seminar report 5nov2013

8/13/2019 Seminar Report 5Nov2013

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CHAPTER 3: BASIC CONCEPTS AND DEFINITIONS

3.1Basic Definitions:3.1.1 Deformation Gradient: Consider a body as shown in Fig. 1 which undergoes some

deformations. Let R and R*be the undeformed and deformed states of the body respectively. Let

x be the position vector of a point P of the body in R. Position of point P in the deformed

configuration is p and its position vector is y. Let dxrepresents a

Figure 1: Body in undeformed and deformed configurationssmall length element in undeformed configuration and it is changed to dy in the deformed

configuration . Then dy and dx are related to each other by

where F is given by

[

]

This F and is called the deformation gradient. Here are the components of displacementu in directions respectively.

(1)

(2)

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3.1.2 Principle Invariants of a Tensor:For any tensor A we can define three terms called as

principle invariants which are denoted by I1, I2, I3and are given by

*,- + where tr denotes trace of tensor and det denotes determinant of tensor.

3.1.3 Indicial notations:The indicial notations we have used can be understood by taking an

example. Let f is a function of then

3.1.4 CauchyGreen Deformation Tensors:

3.1.4.1 Right Cauchy Green deformation tensor:It is defined as

Here T denotes the transpose.

3.1.4.2 Left CauchyGreen deformation tensor:It is defined as

3.1.5 Traction Vector: If dF is the force acting on an area ds of the surface of a body then

traction vector is defined as

3.1.6 Stress At A Point:At any point of the body the state of stress is defined by following stress

tensor

Here represent that the stress is acting in j

th

direction on a plane which is perpendicular to theith

direction.

(3)

(4)

Comment [w1]: We should define ho

calculate tr and det of any tensor. Other

incomplete information. Thissub-section

after indicial notation.


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3.1.7 Relation between t and : If the normal vector to the surface of the body is nthen at thatpoint

and t are related by

3.1.8 Types Of Stress Tensor: When a body undergoes large deformations then we have tomake distinction between undeformed and deformed configuration.Let us assume that traction T

is applied on a small surface element dS in R .N is the normal vector on dS. Whilet and nare

the traction vector and normal vector to the surface element ds in R*. Let force acting on the

body is df. Then we can define following three types of stress tensors

Figure 2: Body with PK-I and Cauchy traction vectors

3.1.9 Cauchy Stress Tensor : In this tensor the area of deformed configuration is used for stress

calculation. It is denoted byand is given by

Here tis the Cauchy traction vector. Cauchy stress tensor is symmetric.

3.1.10 First Piola-Kirchhoff (PK-I) stress Tensor: In this tensor the area of undeformed

configuration is used for stress calculation. It is denoted by

and is given by

Here T is the first Piola-Kirchhoff (PK-I) traction vector. First Piola-Kirchhoff (PK-I) stress

Tensor is generally not symmetric.

3.1.11 Second Piola-Kirchhoff (PK-II) stress Tensor:We define a pseudo force vector suchthat

then second Piola-Kirchhoff stress tensor denoted by S is given by

3.1.12 Relation Between Three Stress Tensors:By equations (6) and (7) we obtain

(5)

(6)

(7)

(8)


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We shall use the Nansons relationwhich relates the normal vector in undeformed and deformed

co-ordinates,

Now

This is the relation between and . J is the third invariant of and its value is 1 forincompressible materials. Again from (7) and (8)

and putting from (9) This is the relation between and S.3.2 Hyper elastic materials and basic governing equations:

3.2.1 Hyperelastic materials:Linear elastic material model assumes that the deformations are

small in the material. There exists some materials which are elastic but exhibits large

deformations upon loading so that linear elastic model is not suitable for such materials.

Hyperelastic model is used to characterize such material behavior. Rubber, vualcaunized

elastomers, unfilled elastomers and biological tissues are some examples whose behaviour can be

best characterized by hyperelastic model. These materials are a special case of Cauchy elastic

materials which means that the stress state depends only on the current state of deformation and

not on the path or history of deformation. For these materials stress-strain relationships are

derived from strain energy density function W and given by

For general hyperelastic material model W is a function of three principle invariants of the left

(or right) Cauchy-Green tensor

The values of are given by

(9)

(10)

(11)

(12)

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I is the unit tensor. Putting values in S

Using (11)

Following are the names ofsome of the widely used hyperelastic material laws:

a. Neo-Hookean

b.Mooney-Rivlin

c. Polynomial form of order 2

d.Reduced Polynomial form of order 2

e. Arruda-Boyce

These all differ in the way that they assume different expressions for W. In this seminar material

is chosen to be Neo Hookean because it gives the simplest expression for W. In addition the

Neo-Hookean material is assumeds the material to be incompressible. So to ensure that

deformation is locally volume preserving (incompressible) we shall assume that an arbitrary

hydrostatic pressure p is acting at all points of the body such that equation (13) converts to

For Neo-Hookean material W is a function ofonly so the derivatives of W with respect to and will be zero hence from equation (14)

and from equation (9)

(13)

(14)

(15)

(16)


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Chapter 4 Problem Considered And Its Solution

Figure 3: Crack with coordinate system

4.1 Solution of the crack problem: Consider a region R which represents the open cross

section of an infinite slab containing a crack of length 2c. The slab is to be deformed in finite anti

plane shear. Such a deformation is characterized by

Our goal is to find which will satisfy the boundary conditions and equilibrium equations andconsistent with the constitutive relation (15) and (16). Now with the above displacementcomponents the values of and from EQs. (???) and (???) will be

[

() ()]

So from the state of stress in the slab in terms of PK-I stress tensor from equation (16) is

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[

]

And in terms of Cauchy stress tensor from equation (15) is

[

() ()]

() () The equilibrium equations in terms of and are

Since the surface of the crack is a free surface i.e. there is no load at that surface so traction will

be zero along the crack faces in both deformed and undeformed configuration. We can not

determine the stress boundary conditions in terms of Cauchy stresses because body has

undergone large deformation so in the deformed configuration we do not know the no rmal vector

in the plane of crack. We can determine the boundary conditions only in terms of PK-I stresses so

we shall solve equation and determine the state of stress . Then from the relation between and we shall determine which we are finally interested in. From (17) and (20)0 1 (21)

(17)

(18)

(19)

(20)


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Comment [w6]: Both places we cant

meaning is different in undeformed and

cases.


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Here and have the range 1, 2. Since left side expression of this equation is a function of and only so p should be linear in . And first two equation requires that should beindependent of so let us assume p to be of the following form Where is an arbitrary constant. Puttingpinto equations (21) and (22)

Solving equation (24) we get

Load along the crack faces is zero so the traction vector along the crack line is

Normal vector of the crack faces in R is

So by the relation between and we get boundary conditions is already zero. By the second boundary condition we get

This equation should be true for all values of so we get From third boundary condition we get

(22)

(23)

(24)

(25)

(26)

(27)

(28)


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So we get the condition

After putting the value of p in equation (22) we get the condition

There is one additional boundary condition

should correspond to simple shear at large

distances from the crack tip i. e.

Here krepresents the amount of shear. Now the problem is to find such that it satisfies (29),(30) and (31). Oone more condition on is that it should be bounded near the crack tips. For the

Neo-Hookean material the value of W is given by

Using this value the equation (30) converts to

And the value of stresses areis,

() ()The global solution of this problem is given by

So the values of stresses are

( )

(29)

(30)

(31)

(32)

(33)


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(

)

Resultant shear stress is

The near tip approximation of near the right crack tip isdenoted by and is given as,

( ) If the problem is solved using the linear theory than near tip approximation of is found to besame as that obtaind above. So we shall use this value (37) to compare with exact value of (36)to get a measure of non linearity of the problem.

4.2 Condition for the nonlinearity to be small scale:

4.2.1 level nonlinear zone (: If represents the specified error tolerance than for thiscrack problem the inequality

represents the set of all points (for which the elastic field is approximately non linear at level (because is non linear and is linear). Putting values from equations (35) and (36) weget Figures below show level nonlinear zone (for different values of k. It can be seen that

(34)

(35)

(36)

(37)

(38)


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different


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Figure 4: level nonlinear zone (, k=0.9

Figure 5: level nonlinear zone (, k=


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Figure 6: level nonlinear zone (, k=For k , contains two bounded, connected subsets and such that +while contains right (left) crack tip. In this case we say that nonlinear effect iscontained. And for is unbounded.4.2.2 level near tip linear zone for right crack tip (:This represents set of all points forwhich relative error commitedcommittedby near tip approximation to the resultant shear stressis in magnitude at most i. e. set of all points for which| | Putting values

./ Figures ??????? below show the

level near tip linear zone

for

and

. The

region between the outer and inner curve is , the middle curve is the locus of all points whereand are exactly equal. The matlab code for the curves of sections 4.2.1 and 4.2.2 havesbeenincluded in the appendix.

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Comment [w8]: Mention numbers


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Figure 7: Near tip linear zone ,

Figure 8: Near tip linear zone , We say that the nonlinear effect is small scale if it is contained and


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Since the problem is symmetric about the

axis so if we satisfy either of the condition of

equation (40) the other will be satisfied automatically so we take first condition

If and represents the intersection of the and with the axis respectively then

by equation (41)

Now we shall determine and explicitly. A point ( belongs to if and only if and equation (38) holds with | | i. e. if and only if

| | | |

And So coincides with the interval

A point ( belongs to if and only if and equation (39) holds with || | |i. e. if and only if

./ | |||

(40)

(41)

(42)

(43)


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.

/ | |||

./ | |||

./ 0 , -1

./ | |||

./ | ||| 0 , -1 So coincides with the interval

0 , -1

0 ,

-1

We can verify that

From equation (42), (43) and (45) it follows that

(44)

(45)

(46)


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So from above two equations we can write

{ } We can show that for all values of between 0 and 1 first entry is smaller than second soequation (47) may be written as

Now we have shown that if the nonlinear effect is small scale at level

then k must satisfy the

condition (48). It can also be shown that equation (48) is sufficient as well as necessary forequation (42) and also that equation (42) implies equation (41). So it follows that equation (48)

supplies necessary as well as sufficient condition to be satisfied by the amount of shear at infinity

k if the nonlinear effect in the crack problem considered to be small scale at level .

(47)

(48)

(49)


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FINITE STRAIN DEFORMATION NEAR THE TIP OF CRACK IN

HYPERELASTIC SOLIDS

Abstract:

Analysis of cracked bodies is a very important subject from the point of view of

fracture. Linear elastic fracture mechanics is used for the analysis of most cracked

bodies. This theory is valid for the elastic bodies undergoing small deformations.

Here in this report, by relaxing the assumption of small strains the crack problem

has been analyzed for hyperelastic material which shows large deformations. The

body has been assumed to be deforming in anti plane shear. A condition has also

been derived under which the error committed by linear approximation of finite

deformation problem will be less than a specified value.

Keywords:Large deformation, Cauchy stresses,Hyperelastic,Neo-hookean


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CHAPTER 1 INTRODUCTION

Finite deformation problems always give a nonlinear solution which is complex to

analyze and in case of cracked bodies it is even more complex. So our motivation is

always to solve such problems by linear theories which are simpler but they may

lead to errors and in some cases they may give ambiguous results. If we have a

condition under which non linearity will be small in the crack problem then we can

apply linear theories without exceeding a specified error level. In this report we

derive such condition for finite anti plane shear of an infinite body containing a

crack of finite length.


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CHAPTER 5 CONCLUSION

From equation (48) it can be inferred if the value of

is higher then the upper limit

on k will also be higher for the nonlinear effect to be of small scale for that level of, it is expected because if the deformation is large then nonlinearity will also belarge and the error due to linaer approximation will be large.

If the crack is not symmetric about axis then two conditions of equation (40) arenot equivalent to each other and the nonlinear effect might be small scale at one

crack tip and not at the other. If the material used is not Neo-hookean then

expression for W will be different and the results derived will not be valid. This

remark also holds if the crack is deformed in mode I or mode II rather than in mode

III (anti plane shear).


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CHAPTER 2 REVIEW OF LITERATURE

Recently much work has been done on the study of deformations and stresses near the tip of

crack for hyperelastic materials.

[4] did asymptotic analysis of deformations and stresses in traction free crack in infinite slab

under plane strain. They assumed uniform axial tension at infinity at right angles to the faces of

crack.

[2] Did analysis of crack in finite anti-plane shear in an infinite slab. He took the material to be

Neo-hookean.

[5] Did the asymptotic analysis of traction plane crack under plane strain condition. He took the

power law materials (which in a special case converts to Mooney-Rivlin material.

[6] Did the asymptotic analysis of traction free interface crack between two dissimilar semi

infinite Neo-hookean sheets.

[1] Did the analysis of anti-plane shear problem in crack and derived the condition for the

nonlineasr effect to be small scale.

[7] Did the asymptotic analysis of traction free crack at interface of two semi infinite slabs

bonded under the conditions of plane strain.

[8] Did the asymptotic analysis of stress and strain near the tip of a crack with generalised Neo-

hookean material. He assumed the crack to be deforming in mode I and mixed mode.

[9] Did analysis of interface crack between, (1) two generalised Neo-hookean sheets having same

hardening characteristics and (2) a generalised Neo-hookean sheet and a rigid substrate.

[10] Did the analysis of motion field surrounding a rapidly propagating crack which is loadedsymmetrically in mode I conditions.

In this seminar anti-plane shear problem for Neo-hookean material has been studied because this

is the simplest problem to solve.


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Appendix I Matlab Code For Section 4.2.1 and 4.2.2

%create the following function files

functioncrak=sem(x,y)syms xycrak=1.5^4*(x.^4+y.^4+2*x.^2*y.^2)-(x.^2+1-2*x+y.^2)*(x.^2+1+2*x+y.^2)

functioncrac2=sem2(x,y)syms xye=0.67crac2=2*(x.^2+y.^2)*(1+e.^2-2*e)-(x.^2+1+2*x+y.^2)^0.5

functioncrac2=sem4(x,y)syms xye=0.67crac2=2*(x.^2+y.^2)-(x.^2+1+2*x+y.^2)^0.5

functioncrac2=sem3(x,y)syms xye=0.67

crac2=2*(x.^2+y.^2)*(1+e.^2+2*e)-(x.^2+1+2*x+y.^2)^0.5%give the following commands

ezplot (sem,[-4,4])

%and separatelyezplot (sem2,[-10,10]);hold on

ezplot (sem3,[-10,10]);hold on

ezplot (sem4,[-10,10])


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REFERENCES

[1] J. K. Knowles, A. J. Rosakis (1986), On the scale of nonlinear effect in a

crack problem, Journal of Applied Mechanics, 53, 545-549.

[2] J. K. Knowles (1977), The finite anti plane shear field near the tip of a crack

for a class of incompressible elastic solids, Journal of Elasticity, 13, 611-639.[3] C. S. Jog (2007) Foundations and applications of mechanics, volume 1:

continuum mechanics, 2E, Narosa Publication House.

[4] Knowles, Sternberg (1973), An asymptotic finite deformation analysis of the

elastostatic field near the tip of a crack, Journal of Elasticity, 3, 67-107.[5] Rodney A. Stephenson (1982), The equilibrium field near the tip of a crack

for finite plane strain of incompressible elastic materials, Journal of

Elasticity, 12, 65-99.

[6] Knowles, Sternberg (1983), Large deformation near a tip of an intrerfacecrack between the Neo-Hookean sheets, Journal of Elasticity, 13, 257-293.

[7] J.M.Herrmann (1989), An asymptotic analysis of finite deformations near thetip of an interface crack, Journal of Elasticity, 21, 227-269.

[8] Philippe H. Geubelle (1994), Finite strains at the tip of a crack in a sheet ofhyperelastic material: I. homogeneous case, Journal of Elasticity, 35, 61-98.

[9] Philippe H. Geubelle (1994), Finite strains atthe tip of a crack in a sheet ofhyperelastic material: II. Special biomaterial case, Journal of Elasticity, 35,

99-137.[10] Angelo Marcello (1999), Large deformation near a tip of an intrerface crack

between the Neo-Hookean sheets, Journal of Elasticity, 57, 85-103.

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