Elasto Plastic Crack

8/2/2019 Elasto Plastic Crack

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ELASTIC BEAM FINITE ELEMENT WITH

TRANSVERSE ELASTO-PLASTIC CRACK

ARINDAM DEY

Y3103010

M.Tech.Geotechnical Engineering

IIT Kanpur


2/18

INTRODUCTION

Cracks occurring in structural elements of machines lead to local changes in thestiffness of these elements and consequently their static, dynamic and stability

behaviour is altered

Cracked structures have been modelled by various methods such as one-dimensional,

discrete-continuous model in which the crack was modelled by a flexibility matrix

(Dimarogonas),where the elements of this matrix have been calculated from relations oflinear fracture mechanics; the method of sub-domains (Lee) for modelling cracked

annular plates; & singular finite elements for the dynamic analysis of a cracked beam

(Shen & Pierre)

The above citations assumed that material around the crack tip behaved in a purely

elastic manner.

In real structures, for many materials, a plastic zone appears around the crack tip & the

flexibility of such structures increases, more than is observed in the case of purely elastic

materials.

The main objective of this work is to develop a model of a cracked beam finite element,

which will take into account the effect of plasticity around the crack tip

A beam finite element model with single-edge crack is developed,which takes into

account the influence of the plastic zone ahead of the crack tip on flexibility of the

element.

The inertia and stiffness matrices of the element are presented in closed forms. Theforms of these matrices are modified by the presence of the crack


3/18

ELASTO-PLASTIC LOCAL FLEXIBILITY OF A

CRACKED BEAM The radius of the plastic zone around the crack tipr

p, for I fracture mode of the

crack evaluation (opening mode) can be calculated from the following

relationship

where, KI

is the elastic stress intensity factor, Y the material yield strength,andd

pdenotes the diameter of the plastic zone around the crack tip

2

112

==

y

pp Krd ..(1)

Fig. 1 Plastic zone around a cracked rip


4/18

Contd

For a crack with the plastic zone around its tip, the stress intensity factor (SIF)

can be expressed as follows

where,a is the crack length,F is a correction function which takes into account

the body and crack geometry, while is the applied nominal stress Substituting the relationship (1) into (2) gives

which allows calculation of the stress intensity factor for an elastoplastic crack

Additional local flexibilities of the body due to the crack cij

are given by the

relation

where Jc

denotes the strain energy density function (SEDF), P is a

corresponding force at the cracked cross-section, whileA is the crack area

pIp raFK += (2)

....(3)

= Ac

ji

ij dAJPPc

2

...(4)

22

1

2

1

21

2

1

+=

+=

YY

Ip

FK

KaFK


5/18

Contd

For I fracture mode of the crack tip evaluation SEDF is given by

where, E'=E for the plane state of stress, E'=E/(1- 2) for the plane state ofstrain,E denotes Young's modulus and is Poisson's ratio.

In the present case the beam is subjected only to the bending momentM. Thus,

SIF may be calculated as

Taking into account the relation (4) the flexibility at the cracked cross-section

of the beam under consideration can be written as

where,B andHdenote the width and height of the beam, respectively

= =

=

6

1'

1 i

i

Ipic KE

J ..(5)

22

2

2

16

+=

Y

Ipi

F

BH

MK

..(6)

+=AYA

adAFHBEadAFHEBc4

2

42

2

42

21672

..(7)


6/18

Contd

The correction functionF is given as follows

Eqn (7) is expressed in a more convenient manner as

where,

( )[ ]( )Ha

Ha

Ha

HaF

2/cos

2/sin1199.0933.0

2/

)2/tan(4

+=

(8)

( ) ( ) adaaFBHE

adaaF

EBH

ckk a

Y

a

+=

0

4

2

2

0

2

2

21672

(9)

H

aa

H

aa kk == ,


7/18

A FINITE ELEMENT MODEL

A cracked beam finite element model has been chosen

Three different parts can be distinguished in the element: left segment, crack

and right segment

The left and right segments are represented by non-cracked beams of length

L /2, whereas the crack segment is represented by a massless spring of length

zero

The element has two nodes with two degrees of freedom (transverse

displacement and rotation) in each node

Fig. 2 A cracked beam finiteelement model


8/18

Contd

The field of displacements for the left and right segments of the element can be written

in the following form

The constants a1-a8 can be expressed by nodal displacements using the elementboundary conditions

where K=EJc, and J denotes the geometrical moment of inertia of the beam cross-section

( )

( )

( )( ) 2

8762

3

8

2

7652

2

4321

3

4

2

3211

32

32

xaxaaxxaxaxaaxw

xaxaax

xaxaxaaxw

++=+++=

++=

+++=

....(10)

( ) ( )( ) ( )

( ) ( )

( ) ( ) ( )( ) ( )

( ) ( )22

22222

22

,0,0

'''

2

'''

1

"

2

"

1

"

221

21

4221

321

LwLw

LwLwLKwLL

LwLw

qLqqLwqw l

==

=

====

......(11)


9/18

Contd

Taking into account the formulas (10) and (11) the shape function matrix for

left and right segments of the element can be determined, as follows

Uusing the matrices of the shape functions and applying the standard finite

element definitions of the characteristic matrices, the mass matrix of the

element can be evaluated as

+

+

+

+

++

+

+

=

+

+

+

+

=

2323

22

2

32

2

2323

222

32

1

1212

)(2

)32(3

)(2

)34(3

00

)(20

)(21

3210

1

1212)(2

)32(3

)(2

)34(30010

0001

3210

1

LLLL

KLL

KL

LKLL

KL

L

KL

K

KL

LK

KL

LK

KL

LK

xx

xxxN

LLLL

KLL

KL

LKLL

KL

Lxx

xxxN

.....(12)

21

420ccc JLMMALM += .(13)


10/18

Contd

The parts of the inertia matrix corresponding to the transverse and rotation

motion, respectively are as follows

( )( )( )

( )

( )

( )

( )

( )

( )( )

( )

( )

+

+++

++

++

+

+

+

+++

+

+

+

=

2

223

2

2

2232

2

223

22

1

16

14819164

4

12388156

)(16

4453163

4

8752

16

14819164

4

875254

4

)12388(156

KL

KLKLLsym

KL

KLLKL

KLKLL

KL

KLL

KL

KLKLL

KL

KLL

KL

LLL

Mc

( )( ) ( )( )

( )( )

+

++

+

+++

++

=

2

22

2

2

22

2

22

22

2

60

1821810

1

5

6

60

18218

10

1

60

1821810

1

5

6

10

1

5

6

KL

KLKLLsym

LL

KLKLKLL

LKLKLKLL

LLLL

Mc


11/18

Contd

The above inertia matrix contain element which are functions of coefficient of

flexibilityKat the crack location. By this, only the terms corresponding to therotational degrees of freedom and those which couple rotational and

translational degrees of freedom are only affected

WhenK=0, the matrix takes the form of the matrix representing a non-cracked

element The stiffness matrix an be represented as

Here alsoK=0 represents the stiffness matrix of a non-cracked element

( ) ( )

( )

+

++

+

+++

++

=

KLL

KLKLsym

LL

KLL

KLKL

LKLL

KLKLLLLL

EJK

22

23

22

2

22

2222

364

612

3626364

612612

..(14)


12/18

NUMERICAL EXAMPLES TO CHECK THE

VALIDITY OF THE PROPOSED MODEL

Investigates influence of the grid density on the first three natural frequencies

of a cantilever beam with a crack

Beam data: length 1 m, height 0.01 m, width 0.01 m, Young's modulus 2.11011

Pa, density 7860 kg/m

3

The crack was located in the middle of the beam length. The depth of the crack

was varying in the range 0.00.6 and no plastic zone around the crack tip was

assumed here

For the non-cracked beam the first free natural frequencies analytically

calculated were: 52.463, 328.741, 920.641 rad/s

In this example several grids: 5 elements, 9 elements, 13 elements and 17

elements were tested

EXAMPLE I


13/18

Table 1. Changes in the first natural frequency (rad/s)

as a function of the element grid density and the crack

depth

Table 2. Changes in the second natural frequency

(rad/s) as a function of the elements grid density and

the crack depth

Table 3. Changes in the third natural frequency (rad/s)

as a function of the elements grid density and the crackdepth

These results show that even for

relatively deep cracks changes in

natural frequencies of the beam

investigated, do not depend on the

grid density


14/18

EXAMPLE II

Shows how the inertia matrix form influences natural frequencies of cracked

beam-like structures Numerical calculation in this example was carried out for a cantilever beam

with dimensions the same as in the example 1

The crack was located 0.05 m from the fixed end of the beam and the Y/

ratio was 0.8 Inertia matrix form does not influence natural frequencies of the cracked beam

under investigation. Thus it is possible and convenient to use in dynamic

analysis of beam-like cracked structures the inertia matrix in the same form as

for non-cracked structures

Table 4.

Relative natural

frequencies of the

cracked beam for

two forms of theinertia matrix


15/18

EXAMPLE III

Illustrates the validity of the proposed model of the cracked beam element

Changes in natural frequencies obtained by applying the elaborated element

are compared to results of experimental investigation and analytical calculation

It is assumed here that the plastic zone around the crack tip does not exist

The calculation in this example was carried out for the following beam data:

length 0.4 m, height 0.02 m, width 0.01 m, Young's modulus 2.11011 Pa,

density 7860 kg/m3

The crack was located 0.08 m from the fixed end of the beam. The depth of the

crack was varying in the range 0.00.6

The beam was modelled by 20 finite elements

Fig. 3. The comparison of changes in the

first bending natural frequency as a

function of the crack depth betweenanalytical, experimental and FEM models

The compatibility obtained between the

results of the experimental investigation

and the results of FEM analysis is much

better than in the case of the analyticalmodel


16/18

EXAMPLE IV

Shows the influence of the plastic zone around the crack tip on the intensity of

changes in natural frequencies The calculations were done with the following beam data: length 1.0 m, depth

0.05 m, width 0.01 m. The material data were: Young's modulus 2.11011 pa,

density 7860 kg/m3

The beam was modelled by 20 finite elements

Fig. 4, Fig. 5 & Fig. 6 shows that the fall in natural frequencies depends not

only on the crack location but also on the size of the plastic zone around the

crack tip

Fig. 7, Fig. 8 & Fig 9 clearly denotes that the fall in natural frequencies depend

on from the crack depth and the size of the plastic zone around the crack tip


17/18

Fig. 4 Fig. 5 Fig. 6

Fig. 7 Fig. 8 Fig. 9

The relative changes of the first bending natural frequency as a function of the crack location and the ratio Y/

The relative changes of the first bending natural frequency as a function of the crack depth and the ratio Y/


18/18

CONCLUSION

A beam finite element with a transverse, one-edge, open, elasto-plastic crack

for static, dynamic and stability analysis of cracked beam-like structures has

been developed in this paper. The model has been successfully employed to

investigate the influence of cracking on changes in natural frequencies of acantilever beam. Numerical examples presented in this paper allow the

following conclusions to be drawn:

1. Crack tip plasticity increases the predictive capabilities of the model.2. From a numerical point of view the form of the inertia matrix does not affect

natural frequencies of the beams analysed. Because of this the inertia matrix in

the same form as for a non-cracked structure can be applied.

Using the method described in the paper it is possible to calculate the full form

of the local flexibility matrix, which also takes into account: axial, torsion and

shear deformations of the element

Elasto Plastic Crack

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