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Chapter 4Chapter 41
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OverviewOverview
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In case of LEFM (Linear elastic fracture mechanics)
is not valid, CTOD and the J integral method will be
adopted – especially for ductile materials. For ductile materials, the crack faces moves prior to
fracture and creates a blunt crack tip, this crack
o enin can be used as a measure of the tou hness of
the material. This parameter is known as Crack Tip Opening
Displacement or commonly in the form of
abbreviation CTOD – this parameter developed byWells.
The displacement at the original crack tip and the
90° intercept. These are equivalent if the crack
blunts in a semicircle.
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Evolution of Crack BluntingEvolution of Crack Blunting
As crack opens under applied load
4Crack advances at critical
value of CTOD
FIGURE 3.2: Estimation of CTOD from the
displacement of the effective crack in the Irwin
plastic zone correction.
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The effect of a plastic zone at the crack tip is to extend the
effective length of the crack by ry - half the diameter of the
plastic zone. Hence the opening of the crack at it’s real tip can be
approximated from the calculated elastic displacements of the
virtual (extended) crack evaluated at a point some r from the
virtual crack tip as shown in Figure 4.1.
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Figure 4.1 : Additional crack opening as a result of plasticity at crack tip
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The CTOD is given by double the displacement uyy in the tensile direction,for plane stress this is given by the equation:
where for plane stress
Evaluating this at r y from the crack tip θ = 180o;
( )
π µ 22
1 y I yy
r K u
+=
−+
=
221
222
2 θ θ
π µ cossin K
r K u I yy
ν
ν
+
−=
1
3K
K
Substituting for the plastic zone size;
Hence6
( )ν µ
+=
12 E
2
2
1
=
ys
I y
K r
σ π ( )
ys
I yy
K u
πσ µ 22
1 2
+=gives
and substituting
ys
I
yy
K
E u
πσ δ
24
2 ==
K
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CTOD,
However from Chapter 2 (LEFM);E K G I
2
=
Or according to Dugdale model;
ys
I yy
K
E u
πσ δ
24
2 ==
7
where m is a constant 1 for plane stress and 2 for plane strain (1< m < 3)
ys ys m
R
m
G
σ σ δ ==
Where; R is work of fracture = G C at fracture
ys ys
C C
m
R
m
G
σ σ δ ==
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CTOD Measurement/TestingCTOD Measurement/Testing
The main objective of the test is to determine the criticalcrack at the onset of crack extension.
This is done by measuring the displacement at the
mouth of the crack using a clip gauge. This procedure is detailed in the ASTM E1820 Standard
provides for CTOD measurements on both compact andSENB specimens.
The machine notch is precracked produced undercarefully controlled fatigue loading conditions (the rateof stress intensity factor is between 0.5 and 2.5 MPa√ms-1).
The specimen is then gradually loaded until themaximum load.
Plot of the load versus the clip gauge displacementsituated close to the mouth opening is made.
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CTOD analysis using ASTM standardsCTOD analysis using ASTM standards
l o a
d Pc
fracture
(a) (b) (c)
PiPu Pm
fracture
Pi
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M outh opening
Figure (a) - Fracture surface does not exhibit tearing and the final fracture occurs
under increasing load.
Figure (b) - Fracture surface exhibits tearing and the final fracture occurs under
increasing load.
Figure (c) - Fracture surface exhibits tearing and the final fracture occurs under
decreasing load.
Three main categories of fracture surface
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Experimental determination of CTOD
Experimental CTOD can be expressed as;
δ = δel + δpl
The plastic displacement at the crack
mouth, V is related to the plastic CTOD,
δpl through similar triangles construction:
W
P
a
r p ( W - a )
z
∆p
Vp
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Experimental determination of CTODExperimental determination of CTOD cont’dcont’d
Where r p is rotational factor. Experiments show that r p lies btween 0.33 to
0.48.
( )1K I ν 22
−
( )
( ) zaaW r
V aW r
E m
K
p
p p
ys
plel
I
++−
−+=+=
σ δ δ δ
2
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⋅=
W
a f
2
3
Bw
PS K I
( )
( ) zaaW r
V aW r
p
p
pl++−
−=
p δ
E 2 ysel σ =
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The plastic component Vp is obtained from the
load-displacement curve by constructing a line
parallel to the elastic loading line as
illustrated
Experimental determination of CTODExperimental determination of CTOD cont’dcont’d
Vp
Load
Mouth opening displacement
ve
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EXAMPLE
A three-point bend specimen with S = 25 cm, W = 6 cm, a = 3 cm, and B = 3 cm isused to determine the critical crack opening displacement δc of a steel plate
according to British Standard BS 5762. The load-versus crack mouth displacement
(P-V) record of the test is shown in figure below. Determine the critical crack
opening displacement, δc when E = 210 Gpa, ν = 0.3, σYS = 800 MPa
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Solution
pl C += el ( )E ys
I el
21K σ
ν δ 22
−=
⋅=W
a f
2
3
Bw
PS K I
For SENB specimen KI can be calculated as follows;
Where P is estimated about 31.6 kN from the graph.
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For = 0. , can e ca cu ate using t e formu a for SENBspecimen or can be obtained from the table in section A 3.5 of ASTM
E399.
W
f
2
3
22
1
1212
7293315219913
−
+
+
−
−−
=
wa
wa
w
a
w
a
w
a
w
a
w
a
W
a f
....
W
; hence = 2.66
W
a f
( )( )
( ) ( ) m MPa .K I 7.4766206.003.0
25.0106.31
2
3
3
=⋅
×
=
And K I can be calculated as;
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( )( )( )
006.010210800
3.07.73
22
=×
−=2
14elδ
Solution cont’d
The elastic component of the crack mouth displacement Vp
is;
The plastic component of the crackmouth displacement Vp is
determined from the test record P –
( ) mm p 286.003.003.04.0
10103.006.04.0 3
=+×
××−=−
δ
mm pl el C 292.0286.0006.0 =+=+= δ δ δ
The critical crack opening displacement δc is;
y raw ng a ne rom e
maximum load parallel to the linear
portion of the curve. We have Vp =
1mm. δp is determined as;
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JJ IntegralsIntegrals The J-integral represents a way to calculate the strain
energy release rate, or work (energy) per unit fracture
surface area, in a material.
The theoretical concept of J-integral was developed in1967 by Cherepanov and in 1968 by Jim Rice
independently, who showed that an energetic contour
ath inte ral called J was inde endent of the ath
around a crack.
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Figure 4.4 : Line integral around the crack tip – J integral
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It can be evaluated experimentally by measuring thestress strain curves for a number of identical specimens
containing cracks of different lengths and plotting the
area under the graph U for each specimen as a function
of the crack length and thus evaluating dU/dA and henceJ.
It is the rate of energy absorbed per unit area as the
JJ IntegralsIntegrals Cont’dCont’d
crack grows; it is not however the energy release ratebecause the plastic energy is not recoverable as it would
be in the elastic case.
There are also specific specimen geometries (deeply
double notched and notched three point bendingspecimens) that allow J to be measured from a single
specimen.
These experiments allow J to be plotted as a function ofthe crack extension.
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Plotting two curves for specimens differing only in thelength of the crack, a and a + ∆a, the energy required to
grow the crack is the difference in the areas under the two
graphs shaded in the Figure 4.5.
Since the area decreases as the crack grows dU/da is
negative and J = ‐dU/da at unit thickness.
Hence;
JJ IntegralsIntegrals Cont’dCont’d
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Figure 4.5 : Energy release rate for nonlinear deformation
Where U is the
potential energy of
the system and A
the area of the
crack.
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Although J = dU/dA is the same as the definition of the energyrelease rate, G used earlier, the J integral for the plastic casedoes not represent the energy released as the crack growsbecause much of the energy used performs plastic deformation.(LEFM, J = G; EPFM, J = R, resistance to crack growth)
The standardised test method for determining JIC materialvalues;
JIC Determination – A Procedure for the Determination of
Techniques. EPFM - On the Determination of Elastic-Plastic Fracture
Material Parameters: A Comparison of Different TestMethods
ASTM-E1820 - Standard Test Method for Measurement ofFracture Toughness.
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Calculation of J for SENB specimen.
Where;
J el = elastic component of J J pl = plastic component of J
pl el J J J +=
( )K J el
22 1 ν −=
Where;
η pl = 1.9 if the load-line displacement is used for Apl .
η pl = 3.667 – 2.199 (a/w) + 0.437 (a/w)2 if the CTOD is
used for Apl .
Validation;
• where20
( )a w B
AJ
pl pl
pl −
= η
Y
Q J
aw B σ 25≥−
, 2
ult YS Y
σ σ σ
+=
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Minimum four specimenswith the same size of
fatigue crack, a/W > 0.5.
Each specimen is loadedto a different point on the
load displacement curve
and then unloaded.
Crack is marked (steelspecimens - heat tinting)
to enable measurement of
stable crack growth.
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