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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).