reentrant corner on crack tip

8/2/2019 Reentrant Corner on Crack Tip

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MODELLING AND ANALYSIS OF

CRACKS IN 2D GEOMETRY USING

FRANC2D

NELSON M

Research Assistant

IIT Bombay


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PART A

GEOMETRY:

The geometry considered here is a rectangular plate with edge crack. The following are the

dimensions of the rectangular plate:

Length 20 inches

Width 10 inches

The thickness of the plate is taken as 1 inch so that plane stress conditions are satisfied. The

material considered is steel (this is default in FRANC2D). It has the following material

properties:

Youngs Modulus 29e6 psi

Poisson ratio 0.25

An edge crack, parallel to X-axis, of length 2.5 inches is introduced on middle on the left side

of the plate.

BOUNDARY CONDITIONS:

The bottom of the rectangular plate is fixed i.e. it is constrained along both X and Y axis. A

normal traction of 25000 psi s applied on the top of the plate. The following are essential

boundary conditions:

( ,0) 0

( ,0) 0

U x

V x

=

=

Uand V denote the displacement along X and Y axis.

Non essential boundary conditions is

( ,20) 25000x x psi =

,x y

andxy

are zero on all the free surfaces(except the bottom surface due to reaction

force) even on the crack tip surface.


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OBJECTIVES:

Study the mesh convergence. To find the stress intensity factor

IK , energy release rate G and study the square root

singularity of the crack tip stress field.

Compare the results obtained with the theoretical values computed from thehandbook.

ANALYTICAL SOLUTION:

The analytical expression forIK * for the edge crack in finite plate loaded under tension is

given by:

2 3 4

( )

( ) 1.12 0.23 10.55 21.72 30.39

IK a f

f

=

= + +

The function ( )f is due to finite plate analysis and also due to crack sitting at the edge of

the plate. The above expression is only valid for 0.6 < .

/a w =

a is the length of the crack and w is the width of the plate.

In this problem 2.5a in= and 10w in= . Hence 0.25 = and the above expression is valid.

Substituting the necessary values in the IK expression we stress intensity factor for the

current problem as

0.51.05 5 /

IK e psi i

=

and strain energy release rate for the above problem G is defined as

2/

IG K E=

Therefore the strain energy release rate is

381.47 / G p i=

* ThisIK expression is obtained from Elements of fracture mechanics by Prashant Kumar


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MESH CONVERGENCE:

The mesh is gradually refined and it is shown that as the mesh gets finer the numerical results

are in good agreement with the theoretical results. The stress intensity factor is the parameter

with which the convergence is studied. The following figures indicate the gradual refinement

of the mesh in order to numerically computeI

K that is close to the theoretical value.

Fig1 Fig2 Fig3

Fig4 Fig5

Fig 1-5 shows snapshots of the meshed geometry. A plot ofIK vs. the mesh refinements is

plotted in the fig 6.


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Fig 6 IK vs. mesh refinement study

Thus the IK (obtained using J-Integral) computed numerically matches with the IK obtained

through the theoretical expression. Henceforth the mesh as shown Fig 5 will be used for other

analysis.

NUMERICAL RESULTS:

The stress intensity factor obtained through FEM analysis is

0.51.051 5 / I

K e psi i

=

And the strain energy release rate is

381.1 / G p i=

Fig 7 shows the square root behaviour displayed byx

andy

plots as function of radius r.

This radius rin the plot corresponds to X-axis i.e. 0 = . The numerical results are compared

with the theoretical stress values neglecting the higher order terms. It is very clear from the

plot that the numerical stress variation is in good agreement near the crack tip with the

theoretical stress values. However as radius rincreases the numerical stress field is deviation

from the theoretical value due to higher order terms.


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Fig 7 Square root singularity of the stress field near the crack tip

COMPARATIVE STUDY OF THE RESULTS:

The following table presents the numerical results obtained through FRANC2D software

(through J-integral approach) against the known theoretical results.

FRANC2D results Theoretical results %Error

Stress Intensity

factor(I

K ),in 0.5/psi i

105100 105178.4 0.0746%

Strain energy release

rate( G ), in /p i 381.1 381.4656 0.0958%

It is very clear from the above table that the results obtained through software are in good

agreement with each other and hence other analysis can be performed on the refined mesh

that is shown in Fig 5.

OBSERVATIONS/EXPLANATIONS:


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MESH ELEMENTS

a) The entire geometry (excluding the region surrounding the crack tip) was meshedwith 8 noded quadrilateral elements so as to capture the stress field that does not vary

linearly.

b) Mesh convergence was only studied based on h refinement on quadratic elementsthough CASCA software had bilinear 3 sided and 4 sided elements.

c) Even with bi quadratic elements the coarser mesh (less than 10 quarter point element),around the crack tip, has difficulty in capturing the stress intensity factor precisely

with respect to the known theoretical values. Therefore the crack tip elements are

made finer which demands the mesh far from the crack tip to be also made finer.

d) The general observation is higher the refinement near the crack tip, more the accuracyof the IK and thus also crack tip stress field.

STRESS INTENSITY FACTOR AND STRAIN ENERGY RELEASE RATE

a) The stress intensity factorI

K can be calculated from the J integral. The contour for J

integral, if it includes crack tip, becomes strain energy release rate G . The J integral in

index notation is given by

( )ki i j jk i

uJ Wn n d

x

=

where W is the strain energy density,i

n is the normal to the contour along i direction

andku is the displacement along k direction

b) Stress intensity factors may be calculated from2

2

/

/

I I

II II

K E G

K E G

=

=

c) The FRANC2D manual says that by default the program selects the contour enclosingthe crack tip elements (quarter point elements) as the domain of integration. However

the option to change the domain of integration is not available.

d) Because of the deformation, the normal applied load is no longer normal to the cracktip surface and hence small values of

IIK and

2G .


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PART B

OBJECTIVES:

To show the effect of re-entrant corner on the crack tip for various /l a (kink lengthto crack length) ratio and kink angle .

To find l (kink length) such that there is no effect of re-entrant corner on the crack tipstress field.

To study the influence of kink angle on the stress concentration factor and crack tipstress field interaction.

EFFECT OF REENTRANT CORNER ON THE CRACK TIP STRESS

FIELD:

One of the ways of studying the effect of re-entrant corner on the crack tip stress field is to

look at the maximum principal stress of the crack tip by varying /l a ratio and kink angle .

The theory behind the study of the effect of the re-entrant corner is that it would affect the

stress field* and thus the stress intensity factor of the crack tip till certain length of the kink.

As the kink length increases the effect of the re-entrant corner on the crack tip decreases.

The stress at the crack tip is found by the following relation

11 12

11

21 22

22

31 32

yy

xx

xy

C CK

C CK

C C

=

Cmatrix relations are obtained from Elements of fracture mechanics by Prashanth Kumar.

C matrix is function of and when 90 = we get the stress the stress along the

Cartesian coordinates. Once if 2D stresses are established along any axis, the maximum

principal stress could be obtained through Mohrs circle.

* Here the maximum principal stress has the unit0.5psii which is nothing but the dimensions

of stress intensity factor. These values are obtained by multiplying the above matrix by

2 r and hence the stresses obtained are rindependent. Thus the maximum principal stress


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obtained through this way is not the actual maximum stress value at the crack tip but a proxy

which is independent of r

Fig 8 Variation of maximum stress for various /l a ratios as function of

Fig 9 Variation of maximum stress for various as function of /l a .


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Fig 8 and Fig 9 shows the variation of the maximum stress at the crack as a function of /l a

(kink length to crack length) ratio and kink angle.

In Fig 8 as increases the maximum stress also increases for increasing /l a (kink length to

crack length) ratio. This trend is clearly seen for higher /l a ratios which is evident from theFig 9. However for smaller /l a ratios (


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Fig 11 Variation of stress concentration factor for various as function of /l a .

L FOR WHICH THERE IS NO EFFECT OF REENTRANT CORNER

ON THE CRACK TIP STRESS FIELD:

It is known that as the crack length increases the stress intensity factor/stress concentration

factor also increases for this problem and it was also seen clearly in the above plots for higher

/l a ratios. However when we look at closely near the smaller /l a ratios, the following trend

is observed.

Fig 12 Variation of stress concentration factor for various smaller /l a ratios at=30.


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The sudden dip in the Fig 12 is attributed towards the rapid decrease in the effect of re-

entrant corner on the crack tip. Fig 13 also shows the dip observed even for =45 degrees.

Fig 13 Variation of stress concentration factor for various smaller /l a ratios at=45.

Similar plots can also be observed for higher angles. Thus by approximation, assuming that

the critical l is actually a function of, can be shown from the above graphs that the critical

lengths lies between /l a ratios of 0.08 and 0.09. More information will be gathered after

plotting the effect of on the crack tip stress fields.

INFLUENCE OF KINK ANGLE ON THE STRESS CONCENTRAION FACTORE

AND CRACK TIP STRESS FIELDS:

A general plot of the influence of kink angle is already plotted in the Fig 9 and Fig 11. Fig

14 and Fig 15 shows the plot of variation of stress concentration factor at smaller /l a ratios.


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Fig 14 Variation of stress concentration factor for various smaller /l a ratios at=45 and 60.

Fig 15 Variation of stress concentration factor for various smaller /l a ratios at=75

It can be seen from above graphs that actually the effect of the re-entrant corner crack tip

stress field is a function of because as the increases the dip occurs at relatively higher

/l a ratio. Hence not only the kink length but also the kink angle plays a major role in relation

to stress concentration factor.

For example at = 45 degrees, it can noted from the above graph that the critical /l a ratio

lies somewhere between 0.08 and 0.09. If rr (Psi) is plotted at 0 = for /l a ratio of 0.1,

then we have to see the detachment of the effect of re-entrant corner on the crack tip stress

field since the detachment occurred recently. Fig 16 shows the snapshot of this plot


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Fig 16rr

vs. length along the X-axis

It is clear from the above figure that there are two peaks one related to re-entrant corner and

other higher peak to crack tip and there is clear detachment.

REPLACEMENT BY OBLIQUE STRAIGHT CRACK:

Consider = 60 degrees and for /l a ratio of 0.1 the detachment is observed to take place. If

the entire crack set is replaced by oblique straight crack whose tip is sitting at kink crack tip

and making an angle 60 degrees (same as that of ) with X-axis as shown in the figure 17,

then the followingIK and IIK values are obtained. Fig 18 shows the original crack with kink.

The kink is so small that it is not clearly seen ( /l a ratio of 0.1).


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Fig 17 Fig 18

The table shows the comparative results of oblique straight when it replaces the original crack

with a kink.

Oblique straight

crack

Crack with angled

kink%Error

Stress Intensityfactor(

IK ),in 0.5/psi i

6.7E4 7.3E4 9.6%

Stress Intensity

factor(IIK ),in

0.5/psi i

4.6E4 4.3E4 6.5%

The reason for relatively high error is due to manual placing of the edge crack (need not be

angle that is desired by the user).

reentrant corner on crack tip

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