,1,345E6 TBDATA,2,7.8E10

y3=L/2+w*tan(beta) x7=a*cos(beta) y7=a*sin(beta)+L/2

K, ,,,, K, ,w,0,, K, ,w,y3,, K, ,w,L,, K, ,0,L,, K, ,0,L/2,, K, ,x7,y7,, K, ,0,L/2,,

L,1,2 L,2,3 L,3,7 L,7,8 L,8,1 L,7,6 L,6,5 L,5,4 L,4,3

AL,1,2,3,4,5 AL,6,7,8,9,3

FLST,2,2,5,ORDE,2 FITEM,2,1 FITEM,2,-2 VEXT,P51X, , ,0,0,H,,,, /USER, 1 LSEL,S,LOC,Z,0 FLST,5,9,4,ORDE,2 FITEM,5,1 FITEM,5,-9 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y LESIZE,_Y1,a/5, , , , , , ,1 KSCON,7,R,1,N/2,0, ALLSEL,ALL MSHAPE,0,2D FLST,5,2,5,ORDE,2 FITEM,5,1 FITEM,5,-2 CM,_Y,AREA

ASEL, , , ,P51X CM,_Y1,AREA CHKMSH,'AREA' CMSEL,S,_Y AMESH,_Y1 CMDELE,_Y CMDELE,_Y1 CMDELE,_Y2 VSWEEP,1,1,3

EXTOPT,ACLEAR,1 EXTOPT,ESIZE,M,0 TYPE, 2 MAT, 1 REAL, ESYS, 0 SECNUM,

VCLEAR,1 VSWEEP,1,1,3 VSWEEP,2,2,9

NSEL,S,LOC,X,0 NSEL,R,LOC,Y,0.0075 D,ALL,UX,,,,,UY,,,,

ALLSEL

NSEL,S,LOC,X,0 NSEL,R,LOC,Y,0.0325 D,ALL,UX ALLSEL

NSEL,S,LOC,X,w NSEL,R,LOC,Y,l/2 NPLOT,1 CM,N_F1,NODE *DIM,TIME,ARRAY,2,1,1, TIME(1,1,1)=0 TIME(2,1,1)=0.01 *DIM,FORCE,ARRAY,2,1,1, FORCE(1,1,1)=0 FORCE(2,1,1)=-5000 EDLOAD,ADD,FX,0,N_F1,TIME,FORCE,0,,,,, ! ALLSEL,ALL ! NSEL,S,LOC,X,w ! NSEL,R,LOC,Y,l/2 ! NPLOT,1 ! CM,N_F2,NODE ! *DIM,TIME1,ARRAY,2,1,1, ! TIME1(1,1,1)=0.01

! TIME1(2,1,1)=0.02 ! *DIM,FORCE1,ARRAY,2,1,1, ! FORCE1(1,1,1)=-5000 ! FORCE1(2,1,1)=-6000 ! EDLOAD,ADD,FX,0,N_F2,TIME1,FORCE1,0,,,,, ALLSEL,ALL EPLOT,1 /AUTO,1 /REP,FAST /NERR,300,100000000,

/SOL TIME,0.01 EDRST,100 EDHTIME,100 EDOUT,SPCFORC EDOUT,MATSUM EDOUT,GLSTAT ALLSEL,ALL SAVE FINISH /STATUS,SOLU !SOLVE !EDWRITE,LSDYNA,'LDT3','K',''

School of Engineering, Department of Mechanical Engineering Telephone: +46 455-38 50 00 Blekinge Institute of Technology E-mail: [email protected] SE-371 79 Karlskrona, SWEDEN