Linear Elastic fracture mechanics report.pdf

7/27/2019 Linear Elastic fracture mechanics report.pdf

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5.2 Fracture Mechanics

Fracture mechanics is a field of solid mechanics that deals with the mechanical behaviour

of cracked bodies. Based on behaviour of Materials, fracture mechanics is classified as

Figure 5.2.1 Classification of fracture mechanics

5.2,1 Linear Elastic Fracture mechanics

The analysis of Linear Elastic Materials is known as Linear Elastic Fracture mechanics. The

assumptions in Lefm are

Material is brittle in nature, i.e., there is no plasticity in the material The thickness of specimen for theory is unity The concept of Griffith, Inglis, Irwin, etc. are valid for the material

5.2.1.1 Crack deformation modes

Consider a plane crack extending through the thickness of a flat plate. Let the crack plane occupiesthe plane xz and the crack front is parallel to the z-axis. Place the origin of the system Oxyz at the

midpoint of the crack front. There are three independent kinematic movements of the upper and

lower crack surfaces with respect to each other. These three basic modes of deformation are

illustrated in Figure 2, which presents the displacements of the crack surfaces of a local element

containing the crack front. Any deformation of the crack surfaces can be viewed as a superposition

of these basic deformation modes, which are defined as follows:

(a) Opening mode, I. The crack surfaces separate symmetrically with respect to the planes xy and

xz.

(b) Sliding mode, II The crack surfaces slide relative to each other symmetrically with respect to the

plane xy and skew-symmetrically with respect to the plane xz.

(c) Tearing mode, III. The crack surfaces slide relative to each other skew-symmetrically withrespect to both planes xy and xz.

FRACTURE

MECHANICS

TIME

INDEPENDENT

LINEAR ELASTICMATERIALS

ELASTIC PLASTIC

MATERIALS

TIME

DEPENDENT


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Figure 5.2.2 The three basic modes of crack extension.

(a) Opening mode, I,

(b) Sliding mode, II, and(c) Tearing (or antiplane) mode, III.

The stress and deformation fields associated with each of these three deformations modes will be

determined in the sequel for the cases of plane strain and generalized plane stress. A body is said to

be in a state of plane strain parallel to the plane xy if

( ) ( ) [5.2.1]where u, v and w denote the displacement components along the axes x, y and z. Thus, the strainsand stresses depend only on the variables x and y. Plane strain conditions are realized in long

cylindrical bodies which are subjected to loads normal to the cylinder axis and uniform in the z-

direction. In crack problems, plane strain conditions are approximated in plates with large thickness

relative to the crack length. A generalized plane stress state parallel to the xy plane is defined by

( ) ( ) () [5.2.2]where , , , and , denote the normal and shear stresses associated with the systemxyz. Generalized plane stress conditions are realized in thin flat plates with traction-free surfaces. In

crack problems, the generalized plane stress conditions are approximated in plates with crack

lengths that are large in relation to the plate thickness. We recall from the theory of elasticity that a

plane strain problem may be solved as a generalized plane stress problem by replacing the value of

Poisson's ratio v by the value v / (1+v).

5.2.1.2Westergaard method

5.2.1.2(a) Description of the method

The Westergaard semi-inverse method constitutes a simple and versatile tool for solving a certain

class of plane elasticity problems. It uses the Airy stress function representation, in which the

solution of a plane elasticity problem is reduced to finding a function U which satisfies thebiharmonic equation,


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[5.2.3]

and the appropriate boundary conditions.

The stress components are given by

[5.2.4]

If we choose the function U in the form

[5.2.5]where the functions

(i = 1,2,3) are harmonic, that is,

[5.2.6]U will automatically satisfy Equation (3).

Following the Cauchy-Riemann conditions for an analytic function of the form

() [5.2.7]we have,

; [5.2.8]and, therefore,

[5.2.9]Thus, the functions (i = 1,2,3) in Equation [5.2.5] can be considered as the realor imaginary part of an analytic function of the complex variable z.

Introducing the notation

[5.2.10]Westergaard defined an Airy function UI for symmetric problems by

[5.2.11]UI automatically satisfies Equation (3). Using Equations [5.2.4] we find the stresses from UI to be

[5.2.12]Using Hooke's law and the strain-displacement equations we obtain the displacement components

[5.2.13]


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where = 3- 4v for plane strain and = (3 - v) / (l + v) for generalized plane stress.

For skew-symmetric problems with respect to the x-axis the Airy function UII is defined by

[5.2.14]and the stresses and displacements by

[5.2.15]and;

[5.2.16]

Figure 5.2.3 A crack of length 2a in an infinite plate subjected to a uniform stress at infinity.

5.2.1.2(b) Crack problems

Consider a crack of length 2a which occupies the segmenta x a along the x-axis in an infinite

plate subjected to uniform equal stresses along the y and x directions at infinity (Figure 5.2.3).

The boundary conditions of the problem may be stated as follows:

[5.2.17]And

( ) [5.2.18]

The function defined by


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() [5.2.19]

satisfies the boundary conditions [5.2.17] and [5.2.18] and therefore is the Westergaard function for

the problem shown in Figure 5.2.3

For the problem of a crack of length 2a which occupies the segment along the x-axisin an infinite plate subjected to uniform in-plane shear stresses T at infinity (Fig.4), the boundaryconditions of the problem may be stated as

( ) [5.2.20]and the Westergaard function of the problem is

() [5.2.21]

Figure 5.2.4 A crack of length 2a in an infinite plate subjected to uniform in-plane shear stresses

at infinity.

5.2.1.3 Singular stress and displacement fields

The study of stress and displacement fields near the crack tip is very important, because these fields

govern the fracture process that takes place at the crack tip.


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5.2.1.3(a) Opening mode

The Westergaard function for an infinite plate with a crack of length 2a subjected to equal stresses

at infinity (Fig.3) is given by Equation [5.2.19]. If we place the origin of the coordinate system atthe crack tip z = a through the transformation

[5.2.22]Equation [5.2.19] takes the form

()() [5.2.23]Expanding Equation [5.2.23] we obtain

()() [ ] [5.2.24]

For small ||,||0, that is near the crack tip at x = a, Equation [5.2.24] may be written

() [5.2.25]Where

[5.2.26]

Using polar coordinates r, we have

[5.2.27]

and the stresses near the crack tip are

[5.2.28]

The quantity KI is the opening-mode stress intensity factor and expresses the strength of the

singular elastic stress field. As was shown by Irwin , Equation [5.2.28] applies to all crack-tip stressfields independently of cracked body geometry and the loading conditions. The stress intensity


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factor depends linearly on the applied load and is a function of the cracked length and the

geometrical configuration of the cracked body. Introducing the value of the Westergaard function ZI

from Equation [5.2.25] into Equations [5.2.13] we obtain the displacements

( )

( ) [5.2.29]where

[5.2.30]

5.2.1.3(b) Sliding mode

The Westergaard function ZII for a crack of length 2a in an infinite plate subjected to uniform in-

plane shear stress rat infinity (Fig.4) is given by Equation [5.2.21]. The stresses and displacements

are obtained from Equations [5.2.15] and [5.2.16]. Following the same procedure as in the previous

case, and recognizing the general applicability of the singular solution for all sliding-mode crack

problems, we obtain the following equations for the stresses and displacements

[5.2.31]

And; ( )

( ) [5.2.32]

The quantity KII is the sliding-mode stress intensity factor and, as for the opening mode, itexpresses the strength of the singular elastic stress field. When the Westergaard function ZII isknown, KII is determined, following the same procedure as previously, by

[5.2.33]For a crack of length 2a in an infinite plate subjected to in-plane shear stress 7 at infinity, we obtain

from Equations [5.2.33] and [5.2.21]

[5.2.34]


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5.2.1.3(c) Tearing mode

For the tearing (or antiplane) mode of crack deformation the in-plane displacements u and v are

zero, while the displacement w is a function of the in-plane coordinates x and y, that is

( ) [5.2.35]Equation [5.2.35] suggests that the movement of the crack surfaces can be related to the warping

action of noncircular cylinders subjected to torsion. Equation [5.2.35] renders

[5.2.36]

Figure 5.2.5 A crack of length 2a in an infinite plate subjected to uniform out-of-plane shear stress at infinity.

and from Hooke's law we have

[5.2.37]

[5.2.38]


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Substituting Equation [5.2.38] into the non-self-satisfied equilibrium equation

[5.2.39]

We obtain for w

[5.2.40]Since w satisfies the Laplace equation it can be put in the form

[5.2.41]

where ZIII is an analytic function. From Equations [5.2.38] we obtain

[5.2.42]

Consider a crack of length 2a which occupies the segmenta x a along the x-axis in an infinite

plate subjected to uniform out-of-plane shear stress T at infinity. The boundary conditions of the

problem may be stated as

( ) [5.2.43]Following the same procedure as in the case of the opening mode we introduce

the function defined by

() [5.2.44]This satisfies the boundary conditions [5.2.43].

Near the crack tip, we obtain for the stresses xz ,yz and the displacement w

[5.2.45]

where

[5.2.46]

Equation [5.2.45] applies to all tearing-mode crack problems near the crack tip. The quantity Km is

the tearing-mode stress intensity factor and expresses the strength of the singular elastic stress field.When the function , is given, is determined by


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[5.2.47]In cases where two or all three deformation modes exist in a crack problem, the singular elastic

stress field in the neighbourhood of the crack tip is obtained by superimposing the solutions

corresponding to each of the three deformation modes and it is characterized by the respective stressintensity factors.

5.2.2 Crack tip Plastic zone

Figure 5.2.6 showing the crack tip plastic zone and Graph of stress vs. r showing crack tip plastic zone

As r= 0 (i.e., at the crack tip) yy approaches infinity. However, in practice, the stress at the crack

tip is limited to at least the yield strength of the material, and hence linear elasticity cannot be

assumed within a certain distance of the crack tip . This nonlinear region is sometimes called the

crack tip plastic zone8. Outside the plastic zone, displacements under the externally applied stress

mostly follow Hookes law, and the equations of linear elasticity apply. The elastic material outsidethe plastic zone transmits stress to the material inside the zone, where nonlinear events occur that

may preclude the stress field from being determined exactly. The strain energy release rate is not

influenced much by events within the plastic zone if the plastic zone is relatively small. It can be

shown that an approximate size of the plastic zone is given by:

[5.2.48]

where ys is the yield strength (or yield stress) of the material. The concept of a plastic zone in the

vicinity of the crack tip is one favoured by many engineers and materials scientists and has useful

implications for fracture in metals. However, the existence of a crack tip plastic zone in brittlesolids appears to be objectionable on physical grounds.


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References

Fracture Mechanics- Fundamentals and Applications by T.L. Anderson. Fracture Mechanics by E.E. Gdoutos. Fracture Mechanics by Dr.ir. P.J.G. Schreurs Lecture notes of Eindhoven University of

Technology.

www.library.veryhelpful.co.uk/ www.google.com
http://www.library.veryhelpful.co.uk/http://www.library.veryhelpful.co.uk/http://www.library.veryhelpful.co.uk/


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Paper1 :- Micrographic Technique For Linear Elastic Fracture Evaluation Of

Crack Initiation Zone

Authors:- K.B.Yeo and E.H. LimCentre of Materials & Minerals, University Malaysia Sabah, 88999 Kota Kinabalu

Sabah Malaysia

Published at Journal of Applied Sciences 10(21) :2663-2667, 2010

Download fromwww.ISSN.org

Summary:-

The specimen used for compact tension test to determine fracture thoughness is separated into two

parts. The fracture is then studied by using Infinite Focus Microscope (IFM) and Scanning Electron

Microscope (SEM). IFM has vertical resolution of 20 nm and SEM has both horizontal and vertical

resolution of 1 to 5 nm. Using both the instruments the crack initiation zone ,i.e., crack tip plastic

zone is measured to be 17 nm.

Such experiments by others using different instruments had the range of crack tip plastic zone 10 to

15 nm. As this experimentation results closer to results of previous experimentation, this

experiment is successfully carried out. And this is verified by the patterns of rivers lines and feather

marks are obtained which is characteristic property of Linear Elastic Material.
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Paper 2:-Linear Elastic Behaviour Of Cylindrical Shell

Authors :- S. S. ANGALEKAR

HOD Civil Engineering Department,

ATSs SBGI, Miraj 416410, Maharashtra, India

[email protected]

And

Dr. A. B. KULKARNI 2

Ex. HOD Applied Mechanics Department,

WCE, Sangli 416415, Maharashtra, INDIA

Published at International Journal of Engineering Science and Technology in 2011

Downloaded fromwww.ISSN.org

Summary:-

In this paper, simulation analysis on concrete pipe is done. Software used for simulation is

ABAQUS. The pattern of deflection in all three directions and corresponding stresses and moments

is simulated and solution for various geometric variations is observed.

As thickness increases the deflection, stress and moments decreases for same span length.

For different geometric parameters, the deflection, stresses and moments at crown and free edge are

opposite in sense.
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Linear Elastic fracture mechanics report.pdf